Engineering Of Chemical Complexity 9789814390460, 9789814390453

Citation preview

ENGINEERING OF CHEMICAL COMPLEXITY

8400_9789814390453_tp.indd 1

21/11/12 11:19 AM

WORLD SCIENTIFIC LECTURE NOTES IN COMPLEX SYSTEMS Editor-in-Chief: A.S. Mikhailov, Fritz Haber Institute, Berlin, Germany B. Huberman, Hewlett-Packard, Palo Alto, USA K. Kaneko, University of Tokyo, Japan Ph. Maini, Oxford University, UK Q. Ouyang, Peking University, China

AIMS AND SCOPE The aim of this new interdisciplinary series is to promote the exchange of information between scientists working in different fields, who are involved in the study of complex systems, and to foster education and training of young scientists entering this rapidly developing research area. The scope of the series is broad and will include: Statistical physics of large nonequilibrium systems; problems of nonlinear pattern formation in chemistry; complex organization of intracellular processes and biochemical networks of a living cell; various aspects of cell-to-cell communication; behaviour of bacterial colonies; neural networks; functioning and organization of animal populations and large ecological systems; modeling complex social phenomena; applications of statistical mechanics to studies of economics and financial markets; multi-agent robotics and collective intelligence; the emergence and evolution of large-scale communication networks; general mathematical studies of complex cooperative behaviour in large systems.

Published Vol. 1 Nonlinear Dynamics: From Lasers to Butterflies Vol. 2 Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems Vol. 3 Networks of Interacting Machines Vol. 4 Lecture Notes on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media Vol. 5 Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology Vol. 6 Frontiers in Turbulence and Coherent Structures Vol. 7 Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics Vol. 8 Granular and Complex Materials Vol. 9 Complex Physical, Biophysical and Econophysical Systems Vol. 10 Handbook on Biological Networks Vol. 11 Engineering of Chemical Complexity

Lakshmi - Engineering of chemical Complexity.pmd 1

11/19/2012, 5:52 PM

World Scientific Lecture Notes in Complex Systems – Vol. 11

editors

Alexander S Mikhailov Gerhard Ertl

Fritz Haber Institute of the Max Planck Society, Germany

ENGINEERING OF CHEMICAL COMPLEXITY

World Scientific NEW JERSEY

•

LONDON

8400_9789814390453_tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

21/11/12 11:19 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

World Scientific Lecture Notes in Complex Systems — Vol. 11 ENGINEERING OF CHEMICAL COMPLEXITY Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4390-45-3

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

Lakshmi - Engineering of chemical Complexity.pmd 2

11/19/2012, 5:52 PM

November 21, 2012

13:3

9in x 6in

b1449-fm

Engineering of Chemical Complexity

PREFACE

How to engineer chemical systems whose operation is self-organized? This question becomes topical in the light of an impressive progress in understanding of complex chemical systems of inorganic and biological origins. The gained knowledge makes it already possible to proceed to the next stage where not only the purposeful control of such natural systems, but also the attempts to design their synthetic counterparts can be made. The phenomenon of life, emerged as a product of a specific biological evolution, may thus be transferred to a broad range of artificial systems which are designed for particular applications. In the long perspective, this would revolutionize the technology, moving it from the present form, essentially based on the mechanistic concepts of the 19th century, to the functional organization characteristic for living beings. Early indications of the approaching transition can be seen in the current research on the design of genetic networks, on the development of artificial or synthetic living cells, and on the construction and manipulation of single-molecule protein machines. This book provides a survey of the current state of the art in the analysis, design and control of complex chemical systems. It is based on the plenary and selected invited talks given at the international conference “Engineering of Chemical Complexity” which had taken place in the summer of 2011 and was hosted by the newly established Berlin Center for Studies of Complex Chemical Systems. The book includes such topics as the manipulation and steering of single molecules and molecular motors, design of the systems based on active soft matter, control of collective dynamics and synchronization behavior, synthetic biology and the development of protocells. While organizing the conference and preparing this publication, we have relied on the assistance by Ingeborg Reinhardt, Ulrike Christine K¨ unkel and Rico Buchholz; we would like to express our sincere gratitude to all of them. We are pleased to acknowledge generous financial support provided for the conference from the German Research Foundation (DFG), from the Collaborative Research Center (SFB 910) “Control of Selforganizing Nonlinear Systems”, the Training Research Group (GRK 1558) “Nonequilibrium Collective Dynamics in Condensed Matter and Biological Systems”, and from the Solvay Company. Alexander S. Mikhailov and Gerhard Ertl Berlin, April 2012 v

November 21, 2012

13:3

9in x 6in

b1449-fm

Engineering of Chemical Complexity

This page intentionally left blank

November 22, 2012

9:2

9in x 6in

b1449-fm

Engineering of Chemical Complexity

CONTENTS

Preface

v

PART I INTRODUCTION

1

1. Analysis, Design and Control of Complex Chemical Systems

3

Alexander S. Mikhailov and Gerhard Ertl PART II SINGLE MOLECULES, NANOSCALE PHENOMENA AND ACTIVE PARTICLES 2.

Imaging and Manipulation of Single Molecules by Scanning Tunneling Microscopy

25 27

Leonhard Grill 3.

Self-Organization at the Nanoscale in Far-FromEquilibrium Surface Reactions and Copolymerizations

51

Pierre Gaspard 4.

Single Molecule and Collective Dynamics of Motor Protein Coupled with Mechano-Sensitive Chemical Reaction

79

Mitsuhiro Iwaki, Lorenzo Marcucci, Yuichi Togashi and Toshio Yanagida 5.

Nanomotors Propelled by Chemical Reactions

101

Raymond Kapral 6.

Biology of Nanobots

125

Wentao Duan, Ryan Pavlick and Ayusman Sen vii

November 21, 2012

13:3

9in x 6in

viii

b1449-fm

Engineering of Chemical Complexity

Contents

PART III REACTION–DIFFUSION SYSTEMS AND NONEQUILIBRIUM SOFT MATTER

145

7.

147

Wave Phenomena in Reaction–Diffusion Systems Oliver Steinbock and Harald Engel

8.

Self-Oscillating Polymer Gels as Smart Materials

169

Ryo Yoshida 9.

Stochastic Fluctuations and Spontaneous Symmetry Breaking in the Chemotaxis Signaling System of Dicyostelium Cells

187

Tatsuo Shibata 10. Mechanochemical Pattern Formation in the Polarization of the One-Cell C. Elegans Embryo

201

Justin S. Bois and Stephan W. Grill PART IV OSCILLATIONS AND SYNCRONIZATION

213

11. Synchronization of Electrochemical Oscillators

215

Mahesh Wickramasinghe and Istv´ an Z. Kiss 12. Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

237

Vladimir Garcia-Morales and Katharina Krischer 13. Quorum Sensing and Synchronization in Populations of Coupled Chemical Oscillators

261

Annette F. Taylor, Mark R. Tinsley and Kenneth Showalter 14. Collective Decision-Making and Oscillatory Behaviors in Cell Populations Koichi Fujimoto and Satoshi Sawai

279

November 21, 2012

13:3

9in x 6in

b1449-fm

Engineering of Chemical Complexity

Contents

15. Synchronization via Hydrodynamic Interactions

ix

299

Franziska Kendelbacher and Holger Stark PART V EVOLUTION, SYNTHETIC BIOLOGY, AND PROTOCELLS

321

16. Emergence and Selection of Biomodules: Steps in the Assembly of a Protocell

323

Susanna C. Manrubia and Carlos Briones 17. From Catalytic Reaction Networks to Protocells

345

Kunihiko Kaneko 18. Constructive Approach Towards Protocells

359

Tadashi Sugawara, Kensuke Kurihara and Kentaro Suzuki 19. Network Reverse Engineering Approach in Synthetic Biology

375

Haoqian Zhang, Ao Liu, Yuheng Lu, Ying Sheng, Qianzhu Wu, Zhenzhen Yin, Yiwei Chen, Zairan Liu, Heng Pan and Qi Ouyang Index

393

November 21, 2012

13:3

9in x 6in

b1449-fm

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Part I INTRODUCTION

1

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

This page intentionally left blank

2

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Chapter 1 ANALYSIS, DESIGN AND CONTROL OF COMPLEX CHEMICAL SYSTEMS Alexander S. Mikhailov∗ and Gerhard Ertl Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany ∗ [email protected] A characteristic property of complex systems, that consist of active autonomous elements, is to self-organize their collective activity. The task of an engineer is to use such self-organization processes, rather than to act against them. In this introductory chapter, we provide a short review of the history of research on self-organization in chemical systems, discuss its current state of the art and consider the perspectives.

Contents 1. Introduction . . . . . . . . . . . 2. A Brief Historical Overview . . 3. Recent Developments and Open References . . . . . . . . . . . . . . .

1.

. . . . . . . . . . . . . . Perspectives . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

3 4 16 22

Introduction

Classical physical chemistry is focused on investigations of individual molecules and of reaction events between them. It is tacitly assumed that the behavior of larger systems, with many kinds of molecules and various reactions, would represent a mere superposition of elementary processes. Once structures of molecules and interactions between them are understood, the prediction of the system’s dynamics should be straightforward. Traditionally, a similar approach is to be followed in chemical engineering. When a multi-component reaction process has to be designed, it is decomposed into individual steps and, for each of them, an appropriate 3

November 21, 2012

4

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

reactor is developed. Then, one needs only to combine the designed reactors into a production chain, so that a required chemical process is implemented. The control of a complex technological process is again reduced to steering of individual reaction steps, assuming that the overall performance should represent a superposition of elementary events. However, it becomes increasingly evident that such reductionist approaches may often fail when dealing with complex chemical systems. The collective behavior of a multi-component chemical system is typically nonlinear and thus different from a sum of elementary behaviors. Through interactions between the components, qualitatively new properties and effects can further emerge. Complex nonlinear systems tend to self-organize their collective dynamics. The self-organization phenomena have strong implications for the analysis, design and control of such systems. If a system is self-organizing, it cannot be cast into an arbitrary shape or operation pattern. Attempting to do this by brute force would destructively interfere with autonomous organization processes and often lead to an outcome which is much different from the original intention. To engineer a self-organizing system, one needs to create the conditions which would promote its intrinsic transition into a desirable autonomous mode. Thus, individual elements need to be chosen in an appropriate way and interactions between them have to be optimized — all based on the understanding how such individual properties contribute to the emergent autonomous collective behavior. If a self-organizing system should be controlled, autonomous organization processes inside it are to be steered through soft perturbations, so that the entire system is tempted to appropriately change its collective performance. We know that chemical complexity can be efficiently managed and controlled. This is obvious if one looks at biological organisms and, in particular, at a living cell. The cell can be viewed as a chemical factory where thousands of chemical reactions involving various biomolecules are running in parallel in a well-coordinated and self-organized way. It is able to operate functionally despite large variations in the environmental conditions (i.e., of the “supply and demand”). The entire factory is packed within a volume of about a cubic micrometer and can completely reproduce itself. Hence, the task is to explore the laws which underlie self-organization in complex chemical systems and to employ them consistently in the design and control of such systems.

2.

A Brief Historical Overview

In the first half of the 20th century, the discussion on whether the laws of physics were applicable to biological systems had been initiated. The

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

5

behavior of living organisms seemed to contradict so much the predictions of classical thermodynamics that many have suspected the existence of peculiar “vitalistic” forces, only valid in biology. In contrast, Ludwig von Bertalanffy, one of the founders of theoretical biology, insisted that physical laws should be universally valid. He pointed out that there are, however, principal differences in the conditions under which biological organisms function. While classical thermodynamics refers to equilibria in closed systems, biological organisms represent open systems which maintain themselves in continuous exchange with the environment. As von Bertalanffy wrote1 in 1940, “true equilibria in closed and stationary “equilibria” in open system show some similarity, in so far as the system in both, considered as a whole and in the view of its components, remains constant. However, the physical situation is fundamentally different in these two cases. The true chemical equilibria in closed systems are based on reversible reactions . . . ; they are, moreover, the consequences of the Second Law and are defined through the minimum of free energy. In open systems, in contrast to this, the steady state as a whole and eventually also many individual reactions are not reversible; moreover the Second Law is immediately applicable only to the closed systems, it does not determine the steady state”. Erwin Schr¨ odinger added further insights to the understanding of biological processes. In emigration in Dublin, he delivered a series of lectures in 1943, which have appeared2 a year later in the book “What is Life?”. His principal attention was paid to the analysis of physical processes responsible for the genetic transfer of information. His brilliant conclusion that information can only be stored at the molecular level was soon confirmed by J. D. Watson and F. Crick in their discovery of the DNA code. One of his lectures was, moreover, devoted to the question why self-organization is possible in biology. As stressed by Schr¨ odinger, all biological organisms represent open systems, maintained in their steady states because of the flow of energy or material passing through them. Together with such flows, entropy may be exported from an open system. The export of entropy can balance its production within the system, which must always take place out of thermal equilibrium. Thus, the entropy content of an open system can stay constant or even get lower as time goes on. Consequently, open systems (and biological organisms specifically) may increase their internal order with time. Such ideas have found their precise expression when Ilya Prigogine published3 in 1947 his book “Thermodynamics of Irreversible Processes”. In his words, “classical thermodynamics is an admirable, but fragmentary doctrine; this fragmentary character results from the fact that it is applicable only to states of equilibrium in closed systems. Therefore, it is necessary to establish a broader theory, comprising states of nonequilibrium as well as those of equilibrium”. Prigogine formulated the principles of the

November 21, 2012

6

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

new theory and showed how they can be applied to a broad variety of chemical systems. While providing a principal explanation why self-organization is consistent with the laws of thermodynamics, the above studies still did not explicitly consider the problems of spatio-temporal pattern formation in nonequilibrium systems. The decisive step came in the publication4 of the mathematician Alan Turing in 1952. The main interests of Turing were in the fields of algorithms, logic and cryptanalysis. During the war, he was the head of a section responsible for breaking German military codes. His work on algorithms has led to the development of digital computers, as they are known today. In addition, Turing was also interested in biological questions. The biological problem which he wanted to tackle was how, in the process of individual development of a biological organism (i.e., of the morphogenesis), patterns may spontaneously develop. Looking for an answer, Turing considered a reaction–diffusion system with two chemical species (morphogens) which were coupled through reactions and had different diffusion rate constants. By performing the linear stability analysis, Turing could demonstrate4 that, under certain conditions, the uniform stationary state can become unstable with respect to the development of stationary waves (periodic in space, but not oscillatory in time). The Turing instability typically occurs when one of the species effectively catalyzes its own production and therefore acts as an activator. The second species may represent an inhibitor, which lowers the activator reproduction rate. The instability is possible if the inhibitor species diffuses faster than the activator. In the same paper,4 Turing has also considered three-species reaction– diffusion models and showed that a different type of instability becomes then possible: periodic traveling waves with a well-defined wavelength and temporal period spontaneously develop starting from the uniform initial state. Today, such behavior is known as the “wave instability”, although it may be more appropriate to talk about the second-type Turing instability in this case. Spontaneous formation of stationary structures in chemical systems, resulting from a Turing instability, could be experimentally confirmed only in 1990 by de Kepper with coworkers5 and, shortly later, by Ouyang and Swinney6 (see Fig. 1). Observations of Turing patterns in biological systems have furthermore been reported.7 Another mathematician, also fascinated in the middle of the 20th century by biological phenomena, was Norbert Wiener. The scientific discipline of cybernetics, which he had put forward, was motivated by his analysis of control and communication in living organisms.8 Together with the Mexican cardiologist Arthur Rosenblueth, Wiener tried to understand

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

Fig. 1.

7

Turing patterns in a chemical reaction. From Ref. 5.

wave propagation phenomena in the cardiac tissue, responsible for the normal heart beat and for the pathological conditions, such as arrhythmia and fibrillation. While electrical excitation waves in the heart have been registered already at the beginning of the 20th century, detailed physical mechanisms of excitability were still unknown. In their theory,9 Wiener and Rosenblueth chose a phenomenological approach, modeling the cardiac excitable medium in terms of automata. They assumed that each element of the medium can exist in the states of rest, excitation and refractoriness. An external perturbation can move the element from the state of rest to the state of excitation, followed by the state of refractoriness and by the eventual return to the rest state. It has turned out that this simple approach could already explain a large number of known effects. The theory predicted9 that excitation waves should propagate at a constant velocity and annihilate under collisions. An excitation wave should be able to circulate around an obstacle, forming

November 21, 2012

8

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

Fig. 2. Spiral waves rotating in opposite directions around two circular obstacles. From Ref. 9.

a spiral wave. A spiral wave should have the shape of an involute of the obstacle around which it rotates. For circular obstacles, the spiral is therefore Archimedian. Figure 2, reproduced from the original paper9 , displays two spiral waves which rotate in opposite directions around two obstacles and annihilate under collision. The work by Wiener and Rosenblueth initiated modern research on mathematical modeling in cardiology. It had also a profound influence on the investigations of chemical systems. A biochemist, Boris Belousov, found in 1952 a chemical reaction which could show oscillations persisting over hours in a closed reactor. His attempts to publish the observations however failed, with the referees invariably pointing out that oscillations were not possible in chemical systems (only a short abstract10 has eventually appeared in press in 1959). Anatol Zhabotinsky began to investigate this reaction in his Ph.D. study, following a suggestion of his supervisor. Soon, Zhabotinsky could reproduce and analyze the chemical oscillations, and improved the reaction’s recipe.11 His Ph.D. thesis was defended in 1965, but he has continued to work on such problems. In parallel, a group of scientists around Israel Gelfand in the Department of Mathematics and Mechanics of the Moscow State University held regular meetings, devoted to mathematical models of the brain activity. The Wiener–Rosenblueth paper9 on excitable media has been known in this community and was extensively discussed. Zhabotinsky was in contact

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

Fig. 3.

9

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

(j)

(k)

(l)

Concentric spreading waves in the Belousov–Zhabotinsky reaction. From Ref. 12.

with this group (and, particularly, with Valentin Krinsky). Looking at the progress of the chemical reaction in thin layers, Zhabotinsky and his student, Zaikin observed the first self-organized wave pattern — a structure of concentric spreading waves, now called a target pattern (Fig. 3). The importance of this observation was immediately recognized and it has been reported12 in Nature in 1970. It is a lesser known fact that Zhabotinsky saw the spiral waves in 1971 (Fig. 4). The observations of spiral waves were first reported in conference proceedings13 and only later published in an international journal.14 At the end of the 1960s, Arthur Winfree entered the field, who subsequently made fundamental contributions to the study of self-organization phenomena in chemical systems. At a scientific conference in Prague in 1968, he had met Zhabotinsky and learnt from him the recipe of

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

10

Fig. 4.

Spiral waves in the Belousov–Zhabotinsky reaction. From Ref. 14.

the Belousov–Zhabotinsky reaction. Subsequently, he modified the recipe, so that an excitable (not oscillatory) medium could be developed. In 1971, Winfree published15 his experimental observations of spiral waves in Science, accompanied by a simple theory. The theory was based on the previous work by Wiener and Roseblueth — Winfree was familiar with their paper (one of the present authors keeps a copy of this paper with many hand-written notes by Winfree). Proceeding further, Winfree soon reported three-dimensional rotating scroll waves (Fig. 5) in the Belousov– Zhabotinsky medium.16

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

11

Fig. 5. Hand drawings of scroll waves by Winfree, provided to explain his experimental observations. From Ref. 16.

November 21, 2012

12

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

For a relatively long time, the Belousov–Zhabotinsky reaction and its modifications remained one of a few examples of chemical systems where complex spatiotemporal self-organization phenomena could be observed. In the beginning of 1990s, observations of self-organization phenomena were however extended to a different class of systems involving heterogeneous catalysis. In surface chemical reactions, molecules arrive via adsorption to the metal surface from the gas phase. Many metals act as catalysts, so that the adsorbed molecules on their surfaces can undergo reactions which do not naturally take place in the gas. Kinetic oscillations of the reaction rates have been observed with polycrystalline samples and supported small catalyst particles. Experiments with well-defined single crystal surfaces under ultrahigh vacuum conditions were first performed with the catalytic CO oxidation on platinum in the laboratory of Gerhard Ertl,17 where rapid oscillations were found with particular crystal planes. The analysis of their mechanism has revealed18 that they resulted from an interplay between adsorption, the oxidation reaction and an adsorbate-induced structural phase transition in the top surface layer of the Pt crystal. The slow surface phase transition was providing a negative feedback needed for the oscillation onset; its characteristic timescale was essentially determining the oscillation period. Depending on the parameters, the regime of an excitable medium could also be realized. To continuously monitor in a spatially resolved way the processes which go on the catalytic surface, a variant of the electron microscopy (PEEM, photo-emission electron microscopy) has been employed in the experiments by Rotermund and Ertl in the Fritz Haber Institute in Berlin. The PEEM observations of the CO oxidation reaction in its oscillatory and excitable regimes have revealed a fascinating picture of self-organization phenomena. Spiral waves, target patterns, standing waves and spatiotemporal chaos (reaction–diffusion turbulence) could be seen.19 As an illustration, Fig. 6 shows an observation of multiple spiral waves (the waves are pinned and rotate around obstacles of different sizes, so that their rotation periods and wavelengths are different). Theoretically, the effects of turbulence in oscillatory reaction–diffusion systems were first considered in 1976 by Yoshiki Kuramoto, who has shown that, near the onset of oscillations, a universal description of such phenomena in terms of the so-called complex Ginzburg–Landau equation is possible21 (see also Ref. 22). It should be noted that Landau and Ginzburg have never worked on chemical systems and their original equation (with real coefficients) referred to equilibrium superconductor phase transitions where oscillations or waves do not exist. Can chemical turbulence be efficiently controlled? In the experiments23 with CO oxidation on Pt in the Fritz Haber institute, the control has been

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

13

Fig. 6. Temporal evolution of a population of spirals with different rotation periods and wavelengths in the CO oxidation reaction on Pt (110). From Ref. 20.

implemented by introducing a global feedback loop, with the reactants’ pressure in the chamber made dependent on the integral monitored PEEM signal. It has been found that, by applying sufficiently strong feedbacks, turbulence could be suppressed and uniform periodic oscillations could be established on the surface. Moreover, at intermediate feedback strengths, intermittent turbulence (with the bursts of activity on the background of uniform oscillations) could also be found (Fig. 7) in such experiments, in agreement with the theoretical predictions made for the complex Ginzburg– Landau equation with global feedbacks.24 While much attention in the studies of self-organization phenomena has been paid to the processes of spatio-temporal pattern formation in continuous media, remarkable effects can be also found in discrete populations of coupled active units. Already Wiener had suggested7 that physiological rhythms might reflect mutual synchronization of myriads of individual oscillatory processes. The question had been further addressed25 in 1967 by Winfree who

November 21, 2012

14

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

Fig. 7. (A,B) Intermittent chemical turbulence observed in oscillatory CO oxidation reaction on Pt(110) under application of global delayed feedback. In the top row (A), three PEEM images of the surfaces are displayed. In (B), temporal evolution along the line (a,b) in the first image is shown (time varies along the horizontal axis). (C) Intermittent turbulence in the complex Ginzburg–Landau equation with global feedback. From Ref. 23.

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

15

had noted that interactions in a population of phase oscillators may result in spontaneous development of synchronized regimes. A systematic theory of synchronization behavior was developed by Kuramoto in 1984.22 He investigated large heterogeneous populations of oscillators whose natural oscillations frequencies could vary considerably. The theory predicted that, as the intensity of global coupling between the oscillators gets stronger, the population should undergo an autonomous transition to the synchronous regime. Near the transition point, a group of oscillators all of which oscillate at the same frequency, different from their various intrinsic frequencies, spontaneously emerges. As the coupling strength is further increased, this group grows, eventually comprising the entire population. For almost 20 years, these important predictions could only be partially and indirectly confirmed in physical experiments. The decisive demonstration has been presented26 in 2002 by Istvan Kiss and John Hudson who investigated collective behavior in populations of globally coupled electrochemical oscillators. Their observations have not only confirmed the existence of the synchronization transition, they have also revealed good quantitative agreement with the theory. As shown in Fig. 8(a), a synchronous group of oscillators with the (collective) oscillation frequency different from their natural frequencies is formed after the transition. Moreover, the synchronization order parameter shows (Fig. 8(b)) a sharp increase above the transition point.

Fig. 8. Experimental observation of synchronization in a population of globally coupled electrochemical oscillators. (a) Oscillation frequencies of different oscillators vs. their natural frequencies in absence of coupling. (b) Dependence of the synchronization order parameter on the coupling strength. From Ref. 26.

November 21, 2012

16

3.

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

Recent Developments and Open Perspectives

Today, investigations of nonequilibrium complex systems, considered esoteric several decades ago, belong to the principal research directions in statistical physics. Specialized journals have been established and thematic international meetings are regularly held. There has also been a strong progress in the field of complex systems of chemical origins. We are not going to review this broad field in this chapter. Indeed, this is extensively achieved by many authors contributing to the present volume. Therefore, we shall provide below only some general comments which express personal attitudes of the authors and are illustrated by examples from our own research. The original analysis of nonequilibrium open systems by Bertalanffy, Schr¨ odinger and Prigogine was applicable to various kinds of systems. Subsequently, most attention has been paid, however, to the investigations of nonequilibrium pattern formation in reaction–diffusion systems, such as the Belousov–Zhabotinsky reaction. This has been certainly justified by the fact that such systems show a wide variety of nonequilibrium patterns, which are readily observable and may also be theoretically understood relatively easily. Such research goes on and, for example, interesting results are obtained when three-dimensional wave patterns in excitable media are considered. However, systems with reaction and diffusion represent only a certain class of complex chemical systems. Waves in classical reaction–diffusion systems are not accompanied by mass transport. Such traveling structures are time-dependent concentration patterns, where the local concentration of a species would grow because it has been produced at higher rate, rather than because this species has traveled from another area. The situation is different when chemical reactions are taking place in the systems collectively known as “soft matter”. The characteristic feature of such systems is that there are weak interaction forces acting between the particles. As a result, such matter goes into a condensed state. The cohesion forces are however weak and therefore the developing structures are soft and labile, easily transforming from one into another when environmental parameters are changed. The presence of various structural transitions is typical for equilibrium soft matter. When chemical reactions are taking place, they can interfere with structural phase transitions in weakly condensed systems, leading to a variety of nonequilibrium phenomena. Both stationary and traveling waves may spontaneously emerge. More complicated time-dependent structures are also possible. There are two principal aspects which distinguish selforganized structures in reactive soft matter from nonequilibrium concentration patterns in reaction–diffusion systems.27 First, such structures are characterized by cohesion and, if they are traveling, mass transfer takes

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

17

place. Second, their characteristic lengths may extend down to nanometers, making nanoscale self-organization phenomena possible. The paradigmatic example of pattern formation in nonequilibrium soft matter is provided by the effect of phase separation in the presence of a chemical reaction.28,29 Nonequilibrium phenomena in gels with chemical reactions, analyzed by Yoshida in the present volume, represent another important class of such systems. Inorganic surface reaction systems can also be often viewed as soft matter. An interplay between the reaction and the surface phase transition is already needed for the emergence of oscillations and excitability in CO oxidation on Pt. Attractive or repulsive interactions between the adsorbed particles on metal surfaces are usually present. Such lateral interactions can be so strong that spinodal decomposition of surface species takes place. When a reaction is added, nonequilibrium stationary structures with the characteristic wavelengths controlled by the reaction rate become possible and can be observed.30 A detailed discussion of nonequilibrium microstructures in reactive soft-matter monolayers can be found in our review paper.31 In soft matter, chemical reactions can, as we have already noted, lead to material flows and transport of molecules. Furthermore, sustained active motion of particles can develop in systems far from thermal equilibrium. As everybody knows, biological organisms, with the exception of plants, are able to move themselves. Animals are using coordinated internal motions to fly in air, swim in water and walk or run over a surface. Active motion is also possible for single cells. Indeed, many bacteria can swim through the medium by cyclically changing their shape. It has been shown that swimming is possible even for single macromolecules. Protein machines, that cyclically change their configuration under ATP hydrolysis reaction, may represent micro-swimmers.32 There is, however, also a different mechanism of active motion in systems with chemical reactions. In an external surfactant’s gradient, a physical particle would move since the surface forces acting on it are not balanced. It may however also be that the particle is itself producing the surfactant species (e.g., through a catalytic reaction) and releasing it into the surrounding medium. In this way, a non-uniform distribution of the surfactant around the particle can be established. If the particle is elongated (i.e., represents a rod) and the surfactant is released closer to one of its ends, surface forces acting on the particle are not mutually compensating and a net force propelling it in a certain direction appears. Thus, active autonomous propulsion becomes possible. A more detailed analysis shows33 that self-propulsion can also take place for spherical particles, when the asymmetry of the spatial surfactant distribution, caused by the motion of a particle, is taken into account.

November 21, 2012

18

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

The mechanism of self-propulsion, involving surface forces and chemical reactions, is so simple that the respective phenomena may be observed at very small length scales, for microscopic particles. As discussed by Sen in this volume, the effects of active motion may be relevant even for the particles representing individual enzyme molecules. For microscopic particles, such as single molecules, propulsion forces would usually be small as compared to thermal fluctuation forces and, therefore, the active motion of a single particle would be weak as compared with its diffusive Brownian motion. Interactions between many actively moving particles can lead however to the emergence of spontaneous active macroscopic flows and to the formation of autonomously traveling clusters, or “swarms”. Investigations of active fluids are an important part of the modern theory of self-organization phenomena. Studies of synchronization effects in chemical systems, already mentioned in Sec. 2, have progressed rapidly in the last years. Here, the research becomes closely linked to novel developments in the mathematical theory of nonlinear dynamical systems. Not only the emergence or breakdown of synchronization, but also the effects of clustering and transition to chaos, are currently investigated. In addition to the experiments with electrochemical oscillators, systems of coupled chemical reactors and populations of interacting biological cells are being considered. A large special class of complex systems is constituted by networks of diffusively coupled reactors. Synchronization effects have been extensively discussed for the networks, see Refs. 34 and 35. However, oscillations represent only one of many interesting aspects of network dynamics. Spreading of infections over the networks is a problem of great importance which has been thoroughly investigated.36 Generally, analogs of almost any self-organization behavior known for continuous reaction–diffusion media can be considered and analyzed for the networks. The Turing instability in the network-organized activator– inhibitor systems has already been studied.37 Taking into account remarkable progress in understanding of complex chemical systems, it could have been expected that various practical applications were already found. Nonetheless, they still remain rare. This can be explained by several reasons. Self-organization phenomena have so far been often investigated in the systems where they are clearly pronounced, but which have no direct use. Concentration waves in the Belousov–Zhabotinsky reaction are fascinating, but this reaction proceeding in a weak aqueoues solution has no significant use beyond serving as a good lecture demonstration. On the other hand, heterogeneous catalysis has many important applications, but waves and self-organized patterns have so far been mostly investigated under very special conditions (ultra-high vacuum, perfect crystals) which are not typical for such applications.

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

19

There are, however, many practically important chemical systems where effects of self-organization should play a principal role, even though this is frequently overlooked. Consider, for example, corrosion of metals. Obviously, the metal is a system far from its equilibrium, which would have been the oxide. Some metals corrode rapidly in certain environments, with the entire material converted to the oxide in a while. Remarkably, corrosion is however strongly retarded in stainless steel. Even under aggressive conditions, this material remains stable and is therefore broadly employed. What is the origin of natural corrosion resistance in the steels? It is not that the oxidation reaction itself is much slowed down in such substances. Exposed to oxygen, the clean steel surface would start to react with it, producing the oxide. However, the corrosion product would quickly form a thin firm oxide film. This film completely covers the surface and protects it by preventing the arrival of oxygen to the underlying clean surface. At a first glance, the mechanism of corrosion protection in stainless steel may look quite simple. Indeed, to prevent metals from corrosion, they would be often painted or smeared with oil, so that a thin corrosion-inhibiting film is brought onto their surface. There is, however, a principal difference: If the surface is scratched and the protective film is locally damaged, it needs to be applied again. Ideally, one needs to monitor the surface and, once a damage is recorded, interfere by locally applying new paint (in practice, surface monitoring is never undertaken and the paint is just periodically refreshed). The situation is different in stainless steel. Here, the protective film is created autonomously, not externally applied. If the steel surface is scratched and the nanometer-thin oxide layer is locally damaged, the oxidation reaction immediately begins and the protective film is recovered (healed) within a fraction of a second. Thus, high corrosion resistance of stainless steel is a consequence of self-organization phenomena taking place on their surfaces. In highly aggressive environments, self-organized surface protection can nonetheless break down and pitting corrosion may develop on stainless steels. Traditionally, the onset of pitting corrosion has been linked to the appearance and growth of individual micrometer-sized corrosion sites (pits). When corrosion onset has been monitored in situ using optical microscopy, a different behavior was however found.38 It could be seen that pits stimulated the development of further pits in their neighborhood. Spreading “infection” fronts were therefore observed on the steel surface and such fronts have turned out to be responsible for the corrosion onset. This example suggests that self-organization may be playing a principal role already in the processes which are commonly found around us. Much can be learned when the knowledge of self-organization processes is appropriately applied. There is also a different aspect of corrosion phenomena on stainless steels. Does not the behavior seen on the steel surface already resemble what

November 21, 2012

20

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

we know for biological organisms? The skin protects the animal body, with delicate and highly tuned processes going within it, from environmental hazards. If an animal is injured and the skin gets scratched, various selforganized healing mechanisms begin to operate. First, blood coagulates densely filling the scission. Then, the adjusting skin cells start to reproduce. Thus, after some time, the skin integrity is recovered and body protection is restored. Although the importance of self-organization in biology has been already recognized by von Bertalanffy, Schr¨ odinger and Prigogine, the concepts of self-organization have only recently started to be applied, on a large scale, to the analysis of biological behavior. Despite its great advances, molecular cell biology has, for a long time, been based on the classical notions of chemical kinetics and equilibrium statistical physics. Traditionally, explanations have been sought at the level of structure of individual involved biomolecules, rather than in the terms of their collective performance — or an emergent function. Today, we know that classical reaction–diffusion phenomena, such as traveling fronts or excitation waves, are possible inside the cells and play an important role there. A living biological cell is a micrometer-size heterogeneous reactor where thousands of different chemical reactions run. Some of the reactions are strongly correlated and mutually tuned, others proceed independently — despite the fact that they are not physically separated. The operation of the entire reactor is self-organized and the whole “factory” is able to completely reproduce (i.e., to replicate) itself. Most of biochemical reactions involve macromolecules, such as proteins, RNA or DNA. Not only chemical structures, but also physical conformations of such large molecules may change as a result of reaction events, such as ligand binding or detachment. In enzyme catalysts, these conformational changes can enable or facilitate reaction events, by bringing the reactants together and/or to the active center in the enzyme molecule. In motor proteins, cyclic conformational changes are converted through ratchet mechanisms to steady translational or rotation motions, making persistent force generation possible. It is a challenge to understand how molecular machines are operating. Extensive research, employing advanced experimental tools, is undertaken at present in this field and significant progress has been already achieved. Next, it should become possible to purposefully modify the existing protein machines, to combine them or to create completely novel active macromolecular devices. Thus, we expect that, in the near future, much attention will be given to biological nanoscale phenomena. Of course, funding agencies have been strongly supporting various nanoscience projects over the last two decades. But, looking closer, one can

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

21

notice that the supported projects usually deal only with static nanostructures approaching equilibrium conditions. Manufacturing of specific rigid nanoscale structures often represents the main purpose of the research. While proceeding from macro- to microscales, the developers still think in a classical way, assuming that individual elements should first be obtained and then combined according to a rational blueprint. There is a deep gap separating current nanotechnology from the operation mechanisms of a biological cell. It is true that all major functions of a cell also involve nanoscale objects. However, both these objects and the manner in which they interplay are different. In biology, individual nanoscale entities are soft; they are moreover subject to intensive thermal fluctuations and genetic variability. Interactions between them are not externally imposed according to a pre-designed blueprint, but arise autonomously through self-organization processes. Despite strong fluctuations, self-organized ensembles of active soft nanoscale objects are capable to operate in a persistent and predictable way under conditions where the operation based on the classical technological approach would have been completely excluded. These profound differences suggest that transition to organic nanotechnology, analogous in its principles to the operation mechanisms of biological systems, would not be simple and straightforward. Not only completely different kinds of elements have to be developed — it is also essential to implement the processes through which the desired active collective behavior of such elements would become self-organized and to create, furthermore, the environments which would allow the constructed artificial systems to “live” inside them. In the framework of bionics, it has been previously attempted to incorporate some biological mechanisms into technological applications. Looking back, we recognize that, too often, this has been done in rather naive ways, by trying to straighforwardly copy or imitate the actual biological systems. Of course, this usually had to be done because deep understanding of the underlying biological behavior was still missing. Even genetic engineering, subject to current public controversy, is essentially based only on a transfer of some existing functional properties from one biological organism to another. In contrast, a different approach is characteristic for modern synthetic (or constructive) biology. Starting from existing molecular components, completely new systems with desired functional properties are being developed. As an example, one can refer here to the current research on synthetic genetic circuits capable of generating required oscillations, operate as switches or give rise to synchronization effects. As the knowledge of the involved phenomena gets progressively improved, such approaches may,

November 21, 2012

13:1

22

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

A. S. Mikhailov & G. Ertl

in the future, gradually transfer to different systems, such as, e.g., the protein signal transduction networks. The ultimate aim of constructive biology would be to obtain artificial cells capable of metabolism and self-reproduction. The research on protocells is undertaken worldwide and remarkable advances have been already reported. Once artificial cells become available, a broad range of novel technological applications shall be opened. However, this would also lead to the necessity of new fundamental research. In most practical applications, artificial cells would need to operate in large ensembles or “organisms”. Hence, one would need to develop and implement the interactions between individual cells which would lead to the desired collective behavior of their large populations. It would be probably very difficult or even impossible to rationally design artificial multi-cellular systems. Here, it should be noticed that all actual biological macroorganisms are products of natural biological evolution. It may be therefore important to engineer evolution processes in synthetic living systems, in such a way that the desired organization and functions are acquired. In order to do this, much better knowledge of evolutionary processes would need to be reached. In this chapter, we have provided a brief survey of the history of research on self-organization in chemical systems, looked at the current state of this research and we have tried to outline its important future directions. Various aspects, essential for engineering of complex chemical systems, are reviewed in detail in the following chapters written by outstanding experts in the respective fields. References 1. L. von Bertalanffy, Naturwissenschaften 33, 34 (1940). 2. E. Schr¨ odinger, What is Life? The Physical Aspect of a Living Cell (Cambridge University Press, Cambridge, 1944). 3. I. Prigogine, Etude Thermodynamique des Phenomenes Irreversibles (Dunod, Paris, 1947). 4. A. Turing, Philos. Trans. R. Soc. Lond. B 237, 37 (1952). 5. V. Castets, E. Dulos, J. Boissonade and P. D. Kepper, Phys. Rev. Lett. 64, 2953 (1990). 6. Q. Ouyang and H. L. Swinney, Nature 352, 610 (1991). 7. S. Sick, S. Reinker and T. Schlake, Science 314, 1447 (2006). See also Ph. Maini, R. E. Baker and Ch.-M. Chuong, Science 314, 1397 (2006). 8. N. Wiener, Cybernetics, or Control and Communication in the Animal and the Machine (MIT Press, Cambridge, 1948). 9. N. Wiener and A. Rosenblueth, Arch. Inst. Cardiol. Mex. 16, 205 (1946). 10. B. P. Belousov, “A periodic reaction and its mechanism,” in: Collection of Short Papers on Radiation Medicine for 1958 (Med. Publ. Moscow 1959). See

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

Analysis, Design, Control of Complex Chemical Systems

11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

23

also the English translation of his original unpublished manuscript of 1951 in Oscillations and Traveling Waves in Chemical Systems, eds. R. J. Field and M. Burger (Wiley, New York, 1985), p. 605. A. M. Zhabotinsky, Biofizika 9, 306 (1964). A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535 (1970). A. M. Zhabotinsky and A. N. Zaikin, “Spatial effects in a self-oscillating chemical system,” in Oscillatory Processes in Biological and Chemical Systems, ed. E. E. Selkov, (Science Publ. Puschino, 1971). A. M. Zhabotinsky and A. N. Zaikin, J. Theor. Biol. 40, 45 (1973). A. T. Winfree, Science 175, 634 (1972). A. T. Winfree, Science 181, 937 (1973). G. Ertl, P. R. Norton and J. R¨ ustig, Phys. Rev. Lett. 49, 177 (1982). K. Krischer, M. Eiswirth and G. Ertl, J. Chem. Phys. 96, 9161 (1992). S. Jakubith, H. H. Rotermund, W. Engel, A. von Oertzen and G. Ertl, Phys. Rev. Lett. 66, 3083 (1991). S. Nettesheim, A. von Oertzen, H. H. Rotermund and G. Ertl, J. Chem. Phys. 98, 9977 (1993). Y. Kuramoto and T. Yamada, Prog. Theor. Phys. 56, 679 (1976). Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984). M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund and G. Ertl, Science 292, 1357 (2001). D. Battogtokh and A. S. Mikhailov, Physica D 90, 84 (1996). A. T. Winfree, J. Theor. Biol. 16, 15 (1967). I. Z. Kiss, Y. Zhai and J. L. Hudson, Science 296, 1676 (2002). A. S. Mikhailov and G. Ertl, Science 272, 1596 (1996). S. C. Glotzer, E. A. D. Marzio and M. Muthukumar, Phys. Rev. Lett. 74, 2034 (1995). M. Motoyama and T. Ohta, J. Phys. Soc. Jpn. 66, 2715 (1997). A. Locatelli, T. O. Mentes, L. Aballe, A. S. Mikhailov and M. Kiskinova, J. Phys. Chem. A 110, 19108 (2006). A. S. Mikhailov and G. Ertl, ChemPhysChem 10, 86 (2009). T. Sakaue, R. Kapral and A. S. Mikhailov, Eur. Phys. J. B 75, 381 (2010). A. S. Mikhailov and V. Calenbuhr, From Cells to Societies: Models of Complex Coherent Action (Springer, Berlin, 2002). S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Phys. Rep. 424, 175 (2006). A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreono and C. Zhou, Phys. Rep. 469, 93 (2008). A. Barrat, M. Bartelemy and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008). H. Nakao and A. S. Mikhailov, Nature Phys. 6, 544 (2010). C. Punkt, M. Blscher, H. H. Rotermund, A. S. Mikhailov, L. Organ, N. Budiansky, J. R. Scully and J. L. Hudson, Science 305, 1133 (2004).

November 21, 2012

13:1

9in x 6in

b1449-ch01

Engineering of Chemical Complexity

This page intentionally left blank

November 22, 2012

9:2

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Part II SINGLE MOLECULES, NANOSCALE PHENOMENA AND ACTIVE PARTICLES

25

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

This page intentionally left blank

26

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Chapter 2 IMAGING AND MANIPULATION OF SINGLE MOLECULES BY SCANNING TUNNELING MICROSCOPY

Leonhard Grill Department of Physical Chemistry, Fritz Haber Institut of the Max Planck Society Faradayweg 4-6, 14195 Berlin, Germany [email protected] The scanning tunneling microscope (STM) is not only used to image single atoms and molecules on a surface, but also to manipulate them in a controlled way. This work aims to summarize the pioneering and most representative examples in this active research field. After an introduction into the basics of the method, the topographic and electronic origin of the images and the resulting “chemical contrast” are discussed. In addition to imaging, molecular orbitals and the chemical nature of adsorbates can be identified by spectroscopy, even if their images are equivalent. Different types of molecular manipulation are presented, including examples for all three possible driving forces: Interatomic forces without a bias voltage, electron-induced manipulation and electric-field induced processes. The lateral manipulation of molecules, including the hopping and rolling of a molecular wheel, and vertical pulling experiments are discussed. The latter ones lead to particular configurations that allow conductance measurements of single molecules between two electrodes.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . Imaging and Spectroscopy of Single Molecules 2.1. Imaging single molecules . . . . . . . . . 2.2. Chemical identification by spectroscopy . 2.3. Imaging of diffusion processes . . . . . . .

27

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

28 31 31 32 35

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

28 3.

Manipulation of Single Molecules . . . . . . . . . . . . . . 3.1. Manipulation without bias voltage . . . . . . . . . . . 3.2. Electron-induced manipulation . . . . . . . . . . . . . 3.3. Electric-field induced manipulation . . . . . . . . . . . 3.4. Lateral manipulation: Hopping vs rolling . . . . . . . 3.5. Vertical manipulation: Pulling single molecules from a References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . . . . . . . . . . . . . . . . . . surface . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

37 39 40 42 43 45 47

Introduction

The invention of the scanning tunneling microscope (STM) by Binnig, Gerber, Weibel and Rohrer1 –4 in 1982 had opened new fascinating possibilities in the investigation of matter at the atomic scale and in real space. Apart from diffraction techniques, which give insight into the reciprocal space of a crystal lattice, and optical microscopes that are limited in the resolution by their large wavelength, only very few techniques obtain truly atomic resolution: One is the field ion microscope (FIM), invented in 1951 by M¨ uller,5 which was the first instrument that allowed to “see” single atoms and molecules6 in an experiment by giving highly magnified images of a specimen, where surface corrugations — down to single atoms — can be imaged due to the enhanced electric field in their vicinity. The other one is the electron microscope, developed in 1930s by Ruska,7 which can also achieve atomic resolution. However, it often requires specific sample preparations and uses high energy electron beams that can damage the sample. STM allows to image surfaces and adsorbates with atomic resolution, and to manipulate single atoms and molecules. It thus offers exciting opportunities to scientists, because it opens the possibility to control things on a small scale. This idea was introduced back in 1959 by Feynman in his speech “There is plenty of room at the bottom”8 when he talked about “the problem of manipulating and controlling things on a small scale”. While its technical realization, the theoretical description and the interpretation of the images are challenging,9 the concept of STM is rather simple: A sharp tip is made to approach close (below 1 nm) to a surface until a tunneling current sets in, due to the small distance and as a consequence of an applied bias voltage between the two electrodes. These two (tip and surface) must, of course, be electrically conducting. Such a procedure requires an extremely high precision in the tip motion, which is provided by the use of Piezo elements that change their shape and dimensions in a reliable way upon applying a bias voltage.10 Moreover, the system must be very stable, which is usually achieved by damping elements that mechanically decouple the instrument from the environment. Further stability is obtained by performing the experiments at very low temperatures that suppress

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

29

molecular and atomic motion and allow the imaging and manipulation of single atoms and molecules on surfaces.11 The most important feature of STM, which is responsible for the high resolution of the instrument, is the strong dependence of the tunneling current on the tip-sample distance. Due to quantum mechanics, it depends exponentially on the electrode distance, thus decreasing (increasing) strongly if the tip is only slightly retracted from (approached to) the surface. In a first approximation, the current I depends on the distance d in the following way (m is the electron mass, φ the effective work function and
is Plancks constant): I ∝ e−2κd with

(1)

κ2 = (2mφ)/
2 .

Due to this exponential dependence, the tunneling current is an extremely sensitive measure for the tip surface junction, both in terms of distance (i.e., topography of the sample) and electronic transparency (i.e., chemical composition of the junction). Once the tip position is stabilized and controlled via the Piezo elements and the very small tunneling current (in the range of nA and below) is measured, the instrument can be operated in two modes: (1) Constant-height mode, where the tip is scanned across the surface at a constant height and the tunneling current variations reflect the surface morphology and (2) the constant-current mode, where the current is kept constant by an electronic feedback loop and the variation of the tip height during scanning is the measured signal that corresponds to the surface structure (as in Fig. 1). The latter is the more common one, because the first holds the risk of crashing the tip for corrugated surfaces, which can be used as an advantage in manipulation experiments of adsorbed molecules as described below. The choice of the tip height plays a fundamental role in imaging and manipulation of single molecules on surfaces. In the case of imaging, any interaction between the tip and the surface or adsorbates must be avoided, because the measurement technique should not alter the properties of the measured object. Thus, the tip is fixed at rather large heights, but still close enough to the surface to ensure a sufficient tunneling current and a high imaging resolution. On the other hand, interaction between the tip and a molecular adsorbate is desired in the case of manipulation and the tip is therefore approached towards the surface to maximize the interaction in such experiments. In general, static conditions (i.e., without adsorbate motion) need to be achievable for a reliable characterization of the surface, which can be challenging in the case of small diffusion barriers, requiring cooled samples to freeze thermal motion and obtain stable adsorbates.12 There

November 21, 2012

30

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 1. “Chemical contrast” of an STM junction with two different atomic or molecular species adsorbed on the surface that appear at different height. The STM tip is scanned over the surface at constant current.

are, however, also cases where adsorbate motion is induced on purpose at elevated temperatures as will be discussed below. An important property in the analysis of STM images is the electronic origin of the tunneling current, which depends not only on the width and height of the tunneling barrier, but also on the electronic structure in the gap between the two metallic electrodes. This so-called “electronic transparency” leads to characteristic apparent heights of adsorbed species in an STM image (taken in the constant-current mode). An example is given in Fig. 1, where two adsorbed species are assumed to be very similar in their dimensions but different in their chemical nature and hence electronic structure. Consequently, their height profiles (i.e., the tip height at constant current during a single scan line as indicated in the figure) show two protrusions with different heights. This effect, which is often called the “chemical contrast”, gives information about the chemical nature of the junction that goes beyond the surface topology and arrangement of adsorbates. It is of particular interest for organic molecules or nonconducting layers that, due to their reduced conductivity, they exhibit a smaller electronic transparency than metal adatoms and are thus imaged at an apparent height that is lower than their real height.13 The chemical contrast can also be used to distinguish different adsorbed species that appear similar in the STM images with a metallic tip, but

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

31

very different if a molecule is attached to the tip apex, because the electron transparency is strongly changed.14 In the following, usage of STM to “see” (i.e., imaging and spectroscopy) and “touch” (i.e., manipulation) matter at the atomic scale, focusing on molecules will be discussed. There are various reasons why such studies at the level of single molecules are of interest, rendering this research field very active: • Fundamental understanding of processes on surfaces: Adsorption and diffusion properties, molecular conformations and electronic structure. • “Single-molecule chemistry”:15 Chemical reactions, dissociation of molecules, isomerization processes and triggering specific molecular functions (e.g. switching as will be discussed below). • Single-molecule vs. ensemble measurements: There are different advantages when studying single molecules instead of ensembles: (i) No averaging, which allows the characterization of rare species, sites, configurations and pathways. (ii) The dependence on the atomic-scale environment (e.g. step edges and defects) can be studied. (iii) Various properties (e.g. conductance, adsorption, dipole moment) are not directly scalable between the single molecule and the ensemble. (iv) Intermolecular linking can be imaged in real space. The knowledge that is gained from such studies is of importance for a number of different possible applications as e.g. nanotechnology, novel materials, sensors and molecular electronics. 2. 2.1.

Imaging and Spectroscopy of Single Molecules Imaging single molecules

After the pioneering studies of Binnig and Rohrer in obtaining STM images of different solid surfaces as CaIrSn4,1 silicon,2 and gold,3 it took only a few years until the first STM images of single molecules were achieved.16 Figure 2 shows one of the first results, where single Cu-phthalocyanine molecules could be imaged on a copper surface.17 Note that these molecules were also among the first species studied by field emission microscopy6 and electron microscopy.18 The STM image (Fig. 2(a)) reveals that individual molecules appear with a characteristic internal structure and can thus be resolved with sub-molecular resolution. The corresponding patterns are very similar to the superimposed sketch in Fig. 2(b) and can thus be assigned to

November 21, 2012

32

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 2. (a) Chemical structure of a Cu-phthalocyanine molecule above a Cu(100) surface. Small (large) open circles are C (Cu) atoms and small (large) filled circles are H (N) atoms. (b) STM image (image size about 29 × 37 ˚ A2 , tunneling parameters: −0.15 V, 2 nA) of single molecules on a Cu(100) surface. A gray scale representation of the HOMO, evaluated 2 ˚ A above the molecular plane, is superimposed. Reprinted with permission from Ref. 17, copyright (1989) by the American Physical Society.

the electronic structure of the molecules. This pioneering work demonstrates that it is not only possible to image single molecules in a stable fashion, but that much more information other than their external shapes and contours can also be obtained by STM. 2.2.

Chemical identification by spectroscopy

It directly follows from the working principle of a scanning tunneling microscope that an image consists of topographic information, and more importantly, electronic information of the studied surface, often named chemical contrast. A characteristic example is the imaging of molecules with different kinds of tips that can completely change the molecular appearance, even from depressions to protrusions in the case of CO molecules on Cu(111).14 Another example of an electronically originating contrast is the observation of electron standing waves in surface states that are observed on noble metal fcc surfaces in (111) orientation. The electrons in the surface state can move rather freely parallel to the surface until they are scattered at a defect (e.g. a step edge or an adsorbate). By constructing artificial atomic structures on a surface, Eigler and co-workers showed for the first time how these standing waves can be controlled in their shape in confined

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

33

areas.19 Furthermore, they also were used to precisely localize the parts of an adsorbed molecule that interact with the surface underneath.20,21 Additional to spatial imaging of the electron standing waves in such a surface state, spectroscopy can be used to detect its energy position. This is done by taking dI/dV curves, which are a measure of the local density of states at the given bias voltage.23 In the same way, molecular orbitals can be detected at a specific tip position within a single molecule. First results could show that the tunneling current depends on the applied bias voltage and the electronic structure in the junction, detecting molecular orbitals in dI/dV spectra.24,25 G. Meyer and co-workers showed in an elegant manner how molecular orbitals of a single molecule can be mapped spatially. This is done by detecting the dI/dV signal at the energy of a specific molecular orbital while scanning over the molecule, in very good agreement with theoretical predictions (Fig. 3(b)).22 When decoupling the molecules from the metal substrate by ultrathin NaCl films (with a large band gap), the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are detected in the dI/dV curves. Consequently, an increase of the I(V) signal is found at these positions (Fig. 3(a)). In addition to the detection of the electronic density on the surface or in the adsorbate, tunneling spectroscopy can also be used to identify inelastic scattering processes of tunneling electrons. In such processes, electrons loose energy during scattering, which can cause electronic or vibrational excitations of the adsorbate. Pioneering results were obtained by W. Ho and co-workers, who showed that vibrations can be induced and detected in single molecules, measuring their characteristic energy.26 Figure 4 shows such an example where specific (C-H and C-D stretch) vibrations are detected in individual acetylene isotopes on Cu(100) that were all deposited together onto the surface under ultra-high vacuum conditions. In STM image (Fig. 4(a)), three individual molecules can be seen, which reveal the same appearance, rendering an identification of the different isotopes impossible. However, spectroscopy allows detection of inelastic electron tunneling processes that give rise to characteristic molecular vibrations. Such vibrations are visible in the second derivative d2 I/dV 2 spectra and indeed peaks are observed at different positions in the spectra (Fig. 4(b)) for the — at first glance — equivalent molecules in STM image. Specifically, three species are identified from the molecular vibrations that depend in a very characteristic way on the mass of the involved atoms: Either C2 H2 , C2 D2 or a mixture of both (C2 HD). Hence, it could be shown that the spectroscopic fingerprints allow a clear identification of the chemical composition of individual molecules. Similar studies were done with a variety of molecular species.15

November 21, 2012

34

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 3. Imaging of molecular orbitals by STM. (a) I/V (current) and dI/dV spectroscopy at the center of a pentacene molecule on NaCl. (b) STM images of a pentacene molecule on a thin NaCl film on Cu(111) and DFT calculations of a free molecule in the lower panel for comparison. The geometry of the free pentacene molecule is displayed in the lower center image. Reprinted with permission from Ref. 22, copyright (2005) by the American Physical Society.

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

35

Fig. 4. (a) STM image (56 × 56 ˚ A2 , 50 mV sample bias, 1 nA tunneling current) and (b) single-molecule vibrational spectra of three acetylene isotopes on Cu(100) at 8 K. Spectra are averaged from 16 scans of 2 mins each. Note that the molecules do not diffuse in the image as measurements were done at cryogenic temperatures. Reprinted with permission from Ref. 26, copyright (1999) by the American Physical Society.

2.3.

Imaging of diffusion processes

STM is typically used to image static situations on a surface, simply because the scanning rate is usually much slower than the dynamic processes, which is not a problem if adsorbates are stable. If on the other hand, the diffusion barrier is too low at the given temperature then adsorbate atoms and molecules can also diffuse during imaging. This might not be visible in the images if it is too fast (e.g. for adatoms with very small diffusion barriers on noble metal surfaces at room temperature27 ) but leads to fuzzy and striped images, if the adsorbate motion is comparable to the scanning speed as individual adsorbates hop away after the STM tip has already partially imaged them.12 In addition to the completion of a close-packed monolayer that suppresses adsorbate diffusion,12 there are two ways to avoid such problems. They allow imaging of dynamic processes, hence add time-resolution to the high spatial resolution of the STM: The first is based on improvements of the scanner head and, in particular, of the STM operation electronics, because data acquisition is a major bottle neck for the time resolution of the instrument, as realized by Rost et al.28 and Esch et al.29 On the other hand, a pump-probe concept has been used to induce and read-out dynamic processes by two subsequent pulses at a defined interval.30 Note that the first method obtains complete images one-by-one in a very fast mode (more than 200 Hz), while the latter is based on a fixed tip position

November 21, 2012

36

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

over the surface, which of course gives a much higher time resolution (in the ns range) in this single point. The second solution, which was realized earlier and is discussed in the following, uses a variation of the sample temperature because cooling reduces adsorbate motion and renders diffusion processes visible. Specifically, a precise control over the sample temperature can be used to properly modify the diffusion and adapt it to the imaging speed, thus

Fig. 5. (a–b) Subsequent STM images (25 × 24 nm2 , T = 76 K) of the same area of O2 molecules on a Ag(110) surface, taken at a time delay of 200 s. (c) Ball model for molecular oxygen adsorbed on Ag(110). Black ovals indicate adsorbed O2 molecules, white and gray circles represent the top and second layer atoms of the substrate, respectively. (d) Hopping rate of single oxygen molecules as a function of the temperature in an Arrhenius plot. Reprinted with permission from Ref. 32, copyright (1997) by the American Physical Society.

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

37

visualizing it directly in STM image series. This was first done by G. Ertl and coworkers who studied in detail the hopping of single N atoms on a Ru(0001) surface.31 They then extended their research to molecules by investigating O2 on the anisotropic Ag(110) surface, imaging individual molecules (Figs. 5(a) and 5(c)) as the sample was cooled to 76 K.32 By taking one image after another of the same surface area, motion of individual molecules on the surface was detected (marked by arrows in Fig. 5(b)). As the diffusion speed of the molecules is in the ideal range for the imaging rate (at 200 s per frame), the hopping rate could be measured for different temperatures. The resulting Arrhenius plot (Fig. 5(d)) gives the characteristic exponential dependence for such thermally induced processes, revealing an activation barrier for diffusion of 0.22 eV. Besenbacher and coworkers have measured the diffusion properties of various more complex molecules and could, for instance, relate them to the precise molecular structure.33 3.

Manipulation of Single Molecules

The tip of an STM is not only used for imaging and spectroscopy, but also to directly interact with adsorbates (and the surface itself). Such manipulation experiments, which offer exciting possibilities to study physical and chemical processes at the level of single molecules,15,34,35 will be discussed in this section with some representative examples. Note that although the STM tip is a local probe, nonlocal reactions can also be induced via hot electrons that are injected into surface states36,37 or the electric field that extends over larger areas.38 During the manipulation, the tip is either moved across a molecule (in a constant-current or constantheight mode) to induce lateral displacement or placed at a fixed position to apply voltage pulses above an adsorbate. A particular interest lies in molecules with a specific function,39 representing, for instance, model systems for molecular nanotechnology, and some of these are discussed here. The incorporation of a specific function typically leads to larger molecules with two important consequences: First, the deposition under ultrahigh vacuum conditions becomes difficult, because thermal sublimation requires higher temperatures and can cause molecular fragmentation, which is usually avoided by spray techniques.40 On the other hand, when inducing lateral motion of a complex molecule on a surface, larger forces are required due to the increased adsorption energy. This can be more easily realized when scanning the tip at a constant height over the molecule (rather than in a constant-current mode), because much stronger repulsive forces can be applied.41

November 21, 2012

38

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 6. Sketch of the different driving forces for STM manipulation of single atoms and molecules on a surface. Three regimes can be distinguished: Without bias voltage, electric current- or electric-field induced (both with a bias voltage).

Depending on the driving forces, different processes can be induced (Fig. 6), which are presented in detail in Secs. 3.1–3.3: Interatomic forces (van der Waals, chemical interaction and the comparably strong Pauli repulsion) are even active in the absence of a bias voltage. On the other hand, an electric current between tip and surface occurs at a finite bias voltage. Although it is very small in absolute values (nA range), the current density in the individual adsorbate atom or molecule is extremely high and even allows to induce processes (as resonant tunneling into/from molecular orbitals42 or electronic and vibrational excitation43 ) that occur with small yields. Finally, strong electric fields are present in the junction, because although the bias voltages are moderate, the electrode distance is very small. This can lead, on the one hand, to field-assisted diffusion and evaporation44 and on the other hand, in the case of molecules to chemical processes38 and the reorientation of polar molecules. A pioneering experiment in the field of molecular manipulation has been reported in 2000 by Hla et al. who succeeded in performing an entire chemical reaction at the level of a single molecule by using STM tip (Fig. 7).45 Starting from two individual iodobenzene molecules at the step edge of a Cu(111) surface, they first dissociated the iodine atoms from the benzene rings by voltage pulses (Figs. 7(b) and 7(c)) and then moved the individual components to desired places on the surface (Figs. 7(d)–7(f)). Subsequently, by applying another voltage pulse over two adjacent benzene rings, they form a biphenyl molecule, in analogy to the Ullmann reaction. Finally, the new chemical bond is proven by a lateral manipulation of the entire molecule when pulling at the front end (Figs. 7(g) and 7(h)).

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

39

Fig. 7. STM images (all 7 × 3 nm2 , 100 mV, 0.53 nA) of a chemical Ullmann reaction performed with single molecules and induced by the STM tip (see text). Superimposed schemes indicate the imaged species. Reprinted with permission from Ref. 45, copyright (2000) by the American Physical Society.

3.1.

Manipulation without bias voltage

Most manipulation experiments with single molecules took advantage of the applied bias voltage between tip and surface and the resulting high current density and intense electric field (see Fig. 6). However, interatomic forces between tip and a molecule adsorbed on the surface are also present in the absence of an applied voltage. Figure 8 shows an example that illustrates this regime by varying the tip height and thus the strength of these forces. In this experiment, a single so-called Lander molecule has been manipulated by rotating only its legs, i.e., side groups, while the central core remained adsorbed on a metallic nanostructure.46 All data points are obtained for the manipulation of one and the same molecule to avoid

November 21, 2012

40

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 8. Different regimes of driving forces in the manipulation of single molecules. (a) Threshold voltages for a successful manipulation for different tip approaches (large tip heights are at the right of the x axis). (b) Sketch of the different driving forces that can be active, depending on the tip height. Reprinted from Ref. 46.

any influence of the atomic-scale environment (which varies between the molecules). At rather large tip heights (i.e., at the right in Fig. 8(a)), a linear relationship between tip height and the threshold voltage indicates electrostatic forces by a dipole–dipole interaction. If the tip is then approached towards the molecule (i.e., moving to the left in Fig. 8(a)), the required voltage decreases down to zero. This represents an interesting point, because at ∆z > 3.6 ˚ A, no bias voltage is required for the manipulation. The tip is simply positioned close to the molecule and, as a result of the chemical attraction between tip and molecular leg, the manipulation and therefore conformational change of the molecule takes place. Hence, a voltage-free manipulation is achieved while repulsion occurs if the tip is approached too closely (at ∆z > 4.2 ˚ A). 3.2.

Electron-induced manipulation

The tunneling electrons in the junction between tip and surface can induce a variety of processes, in particular, by inelastic scattering processes that can lead to electronic and vibrational excitations of a molecule on the surface (see Figs. 4 and 6)15 with a strong dependence on the applied bias voltage. Such excitations can then lead to further processes as for instance rotation47 or lateral displacement48 of molecules.

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

41

Fig. 9. (a) and (b) STM images (2.1 × 2.1 nm2 ) of the same surface area of a single biphenyl molecule on Si(100) at 5 K before and after a voltage pulse from the STM tip (marked by a dot) with the corresponding schemes in (c) and (d). (e) and (f) Yields per electron as a function of the bias voltage for different molecular motions. Reprinted with permission from Ref. 42, copyright (2005) AAAS.

On the other hand, electrons can also tunnel elastically between surface and tip. Depending on their energy (i.e., bias voltage), they can tunnel into or out of molecular states if these match the electron energy. This so-called resonant tunneling leads to an excited state of the molecule, either by removing an electron from an occupied state or adding one to an unoccupied state. Such a modification can cause molecular motion42 or chemical processes,49 depending on the life time of the excited state that is in an adsorption configuration limited by the molecular coupling to the surface (in contrast to gas phase or solution experiments). Figure 9 shows an example of a biphenyl molecule on a Si(100) surface that exhibits atomic dimer rows (visible in STM images as weak vertical stripes). Dujardin and coworkers could then, by applying voltage pulses at different positions of the molecule, induce a rotation of this single molecule as sketched in (Figs. 9(c) and 9(d)). The yield strongly depends on the bias voltage (Figs. 9(e) and 9(f)). Because the threshold voltage matches the orbital structure of the molecule, the process is assigned to resonant tunneling of electrons from the π orbitals of the biphenyl to the STM tip, creating a positively charged molecular species. The precise tip location plays an important role in this process, due to the spatial distribution of the molecular orbitals and the resulting efficiency variations at different positions.

November 21, 2012

42

3.3.

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Electric-field induced manipulation

The STM tip apex shape is typically unknown at the atomic scale, but the very final end needs to be strongly curved in order to achieve a high spatial resolution during scanning (ideally ending in a single atom). Together with the small tip-sample distance of less than 1 nm in STM, this results in a strong inhomogeneous electric field in the tunneling junction (e.g. 107 V/cm for a typical tip-surface distance of 5 ˚ A and 0.5 V bias voltage). It is localized underneath the tip and decays laterally and can be used for the lateral manipulation of atoms44 and the desorption50 or chemical processes38 of molecules at surfaces. In the following, an example for the latter will be presented, where isomerization processes are induced in single molecules via the electric field, i.e., even in absence of tunneling electrons.38,51 Azobenzene derivatives that can exist in two isomers, trans and cis, are deposited onto a Au(111) surface where they form highly ordered islands of molecules in the trans state (Fig. 10(a)). If now voltage pulses are applied to a particular island, many molecules switch in a reversible way from the trans to the cis state, thus leading to bright lobes in Fig. 10(b) that correspond to

Fig. 10. STM image (37 × 37 nm2 ) of an island with about 400 TBA molecules in the trans state on Au(111). After applying a voltage pulse of 2 V for 20 s (in the position marked by a cross), several molecules switch from the trans to the cis state and therefore appear brighter in the subsequent STM image (b) of the same island. (c) Threshold voltage as a function of the tip height for both directions of isomerization. Lines indicate the regime of electric-field induced isomerization. Reprinted from Ref. 38.

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

43

single cis isomers.38 A detailed analysis of this process (Fig. 10(c)) reveals that the threshold voltage is approximately constant at low tip heights, representing the regime of electron-induced isomerization,49 but increases at large tip heights. The latter regime is assigned to an electric-field driven isomerization, because it represents an approximately linear relationship between voltage and electrode distance, i.e., a roughly constant electric field along the threshold line. Moreover, the corresponding data points extend to very large tip heights, where essentially no tunneling current flows in the junction (the exponential decay from Eq. (1) leads to a very fast decrease), thus excluding an electron-induced process in this regime. The spatial distribution of the isomerization processes (voltage pulses were applied at the position of the cross in Fig. 10(a)) is in agreement with this interpretation, because the electric field decays not as fast with the tip-sample distance as the tunneling current and the STM tip behind the sharp apex is usually rather broad. 3.4.

Lateral manipulation: Hopping vs rolling

The first manipulation attempts of matter at the atomic scale focused on their controlled lateral displacement on a flat surface. In this way, atoms,12 molecules,52 and molecular assemblies53 have been dislocated while keeping them intact. Such lateral manipulation procedures of single molecules can also be used to achieve particular molecular configurations, e.g. as model systems for molecular electronics.54 The driving forces for such processes can be of different kinds (see Fig. 6), but can be understood in the most simple model case by considering a Lennard–Jones potential that is valid for noble gas atoms at different distances. It consists of two branches: Attractive forces by van der Waals interaction at large distances and repulsive forces by Pauli repulsion at small distances with the equilibrium distance in between. Hence, even without considering a bias voltage in the STM junction or chemical interactions that appear for other atoms than noble gases, attractive and repulsive forces can appear between the STM tip and an adsorbate, potentially leading to a lateral motion. The measurement of the electric current during a manipulation experiment gives detailed informations on the process in real time. Typical signals are observed, exhibiting a saw-tooth shape either with descending or ascending slope for attractive (“pulling” mode) and repulsive (“pushing” mode) forces, respectively, between tip and adsorbate.52 Although a large number of molecular species has been manipulated by STM in the past years, only hopping motions with the characteristic saw-tooth shape signal were observed. This is in strong contrast to the macroscopic world, where other forces prevail and the rolling motion is

November 21, 2012

44

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

very common. The most obvious molecules to achieve a rolling motion are fullerenes with their spheric shape. Beton and coworkers have shown that a single C60 molecule can be manipulated on Si(100). A fine structure in the pushing signal was interpreted as a rotation of the fullerene during its hopping from one adsorption site to the next.55 On the other hand, so-called nanocars have been synthesized by mounting four C60 molecules as wheels on a non-rigid molecular board, but no manipulation signal was reported to assess the possible rotation of these C60 wheels.55 In terms of “motorized” nanocars, the motion of such a molecular machine on a surface has recently been reported by using the STM tip56 where the use of light as a driving force is of high interest.57 Such complex nano-machines can also show the limits of STM manipulation, where — due to the large adsorption energy — the molecule (or the tip) is rather damaged than inducing a lateral motion when increasing the applied forces step-by-step.58

Fig. 11. (a) Scheme of the rolling process of a wheel-dimer molecule (consisting of two connected triptycene groups), induced by the STM tip. (b) Current signals of different molecular motions, measured during the manipulation process in real time. Reprinted from Ref. 59 (c) Sketch of the rolling motion and the development of the characteristic hat-shaped signal (reproduced with permission from Ref. 60, copyright (2007) Nature Publishing Group).

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

45

The first example of a molecular wheel that exhibits a clear rolling motion during STM manipulation consists of two connected triptycene groups (thus being a wheel-dimer), as sketched in Fig. 11(a). When manipulating this molecule on a corrugated Cu(110) surface, different signals can be found that represent pushing, pulling or rolling motions.59 While the first two exhibit the typical saw-tooth shape signal (with 3.6 ˚ A periodicity equal to the distance of the atomic surface rows), the latter looks very different (Fig. 11(b)). Its shape is rather hat-like, which can be understood from the very characteristic rolling motion (Fig. 11(c)): The process is symmetric and therefore leads to a symmetric signal with a periodicity that does not correspond to the surface, but reflects the molecular dimensions. Two details are important about this example of a molecular wheel: First, it can be freely chosen whether the wheel-dimer hops or rolls on a surface by putting the tip at rather small or large tip height, respectively (in agreement with the macroscopic wheel analogy). Second, the rolling motion cannot be observed on a flat surface, but only if it exhibits a slight corrugation that is apparently required for the triptycene rotation.59 3.5.

Vertical manipulation: Pulling single molecules from a surface

In addition to the lateral displacement in the two-dimensional confinement of a surface, molecules can also be pulled off a surface. This can be, for instance, a transfer to the STM tip15 and subsequent transport across the surface until the molecule is released somewhere else. On the other hand, it is of interest to maintain contact between the molecule and the surface when pulling it upwards, because such a configuration allows to measure the electric current through a single molecule. However, in the first controlled pulling experiment of a single molecule, the molecule was very short with a length of only 1 nm.61 It is, however, difficult to deposit long molecules and polymers in a clean way onto a surface, because heating procedures lead to fragmentation and side groups, which are needed to achieve solubility in a solvent, can disturb the measurement. To avoid these problems, the on-surface synthesis method62 has been used for the formation of polymers, which allows the covalent connection of molecular building blocks directly on a surface via the controlled formation of reactive sites within each individual molecule, even in a hierarchical manner.63 After the sample preparation, an individual polymer can be pulled off the surface as sketched in Fig. 11(a). An important condition for this experiment is that the polymer end is reactive, most likely a radical, as a result of the polymerization process and thus results in a high yield of polymer attachment and strong bond to the STM tip.64

November 21, 2012

46

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

Fig. 12. (a) Sketch of the pulling experiment. STM images of polyfluorene chains on a Au(111) surface before (b) and after (c) a pulling process. (d) Conductance curve of a single polymer as a function of the tip height, i.e., electrode–electrode distance. (e) I(V) curves after pulling a polyfluorene and fixing the tip height (to the indicated values). Reprinted from Ref. 64.

The main advantage of this pulling technique is that the conductance of a single molecule can be measured as a continuous function of its length, which is important to identify the charge transport regime. Figure 12(d) shows such a measurement of a polyfluorene, which reveals an exponential decay with the electrode–electrode distance, thus a tunneling regime. Furthermore, an oscillation with period zo is observed, a result of the sequential detachment of the individual molecular units, i.e., the “mechanical” internal motion.64 During such a pulling process, the polymer-tip bond typically breaks at a certain tip height. The same molecule can therefore be characterized before (Fig. 12(b)) and after (c) the process by STM imaging and spectroscopy — in contrast to break junction techniques for conductance measurements.

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

47

The pulling process can also be stopped at a chosen tip height. Figure 12(e) shows such an example (at different tip heights), where the bias voltage is varied over a large range while measuring the current through the polymer. The resulting curve gives the characteristic I(V) curves with onsets of increasing current at the energy positions of the highest occupied and lowest unoccupied molecular orbitals, thus giving insight into the electronic structure of an individual molecule.64

Acknowledgments This work was supported by the German Science Foundation DFG (SFB projects 658 and 951) and the European Union via projects ARTIST and AtMol.

References 1. G. Binnig, H. Rohrer, C. Gerber and E. Weibel, Phys. Rev. Lett. 49, 57 (1982). 2. G. Binnig, H. Rohrer, C. Gerber and E. Weibel, Phys. Rev. Lett. 50, 120 (1983). 3. G. K. Binnig, H. Rohrer, C. Gerber and E. Stoll, Surf. Sci. 144, 321 (1984). 4. C. F. Quate, Phys. Today 39, 26 (1986). 5. E. W. M¨ uller, Z. Phys. 156, 399 (1959). 6. A. J. Melmed and E. W. M¨ uller, J. Chem. Phys. 29, 1037 (1958). 7. M. Knoll and E. Ruska, Z. Phys. 78, 318 (1932). 8. R. Feynman, Speech at the annual meeting of the American Physical Society (29th December 1959). 9. R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy (Cambridge University Press, Cambridge, 1994). 10. G. Binnig and D. P. E. Smith, Rev. Sci. Instrum. 57, 1688 (1986). 11. D. M. Eigler and E. K. Schweizer, Nature 344, 524 (1990). 12. M. Schunack, E. Laegsgaard, I. Stensgaard and F. Besenbacher, J. Chem. Phys. 117, 8493 (2002). 13. L. Grill, F. Moresco, P. Jiang, C. Joachim, A. Gourdon and K.-H. Rieder, Phys. Rev. B. 69, 035416 (2004). 14. L. Bartels, G. Meyer and K.-H. Rieder, Appl. Phys. Lett. 71, 213 (1997). 15. W. Ho, J. Chem. Phys. 117, 11033 (2002). 16. J. K. Gimzewski, E. Stoll and R. R. Schlittler, Surf. Sci. 181, 267 (1987). 17. P. H. Lippel, R. J. Wilson, M. D. Miller, C. W¨ oll and S. Chiang, Phys. Rev. Lett. 62, 171 (1989). 18. D. J. Smith, J. R. Fryer and R. A. Camps, Ultramicroscopy 19, 279 (1986). 19. M. F. Crommie, C. P. Lutz and D. M. Eigler, Science 262, 218 (1993).

November 21, 2012

48

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

L. Grill

20. L. Gross, F. Moresco, L. Savio, A. Gourdon, C. Joachim and K.-H. Rieder, Phys. Rev. Lett. 93, 056103 (2004). 21. F. Moresco, L. Gross, M. Alemani, K.-H. Rieder, H. Tang, A. Gourdon and C. Joachim, Phys. Rev. Lett. 91, 036601 (2003). 22. J. Repp, G. Meyer, S. Stojkovic, A. Gourdon and C. Joachim, Phys. Rev. Lett. 94, 026803 (2005). 23. N. D. Lang, Phys. Rev. B. 34, 5947 (1986). 24. W. Han, E. N. Durantini, T. A. Moore, A. L. Moore, D. Gust, P. Rez, G. Leatherman, G. R. Seely, N. Tao and S. M. Lindsay, J. Phys. Chem. B. 101, 10719 (1997). 25. K. W. Hipps, D. E. Barlow and U. Mazur, J. Phys. Chem. B. 104, 2444 (2000). 26. B. C. Stipe, M. A. Rezaei and W. Ho, Phys. Rev. Lett. 82, 1724 (1999). 27. J. Repp, F. Moresco, G. Meyer, K.-H. Rieder, P. Hyldgaard and M. Persson, Phys. Rev. Lett. 85, 2981 (2000). 28. M. J. Rost, L. Crama, P. Schakel, E. v. Tol, G. B. E. M. v. VelzenWilliams, C. F. Overgauw, H. t. Horst, H. Dekker, B. Okhuijsen, M. Seynen, A. Vijftigschild, P. Han, A. J. Katan, K. Schoots, R. Schumm, W. v. Loo, T. H. Oosterkamp and J. W. M. Frenken, Rev. Sci. Instrum. 76, 053710 (2005). 29. F. Esch, C. Dri, A. Spessot, C. Africh, G. Cautero, D. Giuressi, R. Sergo, R. Tommasini and G. Comelli, Rev. Sci. Instrum. 82, 053702 (2011). 30. S. Loth, M. Etzkorn, C. P. Lutz, D. M. Eigler and A. J. Heinrich, Science 329, 1628 (2010). 31. T. Zambelli, J. Trost, J. Wintterlin and G. Ertl, Phys. Rev. Lett. 76, 795 (1996). 32. J. V. Barth, T. Zambelli, J. Wintterlin, R. Schuster and G. Ertl, Phys. Rev. B. 55, 12902 (1997). 33. M. Schunack, T. R. Linderoth, F. Rosei, E. Laegsgaard, I. Stensgaard and F. Besenbacher, Phys. Rev. Lett. 88, 156102 (2002). 34. F. Moresco, Phys. Rep. 399, 175 (2004). 35. S.-W. Hla, J. Vac. Sci. Technol. B 23, 1351 (2005). 36. P. Maksymovych, D. B. Dougherty, X.-Y. Zhu and J. J. T. Yates, Phys. Rev. Lett. 99, 016101 (2007). 37. J. M. MacLeod, J. Lipton-Duffin, C. Fu and F. Rosei, ACS Nano 3, 3347 (2009). 38. M. Alemani, M. V. Peters, S. Hecht, K.-H. Rieder, F. Moresco and L. Grill, J. Am. Chem. Soc. 128, 14446 (2006). 39. L. Grill, J. Phys., Condens. Matter. 20, 053001 (2008). 40. L. Grill, J. Phys., Condens. Matter. 22, 084023 (2010). 41. F. Moresco, G. Meyer, K.-H. Rieder, H. Tang, A. Gourdon and C. Joachim, Appl. Phys. Lett. 78, 306 (2001). 42. M. Lastapis, M. Martin, D. Riedel, L. Hellner, G. Comtet and G. Dujardin, Science 308, 1000 (2005).

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

Imaging and Manipulation of Single Molecules

49

43. R. E. Walkup, D. M. Newns and P. Avouris, J. Electron Spectrosc. Relat. Phenom. 64/65, 523 (1993). 44. J. A. Stroscio and D. M. Eigler, Science 254, 1319 (1991). 45. S.-W. Hla, L. Bartels, G. Meyer and K.-H. Rieder, Phys. Rev. Lett. 85, 2777 (2000). 46. L. Grill, K.-H. Rieder, F. Moresco, S. Stojkovic, A. Gourdon and C. Joachim, Nano Lett. 6, 2685 (2006). 47. B. C. Stipe, M. A. Rezaei and W. Ho, Phys. Rev. Lett. 81, 1263 (1998). 48. T. Komeda, Y. Kim, M. Kawai, B. N. J. Persson and H. Ueba, Science 295, 2055 (2002). 49. M. Alemani, S. Selvanathan, F. Moresco, K.-H. Rieder, F. Ample, C. Joachim, M. V. Peters, S. Hecht and L. Grill, J. Phys. Chem. C. 112, 10509 (2008). 50. M. A. Rezaei, B. C. Stipe and W. Ho, J. Chem. Phys. 110, 4891 (1999). 51. C. Dri, M. V. Peters, J. Schwarz, S. Hecht and L. Grill, Nat. Nanotechnol. 3, 649 (2008). 52. L. Bartels, G. Meyer and K.-H. Rieder, Phys. Rev. Lett. 79, 697 (1997). 53. S. Selvanathan, M. V. Peters, J. Schwarz, S. Hecht and L. Grill, Appl. Phys. A. 93, 247 (2008). 54. L. Grill and F. Moresco, J. Phys., Condens. Matter. 18, S1887 (2006). 55. D. L. Keeling, M. J. Humphry, R. H. J. Fawcett, P. H. Beton, C. Hobbs and L. Kantorovich, Phys. Rev. Lett. 94, 146104 (2005). 56. T. Kudernac, N. Ruangsupapichat, M. Parschau, B. Maci´ a, N. Katsonis, S. R. Harutyunyan, K.-H. Ernst and B. L. Feringa, Nature 479, 208 (2011). 57. P.-T. Chiang, J. Mielke, J. Godoy, J. M. Guerrero, L. B. Alemany, C. J. Villag´ omez, A. Saywell, L. Grill and J. M. Tour, ACS Nano 6, 592 (2012). 58. L. Grill, K.-H. Rieder, F. Moresco, G. Jimenez-Bueno, C. Wang, G. Rapenne and C. Joachim, Surf. Sci. 584, L153 (2005). 59. L. Grill, K.-H. Rieder, F. Moresco, G. Rapenne, S. Stojkovic, X. Bouju and C. Joachim, Nature Nanotechnol. 2, 95 (2007). 60. S.-W. Hla, Nature Nanotechnol. 2, 82 (2007). 61. R. Temirov, A. Lassise, F. B. Anders and F. S. Tautz, Nanotechnology 19, 065401 (2008). 62. L. Grill, M. Dyer, L. Lafferentz, M. Persson, M. V. Peters and S. Hecht, Nature Nanotechnol. 2, 687 (2007). 63. L. Lafferentz, V. Eberhardt, C. Dri, C. Africh, G. Comelli, F. Esch, S. Hecht and L. Grill, Nature Chem. 4, 215 (2012). 64. L. Lafferentz, F. Ample, S. H. H. Yu, C. Joachim and L. Grill, Science 323, 1193 (2009).

November 21, 2012

13:1

9in x 6in

b1449-ch02

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Chapter 3 SELF-ORGANIZATION AT THE NANOSCALE IN FAR-FROM-EQUILIBRIUM SURFACE REACTIONS AND COPOLYMERIZATIONS

Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems Universit´e Libre de Bruxelles Campus Plaine, Code Postal 231 B-1050 Brussels, Belgium An overview is given of theoretical progress on self-organization at the nanoscale in reactive systems of heterogeneous catalysis observed by field emission microscopy techniques and at the molecular scale in copolymerization processes. The results are presented in the perspective of recent advances in nonequilibrium thermodynamics and statistical mechanics, allowing us to understand how nanosystems driven away from equilibrium can manifest directionality and dynamical order.

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Aspects of Nonequilibrium Nanosystems 2.1. Structure and function of nanosystems . . . . . . . 2.2. Out-of-equilibrium directionality of fluctuating currents . . . . . . . . . . . . . . . . 2.3. Thermodynamic origins of dynamical order . . . . Heterogeneous Catalytic Reactions in High Electric Fields . . . . . . . . . . . . . . . . . . 3.1. Surface conditions in FEM and FIM . . . . . . . . 3.2. Adsorption–desorption kinetics . . . . . . . . . . . 3.3. Surface oxides of rhodium . . . . . . . . . . . . . . 3.4. The H2 −O2 /Rh system . . . . . . . . . . . . . . . 3.4.1. Kinetic equations . . . . . . . . . . . . . . 3.4.2. Bistability . . . . . . . . . . . . . . . . . . 3.4.3. Oscillations . . . . . . . . . . . . . . . . . . 3.5. Self-organization at the nanoscale . . . . . . . . . 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 53 53

. . . . . . . . . . . . . . . . . . . . . . . .

54 56

. . . . . . . . .

58 58 60 60 62 62 63 63 65

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

52 4.

Copolymerization processes . . . . . . . . . . . . . . . . . 4.1. Information processing at the molecular scale . . . . . 4.2. Thermodynamics of free copolymerization . . . . . . . 4.3. Thermodynamics of copolymerization with a template 4.4. The case of DNA replication . . . . . . . . . . . . . . 5. Conclusions and Perspectives . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

67 67 67 69 71 73 74

Introduction

In macroscopic systems, self-organization arises far-from-equilibrium beyond critical thresholds where the macrostate issued from thermodynamic equilibrium becomes unstable and new macrostates emerge through bifurcations. Such bifurcations may lead to oscillatory behavior and spatial or spatio-temporal patterns, called dissipative structures because they are maintained at the expense of free-energy sources.1 –3 Recent developments have been concerned with the complexity of such nonequilibrium behavior, particularly, in small systems of nanometric size down to the molecular scale.4 –7 The molecular structure of matter largely contributes to the complexity of natural phenomena by the multiplicity and the variety of chemical species and their possible specific actions. Furthermore, the microscopic degrees of freedom manifest themselves at the nanoscale as molecular and thermal fluctuations, which requires a stochastic description for the thermodynamic and kinetic properties of small systems. Remarkably, great advances have been recently achieved, leading to a fundamental understanding of the emergence of dynamical order in fluctuating nonequilibrium systems, as overviewed in Sec. 2. These advances allow us to bridge the gap between the microscopic and macroscopic levels of description, especially, in reactions of heterogeneous catalysis studied by field electron and field ion microscopy techniques (FEM and FIM, respectively).8 –12 In such reactions, dynamical patterns are observed on metallic tips with a curvature radius of tens of nanometers. Accordingly, the crystalline surface is multifaceted so that adsorption, desorption, reaction and transport processes have various speed on different facets. Moreover, the activation barriers are significantly modified by the high electric field present under FEM or FIM conditions, as calculated by quantum electronic ab initio and density functional theories.13 –15 These non-uniform and anisotropic properties of the catalytic surface participate in the generation of nanopatterns in the far-from-equilibrium regimes of bistability and oscillations observed, in particular, during water formation on rhodium, as presented in Sec. 3.16 –20

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

53

Under nonequilibrium conditions, the emergence of dynamical order is already in action at the molecular scale during copolymerization processes. Copolymers are special because they constitute the smallest physicochemical supports of information. Little is known about the thermodynamics and kinetics of information processing in copolymerizations although such reactions play an essential role in many complex systems, e.g., in biology. In this context, the recent advances in the thermodynamics of stochastic processes are shedding light on the generation of informationrich copolymers, as explained in Sec. 4.21 –23 The purpose of this contribution is to present an overview of these recent advances on self-organization at the nanoscale in the perspective of future theoretical and experimental work on these topics, as discussed in following sections.

2. 2.1.

Fundamental Aspects of Nonequilibrium Nanosystems Structure and function of nanosystems

There exist many different processes and systems at the nanoscale: heterogeneous catalysis on nanoparticles,9 –12 electrochemical reactions on nanoelectrodes,24 synthetic molecular machines,25,26 single enzymes,27 linear and rotary molecular motors,28,29 DNA and RNA polymerases responsible for replication and transcription,31,32 or ribosomes performing the translation of mRNAs into proteins.33 Every nanosystem has a specific structure and acquires its function when driven out of equilibrium by some free-energy source. In this regard, the structure and function of a nanosystem can be characterized in terms of its equilibrium and nonequilibrium properties, as shown in Table 1. A nanosystem can be in thermodynamic equilibrium with its environment at given temperature and chemical potentials, as it is the case for a Table 1. Comparison between the equilibrium and nonequilibrium properties of nanosystems. Equilibrium zero affinities zero mean fluxes zero entropy production no free-energy supply needed detailed balancing 3D spatial structure structure

Nonequilibrium non zero affinities non zero mean fluxes positive entropy production free-energy supply required directionality 4D spatio-temporal dynamics function

November 21, 2012

54

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

catalytic surface in contact with a gaseous mixture at chemical equilibrium or for an enzyme in a solution, also at chemical equilibrium. In these equilibrium systems, the ceaseless movements of thermal and molecular fluctuations do not need the supply of free energy to persist. In particular, the catalytic sites are randomly visited by adsorbates or substrates but, on average, there is no flux of matter or energy between the pools of reactants and products. The ratio of partial pressures or concentrations remains at its equilibrium value fixed by the mass action law of Guldberg and Waage. Any movement in one direction is balanced by the reverse movement according to the principle of detailed balancing. At equilibrium, there is no energy dissipation and no entropy production. From a statisticalmechanical viewpoint, the molecular architecture of the nanosystem can be characterized in terms of the average relative positions of its atoms, where their average velocities are vanishing. In this respect, the nanosystem has only a 3D spatial structure at equilibrium. By contrast, if a nanosystem is in an environment containing a mixture which is not in chemical equilibrium for the reactions that it can catalyze, then it will sustain non-vanishing fluxes of matter or energy. These average movements are driven by the free-energy sources of the environment. Energy is dissipated and entropy is produced. For instance, a F1 -ATPase molecular motor rotates in a specific direction if it is surrounded by a solution containing an excess of ATP with respect to the products of its hydrolysis.29,34,35 Therefore, nonequilibrium nanosystems acquire an average directionality, which can be controlled by the external nonequilibrium constraints, and they perform a 4D spatio-temporal dynamics, which is the expression of their function. Accordingly, the function of a nanosystem holds in the specific 4D dynamics, where its 3D structure can be developed when it is driven away from equilibrium under specific conditions. 2.2.

Out-of-equilibrium directionality of fluctuating currents

The kinetics of nanosystems close to or far-from-equilibrium has recently progressed tremendously with the advent of time-reversal symmetry relations, also called fluctuation theorems. These results find their origins in the study of large-deviation properties of chaotic dynamical systems sustaining transport processes of diffusion.36,37 Several versions of such relations have been obtained for systems under transient or stationary nonequilibrium conditions.38 –43 A particular version of the fluctuation theorem concerns the nonequilibrium work on single molecules subjected to the timedependent forces of optical tweezers or atomic force microscopy.26,44 For nonequilibrium systems in stationary states, a general fluctuation theorem

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

55

has been proved for all the currents flowing across open stochastic or quantum systems by using microreversibility.45–51 Nanosystems can be driven out of equilibrium by several independent thermodynamic forces, also called affinities.52 For isothermal systems, they are defined as Aγ =

∆Gγ , kB T

(γ = 1, 2, . . . , c),

(1)

in terms of the Gibbs free-energy differences ∆Gγ = Gγ − Gγ,eq supplied by the nonequilibrium environment to power the mean motion. They are external control parameters that depend on the concentrations or partial pressures of reactants and products. A nanosystem between reservoirs at different temperatures and chemical potentials is characterized by thermal as well as chemical affinities.51 The affinities drive the currents flowing across the system. Examples of such currents are the reaction rates45 or the velocity of a molecular motor.34,35 At the microscopic level of description, the instantaneous currents j(t) = {jγ (t)}cγ=1 can be defined when particles cross fictitious surfaces separating reactants from products,47 as in the reaction rate theory. The instantaneous currents can be integrated over some time interval t to get the numbers of particles
t having crossed the fictitious surface during that time interval: ∆Nγ = 0 jγ (t
)dt
. As long as the time interval t is finite, the currents defined by  1 t ∆Nγ Jγ = = jγ (t
) dt
, (γ = 1, 2, . . . , c) (2) t t 0 are random variables. For given values of the different independent affinities A = {Aγ }cγ=1 , the nanosystem reaches a stationary state in the longtime limit t → +∞. This stationary state is described by a probability distribution PA . Since the currents are fluctuating, they may take positive or negative values J = {Jγ }cγ=1 with certain probabilities PA (J). Now, we compare the probabilities of opposite values for the currents, PA (J) and PA (−J). In general, these probabilites are different but, most remarkably, the time-reversal symmetry of the underlying microscopic dynamics implies that the ratio of these probabilities has a general behavior expressed by the current fluctuation theorem:47 PA (J)  exp(A · Jt) PA (−J)

for t → +∞.

(3)

This result holds for the equilibrium as well as the nonequilibrium stationary states at any value of the affinities in Markovian or semi-Markovian

November 21, 2012

56

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

stochastic processes if the large-deviation properties of the process are well defined in the long-time limit.47 –50 At equilibrium where the affinities vanish, the exponential function takes the unit value and we recover the principle of detailed balancing according to which the probabilities of opposite fluctuations are equal: P0 (J)  P0 (−J). However, out of equilibrium when the affinities do not vanish, the ratio of probabilities typically increases or cdecreases exponentially in time depending on the sign of A · J = γ=1 Aγ Jγ . Therefore, a bias grows between the probabilities of opposite fluctuations and the current fluctuations soon become more probable in one particular direction. Directionality has thus appeared in the system. This directionality is controlled by the affinities because the currents would flow in the opposite direction if the affinities were reversed, as expected from microreversibility. The current fluctuation theorem   has several implications. As a consequence of Jensen’s inequality, e−X ≥ e−X , the thermodynamic entropy production is always non-negative: 1 di S  (4)  = A · JA ≥ 0, kB dt st where JA are the mean values of the currents in the stationary state described by the probability distribution PA . Therefore, the second law of thermodynamics is the consequence of the current fluctuation theorem. Furthermore, this theorem allows us to generalize the Onsager reciprocity relations and the Green–Kubo formulas from the linear to the nonlinear response properties of the average currents with respect to the affinities.45,48 This generalization is the result of the validity of the current fluctuation theorem far-from-equilibrium. In particular, these results apply to the reversible Brusselator model of oscillatory reactions.53 For fully irreversible reactions, in which the rates of the reversed reactions vanish, the corresponding affinities take infinite values so that the entropy production is also infinite, in which case the ratio of the probabilities of opposite fluctuations is consistently either zero or infinite. The fact that the ratio (3) behaves exponentially means that the reversed processes may soon become so rare that their probabilities are negligible and the systems are in a far-from-equilibrium regime which could be considered as fully irreversible. 2.3.

Thermodynamic origins of dynamical order

Another time-reversal symmetry relationship concerns the statistical properties of the histories or paths followed by a system under stroboscopic

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

57

observations at some sampling time ∆t. Such observations generate a sequence of coarse-grained states: ω = ω1 ω2 · · · ωn ,

(5)

corresponding to the successive times tj = j ∆t with j = 1, 2, . . . , n. This ω ) to happen if the system history or path has a certain probability PA (ω is in the stationary state corresponding to the affinities A. Because of the randomness of the molecular fluctuations, these path probabilities typically decrease exponentially as ω ) = PA (ω1 ω2 · · · ωn ) ∼ e−n∆thA , PA (ω

(6)

at a rate hA that characterizes the temporal disorder in the process. Such a characterization concerns stochastic processes as well as chaotic dynamical systems.54,55 In nonequilibrium stationary states, the time-reversed path ω R = ωn · · · ω2 ω1 ,

(7)

is expected to happen with a different probability ω R ) = PA (ωn · · · ω2 ω1 ) ∼ e−n∆thA , PA (ω R

(8)

decreasing at a different rate hR A now characterizing the temporal disorder of the time-reversed paths.56 The remarkable result is that the difference between the disorders of the time-reversed and typical paths is equal to the thermodynamic entropy production:56 1 di S  (9)  = hR A − hA ≥ 0. kB dt st The second law of thermodynamics is satisfied because this difference is known in mathematics to be always non-negative.57 The validity of the formula (9) has been verified in experiments where the nonequilibrium constraints are imposed by fixing the currents instead of the affinities, ω ) and in which case the comparison should be carried out between PJ (ω ω R ).58,59 P−J (ω At equilibrium, detailed balancing holds so that every history and its time reversal are equiprobable, their temporal disorders are equal, and the entropy production vanishes. This is no longer the case away from equilibrium where the typical paths are more probable than their time reversals. Consequently, the time-reversal symmetry is broken at the statistical level of description in terms of the probability distribution PA of the nonequilibrium stationary state. In this regard, the entropy production is a measure of the time asymmetry in the temporal disorders of the typical

November 21, 2012

58

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

histories and their time reversals. As a corollary of the second law, we thus have the following: Theorem of nonequilibrium temporal ordering60 In nonequilibrium stationary states, the typical histories are more ordered in time than their corresponding time reversals in the sense that hA < hR A. This temporal ordering is possible out of equilibrium at the expense of the increase of phase-space disorder so that there is no contradiction with Boltzmann’s interpretation of the second law. The result established by this theorem is that nonequilibrium processes can generate dynamical order, which is a key feature of biological phenomena. In particular, the nonequilibrium ordering mechanism can generate oscillatory behavior in surface reactions or information-rich sequences in copolymers. 3.

Heterogeneous Catalytic Reactions in High Electric Fields

Dynamical order in the form of nonequilibrium patterns or oscillations can arise in heterogeneous catalysis at the nanoscale on metallic tips under FEM or FIM conditions.8 –12 3.1.

Surface conditions in FEM and FIM

Since solid metals are crystalline and the radius of curvature of typical tips may reach 10–30 nm, the surface is multi-faceted, which introduces non-uniformities as shown by Gerhard Ertl and coworkers.9 Moreover, the surface is subjected to high electric fields of about 10 V/nm, which have influence on the surface reactions.12 The electrostatic edge effect tends to concentrate the electric field on the sharpest structures of the metallic needle. This is the case at the edges of the crystalline facets, which provides the atomic resolution of cryogenic FIM.61 At a larger scale, this is also the case at the apex of the tip where the radius of curvature of the average surface is the smallest. If the average shape of the tip is a paraboloid, the electric field varies as F0 , (10) F = 2 1 + Rr 2 as a function of the radial distance r from the axis of cylindrical symmetry of the paraboloid. The quantity R denotes the radius of curvature at the apex where the electric field reaches its maximum value F0 . This high electric field has several effects, which create the conditions of a nanoreactor localized near the apex of the needle under electric tension.

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

59

On the one hand, the electric field polarizes the molecules in the gaseous mixture around the needle. Consequently, the partial pressures increase according to P (F )  P (0) exp

αF 2 , 2kB T

(11)

where α is an effective polarizability of the molecules of a given species and kB Boltzmann’s constant.13 On the other hand, the electric field modifies the activation energies of the different processes taking place on the surface: αx 2 F + ··· (12) Ex (F ) = Ex (0) − dx F − 2 The dependence of the activation energies is in general a nonlinear function of the electric field F . At sufficiently low values of the electric field, such power expansions define the coefficients dx , αx , . . ., which are associated with the transition state in analogy to the situation in the stable states. Since the surface is multi-faceted, its properties also depend on the orientation of every crystalline plane where the reactions proceed. This dependence can be expressed in terms of the corresponding Miller indices (h, k, l) or, equivalently, the unit vector normal to the plane: n = (nx , ny , nz ) = √

(h, k, l) . + k2 + l2

h2

(13)

This anisotropy concerns, in particular, the activation energies, which can be expanded in kubic harmonics as18 –20   (14) Ex (n) = Ex(0) + Ex(4) n4x + n4y + n4z + Ex(6) n2x n2y n2z + · · · for face-centered cubic crystals such as rhodium or platinum. The coefficients of this expansion can be fitted to data collected for different orientations. The knowledge of any activation energy for the three main crystalline planes (001), (011), and (111) determines the three first coefficients of the expansion (14). Finer dependences of the activation energy can be included with experimental data on more crystalline planes. These dependences on the various facets composing the tip are crucial to understand the anisotropy of the surface reactions and the nanopatterns observed under FEM or FIM conditions. These conditions are significantly different from those prevailing on flat crystalline surfaces extending over distances of several hundreds or thousands of micrometers. Such flat surfaces have uniform properties so that the transport mechanisms are diffusive and the patterns observed on flat crystalline surfaces emerge as the result of standard reaction– diffusion processes in uniform media. The wavelengths of such reaction– diffusion patterns are determined by the diffusion coefficients D and the

November 21, 2012

60

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

reaction constants krxn . They are of the order of L ∼ D/krxn ∼ 100 µm, which is much larger than the size R  20 nm of a FIM tip. Therefore, the nanopatterns observed under FEM or FIM conditions are not standard reaction–diffusion patterns and their understanding requires to take into account the non-uniform and anisotropic effects of the electric field and the underlying crystal. On non-uniform surfaces, the transport of adsorbates is not only driven by the gradients of coverages, but also by the surface gradients of desorption energy and electric field.62 –65 3.2.

Adsorption–desorption kinetics

For diatomic molecules A2 such as dihydrogen, adsorption is dissociative. Therefore, the coverage θA of the surface by the atomic species increases at the rate: S 0 as ∂θA  , (15)  = 2 ka PA2 (1 − θA )2 with ka =

∂t ads 2πmA2 kB T where the pressure PA2 is given by Eq. (11), S 0 is the initial sticking coefficient at zero coverage, and as is some reference area. Accordingly, desorption is dissociative and proceeds at the thermally activated rate ∂θA  2 with kd = kd0 e−βEd (n,F,θA) (16)  = −2kd θA ∂t des and β = (kB T )−1 . The desorption energy Ed depends on the electric field (10), the local crystalline orientation (13), as well as the coverage itself if lateral interactions play a role in desorption. Experimental data are available for the adsorption and desorption of hydrogen on rhodium kaH

H2 (gas) + 2∅(ad)
2H (ad). kdH

(17)

Its desorption energy takes values below one electron-Volt so that the mean hydrogen coverage θH is low above 400 K where bistability and oscillations are observed.18 –20 3.3.

Surface oxides of rhodium

The behavior is more complex for oxygen on rhodium. On the one hand, the dissociative adsorption of oxygen involves a precursor state, which explains the dependence of the sticking coefficient on the oxygen coverage.66,67

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

61

On the other hand, oxygen forms surface oxide trilayers on rhodium: O(ad)-Rh-O(sub). Their properties have been systematically investigated, in particular, with DFT calculations.68 –75 By their stoichiometry RhO2 , the surface oxides differ from the bulk oxide Rh2 O3 . Oxygen vacancies can thus exist in either the outer or the inner oxygen layer. Accordingly, the structure of a partially formed surface oxide can be characterized in terms of the occupancies by oxygen atoms of the adsorption and subsurface sites, θO and θs , respectively. These different features of the interaction of oxygen with rhodium are described by the following kinetic scheme:18 –20 ˜aO k

adsorption and desorption: O2 (gas) + surface
O2 (pre) + surface ˜dO k

kaO

dissociation and recombination: O2 (pre) + 2∅(ad)
2O (ad) kdO

kox

oxidation and reduction of Rh: O (ad) + ∅(sub)
∅(ad) + O (sub) kred

(18) The surface oxide trilayer tends to inhibit oxygen adsorption, which is taken into account by the precursor constant:18 –20 K=

O s kaO = K 0 e−β(EK +AK θO +AK θs ) , k˜dO

(19)

with parameters fitted to data on the sticking coefficient.66,67 Besides, the desorption rate constants on the three main surface orientations are fitted to temperature-programmed desorption spectra.18 –20 For rhodium in equilibrium with gaseous dioxygen, the adsorbate and subsurface occupancies satisfy  θO kaO k˜aO kred θs = = PO . (20) 1 − θO kox 1 − θs kdO k˜dO 2 A phase transition between a metallic surface with adsorbed oxygen and a surface oxide trilayer with possible vacancies occurs at about θs  0.5. This condition allows us to determine the ratio of the oxidation and reduction rates of a rhodium layer using the results of DFT calculations on the three main surface orientations.18 –20 Moreover, the dependence of the activation energies on the electric field has been evaluated using DFT calculations,15 which shows that a positive electric field tends to promote the oxidation of rhodium.

November 21, 2012

13:1

9in x 6in

Engineering of Chemical Complexity

P. Gaspard

62

3.4.

b1449-ch03

The H2 −O2 /Rh system

Water formation is catalyzed on rhodium in contact with a gaseous mixture of dihydrogen and dioxygen. If water vapor is not supplied in the mixture, its partial pressure vanishes. Under these conditions, the corresponding affinity (1) is infinite and the overall reaction proceeds in a fully irreversible regime. For the reaction of water formation on rhodium k

r (ad) + H2 O (gas), 2H (ad) + O (ad)→3∅

(21)

the rate constant is taken as kr = kr0 e−β (Er −dr F +Ar θH +Ar θO ) H

O

(22)

O with coefficients AH r and Ar expressing the change of the reaction rate with the hydrogen and oxygen coverages, respectively.18 –20

3.4.1. Kinetic equations Combining together the different processes involving hydrogen and oxygen, the kinetic equations are given by18 –20 ∂θH 2 − 2kr θH θO − divJH , = 2kaH PH2 θ∅2 − 2kdH θH ∂t

∂θO 2 ˜aO KPO θ2 − kdO θ2 k = 2 ∅ O ∂t 1 + Kθ∅2

(23) (24)

−kox θO (1 − θs ) + kred θs θ∅ − kr θH θO , ∂θs = kox θO (1 − θs ) − kred θs θ∅ , ∂t

(25)

with the coverage of empty sites θ∅ = 1 − θH − θO and K = kaO /k˜dO . The current density of hydrogen takes the form   ∇ UH ∇θH + θH∇ θO + θH (1 − θH − θO ) JH = −DH (1 − θO )∇ (26) kB T with the effective energy potential   1 1 2 UH (r) = − EdH (r) − ddH F (r) + αH2 F (r) + cst. 2 2

(27)

The mobility of oxygen is negligible. In contrast, the mobility of hydrogen is very high with a typical diffusion time tdiff = R2 /DH  10−7 s, which is many orders of magnitude shorter than the time scales of the

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

63

other kinetic processes at 500 K. Therefore, the coverage of atomic hydrogen quickly reaches the quasi-equilibrium distribution θH (r, t) =

1 − θO (r, t) , 1 + eβ[UH (r)−µH (t)]

(28)

such that the hydrogen current density is vanishing: JH = 0. This coverage varies in space because the effective potential (27) depends on the electric field (10) and on the normal unit vector (13), which is uniquely determined by the position r on the surface. Therefore, the hydrogen coverage already forms nanopatterns, which reflect the anisotropy of the underlying crystal and the electric field variation. The chemical potential µH is uniform on the surface but slowly varies in time because the population of hydrogen atoms on the surface changes due to adsorption, desorption and the reaction of hydrogen with oxygen. Using multiscale analysis, it has been possible to show that this nonequilibrium chemical potential evolves in time according to  w [(1 − θO )∂t θH + θH ∂t θO ] dµH  = kB T facets , (29) dt facets w θH (1 − θH − θO ) where w = (1 − θO )−1 , while the time derivatives ∂t θH and ∂t θO are given by the kinetic Eqs. (23) and (24).65 This kinetic model reproduces very well the nonlinear dynamics of the system and the spatial dependence of the observed nanopatterns. 3.4.2. Bistability Bistability manifests itself under variation of hydrogen pressure for fixed oxygen pressure, as shown in Fig. 1 as a function of temperature.18 –20 At low (respectively high) hydrogen pressure, the surface is covered with oxygen (respectively hydrogen). A domain of hysteresis appears in between where the two states coexist. The bistability domain depends on the applied electric field. The higher the field, the broader is the coexisting region in the bifurcation diagram of Fig. 1. As temperature increases, so does the water formation rate, leading to the reduction of the oxide and the coverage of the tip by hydrogen. Ultimately, the bistability domain disappears above 550 K. 3.4.3. Oscillations The kinetic model also explains the oscillatory behavior observed in this system in FIM experiments (see Figs. 2 and 3).18 –20 The period of oscillations is about 40 s. In the model, this period is mainly determined by

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

64

(a) 0.02

11.0 V/nm

H2 Pressure (Pa)

0.01

0.00

(b) 0.02

12.3 V/nm

0.01

0.00 400

450 Temperature (K)

500

Fig. 1. The bistability diagram showing hysteresis for PO2 = 5 × 10−4 Pa during catalytic water formation on rhodium at (a) 11.0 V/nm and (b) 12.3 V/nm. The circles and stars indicate the experimental pressures for which the structural transformation occurred when decreasing and increasing the hydrogen pressure, respectively. The area in between marks the coexistence region of bistability. The full lines are the corresponding results for the kinetic model ruled by Eqs. (23)–(29).18 , 19

the rate constants of rhodium oxidation and reduction when oxygen reacts with the first rhodium layer. The feedback mechanism at the origin of the oscillations involves, in particular, the formation of surface oxide and its inhibition of further oxygen adsorption, as taken into account with the rate constant (19). Starting from a quasi metallic surface in Figs. 2(a) and 2(d), an oxide layer invades the topmost plane and grows along the {011} facets forming a nanometric cross-like structure seen in Figs. 2(b) and 2(e). Finally, the oxide front spreads to cover the entire visible surface area in Figs. 2(c) and 2(f). This is associated with a decrease in the overall brightness in Fig. 3(c). The oscillation cycle is closed by a sudden reduction of the surface oxide from

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

65

Fig. 2. Series of FIM micrographs covering the complete oscillatory cycle as well as the corresponding time evolution of the subsurface oxygen distribution on a logarithmic scale as obtained within the kinetic model.18,19 The temperature, electric field and partial pressures of oxygen are T = 550 K, F0 = 12 V/nm, PO2 = 2 × 10−3 Pa, respectively. On the other hand, the hydrogen pressure is PH2 = 2 × 10−3 Pa in the FIM experiments and 4 × 10−3 Pa in the simulation of the kinetic model. For the subsurface site occupation, the white areas indicate a high site occupation value while the dark areas indicate a low site occupation value.

the outskirts towards the top, with a considerable increase of the brightness. During the cycle shown in Fig. 3, rhodium is alternatively covered by the surface oxide when θs  1 and θO  0.5 and, thereafter, by hydrogen at a low coverage since the temperature is 550 K. 3.5.

Self-organization at the nanoscale

In summary, the nonlinear dynamics and the patterns observed in field emission microscopy are determined by the tip geometry, anisotropy from the underlying crystal, and the electric field. The anisotropy can be described by giving to the energy barriers their dependence on the crystalline orientation of the many facets composing the tip of the field emission microscope, in particular, by using the systematic expansions (14) into kubic harmonics for cubic crystals. Another important feature is the ultrafast mobility of hydrogen, which remains in the quasi-equilibrium distribution (28) and

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

66 0

50

1.0

(a)

θO, θS

0.8 0.6 0.4 0.2 0.0

θH

10 10 10 10

(b)

-2 -3 -4 -5

Brightness (a.u.)

40

(c) 20

0 0

50

Time (s) Fig. 3. (a) The time evolution of the oxygen coverage (solid line) and the oxygen subsurface occupation (dashed line) at the (001) plane in the kinetic model in the oscillatory regime. (b) Corresponding oscillations of the hydrogen coverage at the (001) plane. (c) Experimental total brightness during the oscillations.18 The conditions are the same as in Fig. 2. The arrows indicate the transition from a metallic rhodium field emitter tip to one that is invaded by subsurface oxygen.

evolves gradually because of the other kinetic processes. The kinetic model is built on the basis of experimental data about adsorption and desorption of hydrogen and oxygen as revealed by temperature-programmed desorption spectra and the recent studies on the RhO2 rhodium surface oxides.66 –75 By taking into account all these different aspects, the kinetic model provides a comprehensive understanding of the bistability, the oscillations, and the nanopatterns observed in FIM experiments.18 –20

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

67

Chemical nanoclocks have been observed under field emission microscopy conditions in several reactions besides water formation on rhodium.10,11,76 Surprisingly, rhythmic behavior is possible at the nanoscale of 10–30 nm in the population dynamics of the different species reacting on the surface. Indeed, this area may contain up to about 10,000 adsorbates, which is already much larger than the minimum size of a few hundred molecules required to sustain correlated oscillations.77 This behavior is an example of dynamical order, which can manifest itself out of equilibrium as a corollary of the second law.

4. 4.1.

Copolymerization processes Information processing at the molecular scale

If the history of a nonequilibrium system can be recorded on a spatial support of information, the theorem of nonequilibrium temporal ordering60 suggests that dynamical order may generate regular information sequences, which is not possible at equilibrium. At the molecular scale, natural supports of information are given by random copolymers where information is coded in the covalent bonds. This is the idea of Schr¨ odinger’s aperiodic crystal.78 Random copolymers exist in chemical and biological systems. Examples are styrene-butadiene rubber, proteins, RNA, and DNA, with the latter playing the role of informationsupport in biology. Accordingly, dynamical aspects of information are involved in copolymerization processes where fundamental connections with nonequilibrium thermodynamics have been recently discovered.21–23 4.2.

Thermodynamics of free copolymerization

The stochastic growth of a single copolymer proceeds by attachment and detachment of monomers {m} continuously supplied by the surrounding solution, which is assumed to be sufficiently large to play the role of a reservoir where the concentrations of the monomers are kept constant: ω = m1 m2 · · · ml

+ml+1




−ml+1

ω
= m1 m2 · · · ml ml+1 .

(30)

The probability Pt (ω) to find the monomer sequence ω of length l = |ω| at the time t is ruled by the master equation dPt (ω)  = [Pt (ω
)W (ω
|ω) − Pt (ω)W (ω|ω
)], dt
ω

(31)

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

68

where the coefficients W (ω|ω
) denote the rates of the transitions ω → ω
. If attachment and detachment processes are slower than the equilibration of the chain with its environment, the transition rates satisfy the conditions of local detailed balancing G(ω) − G(ω
) W (ω|ω
) = exp ,
W (ω |ω) kB T

(32)

in terms of the Gibbs free energy G(ω) of a single copolymer chain ω in the solution at the temperature T . The enthalpy H(ω) and the entropy S(ω) of the copolymer chain ω can similarly be defined and they are related together by G(ω) = H(ω) − T S(ω). At a given time t, the system may be in different sequences and different configurations so that the total entropy has two contributions:   Pt (ω)S(ω) − kB Pt (ω) ln Pt (ω). (33) St = ω

ω

The first one is due to the statistical average of the phase-space disorder S(ω) of the individual copolymer chains ω and the second is due to the probability distribution itself among the different possible sequences ω existing at the current time t. In the regime of steady growth,79,80 this probability is supposed to be factorized as Pt (ω) = µl (ω) × pt (l),

(34)

into a stationary statistical distribution µl (ω) describing the arbitrarily long sequence which is left behind, multiplied by the time-dependent probability pt (l) of the length l selected in the sequence. In this regime, the mean  growth velocity is constant and is given by v = dlt /dt where lt = l pt (l) × l is the mean length of the copolymer at time t. Mean values per monomer can be defined as 1 µl (ω) X(ω) l→∞ l ω

x = lim

(35)

for entropy s, enthalpy h, and Gibbs free energy g = h − T s. In these circumstances, the total entropy (33) can be calculated and shown to vary in time as de S di S dSt = + , dt dt dt

(36)

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

69

due to the entropy exchange between the copolymer and its surrounding: h de S = v dt T

(37)

di S = kB A v ≥ 0 dt

(38)

and the entropy production:

which is always non-negative according to the second law of thermodynamics. The entropy production is expressed in terms of the affinity21 g + D(polymer), (39) A=− kB T which involves, on the one hand, the free energy per monomer g and, on the other hand, the Shannon disorder per monomer in the sequence composing the copolymer: 1 D(polymer) = lim − µl (ω) ln µl (ω) ≥ 0. (40) l→∞ l ω A prediction of this result is that a copolymer can grow by the entropic effect of disorder D > 0 in an adverse free-energy landscape as long as the affinity (39) is positive.21,22 This is illustrated in Figs. 4 and 5 for free copolymerization with two monomers. The concentration [2] of the second monomer is kept fixed while the concentration [1] of the first one is varied. The growth velocity, as well as the affinity (39) and the entropy production (38), all vanish at equilibrium for [1] = [1]eq , which does not coincide with the concentration [1] = [1]0 where the free-energy driving force ε = −g/(kB T ) is vanishing. Therefore, the growth is possible for intermediate values of the concentration [1]eq < [1] < [1]0 in the entropic growth regime, preceding the growth regime driven by free energy when ε > 0 for [1] > [1]0 . At equilibrium, the Shannon disorder (40) reaches its maximum value D = ln 2 and decreases away from equilibrium, as seen in Fig. 5(a). 4.3.

Thermodynamics of copolymerization with a template

A similar result holds for copolymerizations with a template. If the sequence of the template α is characterized by the statistical distribution νl (α), the Shannon conditional disorder of the copy ω with respect to the template is defined as21 1 D(copy|template) = lim − νl (α)µl (ω|α) ln µl (ω|α) ≥ 0, (41) l→∞ l α,ω

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

...

70

...

time

2121211 21212112 212121121 2121211211 21212112111 212121121111 21212112111 212121121112 21212112111 212121121112 21212112111 212121121111 2121211211111 21212112111111 212121121111111 2121211211111112 21212112111111122 212121121111111222

space Fig. 4. Space-time plot of the stochastic growth of a copolymer composed of two monomers, as simulated by Gillespie’s algorithm with the parameter values: k+1 = 1, k−1 = 10−3 , [1] = 10−3 , k+2 = 2, k−2 = 2 × 10−3 , and [2] = 5 × 10−4 , in the growth regime by entropic effect.23 Under these conditions, the fraction of monomers 1 is p = 0.618, the growth velocity v = 6.17 × 10−4 (i.e., 0.18 monomer per reactive event), the free-energy driving force ε = −g/(kB T ) = −0.265, the Shannon disorder D = 0.665, iS = Av = 2.47 × 10−4 in the affinity A = ε + D = 0.400, and the entropy production ddt units where kB = 1.

and the mutual information between the copy and the template as57 I(copy, template) = D(copy) − D(copy|template) ≥ 0,

(42)

where the Shannon disorder of the copy is defined as in Eq. (40). In this framework, the thermodynamic entropy production is again given by Eq. (38) but with the affinity21 g + D(copy|template) kB T g + D(copy) − I(copy, template) =− kB T

A=−

(43)

which establishes quantitatively a fundamental link between information and thermodynamics at the molecular scale.

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

1

71

p

(a)

0.8

0.6 0.4 0.2

D

0 101

(b)

100

A

–1

10

ε

diS/dt

v

10–2 10–3

diS/dt

10–4 10–5 10–6

10–5

10–4

10–3 [1]

10–2

10–1

10 0

Fig. 5. Comparison between simulation (dots) and theory (lines) for the growth of a copolymer composed of two monomers with the parameter values: k+1 = 1, k−1 = 10−3 , k+2 = 2, k−2 = 2 × 10−3 , and [2] = 5 × 10−4 .23 Several characteristic quantities are depicted versus the concentration [1] of monomers 1: (a) the fraction p of monomers 1 in the copolymer (circles) and the Shannon disorder D (squares); (b) the growth velocity v (triangles), the free-energy driving force ε (open circles), the affinity A = ε + D iS = Av (crossed squares). The entropy (open squares), and the entropy production ddt production vanishes at the equilibrium concentration [1]eq = 5 × 10−4 together with the velocity and the affinity. However, the free-energy driving force vanishes at the larger concentration [1]0 = 1.30453 × 10−3 . The regime of growth by entropic effect exists between these two values of the concentration. The velocity and the affinity are positive for [1] > [1]eq = 5 × 10−4 and negative below (not shown). The free-energy driving force is positive for [1] > [1]0 = 1.30453 × 10−3 and negative below (not shown).

4.4.

The case of DNA replication

The previous results apply to the different living copolymerization processes and, in particular, to DNA replication. In this case, the subunits of the polymers are the four nucleotides N = A, T, C, or G and the monomers are the corresponding nucleoside triphosphates NTP. Assuming that no free energy difference exists between correct and incorrect chains, the

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

72 1

1.5

(a)

(b) I (nt−1)

error (%)

0.1 0.01 0.001

1

0.5

0.0001 −2

0

eq.

2

ε

4

6

8

0 −2

eq.

0

2

ε

4

6

8

Fig. 6. Stochastic simulation21 of DNA replication by polymerase Pol γ of human mitochondrial DNA from GenBank81 using known data on the kinetic constants of Watson–Crick pairing.82 The reversed kinetic constants are taken as k−mn = k+mn e−ε . (a) Percentage of errors in DNA replication versus the driving force ε. (b) Mutual information between the copied DNA strand and the original one versus the driving force. The arrow points to the equilibrium value of the driving force: εeq = − ln 4 = −1.38629.

copolymerization process can be simulated by Gillespie’s algorithm as a function of the driving force ε = −g/(kB T ).21 The results are depicted in Fig. 6 which shows the percentage of replication errors as well as the mutual information between the copy and the template. The error percentage is maximum at equilibrium and it decreases as the growth is pushed away from equilibrium. Similarly, the mutual information vanishes at equilibrium and saturates at Imax = 1.337 nats for high enough values of the driving force. As in the case of free copolymerization, a transition occurs between the regime of growth by entropic effect for εeq < ε < 0 and the growth driven by free energy for ε > 0. At equilibrium, information transmission is not possible between the template and the copy. Fidelity in the copying process becomes possible if enough free energy is supplied during copolymerization. The existence of growth by entropic effect could be experimentally investigated in chemical or biological copolymerizations. In polymer science, methods have not been much developed to perform the synthesis and sequencing of copolymers for the information they may support. However, such methods are already well developed for DNA and under development for single-molecule DNA or RNA sequencing.31 –33 These methods could be used to experimentally test the predictions of copolymerization thermodynamics by varying NTP and pyrophosphate concentrations to approach the regime near equilibrium where the mutation rate increases.

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

5.

73

Conclusions and Perspectives

Nowadays, self-organization has been studied for different phenomena from the macroscopic world down to the molecular scale. At the macroscale, self-organization emerges far-from-equilibrium beyond bifurcations leading to bistability or oscillatory behavior.1 –3 However, on smaller scales, the time evolution of physico-chemical systems is more affected by thermal and molecular fluctuations which are the manifestations of the microscopic degrees of freedom. In the framework of the theory of stochastic processes, the probability to find the system in some coarse-grained state is ruled by a master equation. The macroscopic description in terms of deterministic kinetic equations is only obtained in the large-system limit, in which bifurcations emerge between nonequilibrium macrostates. This emergence concerns in particular oscillations which only become correlated if the system is large enough.77 This is the case for the nanoclocks of heterogeneous catalysis observed by field emission microscopy techniques such as the reaction of water formation on rhodium in high electric field under FIM conditions.18 –20 In the present contribution, methods are described for the modeling of these nonequilibrium processes at the nanoscale on highly non-uniform and anisotropic surfaces. The underlying crystalline structure determines the reactivity of the surface and contributes to the formation of the observed nanopatterns, which cannot be interpreted as standard reaction–diffusion patterns. In spite of their nanometric size, such systems are large enough to undergo self-organization and manifest bistability and rhythmic behavior. This dynamical order is the result of directionality which is induced away from equilibrium by the external constraints and free-energy sources characterized by the empowering affinities. If the principle of detailed balancing holds at equilibrium, this is no longer the case out of equilibrium so that the fluctuating currents are biased and acquire a directionality, which is expressed by the current fluctuation theorem.45 –51 This theorem has, for consequence, the non-negativity of the entropy production in accordance with the second law of thermodynamics, as well as generalizations of the Onsager reciprocity relations and Green–Kubo formula from the linear to the nonlinear response properties.45,48 Furthermore, the entropy production appears as a measure of time asymmetry in the temporal disorder of the typical histories of a system. Out of equilibrium, the typical histories are more probable than their time reversals, which appears as a corollary of the second law.56,60 These new results transcend the known formulation of nonequilibrium thermodynamics, explicitly showing how the second law finds its origin in the breaking of time-reversal symmetry at the mesoscopic level of description in the theory of nonequilibrium systems.

November 21, 2012

13:1

9in x 6in

74

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

These results show that the second law of thermodynamics already governs self-organization at the molecular scale. Thermodynamics can be applied to the stochastic growth of a single copolymer, which is the support of information encoded in its sequence. Away from equilibrium, dynamical order enables information processing during copolymerizations, which is not possible at equilibrium. The statement by Manfred Eigen that “information cannot originate in a system that is at equilibrium”83 is rigorously proved in this framework. The thermodynamics of copolymerization shows that, indeed, fundamental connections exist between information and thermodynamics.21 –23 In particular, the growth of a copolymer can be driven either by the free energy of the attachment of new monomers or by the entropic effect of disorder in the grown sequence. These considerations open new perspectives to understand the dynamical aspects of information in biology. During copolymerization processes with a template as it is the case for replication, transcription or translation in biological systems, information is transmitted although errors may occur due to molecular fluctuations, which are sources of mutations. Metabolism and self-reproduction, which are the two main features of biological systems, turn out to be linked in a fundamental way since information processing is constrained by energy dissipation during copolymerizations. Moreover, the error threshold for the emergence of quasi-species in the hypercycle theory by Eigen and Schuster84 could be induced at the molecular scale by the transition towards high fidelity replication beyond the threshold at zero free energy per monomer between the two growth regimes.85 In this way, prebiotic chemistry could be more closely linked to the first steps in biological evolution. Acknowledgments This research has been financially supported by the F.R.S.-FNRS, the “Communaut´e fran¸caise de Belgique” (contract “Actions de Recherche Concert´ees” No. 04/09-312), and the Belgian Federal Government (IAP project “NOSY”). References 1. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Wiley, New York, 1967). 2. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977). 3. R. Imbihl and G. Ertl, Chem. Rev. 95, 697 (1995).

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

75

4. G. Nicolis and I. Prigogine, Exploring Complexity (W. H. Freeman, New York, 1989). 5. G. Nicolis and C. Nicolis, Foundations of Complex Systems (World Scientific, New Jersey, 2007). 6. R. Feistel and W. Ebeling, Physics of Self-Organization and Evolution (WileyVCH, Weinheim, 2011). 7. P. Gaspard, Nonlinear Dynamics of Nanosystems, eds. G. Radons, B. Rumpf and H. G. Schuster, (Wiley-VCH, Weinheim, 2010), pp. 1–74. 8. M. F. H. van Tol, A. Gielbert and B. E. Nieuwenhuys, Catal. Lett. 16, 297 (1992). 9. V. Gorodetskii, J. Lauterbach, H.-H. Rotermund, J. H. Block and G. Ertl, Nature 370, 276 (1994). 10. N. Ernst, G. Bozdech, V. Gorodetskii, H.-J. Kreuzer, R. L. C. Wang and J. H. Block, Surf. Sci. 318, L1211 (1994). 11. C. Voss and N. Kruse, Appl. Surf. Sci. 94/95, 186 (1996). 12. N. Kruse and T. Visart de Bocarm´e, Handbook of Heterogeneous Catalysis, 2nd edn., eds. G. Ertl, H. Kn¨ ozinger, F. Sch¨ uth and J. Weitkamp (WileyVCH, Weinheim, 2008), pp. 870–895. 13. T. Visart de Bocarm´e, N. Kruse, P. Gaspard and H. J. Kreuzer, J. Chem. Phys. 125, 054704 (2006). 14. J.-S. McEwen and P. Gaspard, J. Chem. Phys. 125, 214707 (2006). 15. J.-S. McEwen, P. Gaspard, F. Mittendorfer, T. Visart de Bocarm´e and N. Kruse, Chem. Phys. Lett. 452, 133 (2008). 16. T. Visart de Bocarm´e, T. B¨ ar and N. Kruse, Ultramicroscopy 89, 75 (2001). 17. T. Visart de Bocarm´e, G. Beketov and N. Kruse, Surf. Interface Anal. 36, 522 (2004). 18. J.-S. McEwen, P. Gaspard, T. Visart de Bocarm´e and N. Kruse, Proc. Natl. Acad. Sci. USA 106, 3006 (2009). 19. J.-S. McEwen, P. Gaspard, T. Visart de Bocarm´e and N. Kruse, J. Phys. Chem. C 113, 17045 (2009). 20. J.-S. McEwen, P. Gaspard, T. Visart de Bocarm´e and N. Kruse, Surf. Sci. 604, 1353 (2010). 21. D. Andrieux and P. Gaspard, Proc. Natl. Acad. Sci. USA 105, 9516 (2008). 22. C. Jarzynski, Proc. Natl. Acad. Sci. USA 105, 9451 (2008). 23. D. Andrieux and P. Gaspard, J. Chem. Phys. 130, 014901 (2009). 24. V. Garcia-Morales and K. Krischer, Proc. Natl. Acad. Sci. USA 107, 4528 (2010). 25. J.-P. Sauvage and P. Gaspard, Editors, From Non-Covalent Assemblies to Molecular Machines (Wiley-VCH, Weinheim, 2011). 26. P. Lussis, T. Svaldo-Lanero, A. Bertocco, C.-A. Fustin, D. A. Leigh and A.-S. Duwez, Nature Nanotechnol. 6, 553 (2011). 27. W. Min, L. Jiang, J. Yu, S. C. Kou, H. Qian and X. S. Xie, Nano Lett. 5, 2373 (2005). 28. T. Funatsu, Y. Harada, M. Tokunaga, K. Saito and T. Yanagida, Nature 374, 555 (1995). 29. H. Noji, R. Yasuda, M. Yoshida and K. Kinosita Jr., Nature 386, 299 (1997).

November 21, 2012

76

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

P. Gaspard

30. K. Adelman, A. La Porta, T. J. Santangelo, J. T. Lis, J. W. Roberts and M. D. Wang, Proc. Natl. Acad. Sci. USA 99, 13538 (2002). 31. W. J. Greenleaf and S. M. Block, Science 313, 801 (2006). 32. J. Eid et al., Science 323, 133 (2009). 33. S. Uemura, C. Echeverr´ıa Aitken, J. Korlach, B. A. Flusberg, S. W. Turner and J. D. Puglisi, Nature 464, 1012 (2010). 34. D. Andrieux and P. Gaspard, Phys. Rev. E 74, 011906 (2006). 35. E. Gerritsma and P. Gaspard, Biophys. Rev. Lett. 5, 163 (2010). 36. P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 (1990). 37. P. Gaspard, New J. Phys. 7, 77 (2005). 38. D. J. Evans, E. G. D. Cohen and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993). 39. G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995). 40. G. E. Crooks, Phys. Rev. E 60, 2721 (1999). 41. J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 (1999). 42. M. Esposito, U. Harbola and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009). 43. C. Jarzynski, Annu. Rev. Condens. Matter Phys. 2, 329 (2011). 44. D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr. and C. Bustamante, Nature 437, 231 (2005). 45. D. Andrieux and P. Gaspard, J. Chem. Phys. 121, 6167 (2004). 46. D. Andrieux and P. Gaspard, J. Stat. Mech., Theory Exp. P01011 (2006). 47. D. Andrieux and P. Gaspard, J. Stat. Phys. 127, 107 (2007). 48. D. Andrieux and P. Gaspard, J. Stat. Mech., Theory Exp. P02006 (2007). 49. D. Andrieux and P. Gaspard, J. Stat. Mech., Theory Exp. P11007 (2008). 50. D. Andrieux and P. Gaspard, J. Stat. Mech., Theory Exp. P02057 (2009). 51. D. Andrieux, P. Gaspard, T. Monnai and S. Tasaki, New J. Phys. 11, 043014 (2009). 52. T. De Donder and P. Van Rysselberghe, Affinity (Stanford University Press, Menlo Park CA, 1936). 53. D. Andrieux and P. Gaspard, J. Chem. Phys. 128, 154506 (2008). 54. P. Gaspard and X.-J. Wang, Phys. Rep. 235, 291 (1993). 55. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge UK, 1998). 56. P. Gaspard, J. Stat. Phys. 117, 599 (2004). 57. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edn. (Wiley, Hoboken, 2006). 58. D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud and A. Petrosyan, Phys. Rev. Lett. 98, 150601 (2007). 59. D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud and A. Petrosyan, J. Stat. Mech., Theory Exp. P01002 (2008). 60. P. Gaspard, C. R. Physique 8, 598 (2007). 61. E. W. M¨ uller and T. T. Tsong, Field Ion Microscopy: Principles and Applications (Elsevier, New York, 1969). 62. A. S. Mikhailov and G. Ertl, Chem. Phys. Lett. 238, 104 (1995). 63. M. Hildebrand, A. S. Mikhailov and G. Ertl, Phys. Rev. E 58, 5483 (1998).

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

Self-Organization at the Nanoscale

77

64. Y. De Decker, A. Marbach, M. Hinz, S. G¨ unther, M. Kiskinova, A. S. Mikhailov and R. Imbihl, Phys. Rev. Lett. 92, 198305 (2004). 65. A. Garc´ıa Cant´ u Ros, J.-S. McEwen and P. Gaspard, Phys. Rev. E 83, 021604 (2011). 66. E. Schwarz, J. Lenz, H. Wohlgemuth and K. Christmann, Vacuum 41, 167 (1990). 67. A. N. Salanov and V. I. Savchenko, Surf. Sci. 296, 393 (1993). 68. M. V. Ganduglia-Pirovano, K. Reuter and M. Scheffler, Phys. Rev. B 65, 245426 (2002). 69. J. Gustafson, A. Mikkelsen, M. Borg, E. Lundgren, L. K¨ ohler, G. Kresse, M. Schmid, P. Varga, J. Yuhara, X. Torrelles, C. Quir´ os and J. N. Andersen, Phys. Rev. Lett. 92, 126102 (2004). 70. C. Africh, F. Esch, W. X. Li, M. Corso, B. Hammer, R. Rosei and G. Comelli, Phys. Rev. Lett. 93, 126104 (2004). 71. L. K¨ ohler, G. Kresse, M. Schmid, E. Lundgren, J. Gustafson, A. Mikkelsen, M. Borg, J. Yuhara, J. N. Andersen, M. Marsman and P. Varga, Phys. Rev. Lett. 93, 266103 (2004). 72. J. Gustafson, A. Mikkelsen, M. Borg, J. N. Andersen, E. Lundgren, C. Klein, W. Hofer, M. Schmid, P. Varga, L. K¨ ohler, G. Kresse, N. Kasper, A. Stierle and H. Dosch, Phys. Rev. B 71, 115442 (2005). 73. C. Dri, C. Africh, F. Esch, G. Comelli, O. Dubay, L. K¨ ohler, F. Mittendorfer, G. Kresse, P. Dubin and M. Kiskinova, J. Chem. Phys. 125, 094701 (2006). 74. E. Lundgren, A. Mikkelsen, J. N. Andersen, G. Kresse, M. Schmid and P. Varga, J. Phys., Condens. Matter 18, R481 (2006). 75. F. Mittendorfer, J. Phys., Condens. Matter 22, 393001 (2010). 76. J.-S. McEwen, P. Gaspard, Y. De Decker, C. Barroo, T. Visart de Bocarm´e and N. Kruse, Langmuir 26, 16381 (2010). 77. P. Gaspard, J. Chem. Phys. 117, 8905 (2002). 78. E. Schr¨ odinger, What is Life? (Cambridge University Press, Cambridge, UK, 1944). 79. B. D. Coleman and T. G. Fox, J. Polym. Sci. A 1, 3183 (1963). 80. B. D. Coleman and T. G. Fox, J. Chem. Phys. 38, 1065 (1963). 81. Homo sapiens DNA mitochondrion, 16569 bp, locus AC 000021, version GI:115315570, http://www.ncbi.nlm.nih.gov. 82. H. Lee and K. Johnson, J. Biol. Chem. 281, 36236 (2006). 83. M. Eigen, Steps towards Life: A Perspective on Evolution (Oxford University Press, Oxford, 1992). 84. M. Eigen and P. Schuster, Naturwissenschaften 64, 541 (1977); Naturwissenschaften 65, 7 (1978); Naturwissenschaften 65, 341 (1978). 85. H.-J. Woo and A. Wallqvist, Phys. Rev. Lett. 106, 060601 (2011).

November 21, 2012

13:1

9in x 6in

b1449-ch03

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Chapter 4 SINGLE MOLECULE AND COLLECTIVE DYNAMICS OF MOTOR PROTEIN COUPLED WITH MECHANO-SENSITIVE CHEMICAL REACTION Mitsuhiro Iwaki∗,‡ , Lorenzo Marcucci∗ , Yuichi Togashi†,‡ and Toshio Yanagida∗,‡,§ ∗

Graduate School of Frontier Biosciences, Osaka University 1-3 Yamadaoka, Suita, Osaka 565-0871, Japan † Graduate School of System Informatics, Kobe University 1-1 Rokkodai, Nada, Kobe, Hyogo 657-8501, Japan ‡ Quantitative Biology Center, RIKEN 6-2-3 Furuedai, Suita, Osaka 565-0874, Japan § Center for Information and Neural Network 1-3 Yamadaoka, Suita, Osaka 565-0871, Japan

Motor proteins such as myosin and kinesin hydrolyze ATP into ADP and Pi to convert chemical energy into mechanical work. This results in various motile processes like muscle contraction, vesicle transport and cell division. Recent single molecule experiments have revealed that external load applied to these motor proteins perturb not only the mechanical motion, but the ATP hydrolysis cycle as well, making these molecules mechano-enzymes. Here, we describe our single molecule detection techniques to reveal the mechano-enzymatic properties of myosin and introduce recent progress from both experimental and theoretical approaches at the single- and multiple-molecule level.

Contents 1. 2. 3.

What is a Motor Protein? . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Myosin at the Single Molecule Level . . . . . . . . . . . Mechanosensitivity of ATP Hydrolysis during the Unidirectional Motion of Dimeric Myosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Mechanosensitive detachment of myosin-V from actin . . . . . . . . 3.2. Mechanosensitive attachment of myosin-VI to actin . . . . . . . . . 79

. . . .

80 81

. . . . . .

84 84 87

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

80 4.

Modeling and Simulating Mechanochemical Coupling and Motor Protein Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . 4.2. Coarse-grained models and dynamics . . . . . . . . . . . . . . . . . 4.3. Quantum mechanics for chemical processes . . . . . . . . . . . . . . 5. Modeling and Simulations of the Collective Behaviour of Motor Proteins 5.1. Huxley’s 1957 model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Huxley and Simmons’ 1971 model . . . . . . . . . . . . . . . . . . . 5.3. Diffusional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . .

. . . . . . . . .

89 91 92 93 94 94 95 97 98

What is a Motor Protein?

Motor proteins are biological molecular machines of nanoscopic size that converts the chemical free energy of Adenosine triphosphate (ATP) into mechanical work. There exist a large number of motor proteins in the cell, where each one of them is responsible for one or several complex mechanical tasks such as muscle contraction, cell division and organelle transport. One certain group of motor proteins, which include myosin, kinesin and dynein (Fig. 1), moves along its biofilaments (cytoskeletal actin filament and microtuble) in a linear fashion. These motors are especially well understood because they have been widely explored by various biophysical techniques. Here, we focus on myosin and review its mechanism for chemomechanical transduction. Myosin V

Kinesin-1

Cargo binding domain

Dynein

Lever arm ATP ADP+Pi Motor domain

Motor domain

Motor domain

Fig. 1. Structure of dimeric motor proteins. Myosin-V, conventional kinesin (Kinesin-1) and cytoplasmic dynein are drawn. These motors constitute homo-dimers. The motor catalytic domains (in which ATP is hydrolyzed into ADP and Pi) are displayed. Boxed areas drawn in dashed line are cargo binding domains.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

81

Myosin constitutes a superfamily of, 24 classes.1 Class II myosin (myosin-II) is best known for its role in muscle contraction, while class V myosin (myosin-V; seen in Fig. 1) is an intracellular vesicle transporter that is particularly popular in single molecule studies. Both myosins are well characterized with regards to their structure, enzymatic properties and physiological roles. All myosins share several structural traits, including a motor domain (traditionally called “head”), which hydrolyzes ATP and directly interacts with an actin filament, making it the fundamental unit for motor function. While the motor domain has a similar structure between classes, the lever arm domain, an which is also an important myosin function, does not. The lever arm domain binds to small proteins such as calmodulin, but its length and number of binding sites differ between classes.

2.

Measurement of Myosin at the Single Molecule Level

To fully comprehend molecular machines, it is necessary to understand the dynamic properties of the biomolecules isolation and when they interact with other molecules. Single molecule detection (SMD) techniques have been developed for this purpose. SMD techniques are based on two key technologies: single molecule imaging (Fig. 2) and single molecule manipulation (Fig. 3). The size of the studied biomolecules and even their assemblies are in the order of nanometers, so they are too small to observe by optical microscopy. To overcome this problem, biomolecules can be fluorescently labeled and visualized using fluorescent microscopy. Single fluorophores were first observed in nonaqueous conditions.3 In 1995, however, we successfully demonstrated that single fluorophores can be seen in aqueous solution by using total internal reflection fluorescence microscopy (TIRFM).4 The major problem to be overcome when visualizing single fluorophores in aqueous solution is the huge background noise caused by numerous sources including Raman scattering from water molecules. In our system, an evanescent field can be formed which totally reflects the laser beam to the diffraction limit of light, causing the light to localize near the glass surface for a penetration depth (∼ 150 nm) (Fig. 2(a)). Consequently, the illumination was restricted to fluorophores either bound to the glass surface or located nearby, thereby reducing the background light. As a result, the background noise could be reduced 2,000-fold compared to that of conventional fluorescent microscopy. Fluorescent measurements from single fluorophores attached to biomolecules and ligands have allowed for the detection of the motility, enzymatic reactions, and other properties at the single molecule level.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

82

(a)

Distance Evanescent field

Light intensity

Incident laser

(b) Evanescent field Cover glass Objective lens

Fluorescent ATP Pi

Fluorescent ADP

Myosin Cover glass

Laser

(c)

Fluorescent ATP

Fluorescent ADP

3 seconds

Fig. 2. Single molecule imaging. (a) The total internal reflection generates an evanescent field where the light intensity exponentially decays depending on the distance from the glass surface. (b) Visualization of ATP hydrolysis by single myosin molecules. (c) Observation of single fluorescent ATP (or ADP) on a myosin molecule. When ATP binds to the myosin, a fluorescent spot appears, and when the ATP is hydrolyzed and ADP dissociates from myosin, the spot disappears.

Along with the above, direct observation of individual ATP hydrolysis cycles could be achieved when using the fluorescent ATP analog, Cy3ATP (Figs. 2(b) and 2(c)). Cy3-ATP in free solution does not produce clear fluorescent spots, because of its rapid Brownian motion. However, when it (or Cy3-ADP) is associated with myosin bound to a surface, the Brownian motion ceases and the labeled nucleotide can be observed as a clear fluorescent spot. Thus, the association and dissociation of Cy3-ATP can be observed by monitoring the flickering of fluorescent spots. The second key technology is single molecule nanomanipulation. Biomolecules, and even single molecules, can be captured on a glass needle or on beads trapped by optical tweezers (Fig. 3). Optical tweezers are used to trap and manipulate dielectric particles of between 25 nm and 25 µm in diameter by using the force created by the laser radiation pressure.5 The

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

(a)

83

(b) Focused laser

Force

Force Dielectric particle

(c)

Actin filament

Focus point

Myosin

Fluorescent bead

Cover glass Etched glass

Infrared laser

(d) Photo sensor A A-B 1 nm B

Expanded image of the bead

Stiffness (pN/nm)

Bead position (nm)

Objective lens

(e) 50 0 -50 0.2 0.1 0 0

5 Time (second)

10

Fig. 3. Optical tweezers assay. (a) A focused laser is refracted at the dielectric particle, generating different force vectors (see grey arrows). The resulting force is directed toward the focus point (black arrow). (b) The optical trapping force is approximately in proportion to the distance from the trap center. (c) An example of the geometry for single molecule force measurements. Two independent infrared focussed lasers are used as a nano-manipulator on the single actin filament. The filament is tautly stretched and moved in the vicinity of a single myosin stuck on the etched glass surface. (d) Detector for the optically-trapped bead at sub-nanometer resolution. The expanded image of the optically-trapped bead is projected onto the separated photo sensor. (e) Typical interaction between a myosin and actin filament. Upper trace, bead position; lower trace, stiffness calculated from the bead fluctuation.

particle is trapped near the focus of the laser light that occurs by passing a microscope objective with a high numerical aperture. The result is that the optical tweezers exert a force in the piconewton range on the particles. Biomolecules are too small to be directly trapped by optical tweezers, so they are generally attached to an optically-trapped bead. The trapping force acts as a spring that expands in proportion to the applied force (Fig. 3(b)). Thus, the force and the displacement caused by the biomolecules can be measured (Fig. 3(c)). The displacement of a bead has been determined with

November 21, 2012

84

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

sub-nanometer accuracy by using a splitted photosensor (Fig. 3(d)), which corresponds to a force sensitivity of sub-piconewtons. Thus, the mechanical property of biomolecules can be determined directly at the single molecule level (Fig. 3(e)). By combining single molecule imaging with optical tweezers, simultaneous measurements of mechanical and chemical reactions by single biomolecules are possible.6 Figure 4 shows the time traces of displacement (upper black trace) and fluorescence intensity of a Cy3-ATP molecule bound to myosin (lower red trace). When the ATP molecule binds to the myosin head, the myosin head dissociates from the actin causing the displacement to reach zero. The dissociated myosin head rebinds to actin at another location. Upon hydrolyzing ATP and releasing the products ADP and Pi, the myosin generates a new displacement and force. Thus, each displacement corresponds to a single ATPase turnover.

3.

3.1.

Mechanosensitivity of ATP Hydrolysis during the Unidirectional Motion of Dimeric Myosin Mechanosensitive detachment of myosin-V from actin

Myosin-V takes many productive mechanical steps upon each diffusional encounter with an actin filament.7 This feature, known as processive movement, makes myosin-V ideal for single molecule studies, because it increases the duration of the interaction, reducing time resolution demands. One of the latest single molecule imaging techniques, FIONA, can detect detailed motion of myosin-V with one nanometer accuracy.8 In FIONA, a single fluorophore is attached to one of the myosin-V motor domains and then has its position tracked. Results have shown that each myosin-V head moves forward along an actin filament with a step size of ∼ 72 nm in an alternating manner with the other (Fig. 5(a)), arguing in favor of the handover-hand mechanism, a popular model for myosin motility in which the rear head detaches from the actin filament, diffuses to the forward helical pitch and attaches there while the other head remains attached (Fig. 5(b)). The hand-over-hand mechanism requires the two myosin-V motor domains to be highly coordinated such that when the rear head is detached from actin, the lead head remains bound. How is the detachment rate regulated to preserve this constraint? As revealed by the experiment shown in Fig. 4, detachment of the myosin head is coupled with ATP binding to the head. Biochemical studies have shown that the head’s attachment period corresponds to the ADP bound state and subsequent ATP waiting state (Fig. 6(a), attached states). Veigel et al. first reported that the ADP

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

85

Force generation

Fluorescent intensity

Stiffness (pN/nm)

Force (pN)

Position of bead (nm)

Dissociation

Time (s) Pi

Cy3-ATP Cy3-ADP

Fig. 4. Direct observation of the chemomechanical coupling in myosin-II. TIRF microscopy is combined with optical tweezers to simultaneously observe mechanical events and ATP hydrolysis. When ATP binds to myosin, the myosin dissociates from actin. When ATP is hydrolyzed and ADP dissociates from actin, the myosin attaches to an actin filament and generates force.

release rate is significantly dependent on the mechanical strain sensed by the motor domain.9 When a forward (the same direction as myosin movement) strain was applied to the motor domain using optical tweezers, ADP release from the head accelerated, shortening the attachment period. On the other hand, a backward strain decelerated the ADP release rate, prolonging the attachment period. As shown in Fig. 6(b), when myosin-V spans the actin helical pitch, there should exist a forward and backward intramolecular

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

86

72 nm

(b)

(a) Position (72 nm per division)

1

2

Fluorophore (–)

72 nm

(+)

3

Time (1 second per division)

Fig. 5. Hand-over-hand motion, as seen in FIONA. (a) Typical trajectory of the center position of a fluorescent image (see inset) attached to a myosin-V motor domain. The step size is approximately 72 nm. (b) Model for the coordinated motion of myosin-V. When the rear head detaches from the actin filament, it swings forward and binds to the actin in a position that puts it in front of the other head.

Detached or Weakly attached state

(a)

M·ATP

(Strongly) Attached state

M·ADP·Pi

M·ADP Pi

M ADP

ATP

(b)

(c)

Intramolecular strain Forward Backward

Head 1

Att Movement

1

2

Rear

Front

Head 2

Det Time

Fig. 6. Load dependent ADP release. (a) Scheme for ATP hydrolysis cycle. M: myosin (b) Schematic drawing of intramolecular strain sensed by a dimeric myosin when both heads are attached to an actin filament. (c) Attachment and detachment states (“Att” and “Det”) for the rear (“Head 1”) and front (“Head 2”) heads during myosin-V motility. The attachment period (gray area) alternately shifts due to the strain dependent ATP hydrolysis cycle.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

87

strain on the rear and front heads, respectively. Estimates have shown the strain to be 1–3 pN, which would correspond to a 2–3 fold acceleration of the detachment rate for the forward strain and 5–30 fold deceleration for the backward strain. Therefore, the rear head dominantly detaches from actin in conjunction with a mechanosensitive ATP hydrolysis cycle. The mechanosensitive kinetics has been acknowledged by other groups such as the Spudich group10,11 and the Ishiwata group.12 The strain-dependent chemical reaction shifts the chemical state between heads, resulting in hand-over-hand motion (Fig. 6(c)). Myosin-VI, another member of the myosin family, operates a little differently as it exhibits an accelerated ATP binding rate when sensing a forward strain. Regardless, this too would promote detachment of the rear head and preserve the hand-over-hand movement. 3.2.

Mechanosensitive attachment of myosin-VI to actin

Once the rear head detaches, it is thought to undergo Brownian motion in the vicinity of the actin filament (the Brownian search). However, to date, the Brownian state has been too short to directly observe by single molecule imaging (see Fig. 5; time resolution of ∼30 ms). We successfully visualized this process by attaching 40 nm gold nano particle to the motor domain and observing scattered light from gold nano particle with 37 µs time resolution (∼1000-fold improvement compared to FIONA).13 This technique, dark-field imaging, involves only scattered light produced by the laser illumination reaching the camera. These direct observations showed that the detached rear head undergoes Brownian motion before catching a forward actin target (Brownian search-and-forward catch) (Fig. 7(a)). How does the Brownian head attach in the forward actin target? (Fig. 7(b)) Various biochemical studies14 and Fig. 4 suggest that the inorganic phosphate (Pi) release that precedes ADP release from the myosin head is coupled with strong attachment between the myosin head and actin (Fig. 6(a)). When the Brownian head catches, both heads are simultaneously attached to the actin filament, which should create intramolecular strain (Fig. 6(b)). Therefore, we hypothesized that the Pi release is also mechanosensitive and that this sensitivity controls the attachment of the myosin head to actin. To test this hypothesis, we developed an optical tweezers system in which we can apply a mechanical strain onto the myosin head during Brownian motion (Fig. 8). One notable observation from the experiments is that the attachment period prior to Pi release is quite short (1–2 ms), therefore we improved our optical tweezers system up to a micro second (∼ 50 µs) manipulation range.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

88

Gold nano particle

Position of gold along actin (nm)

Catch Catch

100 Probability of attachment

Probability density of Brownian search

0 Brownian search

Time (50 ms per division)

(a)

(b)

Fig. 7. Visualization of the Brownian motion in the rear head during hand-overhand motion. (a) The position of a 40 nm gold nano particle attached to a myosin-VI motor domain at 37 µs time resolution. Myosin-VI is a vesicle transporter with similar mechanosensitive detachment properties to myosin-V. Fluctuation increased during the Brownian search, but was suppressed when the catch occurred (see arrow termed “catch”). (b) Probability density of the Brownian search (grey) and strong attachment by the head (red).

Figure 8 shows the experiment of our optical tweezers system for applying forward and backward force to the ADP·Pi·myosin complex and the force application to the weak binding. We found the attachment rate (strong binding), and therefore Pi release, was greatly accelerated by a backward strain, indicating mechanosensitivity is utilized for the rectification of Brownian motion.15 In summary, mechanosensitivity of the transient chemical steps (Pi release, ADP release and ATP binding) in ATP hydrolysis plays an important role in the unidirectional motion done by myosin. A phenomenological model for the unidirectional motion of myosin that can explain the mechanical properties of muscle contraction was proposed by Andrew Huxley more than 50 years ago16 (Fig. 9). In that model, the detachment rate is high in the backward region, while the attachment rate is high in the forward region. The driving force for movement is Brownian motion, while asymmetric detachment and attachment produces unidirectional motion and force generation. Recent findings on the mechanosensitive chemical reaction described here can well explain this model at the substantial level. When a myosin head binds to the backward region and forward strain is applied, either the ADP release rate or ATP binding rate accelerates in a strain-dependent manner, which results in a high dissociation rate from actin (Fig. 6). On the other hand, when the Brownian head temporarily binds to the forward region and backward strain is applied, Pi release

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

Force by optical tweezers mimicking intramolecular strain Forward strain

89

Pi

Backward strain

ADP

Myosin

+

–

+

–

Weak binding

Actin filament

Pi is not released

Pi is released

Dissociation

(a)

Weak binding

Strong binding (Catch)

Pi release was accelerated (b)

Fig. 8. Force application to a single myosin head with ADP and Pi. A single myosin head was attached to an optically-trapped bead, which was scanned on a micro-second time scale while moving along a single actin molecule bridge etched to the glass surface. The actin filament has polarity, as indicated by “+” and “−” at the ends of the filament. (b) When a myosin head with ADP and Pi bound was scanned backward, strong binding was frequently observed, indicating Pi release was accelerated. (a) On the other hand, when the head was scanned forward, it dissociated from actin, suggesting Pi was not released.

accelerates in a strain-dependent manner, which results in a high attachment rate (Fig. 8). 4.

Modeling and Simulating Mechanochemical Coupling and Motor Protein Motion

As seen in previous sections, single molecule experimental techniques have enabled us to observe the motion and chemical cycles of single motor molecules under various conditions, helping reveal the mechanosensitivity of their chemical reactions to provide a chemical basis for the phenomenological model by A. F. Huxley.16 Another important question about motor molecules is, how do they sense mechanical stimuli like a force applied to the myosin tail, to accordingly alter their physical and chemical properties like the affinity to

November 21, 2012

13:1

9in x 6in

b1449-ch04

M. Iwaki et al.

90

(a)

Engineering of Chemical Complexity

Myosin filament Brownian motion

Myosin head

Actin filament (b)

Backward

Dissociation rate

Equillibrium position of myosin head

Forward

Attachment rate Position

Fig. 9. Model for the muscle contraction by A. F. Huxley in 1957. (a) A myosin head is connected to a myosin filament backbone via an elastic component, enabling the head to undergo Brownian motion back and forth around an equilibrium position. (b) Assuming the attachment and dissociation rates are asymmetric can explain the mechanical property (force-velocity curve) of muscle contraction.

a molecular track and the ATP hydrolysis cycle reaction rates? As these properties depend on the internal conformation of the molecule, responses to mechanical inputs should be complemented by conformational changes. It is theoretically possible then that a mechanical force can directly induce conformational changes in locations like the actin-binding cleft that would change affinity. However, some of the changes like those in the cleft are so far from where the external force is applied (several to more than 10 nanometers) that some sort of information transmission mechanism inside the molecule is required. Recently, the structures of different chemical states of motor proteins bound to ligands like ATP-analogs and ADP have been resolved by X-ray crystallography, NMR or electron micrography. These structures provide insights into how conformational changes couple with the chemical cycle and mechanical forces, though they are basically static structures determined by a large ensemble of molecules. Because a mechanical deformation is a dynamical process, information on the molecular dynamics or motions is needed to fully understand the motors’ mechanosensitive reaction cycles.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

91

Though single-molecule measurements of conformational changes are possible with fluorescent resonance energy transfer (FRET) or atomic force microscopy (AFM) technologies, either the number of measurement points or the time resolution is limited, meaning conformational changes of a whole molecule cannot be observed in sufficient detail. 4.1.

Molecular dynamics simulations

When experiments are not an option, we turn to computer simulations. In particular, molecular dynamics (MD) simulations are widely used for the investigation of macromolecular motion. In MD simulations, the position and velocity of each element (atom or coarse-grained particle) is defined as a variable and the motion of every part of the molecule is solved numerically. Particularly, in classical MD simulations of macromolecules, the force on each atom or the potential energy of the molecule is approximated as a function of positions of the elements.17 The potential function, often called the force-field, in all-atom MD typically depends on distances between atoms and angles between chemical bonds, with parameters determined empirically or by calculating the quantum mechanics. One direct way to observe conformational changes induced by external forces is to apply virtual forces to a designated set of atoms in an MD simulation, known as steered molecular dynamics.18 An example of steered MD for myosin was done by the same group.19 This technique is analogous to AFM or optical tweezers experiments, but with the benefit that we can arbitrarily choose the direction and intensity of the force. Additionally, we can record the stress-strain curve in the same manner as in AFM. Molecular motors show large conformational changes in their working cycle that may involve transitions over relatively high energetic barriers. Since the intervals between such transitions are long, of the order of microseconds to milliseconds, it is inefficient to wait until a transition event occurs during a simple MD simulation. Rather, sampling methods for such rare transitions such as replica-exchange molecular dynamics20 and multicopy enhanced sampling21 can be used, though their application is still limited to a relatively small part of the motor protein. Another way to observe such transitions is to guide the molecule to a certain direction by adding a fictitious potential energy or a moving constraint that converges to the final structure.22 Such adjustments are referred to as targeted or guided molecular dynamics. The most limiting factor to MD simulations is the computational cost, as the time-step must be of the order of femtoseconds (typically 1–2 fs) when adopting an all-atom model and a number of forces must be recalculated for each step, calling for continuous development of peta- to exa-FLOPS

November 21, 2012

92

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

supercomputers and specialized hardware for molecular dynamics.23,24 At this point in time, though, only a simulation of just above a millisecond has been obtained for a small enzyme bovine pancreatic trypsin inhibitor (BPTI) consisiting of 58 amino-acid residues.24,25 Thus, while conformational transitions of a single motor may be possible, understanding its mechanism requires not only a simulation of the entire cycle but also statistical analysis on a number of trials or an array of motors, which remains as a great challenge. 4.2.

Coarse-grained models and dynamics

To mitigate the burden of computation, a variety of coarse-grained and simplified models have been invented. A well-known example is the G¯ omodel27,28 (and its variations, so-called G¯ o-like models), in which an aminoacid residue is represented by a point particle, and the potential function is defined by using a reference (native state) conformation and introducing interactions between neighboring residues in the reference (so-called native contacts) so that the state is most stable (i.e., the global minimum of the potential). The original G¯ o-model was on-lattice, i.e., the particle-chain moved only on a grid, intended for the study of protein folding, but was later extended to continuous, three-dimensional space (off-lattice)29 and applied also to the conformational motion of folded proteins. The elastic network model (ENM)30 –32 in which all interactions are simplified as linear elastic springs that have natural lengths equal to their distances in the reference conformation (an example is shown in Fig. 10) is one of the simplest variations. Because of its simplicity, ENM has often been used with linearized normal mode analysis around the reference to explain conformational changes of proteins between different chemical states, and is also applied to molecular motors.33 However, it is limited because the model is nonlinear for perturbations as small as thermal fluctuations and molecular motors exhibit much larger conformational changes, in which case MD simulations of ENM should be considered.34,35 Manipulation techniques such as steered and guided MD can be used in combination with these simplified models. Coarse-grained models have been applied to motor proteins, revealing an array of insight on chemical cycle of different molecular motors including that of F1 -ATPase, a rotatory motor,36 and of HCV helicase, a DNA-based motor.37 Using a variation of ENM for the myosin head, it was seen that electrostatic interactions may unidirectionally bias the Brownian motion of myosin when it approaches an actin filament.38 We have demonstrated force-induced conformational changes in myosin V by sampling responses to many different stimuli, i.e., forces to different directions on each residue,

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

(a)

(b)

93

(c)

Tail

Actin-binding cleft

Fig. 10. Models with different resolutions. (a) Structure of scallop myosin S1 (Protein Data Bank ID: 1KK726 ), (b) its all-atom representation and (c) coarse-grained elasticnetwork model.

by steered MD of ENM.39 This method drastically reduced computational costs. 4.3.

Quantum mechanics for chemical processes

Another shortcoming of classical MD is that the fundamental reaction event, the reconfiguration of chemical bonds, cannot be considered in the framework of the simulation. To understand the basis of mechanochemical coupling, it is important to unveil the details of ATP binding, hydrolysis and product release to show how conformational changes affect such reconfigurations. For this purpose, we need quantum mechanics. However, considering the quantum mechanics over the whole molecule has an extreme computational cost. Even if calculation methods like the fragment molecular orbital method40 improve the situation, they cannot be done for each step when simulating a large molecule. The most currently acceptable method, despite its limitations, is the QM/MM,41 which combines quantum mechanics (QM) of a small part of a protein with classical molecular mechanics (MM) of the entire molecule. When examining ATP hydrolysis, QM/MM is sufficiently practical since the reaction occurs within a relatively small nucleotide-binding pocket so that classical MD alone can address the nonreactive parts of the molecule.42

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

94

Finally, once we have obtained the dynamical behavior of a motor protein for its entire ATP hydrolysis cycle, we can define the reaction coordinate, or conformational states, to construct a simple model with one or few variables that can then be used as the physical basis for phenomenological models.

5.

Modeling and Simulations of the Collective Behaviour of Motor Proteins

The methods presented in the previous section are useful to explore the mechanics of a single or few molecular motors, but their computational cost becomes a limit in simulating cooperative motors, where several motors act synchronously. To describe the collective behaviour of such motors, as is the case in muscle contraction, a less detailed approach is commonly used. Nevertheless, thermal fluctuations still have an important role in such models. 5.1.

Huxley’s 1957 model

Before 1954, most theories of muscle contraction were based on the idea that muscle shortening and force production were the result of some kind of folding or coiling of large proteins. In 1954, Huxley and Hanson43 as well as Huxley and Niedergerke44 demonstrated that muscle contraction is not associated with any change of length inside the muscle microstructure. Instead, the authors postulated that force is generated through an interaction between actin and myosin filaments. Using this conclusion, Huxley, in 1957, developed a new theory of muscle contraction16 (Fig. 9). The thick myosin filament is assumed to be fixed in space while the thin filament is assumed to slide parallel to the myosin filament with velocity v. Movement is generated by a mechanical structure (now known to be myosin motor) that can occupy different positions along the actin filament, and whose movement is limited by an elastic element. When the structure is attached to actin, a force between actin and myosin arises that depends on the position of the motor head, with thermal fluctuations constraining attachment to the axial position. The motors exert a force when attached to actin and the elastic element is stretched, but a source of asymmetry is needed to generate a net force in one particular direction.16 To calculate the total force generated by muscle, one needs to know the total number of attached motors and their position x relative to the reference position of the motor itself at each time t.

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

95

If n(x, t) is the fraction of attached motors whose distance from their reference position is x at time t, then its time evolution can be found from a first order kinetic equation:16 ∂n(x, t) ∂n(x, t) −v = (1 − n(x, t))f (x) − n(x, t)g(x), ∂t ∂x

(1)

where f (x) and g(x) are the attachment and dissociation rates shown in Fig. 9. Huxley limited the analysis to the steady state case, when the solution n(x) is constant on time, so the first term in the left-side of equation is zero. Equation (1) allows for the computation of n(x) at different v. For example, at v = 0, n(x) reaches the constant value f /(f +g), while at higher values of v there are two factors that reduce n(x): less time for the motors to attach, and a faster rate at which the motors take negative x. Optimizing the free parameters, Huxley obtained an excellent fit of the force-velocity curve, but to fit other experimental behaviours, a more detailed description of actin and myosin interaction were needed. 5.2.

Huxley and Simmons’ 1971 model

The analysis of the fast equilibrium achieved by the actomyosin complex while in the attached state has been intensively studied since the pioneering work proposed by Huxley and Simmons in 1971.45 Applying a fast (microsenconds) and small (few nanometers per half sarcomere) increment of length δ to a muscle during isometric contraction, results in a clear tension transient (see Fig. 11(b), inset, for a schematic description). Initially, there is an almost instantaneous change in isometric tension from T0 to T1 (δ). This change is usually related to the change in length of an elastic element in series between the actin filament and the myosin backbone. This is followed by a slower (milliseconds) recovery of tension towards T0 but plateaus beforehand at T2 (δ). Because this behavior cannot be easily explained by the 1957 Huxley’s model, Huxley and Simmons proposed a new model, that deals only with force generation in attached motors, giving no consideration to the detachment process. The model assumes that the motor contains a linear elastic spring that is linked to the myosin head. When attached to the actin filament, the head can be in (at least) two states, switching between them in a discrete manner (jump process). The switching can stretch or relax the elastic element, so that we can refer to the states as “low” force generating (l ) and “high” force generating state (h). Changes in δ will affect the tension of the linear spring, and therefore affect T1 . This also causes a change in the energy minima of the two force generating states, because of the elastic energy,

November 21, 2012

96

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

Fig. 11. Adapted from Ref. 57. (a) Schematic description of the model. Myosins are modelled as over-damped particles attached through an elastic element to a rigid backbone and subjected to thermal fluctuations. In the attached state, the myosin head is also subject to a multi-stable potential. Attachment and detachment rate functions are shown. (b) Simulation of the fast increment experiments. Time traces (inset) and T1 T2 curves. Simulations (triangles) fit well the experimental data (points, from Ref. 54). (c) Simulation of contraction at constant loads. Time traces (inset) and force-velocity curve. Simulations (triangles) and experimental data (points, from Ref. 55).

and changes the total number of motors in each state, since the ratio of the transition rates are controlled by the relative energies of the two states. These changes are driven by the kinetics of the discrete transitions, which occur on a time scale slower than that of the T1 response. The steady state of these transitions leads to T2 . The differential equation describing the number of motors in state h during the transients is: dnh (t) = k+ nl (t) − k− nh (t) = −(k+ + k− )nh (t) + k+ , (2) dt where nlh is the number of motors in each state and k+ and k− are rate constants. Experimental data for T1 (δ) and T2 (δ) are shown in Fig. 11(b). An important aspect of the model is that despite very clear mathematical interpretation of the mechanical elastic element, the bi-stable chemical

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

97

energy mathematically degenerates because the energy wells are infinitely narrow. This leads to a description of the state transition in terms of a jump process, which requires an empirical definition of chemical rate constants. 5.3.

Diffusional model

The 1957 and 1971 models have been since refined by several works and extended to the entire cross-bridge cycle.46 –51 As informative as these contemporary models are, they are limited by the lack of direct information on state transition dynamics. While they introduce an explicit definition of the energy in each state, transitions between states are defined as rate functions with ad-hoc dependencies on the chemical reaction coordinate or on hidden assumptions about the actomyosin potential energy. In other words, actomyosin properties are deduced by fitting the macroscopic behavior of the muscle fiber even though the goal of the model is to interpret the muscle behavior from the actomyosin properties. This approach reduces the predictive power of such models and blurs the mechanical relationship between force and the conformational states. The oscillatory behavior of the actomyosin complex is a fundamental property of muscle function such that its description cannot be restricted to phenomenological rate functions. With the improved SMD techniques, explained in the previous sections, one can now obtain quantitative information on these actomyosin properties to deduce the macroscopic collective behavior of myosin motors. For example, oscillating sub-steps during the force generating state have been observed experimentally by attaching a single myosin-II molecule to a large microneedle and associating it with an actin bundle,52,53 showed several 5.5 nm steps per ATP cycle biased in one direction. Using the microscopic proprieties revealed from single myosin-II experiments, we created a “diffusional model” in which the myosin takes only two states, a force generating state and a non-force generating state (zero in average), roughly related to the attached and detached states. In the detached state, the myosin head is subjected only to thermal fluctuations and to force generated by the elastic element. In the attached state, the myosin head is also subject to a piecewise linear, multi-stable potential, Ea (x), with a periodicity of L = 36 nm as shown in Fig. 11(a). This results in a flashing ratchet that can be described by the following system of Langevin equations:  √ ηx x˙ i = −ωi (t)Ea (xi ) + Ee (xi − li ) + ηx kB θΓ(t) (3) Γ(t )Γ(t ) = 2δ(t − t ). 1 2 1 2

November 21, 2012

13:1

98

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

The continuous behavior of the myofibril is ensured by the uniform distribution of the variable l in the range [0 − L] for N different myosin heads, and by setting li (t) = z(t) + iL/N where z(t) represents the position of the backbone. ωi is a stochastic variable which fluctuates between the values 0 and 1 determined by the rate functions k+ and k− described in Fig. 11(a). Due to the periodicity of the potential, net movement of the backbone cannot happen without breaking the global equilibrium.56 We introduce breaking by considering rate functions out of balanced equilibrium. Furthermore, attachment of the myosin heads is prevented between two consecutive four-minima regions, mimicking the geometry of the actin filament. The attachment rate k+ is based on the preferential attachment of the myosin VI when stressed (see Sec. 1.3.2), as we assume the same mechanism is likely to be active in skeletal myosin. The detachment rate k− is slightly modified from the one originally proposed in Huxley’s 1957 model in order to consider the detailed geometry of our potential. We have tested this model on data from two classical skeletal muscle experiments: the fast tension recovery after a small and fast increment in isometric length, and velocity of contraction against a constant load. Simulations for both the V /Vmax vs. T /T0 curve (Fig. 11(c)) and for T1 (δ) and T2 (δ) curves (Fig. 11(b)) show very good fitting. Although such models are less precise than MD simulations, they offer the ability to simulate behavior over a long time, and therefore can more comprehensively recreate experimental observations even of several motor proteins interacting together, to explain collective behavior. This new approach also has the important benefit of reducing the number of free parameters and makes them constant rather than complex empirical functions of chemical reaction coordinates.

References 1. B. J. Foth, M. C. Goedecke and D. Soldati, Proc. Natl. Acad. Sci. USA 103(10), 3681 (2006). 2. R. D. Vale, Cell 112(4), 467 (2003). 3. W. E. Moerner and L. Kador, Phys. Rev. Lett. 62(21), 2535 (1989). 4. T. Funatsu, Y. Harada, M. Tokunaga, K. Saito and T. Yanagida, Nature 374(6522), 555 (1995). 5. A. Ashkin, J. M. Dziedzic and T. Yamane, Nature 330(6150), 769 (1987). 6. A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga et al., Cell 92(2), 161 (1998). 7. A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich et al., Nature 400(6744), 590 (1999).

November 21, 2012

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

Single Molecule and Collective Dynamics of Motor Protein

99

8. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha et al., Science 300(5628), 2061 (2003). 9. C. Veigel, S. Schmitz, F. Wang and J. R. Sellers, Nature Cell Biol. 7(9), 861 (2005). 10. T. J. Purcell, H. L. Sweeney and J. A. Spudich, Proc. Natl. Acad. Sci. USA 102(39), 13873 (2005). 11. D. Altman, H. L. Sweeney and J. A. Spudich, Cell 116(5), 737 (2004). 12. Y. Oguchi, S. V. Mikhailenko, T. Ohki, A. O. Olivares et al., Proc. Natl. Acad. Sci. USA 105(22), 7714 (2008). 13. S. Nishikawa, I. Arimoto, K. Ikezaki, M. Sugawa et al., Cell 142(6), 879 (2010). 14. C. M. Yengo, E. M. De La Cruz, D. Safer, E. M. Ostap and H. L. Sweeney, Biochemistry 41(26), 8508 (2002). 15. M. Iwaki, A. H. Iwane, T. Shimokawa, R. Cooke and T. Yanagida, Nature Chem. Biol. 5(6), 403 (2009). 16. A. F. Huxley, Prog. Biophys. Biophys. Chem. 7, 255 (1957). 17. J. A. McCammon, B. R. Gelin and M. Karplus, Nature 267, 585 (1977). 18. S. Izrailev, S. Stepaniants, M. Balsera, Y. Oono and K. Schulten, Biophys. J. 72(4), 1568 (1997). 19. Y. Liu, J. Hsin, H. Kim, P. R. Selvin and K. Schulten, Biophys. J. 100(12), 2964 (2011). 20. Y. Sugita and Y. Okamoto, Chem. Phys. Lett. 314(1–2), 141 (1999). 21. M. Cecchini, Y. Alexeev and M. Karplus, Structure 18(4), 458 (2010). 22. R. Elber, J. Chem. Phys. 93(6), 4312 (1990). 23. T. Narumi, Y. Ohno, N. Okimoto, T. Koishi et al., A 55 TFLOPS simulation of amyloid-forming peptides from yeast prion Sup35 with the special-purpose computer system MDGRAPE-3, Proc. 2006 ACM/IEEE conf. Supercomputing (SC06) (2006). 24. D. E. Shaw, R. O. Dror, J. K. Salmon, J. P. Grossman et al., Millisecond-scale molecular dynamics simulations on Anton, Proc. Conf. High Performance Computing, Networking, Storage and Analysis (SC09) (2009). 25. D. E. Shaw, P. Maragakis, K. Lindorff-Larsen, S. Piana et al., Science 330(6002), 341 (2010). 26. D. M. Himmel, S. Gourinath, L. Reshetnikova, Y. Shen et al., Proc. Natl. Acad. Sci. USA 99(20), 12645 (2002). 27. N. G¯ o, Theoretical studies of protein folding, Annu. Rev. Biophys. Bioeng. 12, 183 (1983). 28. S. Takada, Proc. Natl. Acad. Sci. USA 96(21), 11698 (1999). 29. C. Clementi, H. Nymeyer and J. N. Onuchic, J. Mol. Biol. 298(5), 937 (2000). 30. I. Bahar, A. R. Atilgan and B. Erman, Fold. Des. 2(3), 173 (1997). 31. P. Doruker, A. R. Atilgan and I. Bahar, Proteins 40(3), 512 (2000). 32. Q. Cui and I. Bahar (eds.), Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems, (Chapman & Hall/CRC, Boca Raton, 2006). 33. W. Zheng and S. Doniach, Proc. Natl. Acad. Sci. USA 100(23), 13253 (2003).

November 21, 2012

100

13:1

9in x 6in

b1449-ch04

Engineering of Chemical Complexity

M. Iwaki et al.

34. Y. Togashi and A. S. Mikhailov, Proc. Natl. Acad. Sci. USA. 104(21), 8697 (2007). 35. Y. Togashi, T. Yanagida and A. S. Mikhailov, PLoS Comput. Biol. 6(6), e1000814 (2010). 36. N. Koga and S. Takada, Proc. Natl. Acad. Sci. USA 103(14), 5367 (2006). 37. H. Flechsig and A. S. Mikhailov, Proc. Natl. Acad. Sci. USA 107(49), 20875 (2010). 38. M. Takano, T. P. Terada and M. Sasai, Proc. Natl. Acad. Sci. USA 107(17), 7769 (2010). 39. M. D¨ uttmann, Y. Togashi, T. Yanagida and A. S. Mikhailov, Biophys. J. 102(3), 542 (2012). 40. K. Kitaura, E. Ikeo, T. Asada, T. Nakano and M. Uebayasi, Chem. Phys. Lett. 313(3–4), 701 (1999). 41. A. Warshel and M. Levitt, J. Mol. Biol. 103(2), 227 (1976). 42. G. Li and Q. Cui, J. Phys. Chem. B 108(10), 3342 (2004). 43. H. Huxley and J. Hanson, Nature 173(4412), 973 (1954). 44. A. F. Huxley and R. Niedergerke, Nature 173(4412), 971 (1954). 45. A. F. Huxley and R. M. Simmons, Nature 233(5321), 533 (1971). 46. E. Pate and R. Cooke, J. Muscle Res. Cell Motil. 10(3), 181 (1989). 47. G. Piazzesi and V. Lombardi, Biophys. J. 68(5), 1966 (1995). 48. A. F. Huxley and S. Tideswell, J. Muscle Res. Cell Motil. 17(4), 507 (1996). 49. T. A. Duke, Proc. Natl. Acad. Sci. USA 96(6), 2770 (1999). 50. D. A. Smith, M. A. Geeves, J. Sleep and S. M. Mijailovich, Ann. Biomed. Eng. 36(10), 1624 (2008). 51. A. M˚ ansson, Biophys. J. 98(7), 1237 (2010). 52. K. Kitamura, M. Tokunaga, A. H. Iwane and T. Yanagida, Nature 397(6715), 129 (1999). 53. K. Kitamura, M. Tokunaga, S. Esaki, A. H. Iwane and T. Yanagida, Biophysics 1, 1 (2005). 54. G. Piazzesi, L. Lucii and V. Lombardi, J. Physiol. 545(1), 145 (2002). 55. G. Piazzesi and V. Lombardi, J. Muscle Res. Cell Motil. 17(1), 45 (1996). 56. J. Prost, J.-F. Chauwin, L. Peliti and A. Ajdari, Phys. Rev. Lett. 72(16), 2652 (1994). 57. L. Marcucci and T. Yanagida, PLoS One 7(7), (2012).

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Chapter 5 NANOMOTORS PROPELLED BY CHEMICAL REACTIONS

Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada [email protected] Molecular motors, like their macroscopic counterparts, consume energy and convert it to work; however, unlike macroscopic motors, they are subject to strong fluctuations and do not rely on inertia for their operation. In this chapter, the dynamics of synthetic chemically-powered nanomotors and mechanisms by which they operate are described. The focus is on motors that propel themselves by utilizing fuel in the environment to generate their own concentration gradients through chemical reactions. Macroscopic diffusiophoretic mechanisms for such motions are discussed, as well as microscopic and mesoscopic descriptions of motor dynamics.

Contents 1. Introduction . . . . . . . . . . . . . . . . 2. Propulsion by Phoretic Mechanisms . . . 3. Microscopic and Mesoscopic Dynamics of 4. Sphere Dimer Motors . . . . . . . . . . . 5. Collective Dynamics . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . . . . . . . . . Nanomotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

101 103 106 109 119 122 123

Introduction

Molecular machines typically have units with nanoscale dimensions and possess structures that allow them to perform useful functions.1 When functioning as molecular motors, molecular machines consume energy and 101

November 21, 2012

102

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

convert it to work. Our interest in this chapter is on molecular motors that use chemical energy to produce directed motion. A diverse array of molecular motors are found in biological systems and are essential for biological functions, such as active transport of organelles, vesicles and other materials in the cell, cell division, muscle contraction, etc. Chemical energy, obtained from the conversion of adenosine triphosphate to adenosine diphosphate, is used often to induce the conformational changes that underlie the mechanisms by which these motors operate.2,3 Such small machines carry out their functions in condensed phase environments in the presence of strong molecular fluctuations and are designed to operate effectively in spite of strong perturbations. They function in the regime of small Reynolds numbers where inertia is unimportant.4 In addition to naturally-occurring biological molecular machines and motors, synthetic motors with different geometries, using various materials, have been constructed.5 These synthetic motors use chemical, light or other energy sources. Some motors depend on asymmetric molecular motions for propulsion while others have no moving parts. Some of the simplest synthetic motors, such as metallic nanorods,6 –9 do not depend on conformational changes for their operation. In addition, sphere dimer10 and Janus particle11 Silica-Pt motors have been constructed and studied. Colloidal Janus particles have been investigated theoretically and experimentally.12 –15 The motions of self-propelled sphere dimer motors have been simulated using mesoscopic models for their dynamics.10,16 –18 The basic mechanisms underlying particle motion arising from chemical gradients have been known for some time;19,20 however, recent developments, especially experimental work on the design of synthetic molecular, nano and micron-scale motors, the ability to probe their dynamics in considerable detail, their significant potential applications, and the theoretical challenges posed by the full description of their dynamics, have made this an important topic of current research. In this chapter, we confine our attention to chemically self-propelled motors without moving parts; i.e., they derive their directed motion from asymmetry in chemical activity rather than through asymmetrical conformational changes.a After briefly describing the basis of classic diffusiophoretic mechanisms for self-propulsion, we turn our attention to microscopic and mesoscopic descriptions of the dynamics of such motors. Many of the phenomena are illustrated by considering a simple motor geometry, — a sphere dimer motor, which consists of linked catalytic and non-catalytic spheres.

a Self-propulsion can also arise in systems with symmetrical chemical activity through symmetry-breaking bifurcations.21

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

2.

103

Propulsion by Phoretic Mechanisms

It is well known that colloidal particles can move in the presence of concentration or temperature gradients giving rise to diffusiophoresis or thermophoresis. There is large literature on this topic dating to work by Derjaguin et al.,19 and comprehensive reviews on the topic exist.20 Many principles that underlie the phenomenon of diffusiophoresis can be used to understand the propulsion of particles by self-generated concentration gradients, so, we briefly comment on the origins of this effect and the assumptions that underlie its description. Diffusiophoresis arises from the coupling between interfacial forces and the fluid fields in the vicinity of the particle surface. The derivations of this phenomenon rely on a macroscopic continuum description of the solvent.20,22 One of the simplest cases to consider is a large, hard, colloidal particle with radius a in a fluid composed of solvent and a dilute solute with an inhomogeneous concentration field c(r). The surrounding fluid is considered to be a viscous continuum. The solute molecules interact with the surface of the particle through a short-range potential-of-mean-force W , with characteristic range L, obtained by coarse graining over all solvent molecules. The length L = R0 − a can be used to define an interfacial zone around the particle whose outer radius is R0 (see Fig. 1). If the colloidal particle is very large, the surface will appear to be flat locally and a local coordinate frame can be introduced with origin on the surface ˆ is along the normal n ˆ and y takes the of the particle. The vector y = y n ˆ. value zero on the particle surface, while s is the vector orthogonal to n In addition to variations in the concentration field in the normal direction, we suppose the solute concentration field also varies along the orthogonal s-direction.

Fig. 1. Schematic diagram showing the colloidal sphere with radius a, and the interfacial zone of thickness L and outer radius R0 . The local coordinate frame with axes y and s is also shown.

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

104

The concentration field in the interfacial zone can be written as the product of the concentration field at the outer boundary of the interfacial zone times a Boltzmann factor that accounts for the nonzero interactions between the solute particles and the colloidal particle within the interfacial zone: c(s, y) = c(s) exp(−βW (y)), with β −1 = kB T . The potential W gives rise to a body force −c(s, y)dW (y)/dy acting on the particle and this leads to an inhomogeneous pressure field in the interfacial zone to compensate for this force. The pressure can be determined from dW (y) ∂ ∂p + c(s, y) = (p(s, y) − kB T c(s, y)) = 0, (1) ∂y dy ∂y
 which yields p(s, y) = p∞ + kB T c(s) e−βW (y) − 1 . Because of the dependence of the pressure on s, there is a pressure gradient along this direction in the interfacial zone, which is balanced by the viscous stress: η

∂ 2 vs − ∇s p = 0, ∂y 2

(2)

ˆn ˆ ) · v is the tangential component where η is the shear viscosity, vs = (1 − n ˆn ˆ ) · ∇. Solution of this equation, of the fluid velocity and ∇s = (1 − n subject to the no-slip boundary condition at the surface, vs (y = 0) = 0, and the boundary condition of no pressure gradient far from the surface, limy→∞ (∂vs /∂y) = 0, yields the velocity field,  y  ∞  

kB T vs (s, y) = − dy  dy  e−βW (y ) − 1 . (3) (∇s c(s)) η 0 y
While the velocity field is taken to be zero at the surface of the particle, its value at the outer edge of the interfacial zone where W vanishes gives an apparent slip velocity v(s) ,  ∞   k T kB T B (s) −v (s) = dy y e−βW (y) − 1 = (∇s c(s)) (∇s c(s))λ2D , η η 0 (4) where the Derjaguin length λD has been introduced in the last identity. The slip velocity can be used to determine, V, the particle velocity.15,22,23 The flow field in the region outside the interfacial zone is given by the solution of the force-free incompressible Stokes’ equation, ∇ · Π = 0,

∇ · v = 0,

(5) (s)

subject to the boundary conditions, v|R0 = V + v and limr→∞ v(r) = 0. The pressure tensor is given by Π = −p1+η(∇v)S , where the superscript S refers to the symmetrized product. The velocity and pressure tensor fields

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

105

may be written as the sums of new fields, v = v + v and Π = Π + Π , where v satisfies the boundary condition v |R0 = v(s) and v satisfies v |R0 = V. Next, we make use of two results from the rheology of colloids.15,24 (1) The fields defined above satisfy the reciprocal relation,   ˆ · Π · v = ˆ · Π · v , dS n dS n (6) S0

S0

  where the surface integrals are defined by So dSf (r) = drδ(r − R0 )f (r). (2) The pressure tensor on the surface of a sphere translating with velocity ˆ · Π |R0 = −6πηR0 V/4πR02 . V in an unbounded fluid at rest is given by n Given the boundary conditions on the velocity fields, the reciprocal relation, in conjunction with the expression for the pressure tensor on the surface, yields     ˆ · Π · V = ˆ · Π · v(s) = 6πηR0 V · v(s) , dS n dS n (7) S0

S0

S



 where the angle brackets signify a surface average, v(s) = S0 dSv(s) / S 4πR02 . In view of Eq. (5), we have    ˆ ·Π = − ˆ · Π , dS n dS n (8) S0

S0

and using this result in Eq. (7), we find V = −
v(s) S . Substituting this result into the expression for the slip velocity given in Eq. (4), we find V=

kB T
(∇s c0 (s))S λ2D , η

(9)

which expresses the particle velocity in terms of the viscosity of the solution, the Derjaguin length and the gradient of the concentration field along the surface. These ideas also apply to the situation where the concentration gradient is generated by the particle itself through chemical reactions at its surface.13,15,25 We then have chemically-powered self-propulsion through a diffusiophoretic mechanism. Consider the case of a Janus particle, where one hemisphere is catalytically active and the other hemisphere is chemically inactive. The catalytic hemisphere consumes or produces solute particles. The inhomogeneous concentration field can be found from the solution of the diffusion equation subject to suitable boundary conditions. We suppose that the Peclet number, P e = V a/D  1, where D is the solute diffusion coefficient. In this circumstance, we can neglect the convective transport of solute particles. In the region outside the interfacial zone, the concentration

November 21, 2012

106

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

field satisfies the steady state diffusion equation, D∇2 c(r) = 0, subject to the “radiation” boundary condition, −Dˆ n · ∇c(r)|R0 = αr H(θ), where αr is the reaction rate per unit area and H(θ) = 1 for π/2 ≤ θ ≤ π and zero otherwise. The polar axis is taken to be along ˆz. In addition, we ∞ assume limr→∞ c(r) = c0 . Writing H(θ) = =0 H P (cos θ), where P (x) is a Legendre polynomial, the solution of the diffusion equation for these boundary conditions is15,25

+1 ∞ R0 αr R0 H c(r, θ) = c0 − P (cos θ). (10) D +1 r =0

We can compute
(ˆ z · ∇s c0 (s))S = −αr /4D, and this, in turn, yields the z-component of the particle velocity in the reaction-controlled limit, Vz = −

3.

kB T αr 2 λ . η 4D D

(11)

Microscopic and Mesoscopic Dynamics of Nanomotors

A full analysis of the dynamics of chemically-powered nanomotors requires a molecular-level description of the motor and its environment. Molecular dynamics (MD), based on the solutions of Newton’s equations of motion for the entire system, can account for all conservation laws that underlie macroscopic descriptions relying on Navier–Stokes and reaction–diffusion equations. While full molecular dynamics simulations of motors are feasible, the lengths of the relevant space and times scales prohibit such a direct approach for all but the smallest systems. Consequently, it is fruitful to consider coarse-grain mesoscopic methods that allow one to extend the range of accessible space and time scales. At a basic level, a molecular motor is built from atoms linked by chemical bonds. Except for the smallest molecular motors where such a description is required, for many nanoscale motors we can coarse grain over collections of atoms that constitute functional groups in the motor. Following this strategy, the molecular motors considered here are taken to be built from constituent molecular or atomic groups, termed beads, linked by bonds. Specifically, we consider motors made from Nm molecular groups m m m or beads with coordinates rN = (rm m 1 , r2 , . . . , rNm ). Of these, a certain C number Nm are able to catalyze chemical reactions, while the remaining N Nm groups are chemically inactive. The beads can then be linked in various ways to construct motors with different geometries and properties. Two examples of coarse-grained motors are shown in Fig. 2. Figure 2(a) shows a Janus motor built from beads, half of which are catalytically active,26

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

(a)

107

(b)

Fig. 2. Examples of coarse-grain models for chemically-powered molecular motors. (a) An irregular Janus-like motor composed of catalytic (dark) and noncatalytic (light) spheres, and (b) a polymer motor with a chemically active head and a noncatalytic tail. For the polymer motor, the nonequilibrium field of product molecules generated by chemical reactions at the head is shown.

and the right figure shows a polymer motor where the head is catalytically active and polymer tail is inactive.27 Such motors move in a condensed phase environment that consists of solvent molecules as well as chemical species that participate in chemical reactions that occur at active motor sites and possibly also in the bulk of the solution. There are Nb molecules in the environment with coordinates  Nb b b rb = r1 , r2 , . . . , rbNb . These molecules may be inert solvent as well as other chemically active species. The total potential energy of the system is Nb m V (rN m , rb ), which we assume to be pairwise additive. The total potential energy can be written as the sum of the potential energy of the motor, Vm , the environment, Ve , and the interaction energy between the motor and the environment, Vme : V = Vm + Ve + Vme . The interactions between the motor beads and the chemical species in the environment are also responsible for chemical reactions. Since momentum is conserved, the total force acting on the system is zero. The motion of the motor is governed by forces that Nthe m Mi rm act on it. The force acting on its center of mass, rCM = i=1 i /Mm , where Mi is the mass of motor bead i and Mm is the total mass of the motor, is given by Fm = −

Nm     ∂ ∂ Nb Nb m m = − . V rN , r Vme rN m m , rb b m ∂rCM ∂ri i=1

(12)

The last equality follows from the fact that there is no contribution to the force on the motor center of mass from Vm . Since the total force on

November 21, 2012

108

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

the system is zero, the motor exerts an equal and opposite force on the molecules in the environment, which results in fluid flow. When the entire system is in equilibrium T , the   at temperature −1 Nm Nb , where Z is canonical distribution function is Z exp −βV rm , rb the canonical partition function. In equilibrium, the average force on the motor is zero, since  “ ” m Nb 1 −βV rN Nb m ,rb m
Fm eq = drN dr F e m m b Z “ ” Nm  m Nb ∂ −βV rN 1 Nb Nm m ,rb = e = 0. (13) drm drb βZ i=1 ∂rm i However, chemically-powered motors operate in the nonequilibrium regime and we shall be interested in systems that are maintained in farfrom-equilibrium steady states by flows of reagents into and out of the system. For an equilibrium system containing many Brownian particles in a solvent, generalized Langevin equations can be derived for the momenta of the Brownian particles.28 A calculation in a similar spirit can be carried out for the beads of a motor in a nonequilibrium steady state.29 The resulting equation for the time evolution of the center of mass velocity of the motor is,   dV(t) m =
Fm  − V(t) · ζ rN (t) + K∗ (t), (14) Mm m dt   m is a generalized friction where K∗ (t) is a random force and ζ rN m (t) tensor which depends on the bead coordinates. The angle brackets
· denote a nonequilibrium steady state average. The microscopic expression for the friction tensor involves the autocorrelation, in the nonequilibrium steady state ensemble, of the deviation of the force on the motor from its nonequilibrium average value. This equation may then be averaged over fluctuations and used to compute, V, the average velocity of the motor. If inertial terms are neglected in the low Reynolds regime of interest here, the velocity of the center of mass of the motor is given by V =
Fm  · ζ −1 . In addition to a coarse-grain model of the motor, the environment may also be modeled at a mesoscopic level of description. The simulation results reported below were obtained using multiparticle collision (MPC) dynamics.30,31 In this method, the effects of the environmental potential, Ve , are replaced by multi-particle collisions that occur at discrete times and account for the effects of many real collisions. Full details of the implementation of the method, along with examples of applications, can

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

109

be found in the original references and in reviews.32,33 Since this hybrid MD-MPC dynamics respects the mass, momentum and energy conservation laws, the coupling between the motor and fluid flows in the environment are taken into account properly; in addition, molecular fluctuations are automatically included in the dynamics. 4.

Sphere Dimer Motors

A simple nanomotor configuration that illustrates the important ingredients of chemically-powered self-propulsion is a sphere dimer motor that consists of linked catalytic and noncatalytic spheres with bond length R. (see Fig. 3). The C sphere catalyzes the chemical reactions that are responsible for the nonequilibrium species concentration gradients, while the N sphere is chemically inert. We suppose that the C sphere catalyzes the reversible reaction A
B. It is not difficult to simulate more complicated reactionsb ; however, this simple reaction scheme will serve to capture the effects we wish to consider. In addition, we assume that these A and B species are the only types of molecule in the environment. At the expense of increased computational cost, chemically inert solvent molecules can be included in

(a)

(b)

Fig. 3. (a) Diagram showing the catalytic (small) and noncatalytic (large) spheres. The dashed lines indicate the range of the intermolecular potentials. The chemical reaction A → B (and its reverse, not shown) occurs within the dashed region around the C sphere. Only nonreactive interactions occur with the N sphere. As an example, a nonreactive collision of A with N is shown. Outside these dashed regions the species densities may be described macroscopically by the diffusion equation (see text). (b) A sphere dimer from the simulation along with the instantaneous concentration field (small dots) of product (B) molecules produced in the vicinity of the motor as result of the chemical reaction. b More

detailed versions of reaction dynamics have been used to study enzyme kinetics.34

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

110

the description.34 Because the particle number is conserved on reaction, no pressure gradient is generated as a result of reaction, in contrast to the diffusiophoretic mechanism described earlier. However, a chemical potential gradient as well as a pressure gradient can give rise to a slip velocity, so the main elements of the macroscopic analysis are unchanged for a binarymixture environment.35 The A and B species are taken to interact with the spheres in the dimer motor through pairwise additive intermolecular potentials VαI , where α = A, B and I = C, N . In the simulations reported below, these potentials are taken to be repulsive Lennard–Jones (LJ) potentials. The repulsive LJ potentials were cut off at R0S for the S = C, N monomers. For simplicity, we take VAC = VBC = VAN and VBN = VAN , although other choices are possible. The general formalism given in Sec. 3 takes a simple form for a sphere dimer with these interactions. We use r to denote coordinates measured with the catalytic C sphere as the origin, while ˆz denotes the unit vector along the dimer internuclear bond, pointing in the direction of the catalytic C sphere. The r coordinate, defined with the noncatalytic N sphere as the origin, is related to r by r = r − Rˆz. The total potential energy of the system is V (rNA , rNB ) =

Nα B

 [VCα (riα ) + VN α (riα )],

(15)

α=A i=1

where rNα = (r1α , r2α , . . . , rNα α ), with riα the vector distance to solvent  molecule i of species α and riα its magnitude, and riα = |riα − Rˆz|. Rather than using an intermolecular interaction between the C and N spheres to form a nanodimer, we employ a holonomic constraint to fix the bond length at R. Since the total momentum is conserved, the instantaneous force on the dimer center of mass along the dimer axis may be expressed in terms of the force exerted on the solvent, which is given by ˆ z · Fm =

Nα  B α=A

i=1

  ) dVCα (riα ) d VN α (riα  + (ˆz · ˆriα ) (ˆ z · ˆriα ) . (16)  driα driα

The force may also be written in a more convenient form by using the definition of the microscopic local density Nα field of A and B particles at point r in the fluid, ρα (r; rNα ) = i=1 δ(riα − r). Introduction of this definition into Eq. (16) allows one to write the force in terms of an integral over space of the product of the microscopic expressions for the local particle density and local force. As discussed above, when averaged over a canonical equilibrium distribution, this force vanishes; however, it is not zero when averaged over

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

111

a nonequilibrium distribution that results when there are fluxes of reactants and products into and out of the system to maintain it in a nonequilibrium steady state. This nonequilibrium average force is Fp =
ˆz · Fm , where the angle brackets again denote the steady state nonequilibrium It    average. Nα may be computed in the following way. By letting ρα (r) = ρα r; r be the nonequilibrium average of the microscopic density fields, the average force may be written as Fp =

B 

dr ρα (r) (ˆz · ˆr)

α=A

+

 B α=A

dVCα (r) dr

dr ρα (r + Rˆz) (ˆz · ˆr )

dVN α (r ) . dr

(17)

An analytical microscopic calculation of the nonequilibrium concentration fields ρα (r) in this formula is a difficult task, although they are easily determined by simulation. We may estimate these fields by making a local equilibrium approximation and adopting a macroscopic diffusion equation description outside a microscopic boundary layer around the dimer spheres. We take the outer radii of the boundary layers to be R0C ≡ R0 and R0N for the C and N monomers, respectively. In particular, we assume that the local concentration field can be written as ρα (r) = nα (R0 ˆr)gαC (r), with origin at the C monomer, for r ≤ R0 , and ρα (r) = nα (r) for r > R0 . Here gαC (r) is the equilibrium radial distribution function for species α at point r in the fluid containing fixed C and N spheres, while nα (r) is the local concentration of α determined from the solution of a diffusion equation with suitable boundary conditions at the dimer monomers and boundaries of the system. Thus, ρα (r) has the same form as the radial distribution function with the bulk number density replaced by its nonequilibrium analog. An analogous expression applies when the origin is taken at the N monomer. To estimate the concentration fields, we make use of the geometry shown in Fig. 3, where a sketch of the sphere dimer is given with the microscopic boundary layer zones indicated by dashed lines. We assume that the short range potentials between the A and B species and the C and N spheres are nonzero only within these regions. Note that we have chosen the bond length R of the dimer to be sufficiently long so that the molecules do not simultaneously interact with both spheres. Although unnecessary, this simplifies the description. We focus on the C sphere which is the source of the nonequilibrium concentration gradient. In the steady state, outside the diffusive interfacial zone, the concentration fields satisfy the diffusion equation, D∇2 nα (r) = 0, where D is the common diffusion

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

112

coefficient of species A and B.c The reversible reaction at the C sphere, k1

A + C
B + C, is accounted for by employing a “radiation” boundary k−1

condition at R0 , the outer radius of the dashed boundary layer region:   c 2 d Rα (n(R0 )) + 4πDR0 dr nα (r) = 0, where the reaction rates are given R0

0 nB (R0 ) = −RcB (n(R0 )), and k10 and by RcA (n(R0 )) = −k10 nA (R0 ) + k−1 0 k−1 are intrinsic reaction rate constants that characterize the forward and reverse reactions in the boundary layer around the catalytic monomer. The system is maintained out of equilibrium by controlling the concentrations of these species far from the sphere dimer. Here, we take nA (r = ∞) = n0 and nB (r = ∞) = 0. With these boundary conditions, the solution of the diffusion equation isd

nB (r) =

rf R02 1 k10 kD n0 = , 0 + k 4πDr k10 + k−1 Dr D

(r ≥ R0 ),

(18)

where kD = 4πDR0 is the Smoluchowski rate constant for a diffusion controlled reaction and rf =

k10 1 dNB Dn0 = 0 , 2 0 4πR0 dt k1 + k−1 + kD R0

(19)

is the initial forward reaction rate per unit area of the C sphere. Inside the boundary layer, nB (r) = nB (R0 ). Also, nA (r) = n0 − nB (r), since the total local density is not affected by this reaction. We may now use these results to approximately compute Fp . Taking the low density forms for the equilibrium radial distribution functions, gαS (r) = e−β(VαS (r) , S = C, N (these are exact for the system described by MD-MPC dynamics for the sphere dimer model) and using nB (r) in Eq. (18) we find that Fp can be written in the form Fp = −

c Since

rf R02 Dβ



dˆr

(ˆ z · ˆr ) N |R0 ˆr + Rˆz|



dr r2


d  −βVN B (r) e − e−βVN A(r ) .  dr (20)

the sphere dimer motion generates a fluid flow in the surroundings that convect the chemical species, a diffusion-advection equation describes concentration fields.36 For small Peclet numbers, we can neglect the advective terms. d This solution neglects the presence of the N sphere. Boundary conditions should be applied on its surface, outside a microscopic boundary layer, but this precludes an analytical solution of the equation. Comparisons of the theoretical and simulated density fields show good agreement.

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

113

Performing the angular integration and integrating the radial integral by parts, we obtain Fp = −

8π R2 RN rf kB T 0 20 λ2 , 3 DR

where the length λ has been defined as 
λ2 = dr r (e−βVN B (r) − e−βVN A (r ) ).

(21)

(22)

For the sphere dimer, the nonequilibrium average of the force is directed along ˆ z; thus, the average velocity of the sphere dimer along the bond is given by Vz =−

8π kB T R02 R0N 2 r λ , 3 ζ f DR2

(23)

where ζ is the zz-component of the generalized friction tensor. From this formula, we may deduce a number of factors that determine the dimer velocity. Suppose, the VαN , (α = A, B) are repulsive potentials controlled by the energy parameters α . From the form of λ2 in Eq. (22), we see that λ2 > 0 if B < A and vice versa. Thus, for B < A the sphere dimer will move in a direction with the C sphere at its head, while if B > A it will move in the opposite direction. Motor efficiency Chemically powered molecular motors convert chemical energy into mechanical work, driving the self-propelled directed movement. A measure of the thermodynamic efficiency can be determined from the power transduction of the motor. In this case, the efficiency may be defined as the ratio between the power associated with the work done by the motor against an external conservative force Fex and the power input due to chemical reaction.3,37 For the reversible chemical reaction A + C
B + C considered above, the thermodynamic efficiency can be computed from38 ηT D = −

V z Fex . ∆µR

(24)

Here, R is the net chemical reaction rate, and ∆µ is the change in the chemical potential in the reaction. In the presence of an external force, the sphere dimer velocity is given by V z = (Fp + Fex )/ζ so that the efficiency

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

114

can be rewritten as ηTD = −

2 + Fp Fex Fex . ζ∆µR

(25)

From this formula, we see that the efficiency has its maximum at ηmax = Fp2 /4ζ∆µR, and Fex = −Fp /2. The efficiency goes to zero at the stall point where the external force is equal to the propulsion force. In order to determine the efficiency from Eq. (25), the net chemical reaction rate R was calculated by counting the number of A → B and B → A reactive events at the catalytic sphere as a function of time and subtracting these rates to obtain the net rate. In the MD-MPC simulations, the change in eq the chemical potential is given by ∆µ = µB − µA = −kB T ln nB neq A /nB nA , eq eq where nA and nB denote the equilibrium number densities of A and B species, respectively, while the steady state densities are again nA and nB . The maximum efficiency of the sphere dimer motor is small. For a sphere dimer motor with dC = 2 and dN = 4, subject to repulsive LJ forces, and a reversible A
B reaction with reaction probability pR = 0.5 upon encounter with the catalyic sphere, we find ηT D ≈ 0.0004.38 Other measures of efficiency can be used. The Stokes efficiency,37 2

ηS =

ζV z , ∆µR

(26)

gauges how well the motor can utilize the free energy to drive a load through a viscous medium and this measure has been used to compute the efficiency of chemically-powered motors.36,39 The Stokes efficiency is also small. For the same parameters, using the existing simulation data for this motor,38 we find ηS ≈ 0.0015 . This is about three times larger than the maximum thermodynamic efficiency, but still very small. Many biological molecular motors have higher efficiencies, but synthetic chemically-powered motors also have small efficiencies.40 Motor dynamics in chemically active media Thus far, we have considered situations where chemical reactions only occur at the catalytic sphere. However, the more common situation is that chemical reactions, α

kj

ναj Xα


k−j



ν¯αj Xα

(j = 1, . . . , r),

(27)

α

may also take place in the environment in which the motor moves. Here, a reaction is labeled by the index j, ναj and ν¯αj are the stoichiometric

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

115

coefficients for reaction j and kj and k−j are the rate constants that characterize the forward and reverse reactions. Such bulk phase reactions can change the propulsion properties of the motor. In particular, if the reactants or products of the reaction at the C sphere also participate in bulk reactions which change the nonequilibrium concentration gradients in the system, the motor dynamics will be affected.41 The computation of the force on a sphere dimer motor may still be based on Eq. (17), except that the nonequilibrium concentration fields must now be determined from the solution to a reaction–diffusion equation. An example will serve to illustrate the types of phenomena that arise when bulk reactions are present. In addition to the A
B reactions at the C surface, we suppose that the cubic autocatalytic chemical reactions, k2

B + 2A
3A, take place in the bulk phase. The bulk phase reaction rates k−2

of the A and B species are given by RA = −RB = k2 nB (r, t)n2A (r, t) − k−2 n3A (r, t).

(28)

We see that the local concentrations A and B, which are also involved in the catalytic reactions at C, are both changed by the cubic autocatalytic reactions in the environment. The concentration fields of the A and B species can be determined from the solution of the steady state reaction– diffusion equation, D∇2 nB (r, t) = k2 nB (r, t)n2A (r, t) − k−2 n3A (r, t),

(29)

0 subject to the boundary condition −k10 nA (R0 ) + k−1 nB (R0 ) = 4πDR02  d , at the surface with radius R0 around the C sphere, and a dr nB (r) R0

reflecting boundary condition at the N sphere. As r → ∞, we assume that the concentration fields are given by the steady state values of the cubic autocatalytic reaction in the bulk phase, which are determined from the steady state condition k2 n ¯B n ¯ 2A = k−2 n ¯ 3A . Again, the spatial variations in the total density can be neglected so that nA (r, t) + nB (r, t) = n0 and we have n ¯ B = n0 (1 + k2 /k−2 )−1 and n ¯ A = n0 (1 + k−2 /k2 )−1 . As for the case of a chemically inactive environment, we neglect the presence of the N sphere in order to facilitate the analytical analysis. An analytical approximation to the solution of this nonlinear equation can be obtained by linearizing the asymptotic values of the concentration ¯ B +δnB (r) and nA (r) = n ¯ A −δnA (r), so that the linearized fields, nB (r) = n steady state reaction–diffusion is, n2A δnB (r, t). D∇2 δnB (r, t) = (k2 + k−2 )¯

(30)

November 21, 2012

116

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

Fig. 4. (a) Species B concentration field nB (r) versus r for (red) forward irreversible reactions at the C sphere and in the bulk phase, (blue) reversible reactions that violate detailed balance and (green) reversible reactions that satisfy detailed balance. (b) Probability distribution function p(Vz ) of the center-of-mass velocity of the dimer projected along the internuclear axis for the same cases with the same color coding as in 4(a).

The steady state solution satisfying the boundary conditions is 
0 0 k1 n ¯ A − k−1 n ¯ B kD e−κ(r−R0 ) , nB (r) = n ¯B + 0 0 + k (1 + κR ) k1 + k−1 4πDr D 0

(31)

 where κ = (k2 + k−2 )¯ n2A /D is an inverse length. The A concentration field is given by nA (r) = n0 − nB (r). The nB concentration field is shown in Fig. 4(a) for three different situations: forward irreversible reactions at the C sphere and in the bulk phase, reversible reactions that violate detailed balance and reversible reactions that satisfy detailed balance. The expression in Eq. (31) is in almost quantitative agreement with these simulation results,41 which indicates that the simplifying assumptions in the calculation capture the main physics of the phenomenon. Self-propulsion only occurs when the system is out of equilibrium. If we suppose the system is in chemical equilibrium, detailed balance requires that the rates of the reactions at the dimer and in the bulk are individually eq eq eq 2 eq 3 equal to zero, −k1 neq A + k−1 nB = 0 and k2 nB (nA ) − k−2 (nA ) = 0. This detailed balance condition places the following restrictions on the eq rate constants: k1 /k−1 = k−2 /k2 = neq B /nA . From Eq. (31) and the 0 0 fact that k1 /k−1 = k1 /k−1 , we observe that nB (r) = neq B and there is no concentration gradient to induce propulsion. This is confirmed by the simulation results for nB (r) in Fig. 4(a) where we see that nB (r) ≈ neq B and is almost independent of r.

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

117

Far-from-equilibrium conditions give rise to very different results. For the simple case where the irreversible reaction A → B occurs at C and the irreversible autocatalytic reaction B + 2A → 3A occurs in the bulk, the reaction product B of the C catalytic reaction is converted in the bulk back to the “fuel” A needed for the dimer propulsion. In Fig. 4(a), we see that the nB concentration field falls sharply with distance and the rate of this decrease is governed by the characteristic inverse length κ. Similar considerations apply for reversible reactions that do not satisfy detailed balance and nB (r) for this case is also shown in the figure. The velocity of the sphere dimer depends on the rates at which the reactions in the bulk phase occur; in addition, fluctuations are an integral aspect of the motion of these nanoscale motors. The velocity probability distribution functions p(Vz ) of the sphere dimer are shown in Fig. 4(b) for the three situations discussed above. For a reversible reaction satisfying detailed balance, there is no nonequilibrium B concentration gradient and directed sphere dimer propulsion cannot occur. We see that the velocity probability distribution is centered at zero. For the other two nonequilibrium cases, the probability distributions are centered at positive nonzero values indicating directed motion along the bond in the direction of the C sphere. Motors can interact with chemical patterns Once the possibility of a far-from-equilibrium chemically-active environment is admitted, a variety of environmental states becomes possible. These include states where the environment oscillates periodically or chaotically, or exists in a form where there are chemical patterns.42 Chemical waves may interact with inactive and self-propelled particles and change their motion.43 For example, the cubic autocatalytic reaction considered above admits traveling chemical wave solutions. Consider the irreversible reaction k B + 2A →2 3A and suppose that the system initially contains the autocatalyst A in the left part of the system and the fuel B in the right part. As the autocatalyst consumes the fuel, a moving chemical front will form which propagates to the right with speed c. Macroscopically, the front dynamics can be determined from the solution of the reaction–diffusion equation for the B species density field, D

dnB (u) d2 nB (u) − k2 nB (u)n2A (u) = 0, +c du2 du

(32)

in a frame moving with the front velocity, u = x−ct. In these equations, D is the mutual diffusion coefficient. The equation for nA (r, t) is not independent and follows from number conservation nA (r, t) + nB (r, t) = n0 as discussed

November 21, 2012

118

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

Fig. 5. Trajectory of a self-propelled nanodimer motor showing the reflection from a chemical wave in a system with cubic autocatalysis involving A and B species.

earlier. The solution of the equation is, nB (u) = n0 (1 + e−cu/D )−1 , with the front speed given by c = (Dk2 n20 /2)1/2 . Suppose a sphere dimer is initially in the region containing the autocatalyst (fuel for the motor) and is moving toward the chemical wave which is propagating with speed c. If the dimer motor velocity is greater than that of the wave, the dimer will encounter the chemical wave and interact with it.43 Figure 5 shows the trajectory of the dimer as it encounters the wave. The self-generated dimer concentration field interacts with that of the chemical wave and, due to orientational Brownian motion in the concentration field, forces are induced that lead to reflection from the chemical wave as shown in the figure. Extensions of this scenario may involve chemically patterned surfaces, where the patterns may adopt stationary regular or labyrinthine forms, or exhibit time evolving structures. The reflection mechanism could be used to enable self-propelled particles to travel along predetermined paths, akin to that of nanomotors in microchannels. Diffusion and mean square displacement As we have seen, small self-propelled motors experience strong fluctuations and will not simply move ballistically in a given direction. Instead, although the motion will be ballistic at short times, Brownian motion will cause the orientation of the motor to change, resulting in diffusive motion on long time scales. The self-diffusion coefficient of the motor is given by the velocity autocorrelation function,  1 ∞ dt
V(t) · V, (33) DM = d 0

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

119

where V is again the velocity of the center of mass of the motor and d is the dimension. For self-propelled sphere dimers, the velocity can be decomposed into its average value along the instantaneous bond direction zˆ(t) and fluctuations as, V = ˆ z(t)V z + δV(t). The diffusion coefficient may then be written as   1 ∞ 1 2 ∞ dt
δV(t) · δV + V z dt
ˆz(t) · ˆz. (34) DM ≈ d 0 d 0 The first term on the right can be identified as the diffusion coefficient in the absence of propulsion, D0 , while the second term can be evaluated from a knowledge of the decay of the orientation correlation function. Assuming exponential decay,
ˆz(t) · ˆz = exp (−t/τR ), characterized by the orientational relaxation time τR , we find 1 2 DM = D0 + V z τR . d

(35)

Thus, active self-propelled particles exhibit enhanced diffusion compared to their inactive counterparts. The diffusion coefficient can equivalently be determined from the mean square displacement (MSD)

(36) ∆L2 (t) = |rCM (t) − rCM (0)|2 , from DM = limt→∞ ∆L2 (t)/dt. Two characteristic times are especially important in an analysis of the diffusive dynamics: τD = Rd2 /D, the time that gauges how long it takes solvent molecules to diffuse a distance equal to the dimer size Rd , where D is again the diffusion coefficient of solvent molecules; and τR , the orientational relaxation time. The MSD and diffusion coefficient for a single chemically-powered particle have been studied by Golestanian.44 Consider the regime where τD  τR . For short times (t  2 τD ), there is a ballistic regime where ∆L2 (t) ∼ V z t2 , while  for long times,  2

t τR , the MSD is a linear function of time, ∆L2 (t) ∼ 4 D0 + V z τR /2 t, which can be used to find the expression for the diffusion coefficient of the self-propelled sphere dimer given in Eq. (35). These expressions for the MSD and motor diffusion coefficient have been verified in simulations on sphere dimer motors.41

5.

Collective Dynamics

If more than one chemically-powered motor is present in the system, interactions among the motors must be taken into account. Active particles can interact through direct intermolecular interactions among neighbors,

November 21, 2012

120

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

chemical concentration gradients, hydrodynamic interactions, as well as other types of interactions. It is well known that the dynamics of an ensemble of interacting active objects can display collective behavior such as the flocking of birds and the schooling of fish, the concerted actions of molecular motors involved in intracellular, intraflagellar and axonal transport, as well as swarming, clustering and giant number fluctuations seen in suspensions of active colloidal particles and microorganisms.45–47 The collective dynamics of synthetic chemically-powered motors has been observed in the laboratory where phenomena such as the self-assembly of chemically active Janus colloidal particles14 and schooling of light-powered micromotors48 have been seen. A simple case to consider is the interaction between two sphere dimers.18 Suppose the monomers in each dimer have catalytic and noncatalytic sphere diameters dC and dN , respectively, and that the monomers in different dimers interact through repulsive LJ interactions with strength D . If the dimers are initially targeted to collide, the post-collision state depends on the values of D and the monomer diameters. The post-collision states that are observed when dC and D vary for fixed dN are presented in the phase diagram in Fig. 6. This phase diagram has four regions: In addition to simple collision dynamics where the dimers encounter each other and move apart after collision (labeled IP), bound dimer pairs exist

Fig. 6. Phase diagram showing the post-collision nature of the pair of dimers in the
D dC -plane. Different phases are: (BP) Brownian dimer pairs, (RP) rotating dimer pairs, (MP) moving dimer pairs and (IP) independently moving dimers. The two dimers are labeled as 1 and 2.

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

121

as a result of depletion forces — when the dimers approach closely, solvent is excluded and the resulting force imbalance leads to the formation of bound pairs with very long lifetimes. Three types of bound dimer pairs with distinctive properties were observed: Brownian pairs (BP) where directed motion is absent and only translational Brownian motion occurs, rotating pairs (RP) where self-propelled rotational motion occurs with zero average translational velocity and moving pairs (MP) where the bound pair executes self-propulsion as a unit. The dynamics of an ensembles of ND sphere dimer motors is more complicated.49 In order to avoid situations where the dimers cluster due to depletion forces, we restrict our attention to the IP region of the phase diagram where collisions between pairs of dimers do not result in longlived dimer pairs. We consider situations where the sphere dimers are confined to three-dimensional slab geometries bounded by two parallel walls perpendicuar to the z-axis of the system. In such geometries, the dimer motion can be restricted to lie primarily in the xy-plane lying midway between the walls by wall-dimer-momomer interactions. In this circumstance, three-dimensional orientational Brownian motion may be suppressed and the resulting quasi-two-dimensional motion has a simpler character than that in an unconfined three-dimensional system. The system is maintained in a far-from-equilibrium steady state by fluxes of chemical species at the walls. Figure 7(a) shows snapshots from the evolution of an ensemble of ND = 5 dimers. Dimer motors self-assemble and propagate as a unit for long times before they break up as a result of collisions with other clusters and single motors or as a consequence of thermal fluctuations. The B particle concentration field, resulting from chemical reactions at the surfaces of the catalytic spheres is also shown. The concentration fields from individual dimers combine to form a much more complex concentration pattern, which then modifies the collective dynamics of the motors in the ensemble. As well, it is interesting to consider the dynamics of individual motors in the ensemble. An isolated nanodimer is subject to strong thermal fluctuations and the probability density of dimer velocities along the bond is approximately Maxwellian with mean V z .16,17,27 In contrast, in an ensemble of interacting motors, the distribution is no longer Maxwellian (see Fig. 7(b)). The peak of the distribution shifts to smaller velocities and the distribution has a long tail toward smaller or even negative velocities. As ND increases, the peak shifts to very small velocities and the distribution has a long tail towards higher velocities. Clusters of varying sizes move with different velocities and dimers within such clusters move with the speed of the cluster and may have orientations at angles different from their directed motion. This effect leads to the large dispersion of Vz velocities.

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

122

50

ND = 1 ND = 5 ND = 10

p(Vz)

40 30 20 10 0 -0.04

(a)

-0.02

0

0.02

Vz

0.04

0.06

0.08

(b)

Fig. 7. (a) An instantaneous configuration of a small ensemble of five sphere dimers showing the formation of transient propagating clusters. The concentration field of the point-like B particles is also shown. (b) Plots of p(Vz ), the unnormalized probability density of Vz , for ND = 1, 5 and 10 dimers. The monomers in the dimers have diameters dC = 4 and dN = 8.

6.

Conclusion

Investigations of the dynamics of synthetic chemically-powered motors are at an early stage of development. Experimental studies have shown how new types of chemically-powered motors can be made, and theoretical studies have elucidated the mechanisms by which they operate. In addition, the first steps towards realizing applications for their uses have been taken. Challenges in this area remain. Chemically-powered motors operate in the far-from-equilibrium domain. A complete microscopic analysis requires a statistical mechanical theory applicable to far-from-equilibrium systems. In particular, since fluctuations are important for small motor dynamics, the study of nonequilibrium fluctuations in the steady state motion nanomotors is a topic that merits investigation. Future studies will almost certainly consider more fully situations where motors move in complex environments, which contain many reactive species whose reactions can be used to tune motor dynamics, or even exist in nonequilibrium states where chemical patterns form. Other applications, such as delivery of cargo to a specific location, not only entail a knowledge of how to control the dynamics of an individual motor, but may also require information about the collective dynamics of many motors. This, in turn, involves an understanding of how direct interactions between motors, chemical gradients and hydrodynamic interactions conspire to influence collective motion. Processes such as swarming and active self-assembly have

November 21, 2012

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

Nanomotors Propelled by Chemical Reactions

123

already been demonstrated. Thus, research on the dynamics of nanomotors has potentially important applications in nanotechnology, and presents opportunities to advance our knowledge of basic principles related to the far-from-equilibrium statistical mechanics of active media. Acknowledgments The work of R. K. was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. I would like to express my thanks to my coworkers, G. R¨ uckner, Y.-G. Tao, S. Thakur and P. de Buyl, whose work is described in this chapter. References 1. V. Balzani, A. Credi, and M. Venturi, Molecular Devices and Machines — A Journey into the Nano World (Wiley VCH, Weinheim, 2002). 2. R. D. Vale and R. A. Milligan, Science 288, 88 (2000). 3. F. J¨ ulicher, A. Ajdari and J. Prost, Rev. Mod. Phys. 69, 1269. 4. E. M. Purcell, Am. J. Phys. 45, 3 (1977). 5. E. R. Kay, D. A. Leigh and F. Zerbetto, Angew. Chem., Int. Ed. 46, 72 (2007). 6. W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. S. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert and V. H. Crespi, J. Am. Chem. Soc. 126, 13424 (2004). 7. S. Fournier-Bidoz, A. C. Arsenault, I. Manners and G. A. Ozin, Chem. Commun. (4), 441 (2005). 8. W. F. Paxton, S. Sundararajan, T. E. Mallouk and A. Sen, Angew. Chem., Int. Ed. 45, 5420 (2006). 9. R. Laocharoensuk, J. Burdick and Y. Wang, ACS Nano 8, 1069 (2008). 10. L. F. Valadares, Y.-G. Tao, N. S. Zacharia, V. Kitaev, F. Galembeck, R. Kapral and G. A. Ozin, Small 6, 565 (2010). 11. H. Ke, S. Ye, R. L. Carroll and K. Showalter, J. Phys. Chem. A. 114, 5462. 12. R. Golestanian, T. B. Liverpool and A. Ajdari, Phys. Rev. Lett. 94, 220801 (2005). 13. J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh and R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007). 14. S. Ebbens, R. A. L. Jones, A. J. Ryan, R. Golestanian and J. R. Howse, Phys. Rev. E 82, 015304 (2010). 15. M. N. Popecu, S. Dietrich, M. Tasinkevych and J. Ralston, Eur. Phys. J. 31, 351 (2010). 16. G. R¨ uckner and R. Kapral, Phys. Rev. Lett. 98, 150603 (2007). 17. Y.-G. Tao and R. Kapral, J. Chem. Phys. 128, 164518 (2009). 18. S. Thakur and R. Kapral, J. Chem. Phys. 133, 204505 (2010).

November 21, 2012

124

13:2

9in x 6in

b1449-ch05

Engineering of Chemical Complexity

R. Kapral

19. S. S. Dukhin and B. V. Derjaguin, Electrokinetic Phenomea, Surface and Colloid Sicence, in ed. E. Matijevic, Vol. 7 (Wiley, New Yok, 1974), p. 365. 20. J. L. Anderson and D. C. Prieve, Sep. Pur. Reviews 13, 67 (1984). 21. A. Mikhailov and D. Meink¨ ohn, Lect. Notes Phys. 484, 334 (1997). 22. J. L. Anderson, Annu. Rev. Fluid Mech. 21, 61 (1989). 23. M. N. Popecu, S. Dietrich and G. Oshanin, J. Chem. Phys. 130, 194702 (2009). 24. H. Brenner, Chem. Eng. Sci. 18, 1 (1963). 25. R. Golestanian, T. B. Liverpool and A. Ajdari, New J. Phys. 9, 126 (2007). 26. P. de Buyl and R. Kapral, unpublished (2012). 27. Y.-G. Tao and R. Kapral, ChemPhysChem 10, 770 (2008). 28. J. M. Deutch and I. Oppenheim, J. Chem. Phys. 54, 3547 (1971). 29. R. Kapral, unpublished (2011). 30. A. Malevanets and R. Kapral, J. Chem. Phys. 110, 8605 (1999). 31. A. Malevanets and R. Kapral, J. Chem. Phys. 112, 72609 (2000). 32. R. Kapral, Adv. Chem. Phys. 140, 89 (2008). 33. G. Gompper, T. Ihle, D. M. Kroll and R. G. Winkler, Adv. Polym. Sci. 221, 1 (2009). 34. J.-X. Chen and R. Kapral, J. Chem. Phys. 134, 044503 (2011). 35. F. J¨ ulicher and J. Prost, Eur. Phys. J. 29, 27 (2009). 36. B. Sabass and U. Seifert, J. Chem. Phys. 136, 064508 (2012). 37. H. Wang and G. Oster, Europhys. Lett. 57, 134 (2002). 38. Y.-G. Tao and R. Kapral, J. Chem. Phys. 131, 024113 (2009). 39. B. Sabass and U. Seifert, Phys. Rev. Lett. 105, 218103 (2010). 40. W. F. Paxton, A. Sen and T. E. Mallouk, Chem. Eur. J. 11, 6462 (2005). 41. S. Thakur and R. Kapral, J. Chem. Phys. 135, 204509 (2010). 42. R. C. Desai and R. Kapral, Dynamics of Self-Organized and Self-Assembled Structures (Cambridge University Press, Cambridge, 2009). 43. S. Thakur, J-X. Chen and R. Karpal, Angew. Chem. Int. Ed. 50, 10165 (2011). 44. R. Golestanian, Phys. Rev. Lett. 102, 188305 (2009). 45. R. A. Simha and S. Ramaswamy, Phys. Rev. Lett. 89, 058101 (2002). 46. F. Ginelli, F. Peruani, M. B¨ ar and H. Chat´e, Phys. Rev. Lett. 104, 184502 (2010). 47. J. P. Hernandez-Ortiz, C. G. Stoltz and M. D. Graham, Phys. Rev. Lett. 95, 204501 (2005). 48. M. Ibele, T. E. Mallouk and A. Sen, Angew. Chem., Int. Ed. 48, 3308 (2009). 49. S. Thakur and R. Kapral, Phys. Rev. E 85, 026121 (2012).

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

Chapter 6 BIOLOGY OF NANOBOTS

Wentao Duan, Ryan Pavlick and Ayusman Sen Department of Chemistry The Pennsylvania State University University Park, Pennsylvania 16802 (USA) [email protected] One of the more interesting recent discoveries has been the ability to design nano/microbots which catalytically harness the chemical energy in their environment to move autonomously. Their potential applications include delivery of materials, self-assembly of superstructures, and roving sensors. One emergent area of research is the study of their collective behavior and how they emulate living systems. The aim of this chapter is to describe the “biology” of nanobots, summarizing the fundamentals physics behind their motion and how the bots interact with each other to initiate complex emergent behavior.

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of Motility . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Non-electrolyte diffusiophoresis . . . . . . . . . . . . . . . . . . . . 2.2. Electrolyte diffusiophoresis . . . . . . . . . . . . . . . . . . . . . . 2.3. Electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Other phoretic mechanisms . . . . . . . . . . . . . . . . . . . . . . 2.5. Bubble propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Non-reciprocal swimmers . . . . . . . . . . . . . . . . . . . . . . . 2.7. Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emergent Collective Behavior of Nanobots . . . . . . . . . . . . . . . . 3.1. Interaction between colloid particles based on self-diffusiophoresis 3.1.1. Diffusiophoretic interaction between the central particle and nearby positive particles . . . . . . . . . . . . . . . . . 3.1.2. Diffusiophoretic interaction between the central particle and nearby negative particles . . . . . . . . . . . . . . . . .

125

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

126 127 128 129 131 131 132 132 135 135 136

. . .

136

. . .

137

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

126

3.2. Examples of diffusiophoresis-based emergent systems . . . . 3.3. Motion analysis of particles in collective emergent systems 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

138 140 141 142

Introduction

The creation and application of nano and micron-sized self-powered objects are areas of great interest.1 –8 Coincidently, these systems operate in the same size regime as many biological systems including bacteria, cells, and proteins, and the physics applicable to the synthetic systems is the same. As objects become smaller, the physics that dominates their motion begins to deviate from the everyday classical Newtonian physics. At these scales, inertial forces become less important to motion and viscous forces begin to dominate. The Reynold’s number (Re) describes the ratio of inertial forces to viscous forces and its magnitude provides information on which of these two forces dominates the motion of the object of interest.9 Re =

ρV l , η

(1)

where ρ is the density of the medium, V is velocity, l is length, and η is viscosity. Other forces that affect the motion of objects at low Reynold’s numbers are the rapid thermal “bumping” of the solvent particles against the objects’ surface. This effect creates what is known as Brownian motion where the objects are driven to “move” and diffuse in a solution. The diffusion coefficient for an object is then defined as: D=

kT , f

(2)

where k is the Boltzmann constant, T is the temperature in Kelvin, and f is the viscous drag coefficient which changes based on the geometry of the system (for a sphere this is 6πηa, where a is the radius of the sphere).10 Thermal “bumping” also causes an object to rotate, known as Brownian rotation. The frequency of this rotation for a sphere is defined by the equation: τR =

kT , 8πηa3

(3)

where τR is the rotational coefficient, k is the Boltzmann constant, T is the temperature in Kelvin, η is the viscosity of the solution, and a is the radius of the particle. The above is important in creating self-powered motors, for if the motor is rotating (tumbling) too fast it will move in a more random fashion and will be less directional.

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

127

Often, when examining the nature of particle motion the question is: how far does the particle travel in a given time interval, τ ; or in a more mathematical sense, what is the objects’ root-mean-squared-displacement (RMSD) over that time interval? For convenience, this value is typically squared, and the mean-squared-displacement (MSD) of the collection of particles is examined. For several idealized types of motions, the MSD has been shown to increase as a function of τ raised to some power, α (Eq. (4)).11 In the case of a particle with a constant ballistic velocity, α is simply 2. For particles undergoing a purely diffusive, random Brownian walk, Einstein proved that the MSD increases linearly with τ (i.e., α = 1). Another kind of motion observed in a variety of living and non-living systems is the Levy-walk, where during most of the identically-sized time intervals, the displacements encountered is relatively small, but occasionally a much longer displacement is observed. Accordingly, a Levy-walk type motion is characterized by an α value which is between 1 and 2. Since ordinary Brownian diffusion is by far the most commonly observed motion, systems in which α does not equal 1 are often deemed as having “anomalous” diffusive behavior, with values greater than 1 corresponding to “superdiffusive” systems and values less than 1 corresponding to “subdiffusive” systems. MSD = Kα τ α ,

(4)

where Kα is related to the diffusion coefficient of the particles. For particles that undergo two-dimensional Brownian motion, Kα is four times the diffusion coefficient and for particles that are powered by external forces, a higher Kα will be achieved. Creating powered motion at low Reynold’s number regimes requires methods to overcome both viscous forces and Brownian effects. This chapter will highlight methods that have been utilized to make motors with enhanced motility and directionality, as well as those that exhibit collective behavior. First, classical bulk methods for creating motion in fluids will be discussed which will lead to how motors have been fabricated based on these principles. Then, some details that are emerging from the analysis of the motion of these bots will be reviewed. Finally, the chapter will conclude with how these motors can be designed to exhibit collective behavior. Through these discussions, we hope to outline a clearer picture of the “biology” of nanobots. 2.

Mechanisms of Motility

Most motility mechanisms require the creation of a gradient across the particle surface, where there is a change in some property (temperature,

November 21, 2012

128

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

surface tension, concentration, etc.) over a given distance. In other words, a nonequilibrium condition has to be applied to the particle to generate directed motion. The gradient gives rise to a force that acts on the fluid at the surface of the object which is then made to flow causing the object to move in the opposite direction. 2.1.

Non-electrolyte diffusiophoresis

Non-electrolytic diffusiophoresis is a movement which is caused by a gradient of uncharged solutes across a surface. These solutes interact with the surface with a certain potential which is determined by the Gibbs’ absorption length, K, and the length of the particle-solute interaction, L. The overall equation that defines the velocity of a particle in the gradient is, U=

kT KL∇C, η

(5)

where k is the Boltzmann constant, T is the temperature of the solution, η is the solution’s viscosity, and ∇C is the concentration gradient of the solute.10 The direction of particle movement depends on whether the particle-solute interaction is repulsive or attractive. Anderson gives an estimation for a typical velocity based on a gradient of small neutral particles. A good approximation for KL for small molecules to be 5.8 × 10−16 cm2 and, assuming a reasonable solute gradient of 0.1 mole/cm−4 , this results in a velocity of 1.4 µm/s in water at room temperature. If the solute molecules are bigger, an approximation for KL is given by the equation, KL =

a2 , 2

(6)

where a is the solute’s radius.10 This mechanism was demonstrated experimentally by Staffeld and Quinn who tracked the motion of particles in a stop-flow cell in a gradient of dextran and polymer nanoparticles. They showed that the particles traveled down the gradient. The reason behind this movement is that the solute sterically excludes solvent from the particles surface creating areas of high and low solvent pressure which caused the fluid to flow. In this case, the fluid flowed up the gradient allowing the particles to move down the gradient.12 The above mechanism, as first postulated, affected a relatively large area. However, the same mechanism can function in some form on a smaller scale and stems from bots generating their own gradients via catalytic

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

129

Fig. 1. Diagram showing the active mechanism for the polymerization powered motor. Fluid flows from low concentration to high concentration causing the motor to move in the opposite direction.14

reactions. The osmotic mechanism functions by a catalytic motor depleting surrounding reactant molecules at the catalyst end creating an osmotic force around the motor. This force then drives the motion of the bot.13 A bot that is believed to function by this mechanism is the polymerization motor reported by Pavlick et al. This motor was fabricated from an asymmetric gold/silica Janus particle that was coated with a polymerization catalyst on the silica face. This motor exhibited a 70% increase in diffusion when placed in a solution of monomer. Since the motor depletes monomer molecules at the catalyst face and releases very few polymer molecules, this creates a gradient of monomer around the particle. Fluid will then flow towards the area with a high concentration of monomer (silica face) propelling the particle catalyst end forward (Fig. 1).14 There are other proposals on how non-electrolyte motor systems can function. The inverse of the above osmotic motor has been explored where one side of the particle creates more reactants than products.15 One example of a motor that functions this way involves platinum/silica Janus particles.16,17 The platinum side of these particles decomposes hydrogen peroxide into water and oxygen. This decomposition creates an oxygen gradient across the particle. The gradient then creates a force that acts on the motor causing it to move in a directed fashion. 2.2.

Electrolyte diffusiophoresis

Another powerful transport mechanism is electrolyte diffusiophoresis. This mechanism works when a gradient of electrolytes is formed across a charged

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

130

surface. For diffusiophoresis near a wall, there are two effects contributing to the movement of a particle: an electrophoretic effect and a chemophoretic effect, and the speed of the diffusiophoretic movement can be approximated by Eq. (7).18
U =

d ln(C) dx




DC − DA DC + DA




kB T e




(ζP − ζW ) η

 Electrical Field Effect 

2 2 d ln(C) 2kB T [f (ζW ) − f (ζP )] + dx ηe2   , Chemophoretic Effect



 

(7)

where U is the particle velocity, kB is the Boltzmann constant, T is temperature, κ−1 is the Debye length, e is the charge of an electron, d ln(C)/dx is the gradient of the electrolyte, DC is the diffusion coefficient of the cation, DA is the diffusion coefficient of the anion, ζP is the zeta potential of particles, ζW is the zeta potential of the wall and f (t) is defined as f (t) = ln[1 − tanh2 et/(4kB T )]. As shown in Eq. (7), electrophoresis results from a difference in diffusion between the cation and anion which contributes to the ion gradient in a given direction. This difference leads to a net electric field, which acts both electrophoretically on the nearby particles and osmophoretically on the ions adsorbed in the double layer of the wall. Also, the concentration gradient of the electrolytes causes a gradient in the thickness of the electric double layer, and thus a “pressure” difference along the wall is created. As a result, the solution will flow from the area of higher electrolyte concentration to that of lower concentration, known as the chemophoretic effect. The combination of electrophoretic and chemophoretic effects leads to an overall diffusiophoretic flow, which powers the movement of particles. The enhanced movement of the enzyme urease in the presence of its substrate is thought to result from this mechanism. In the presence of the enzyme, urea is converted to ammonium and hydroxide ions, the latter with significantly higher diffusion coefficient.19 The diffusiophoresis mechanism can lead to many biomimetic collective emergent patterns, which will be further discussed in Sec. 3. Each of the two previously mentioned mechanisms have their own set of advantages. Non-electrolyte diffusiophoresis has no dependence on surface charge and is able to function in high ionic strength media. Electrolyte diffusiophoresis, however, is not effective in high ionic strength media because of the collapse of the double layer on the particle surface. Conversely, in a low ionic strength medium, electrolyte diffusiophoresis is a more powerful mechanism resulting in higher speeds. This

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

131

is shown qualitatively by considering that the chemophoretic component of electrolyte diffusiophoresis has similar origins as non-electrolyte diffusiophoresis. Both of these mechanisms occur by the chemical species responsible for the gradient being attracted to the surface either by electrostatic (electrolyte) or through van der Waals (non-electrolyte) interactions. If these two effects are comparable, the electrolyte diffusiophoresis is stronger because it has an additional electric field term (Eq. (7)). 2.3.

Electrophoresis

Electrophoresis is the movement of charged objects in an electric field. Typically, an electric field is applied across a suspension of particles in a fluid. This electric field drives the motion of the charges on the surface of the object creating a slip velocity whereby the fluid is allowed to flow around the particle. The particle is thus driven in the opposite direction at a velocity U governed by the equation, U=

ζE , η

(8)

where  is the dielectric constant of the solution, ζ is the zeta potential of the particles, E is the magnitude of the electric field, and η is viscosity.10 Redox reactions occurring at the two ends of an object can result in an electric field similar to an externally applied field resulting in motion. The Pt/Au nanorod motor discovered by Paxton et al. functions by this mechanism; the catalytic decomposition of hydrogen peroxide localizes the field to the length of the rod (Fig. 2).20 This concept was also utilized by Mano et al.21 who attached two electrochemically coupled enzymes to the two ends of a carbon fiber to form a bioelectrochemical motor. Recently, Liu et al. synthesized a copper–platinum nanowire battery that was selfpropelled in bromine and iodine solutions via the oxidation of copper and the concomitant platinum catalyzed reduction of the halogen.22 2.4.

Other phoretic mechanisms

There are also less common yet still significant phoretic mechanisms that have been used to engineer motion of nanoscale objects. A gradient of

Fig. 2.

Scheme for Pt/Au bimetallic rod movement in hydrogen peroxide solution.

November 21, 2012

132

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

a magnetic field known as magnetophoresis has been used extensively in biological applications such as separations.23 Thermal gradients, also known as thermophoresis24 and/or Soret effect, have been employed in some systems, and finally, movement due to surface tension gradients known as Marangoni effects have been described in droplet and bubble systems.25 –27 The general concept here is that by creating a gradient across an object, an entropic force is generated causing directed movement. More recently, a gradient of light has been used to direct the motion of a particle coated by azobenzene molecules.28 2.5.

Bubble propulsion

Another way to create motion is via bubble propulsion. Motors that utilize this type of motion create bubbles on their catalytic side and the force from the release of the bubble causes the motion, also known as bubble recoil. This is a gradient-like mechanism since bubbles need to be generated only on one side, so there is a change in bubble concentration with distance. Building on earlier work by Whitesides,29 the most effective motor to utilize this mechanism was first synthesized by Schmidt and Sanchez and later improved upon by Wang’s lab.1,30,31 The motor consists of a rolled up microtube with an exposed platinum surface on the inside. The platinum decomposes hydrogen peroxide present in solution to form oxygen, and in the confined volume of the tube the oxygen molecules coalesce to form oxygen bubbles. These bubbles are then released out of the larger open end (one end always ends up being slightly larger than the other). Motor speeds of hundreds of microns per second have been reached using this method. An advantage of this motor is that it is able to work in high ionic strength systems, where other motors often fail. This allows for their use in biological systems and by adding recognition sites to these motors, it is possible to transport specific cargo including cancer cells.31 2.6.

Non-reciprocal swimmers

A consequence of the low momentum transfer experienced at low Reynolds number regime is that reciprocal motion in opposing directions leads to no net movement. A common example used to express this concept is the Scallop Theorem. As an actual scallop moves, it closes its shell fast ejecting fluid to create momentum and opens its shell slowly in order to stop itself while gaining a net displacement. The same scallop in a low Reynolds number environment would experience no net displacement regardless of the speeds at which it opens or closes the shell because of the relative lack of momentum transfer.

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

Biology of Nanobots

133

Fig. 3. (a) The reciprocal swim cycle of an unperturbed scallop opening and closing its shell. (b) The motion path of a similar swimmer which undergoes a single rotation event after it has travelled a distance (d) into its cycle. The forward paths are shown in red, the reverse paths are shown in blue.

Table 1. Enhanced translational diffusion constants (Denh ) of a series of hypothetical reciprocal swimmers based on swimmer size, reciprocal swim speed (U ) and half-stroke length (λh ). Rotational (DRot ) and translational (DT rans ) diffusion constants are derived from the particle size based on Eqs. (2) and (3), assuming the objects are spherical and in water at 298K. For swimmers with radius of 1µm, optimized swimming speeds and reciprocal time periods can lead to a fourfold enhancement in diffusion coefficient (0.9µm2 /s compared to the theoretical 0.22µm2 /s). Size (µm)

U (µm/s)

λh (µm)

DRot (rad2 /s)

DTrans (µm2 /s)

Denh (µm2 /s)

1 1 1 1 0.5 0.5 0.5 0.5 0.25 0.25 0.25 0.25

10 10 1 1 5 5 0.5 0.5 2.5 2.5 0.25 0.25

10 1 10 1 5 0.5 5 0.5 2.5 0.25 2.5 0.25

0.16 0.16 0.16 0.16 1.3 1.3 1.3 1.3 10.5 10.5 10.5 10.5

0.22 0.22 0.22 0.22 0.44 0.44 0.44 0.44 0.87 0.87 0.87 0.87

9.00 × 10−1 1.01 × 10−2 6.30 × 10−1 8.92 × 10−3 1.25 1.51 × 10−2 6.55 × 10−2 1.21 × 10−2 2.38 × 10−1 2.41 × 10−2 2.60 × 10−2 2.42 × 10−2

November 21, 2012

134

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

To overcome the constraints of the scallop theorem, small objects like bacteria or micro-swimmers can take advantage of Brownian rotation that frequently changes their moving direction (Fig. 3). Lauga32 showed that micro-swimmers undergoing reciprocal actuation experience diffusive motion with enhanced diffusivities on time scales larger than that of rotational diffusion. Ibele33 also demonstrated that a four fold enhancement in translational diffusion coefficient can be achieved with optimized swimming speeds and reciprocal time periods (Table 1). Another way to circumvent the Scallop Theorem is to create a motor that functions by moving in a non-reciprocal manor, just like swimming micro-organisms that use wavelike deformations of their appendages or bodies to create self-propulsion. This method was explored theoretically by Mikhailov, Kapral, and Golestanian34 –37 (Fig. 4). The enhanced motion

Fig. 4. Scheme showing the non-reciprocal motion of an enzyme as it goes through a catalytic reaction.

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

Biology of Nanobots

135

of enzymes in the presence of substrates may be an example of this mechanism.19,38 2.7.

Chemotaxis

Chemotaxis is a common phenomenon observed in biological systems39 and recently it has been shown in some non-biological systems as well. Unlike biological systems, the mechanism for chemotaxis is not well understood in the latter systems. According to a hypothesis put forth by Hong et al., when a catalytic motor experiences different diffusivities at different substrate (fuel) concentrations, it will chemotaxis towards areas of higher diffusivity. Movement occurs in this direction because with higher diffusivity the motor experiences a higher average displacement and will continue to move farther as it travels up the gradient.40 This was demonstrated experimentally by the authors by placing Pt/Au nanorods in a gradient of hydrogen peroxide. Over time, the density of rods began to increase in the area of the highest concentration of hydrogen peroxide (where the diffusivity was highest). Another example comes from Pavlick et al. who observed that when their polymerization motor was placed in a concentration gradient of the monomer, the density of motor particles increased with time in the area of highest monomer concentration.14

3.

Emergent Collective Behavior of Nanobots

Schools of fish, flocking birds, and colonies of ants are examples of selforganization in living systems. These self-organizing emergent behaviors are based on the interactions between individual units in response to a change in local environment. Ants, for example, communicate with each other and work cooperatively by reacting to chemical stimuli generated by other ants and in turn, leave behind chemical trails which work as stimuli for other ants. This phenomenon has inspired the design of artificial intelligent nano- and microbots that can communicate with each other and work cooperatively with potential applications in the fields of MEMS/NEMS,41 –44 cargo delivery, particle assembly and chemical sensing.4,6,20,29,31,45 –54 Collective emergent behaviors have been discovered in non-biological systems of micro/nanomotors. Ibele et al.18 found that silver chloride microparticles form “schools” when exposed to UV light, and Kagan et al.55 observed that gold microparticles swarm in the presence of hydrogen peroxide with the addition of hydrazine. These collective emergent behavior of micro/nanoscale active particles are explained by the self-diffusiophoretic

November 21, 2012

136

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

mechanism as discussed in Sec. 2. In these cases, each active particle “secretes” chemicals that serve as signals to other nearby particles. When the neighboring particles are close enough to sense the signal, the diffusiophoretic flow that is caused by the chemical concentration gradient pushes/pulls nearby particles away/towards the active particle, and leading to different kinds of collective emergent behaviors. 3.1.

Interaction between colloid particles based on self-diffusiophoresis

According to the classical DLVO theory,56,57 interactions between colloidal particles are a combination of van der Waals forces (generally attractive) and electrostatic interaction between double layers. However, for an active particle that generates a concentration gradient of ions, it will have additional interactions with the other nearby particles (active or inert) based on electrolyte diffusiophoresis. Whether these diffusiophoretic interactions are attractive or repulsive depend on several factors, including the zeta potentials of particles and the wall (in many cases, negatively charged glass slides), the mobility of ions, and distances between particles. 3.1.1. Diffusiophoretic interaction between the central particle and nearby positive particles If the active particles are secreting cations which diffuse faster than the anions, the different diffusion rates will result in a net electric field pointing back towards the particles. For nearby positively charged particles, the electric field acts phoretically and pulls them towards the central particle. The electric field also acts osmotically on the adsorbed cations in the double layer of the negatively charged glass wall pumping the fluid along the wall’s surface together with the nearby particles towards the central particle. In this case, the electric field effect, which is the combination of electrophoresis and electro-osmosis, leads to a diffusiophoretic attraction between the central particle and nearby positive particles (Fig.5(a)). The chemophoretic effect counteracts this slightly, and attempts to pump the fluid, along with nearby particles, away from the central particle, where the double layer is thinner at the glass surface. In general, an inward electric field leads to diffusiophoretic attraction between central particles and nearby positive particles, and this attraction results in the formation of “schools”, i.e., regions in which the number density of particles is significantly higher than the global average. If the direction of the electric field is reversed when the active particles are releasing faster anions and slower cations, the diffusiophoretic interaction

November 21, 2012

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

Biology of Nanobots

137

Fig. 5. Schemes for diffusiophoretic interaction between the central particle and nearby particles with an inward electric field. (a) When the nearby particles are positively charged, the directions of electrophoresis and electroosmosis are both inwards, the diffusiophoretic interactions are attractive, and the system shows a “schooling” pattern. (b) When the nearby particles are negatively charged and the magnitude of ζP is larger than that of ζW , the outward electrophoresis dominates over the inward electroosmosis, the diffusiophoretic interactions are thus repulsive, and the system shows exclusion patterns with exclusion zones between them. (c) When the nearby particles are negatively charged and the magnitude of ζP is smaller than that of ζW , the inward electroosmosis dominates the diffusiophoretic interactions are attractive, and the system shows “schooling” patterns. However, when the particles come close enough to the central one, a repulsive electrophoretic force dominates again and small exclusion zones are formed between the central particle and nearby particles.

will become repulsive, and the system will show “exclusion” patterns with exclusion zones formed between one another, i.e., regions in which particles are absent. The size of exclusion zones depends on the strength of the diffusiophoretic repulsion. 3.1.2. Diffusiophoretic interaction between the central particle and nearby negative particles So what if the nearby particles are negatively charged? Again, for the inward electric field, the electro-osmotic flow along the glass slide carries them towards the central particle, and the relatively negligible chemophoretic flow directs them outwards. However, in this case, the electrophoretic interaction turns repulsive, and the net direction of movement depends on the difference in zeta potential between the particles and the wall (Eq. (7)). If the magnitude of the particle zeta potential (ζP ) is larger than that of the wall (ζW ), then electrophoresis will dominate, diffusiophoretic interaction will be repulsive, and the negative particles will be pumped away from the central one. (Fig. 5(b)) On the other hand, if the magnitude of the wall’s zeta potential is larger, electro-osmotic flow will dominate and will be sufficient to overcome the outward electrophoretic and chemophoretic effects. In this case, the diffusiophoretic interaction will be attractive. The dominance of electro-osmosis is valid as long as the particles are on the same plane along the wall. However, fluid continuity dictates that the inward

November 21, 2012

138

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

electro-osmotic flow along the wall will approach zero, very near to the central particle, as the fluid is forced upwards and away from the wall. Therefore, repulsive electrophoretic interaction dominates very near to the central particle. Thus, the diffusiophoretic interaction will be attractive over long distances but repulsive at close range, and the nearby negative particles will be pumped towards the central one, but with a small exclusion zone left between them. (Fig. 5(c)). Finally, if the electric field is reversed with faster anions and slower cations, the diffusiophoretic interaction will be reversed as well, i.e., from attraction to repulsion. Based on these diffusiophoretic interactions, it is possible for particles to show collective emergent behaviors that can be modeled as discussed by Sen et al.20 Increases in the particle population, their zeta potential, ion secretion rate, as well as a decrease in the particles’ Brownian motion, will aid in the formation of collective emergent patterns. 3.2.

Examples of diffusiophoresis-based emergent systems

Currently, several diffusiophoresis-based collectively emergent systems are available. They utilize micro/nano-sized particles which produce attractive/ repulsive diffusiophoretic interactions between them, allowing them to show corresponding “schooling”/“exclusion” patterns. Ibele et al.18 discovered that silver chloride microparticles, when placed under UV light, will have the following reaction at their surface: UV, Ag+ 4AgCl + 2H2 O −−−−−−−→ 4Ag + 4H+ + 4Cl− + O2 .

(9)

The production of protons and chloride ions in the solution generates an electrolyte concentration gradient that powers the motion of AgCl particles. Because protons diffuse much faster than chloride ions (DH + = 9.311 × 10−5 cm2 s−1 , DCl− = 1.385 × 10−5cm2 s−1 ), the electric field points towards the AgCl particles, the net diffusiophoretic interaction between the particles (ζP = −50 mV, and ζW = −60 mV) is attractive, as discussed in 3.1.2., and the particles form “schools” as shown in Fig. 6. The inert silica particles can also respond to the chemical secretion of active silver chloride particles, and swim towards the latter, a predator-prey behavior as shown in Fig. 7. If hydrogen peroxide is added to the system, a chemical oscillating system is formed, in which silver chloride is reversibly converted to silver metal under UV irradiation.58 The oscillatory reaction couples to particle motion by means of diffusiophoretic interactions between particles, and silver chloride particles display oscillatory attach-release

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

139

Fig. 6. Time-lapse optical microscope images of AgCl particles in deionized water (a) before UV illumination, (b) after 30 s of UV exposure, and (c) after 90 s. Scale bars: 20 µm.

Fig. 7. “Predator-prey” behavior of two different particles, AgCl particles (darker objects) and silica spheres. (a) Without UV light (b,c) When illuminated with UV light the silica spheres actively seek out the AgCl particles and surround them. While the UV is on, an exclusion zone is seen around the AgCl particles. (d) The exclusion zone disappears when UV light is turned off. Times (t; seconds) are given in the upper right corner. Scale bars: 20 µm.

motion with nearby silica spheres, as shown in Fig. 8. Besides the AgCl “schooling” system, gold microparticles were also found by Wang and coworkers to form “schools” in 10% hydrogen peroxide aqueous solution after the addition of 0.1% hydrazine.55

November 21, 2012

140

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

Fig. 8. Time-lapse optical microscope images of a AgCl particle with silica spheres in 1% (v/V) H2 O2 solution under UV light. The AgCl particle alternates between attracting and binding to nearby silica particles for several seconds, then releasing them and engaging in a brief period of supra-Brownian diffusion before the next binding event.

Fig. 9. Motion analysis of three groups of AgCl particles: isolated, coupled and schooled. (a) Optical microscope image showing examples of the three groups of AgCl particles. (b) Mean squared displacements of the AgCl particles at different time intervals, suggesting different diffusive behaviors among the three groups.

3.3.

Motion analysis of particles in collective emergent systems

In addition to purely phenomenological experiments on collective emergent systems, the diffusive behaviors of particles in such systems have recently been analyzed,59 specifically, silver chloride particles in the AgCl “schooling” system. Three distinct classes of particles in this system were analyzed, examples of which are shown in Fig. 9(a): (1) “Isolated” AgCl particles with no other particle within a radius of five body lengths. (2) “Coupled” AgCl particles that only have 1–3 other particles within a radius of five

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

141

body lengths. (3) “Schooled” AgCl particles that are surrounded by more than three other particles within a radius of five body lengths and whose nearest neighbors are roughly symmetrically distributed in space (i.e., the “schooled” particles were selected from near the center of the particle schools, not the edges). In Fig. 8, for each class of AgCl particles, mean squared-displacements across consecutive time intervals of a given length, τ , were recorded. These displacements were averaged (arithmetic mean) together to generate the MSD for each τ value ranging from 0.1 s to 15 s. These MSD values were plotted versus τ to analyze the motion of particles in each group as shown in Fig. 9(b). The data showed that the different classes of AgCl particles have their own distinctive diffusive behaviors, as shown in Fig. 5 (b): (1) Isolated, active particles exhibit a transition between ballistic motion at short time intervals to normal diffusion with an enhanced diffusion constant at greater time intervals. (2) Coupled active particles exhibit primarily enhanced normal diffusive motion over all the timescales measured. (3) Schooled active particles exhibit enhanced normal diffusive behavior that transits to a sub-diffusive behavior at longer timescales. The transition between the different diffusive behaviors is due to the influence of interparticle interactions with increasing number of nearby particles. The motion of the particles can be described as the summation of four effectively independent behaviors: (1) A translational diffusive motion, in which particles advance with a given speed, but at a randomly chosen direction. (2) A rotational diffusive motion, in which particles change their orientation. (3) A ballistic motion, in which particles are propelled with a given speed along a certain direction. (4) The interaction of the particle with its neighbors, in which particles, without changing their orientations, attract and repel others according to the distances between them. Without neighboring particles, the particle shows superdiffusion with a combination of ballistic motion at short time intervals and enhanced diffusion at long time intervals16 (when rotational diffusion dominates). As the number of neighboring particles is increased, the transition to enhanced diffusion occurs at successively shorter time intervals due to particle– particle interactions. If the number of neighboring particles is further increased, the particle becomes “trapped” inside the formed “school” and starts to show subdiffusive behavior. Thus, with increasing number of neighboring particles, the behavior of the active particle changes from superdiffusion, to normal enhanced diffusion and then subdiffusion.

4.

Conclusion

A grand challenge in science is the ability to control energy and information at the nano/microscale. Mastering this challenge will allow us to create

November 21, 2012

13:2

9in x 6in

142

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

technologies that rival those of living systems. As with living systems, this level of control requires the design of intelligent systems that are driven far from equilibrium through the use of free energy derived from chemical reactions. A functioning intelligent system requires (a) information and (b) information processors that act on the information. For most of the systems described above, the information is a chemical or light gradient, and the information processors are self-powered nano/microbots. In the collective emergent systems discussed, individual active particles move randomly, but collectively they show complex behaviors similar to chemotaxis or predator-prey phenomena normally seen only in living systems. The close coupling between gradient sensing and transport is likely to lead to novel applications, such as dynamic spatio-temporal distribution of materials and directed cargo (e.g. drug) delivery. Freed of usual biological constraints, we now have the unprecedented opportunity to probe the ultimate limits of self-organization in these dynamic systems that operate far from equilibrium.

Acknowledgment We gratefully acknowledge NSF (DMR-0820404, CBET-1014673), and AFoSR (FA9500-10-1-0509) for supporting our research.

References 1. Y. Mei, A. A. Solovev, S. Sanchez and O. G. Schmidt, Chem. Soc. Rev. 40, 2109 (2011). 2. T. Mirkovic, N. S. Zacharia, G. D. Scholes and G. A. Ozin, Small 6, 159 (2010). 3. T. Mirkovic, N. S. Zacharia, G. D. Scholes and G. Ozin, ACS Nano. 4, 1782 (2010). 4. Y. Hong, D. Velegol, N. Chaturvedi and A. Sen, Phys. Chem. Chem. Phys. 12, 1423 (2010). 5. S. J. Ebbens and J. R. Howse, Soft Matter 6, 726 (2010). 6. J. Wang, ACS Nano 3, 4 (2009). 7. S. S´ anchez and M. Pumera, Chem. Asian J. 4, 1402 (2009). 8. W. F. Paxton, S. Sundararajan, T. E. Mallouk and A. Sen, Angew. Chem., Int. Ed. 45, 5420 (2006). 9. E. M. Purcell, Am. J. Phys. 45, 9 (1977). 10. J. L. Anderson, Ann. Acad. Sci. (NY) 469, 166 (1986). 11. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2009). 12. P. O. Staffeld and J. A. Quinn, J. Colloid Interface Sci. 130, 88 (1989). 13. U. M. C´ ordova-Figueroa and J. F. Brady, Phys. Rev. Lett. 100, 158303 (2008).

November 21, 2012

13:2

9in x 6in

b1449-ch06

Biology of Nanobots

Engineering of Chemical Complexity

143

14. R. A. Pavlick, S. Sengupta, T. McFadden, H. Zhang and A. Sen, Angew. Chem., Int. Ed. 50, 9374 (2011). 15. R. Golestanian, Phys. Rev. Lett. 102, 188305 (2009). 16. J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh and R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007). 17. H. Ke, S. Ye, R. L. Carroll and K. Showalter, J. Phys. Chem. A 114, 5462 (2010). 18. M. Ibele, T. E. Mallouk and A. Sen, Angew. Chem., Int. Ed. 48, 3308 (2009). 19. H. S. Muddana, S. Sengupta, T. E. Mallouk, A. Sen and P. J. Butler, J. Am. Chem. Soc. 132, 2110 (2010). 20. A. Sen, M. Ibele, Y. Hong and D. Velegol, Faraday Discuss. 143, 15 (2009). 21. N. Mano and A. Heller, J. Am. Chem. Soc. 127, 11574 (2005). 22. R. Liu and A. Sen, J. Am. Chem. Soc. 133, 20064 (2011). 23. M. Suwa and H. Watarai, Anal. Chim. Acta. 690, 137 (2011). 24. S. Iacopini, R. Rusconi and R. Piazza, Eur. Phys. J. E, 19, 59 (2006). 25. F. Brochard, Langmuir 5, 432 (1989). 26. R. Eli, J. Colloid Interface Sci. 83, 77 (1981). 27. K. Nagai, Y. Sumino and K. Yoshikawa, Colloids Surf. B, Biointerfaces 56, 197 (2007). 28. J.-P. Abid, M. Frigoli, R. Pansu, J. Szeftel, J. Zyss, C. Larpent and S. Brasselet, Langmuir 27, 7967 (2011). 29. R. F. Ismagilov, A. Schwartz, N. Bowden and G. M. Whitesides, Angew. Chem., Int. Ed. 41, 652 (2002). 30. W. Gao, S. Sattayasamitsathit, J. Orozco and J. Wang, J. Am. Chem. Soc. 133, 11862 (2011). 31. S. Balasubramanian, D. Kagan, C.-M. JackHu, S. Campuzano, M. J. LoboCastaon, N. Lim, D. Y. Kang, M. Zimmerman, L. Zhang and J. Wang, Angew. Chem., Int. Ed. 50, 4161 (2011). 32. E. Lauga, Phys. Rev. Lett. 106, 178101 (2011). 33. M. Ibele, unpublished results. 34. M. Iima and A. S. Mikhailov, Europhys. Lett. 85, 44001 (2009). 35. T. Sakaue, R. Kapral and A. S. Mikhailov, Eur. Phys. J. B. 75, 381 (2010). 36. A. Cressman, Y. Togashi, A. S. Mikhailov and R. Kapral, Phys. Rev. E. 77, 050901 (2008). 37. R. Golestanian, Phys. Rev. Lett. 105, 018103 (2010). 38. H. Yu, K. Jo, K. L. Kounovsky, J. J. d. Pablo and D. C. Schwartz, J. Am. Chem. Soc. 131, 5722 (2009). 39. H. C. Berg and D. A. Brown, Nature 239, 500 (1972). 40. Y. Hong, N. M. K. Blackman, N. D. Kopp, A. Sen and D. Velegol, Phys. Rev. Lett. 99, 178103 (2007). 41. T. R. Kline, W. F. Paxton, Y. Wang, D. Velegol, T. E. Mallouk and A. Sen, J. Am. Chem. Soc. 127, 17150 (2005). 42. M. E. Ibele, Y. Wang, T. R. Kline, T. E. Mallouk and A. Sen, J. Am. Chem. Soc. 129, 7762 (2007). 43. W. F. Paxton, P. T. Baker, T. R. Kline, Y. Wang, T. E. Mallouk and A. Sen, J. Am. Chem. Soc. 128, 14881 (2006).

November 21, 2012

144

13:2

9in x 6in

b1449-ch06

Engineering of Chemical Complexity

W. Duan, R. Pavlick & A. Sen

44. J. M. Catchmark, S. Subramanian and A. Sen, Small 1, 202 (2005). 45. S. Sanchez, A. A. Solovev, S. Schulze and O. G. Schmidt, Chem. Commun. (Cambridge, UK). 47, 698 (2011). 46. D. Kagan, S. Campuzano, S. Balasubramanian, F. Kuralay, G.-U. Flechsig and J. Wang, Nano Lett. 11, 2083 (2011). 47. S. Sundararajan, S. Sengupta, M. E. Ibele and A. Sen, Small. 6, 1479 (2010). 48. A. A. Solovev, S. Sanchez, M. Pumera, Y. F. Mei and O. G. Schmidt, Adv. Funct. Mater. 20, 2430 (2010). 49. K. M. Manesh, S. Balasubramanian and J. Wang, Chem. Commun. (Cambridge, UK) 46, 5704 (2010). 50. D. Kagan, R. Laocharoensuk, M. Zimmerman, C. Clawson, S. Balasubramanian, D. Kang, D. Bishop, S. Sattayasamitsathit, L. Zhang and J. Wang, Small 6, 2741 (2010). 51. Y. Hong, M. Diaz, U. M. C´ ordova-Figueroa and A. Sen, Adv. Funct. Mater. 20, 1568 (2010). 52. S. Gangwal, A. Pawar, I. Kretzschmar and O. D. Velev, Soft Matter 6, 1413 (2010). 53. Y. Wang, S.-t. Fei, Y.-M. Byun, P. E. Lammert, V. H. Crespi, A. Sen and T. E. Mallouk, J. Am. Chem. Soc. 131, 9926 (2009). 54. S. Gupta, R. G. Alargova, P. K. Kilpatrick and O. D. Velev, Langmuir 26, 3441 (2009). 55. D. Kagan, S. Balasubramanian and J. Wang, Angew. Chem., Int. Ed. 50, 503 (2011). 56. B. V. Derjaguin and L. Landau, Acta Physicochim. (URSS) 14, 633 (1941). 57. E. J. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). 58. M. E. Ibele, P. E. Lammert, V. H. Crespi and A. Sen, ACS Nano 4, 4845 (2010). 59. W. Duan, M. Ibele, R. Liu and A. Sen, Eur. Phys. J. E 35, 77 (2012).

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Part III REACTION–DIFFUSION SYSTEMS AND NONEQUILIBRIUM SOFT MATTER

145

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

This page intentionally left blank

146

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Chapter 7 WAVE PHENOMENA IN REACTION–DIFFUSION SYSTEMS Oliver Steinbock∗ and Harald Engel† ∗

Department of Chemistry and Biochemistry, Florida State University Tallahassee, Florida 32306-4390, USA † Institute f¨ ur Theoretische Physik, Technische Universit¨ at Berlin Hardenbergstrasse 36, EW 7-1, 10623 Berlin, Germany Pattern formation in excitable and oscillatory reaction–diffusion systems provides intriguing examples for the emergence of macroscopic order from molecular reaction events and Brownian motion. Here we review recent results on several aspects of excitation waves including anomalous dispersion, vortex pinning, and three-dimensional scroll waves. Anomalies in the speed-wavelength dependence of pulse trains include nonmonotonic behavior, bistability, and velocity gaps. We further report on the hysteresis effects during the pinning–depinning transition of twodimensional spiral waves. The pinning of three-dimensional scroll waves shows even richer dynamic complexity, partly due to the possibility of geometric and topological mismatches between the unexcitable, pinning heterogeneities and the one-dimensional rotation backbone of the vortex. As examples we present results on the pinning of scroll rings to spherical, C-shaped, and genus-2-type heterogeneities. We also review the main results of several experimental studies employing the Belousov–Zhabotinsky reaction and briefly discuss the biomedical relevance of this research especially in the context of cardiology.

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . The Belousov–Zhabotinsky Reaction . . . . . . Anomalous Dispersion of Periodic Pulse Trains Pinned Two-dimensional Spiral Waves . . . . .

147

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

148 148 150 155

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

148

5. Scroll Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

159 164 166

Introduction

Nonequilibrium processes are essential to all living systems and closely associated with the emergence of dynamical and spatial complexity. The resulting functionalities and patterns are often perplexing and seemingly transcend, like in the cases of biological clocks and neuronal systems, the realm of chemistry. Clearly, however, they are caused by molecular events and transport processes and as such provide exciting opportunities for modern cross-disciplinary research. Excitable and oscillatory reaction– diffusion systems are another important example illustrating the close link between chemistry and biology. This system class shows a remarkable spectrum of macroscopic phenomena such as travelling wave trains, rotating spiral waves, stationary Turing patterns, and spatio-temporal chaos. Its members are extremely diverse and include catalytic surface reactions, corrosion systems, reactive micro-emulsions, self-aggregating microorganisms, cytosolic processes (e.g. calcium-induced calcium release), neurons, and the human heart. All of the latter examples can show rotating spiral wave patterns in some of their key variables. The rotation is driven by an autocatalytic reaction zone that spreads via molecular diffusion. This forest-fire-like process is typically isothermal and involves no fluid motion. The locations of highest reaction rate define a continuous wave front which traces an Archimedean spiral of constant pitch. The spiral tip in the center of the pattern is a unique point as it defines a zerodimensional phase singularity of the oscillating concentration fields. During the past two decades, numerous groups have studied pulse trains and spiral waves systematically. This research revealed a surprising wealth of dynamics and resulted in a deep theoretical understanding of their structure and dynamics. In this chapter we review some recent progress in this field with an emphasis on the dynamics and instabilities of onedimensional pulse trains and the pinning of two- and three-dimensional vortex structures. 2.

The Belousov–Zhabotinsky Reaction

The Belousov–Zhabotinsky (BZ) system is among the most widely studied examples of chemical self-organization. It comprises a family of reactions that involve the oxidation of an organic compound by bromate in acidic,

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

149

aqueous solution. These reactions are catalyzed by redox couples such as ferroin/ferriin (Fe(II/III)(phen)3 ), Ce(III/IV), and Ru(II/III)(bpy)3 .1 Chemical oscillations and traveling waves in this system were first observed by Boris Pavlovich Belousov in the 1950s.2 However, until the late 1960s the reaction was not noticed in wider scientific circles. A key publication — marking the end point of the reaction’s scientific prehistory — is the paper by Albert Zaikin and Anatol Zhabotinsky in 1970.3 Today the BZ reaction has become a standard model system for the study of pattern formation in excitable and oscillatory reaction–diffusion media. Both excitable dynamics and Hopf oscillations in the BZ reaction are semi-qualitatively described by the Oregonator model,4 which itself is based on a more complex reaction mechanism suggested by Field, Kor¨os, and Noyes in 1974.5 In 1990, Krug et al. modified the Oregonator to include the inhibitory effects of visible light (λmax = 460 nm) in the Ru(bpy)3 catalyzed BZ reaction.6 This modified Oregonator involves three variables, x, y, and z, which are proportional to the concentrations of the activator species HBrO2 , the inhibitor species Br− , and the oxidized form of the catalyst, respectively:  dx/dt = x(1 − x) + y(q − x), 

 dy/dt = φ + f z − y(q + x), dz/dt = x − z.

(1(a)) (1(b)) (1(c))

Here, ,  , and q are parameters that depend on various rate constants and the pseudo-stationary concentration of bromate ion (BrO− 3 ). The parameter f is a stoichiometric factor which controls the strength of bromide production in the “dark” reaction and t is rescaled, dimensionless time. Lastly, φ denotes the additional bromide flow induced by illumination of the system. From an experimental point of view, the BZ reaction has several advantageous features that distinguish it from other self-organizing reaction–diffusion systems. Foremost, the system is relatively easy to prepare, inexpensive, and highly reproducible. In addition, wave patterns can be monitored conveniently by following the spatio-temporal changes in the absorption of monochromatic, visible light. The resulting data are qualitatively related to the concentration ratio of the oxidized to the reduced catalyst. For the ferroin-catalyzed reaction, these changes can also often be detected with the unaided eye to reveal striking blue patterns on a red background. Moreover, the wave patterns and their dynamics involve convenient length and time scales. For typical preparations, rotating spiral waves have rotation periods of several seconds to minutes and wavelengths in the millimeter range. We also

November 21, 2012

150

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

emphasize that numerous methods for the initiation and perturbation of BZ patterns are available. These include gradients in reactant concentrations, temperature, and light-intensity7 as well as externally applied electric fields.8 Another feature of the BZ reaction is that, dependent on the choice of initial concentrations, closed systems sustain chemical oscillations and wave patterns up to several hours. During this long life time of the system, important system parameters, such as the oscillation period, wave velocity, or rotation frequency of spiral waves, undergo slow changes. For some studies these transients can be tolerated, while others require the use of chemical reactors. Spatially homogeneous conditions can be provided by continuously-stirred tank reactors (CSTRs); transient free, quasi-twodimensional systems can be studied under stationary nonequilibrium conditions in continuously fed unstirred reactors (CFURs, see for e.g. Ref. 9). In most CFURs, the pattern forming system is confined to a thin gel layer or porous glass plate, which is exposed to well-stirred, continuously fed tanks containing incomplete sets of the reactants. In the simplest case, the catalyst is immobilized in the reaction layer, while all other reagents enter from one tank. Most BZ studies employ malonic acid (CH2 (COOH)2 ) as the sole organic substrate of the reaction. This substrate is brominated and oxidized to a variety of intermediates including bromomalonic acid. These processes cause the formation of CO2 and eventually form macroscopic gas bubbles. These bubbles are highly undesired because they disturb the spatial homogeneity of the system and, in some situations, create convection. Kurin-Cs¨ orgei et al. noticed that 1,4-cyclohexanedione (C6 H8 O2 ) is a useful substitute for malonic acid.10 Its reaction products are nongaseous and the CHD-BZ reaction is, hence, free of bubbles. Later, Steinbock et al. reported the existence of anomalous wave dispersion in the CHDBZ reaction.11 These interesting findings will be further discussed in Sec. 3.

3.

Anomalous Dispersion of Periodic Pulse Trains

From experimental observations in a variety of excitable media it is well-known that excitation pulses can form periodic pulse trains. Their propagation speed is defined by the interaction between consecutive pulses within the train, which in turn depends on the peculiarities of recovery from the excited into the rest state during the refractory phase of the excitation cycle. If excitable media recover monotonically, the larger the

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

151

pulse spacing the lower will be the excitation threshold that the next-coming pulse has to overcome. Therefore, in such media the dispersion curve that plots propagation speed, c(L), of the pulse train against wavelength, L, is a monotonically increasing function of L. It saturates at the speed c∞ of the solitary pulse propagating into the fully recovered medium, and displays an instability at some minimal pulse spacing, below which stable wave trains cease to exist. This type of dispersion resulting from repulsive pulse interaction is referred to as normal dispersion. This has been observed in the BZ reaction and many other (weakly) excitable media.12 A dispersion relation that exhibits negative slope within some interval of wavelength is referred to as anomalous dispersion. The negative slope corresponds to attractive pulse interaction, and leads to unstable pulse trains (at least in the large wavelength limit). While most excitable media exhibit normal dispersion, some examples of anomalous dispersion are frequent as we will see below. Numerical simulations with the FitzHugh–Nagumo model revealed that normal dispersion at weak excitability transforms with decreasing excitability threshold into anomalous dispersion initially with oscillating and after this with bistable dispersion curves.13 The same transition from normal via oscillatory to bistable dispersion was observed previously for the three-variable Oregonator model of the light-sensitive BZ reaction using the intensity of applied illumination as a measure of the local excitation threshold.14 This transition scenario is typical for excitability type II models (according to a classification by Izhikevich15 ) which involve both a soft transition from excitable to oscillatory local kinetics via a supercritical Hopf bifurcation and Canard behavior due to strong time scale separation between activator and inhibitor. We would also like to mention early work by Elphick et al. about consequences of oscillatory recovery for the dispersion of periodic pulse trains,16 and the paper by Winfree17 demonstrating that excitable media with oscillatory dispersion may support coexisting stable spiral waves rotating at different frequency. Remarkably, in the parameter region with oscillatory dispersion colliding excitation pulses do not annihilate but display reflective collision.18 Such unusual behavior of excitation pulses in head-on collision resembles colliding solitons in conservative systems. Bistable dispersion gives the simplest example where propagation speed is multi-valued over a range of wavelength. In the corresponding wavelength bands alternative stable pulse trains coexist that propagate at different speeds although they have the same wavelength. As a result the dispersion relation contains hysteresis loops. R¨oder et al. demonstrated

November 21, 2012

152

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

Fig. 1. Bistable dispersion curve of the FitzHugh–Nagumo model according to ut = uxx + u(1 − u)(u − a) − v; vt =
(u − bv) for b = 0.25,
= 0.3 10−4 , and decreasing values of the excitation threshold, a. At a = −0.002634 (a), due to the break-up of the dispersion curve separated isola-like fragments emerge. A velocity gap opens up that expands as the isolas shrink (b: a = −0.002637), and finally wavelength bands exist wherein pulse train propagation is impossible (c: a = 0.002650). Solid (dashed) lines indicate stable (unstable) pulse trains.13

that the unstable branches of the hysteresis loops break-up close to the Canard-point.13 Thus, the dispersion curve splits into disconnected fragments, and isola-like closed curves are formed. Fragmentation of the bistable dispersion curve and isola formation lead to velocity and wavelength gaps within each hysteresis loop. The appearance of a velocity gap in the dispersion relation has been reported previously for a model of intracellular Ca2+ -dynamics in Ref. 19. When the excitability threshold decreases further, isolas shrink and finally disappear, (cf. Fig. 1). The results regarding the transition from normal to oscillatory and bistable dispersion due to an increase in excitability improved the qualitative understanding of the related transition from trigger waves to phase waves.20,21 Another interesting case of anomalous dispersion is a dispersion curve with a single maximum separating normal dispersion at short wavelength from anomalous dispersion at long wavelength. Experimentally, anomalous dispersion with a single overshoot was found in the catalytic reduction of NO with CO on platinum single crystal surfaces,22 in heart tissue (where it is known as supernormal conduction23 ) and in particular in the 1,4cyclohexanedione (CHD)–BZ reaction in which malonic acid is replaced by CHD.11,24 The last example was studied in detail both experimentally as well as by means of numerical simulations and bifurcation analysis.11,24,25 The experiments with the CHD-BZ reaction are performed in thin, quasi one-dimensional capillary tubes. It was found that single-overshoot dispersion induces at least three different regimes of pulse dynamics referred to as stacking, merging, and tracking (Fig. 2).

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

153

Fig. 2. (A) Experimentally obtained space-time plots and (B) corresponding dispersion relation of stacking, merging and tracking pulse dynamics. Experiments are carried out with the CHD–BZ reaction in thin glass capillaries of inner diameter and length 0.63 mm and 64 mm, respectively.11 ,24 Dispersion relations are calculated numerically from Eqs. (2). Stable (unstable) pulse trains correspond to full blue (broken red) lines. Parameter values:
= 10−2 , a = 0.7, β = 0.3, γ = 5.0, and δ = 0.20 (a), 0.28 (b), 0.30 (c).24 ,25

November 21, 2012

154

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

Qualitatively most of the observed dynamic phenomena are reproduced by the following three-variable reaction–diffusion model
 ∂u ∂2u 1 v+w = u(1 − u) u − + , (2(a)) ∂t ∂x2  a ∂v = u − v, ∂t ∂w = β(δ − w) − γuw. ∂t

(2(b)) (2(c))

This model was used for numerical simulations, in particular for a stability and bifurcation analysis of the obtained travelling wave solutions25 employing continuation methods of numerical bifurcation analysis.26 Here, δ plays the role of the bifurcation parameter. The transition from normal dispersion (δ < 0) to dispersion with a single overshoot (δ > 0) was shown to result from a bifurcation of the homoclinic orbit representing the solitary pulse known as orbit flip.27 Pulse stacking (Fig. 2(a)) corresponds to a situation in which slow solitary pulses coexist with faster periodic pulse trains, and in which pulse trains with wavelength L0 propagating at the speed of the solitary pulse are stable. In the merging regime (Fig. 2(b)) pulse trains break-up in the wake of the slow solitary pulse because a small wavelength instability destabilizes pulse trains propagating at the speed of the solitary pulse. Finally, in the tracking regime (Fig. 2(a)) no solitary pulses exist, however, periodic pulse trains within finite wavelength limits can propagate through the medium. Consequently, the dispersion relation forms a closed loop covering the corresponding band of wavelengths. We note that the stability of periodic pulse trains at small and moderate wavelengths cannot be derived from the simple criterion relating stable (unstable) solutions to segments of the dispersion relation with positive (negative) slope. Instead, the corresponding spectra of the linearized stability operator have to be calculated. The results obtained for Eqs. (2) are presented in Ref. 25. In summary, the theoretical analysis shows that the described regimes of pulse dynamics are not confined to the specific model used here but are generically expected for all excitable media that exhibit single-overshoot dispersion. Recently, Echebarria et al. studied the propagation of action potentials in simple ionic models of excitable cardiac tissue. It was demonstrated that supernormal conduction promotes concordant alternans (period doubling) and leads to straight defect lines in spiral waves undergoing an alternans transition. Period doubled spiral waves with straight defect lines have been observed in cultured rat cells.28

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

4.

155

Pinned Two-dimensional Spiral Waves

It is well known that two-dimensional spiral waves can be pinned to unexcitable heterogeneities as long as the size of the pinning domains is sufficiently large. This critical size is approximately the same as the size of the spiral core in the homogeneous system. In systems with normal dispersion, larger anchors will increase the rotation period of the spiral. For very large anchors, the velocity of the spiral tip is constant (c∞ ) and the rotation period T is expected to obey T = U/c∞ where U is the perimeter of the anchor. We note that spiral wave pinning in the photosensitive Ru(bpy)3 -catalyzed BZ reaction allows the convenient repositioning of spiral tips, their pairwise annihilation, as well as their pairwise creation. Pertsov et al.30 in 1984 studied a spiral wave pinned to a circular inhomogeneity of some diameter dc with no flux boundary conditions along the circumference of the defect. In numerical simulations with the FitzHugh–Nagumo model, the authors continuously decreased dc until at some critical value dcr c the spiral wave detached from the defect and transformed into a freely rotating spiral wave in which core radius, shape and rotation frequency in media with normal dispersion are uniquely selected by the properties of medium. If the defect diameter was again increased, at some value dc > dcr c the spiral wave anew became anchored. Consequently, a range of defect diameters existed where a pinned and an unpinned spiral wave coexist. The hysteresis phenomenon in the dependence of the rotation frequency on the hole radius corresponding to this bistability was found in a broad class of models, and it was realized that it expresses basic features of excitable media. For example, by continuation of spiral wave solutions to the Barkley model precise numerical data for the pinning– depinning transition were obtained.29 In these numerical experiments the transition between coexisting branches of stable (pinned and unpinned) spiral waves connected by a branch of unstable spiral wave solutions was associated with hysteresis (Fig. 3). Later on, the same behavior was found for the Fitz–Hugh–Nagumo31 and for the Oregonator model, compare Fig. 5.32 In general, two basic mechanisms are crucial for existence, stability and dynamics of spiral waves in excitable media: The curvature dependence of the front velocity (encoded in the eikonal equation) and the (possibly incomplete) relaxation of the inhibitor to the rest state before the next wave front arrives at a given point in the medium due to refractoriness (given by the dispersion relation of periodic wave trains). Within a kinematical approach, recently it was possible to decouple curvature and dispersion effects in order to understand the role played by both

November 21, 2012

156

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

Fig. 3. Hysteretic transition between pinned and unpinned spiral waves. Continuation results for the Barkley model in an annulus with inner and outer radius r− and r+ , respectively. (a) Dependence of the rotation frequency on the inner radius. Full (broken) lines indicate stable (unstable) solution branches within the bistability region. The transition between the two stable regimes was found to be accompanied by hysteresis. (b) Snapshots of wave solutions corresponding to labels 1–3 above. Parameters of the Barkley model 4(b):
= 0.05, a = 0.75, r+ = 10.0, b = 0.045. (Figure taken from Ref. 29.)

mechanisms in the hysteresis phenomenon separately.31 First, adopting a linear curvature-velocity relation and neglecting dispersion effects, the spiral wave remained pinned to the hole for arbitrary small hole radius. In a second step, a kinematical model that includes a pulse front of some finite thickness, d, (instead of considering a jump-like transition from the rest to the excited state) leads to a nonlinear curvature-velocity relation with a critical front curvature beyond which stable wave propagation becomes impossible. Within such a boundary layer kinematical model, a pinning–depinning transition with hysteresis was reproduced qualitatively.

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

157

Fig. 4. Frequency of rotation vs. the hole radius in the Fitz–Hugh–Nagumo model. The full line shows the numerical results obtained from the reaction–diffusion equations. Asterisks correspond to the results from a boundary layer kinematical model with nonlinear velocity-curvature relation and by taking dispersion into account.31

However, the quantitative agreement with the numerical data turned out to be weak. To reach quantitative agreement, finally dispersion effects were taken into account. Because of medium’s refractoriness, the inhibitor level straight ahead of the boundary layer changes periodically and cannot be considered constant as was assumed so far. Taking this into account one obtains fairly good agreement between the predictions of the kinematical approach and the numerically obtained data as shown in Fig. 4. Recently, hysteresis in the pinning–depinning transition of spiral waves in the presence of circular inhomogeneities was studied experimentally in an open gel reactor for the light-sensitive BZ reaction.32 Here, first a rigidly rotating spiral wave was pinned to a circular light spot of intensity well beyond the critical value for wave propagation inside. The idea was to appropriately change the spot diameter in order to observe escape of the spiral tip from the spot at sufficiently low spot size and subsequent anchorage to the spot under increase of the spot size. However, soon it turned out that the corresponding transitions were not accompanied by hysteresis due to the absence of no-flux conditions at the spot boundary. Numerical experiments based on simulations of the underlying modified Oregonator model Eq. (1) revealed to be concordant with these experimental findings. Without no-flux boundary conditions imposed on the circumference of the heterogeneity, no hysteretic transition appeared. In an attempt to realize the necessary no-flux boundary condition a small glass cylinder approximately of radius 0.1 mm was fixed on the plane glass support before the gel solution with the photosensitive catalyst was added and gelation started. The glass cylinder was only slightly higher than the thickness of the catalyst-loaded gel layer (0.2 mm) where the reaction

November 21, 2012

158

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

Fig. 5. Pinning–depinning transition of two-dimensional spiral waves. (a) Results obtained from numerical simulations of the Oregonator equation (1). The full line (circles) is an exponential fit of the numerically calculated increase of the core radius with φ while the full horizontal line marks the defect radius. Insets show snapshots of the spiral waves. Oregonator parameters:
= 6.99 × 10−2 ,

/
= 3.99 × 10−3 , q = 2 × 10−3 , f = 1.4, Du = 1.0, Dv = 1.12. (b) Illustration of the experimental results for the light-sensitive BZ reaction: The full line shows the dependence of the core size of the free spiral wave on the applied light intensity. In the experiment, the radius of the cylindrical Neumanndefect was 0.5 mm (horizontal line). Stimulated (at I < I ∗ ) and spontaneous (at I ∗ ) detachment of the spiral tip from the defect are marked by full and broken vertical arrows, respectively.33

takes place. In light-sensitive BZ media the local excitation threshold and the core diameter of rigidly rotating spiral waves increase under increasing applied light intensity, while the rotation frequency becomes smaller. A spiral wave pinned to the cylindrical defect should detach from it as soon as under increased globally applied illumination the core of the free spiral exceeds the defect diameter sufficiently enough. Preliminary numerical simulations with the Oregonator equations confirmed this expectation. In these simulations the photochemically induced flow of the inhibitor bromide, φ, was assumed to be proportional to the applied light intensity. Rigid rotation of free spiral waves covered the φ interval between 0.030 (meandering spiral waves below φ = 0.030) and 0.036 (up to propagation failure close-by), see left panel of Fig. 5. The escape of the pinned spiral from the defect occurred along the broken vertical arrow. Thus, within the parameter range 0.030 < φ < 0.0348 pinned and free spiral wave solutions coexisted. In agreement with the numerical predictions, experiments performed within an open gel-reactor for the light-sensitive BZ reaction revealed an extended bistability domain. For example, for light intensity I1 in the right panel of Fig. 5, a spiral wave with a period of rotation equal to

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

159

Trot = (135 ± 5)s pinned to a circular Neumann defect (radius 0.5 mm) coexisted with a free spiral wave rotating at period Trot = (200 ± 10)s around a core of radius rc = 0.9 mm.33 Further details of the experimental verification of this phenomenon generic to excitable media will be published elsewhere.33 5.

Scroll Waves

Scroll waves are the three-dimensional analog of spiral waves. Their rotation occurs around one-dimensional curves that are usually referred to as filaments. For topological reasons, these filaments can terminate only at the system boundaries or they must close in on themselves to form loops, knots, and chain links. B´ ans´ agi et al. reported the only known exception to this rule, which seems to require a specific type of anomalous dispersion. In their experiments and simulations, filaments terminate in the back of nonrotating, travelling wave front.34 A simple example for this wavetermination is shown in Fig. 6. The data are obtained by optical tomography from experiments with the 1,4-CHD-BZ reaction. In general, the filaments of scroll waves are not stationary but move with velocities that depend strongly on the filament’s local curvature K. Theoretical analyses as well as experiments with the BZ reaction show that this motion is often well described by the equation ds ˆ ˆ + β B), = K(αN dt

(3)

ˆ denote the local filament position and the corresponding ˆ , and B where s, N normal and binormal unit vectors, respectively. The constant values of

Fig. 6. Three-dimensional, tomographic reconstruction of a wave-pinned scroll wave in the CHD-BZ reaction. The initial reagent 3 ] = 0.18 M, i h concentrations are [NaBrO [1,4-CHD] = 0.19 M, [H2 SO4 ] = 0.6 M, and [Fe(batho(SO3 )2 )3 ]4− = 0.475 mM.

November 21, 2012

160

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

the parameters α and β depend on the specific point dynamics and the involved diffusion coefficients. For two-variable systems with equal diffusion coefficients, the system parameter β equals zero. The same holds for the ˆ complex Ginzburg–Landau equation. The direction of the normal vector N is defined so that it points towards the center of a circular filament loop. This choice is independent of the directional orientation of the filament, which is usually determined according to the “right-hand rule” for which the fingers point in the direction of scroll wave rotation and the thumb indicates the filament direction. If these conventions are followed, positive filament tension (α > 0) causes filament loops to shrink and self-annihilate in finite times tL . For a circle of initial radius R0 , Eq. (1) yields tL = R02 /(2α). This simple relation has been verified in several experimental and numerical studies (see e.g. Ref. 35). The recent years have seen an increasing interest in the study of three-dimensional, excitable systems with negative filament tension (“NT”). Values of α < 0 are one of the possible sources of scroll wave instabilities36 and induce spatio-temporal chaos that is sometimes referred to as NT or Winfree turbulence (after the late theoretical biologist Art Winfree).37,38 This turbulence involves the continuous bending and buckling of the filament which typically causes frequent collisions with the system boundaries and other filaments. For the case of infinity long, initially linear, and untwisted filament, linear stability analysis reveals a finite band of positive growth rates spanning from zero to a finite wave number.36 Numerical studies of the Fitz–Hugh–Nagumo model by Zaritski et al. show that the subsequent, nonlinear evolution of the filament can induce temporarily local order that is perhaps best described as tightly spaced, triple filament strands.39 All of these NT-related phenomena are also of great biomedical interest because similar electric patterns in the human heart have been linked to ventricular fibrillation and sudden cardiac death.40 In the following, we will limit our discussions to cases with positive ˆ where filament tension and β = 0 for which Eq. (3) yields ds/dτ = K N τ = αt. Notice that, under these conditions, planar filaments remain planar, which ideally allows the discussion of filament dynamics in R2 . This seemingly simple, contracting curvature flow has been studied prior to its application to scroll waves and has many interesting features and solutions.41,42 For instance, using the Gauss–Bonnet theorem, it can be shown that the area A enclosed by any (nonintersecting) curve decreases at a rate of dA/dτ = −2π. In addition, interesting stationary filament solutions exist some of which can be described by analytical solutions. A recently reported example are planar, hairpin-shaped filaments of constant velocity v.43 These self-reinforcing filament shapes trace the functions

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

161

Fig. 7. Scroll ring in the MA-BZ reaction pinned to an inert, torus-shaped heterogeneity from which a small segment was cut out prior to the experiment. This image sequence spans approximately one rotation period and shows the 8 mm thick BZ system as viewed from the top. Black arrows indicate the local direction of wave propagation. The images show an area of 1.5 cm2 . The initial reagent concentrations are [NaBrO3 ] = 0.04 M, [MA] = 0.04 M, [H2 SO4 ] = 0.16 M, and [Fe(phen)3 ] = 0.5 mM.

y(x) = − α/v ln(cos(vx/α) and their speeds obey v = πα/w where w denotes the width of the hairpin. The latter conditions are an ideal platform for the exploration of more complicated phenomena such as filament reconnections, filament–filament interaction, and scroll wave pinning. The first experimental demonstration of a pinned scroll wave was reported by Jim´enez, Marts, and Steinbock in 2009.44 A typical example from this BZ study is shown in Fig. 7. Here a scroll ring has been pinned to an inert heterogeneity that in the figure can be discerned as a black C-shaped area. This inert, impermeable object consist of the fluoropolymer viton and has the overall shape of a thin (partial) torus. Detailed analyses show that the scroll ring is pinned to this wave obstacle,44

November 21, 2012

162

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

thus, preventing the vortex collapse expected for homogeneous systems. Moreover, interesting effects are induced by the topological mismatch between the circular rotation backbone of the scroll wave and the C-shaped anchor. The pinned segments of the vortex rotate with a slightly lower frequency than the unpinned part in the obstacle gap. Accordingly, the vortex structure undergoes a twisting process, which results in detectable phase variation along the outer and inner obstacle perimeter. However, the build-up of twist is limited by the intrinsic reaction–diffusion dynamics of the BZ system. This interplay induces the formation of a stationary twist pattern and yields a constant rotation period. The period T depends on the frequency difference between the pinned and the free rotation ∆ω as well as on the cut angle θ. Detailed measurements of T (θ) and T (∆ω) reveal very good agreement with (implicit) analytical equations derived from the forced Burgers equation. The Burgers equation, a nonlinear diffusion equation, had been previously suggested as an appropriate description for twist dynamics in excitable reaction-diffusion systems (see e.g. Refs. 45 and 46). Scroll pinning has also been accomplished using small spherical heterogeneities. Figure 8 shows an example from work by Jim´enez and Steinbock, in which a scroll ring pinned to three millimeter-sized glass beads located

Fig. 8. Scroll ring in the MA-BZ reaction pinned to three inert, spherical heterogeneities. This snapshot shows a top-view of the approx. 8 mm thick BZ system. The small, dark spots are gas bubbles. The initial reagent concentrations are [NaBrO3 ] = 0.04 M, [MA] = 0.04 M, [H2 SO4 ] = 0.16 M, and [Fe(phen)3 ] = 0.5 mM.

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

163

Fig. 9. Numerical simulation of an excitation vortex pinned to an unexcitable double torus. The two snapshots show domains of high inhibitor values v > 0.4 (solid, gold color) and the unexcitable anchor (solid, blue). Only the posterior half of the system is shown. The omitted anterior half is symmetric to the depicted volume.

at the vertices of a nearly equilateral triangle. This pinning scenario is sufficient to prevent the curvature-induced collapse of the vortex and yields a stationary filament that roughly connects the pinning sites with linear (slightly convex) segments. Similar experiments with up to four pinning sites have been reported by the Steinbock group.47 The vortex stabilization described above does not occur if only one spherical heterogeneity is used. In the case of only one small pinning site, the filament loop collapses toward the heterogeneity. During that process, it acquires a drop-like shape that is well described by the curvature flow in Eq. (3) with β = 0 if the pinning site is modelled as a Dirichlet boundary. The life-time of initially circular filament loops is increased by 25% (28% in simulations) relative to the life time of the free scroll ring. This factor is independent of the initial filament radius.47 The case of scroll ring pinning to two spherical anchors is still under investigations. We have already mentioned that scroll wave pinning can involve interesting topological mismatches between the filament and the pinning heterogeneity. A striking example is the stabilization of scroll rings at genus-2 objects that was demonstrated experimentally by Dutta and Steinbock in 2011.48 Figure 6.4 shows the result of comparable numerical simulation based on the Barkley model
 v+b ∂u 1 = ∇2 u + u(1 − u) u − , (4(a)) ∂t  a ∂v = ∇2 v + u − v, ∂t

(4(b))

November 21, 2012

164

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

where  = 0.02, a = 1.1, and b = 0.18 are constant system parameters. The wave structure illustrated in the figure is periodic and stable. It is one of several possible vortex states associated with pinning double tori and involves a mixture of one- and two-armed scroll waves that is easy to discern in frame (a). Accordingly, the rotation frequency is approximately twice as high as the frequency of a scroll wave pinned to a linear anchor of identical thickness. Also notice that the waves traverse the holes in opposite directions. For the example shown in Fig. 9, these directions are upward for the right hole and downward for the left. These directions can be also identical but then the vortex frequency is reduced by a factor of one half. This frequency is also observed for vortex states created by initializing the system with a half-spherical wave capping only one of the two holes. The latter state has a four-fold degeneracy because the initial state can cap either hole on its top or bottom side. Additional states should exist for larger numbers of topological charges but have not been investigated yet.

6.

Summary and Conclusions

Since its discovery, the BZ reaction has been modified and optimized in several directions. Improvements include the development of open gel reactors to maintain stationary nonequilibrium conditions, the confinement of the reaction to a transparent gel layer or porous glass plate in order to prevent convection, and the substitution of the organic substrate malonic acid by 1,4-cyclohexanedione to inhibit formation of disturbing CO2 bubbles during the reaction. Another important step was the replacement of the “classical” catalytic Fe- and Ce-complexes by the light-sensitive catalyst Ru(bpy)3 . Now, the medium’s excitability depended on the externally applied light intensity, i.e., the local excitation threshold varies according to a given space- and/or time-dependent illumination pattern. This has significantly extended the possibilities to perform experiments under external forcing, and has opened-up entirely new options for light-mediated feedback control of nonlinear waves. Altogether, with the described modifications the BZ reaction still plays an important role in the study of reaction–diffusion patterns, in particular in the verification of theoretical predictions about nonlinear wave dynamics in excitable and oscillatory media under well-controlled experimental conditions. In this chapter, we have reviewed recent results on anomalous dispersion of periodic wave trains and pinning of two- and three-dimensional vortex structures to heterogeneities of the medium. Anomalous dispersion has far reaching consequences for wave pattern selection. Usually, trigger waves select a distinct spatial wave profile and

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

165

a corresponding unique propagation velocity. With anomalous dispersion, several stable wave solutions can coexist in one and the same medium. Interestingly, in the parameter range of coexisting pulse trains, colliding pulses do not annihilate each other but undergo reflective collision. Thus, under certain conditions trigger waves that represent wave solutions to dissipative reaction–diffusion equations may survive head-on collision as do wave solutions to conservative nonlinear equations (so-called solitons), for example, shallow water waves described by the Korteweg-de Vries equation or optical solitons described by the nonlinear Schr¨ odinger equation. Another important finding is that anomalous dispersion with a single overshoot, as observed in the CHD-BZ medium, is closely related to a novel type of scroll ring nucleation in excitable media. Practically, more often than not excitable media are heterogeneous. In this review we have discussed the effect of obstacles on the dynamics of two- and three-dimensional vortices. First, the pinning of two-dimensional spiral waves to a circular unexcitable defect is studied in some detail. Both dispersion and curvature effects turn out to be responsible for the experimentally observed hysteresis in the pinning–depinning transition, whereby the curvature-velocity relation was found to be nonlinear. The last observation is interesting because still it remains as an unsolved problem to analytically predict the selected rotation frequency and core radius of a rigidly rotating spiral wave within a freeboundary approach even if the linear eikonal equation is taken as the basis. Our result demonstrates that the correct description of a relatively simple, generic phenomenon as the hysteresis in the pinning–depinning transition already requires a nonlinear eikonal equation. Second, we have reported on scroll waves pinned to different heterogeneities. Pinning prevents a scroll ring with positive filament tension from collapse, thus creating a pacemaker within the medium. Scroll wave pinning to small spherical heterogeneities can involve interesting topological mismatches between the filament and the pinning heterogeneity. In conclusion, the interaction between spiral and scroll waves with heterogeneities of the medium substantially affects the dynamics of vortices. This should be important for biological systems as, for example, the heart where heterogeneities are abundant. Understanding dynamics and stability of nonlinear waves is a first step in designing efficient and robust control methods aimed at selecting and stabilizing desired wave patterns. This will open new possibilities for the control of wave phenomena in a variety of quite diverse applications including cardiac arrhythmia as well as catalytic fixed bed reactors, frontal polymerization, and solid fuel combustion, to mention only a few.

November 21, 2012

13:2

9in x 6in

166

b1449-ch07

Engineering of Chemical Complexity

O. Steinbock & H. Engel

Acknowledgments We are grateful to Jan Totz for valuable discussions about the escape of a spiral wave from a cylindrical defect and providing data presented in Fig. 5. This material is based upon work supported by the National Science Foundation under Grant No. 0910657 and by the German Research Foundation within the framework of the CRC 910.

References 1. A. M. Zhabotinsky, Scholarpedia 2(9), 1435 (2007). 2. I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics (Oxford University Press, New York, 1998). 3. A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535 (1970). 4. R. J. Field and R. M. Noyes, J. Chem. Phys. 60, 1877 (1974). 5. R. J. Field, E. Kors and R. M. Noyes, J. Amer. Chem. Soc. 94, 8649 (1972). 6. H.-J. Krug, L. Pohlmann and L. Kuhnert, J. Phys. Chem. 94, 4862 (1990). 7. M. Braune, H. Engel, Phys. Rev. E 62, 5986 (2000). 8. O. Steinbock, J. Sch¨ utze and S. C. M¨ uller, Phys. Rev. Lett. 68, 248 (1992). 9. H. Brandtst¨ adter, M. Braune, I. Schebesch and H. Engel, Chem. Phys. Lett. 323, 145 (2000). 10. K. Kurin-Csorgei, A. M. Zhabotinsky, M. Orban and I. R. Epstein, J. Phys. Chem. A 101, 6827 (1997). 11. C. T. Hamik, N. Manz, O. Steinbock, J. Phys. Chem. A 105, 6144 (2001). 12. J.-M. Flesselles, A. Belmonte and V. Gspr, J. Chem. Soc. Faraday Trans. 94, 851 (1998). 13. G. R¨ oder, G. Bordyugov, H. Engel and M. Falcke, Phys. Rev. E 75, 036202 (2007). 14. G. Bordiougov and H. Engel, Phys. Rev. Lett. 90, 148302 (2003). 15. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT press, 2007). 16. C. Elphick, E. Meron, J. Rinzel and E. A. Spiegel, J. Theor. Biol. 146, 249 (1990). 17. A. T. Winfree, Physica D49, 125 (1991). 18. G. Bordyugov and H. Engel, Chaos 18, 026104 (2008). 19. M. Falcke, M. Or-Guil and M. Br, Phys. Rev. Lett. 84, 4753 (2000). 20. G. Bordiougov and H. Engel, Physica D 215, 25 (2006). 21. E. J. Reusser and R. J. Field, J. Amer. Chem. Soc. 101(5), 1063 (1979). 22. J. Christoph, M. Eiswirth, N. Hartmann, R. Imbihl, I. Kevrekidis and M. Br, Phys. Rev. Lett. 82, 1586 (1999). 23. F. H. Fenton, E. M. Cherry, H. M. Hastings and S. J. Evans, Chaos 12, 852 (2002). 24. N. Manz, C. T. Hamik and O. Steinbock, Phys. Rev. Lett. 92, 248301 (2004).

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

Wave Phenomena in Reaction–Diffusion Systems

167

25. G. Bordyugov, N. Fischer, H. Engel, N. Manz and O. Steinbock, Physica D 239, 766 (2010). 26. B. Sandstede, in Handbook of Dynamical Systems, Vol. 2, ed. by B. Fiedler (North-Holland, 2002). 27. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, Berlin, 1995). 28. B. Echebarria, G. R¨ oder, H. Engel, J. Davidsen and M. Br, Phys. Rev. E 83, 040902(R) (2011). 29. G. Bordyugov and H. Engel, Physica D 228, 49 (2007). 30. A. M. Pertsov, E. A. Ermakova and A. V. Panfilov, Physica D 14, 117 (1984). 31. V. Zykov, G. Bordyugov, H. Lentz and H. Engel, Physica D 239, 797 (2010). 32. J. Totz, Bachelor thesis, TU Berlin (2011) 33. J. Totz et al., in preparation. 34. T. B´ ans´ agi Jr., K. J. Meyer and O. Steinbock, J. Chem. Phys. 128, 094503, (2008). 35. T. B´ ans´ agi Jr. and O. Steinbock, Phys. Rev. Lett. 97, 198301 (2006). 36. H. Henry and V. Hakim, Phys. Rev. E 65, 046235 (2002). 37. A. T. Winfree, Science 299, 1722 (2003). 38. S. Alonso, F. Sagues and A. S. Mikhailov, Science 181, 937 (1973). 39. R. M. Zaritski, S. F. Mironov and A. M. Pertsov, Phys. Rev. Lett. 92, 168302 (2004). 40. S. Luther et al., Nature 475, 235 (2011). 41. W. W. Mullins, J. Appl. Phys. 27, 900 (1956). 42. I. Bakas and C. Sourdis, J. High Energy Phys. 27, 057 (2007). 43. S. Dutta and O. Steinbock, Phys. Rev. E 81, 055202 (2010). 44. Z. A. Jim´enez, B. Marts and O. Steinbock, Phys. Rev. Lett. 102, 244101 (2009). 45. J. P. Keener and J. J. Tyson, SIAM Rev. 34, 1 (1992). 46. D. Margerit and D. Barkley, Chaos 12, 636 (2002). 47. Z. A. Jimnez and O. Steinbock, Europhys. Lett. 91, 50002 (2010). 48. S. Dutta and O. Steinbock, J. Phys. Chem. Lett. 91, 50002 (2010).

November 21, 2012

13:2

9in x 6in

b1449-ch07

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Chapter 8 SELF-OSCILLATING POLYMER GELS AS SMART MATERIALS

Ryo Yoshida Department of Materials Engineering, School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan [email protected] Stimuli-responsive polymer gels and their application to smart materials have been widely studied. On the other hand, as a novel biomimetic gel, we have been studying gels with an autonomous self-oscillating function, since first reported in 1996. We succeeded in developing novel self-oscillating polymers and gels by utilizing the oscillating reaction, called the Belousov–Zhabotinsky (BZ) reaction. The self-oscillating polymer is composed of a poly(N-isopropylacrylamide) network in which the catalyst for the BZ reaction is covalently immobilized. In the presence of the reactants, the polymer gel undergoes spontaneous cyclic swelling–deswelling changes or soluble–insoluble changes (in the case of uncrosslinked polymer) without any on–off switching of external stimuli. Potential applications of the self-oscillating polymers and gels include several kinds of functional material systems, such as biomimetic actuators and mass transport surface.

Contents 1. 2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemomechanical Behavior of Self-Oscillating Polymer Gel . Application to Autonomous Mass Transport Systems . . . . 3.1. Mass transport surface utilizing peristaltic motion of gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Modeling of mass transport on gel surface . . . . . . . . 3.3. Surface design of gel conveyer . . . . . . . . . . . . . . . 3.4. Effect of physical interactions between loaded cargo and 3.5. Effect of surface roughness . . . . . . . . . . . . . . . . Self-oscillation of Polymer Solution and Microgel Dispersion 169

. . . . . . . . . . . . . . . . . . . . . . . . . . .

170 172 174

. . . . . . gel . . . .

174 175 176 176 178 179

. . . . . . . . . . . . . . . surface . . . . . . . . . .

. . . . . .

. . . . . .

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

170

4.1. Self-flocculating/dispersing oscillation of microgels . . . . . . . . . . 4.2. Viscosity oscillation of polymer solution and microgel dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Autonomous viscosity oscillation by reversible complex formation of terpyridine-terminated poly(ethylene glycol) in the BZ reaction . . . 5. Attempts of Self-oscillation under Physiological Conditions . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. .

179

. .

179

. . . .

182 182 184 184

. . . .

Introduction

Many researchers have developed several kinds of stimuli-responsive polymer gels that exhibit reversible swelling–deswelling change in response to environmental changes such as solvent composition, temperature, pH change, etc. In contrast, we developed a novel polymer gel that cause autonomous mechanical oscillation without an external control in a completely closed solution (Fig. 1). In order to realize the autonomous polymer system, the Belousov– Zhabotinsky (BZ) reaction, which is well-known for exhibiting temporal and spatiotemporal oscillating phenomena,1,2 was focused. We attempted to convert the chemical oscillation of the BZ reaction into a mechanical change in gels and generate an autonomous swelling–deswelling oscillation under non-oscillatory outer conditions. A copolymer gel consisting of N-isopropylacrylamide (NIPAAm) and ruthenium tris(2,2-bipyridine) (Ru(bpy)3 ) was prepared (Fig. 2(a)). Ru(bpy)3 , acting as a catalyst for the BZ reaction, is pendent to the polymer chains of NIPAAm. Poly(NIPAAm) is a well known thermosensitive polymer which exhibits a lower critical solution temperature (LCST) of approximately 32◦ C, and the homopolymer

Fig. 1.

Stimuli-responsive and self-oscillating gels.

November 21, 2012

13:2

9in x 6in

b1449-ch08

171

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

Mechanism of self-oscillation and several aspects of self-oscillating behavior. Fig. 2.

November 21, 2012

172

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

gel undergoes a volume phase transition at that temperature. For the poly(NIPAAm-co-Ru(bpy)3 ) gel, the oxidation of Ru(bpy)2+ 3 moiety caused not only an increase in the swelling degree of the gel, but also a rise in the volume phase transition temperature (Fig. 2(b)). As a result, it is expected that the gel would undergo a cyclic swelling–deswelling change when the Ru(bpy)3 moiety is periodically oxidized and reduced under constant temperature. When the poly(NIPAAm-co-Ru(bpy)3 ) gel is immersed in the catalystfree BZ solution, the reaction occurs in the gel by the catalytic function of the polymerized Ru(bpy)3 (Fig. 2(c)). The redox changes of the polymerized → catalyst moiety (Ru(bpy)2+ ← Ru(bpy)3+ 3 3 ) change the volume phase transition temperature as well as the swelling ratio of the gel because the hydrophilicity of the polymer chains increases at the oxidized Ru(III) state and decreases at the reduced Ru(II) state. As a result, the gel exhibits autonomous swelling–deswelling oscillation with the redox oscillation in the closed solution under constant condition. Since being first reported in 1996 as a “self-oscillating gel”,3 we have been systematically studying the selfoscillating polymer and gel as well as their applications to biomimetic or smart materials4 –25 (Fig. 3). In this chapter, these recent progresses on the self-oscillating polymers and gels and the design of functional material systems were summarized.

2.

Chemomechanical Behavior of Self-Oscillating Polymer Gel

As shown in Fig. 2(d), when the bulk gel is small enough than the wavelength of chemical wave, redox changes occur homogeneously in the gel without pattern formation.7 However, a train of excited pulses of the oxidized state (i.e., chemical waves) spontaneously evolves and propagates in the gel by the reaction–diffusion mechanism when the gel size is much larger than the wavelength of the chemical wave. In the case of a twodimensional gel sheet, concentric or spiral waves can be observed. With the propagatin of chemical waves, the self-oscillating gel exhibits peristaltic motion,8 that is, the locally swollen (or shrunken) region corresponding to a locally oxidized (or reduced) state propagate in the gel, similar to the locomotion of a living worm. Many applications to autonomous chemomechanical actuators can be expected. As applications to autonomic biomimetic actuators, ciliary motion actuators (artificial cilia)9 and self-walking gels,10 etc. were realized (Fig. 3). The self-oscillating gel constructs a unique model of material systems, i.e., coupling system of reaction–diffusion and mechanical motion.

November 21, 2012

13:2

9in x 6in

b1449-ch08

173

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

Development of self-oscillating polymers and gels. Fig. 3.

November 21, 2012

174

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

If the chemical oscillation and mechanical oscillation affect each other by feedback, it would be difficult to predict and explain the oscillating behaviors. Then, theoretical simulation becomes effective tool. Balazs et al.26 developed a mathematical model for simulating chemomechanical behaviors of the self-oscillating gels. They have demonstrated several aspects of the self-oscillating behavior for the gel by theoretical simulation. Many interesting phenomena are demonstrated theoretically. For example, selfpropelled motion of gels was theoretically demonstrated and experimentally realized by utilizing the gels.11 The findings from these studies provide guidelines for creating autonomously moving objects, which can be used for robotic or microfluidic applications. In the case of the uncrosslinked linear polymer, as shown in Fig. 2(d), the polymer undergoes spontaneous cyclic soluble–insoluble changes and the transmittance of the polymer solution oscillates autonomously.12,13 In addition, submicron-sized self-oscillating microgel beads were prepared by a precipitation polymerization method.14 –18 As a nanoactuator to exhibit autonomous oscillation on a nanometer scale, the oscillating behavior was investigated through the optical transmittance or viscosity changes17 –20 of the polymer solution or microgel dispersions. 3. 3.1.

Application to Autonomous Mass Transport Systems Mass transport surface utilizing peristaltic motion of gel

Further, in order to realize the self-driven gel conveyer as a novel autonomous mass transport system, we attempted to transport an object by utilizing the peristaltic motion of the self-oscillating gel (see Fig. 3).21 –24 For a control of the transportability, it is necessary to enhance the driving force as a conveyer. It was found to be effective to copolymerize 2-acrylamido2-methylpropanesulfonic acid (AMPS) to poly(NIPAAm-co-Ru(bpy)3 ) gel network to generate large amplitude of volume change of the self-oscillating gel.21 The gel had a microphase-separated structure when the AMPSs feed ratio was less than 5 mol% due to the effect of the poor solvent in the polymerization process. On the other hand, when the AMPSs feed ratio is more than 10 mol%, the gel had a homogeneous structure. The velocity of the chemical wave for the microphase-separated gel was faster than that for the homogeneous gel. Further, the microphase-separated structure highly improved the swelling–deswelling kinetics and generated swelling–deswelling amplitude more than 10% of the gel thickness, that was approximately 10 times larger than that of the gel with a homogeneous network structure.

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

175

Fig. 4. (a) Phase diagram of the transportable region given by the velocity and the inclination angle of the wave front: ◦ transported, × not transported. (b) Model of the rolling cylindrical gel on the peristaltic gel surface (RC = radius of curvature, W = load of the PAAm gel, b = contact half-width).21

As a model object, a cylindrical poly(acrylamide) (PAAm) gel was applied on the self-oscillating gel surface. It was observed that the PAAm gel was transported on the gel surface with the propagation of the chemical wave as it rolled (see the photograph in Fig. 3) when the AMPSs feed ratio was low (less than 2.5 mol%). Figure 4 shows a phase diagram of transportable conditions given by the velocity and the inclination angle θ of wave front. The velocity and the inclination angle of the chemical wave were changed by changing the concentratons of the outer solution. For the controlled chemical waves with several inclination angles and velocities, whether the cylindrical gel could be transported or not was estimated. It was found that the cylindrical PAAm gel was not transported if the inclination angle was less than approximately 3◦ . The mass transportability did not depend on the velocity of the chemical wave but on the diameter of the cylindrical PAAm gel and the inclination angle of the wave front. 3.2.

Modeling of mass transport on gel surface

For the analysis of this result, we have proposed a model to describe the mass transport phenomena based on the Hertz contact theory, and the relation between the transportability and the peristaltic motion was investigated.21 Figure 4(b) shows the contact model of the cylindrical

November 21, 2012

176

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

PAAm gel and the self-oscillating gel sheet. Here, it is assumed that they are a non-adhesive rigid cylinder and an elastic plane, respectively, and there is no physical interaction between them. If the inclination angle was smaller than the slope of the tangent at point A, sufficient contact force from the strain does not act to rotate the cylindrical PAAm gel. Therefore, the condition for the transport of the cylindrical PAAm gel is written as RC sin θ ≥ b. The contact half-width, b, can be calculated from the Hertz contact theory by using the actual values of physical property for the loaded gel and the gel sheet such as Poisson ratio, the Young modulus, etc.21 As a result of calculation from the above equation, the minimum inclination angle was 3 degree and it was the same as the angle resulted from the experiment. It was found experimentally and supported by the model that the sheer wave front of the peristaltic motion was necessary to transport cylindrical gels. If the cargo materials are not adsorbed to the self-oscillating gel surface by an attracting force like hydrophobic interaction, mass transportation depends on the performance of the peristaltic motion of the self-oscillating gel surface. It was found that the microphase-separated structure was a key to transport cylindrical gel because the enlarged peristaltic motion could be useful for generating rotational movement of the cylindrical gel. 3.3.

Surface design of gel conveyer

Further, the surface figure capable of transporting microparticles in one direction was designed to fabricate more versatile self-driven gel conveyer. The self-oscillating gel having a grooved surface was fabricated by using PDMS template and the effectiveness of the surface design was investigated. Poly(AAm-co-AMPS) gel beads with the diameter of several hundred µm to several mm were transported on the grooved surface of the self-oscillating gel by its autonomous peristaltic motions.22 It was found that the traveling direction of the peristaltic motion could be confined to the direction along the grooves by designing the groove-distance shorter than the wavelength of the chemical wave. Consequently, several gel beads were transported in parallel. 3.4.

Effect of physical interactions between loaded cargo and gel surface

For a wider use of the autonomous transport system, it is important to investigate the influence of the physical interaction between the self-oscillating gel and the loaded cargo on its transporting ability.

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

177

Properties such as charge state, hydrophobicity, surface roughness of the gels, and the ionic strength of the surrounding solution are considered to affect the adhesive interactions. Hence, instead of the homopolymer gel of PAAm, several kinds of copolymer gels consisting of AAm with AMPS, N-(3-aminopropyl)methacrylamide hydrochloride (APMA), N-(hydroxymethyl)acrylamide (HMAAm), and methyl methacrylate (MMA) were prepared as model cargos with different surface properties; positive or negative charge, more hydrophilicity or hydrophobicity, respectively. The influences of these surface properties on the transport behaviors were investigated.23 Figure 5 shows whether or not the cylindrical copolymer gel cargo (AAm = 50 mol%) was able to be transported on the self-oscillating gel sheet when the inclination angle of the chemical wave front was changed by changing the concentrations of the outer solution. Various inclination angles of the wave front of the chemical wave (ca. 0–8 degrees) were successively generated with the appropriate choice of the concentration of the substrates. As mentioned before, the theoretical model for the transport of a cylindrical gel on the self-oscillating gel sheet was proposed based on the Hertz contact theory. The model demonstrated that the inclination angle of the wavefront

Fig. 5. Dependence of the transportability of the cylindrical copolymer gel (AAm = 50 mol%) on the inclination angle of the wave front of the self-oscillating gel. (a) the PAAm gel, (b) the poly(AAm-co-AMPS) gel, (c) the poly(AAm-co-APMA) gel, (d) the poly(AAm-co-HMAAm) gel and (e) the poly(AAm-co-MMA) gel (: transported, : not transported).23

November 21, 2012

178

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

larger than a certain critical value was necessary to transport the cylindrical gel cargo, and the minimum angle was approximately 3 degree in the case of the PAAm cylinder gel with the diameter of 750 µm. As shown in Fig. 5, all the copolymer gels except for the poly(AAm-co-MMA) gel were transported when the inclination angle of the wave front was larger than approximately 3 degree, as expected from the theoretical model. The cylindrical poly(AAm-co-MMA) gel was not transported even when the inclination angle was larger than 5 degree, and this result was the same also for the gels with AAm = 70 and 90 mol%. Therefore, it was suggested that the adhesion force between the cylindrical poly(AAm-coMMA) gel and the self-oscillating gel sheet was too strong to apply the transport model because of their hydrophobic interaction. The adhesive force to prevent transportation is not significant for the other gels, which agrees with the prediction from swelling, zeta potential, and contact angle measurements.23 3.5.

Effect of surface roughness

Then, we investigated the effect of surface roughness on the transportation. The feasibility of transportation for the cylindrical poly(NIPAAm-coHMAAm) gels prepared at two different polymerization temperatures, 4 and 70◦ C, was evaluated. As the content of HMAAm increased, the gel was able to be transported because of the increased hydrophilicity of the gel due to the hydroxyl group. However, the lower limit of the HMAAm content for transportation is different between the two gels. The difference may be attributed to the surface morphology. All the gels polymerized at 4◦ C were transparent regardless of the HMAAm content. In contrast, the gels polymerized at 70◦ C were milky white for the HMAAm contents = 0, 10, 20, and 30 mol% and opaque for 50 mol%. Therefore, the poly(NIPAAm-coHMAAm) gels polymerized at 70◦ C are considered to have the aggregative structure of their microgels by phase separation during the polymerization, and the surface roughness is considered to be higher compared with the transparent gels prepared at a lower polymerization temperature below the LCST. The self-oscillating gel also has aggregative structure of microgels due to the effect of poor solvent during polymerization.21 Therefore, the contact area between the loaded milky white or opaque gel and the selfoscillating gel surface is considered to be larger than that between the loaded transparent gel and the self-oscillating gel surface, which is effective in inducing the rotational motion of the loaded gel. It becomes apparent that higher surface roughness is more effective in transporting the loaded gel because frictional force increases and the moment of force of the rotational motion also increases.

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

4.

4.1.

179

Self-oscillation of Polymer Solution and Microgel Dispersion Self-flocculating/dispersing oscillation of microgels

As mentoned before, the periodic changes of linear and uncrosslinked polymer chains can be easily observed as cyclic transparent and opaque changes for the polymer solution with color changes due to the redox oscillation of the catalyst.12 Then, we prepared submicron-sized poly(NIPAAm-coRu(bpy)3 ) gel beads by surfactant-free aqueous precipitation polymerization, and analyzed the oscillating behavior.14,15 At low temperatures (20–26.5◦C), on raising the temperature, the amplitude of the oscillation became larger. The increase in amplitude is due to increased deviation of the hydrodynamic diameter between the Ru(II) and Ru(III) states. Furthermore, a remarkable change in waveform was observed between 26.5 and 27◦ C. Then, the amplitude of the oscillations dramatically decreased at 27.5◦ C, and finally, the periodic transmittance changes could no longer be observed at 28◦ C. The sudden change in oscillation waveform should be related to the difference in colloidal stability between the Ru(II) and Ru(III) states. Here, the microgels should be flocculated due to lack of electrostatic repulsion when the microgels were deswollen (see Fig. 1). The remarkable change in waveform was only observed at higher dispersion concentrations (greater than 0.225 wt.%). The self-oscillating property makes microgels more attractive for future developments such as microgel assembly, optical and rheological applications, etc.

4.2.

Viscosity oscillation of polymer solution and microgel dispersion

In the case of the self-oscillating polymer solution or the mircogel dispersion, the solubility or the swelling–deswelling changes can be measured as viscosity as well as optical transmittance changes of the solution or the dispersion. Actually, we succeeded in observing the viscosity selfoscillation for the polymer solution induced by the BZ reaction at constant temperature.13 The viscosity self-oscillation was originated by the difference between solubilities of the polymer chain in the reduced and oxidized states. Recently, we have achieved autonomously oscillating viscosity in a microgel dispersion using autonomously oscillating microgels.17,18 We found out that viscosity oscillation occurs in two different manners, exhibiting a simple pulsatile waveform or a complex waveform with two peaks per period (Fig. 6). It was suggested that the difference in

November 21, 2012

180

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

Fig. 6. Two different types of oscillating waveforms observed at 20◦ C and 23◦ C. The numbers in each oscillating profiles refer to the corresponding cartoons.17

waveform is due to the difference in the oscillating manner of the microgels: swelling/deswelling or dispersing/flocculating oscillation, as mentioned before. We can control rhythm and amplitude of the oscillation using these two phenomena of the microgels. In order to characterize the viscosity oscillation, two types of the microgels were synthesized by changing the feed ratio of Ru(bpy)3 and crosslinker. Viscosity of the microgel dispersions at high salt concentrations could be controlled by changing concentrations of the microgels. Autonomously oscillating viscosity was only measured when concentration of the microgels was high. The amplitude of the oscillation became bigger with increasing concentrations of the microgels. By adjusting the concentration of the substrates for the BZ reaction, we could achieve the constant oscillation for a long time. Moreover, with increasing Ru(bpy)3 and decreasing the crosslinker, microgels showed a high degree of swelling/deswelling oscillation, resulting in bigger amplitudes of autonomously oscillating viscosity. These technologies could be applied in many applications as electro- or magnetic-rheological (ER or MR) fluids have been done. In particular, the dispersion with autonomously

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

181

Fig. 7. Concept of autonomous viscosity oscillation by reversible complex formation of Ru(terpy)2 -tetra PEG in the BZ reaction. Oscillating profiles of viscosity of the aqueous solution containing Ru(terpy)2 -tetra PEG (Mw :16 k c:3.0 wt%), HNO3 , NaBrO3 and MA at 25◦ C.20

November 21, 2012

182

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

oscillating viscosity may be used as a micropump, which could realize novel microfluidics devices. 4.3.

Autonomous viscosity oscillation by reversible complex formation of terpyridine-terminated poly(ethylene glycol) in the BZ reaction

Further, we realized viscosity oscillation of polymer solution based on different mechanisms.20 It is known that a terpyridine ligand binds or dissociates with a Ru metal ion depending on the redox states of the Ru metal ion.27 Generally, when the Ru metal ion is in the reduced Ru(II) state, the Ru(II) metal ion forms bis-complexs with terpyridine (Ru(terpy)2 ). However, when the Ru metal ion is in the oxidized Ru(III) state, the Ru(III) metal ion forms monocomplex with terpyridine (Ru(terpy)). Therefore, supramolecular block copolymers have been made by using Ru(terpy)2 as a junction point.27 If the Ru-terpyridine complex acts as a catalyst of the BZ reaction, the redox oscillation may cause periodical binding/dissociation of the Ru-terpyridine complex. Recently, a theoretical computational simulation in the case that the Ru-terpyridine complex acts as a reversible cross-linking point of polymer network during the BZ reaction has been reported by Balazs et al.28 The swelling–deswelling oscillating behaviors of the gel were theoretically demonstrated by the simulation. When the crosslinking density is not enough high to form the gel, it is expected that the reversible complex formation cause viscosity oscillation of the polymer solution due to a change in molecular weight. Actually, we achieved autonomous viscosity oscillation by reversible complex formation of terpyridine-terminated PEG and/or terpyridineterminated Tetra PEG in the BZ reaction.20 Then the BZ reaction induces the periodical binding/dissociation of the Ru-terpyridine complex and causes periodical molecular changes to results in viscosity changes (Fig. 7). Differently from the viscosity oscillation we reported before, this mechanism based on complex formation may be advantageous in terms of remarkable change in molecular weight. Although the amplitude of viscosity oscillation in this experiment was still small for practical applications, we believe that remarkable changes like sol-gel transition could be possible by controlling molecular design. 5.

Attempts of Self-oscillation under Physiological Conditions

So far, the author had succeeded in developing a novel self-oscillating polymer (or gel) by utilizing the BZ reaction. However, the operating

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

183

conditions for the self-oscillation are limited to conditions under which the BZ reaction occurs. For potential applications as functional bio- or biomimetic materials, it is necessary to design a self-oscillating polymer which acts under biological environments. To cause self-oscillation of polymer systems under physiological conditions, BZ substrates other than organic ones, such as malonic acid and citric acid, must be built into the polymer system itself. For this purpose, we have synthesized a quarternary copolymer which includes both pH-control and oxidant-supplying sites in the poly(NIPAAm-co-Ru(bpy)3 ) chain at the same time.13 By using the polymer, self-oscillation under conditions where only the organic acid (malonic acid) exists was actually achieved. Further, it is desirable that the self-oscillation can be induced around body temperature. Typically, the volume-phase transition temperature of the poly(NIPAAm-co-Ru(bpy)3 ) gel is around 25◦ C, and above that temperature the gel shrinks for both the reduced and oxidized states. As a result, it is difficult to induce self-oscillation near body temperature. For self-oscillation at higher temperatures, it is necessary to avoid the

Fig. 8. (a) Strategy for achievement of self-oscillation at higher temperature while maintaining a large amplitude by utilizing a polymer with higher LCST. (b) Comparison of chemical wave velocity, oscillation period and amplitude of swelling–deswelling amplitude at 18◦ C and 37◦ C among the poly(NIPAAm-co-Ru(bpy)3 ) gel, the poly(EMAAm-coRu(bpy)3 ) gel and the poly(DMAAm-co-Ru(bpy)3 ) gel.25

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

R. Yoshida

184

collapse of the polymer at those temperatures. One possible method may be to utilize a non-thermosensitive polymer without an LCST. In this case, the difference in swelling ratios between the reduced and oxidized states rely only on a change in hydrophilicity due to the charge number of the redox site without the help of an attractive intermolecular force by phase transition. However, it would be difficult to maintain a large difference in the swelling ratio between the reduced and oxidized states. Otherwise, it would be better to use a thermosensitive polymer with a higher LCST to maintain a large difference between the reduced and oxidized states by utilizing the phase transition at higher temperatures. In order to induce selfoscillation while maintaining a larger amplitude at higher temperatures and around body temperature, we prepared a self-oscillating gel composed of a thermosensitive N,N-ethylmethylacrylamide (EMAAm) polymer exhibiting a higher LCST than that of the NIPAAm polymer.25 The self-oscillating behavior of the poly(EMAAm-co-Ru(bpy)3 ) gel was investigated by comparing against gels composed of a thermosensitive NIPAAm polymer with a lower LCST or non-thermosensitive N,N-dimethylacrylamide (DMAAm) polymer. It was shown that the poly(EMAAm-co-Ru(bpy)3 ) gel can induce swelling–deswelling self-oscillation while maintaining a larger amplitude near body temperature, while the other two gels do not undergo swelling– deswelling oscillation at that temperature (Fig. 8). The design concept of self-oscillation at higher temperatures without a decrease in swelling– deswelling amplitude was demonstrated by utilizing a thermosensitive polymer exhibiting a higher LCST. 6.

Conclusions

We proposed novel chemomechanical systems to convert chemical oscillation of the BZ reaction to mechanical changes of polymer and gel, and succeeded in realizing such an energy conversion system producing autonomous self-oscillation of polymer gel. Here, these recent progresses on the selfoscillating polymer and gels and the design of functional material systems were summarized. As an innovative study to propose novel potential of polymer gels and achieve an autonomous behavior by coupling chemical and mechanical oscillations in polymer systems, the study has attracted much attention in the research field of polymer science, material science, physical chemistry, theoretical simulation, etc. Further development on the self-oscillating polymer and gel will be expected in future. References 1. R. Field and M. Burger, Oscillations and Traveling Waves in Chemical Systems (John Wiley & Sons, New York, 1985).

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

Self-oscillating Polymer Gels as Smart Materials

185

2. I. Epstein and J. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Oxford University Press, New York, 1998). 3. R. Yoshida, T. Takahashi, T. Yamaguchi and H. Ichijo, J. Am. Chem. Soc. 118, 5134 (1996). 4. R. Yoshida, Adv. Mater. 22, 3463 (2010). 5. R. Yoshida, Colloid Polym. Sci. 289, 475 (2011). 6. R. Yoshida, T. Sakai, Y. Hara, S. Maeda, S. Hashimoto, D. Suzuki and Y. Murase, J. Controlled Release. 140, 186 (2009). 7. R. Yoshida, M. Tanaka, S. Onodera, T. Yamaguchi and E. Kokufuda, J. Phys. Chem. A. 104, 7549 (2000). 8. S. Maeda, Y. Hara, R. Yoshida and S. Hashimoto, Angew. Chem., Int. Ed. 47, 6690 (2008). 9. O. Tabata, H. Kojima, T. Kasatani, Y. Isono and R. Yoshida, Chemomechanical actuator using self-oscillating gel for artificial cilia, Proc. Int. Conf. on MEMS 2003, pp. 12–15 (2003). 10. S. Maeda, Y. Hara, T. Sakai, R. Yoshida and S. Hashimoto, Adv. Mater. 19, 3480 (2007). 11. O. Kuksenok, V. Yashin, M. Kinoshita, T. Sakai, R. Yoshida and A. Balazs, J. Mater. Sci. 21, 8360 (2011). 12. R. Yoshida, T. Sakai, S. Ito, and T. Yamaguchi, J. Am. Chem. Soc. 124, 8095 (2002). 13. Y. Hara and R. Yoshida, J. Phys. Chem. B. 112, 8427 (2008). 14. D. Suzuki, T. Sakai and R. Yoshida, Angew. Chem., Int. Ed. 47, 917 (2008). 15. D. Suzuki and R. Yoshida, Macromolecules 41, 5830 (2008). 16. D. Suzuki and R. Yoshida, Polymer J. 42, 501 (2010). 17. D. Suzuki, H. Taniguchi and R. Yoshida, J. Am. Chem. Sci. 131, 12058 (2009). 18. H. Taniguchi, D. Suzuki and R. Yoshida, J. Phys. Chem. B. 114, 2405 (2010). 19. Y. Hara and R. Yoshida, J. Chem. Phys. 128, 224904 (2008). 20. T. Ueno, K. Bundo, Y. Akagi, T. Sakai and R. Yoshida, Soft Matter 6, 6072 (2010). 21. Y. Murase, S. Maeda, S. Hashimoto and R. Yoshida, Langmuir 25, 483, (2009). 22. Y. Murase, M. Hidaka and R. Yoshida, Sensors and Actuators B, Chemical 149, 272 (2010). 23. Y. Murase, R. Takeshima and R. Yoshida, Macromol. Biosci. 11, 1713 (2011). 24. R. Yoshida and Y. Murase, Colloids Surf. B, Biointerfaces 99, 60 (2012). 25. M. Hidaka and R. Yoshida, J. Controlled Release 150, 171 (2011). 26. V. Yashin, O. Kuksenok and A. Balazs, Prog. Polym. Sci. 35, 155 (2010). 27. B. Lohmeijer and U. Schubert, Angew. Chem., Int. Ed. 41, 3825 (2002). 28. V. Yashin, O. Kuksenok and A.C. Balazs, J. Phys. Chem. B. 114, 6316 (2010).

November 21, 2012

13:2

9in x 6in

b1449-ch08

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Chapter 9 STOCHASTIC FLUCTUATIONS AND SPONTANEOUS SYMMETRY BREAKING IN THE CHEMOTAXIS SIGNALING SYSTEM OF DICYOSTELIUM CELLS

Tatsuo Shibata Laboratory for Physical Biology, RIKEN Center for Developmental Biology 2-2-3 Minatojima-minamimachi, Chuo-ku, Kobe 650-0047, Japan [email protected] The signal transduction system of eukaryotic chemotaxis is one of the most well-characterized systems with respect to its molecular components and their interactions. Because cells are such tiny systems, the stochastic fluctuations of reactions are prominent. We first consider the noise in the chemotaxis of Dictyostelium cells. The theoretical signal-to-noise ratio can explain the accuracy of chemotaxis obtained experimentally. Then, we show self-organization in the chemotaxis signal transduction system in Dictyostelium cells. A stochastic reaction– diffusion model can reproduce the observed behavior.

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eukaryotic Chemotaxis is Limited by Noise . . . . . . . . . . Signal Transduction and Spontaneous Cell Motility . . . . . . Spontaneous Pattern Formation in Phosphatidylinositol Lipid Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187

. . . . . . . . . . . . . . . . . . Signaling . . . . . . . . . . . . . . . . . .

. . . . . .

188 189 194

. . . . . .

195 199 199

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

T. Shibata

188

1.

b1449-ch09

Introduction

The ability of cells to sense an extracellular chemical gradient and generate a directional motion along the gradient is called chemotaxis. Such chemicals are called chemoattractants. Chemotaxis behaviors are found in cell types from bacteria to eukaryotic cells. In bacterial chemotaxis, cells show random cell motions, which are modulated according to the temporal differences in the chemoattractant concentration that they perceive. As a result, the random motion is biased up the gradient. Chemotaxis behaviors are also found in many typs of eukaryotic cells, such as neutrophils, immunocytes, and neuronal cells. In these cells, chemotaxis plays an important role in environmental foraging behavior, morphogenesis, and immune responses. In eukaryotic chemotaxis, cells can sense the spatial chemical gradient directly. Thus, the mechanism of eukaryotic chemotaxis is different from bacterial chemotaxis. The social amoebae Dictyostelium discoidium is the most widely studied model organism of such eukaryotic chemotaxis, as it shares the chemotaxis mechanisms of mammalian cells. The chemotaxis signal transduction system of Dictyostelium cells consists of a G-protein coupled receptor with seven-transmembrane domain and its binding partner, a trimeric G-protein at the start of the pathway, which is followed by a several parallel pathways (Fig. 1). Among the downstream pathways of the trimeric G-protein, the phosphatidylinositol (PtdIns) lipid signaling reaction is the most well-studied reaction and plays an essential role for the chemotaxis signal transduction. The PtdIns lipid is one of the phospholipids that form the plasma membrane. Depending on the number and the position of phosphate group on the inositol ring, PtdIns species have different physiological roles.1 Among the PtdIns lipids, PtdIns 4,5-disphosphate

cAMP

PI(4,5)P2

cAMP

PI(3,4,5)P3

Ras Gα

Gβ

PI3K

Gγ PTEN

trimeric G

Fig. 1. A schematic illustration of signal transduction system of chemotaxis in Dictyostelium cells. The cAMP receptor can associate with the trimeric G-protein to transmit a signal toward downstream signaling processes. One of the key mediators of chemotaxis signal is the phosphatidylinositol lipid reaction, which consists of a kinase PI3K that produces PtdIns(3,4,5)P3 from PtdIns(4,5)P2 by phosphorylation and a phosphatase PTEN that catalyzes the reverse reaction. Upon cAMP stimuli, PI3K is activated and increases the level of PtdIns(3,4,5)P3 on the plasma membrane, which induces actin polymerization.

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 189

(PtdIns(4,5)P2) and PtdIns 3,4,5-trisphosphate (PtdIns(3,4,5)P3) play important roles in the chemotaxis signal transduction pathway. When the level of the extracellular chemoattractant of Dictyostelium cyclic adenosine 3,5-monophosphate (cAMP) is elevated, phosphoinositide-3-kinase (PI3K) produces PtdIns(3,4,5)P3 on the plasma membrane, leading to a transient increase in the membrane PtdIns(3,4,5)P3 level. The phosphatase of PtdIns(3,4,5)P3 PTEN (phosphatase and tensin homolog) produces PtdIns(4,5)P2 from PtdIns(3,4,5)P3 to keep the level of PtdIns(3,4,5)P3 on the plasma membrane constant. Under an extracellular cAMP gradient, PI3K accumulates on the membrane region facing the higher concentration side of the cAMP gradient, while PTEN is localized in the other membrane region.2 As a result, PtdIns(3,4,5)P3 accumulates in the front region of the cell. PtdIns(3,4,5)P3 induces actin polymerization and protrusive activities on the plasma membrane. Therefore, this PtdIns(3,4,5)P3 accumulation leads to directional cell migration mediated by the extracellular signal.

2.

Eukaryotic Chemotaxis is Limited by Noise

To detect the direction of a gradient, a single chemotaxis cell may have to detect and amplify the difference in the concentration at the front and back of the cell, so that the cell can activate the motile apparatus. Chemotaxis cells can show directional motion even in a 1% to 5% gradient across their cell length, which is typically 20 to 30 µm. Dictyostelium cells can exhibit chemotaxis in a wide concentration range from 10 pM to 1 mM. Chemotaxis is the most accurate at a chemoattractant concentration in the range of tens of nM. At this concentration, the number of cAMP molecules on the surface of a single cell is calculated to be approximately 16,000, according to the affinity between the receptor and ligand, and the difference between the front and back halves of the cell is approximately 100 for a 2% gradient across 10 µm. At 1 nM cAMP, the accuracy of chemotaxis is about half of the maximum value. At this concentration, the number of cAMP molecules on the surface of a single cell is approximately 300, and the difference between the front and back is less than 10 for a 2% gradient. These small differences between front and back are the chemotaxis signal that the cell has to detect. Through the signal transduction system, such a small signal may increase the frequency of protrusive activities in the membrane region facing the higher chemoattractant concentration side, leading to directional cell migration. One of the characteristics that distinguish living systems from manmade systems is their stochastic nature. Molecular machines of proteins perform the processes in living systems. Thermal noises are essential for

November 21, 2012

190

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

the function of these molecular machines, but the signal transduction mechanisms operated by these stochastic molecular machines inevitably exhibit a stochastic nature. This characteristic originates from the large number of degrees of freedom as in the case of thermodynamics and statistical mechanics. As a result of this stochastic nature, the concentration of proteins and other factors exhibit stochastic fluctuations with time in cells and tissues. Therefore, living systems are considered to be inherently stochastic systems. The association and dissociation reactions between receptors and chemoattractant ligands are also stochastic processes. Consequently, the number of ligand molecules on the surface of a cell shows stochastic fluctuations (noise) over time. The standard deviation of the probability distribution of the binding-ligand number is estimated to be approximately 120 at 25 nM cAMP and approximately 20 at 1 nM. Comparing these numbers with the average difference between the front and back calculated above, the noise strength is comparable to or even larger than the average numbers. In fact, single cell molecular imaging has demonstrated the presence of time intervals during which the number of ligand molecules in the back region could be larger than in the front region.3 The signal perceived by cells is transduced in downstream reactions. Because these reactions are also stochastic processes, the sufficiently large signal perceived at the receptor from the environment may deteriorate in the downstream stochastic signaling processes. In the case of the Dictyostelium cell, it is known that motile activity is not necessary for gradient information processing. Even when the motile activity is restricted by inhibiting actin polymerization, some molecules form a gradient inside the cell along the external chemoattractant gradient. For example, PtdIns(3,4,5)P3 forms a positive gradient, while phosphatase and tensin homolog (PTEN) form a negative gradient along the external gradient.2,4,5 The formation of internal gradient indicates that this chemotactic system can detect the spatial difference in the chemoattractant concentration without motile activities through a distribution of the chemoattractant ligand on the cell surface, i.e., a distribution of receptor occupancy. If the gradient of the chemoattractant ligands bound to the cell surface is almost constant over time, the cell can certainly detect the gradient information. In contrast, if the gradient on the cell surface shows large stochastic fluctuations, detection of the signal could be difficult. The difficulty may be roughly proportional to the ratio of the strength of stochastic fluctuations in the gradient to the average value. For the downstream reactions, the same consideration may be applicable.

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 191

Here, we study the signal and noise propagation based on the spatial sensing mechanism, in which the chemotactic signals are the spatial differences in the receptor occupancy of receptor (cAMP receptor cAR1) and the subsequent activation of second messenger on membrane along cell body.6 The chemoreceptor is activated upon binding of the chemoattractant ligand, which leads to the production of activated second messenger. The occupied and activated receptor and the activated second messenger are denoted by R* and X, respectively (Fig. 2(a)). The concentration of occupied receptor and the amount of second messenger are given by S and X, respectively. For the case of Dictyostelium cells, we suppose that X is the activated G-protein on the membrane. The external chemoattractant gradient ∆L induces a gradient of receptor occupancy ∆R∗ between the anterior and posterior halves of chemotactic cells, which then leads to the difference in the amount of the second messenger X, ∆X = Xa − Xp where Xa and Xp are the amounts of the two regions (Fig. 2(a)). The difference ∆X formed inside the cell is considered as an internal signal, which induces motile activity such as actin polymerization. We performed a stochastic numerical simulation of this signaling process for the case of a Dictyostelium cell. The time series of the difference ∆X obtained numerically is plotted in Fig. 2(a). The difference sometimes exhibits a negative value, indicating that the spatial signal can be reversed against the external chemoattractant gradient by stochastic fluctuations in the process of ligand binding and second messenger activation. To consider the accuracy of gradient sensing, we first study the stochastic fluctuation in the signal ∆X.8 –10 For a shallow chemoattractant gradient, let us consider a small temporal deviation from the average ∆X, ∆x , i.e., ∆X = ∆X + ∆x. The linear evolution equation for ∆x can then be written as ∆x = γ∆r − Γ∆x + σζ ∆ζ(t), dt

(1)

where ∆r is the difference in the receptor activation in the front and back halves of the cell and ∆ζ = ζa − ζp with the noises in the front and back halves of the cell, respectively. Here, γ and Γ are coefficients, and σζ is the noise strength. The noise ∆ζ is delta-correlated with zero mean and 2 ∆ζ(t)∆ζ(t ) = 2δ(t − t ). The noise intensity σ∆X of ∆X is approximately calculated as the variance given by
2 σ∆X = gX + g 2

X R∗

2

τR σ2 ∗ , τ + τR ∆R

(2)

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

192

(a)

L

ligand (L) receptor-ligand (S)

1

receptor

ΔL

Ligand

2nd messenger (Y)

2

position

X activated 2nd messenger (X)

Activated 2nd messenger

ΔX

2000 1000

∆X

0

position

-1000 0

10

20

30

40

50

60

70

80

90

Posterior

Anterior

100

time (sec)

Signal

:

Signal-to-noise ratio (SNR) :

0.5 0.4

SNR

ΔX* ΔX*/σΔX* 1.0

SNR (theoretical) Experimental data

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0

10-12

10-11

10-10

10-9

10-8

10-7

10-6

Accuracy

(b)

ΔR* ΔR*/σΔR*

0

(c) 1

SNR (extrinsic) SNR (intrinsic) SNR (total)

SRN

0.8 0.6 0.4 0.2 EC50

10-12

10-11

10-10

10-9

10-8

Kd

10-7

10-6

L(M) Fig. 2. The Signal-to-noise ratio of chemotaxis signal. (a) (Upper left) A stochastic model of chemotaxis signal transduction. L is the chemotaxis ligand, R is the inactive receptor, R* is the activated receptor, X is the inactive second messenger, and X* is the activated second messenger. (Right) A schematic illustration of the gradient of chemoattractant and the distribution of the second messenger. (Lower left) The time series of the difference in the second messenger concentration between front and back of the cell obtained by numerical simulation. The difference sometimes exhibits a negative value. (b) The signal-to-noise ratio (line) and the accuracy of chemotaxis (circle) obtained experimentally,7,8 showing a good agreement with each other without curve fitting.

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 193

where X = Xa + Xp and R∗ = (Ra∗ + Rp∗ )/2. Here, g is the gain of G-protein reaction, τR is the time constant of the noise in the activated receptor, and τ is the time constant of the G-protein reaction. Noting that ∆X = gX∆R∗ /R, the relative noise intensity defined by the ratio between the variance and its squared average (σ∆X /X)2 is given by


σ∆X ∆X

2 =

1 gX




R∗ ∆R∗

2 +

τR τ + τR




σ∆R∗ ∆R∗

2 .

(3)

The second term is the contribution of the noise generated at the receptor-ligand reaction, and the first term is the noise generate at the G-protein reaction even if the receptor-ligand noise would be absent. The accuracy of gradient sensing at the level of the second messenger is considered as the signal-to-noise ratio (SNR) ∆X/σ∆X , which is obtained by the inverse of the square root of Eq. (3). Here, the first and second terms are related to noises in the G-protein reaction and the receptor-ligand reaction, respectively. In Fig. 2(b), the SNR ∆X/σ∆X is plotted as a function of the average chemoattractant L. To calculate the SNR, the parameter values for Dictyostelium cells were used.8 The SNR of chemotactic signals attains a maximum at the ligand concentration between the affinity of the receptor, Kd , and the EC50 concentration, where the G-protein activation reaches its half-maximum. In Fig. 2(c), the intrinsic and extrinsic noise contributions to the SNR are plotted. In the lower ligand concentration range, the SNR is determined mainly by the contribution of the extrinsic noise. This result indicates that the fluctuations in active receptor predominantly affect the quality of the chemotactic signals. In the higher ligand concentration range, the SNR deteriorates with an increase in ligand concentration because receptors are gradually saturated. This makes them unable to produce the large differences in second messenger concentrations between the anterior and posterior halves of cells, leading to an increase in intrinsic noise. The chemotactic accuracy of Dictyostelium cells has been measured experimentally by Fisher et al.7,11 The dependence of chemotactic accuracy on the ligand concentration exhibits a profile quite similar to our calculated SNR, shown in Fig. 2(b). In the experiment, the accuracy of chemotaxis attained a maximum value at 25 nM cAMP. This optimal value is almost the same as the concentration at which the SNR reaches the maximum. The agreement between the SNR and chemotactic accuracy indicates that directional sensing is limited by the inherently generated stochastic noise during the transmembrane signaling of receptors. Note that Eq. (3) does not depend on a particular detail of the spatial sensing mechanism and

November 21, 2012

194

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

can be applied to other systems. In fact, similar dependence of chemotactic accuracy has been observed in mammalian leukocytes and neurons.12,13 When the chemoattractant concentration L is sufficiently small compared to the receptor’s √ dissociation constant, the SNR proportionally changes to SN R ∝ ∆L/ L. If the cell requires a signal exceeding a threshold SNR to detect chemical gradients, there exists a threshold gradient ∆Lthreshold for chemotaxis, which can be dependent on L. Suppose that this threshold SNR is independent √ of the ligand concentration L. Then, we obtain the relation ∆Lthreshold ∝ L, which has also been obtained experimentally.14 The result indicates that stochastic properties of receptors at the most upstream stages of the signaling pathway determine the chemotactic accuracy of the cells. The noise generated at the receptor level limits the precision of directional sensing, suggesting that receptor-G protein coupling and its modulation have an important role in the chemotaxis efficiency of the cells. 3.

Signal Transduction and Spontaneous Cell Motility

As shown so far, a chemoattractant gradient elicits chemotactic motions by activating motile apparatus at the cell side facing the source of the chemoattractant cAMP. However, the chemoattractant gradient is not necessary for cell motion. In the case of Dictyostelium cells, in the absence of cAMP, cells exhibit spontaneous motility in random directions. Although the preferential direction is absent in isotropic conditions, the protrusive activities of individual cells are not disorganized. The protrusive activities have specific properties, such as the lifetime of the activities and the time interval between the activities. Although these properties may show a slight dependence on the cAMP concentration and the steepness of the gradient, these activities should have their own intrinsic dynamics that may be mostly independent of the extracellular conditions. That the protrusive activities of cells continue to work in the absence of an extracellular gradient implies the presence of some signals that initiate the activities at some particular position and time inside the cell. In particular, a mutant cell lacking the trimeric G-protein (Fig. 1) shows normal motility but not the directional motion of chemotaxis.15 This phenomenon indicates that motility is not necessarily induced by the extracellular chemoattractant gradient, and the signal transduction network downstream of trimeric G-protein may be responsible for this spontaneous motility. What is the mechanism that enables cell motility but does not depend on extracellular information?

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 195

4.

Spontaneous Pattern Formation in Phosphatidylinositol Lipid Signaling Reaction

Under a cAMP gradient, PtdIns(3,4,5)P3 has been shown to localize in the membrane region facing the source of cAMP and induce actin polymerization at the cell periphery. What is the distribution of PtdIns(3,4,5)P3 along the membrane in the absence of a cAMP gradient, when cells exhibit spontaneous cell motility? We observed the dynamics of the spatiotemporal distributions of PtdIns(3,4,5)P3 and PTEN within individual cells in the absence of cAMP using fluorescently labeled probes.16 In Figs. 3(a) and 3(b), the periphery of a cell is the membrane region, along which PtdIns(3,4,5)P3 and PTEN were distributed non-uniformly and spontaneously produced a localized pattern. Moreover, the domain was not stationary over time but traveled along the plasma membrane. The traveling direction was randomly distributed over the cell population and lasted for a long time. Here, to avoid the effect of cell shape changes on the reaction–diffusion field, the cells were treated with the actin polymerization inhibitor Latrunculin A, which made the cell shape spherical, and the effect of the temporal dependence of boundary conditions could be eliminated. The temporal auto-correlation function at individual spatial points for PtdIns(3,4,5)P3 and PTEN indicated that the period of changes in these concentrations was approximately 200 seconds. The cross-correlation function between two intensities exhibited a negative value when the time delay was 0, indicating that the two concentrations are anti-correlated. Moreover, the lowest value was found not at t = 0 but at t = 10 sec. This result indicates a delay between the changes in PtdIns(3,4,5)P3 and PTEN. Because the cross-correlation was calculated between the PtdIns(3,4,5)P3 intensity at a given time and the PTEN intensity at the time with delay t, the shift of the peak to 10 seconds indicates that the change in the PTEN intensity 10 seconds after the change in the PtdIns(3,4,5)P3 intensity. Considering that PtdIns(3,4,5)P3 is the substrate for the phosphatase PTEN, the result that the change in the PtdIns(3,4,5)P3 concentration occurred first may be counterintuitive. The result may not be explained by a simple relationship between an enzyme and its substrate. As shown in Fig. 3(e), this delay in the concentration changes may indicate that in the phase space of the PtdIns(3,4,5) concentration in the horizontal axis and the PTEN concentration in the vertical axis, the trajectory moves in the clockwise direction on a closed orbit. As shown in Fig. 3(f), we obtained the average dynamics, which showed a characteristic crescent shape in the phase space of the two concentrations and moved in the clockwise direction.16 The

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

196

(b)

0:00

0:25

0:50

1:15

1:40

2:05

2:30

2:55

3:20

3:45

4:10

4:35

Time (min)

(a)

Angle (π) 00

0

1

2

20

PTEN-TMR

PHAkt/PKB-EGFP

(e)

Autocorrelation

1

Angle (π) 2

10

30

(c)

1

0

-0.5

PHAkt/PKB(a.u.)

900

1200

700 600 500

PtdIns(3,4,5)P3

150

60

800 800 400 150 300 Time (s)

400 0

40 20

0s

400 500 600 700 800 900 1000 1100 1200

PH

(a.u.)

(g)

Direction of wave propagation 1200 back 1100 1000 900 front 800 700 600 500 0

900 front

800 700

back

600 500 1

2

PTEN (a.u.)

0 Time (s)

800

PTEN (a.u.)

PTEN

0

-0.5 -150

(f)

400

Cross-correlation

0.5

C(t)

200 Time (s)

PHAkt/PKB(a.u.)

(d)

0

PTEN(a.u.)

C(t)

0.5

400

Angle (π)

Fig. 3. Self-organization signaling in phosphatidylinositol lipid signaling reaction. (a) The temporal change in the spatial pattern of PtdIns(3,4,5)P3.16 (b) The spatiotemporal plot (kymograph) of PTEN (left) and PtdIns(3,4,5)P3 (right) intensities.16 (c) The auto-correlation functions of PTEN and PtdIns(3,4,5)P3 intensities.16 (d) The crosscorrelation function between PtdIns(3,4,5)P3 and PTEN. (e) A schematic illustration of trajectory expected from the correlation functions in the phase space of PtdIns(3,4,5)P3 and PTEN. (f) The average dynamics in the phase space constructed from the experimental data. (inset) The average time series.

anti-correlation between the two intensities could be well described by a reciprocal relation. What is the mechanism that induces instability in the uniform state on the membrane without an extracellular stimulus and produces an asymmetric distribution?

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 197

On the basis of our data analysis and what is known about molecular biology, we developed a reaction-diffusion model of the self-organized phosphatidylinositol reaction.16,21 The reactions consist of the enzymatic activities of PI3K and PTEN, which catalyze PtdIns(3,4,5)P3 and PtdIns(4,5)P2 production, respectively. With the supply reaction for PtdIns(4,5)P2 and the degradation pathways for both lipids, the reaction scheme is given by k

− → PtdIns(4, 5)P2 PtdIns(3, 4, 5)P3

VPI3K K [PIP2] PI3K +[PIP2]

−−−−−−−−−−−−→ VPTEN [PTEN]

[PIP3] KPTEN +[PIP3]

−−−−−−−−−−−−−−−−−−−→

PtdIns(4, 5)P2 , PtdIns(3, 4, 5)P3 , PtdIns(4, 5)P2 ,

λPIP2 [PIP2]

PtdIns(4, 5)P2

−−−−−−−−→

PtdIns(3, 4, 5)P3

−−−−−−−−→

λPIP3 [PIP2]

(4) ,

KPIP3 [PIP2] Vass [PTENglycol ] K PIP3 +[PIP3] KPIP2 +[PIP2]

PTENcytosol −−−−−−−−−−−−−−−−−−−−−−−−−−−−→ PTENmembrane , PTENmembrane

λPTEN [PTEN]

−−−−−−−−−→

PTENcytosol .

Here, k, VPTEN , KPTEN , VPI3K , KPI3K , λPIP2 , λPIP3 , Vass , KPIP3 , KPIP2 , and λPTEN are reaction constants; [PIP3], [PIP2], and [PTEN] are the membrane concentrations of PtdIns(3,4,5)P3, PtdIns(4,5)P2, and PTEN concentrations, respectively; and [PTENcytosol ] is the cytosol PTEN concentration. For the reaction between the cytosol PTEN (PTENcytosol ) and the membrane PTEN, because PTEN has a PtdIns(4,5)P2 binding domain, we suppose that the PTEN membrane binding rate increases with the concentration of PtdIns(4,5)P2. Furthermore, a statistical analysis suggested a reciprocal relation between the PtdIns(3,4,5)P3 and PTEN membrane concentrations.16 Thus, we postulated that PtdIns(3,4,5)P3 negatively regulates the membrane concentration of PTEN (PTENmembrane). The diffusion constant in the cytosol is much faster than that on the membrane. Thus, the PTEN concentration in the cytosol is considered to be uniform. Then, [PTENcytosol ] = [PTENtotal ]− χ[PTEN], where [PTENtotal ] is the total PTEN concentration, [PTEN] is the average of membrane PTEN concentration and χ is the constant used to change the membrane concentration to the cytosol concentration. From a mathematical point of view, this conservation in the total PTEN concentration introduces a global coupling effect on the model. The fast diffusion in the cytosol can be considered to coordinate signaling reactions at individual local regions on the membrane. In the model, we consider the diffusion effect

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

198

PtdIns(4,5)P2

PtdIns(3,4,5)P3 plasma membrane

positive regulation PI3K PI P

P

P

P

PTEN

positive effect

P

cytosol

negative regulation PH domain

PTEN

60 40

20

30 20 10 100 200 Time (s)

20 0s

0

Time (min)

30

400 300 200 100 0

PTEN

PTEN (molecules/µm2)

T=176 s

PtdIns(3,4,5)P3

(a) Angle (π) 0 1 2

10

20

10 100

200

300

400

PtdIns(3,4,5)P3(molecules/µm2) (b)

30 PtdIns(3,4,5)P3

(c)

Fig. 4. A mathematical model of the self-organization in phosphatidylinositol lipid signaling reaction. (a) The reaction scheme used in the mathematical model. (b) The average dynamics obtained from the data of numerical simulation, which was obtained by the same statistical method that was applied for the experimental data as shown in Fig. 3(f). (c) The space and time plot of the PtdIns(3,4,5)P3 intensity obtained by the numerical simulation.

for PtdIns(4,5)P2 and PtdIns(3,4,5)P3 with diffusion constant D, whereas we neglect the diffusion process for PTEN on the membrane because it is much slower than that of the lipid molecules. Figures 4(b) and 4(c) show the results of stochastic numerical simulations for the reaction scheme (4). As shown in Fig. 4(b), the dynamics of the PtdIns(3,4,5)P3 and PTEN concentrations obtained numerically exhibited the same characteristics that we found experimentally. In the numerical simulation, the uniform state is destabilized by changes in some parameter values, and a variety of spatiotemporal dynamics, such as traveling wave patterns and standing wave patterns, were obtained. Moreover, even when the uniform state is stable, stochastic noises trigger

November 21, 2012

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

Fluctuations and Symmetry Breaking in Signaling System of Dicyostelium Cells 199

an increase in the PtdIns(3,4,5)P3 concentration, leading to the formation of localized domains, suggesting that the system is an excitable media.

5.

Conclusion

The spontaneous symmetry breaking observed in the PtdIns(3,4,5)P3 dynamics and the formation of a localized domain on the plasma membrane are the potential candidates that induce the protrusive activities of cells in the absence of a chemoattractant. Similar spontaneous activities have been reported in Dyctiostelium cells under different conditions17,18 and mammalian cells.19,20 Under a shallow gradient of extracellular cAMP, the properties of the formation of the PtdIns(3,4,5)P3 domain on the plasma membrane in Dyctiostelium cells may be almost the same as that in the absence of a cAMP gradient. However, the extracellular gradient may modulate the probability distribution of formation. The probability of domain formation in the membrane region facing the higher cAMP concentration could be higher, while the probability could be lower on the opposite side. As a result, the motion of cells could be biased toward the cAMP source. The spontaneous symmetry breaking may lead the formation of a new variable that specifies the direction. Such a new variable formed through spontaneous symmetry breaking could have a robust property against stochastic fluctuations. Thus, the spontaneous formation of a PtdIns(3,4,5)P3-enriched domain could stably store the directional information, which may provide a basis for the directional sensing of chemotactic cells.

References 1. G. Di Paolo and P. De Camilli, Nature 443, 651 (2006). 2. P. J. Van Haastert and P. N. Devreotes, Nat. Rev. Mol. Cell Biol. 5, 626 (2004). 3. Y. Miyanaga, S. Matsuoka, T. Yanagida and M. Ueda, BioSystems 88, 251 (2007). 4. C. A. Parent and P. N. Devreotes, Science 284, 765 (1999). 5. C. A. Parent, Curr. Opin. Cell Biol. 16, 4 (2004). 6. C. Janetopoulos, T. Jin and P. N. Devreotes, Science 291, 2408 (2001). 7. P. R. Fisher, R. Merkl and G. Gerisch, J. Cell Biol. 108, 973 (1989). 8. M. Ueda and T. Shibata, Biophysical J. 93, 11 (2007). 9. T. Shibata, Noisy signal transduction in cellular systems, Cell Signaling Reactions: Single-Molecule Kinetic Analysis, in ed. Y. U. M. Sako (SpringerVerlag, 2010).

November 21, 2012

200

13:2

9in x 6in

b1449-ch09

Engineering of Chemical Complexity

T. Shibata

10. T. Shibata and M. Ueda, BioSystems 93, 126 (2008). 11. L. Song, S. M. Nadkarni, H. U. Bodeker, C. Beta, A. Bae, C. Franck, W. J. Rappel, W. F. Loomis and E. Bodenschatz, Eur. J. Cell Biol. 85, 981 (2006). 12. S. H. Zigmond, J. Cell Biol. 75, 606 (1977). 13. W. J. Rosoff, J. S. Urbach, M. A. Esrick, R. G. McAllister, L. J. Richards and G. J. Goodhill, Nat. Neurosci. 7, 678 (2004). 14. P. J. Van Haastert and M. Postma, Biophysical J. 93, 1687 (2007). 15. C. L. Manahan, P. A. Iglesias, Y. Long and P. N. Devreotes, Annu. Rev. Cell Dev. Biol. 20, 223 (2004). 16. Y. Arai, T. Shibata, S. Matsuoka, M. J. Sato, T. Yanagida and M. Ueda, Proc. Natl. Acad. Sci. USA. 107, 12399 (2010). 17. M. Postma, J. Roelofs, J. Goedhart, H. M. Loovers, A. J. Visser and P. J. Van Haastert, J. Cell Science 117, 2925 (2004). 18. M. Postma, J. Roelofs, J. Goedhart, T. W. Gadella, A. J. Visser and P. J. Van Haastert, Mol. Biol. Cell 14, 5019 (2003). 19. C. Arrieumerlou and T. Meyer, Dev. Cell 8, 215 (2005). 20. M. C. Weiger, C. C. Wang, M. Krajcovic, A. T. Melvin, J. J. Rhoden and J. M. Haugh, J. Cell Science 122, 313 (2009). 21. T. Shibata, M. Nishikawa, S. Matsuoka and M. Ueda, J. Cell. Sci. (2012) in press. doi:10.1242/jcs.108373.

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Chapter 10 MECHANOCHEMICAL PATTERN FORMATION IN THE POLARIZATION OF THE ONE-CELL C. ELEGANS EMBRYO Justin S. Bois∗ and Stephan W. Grill† ∗

Department of Chemistry and Biochemistry, University of California, Los Angeles, CA, 90095, USA † Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Straße 38, D-01187 Dresden, Germany † Max Planck Institute of Molecular Cell Biology and Genetics Pfotenhauerstraße 108, D-01307 Dresden, Germany Cellular polarity refers to the uneven distribution of certain proteins and nucleic acids on one half of a cell versus the other. Polarity establishment is often an essential process in the development, being responsible for cell differentiation upon division of the polarized cell. The one cell embryo of the nematode Caenorhabditis elegans is a classic model system for the study of polarity. Interestingly, distribution of polarity proteins is accompanied by directional movements of the cell cytoskeleton in this system. In addition to undergoing diffusion, the polarity proteins are transported by these movements. Thus, polarization is achieved by both mechanical and chemical means. We discuss our current understanding of this process in the C. elegans model system. We also discuss more general consequences of mechanochemical coupling in morphogenesis.

Contents 1. Introduction . . . . . 2. Polarization of the C. 3. Mechanics . . . . . . 4. Biochemistry . . . . . 5. Mechanochemistry . 6. Conclusions . . . . . References . . . . . . . . .

. . . . . elegans . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . zygote: The . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

. . . . Basics . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

202 203 204 207 208 211 211

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

202

1.

b1449-ch10

Introduction

The spatiotemporal patterning of cells and tissues is a key process in morphogenesis (Greek for “form generation”) of a developing organism. Patterns of molecular components can arise through processes that can be described in a reaction–diffusion framework.1 However, morphogenesis goes beyond simple patterning of molecules, since it involves the mechanical reshaping and restructuring of cells and tissues. Importantly, patterned molecules can regulate the mechanical properties of the cells and tissues in which they reside through active nonequilibrium processes, e.g. by exertion of stress by motor proteins. Furthermore, components of these regulatory pathways are transported by elastic deformation and viscous flow arising from these active mechanical processes.2 Thus, the establishment of form in a developing organism depends on an interplay between molecular biochemistry1 and mechanics.3,4 In some instances one can successfully decouple the biochemistry from the mechanics,1 but we are learning more and more that this is not generally possible.2 Therefore, characterization of biochemical regulation, mechanical stresses, and the interplay between the two are crucial to fully understand the fundamental mechanisms by which patterning, structure, and form arise in development. In this chapter, we review our understanding of the mechanochemical establishment of cellular polarity in the one-cell Caenorhabditis elegans (C. elegans) embryo, a classic model system for study of morphogenesis. Polarity establishment in this system occurs in a highly stereotyped and reproducible fashion, ideal for quantitative investigations. PAR proteins (PARtitioning-defective, conserved throughout the animal kingdom, originally identified in C. elegans 5 ) segregate into two groups that inhabit mutually exclusive membrane domains, thereby establishing the directional anterior–posterior (AP) axis of the organism. These PAR proteins are in turn responsible for the spatial regulation of downstream polarity pathways, including those responsible for maintaining the stem-cell character of the germ-line cell lineage and establishing the major body axis of the organism. Indeed, much of what we know about polarity establishment in general comes from studies of C. elegans,6,7 but only in recent years have the coupling between mechanics and biochemistry been elucidated. We first give an overview of the polarity establishment process in C. elegans. Next, we describe the mechanical and biochemical processes of polarization, insofar as they may be decoupled. Finally, we incorporate mechanochemical coupling and discuss further steps necessary to complete the analysis of polarity establishment. Throughout, we demonstrate that a combination of quantitative experimental measurement and theoretical

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Mechanochemical Pattern Formation in Polarization

A

203

P

10 m

PAR-6/PAR-2

Fig. 1. A one-cell C. elegans embryo that has undergone polarization. The embryo has the shape of an egg, and polarization occurs through asymmetric localization of PAR proteins on its surface at the membrane. The anterior PAR domain is characterized by the localization of PAR-6 (red), the posterior PAR domain by the localization of PAR-2 (green). The anterior–posterior (AP) axis is drawn in white. Courtesy Nathan Goehring.

modeling of both biochemistry and mechanics elucidate the means by which polarity is established.

2.

Polarization of the C. elegans zygote: The Basics

Much of the dynamics relevant to polarization occur at the cell membrane and in a thin (≈1–2 µm) membrane-associated layer of crosslinked polymers known as the cell cortex. The cortex consists of a meshwork of actin on which the motor protein myosin exerts contractile stresses. For the purposes of this discussion, we focus on the cortical region and break down the first embryonic cell cycle into three phases.6 –9 In the initial establishment phase, the anterior PAR proteins (PAR-3, PAR-6, and atypical protein kinase C, collectively aPAR) occupy the entire cell membrane, while the posterior PAR proteins (PAR-1, PAR-2 , and LGL, collectively pPAR) are present in the cytoplasm, but absent from the membrane. The actomyosin cortex is uniformly contractile and absent of concerted flow. A cue from the sperm-donated microtubule organizing center at the posterior pole of the zygote, serves to locally down-regulate myosin activity, though the details of this cue are unknown.10 This results in a stress imbalance in the cortex, giving rise to an anteriorly-directed cortical

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

204

flow.8,11,12 The flow is long-ranged, persisting over much of the periphery of the zygote,13 and transports aPARs toward the anterior, independent of the biochemical interactions between anterior and posterior PARs.11,14 A small region of the membrane on the posterior pole, freshly depleted of aPAR, becomes inhabited with a small domain of the pPAR complex. Flows of the cortex continue as the pPAR domain expands to occupy approximately half of the cellular surface (the posterior side), with the aPAR complex occupying the other half. Once polarity is established, the zygote enters a maintenance phase during which the two PAR domains persist in a stable state, each occupying roughly half of the cell surface. Actively generated stresses of the cortex are greatly reduced in this phase, with the consequence that long-ranged flows cease. At this point, the cortex becomes dispensable for maintaining the segregation of the PARs.8,9 Instead, persistent asymmetry over the maintenance phase depends on biochemical interactions between aPARs and pPARs.6 –9 Finally, the correction phase coincides with cytokinesis, during which the boundary between the PAR domains is displaced to match the ingressing cleavage furrow, thereby ensuring differential inheritance of PAR proteins by the two daughter cells. Actomyosin contraction and cortical flows reappear to drive cytokinesis and alignment of the PAR boundary with the ingressing furrow.

3.

Mechanics

We first describe the mechanics governing cortical flows of establishment phase, independent of PAR biochemistry. The cortex can be described at different length, time, and force scales.15 Although its molecular constituents are very small (≈5 nm for an actin monomer, the microscopic scale), individual actin filament length is of intermediate order (hundreds of nanometers to a few microns, the mesoscopic scale), and the components collectively interact to form a higher-order network that spans the entire cell (several tens of microns, the macroscopic scale). At a sufficiently coarse-grained level, the cortex can be described as a bulk material using a continuum description, omitting reference to details of molecular origin and structure. From this point of view, the cortex is a thin film of a highly dynamic viscoelastic gel, which actively generates forces (microscopically, due to the presence of ATP-consuming myosin motors and ATP-dependent treadmilling).16,17 The active polar gel theory13,17 –20 represents such a coarse-grained physical description. This theory is a versatile tool to describe the physical principles governing the cortex as

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Mechanochemical Pattern Formation in Polarization

205

a whole. Most importantly, it has brought understanding concerning the mesoscale biophysical laws for the mechanics of actomyosin deformation and flow in development.21 In what follows, we apply a simplified version of the theory to characterize cortical dynamics. The alignment of the filaments in the cortex during establishment phase is essentially isotropic in the plane. Because the cortex is confined to a thin sheet, active stresses result in in-plane contraction.20 Gradients in these active stresses drive deformation of the cortex. These deformations correspond to elastic relaxations and viscous flow on short and long time scales, respectively. In particular, cortical flows occur over long time scales and low shear rates, so the flow is viscous. Noting that the observed dynamics are azimuthally symmetric and neglecting curvature, the governing equations are σ(x) = σactive (x) + η ∂x v, ∂x σ = γv ⇒ ∂x σactive =

(1) −η∂x2 v

+ γv,

(2)

where x is the coordinate along the AP axis, η is the cortical viscosity, v is the flow velocity, σ is the total stress present in the cortex, and σactive is the contractile active stress in the cortex. The constitutive relation (1) says that the total stress is given by the sum of active and viscous stresses. The equation of motion (2) results from being in the low Reynolds number regime with frictional resistance to motion of the cortex against the membrane and cytoplasm (with friction coefficient γ). Characterization of the driving forces of the cortical flows requires quantification of active stress gradients, which necessitates measurements of the sub-cellular distribution of cortical stress. This is achieved by ablation of a line in the cortex with a pulsed UV laser (COrtical Laser Ablation (COLA), Fig. 2) and measurement of the ensuing recoil of the cortex. The recoil dynamics are rapid, occurring within a few seconds of ablation, and are fast compared to the viscoelastic relaxation time, so the cortex is essentially elastic for a COLA experiment. The simplified elastic active polar gel theory describes the recoil: ζ y˙ = −ky − σactive + σ,

(3)

where y is the position of the edge of the ablated region, σ is the total stress present in the system, and σactive is the contractile active stress. The elastic stiffness of the cortex is given by k, and ζ is the damping coefficient describing frictional resistance to the recoil. In a COLA experiment, the stress is given by σ(t) = σ0 (1 − U (t)),

(4)

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

206

(a)

(b)

Fig. 2. Laser ablation of the C. elegans cortex is used to measure cortical tension for the purpose of characterizing its local mechanical state. The cortex is cut along the blue line. (a) Pre-cut (top) and post-cut (bottom) image in a zygote with GFP-labeled non-muscle myosin (NMY-2::GFP). A denotes anterior and P posterior. (b) Enlarged overlay. Arrows indicate the resultant recoil and show displacements between the pre-cut (purple) and post-cut (green) frames. Scale bars, 5 µm. Adapted from Ref. 13.

where σ0 is the stress prior to ablation and U (t) is the unit step function, indicating that upon ablation the cortex can no longer support stress. We may solve (3) and (4) for the velocity of the recoil as v(t) = y(t) ˙ =−

σ0 −t/τ e , ζ

(5)

with τ ≡ ζ/k. Therefore, the initial (t = 0) velocity of the recoil is proportional to the total stress present in the cortex immediately prior to ablation. Varying the position and direction of the ablation line enables measurement of positional and directional COLA recoil velocities, providing a comparison of stresses. Ablation of a line along the AP axis reveals that the azimuthal component of the stress is greater in the anterior than in the posterior. Since the cortical flow is mostly directed anteriorly with negligible azimuthal velocity or gradients, there are no viscous stresses in that direction, so the total stress is equal to the active stress. This verifies existence of a gradient in contractile active stress that serves to pull the flowing cortex toward the anterior. By performing ablations in the orthogonal direction, we find

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Mechanochemical Pattern Formation in Polarization

207

that there are no gradients in the component of the stress along the AP axis. The lack of a total stress gradient indicates that viscous dissipation dominates over frictional losses, by (2). Furthermore, (2) reveals that the distance over which
the flow velocity decays as a result of a local stress gradient is  ≡ η/γ. Since viscous stresses are dominant over friction, this length scale is large. Therefore, local regulation of active stress results in large-scale motion.

4.

Biochemistry

Much has been learned in the past decade concerning the molecular constituents of both the anterior and the posterior PAR complexes, and their modes of interaction.6 –9 Key to the ability to form stable domains appears to be the mutual antagonism between these two groups of proteins; aPARs on the membrane exclude pPARS from the membrane, and vice versa. This yields a cell membrane that tends to exist in one of two states: an anterior-like state enriched in aPARs and a posterior-like state with pPARs. To characterize the PAR dynamics on the membrane, we may write down their respective reaction–diffusion-advection (RDA) equations. ∂t A = DA ∂x2 A − ∂x (vA) + kon,A Acyto − koff,A A − kAP P α A,

(6)

∂t P = DP ∂x2 P − ∂x (vP ) + kon,P Pcyto − koff,P P − kP A P Aβ ,

(7)

where A and P denote the concentration of the aPAR and pPAR complexes, respectively. Going from left to right, the terms on the righthand side represent diffusive transport, advective transport, association to the membrane, dissociation from the membrane, and mutual antagonism, respectively. The combined total concentration of the PAR complexes in the cytosol and on the membrane is conserved and is used to determine the cytoplasmic concentrations, Acyto and Pcyto , of the complexes. Naturally, we seek to measure as many parameters as possible in the RDA equations. The relative total amounts of aPARs and pPARs may be determined by measuring intensities of fluorescently labeled species.22 By waiting until maintenance phase, when the cortical flows have ceased and the advection terms are negligible, and applying RNA interference23 to effectively remove the pPAR complex from the system, measurement of DA , kon,A , and koff,A is possible. This is achieved using a quantitative Florescence Recovery After Photobleaching (FRAP) technique. Molecules in the plane of the membrane are photobleached in squares of varying sizes, and spatiotemporal characteristics of the return of fluorescence into the

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

208

bleached area are recorded and fit with an appropriate reaction–diffusion model.24 Analogous measurements are made for the pPAR parameters. With these measurements in hand, we can approximate the parameters relating to the mutual antagonism as those commensurate with observed domain shapes in maintenance phase. Finally, the cortical flow velocity during establishment phase is measured using particle image velocimetry on GFP-labeled myosin.13 We therefore have a complete picture of the RDA dynamics, which compares well with experimental measurements (Fig. 3).22,25 The RDA equations elucidate several phenomena about the PAR polarization system. First, stability analysis about the initial steady state in which the aPAR complex homogeneously occupies the membrane reveals stability in the absence of advection.22,26 This implies that a large perturbation, in this case by strong cortical flows accompanied by advective transport, is necessary for polarization. Second, the P´eclet number, the ratio of the time scales of diffusive and advective transport, is close to unity for both the aPAR and pPAR complexes. Therefore, both diffusive and advective transport are important to the dynamics. Finally, the existence of a single nonhomogeneous steady state in which the whole zygote is split in two (as opposed to other patterns such as spots or stripes) relies crucially on the conserved pools of total PAR protein amounts.22,27 Further, the relative size of the limiting pools determines the size of the respective PAR domains; the more anterior PARs or posterior PARs present in the zygote, the larger the size of the respective PAR domain. This model prediction was tested using RNAi rundown experiments together with over- and under-expression studies, which revealed that PAR amounts indeed specify domain size in the embryo.22 5.

Mechanochemistry

By analyzing the PAR dynamics at maintenance phase and measuring the cortical flow velocity of PAR biochemistry independently, we could decouple the biochemistry from the mechanics. We artificially coupled the two by inserting the measured flow velocity into the RDA equations to get the dynamics depicted in Fig. 3. Indeed, Eqs. (2), (6) and (7) are coupled through the cortical flow velocity, which appears in all equations. However, it is known that the anterior PAR complex serves to up-regulate active stress, implying that σactive = σactive (A).11,28,29 Furthermore, as mentioned before, we know that the active stress is a function of a down-regulating cue from the sperm-donated microtubule organizing center. A full treatment of the dynamics requires the functional dependence of the active stress on the PAR complexes and on the cue.

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Mechanochemical Pattern Formation in Polarization

A/P

ν(x,t )

0

00:00

PAR−2/6

A/P

ν(x,t )

0

05:00

05:00

A/P

ν(x,t )

PAR−2/6

1

*

0

07:30

*

07:30

A/P

ν(x,t ) Posterior

(a)

*

0

13:00

Anterior

PAR−2/6

1

1

0

03:00

1

1

0

PAR−2/6

A/P

ν(x,t )

0

03:00

1

0

00:00

1

1

0

PAR−2/6

1

1

0

209

Anterior

*

13:00 Posterior

(b)

Fig. 3. PAR polarization in C. elegans proceeds by advective triggering of a patternforming system. This figure provides a comparison between flow-triggered polarization in theory and experiment. (a) Theory, calculated anterior (A, red) and posterior (P , cyan) PAR profiles with the unpolarized embryo subject to measured wild-type flow velocities (dashed black line; positive velocities are toward the anterior). Shaded lines in last time point indicate the final (steady state) distributions that are achieved when flow ceases. (b) Experiments, GFP::PAR-6 (red) and mCherry::PAR-2 (cyan) fluorescence profiles (left) with still images (right) of a single cell stage C. elegans embryo undergoing polarization. The small PAR-2 peak in the anterior (*) is due to the polar body. Adapted from Ref. 22.

We can gain insight on how biochemistry and mechanical flow could interact for the purpose of mechanochemical pattern formation through the following thought experiment.30 Imagine a class of molecules that are bound to the membrane and that locally up-regulate active stress in the cell cortex. The regulator has concentration c and is assumed to diffuse

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

210

freely in the plane of the membrane. Then, the coupled mechanical/RDA equations are, in the viscous limit, ∂x σactive (c) = −η∂x2 v + γv, ∂t c =

D∂x2 c

(8)

− ∂x (vc).

(9)

At steady state, integration of (9) gives v = D∂x ln c. Substitution of this expression for the velocity into (8) gives ∂x2 ln c =

σ  − σactive (c) γ ln c + , η Dη

(10)

where the boundary stress σ  is a constant of integration. This is the equation for an anharmonic oscillator where ln c is the “position” of the oscillator and x is “time.” Nonhomogeneous solutions exist if the active stress is sufficiently large. The nonhomogeneous steady state consists of peaks in concentration of regulator with active stress-driven flow of material

(a)

(b)

Fig. 4. Representative steady states for an active stress regulator. In this case, the solution to (10) may feature one (a) or two (b) peaks. Each peak corresponds to a peak in both active stress and total stress and features concerted flow which delivers more regulator by advective transport. When transport by diffusion down the concentration gradient of the peaks matches this advective influx, steady state is achieved. Velocity is plotted in units of U = σactive ∂ 2 v, where σactive is the characteristic active stress. Modified from Ref. 30.

November 21, 2012

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

Mechanochemical Pattern Formation in Polarization

211

into them (Fig. 4). These patterns rely on cortical flow to counteract diffusion, maintaining high concentration of the regulator in those regions toward which the cortex flows.30 6.

Conclusions

The coupling of mechanics and biochemistry is a central theme in morphogenesis. As we have seen in the case of the C. elegans zygote and in our thought experiment, unevenly distributed mechanical stresses result in material movement which advectively transports biochemical components. This affects local concentrations and therefore biochemical reaction rates. Furthermore, as is the case for the aPAR complex, these stress gradients transport the very components that regulate active stress. This results in a positive feedback loop, which is itself modulated by biochemistry with other species, such as in the case with the aPAR and pPAR complexes. Clearly, mechanics and biochemistry are inextricably linked. Much remains to be learned about how biochemical pathways interact with mechanical ones, and the importance of these interactions make mechanochemical processes in morphogenesis a pertinent research topic. In the example of the polarizing C. elegans zygote, we need to understand how the microtubule organizing center and PAR proteins regulate active stress. Additionally, the polarization mechanism is complemented by a rescue mechanism that appears to operate through microtubule-induced formation of pPAR domain that can serve to polarize the zygote under conditions where flows are absent.31 This motivates inclusion of microtubule dynamics and their mechanics in future models. Whether we are studying C. elegans or any other developmental system, one thing is as clear today as it was to D’Arcy Thompson nearly a century ago:3 mechanics need to be included in descriptions that seek to explain the ways by which patterning, structure, and form arise in development. References 1. A. M. Turing, Philos. Trans. R. Soc. B. 237(641), 37 (1952). 2. J. Howard, S. W. Grill and J. S. Bois, Nat. Rev. Mol. Cell Bio. 12(6), 392 (2011). 3. D. Thompson, On Growth and Form (Cambridge University Press, 1917). 4. L. V. Beloussov, The Dynamic Architecture of a Developing Organism (Springer-Verlag, 1998). 5. K. J. Kemphues, J. R. Priess, D. G. Morton and N. Cheng, Cell 52, 311 (1988). 6. J. Betschinger and J. A. Knoblich, Current Biol. 14(16), R674 (2004).

November 21, 2012

212

13:2

9in x 6in

b1449-ch10

Engineering of Chemical Complexity

J. S. Bois and S. W. Grill

7. P. G¨ onczy, Nat. Rev. Mol. Cell Bio. 9(5), 355 (2008). 8. C. R. Cowan and A. A. Hyman, Development 134, 1035 (2007). 9. E. Munro and B. Bowerman, Cold Spring Harbor Perspect. Biol. 1(4), a003400 (2009). 10. J. Nance and J. A. Zallen, Development 138(5), 799 (2011). 11. E. Munro, J. Nance and J. R. Priess, Development Cell 7(3), 413 (2004). 12. C. Cowan and A. Hyman, Nature 431, 92 (2004). 13. M. Mayer, M. Depken, J. S. Bois, F. J¨ ulicher and S. W. Grill, Nature 467(7315), 617 (2010). 14. N. W. Goehring, C. Hoege, S. W. Grill and A. A. Hyman, J. Cell Biol. 193(3), 583 (2011). 15. D. A. Fletcher and P. L. Geissler, Annu. Rev. Phys. Chem. 60, 469 (2009). 16. A. Bausch and K. Kroy, Nat. Phys. 2(4), 231 (2006). 17. J.-F. Joanny and J. Prost, HFSP J. 3(2), 94 (2009). 18. R. Simha and S. Ramaswamy, Phys. Rev. Lett. 89(16), 058101 (2002). 19. K. Kruse, J. Joanny, F. Julicher, J. Prost and K. Sekimoto, Eur. Phys. J. E. 16, 5 (2005). 20. G. Salbreux, J. Prost and J. F. Joanny, Phys. Rev. Lett. 103(5), 1 (2009). 21. S. W. Grill, Curr. Opin. Genet. Dev. 21(5), 647 (2011). 22. N. W. Goehring, P. K. Trong, J. S. Bois, D. Chowdhury, E. M. Nicola, A. A. Hyman and S. W. Grill, Science 334(6059), 1137 (2011). 23. A. Fire, X. SiQun, M. K. Montgomery, S. A. Kostas, S. E. Driver and C. C. Mello, Nature 391, 806 (1998). 24. N. W. Goehring, D. Chowdhury, A. A. Hyman and S. W. Grill, Biophys. J. 99(8), 2443 (2010). 25. A. T. Dawes and E. M. Munro, Biophys J. 101(6), 1412 (2011). 26. P. Khuc Trong, E. M. Nicola, J. S. Bois, N. W. Goehring and S. W. Grill, in perparation. 27. Y. Mori, A. Jilkine and L. Edelstein-Keshet, Biophys. J. 94(9), 3684 (2008). 28. R. J. Cheeks, J. C. Canman, W. N. Gabriel, N. Meyer, S. Strome and B. Goldstein, Curr. Biol. 14(10), 851 (2004). 29. A. Suzuki and S. Ohno, J. Cell Sci. 119(6), 979 (2006). 30. J. S. Bois, F. J¨ ulicher and S. W. Grill, Phys. Rev. Lett. 106, 028103 (2011). 31. F. Motegi, S. Zonies, Y. Hao, A. A. Cuenca, E. Griffin and G. Seydoux, Nat. Cell Biol. 13(11), 1361 (2011).

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Part IV OSCILLATIONS AND SYNCHRONIZATION

213

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

This page intentionally left blank

214

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Chapter 11 SYNCHRONIZATION OF ELECTROCHEMICAL OSCILLATORS Mahesh Wickramasinghe and Istv´ an Z. Kiss∗ Department of Chemistry, Saint Louis University, 3501 Laclede Ave St Louis, MO 63103, USA ∗ [email protected] Oscillatory electrochemical reactions on multi-electrode arrays can be described with the theory of coupled oscillators. Complex dynamical features are often interpreted using synchronization concepts. In this chapter, we review experimental investigations related to the description and design of complex responses of small sets and large populations of electrochemical oscillators with coupling, forcing, and feedback. It is demonstrated that dynamical behavior of chemical oscillator assemblies can be effectively tuned with designed coupling structure and symmetry. The inherent dynamical features (e.g. phase and identical synchronization, anomalous phase synchronization, stable and itinerant clustering) can be further tuned by carefully engineered open-loop or closed-loop input signals. The knowledge of engineered self-organized macro- and micro-scale electrochemical systems could be used for design of electrochemical oscillator networks with prescribed temporal and spatial correlations of chemical reaction rates.

Contents 1. 2.

3. 4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Experimental methodologies . . . . . . . . . . . . . . . . 2.2. Data processing . . . . . . . . . . . . . . . . . . . . . . . Synchronization with External Forcing . . . . . . . . . . . . . Coupled Electrochemical Oscillator Systems . . . . . . . . . . 4.1. Synchronization of two electrochemical oscillators . . . . 4.2. Synchronization in globally coupled oscillator populations Synchronization Engineering . . . . . . . . . . . . . . . . . . .

215

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

216 217 217 218 221 222 222 225 227

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

216

6. Electrochemical Oscillators in Networks . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

228 232 233

Introduction

Complex dynamic behavior in chemical systems often arises from the interplay of chemical and physical processes.1 –3 In electrochemical systems periodic, chaotic, and quasiperiodic oscillatory behavior can occur due to interaction of charge transfer chemical reactions, electric effects (e.g. potential drop in the electrolyte) and mass transfer contributions.4 –7 Typical experiments are carried out at a constant potential (driving force) which maintains the cell in a far-from-equilibrium state for the time of the experiment. The oscillations are then recorded in the measured current, which is proportional to the rate of the chemical reactions. The charge transfer reactions take place on the surface of the electrode; the surface can be envisioned as an ensemble of reaction sites. For macroscopic oscillation to occur, a synchronization mechanism that co-ordinates the phases of the individual oscillations must exist. Consequently, engineering the complex oscillatory dynamics of the electrochemical system is possible by tuning the synchronization features of the local oscillations. Outside electrochemistry, synchronization of oscillatory chemical systems has been studied in Belousov–Zhabotinsky (BZ) reaction,8 –11 pH oscillators,12 and biochemical reaction13 in coupled continuously fed, stirred tank reactors, in CO oxidation on heterogeneous catalyist,14 and in BZ beads,15,16 microwell arrays,17 and micro18 and nanoscale19 droplets. In these systems a wide range of synchronization behavior was observed (in-phase and anti-phase synchronization, amplitude death, clustering, quorum transition). With development of synchronization theories,20 the field of coupled electrochemical oscillators has been rejuvenated because comprehensive investigations of synchronization features are possible with coupled multi-electrode arrays (up to 100 discrete units21 ) that produce highly reproducible, long time series (thousands of oscillations22 ) with tunable inherent local dynamics by experimental conditions. The electrochemical systems can also be conveniently controlled by external perturbation of circuit potential, as originally demonstrated with electrochemical chaos control applications,23,24 which introduces an additional level of freedom for tuning complex dynamic structures. In this chapter, we review some results on engineering characteristics of synchronization of electrochemical oscillators through coupling, forcing, and feedback. First, an account of general experimental and data processing

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

217

methodology is given in Sec. 2, which is followed by a review of effects of forcing, coupling, and feedback, in Secs. 3–5, respectively. Dynamics of electrochemical networks are discussed in Sec. 6. Finally, we give some general remarks and outline some future directions.

2.

Methods

Synchronization of electrochemical oscillators is commonly studied with the use of electrode arrays.25,26 In this section we briefly summarize experimental methodologies and data processing techniques related to measurements of oscillatory electrochemical reactions with electrode arrays. 2.1.

Experimental methodologies

A schematic of a standard electrochemical cell accommodating working (W), reference (R), and counter (C) electrodes is shown in Fig. 1(a). From traditional electroanalytical chemistry perspective the instrumentation encompasses a potentiostatic measurement using multi-electrode chronoamperometry with common reference and counter electrodes. The counter electrode is usually Pt and the reference electrode (e.g. calomel electrode) is used for definition of a constant potential point in the electrochemical cell. The working electrode is usually an array of metal wires embedded in an insulating material (e.g. epoxy or Teflon) so that the reaction takes place only at the ends (see Figs. 1(a)–(b)); the currents of individual electrodes are measured by multi-channel current meters.26 The potentiostat polarizes the electrodes at a given circuit potential V (t) relative to the reference electrode. In simple coupling experiment the circuit potential is kept at a constant value, i.e., V (t) = V0 . In experiments with external control the potential is perturbed: V (t) = V0 + δV (t), where δV (t) is an externally imposed potential perturbation. The establishment of well-defined mass transfer conditions for providing fresh electrolyte solutions to the surface and the flexible design of electrodes of various sizes and spacings are major experimental difficulties of electrode array studies. When the number of electrodes are small (e.g. two) or local measurements are not required, rotating disks can be used. An impinging jet system was developed for the investigation of oscillations in the mass transfer limited region of dissolution of arrays of relatively large (>10) number of electrodes.28 However, investigators have chosen electrochemical reactions under kinetic control where critical phenomena occurs at stagnant or weakly stirred conditions because of the difficulty of providing laminar flow in macro-electrode cells.29,30 Recently, on-chip

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

218

(a)

(c)

(b)

(d)

Fig. 1. Electrode arrays for studying synchronization of electrochemical oscillators. (a) Schematic of a multi-electrode experimental setup. (b) 8 × 8 Ni electrode array embedded in epoxy. (c) Schematic of experimental setup for on-chip fabricated microfluidic flow cell.69 Wf , Wr , C, R: front and rear working, counter, and reference electrodes, respectively. (d) Microfluidic dual electrode setup with 100 µm Pt band electrodes over which a 100 µm wide flow channel is placed.27

fabrication technologies with microfluidic flow cells (see Figs. 1(c) and 1(d)) provided alternative means for flexible cell design with well-controlled mass transfer conditions.27,31 2.2.

Data processing

For each oscillator, time series data is collected in the experiments. In addition to standard techniques of chemical nonlinear dynamics1,7 (e.g. Fourier transform analysis, time-delay embedding, cross correlations), several measures can be calculated in order to characterize the extent of synchrony. The nature of data processing strongly depends on the type of synchronization being studied. Phase synchronization. In phase synchronization, the phase difference between two oscillators is locked, or, in a more general sense, is

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

219

bounded.20,34 For data analysis, the signal obtained from a single oscillator, x(t), is decomposed into a time dependent phase φ(t) and amplitude A(t). These two quantities can be obtained by plotting the data in a 2D state space where at each time the system is represented as a point; the phase and amplitude of the oscillations are the angle and the magnitude of the state vector, respectively, pointing from the origin to the phase point. The 2D state space is often constructed using the Hilbert transform
∞ x(τ ) −
x 1 dτ, (1) H(t) = PV π t−τ −∞ where
 denotes temporal average and the integral should be evaluated in the sense of Cauchy principle value (PV). The state space is thus reconstructed from the H(t) vs. x(t) −
x plot; phase and the amplitude are obtained as follows: φ(t) = arctan and

H(t) x(t) −
x

 2 2 A(t) = H(t) + (x(t) −
x) .

(2)

(3)

These quantities are meaningful only when the phase space trajectories have proper rotation around the origin. After the phase of the signal is obtained, various techniques can be applied to obtain synchronization measures20 that usually depend on the phase difference ∆φ(t) between the oscillators. For example, a commonly used measure is the synchrony index ρ |ρ| = |
ej∆φ(t) |.

(4)

Close to zero value of |ρ| indicates drifting phases (lack of synchrony) while |ρ| ≈ 1 indicates fixed phase difference (strong synchrony). Identical synchronization. At very strong coupling the two oscillators could follow the same trajectories, therefore, for every variable the variation is the same and identical synchronization sets in.20 A simple test for identical synchronization of two signals is a plot of the two signals vs. each other; for identical synchronization the behavior is expected to be along the diagonal line (synchronization manifold) and thus standard deviation from the diagonal line is one measure of identical synchronization. To reflect higher-dimensional motion of the oscillators, often at a given time pair distance between the (time delay) reconstructed phase points is calculated; then the oscillators are defined in identical synchronized

November 21, 2012

220

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

state when the pair distances are less than a small threshold value that characterizes experimental noise and heterogeneities.35,36 A suitable measure of identical synchronization is the fraction of time in which the oscillator pairs are in identically synchronized state. Some other types of synchronization also exist. For lag synchronization, identical synchronization is established with a certain nonzero lag time.37 Generalized synchronization can be characterized with existence of continuous function relationships between the two reconstructed attractors in the state space.38 An algorithm based on false nearest neighbors can be used to characterize functional relationships between signals; a fraction of data points is calculated for which a unique nearest-neighbor “mirror” image points exist on the other attractor for the entire time series within a given interval.39 These various types of synchronization methods can be studied with different number and types of oscillators in various coupling configurations. Electrochemical systems provide a wide variety of oscillatory waveform/inherent dynamics for synchronization. Figure 2 shows smooth periodic oscillators close to Hopf bifurcation, relaxation oscillations close to homoclinic bifurcation, phase coherent and non-phase coherent chaotic oscillations, and complex bursting oscillations.

Fig. 2. Wide variety of dynamical behavior of single electrochemical oscillators. (a–d) Nickel electrodissolution. (a) Smooth periodic oscillation close to Hopf bifurcation.29 (b) Relaxation oscillation close to a homoclinic bifurcation.29 (c) Phase coherent chaotic oscillations.32 (d) Non-phase-coherent chaotic oscillations.22 (e) Chaotic bursting during iron electrodissolution.33

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

3.

221

Synchronization with External Forcing

The description of the effect of a periodic forcing signal to an oscillatory system can be considered as unidirectional coupling between the system and the external signal; phase synchronization in this configuration constitutes a standard theoretical problem of nonlinear science.20 The periodic forcing signal (of frequency ωf ) above a critical forcing amplitude Ac can entrain the oscillatory system: The natural frequency of the system (ω0 ) becomes adjusted to the forcing frequency. In the forcing amplitude (A) vs. forcing frequency phase diagram, entrained states form long horizontal lines at resonant frequencies (kωf = mω0 , where k and m are integers) called Arnold tongues. In addition, in many periodically forced systems, bifurcations to chaotic behavior were predicted from simple oscillator models.20 The electrochemical experiments are typically performed by superimposing a forcing signal on the constant circuit potential, e.g., for sinusoidal forcing δV (t) = A sin(2πωf t). The appearance of Arnold tongues has been confirmed in periodically forced iron electrodissolution.40 In iron dissolution, harmonic forcing of periodic electrochemical oscillators results in entrainment, spike generation, and quasiperiodicity,41 harmonic, subharmonic, and super-harmonic entrainment.40 Regular oscillations were transformed to chaotic by periodic forcing of the potential in reduction of 3− Fe(CN)6 on glassy carbon electrode.42 With harmonic forcing of periodic Ni electrodissolution, complex oscillation waveforms were observed.43 In Ni electrodissolution, periodic forcing was also applied to a single chaotic oscillator.44 When the forcing frequency was similar to the natural frequency of the oscillator, a critical forcing amplitude was observed above which the chaotic oscillations were entrained. The entrained states in the forcing amplitude vs. frequency space, shown in Fig. 3, formed an Arnold tongue. Local forcing was also applied to iron electrodissolution for a single electrode (with laser pertubation45,46 ) and to one element in an array.47 Pacemaker activity was recorded in both examples when the entire system could be entrained by the local perturbations. Global forcing applied to electrode arrays resulted in global phase synchronization,48 stabilization of various cluster configurations,49 and appearance of resonance clusters states.50 An engineering aspect of external forcing is the existence of optimal forcing waveform by which most efficient entrainment could be achieved. It was shown that the optimal waveform is often closely related to the phase response curve of the oscillator.51 Therefore, oscillators close to Hopf bifurcations, which often exhibit sinusoidal phase response curves, are optimally entrained by sinusoidal signals, however, more nonlinear relaxation

November 21, 2012

222

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

Fig. 3. Effect of periodic forcing on chaotic dissolution of Ni in sulfuric acid.44 Phase locked region (Arnold tongue) in forcing amplitude forcing frequency parameter space.

oscillators with higher harmonics have non-trivial optimal waveform that produce 50–90% enhancement in the entrainabilities.51

4.

Coupled Electrochemical Oscillator Systems

The fundamental phenomenon related to synchronization of electrochemical cells dates back (at least) to the 1950s report of Franck and Meunier,52 however, clear description and understanding of the processes became possible only in the 1990s with development of synchronization theories.20 4.1.

Synchronization of two electrochemical oscillators

Different cell designs for dual electrode setups employ different methods to induce communication between the oscillators. The communication can be either electrical (through potential drop in the electrolyte or external circuitry) or chemical through the diffusion of substances. Important finding of the past two decades is that the coupling is predominantly electrical in the electrochemical systems.4 The ability of tuning the communication (electrical coupling strength) between electrodes is an important challenge in these cell designs. When the electrodes are spaced in large distances and the potential drop in the electrolyte is small, the inherent coupling between the electrodes in a cell design shown in Fig. 1(a) is weak. Externally added electrical

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

223

coupling can be achieved with a combination of collective resistance Rcoll connected to both electrodes and individual electrodes Rind in parallel.29 The coupling strength is K = Nel Rcoll /Rind ,

(5)

where Nel = 2 is the number of electrodes. The experiments are carried out by varying K while keeping the total cell resistance (Rtot = Rind +Nel Rcoll ) constant. Two smooth oscillators (e.g. see Fig. 2(a)) without any added coupling in nickel electrodissolution exhibit no synhcrony, as shown in Fig. 4(a); because the two electrodes are slightly different (e.g. because of slight heterogeneities of surface conditions) there is about 2% difference in the natural frequency of the oscillations. With weak added coupling in Fig. 4(b), the frequencies become identical and in-phase synchrony develops.29,53 In contrast, two coupled relaxation oscillators close to

(a)

(b)

(d)

(c)

(e)

Fig. 4. Effect of electrical coupling in dual electrode setup on synchronization. Top row: Effect of collective (series) resistance in oscillatory duel electrode Ni electrodissolution in sulfuric acid.66 (a) Desynchronized current oscillations without added electric coupling with smooth oscillations (K = 0). (b) In-phase synchronized current oscillations with added electric coupling with smooth oscillations (K = 0.03). (c) Anti-phase synchronized current oscillations with added electric coupling with relaxation oscillations (K = 1.50). Bottom row: Effect of working-to-reference electrode placement in oscillatory dual electrode formic acid oxidation on Pt.69 (d) Desynchronized current oscillation with near (2.4 mm) working-to-reference electrode placement. (e) Synchronized current oscillations with far (11 mm) working-to-reference electrode placement.

November 21, 2012

224

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

homoclinic bifurcation exhibit anti-phase synchrony (see Fig. 4(c)) with weak coupling. If the coupling strength is increased further, complex oscillations develop but at very strong coupling (nearly) identical synchronization is achieved. Iron dissolution in H2 SO4 produces relaxation oscillations5 in which the coupling between the electrodes is due to the inherent potential drops in the electrolyte because of the large current density of the reaction. The electrical coupling strength can be tuned by varying the distance between the two working electrodes and the distance between the working and the reference electrodes.54 By proper cell design in-phase, outof-phase, or anti-phase synchrony have been observed.55 The effect of position of reference/counter on synchronization features was also explored in H2 O2 reduction with macroscopic electrodes56 and formic acid electro-oxidation in a microchip-based dual-electrode flow cell (Figs. 1(c) and (d)). As shown in Fig. 4(d), the synchronization of current oscillations in microchip integrated cell does not occur with close placement of working/reference electrode. However, when the reference/counter electrode is far (e.g. 11 mm) from downstream working electrode, the oscillations become synchronized (see Fig. 4(e)). Chaotic oscillations exhibit more domains of qualitatively different synchronization that is obtainable with systematic variation of coupling strength. In nickel electrodissolution, as electrical coupling is added with a combination of collective and individual resistors a transition is seen from absence of synchrony, through phase synchrony, to identical synchrony.57 Without any added coupling (see Fig. 5(a)) the phase analysis reveals that similar to that observed with periodic oscillators the frequencies are slightly different. With very weak coupling added phase synchronization sets in (see Fig. 5(b)); the frequencies become identical but the amplitudes are

Fig. 5. Synchronization of chaotic oscillations during nickel electrodissolution of two coupled electrodes.57 (a) Asynchronous oscillation without added coupling. (b) Phase synchronization with weak added coupling. (c) Identical synchronization with strong added coupling.

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

225

not correlated. As it is shown in Fig. 5(c) at very strong coupling the currents of the electrodes exhibit identical variation (similar to that of a single electrode) thus identical synchronization sets in. With iron dissolution a transition sequence “absence of synchronization → phase synchronization → lag synchronization → identical synchronization” of two coupled non-identical chaotic oscillators has been observed as the diminishing distance between the electrodes intensified the inherent electrical coupling strength.58 Even more complex behavior can be observed with non-phase coherent chaotic and bursting oscillators. In contrast to phase coherent chaos with nickel eletrodissolution, with the non-phase coherent chaos of iron electrodissolution phase and, generalized synchronization occurs simultaneously at strong coupling strengths.39 Studies on synchronization of electrochemical bursters showed that there can be three different regimes of synchronization, namely, synchronization of individual spikes, synchronization of bursts, and complete synchrony where both spike and burst synchronization occurs simultaneously.55,59,60 When two autonomous electrochemical Fe/H2 SO4 , Cl− bursters are coupled, the type of synchrony is determined by the type of bursting oscillations. For instance, for electrochemical bursting of elliptic type, both burst and spike synchronization can be observed whereas for square wave type bursting, spike or burst synchronization is very difficult to achieve.55,59,60

4.2.

Synchronization in globally coupled oscillator populations

A population of oscillators can exhibit collective dynamical features.20 In 1960s and 1970s Winfree61 and Kuramoto62 predicted that in a large population of oscillators the transition to synchrony will take place through a second-order phase transition: There exists a critical coupling strength (Kc ) below which the population will be fully desynchronized. Above the coupling strength the synchrony will quickly increase by forming a group of synchronized oscillators; the number of elements in the synchronized group is expected to increase with increase in K. The Kuramoto transition was experimentally confirmed with an array of 64 Ni electrodes (8 × 8 configuration).63 Figure 6 shows that as the coupling is increased the second-order phase transition takes place such that the Kuramoto order parameter, r, (related to the amplitude of the mean current oscillations) starts to increase above Kc . With strong relaxation oscillators instead of single synchronized group, two or three synchronized groups having a constant phase difference are observed.29,53

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

226

(a)

(b)

(c)

(c) Fig. 6. Emergence of coherence (Kuramoto transition) in oscillatory Ni electrodissolution with an 8×8 electrode array.63 (a) Frequency distribution without added electric coupling (K = 0). (b) Frequency distribution just above the phase transition point. (K = 0.29). (c) Frequency distribution at strong electric coupling strength (K = 0.52). (d) Kuramoto order parameter as a function of coupling strength. Insets illustrate behavior in 2D state space below and above the phase transition. The frequencies in panels a–c are renormalized with the mean value of the natural frequency distribution.

In a population of chaotic oscillators in addition to the desynchronized36 (Fig. 7(a)), phase synchronized64 (Fig. 7(b)), and identically synchronized states (Fig. 7(d)) there exists a coupling strength region below identical synchronization where chaotic clustering (or dynamical differentiation) takes place.36 The array splits into groups; the elements in each group have identical dynamics different from that of the other group. Two clusters with a large number of possible cluster configurations have been observed with chaotic Ni dissolution. A representative cluster configuration is shown in Fig. 7(c). At coupling strengths slightly weaker and stronger than that required for clustering dynamics, itinerant clustering was observed.65 The cluster configurations varied with time: spontaneous changes of number of clusters and their configurations were detected. In 1D ring and 2D arrays of iron electrodissolution in sulfuric acid traveling waves develop in both mass-transfer and active–passive oscillatory regions.26,28

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

(a)

(b)

(c)

227

(d)

Fig. 7. Synchronization of chaotic oscillations in chaotic Ni electrodissolution in sulfuric acid on an 8 × 8 electrode array.36,64 Top: snapshots of currents on the electrodes. Bottom: Snapshot of position in state space constructed from the currents (mA) of all 64 electrodes. (a) Desynchronized chaos, K = 0. (b) Phase synchronized chaos, with weak added electrical coupling, K = 0.12. (c) Dynamical differentiation (clustering) with moderate electrical coupling, K = 3. (d) Identical synchronization with strong added electrical coupling, K = ∞.

5.

Synchronization Engineering

A major question of both theoretical and practical importance is how to bring the collective behavior of a rhythmic system to a desired condition or, equivalently, how to avoid a deleterious condition without destroying the inherent behavior of its constituent parts. The efficient design of a complex dynamic structure is a formidable task that requires simple yet accurate models incorporating integrative experimental and mathematical approaches that can handle hierarchical complexities and predict emergent, system-level properties. The kinetics-mass transfer type models often used in electrochemistry are often not detailed enough for use in design of collective behavior. It was shown that nonlinear feedback loops can be rigorously designed using experiment-based phase models53 to dial up a desired collective behavior without requiring detailed knowledge of the underlying physiochemical properties of the target system.66 Weak feedback signals can be designed so as to have a minimal impact on the dynamics of the individual electrodes while producing a collective behavior of the population that is both qualitatively and quantitatively different than the dynamic behavior of an uncontrolled system. The method, termed as synchronization engineering,66 –68 was demonstrated to create phase locked oscillators with arbitrary phase

November 21, 2012

228

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

Fig. 8. Synchronization engineering: designed sequential cluster patterns in Ni electrodissolution in sulfuric acid in a four electrode array with a cubic feedback of electrode potential to the circuit potential.66 (a) Time series of the Kuramoto order parameter along with selected cluster configurations. (b) and (c) Trajectory in state space of phase differences. The black lines represent theoretically calculated heteroclinic connections between cluster states (black fixed points). The red surface in (c) is the set of trajectories traced out by a heterogeneous phase model. The experimental trajectory is colored according to its phase velocity.

difference, subtle dynamical structures such as itinerant cluster dynamics, desynchronization, and various cluster states. For example, Fig. 8 shows a slow switching state where under the feedback the four-oscillator system itinerates among two-cluster states along heteroclinic orbits. The methodology was also extended for population of chaotic oscillators.69 Closed-loop feedbacks with global feedback to the circuit potential49,70 and local feedback to individual resistors71 have been applied to induce identically synchronized, two-cluster, and multi-cluster states in chaotic Ni electrodissolution. 6.

Electrochemical Oscillators in Networks

Results discussed in Sec. 4 dealt with a pair of oscillators or with a population of oscillators in global, “all-to-all” coupling among the electrodes. Many

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

229

biological and engineered systems consisting of discreet dynamical units self-organize into complex networks72,73 ; integration of electrochemical oscillators into complex networks could reveal novel dynamical features that are not seen in simple coupling topologies. Small networks of cathode–anode relaxational oscillator electrochemical units have been built with iron electrodissolution in Fe|H2 SO4 , Cu|CuSO4 system.30,55,74 –76 In-phase synchronization is possible in three cells arranged in a chain configuration where interaction between two boundary electrodes is minimal as well as in a ring where cells experience all-to-all coupling effects.55 These results suggest that in the given experimental setup chains and rings consisting of the same number of cells synchronize in a similar manner since boundary effects are not important. The synchronization of six cells in a regular hexagon configuration is inphase when the node consists of electrodes of a single kind (all anode or all cathode). Arrangements of nearby electrodes of different kind (e.g. alternating anode and cathode) results in out-of-phase synchronized clusters. Cells within a group oscillate in-phase, but different groups oscillate out-of-phase. This example reveals that the synchronization modes and the resulting spatiotemporal patterns are determined by the architecture of the network. Complex clustering dynamics have been observed in symmetric geometrical configurations consisting of 24 cells in star arrangement.55 In the nickel electrodissolution system small networks of oscillators can be built with a hybrid chemical-resistive device shown in Fig. 9(b). In this system the coupling topology is controlled by cross-resistors R among the electrodes; the coupling strength is proportional to 1/R. The device

Fig. 9. Identical synchronization of small networks of electrochemical oscillators. (a) Order parameter (r), a measure of extent of identical synchronization, vs. coupling strength (K = 1/R) for two coupled oscillators (solid circles) and a small network of four oscillators in star configuration (hollow circle). (b) Schematic diagram of a network of four coupled electrodes in star configuration in a hybrid resistive-chemical network. Coupling strength between two oscillators is tuned by cross-resistance (R). Rext : Individual resistance. W: Working electrode.

November 21, 2012

230

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

can be used to study the effect of geometrical configuration on the dynamical behavior of the chemical system. For example, for two coupled chaotic oscillators identical synchronization brings the oscillators to one group, and the order parameter for identical synchronization r increases to 1 at a critical coupling strength (see Fig. 9(a)). In a small network of four oscillators arranged in star configuration (Fig. 9(b)) identical synchronization can be reached as well, however, at much stronger critical coupling strength than that required for two oscillators. The architecture of the network seems to be a dominant factor determining the critical coupling strength of identical synchronization. Theoretical studies indicate that it could be possible to interpret such dependency using master stability function theory.77,78 Understanding the characteristic behavior of a network requires inferring the connection topology of its constituting parts. Synchronization properties based on Granger casuality principles of the network can be useful in deducing the connection topology.81 For instance, it was shown that phase synchronization properties of networks can be applied to differentiate direct and indirect connections in small networks of chaotic oscillations.81 A synchronization matrix can be constructed from the pairwise synchronization indices (Eq. 4); the corresponding elements in the inverse of synchronization matrix can characterize synchronization from elements that are directly coupled. The technique was confirmed in a small network of three linearly coupled electrochemical oscillators in both phasecoherent and non-phase-coherent chaos of nickel electrodissolution.79,82 Figure 10(a) shows that the pairwise bivariant (traditional) synchronization indices all increase in the network with increase of the coupling strength; therefore, inferring the coupling topology from these measures is troublesome. However, the bivariant synchronization index in Fig. 10(b) obtained from the inverse of the synchronization matrix correctly predicts high level of synchrony for the directly coupled oscillator pairs and low level of synchrony for the indirectly coupled (edge) oscillator pair at strong coupling strength. In network dynamics, coupling asymmetry could play an important role; for example, anomalous phase synchronization effects could occur.83 In a two-oscillator system the anomaly is typically contrasted to the classical route of synchrony; that is, in the limit of weak coupling, the frequency difference between two oscillators (∆ω) decreases with increasing coupling strength (K) following the classical square root formula ∆ω = ∆ω0 (1 − K 2 /Kc2 )0.5 , where ∆ω0 is the natural (uncoupled) frequency difference, and Kc is the critical coupling strength at which synchronization occurs.83 As it is shown in Fig. 10(c), with two electrodes of identical size such square-root relationship can be experimentally confirmed in

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

(a)

231

(b)

(c)

Fig. 10. Network effects on phase synchronization. Top row: Analysis of network connectivity using phase synchrony.79 (a) Bivariant phase synchronization index and (b) partial phase synchronization index for direct coupling between 1–2 (closed circle), 2–3 (closed triangle), and for indirect coupling between 1–3 (open square) pairs. Bottom row: Asymmetrical coupling induced anomalous phase synchronization effects in frequency difference vs. normalized coupling strength (Kr ) plots.80 (c) Phase synchronization under symmetric coupling (closed circle), advanced phase synchronization (dashed line) and delayed phase synchronization (open circle with solid line) under asymmetric coupling. Slow (hollow) driver (large) electrode causes advanced synchronization, fast (grey) driver electrode causes frequency difference enhancement and delayed synchronization compared to symmetrical coupling.

nickel electrodissolution. Anomalous phase synchronization (states that exhibits large deviation from classical route of phase synchrony) can be observed in strongly non-isochronous oscillators with asymmetrical coupling.83 The most prominent forms are advanced/delayed anomalous synchronization: with asymmetrical coupling the critical coupling strength is weaker/stronger than with symmetrical coupling. Such asymmetrical coupling can be induced in dual-electrode electrochemical systems that consist of electrodes of different surface area: the large electrode drives

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

232

the small electrode. As shown in Fig. 10(c) with slow driver electrode (large surface area) advanced phase synchronization, while with fast driver electrode delayed phase synchronization, and, to a lesser extent, frequency difference enhancement were observed.80

7.

Conclusions

The investigation of synchronization of electrochemical oscillators served two major purposes: • Exploration of nonlinear dynamics of electrochemical systems. Electrode arrays with small wires and close spacing have been shown to exhibit similar dynamical behavior to that observed with one large electrode in iron electrodissolution.26 Therefore, many of the dynamical features (e.g. emerging coherence, cluster formation) are expected to arise in the behavior of the equivalent “distributed” system, i.e., with one large electrode. In particular, weak coupling effects could greatly affect the dynamical features of oscillatory systems. The understanding of the coupling effects of multi-electrode charge transfer systems could provide useful information for decrypting the complex response of multielectrode collector-generator devices in electroanalytical chemistry. • Experimental testbed for synchronization theories. Experimental confirmation of several predictions of synchronization theories has been accomplished with electrochemical systems. The results have shown that by tuning the interaction strengths among the electrodes different types and levels of strongly nonlinear correlations can be observed. The coupling, forcing, and feedback signals induce novel time scales that regulate the dynamics and functional relationship among the units of the entire systems. Many of the synchronization phenomena previously have been associated with biological systems.20 Therefore, the experiments confirm that abiotic physicochemical systems are capable of reproducing some of the complex dynamic response of intelligent biological systems. Along this line, an important future effort could be the development of an electrochemical computing device.84 The introduction of sub-micro and nano-scale structures could challenge experimentation and theoretical description electrochemical oscillations. Important progress has been made in the description of oscillatory nanoscale solid–gas catalytic systems.85 Electrochemical experiments of electrodes with nanoscale features have been reported86 ; the synchronization characteristics of such electrode systems could be further complicated

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

233

by internal fluctuation effects that could induce nontrivial temporal and spatial correlations of the observed current oscillations.87

Acknowledgment This material is based upon work partially supported by the National Science Foundation under CHE-0955555.

References 1. I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos, Topics in Physical Chemistry (Oxford University Press, Oxford, 1998). 2. I. Z. Kiss and J. L. Hudson, AIChE J. 49(9), 2234 (2003). 3. R. Kapral and K. Showalter, Chemical Waves and Patterns, Understanding Chemical Reactivity, (Kluwer, Dordrecht, 1995). 4. K. Krischer, Principles of temporal and spatial pattern formation in electrochemical systems, in Modern Aspects of Electrochemistry, Vol. 32, eds. B. E. Conway, O. M. Bockris, and R. E. White, (Kluwer Academic, New York, 1999), p. 1. 5. J. L. Hudson and T. T. Tsotsis, Chem. Eng. Sci. 49(10), 1493 (1994). 6. M. T. M. Koper, Adv. Chem. Phys. 92, 161 (1996). 7. I. Z. Kiss, T. Nagy and V. G´ asp´ ar, Dynamical instabilities in electrochemical processes, in Solid State Electrochemistry II, ed. V. V. Kharton (Wiley, Weinheim, Germany, 2011). 8. M. Marek and I. Stuchl, Biophys. Chem. 3(3), 241 (1975). 9. M. Crowley and I. Epstein, J. Phys. Chem. 93(6), 2496 (1989). 10. J. Weiner, R. Holz, F. W. Schneider and K. Bar-Eli, J. Phys. Chem. 96(22), 8915 (1992). 11. M. J. B. Hauser and F. W. Schneider, J. Chem. Phys. 100(2), 1058 (1994). 12. O. Peˇsek, L. Schreiberov´ a and I. Schreiber, Phys. Chem. Chem. Phys. 13, 9849 (2011). 13. A. Lekebusch and F. W. Schneider, J. Phys. Chem. B 101(47), 9838 (1997). 14. M. M. Slinko, A. A. Ukharskii and N. I. Jaeger, Phys. Chem. Chem. Phys. 3(6), 1015 (2001). 15. H. Fukuda, H. Morimura and S. Kai, Physica D 205(1–4), 80 (2005). 16. A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang and K. Showalter, Science 323(5914), 614 (2009). 17. T. Okano and K. Miyakawa, Phys. Rev. E 80(2), 026215 (2009). 18. M. Toiya, V. K. Vanag and I. R. Epstein, Angew. Chem. Int. Ed. 47(40), 7753 (2008). 19. V. K. Vanag and I. R. Epstein, Phys. Rev. Lett. 87(22), 228301 (2001).

November 21, 2012

234

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

20. A. S. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, Cambridge, 2001). 21. Y. Zhai, I. Z. Kiss, H. Daido and J. L. Hudson, Physica D 205, 57 (2005). 22. M. Wickramasinghe and I. Z. Kiss, Chaos 20(2), 023125 (2010). 23. I. Z. Kiss, V. G´ asp´ ar, L. Nyikos and P. Parmananda, J. Phys. Chem. A 101(46), 8668 (1997). 24. P. Parmananda, P. Sherard, R. W. Rollins and H. D. Dewald, Phys. Rev. E 47(5), R3003 (1993). 25. A. J. Bard and L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980). 26. Z. Fei, R. Kelly and J. L. Hudson, J. Phys. Chem. 100, 18986 (1996). 27. I. Z. Kiss, N. Munjal and R. S. Martin, Electrochimi. Acta 55(2), 395 (2009). 28. Z. Fei and J. L. Hudson, Ind. Eng. Chem. Res. 37(6), 2172 (1998). 29. I. Z. Kiss, W. Wang and J. L. Hudson, J. Phys. Chem. B 103(51), 11433 (1999). 30. A. Karantonis, M. Pagitsas, Y. Miyakita and S. Nakabayashi, J. Phys. Chem. B 108, 5836 (2004). 31. A. G. Cioffi, R. S. Martin and I. Z. Kiss, J. Electroanal. Chem. 659(1), 92 (2011). 32. I. Z. Kiss, W. Wang and J. L. Hudson, Phys. Chem. Chem. Phys. 2, 3847 (2000). 33. I. Z. Kiss, Q. Lv, L. Organ and J. L. Hudson, Phys. Chem. Chem. Phys. 8(23), 2707 (2006). 34. M. Rosenblum, A. Pikovsky and J. Kurths, Phys. Rev. Lett. 76(11), 1804 (1996). 35. D. H. Zanette and A. S. Mikhailov, Phys. Rev. E 57(1), 276 (1998). 36. W. Wang, I. Z. Kiss and J. L. Hudson, Chaos 10(1), 248 (2000). 37. M. G. Rosenblum, A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 4193 (1997). 38. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring and H. D. I. Abarbanel, Phys. Rev. E 51(2), 980 (1995). 39. I. Z. Kiss, Q. Lv and J. L. Hudson, Phys. Rev. E 71, 035201 (2005). 40. M. Pagitsas, D. Sazou, A. Karantonis and C. Georgolios, J. Electroanal. Chem. 327, 93 (1992). 41. A. Karantonis, M. Pagitsas and D. Sazou, Chaos 3, 243 (1993). 42. V. Varma and P. Upadhyay, J. Electroanal. Chem. 271, 345 (1989). 43. F. Berthier, J. Electroanal. Chem. 572(2), 267 (2004). 44. I. Z. Kiss and J. L. Hudson, Phys. Rev. E 64(4), 046215 (2001). 45. A. Karantonis, Y. Shiomi and S. Nakabayashi, Chem. Phys. Lett. 335, 221 (2001). 46. Y. Shiomi, A. Karantonis and S. Nakabayashi, Phys. Chem. Chem. Phys. 3, 479 (2001). 47. Z. Fei and J. L. Hudson, J. Phys. Chem. B 101(49), 10356 (1997).

November 21, 2012

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

Synchronization of Electrochemical Oscillators

235

48. M. G. Rosenblum, A. S. Pikovsky, J. Kurths, G. V. Osipov, I. Z. Kiss and J. L. Hudson, Phys. Rev. Lett. 89(26), 264102 (2002). 49. W. Wang, I. Z. Kiss and J. L. Hudson, Phys. Rev. Lett. 86(21), 4954 (2001). 50. I. Z. Kiss, Y. Zhai and J. L. Hudson, Phys. Rev. E 77(4), 046204 (2008). 51. T. Harada, H.-A. Tanaka, M. J. Hankins and I. Z. Kiss, Phys. Rev. Lett. 105(8), 088301 (2010). 52. U. F. Franck and L. Meunier, Z. f¨ ur Natur. 8b, 396 (1953). 53. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Phys. Rev. Lett. 94(24), 248301 (2005). 54. S. Nakabayashi, K. Zama and K. Uosaki, J. Electrochem. Soc. 143(7), 2258 (1996). 55. A. Karantonis, Y. Miyakita and S. Nakabayashi, Phys. Rev. E 65(4), 046213 (2002). 56. Y. Mukouyama, H. Hommura, T. Matsuda, S. Yae and Y. Nakato, Chem. Lett. 463 (1996). 57. I. Z. Kiss and J. L. Hudson, Phys. Chem. Chem. Phys. 4(12), 2638 (2002). 58. J. Cruz, M. Rivera and P. Parmananda, Phys. Rev. E 75(3), 035201 (2007). 59. A. Karantonis, D. Koutsaftis and N. Kouloumbi, Electrochim. Acta 55, 374 (2009). 60. A. Karantonis, D. Koutsaftis and N. Kouloumbi, J. Appl. Electrochem. 40(5), 989 (2010). 61. A. T. Winfree, J. Theor. Biol. 16, 15 (1967). 62. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984). 63. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Science 296(5573), 1676 (2002). 64. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Ind. Eng. Chem. Res. 41, 6363 (2002). 65. I. Z. Kiss and J. L. Hudson, Chaos 13, 999 (2003). 66. I. Z. Kiss, C. G. Rusin, H. Kori and J. L. Hudson, Science 316(5833), 1886 (2007). 67. C. G. Rusin, I. Z. Kiss, H. Kori and J. L. Hudson, Ind. Eng. Chem. Res. 48(21), 9416 (2009). 68. H. Kori, C. G. Rusin, I. Z. Kiss and J. L. Hudson, Chaos 18(2), 026111 (2008). 69. C. G. Rusin, I. Tokuda, I. Z. Kiss and J. L. Hudson, Angew. Chem., Int. Ed. 50(43), 10212 (2011). 70. Y. Zhai, I. Z. Kiss and J. L. Hudson, Ind. Eng. Chem. Res. 47(10), 3502 (2008). 71. I. Z. Kiss, V. G´ asp´ ar and J. L. Hudson, J. Phys. Chem. B 104, 7554 (2000). 72. M. E. J. Newman, SIAM Rev. 45(2), 167 (2003). 73. R. Albert and A. L. Barabasi, Rev. Mod. Phys. 74(1), 47 (2002). 74. A. Karantonis, M. Pagitsas, Y. Miyakita and S. Nakabayashi, Electrochim. Acta 50, 5056 (2005). 75. Y. Miyakita, S. Nakabayashi and A. Karantonis, Phys. Rev. E 71(5), 056207 (2005).

November 21, 2012

236

13:2

9in x 6in

b1449-ch11

Engineering of Chemical Complexity

M. Wickramasinghe & I. Z. Kiss

76. A. Karantonis, M. Pagitsas, Y. Miyakita and S. Nakabayashi, J. Phys. Chem. B 107, 14622 (2003). 77. L. Pecora and T. Carroll, Phys. Rev. Lett. 80(10), 2109 (1998). 78. T. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, Int. J. Bifur. Chaos 15(11), 3423 (2005). 79. M. Wickramasinghe and I. Z. Kiss, Phys. Rev. E 83(1), 016210 (2011). 80. M. Wickramasinghe, E. M. Mrugacz and I. Z. Kiss, Phys. Chem. Chem. Phys. 13(34), 15483 (2011). 81. B. Schelter, M. Winterhalder, R. Dahlhaus, J. Kurths and J. Timmer, Phys. Rev. Lett. 96(20), 208103 (2006). 82. J. Nawrath, M. C. Romano, M. Thiel, I. Z. Kiss, M. Wickramasinghe, J. Timmer, J. Kurths and B. Schelter, Phys. Rev. Lett. 104(3), 038701 (2010). 83. B. Blasius, Phys. Rev. E 72(6), 066216 (2005). 84. S. Sadeghi and M. Thompson, BioSystems 102, 99 (2010). 85. J.-S. McEwen, P. Gaspard, T. V. de Bocarme and N. Kruse, Proc. Natl. Acad. Sci. USA. 106(9), 3006 (2009). 86. Y. E. Seidel, Z. Jusys, R. W. Lindstrom, M. Stenfeldt, B. Kasemo and K. Krischer, ChemPhysChem 11(7), 1405 (2010). 87. V. Garcia-Morales and K. Krischer, Proc. Nat. Acad. Sci. 107, 4528 (2010).

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Chapter 12 TURBULENCE AND SYNCHRONY IN SPATIALLY EXTENDED ELECTROCHEMICAL OSCILLATORS Vladimir Garcia-Morales∗,† and Katharina Krischer†,‡ ∗

Institute for Advanced Study, TU M¨ unchen Lichtenbergstr. 2a, D-85748 Garching, Germany ∗ [email protected] †

Non-Equilibrium Chemical Physics, Physik-Department, TU M¨ unchen, James-Franck-Str. 1, D-85748 Garchen, Germany ‡ [email protected] We review recent progress made in the study of spatially extended electrochemical oscillatory media. Experiments and theoretical modeling of the nonlocal (migration) spatial coupling and both forms of linear and nonlinear global coupling are discussed with special emphasis in normal forms that we have recently derived to deal with these systems.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrochemical Oscillators Under Nonlocal (Migration) Coupling (NLC) 2.1. NLC in electrochemical systems . . . . . . . . . . . . . . . . . . . . 2.2. Electrochemical oscillations and the Nonlocal Complex Ginzburg–Landau Equation (NCGLE) . . . . . . . . . . . . . . . . . 2.3. Electrochemical turbulence and the NCGLE . . . . . . . . . . . . . 3. Electrochemical Oscillators Under Global Coupling (GC) . . . . . . . . . 3.1. Physical origin of the global coupling . . . . . . . . . . . . . . . . . 3.2. Global coupling induced pattern formation in experiments . . . . . . 3.3. Linear global coupling in electrochemical systems and the normal form approach . . . . . . . . . . . . . . . . . . . . . 3.4. Nonlinear global coupling (NGC) and its normal form . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

. . . . . .

238 239 239

. . . . .

. . . . .

242 243 247 247 249

. . . .

. . . .

252 254 258 259

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

238

1.

b1449-ch12

Introduction

Self-sustained oscillations1 are a common phenomenon in electrochemical systems.2 Understanding the nonlinear dynamics associated with the oscillations is an integral part of a comprehensive picture of the electrochemistry of these systems. The probably most famous example is the anodic oxidation of iron whose dynamic properties were already studied intensively more than 100 years ago. Most noticeable are studies by Ostwald3 –5 and Lillie6,7 who already had a notion of the universal aspects of chemical waves and used the propagation of electrical excitations along iron wires as a model system for nerve impulse propagation. This double role in the study of dynamic instabilities in electrochemical systems, namely to study universal properties of dynamical systems and to gain a better understanding of the electrochemistry itself, has continued since. Many of the oscillating systems are linked to technical applications or economic issues. Examples are oscillations during electrocatalytic reactions or the already mentioned iron corrosion. Concerning model studies on oscillatory systems,1,8,9 electrochemical oscillators are especially suited to investigate the effect of nonlocal10 –17 and global18 –27 coupling on pattern formation and synchronization phenomena. This is because in most electrochemical systems the activatory variable is the electrode potential, i.e., an electric variable. If at some position the electrode potential, or, equally, the potential drop across the double layer between the electrode and the electrolyte, differs from the average potential drop, this deviation decays with 1/r, where r is the distance to a reference position.2,14,16 Hence, an entire range of neighboring positions of the electrode experiences a different electrostatic potential because of a local potential variation. Therefore, the spatial coupling of the activatory variable is nonlocal. Moreover, as we will explain below in more detail, the range of the coupling depends on the cell geometry and, in fact, it can be experimentally tuned between a situation where the coupling affects the entire electrode and a local, i.e., diffusional coupling. This property makes electrochemical systems ideal candidates for studying the impact of nonlocal spatial coupling on pattern formation. Furthermore, a global coupling can be realized very easily in an electrochemical experiment,23 and, in fact, it is naturally present in the two prevalent experimental conditions, potentiostatic experiments with a so-called Haber–Luggin capillary, with which the IR-drop through the electrolyte is minimized, and constant current conditions. The use of a Haber–Luggin capillary introduces a negative (desynchronizing) global coupling into the system, while the galvanostatic control exerts a positive (synchronizing) global coupling. In studies of the impact of global coupling

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

239

on pattern formation, one usually inserts an impedance in the external circuit between the working electrode and the potentiostat, which can be chosen to be positive (in which case it is simply an ohmic resistor) or negative. In this way, strength and sign of the global coupling can be easily controlled. When studying general aspects of pattern formation, the final goal is to find a universal, system-independent mathematical description with a minimal number of parameters and variables. Close to the onset of oscillations, all oscillatory media with diffusion coupling can be mapped to the complex Ginzburg–Landau equation.1,28,29 For electrochemical oscillators, a nonlocal variant of the CGLE, the nonlocal complex Ginzburg– Landau equation (NCGLE), has recently been derived.16 Moreover, an extension of the NCGLE was introduced to also account for the global coupling.25,27 In this review we summarize the dominant patterns observed in electrochemical experiments that can be described by the NCGLE.

2.

2.1.

Electrochemical Oscillators Under Nonlocal (Migration) Coupling (NLC) NLC in electrochemical systems

Electrochemical systems cannot be, in general, modeled by reaction– diffusion equations, because of the coupling through the electrostatic potential which is long-range.2,16,17 Let us consider the impact of an inhomogeneous profile of the electrostatic potential on the dynamics of an electrochemical system. First, observe that any deviation of the electrostatic potential from its average value decays ∼1/r, where r = |r| is the distance to a reference position. If we consider a conducting medium inside a box where the potential is fixed at one side whereas at the opposing side we assume that the electrostatic potential follows a given non-uniform profile, we are close to a situation as present in an electrochemical cell. The two faces with a given inhomogeneous and constant potential distribution represent the working and the counter electrode, respectively. The conducting medium is the electrolyte, the other four faces symbolize the insulating walls of the electrochemical cell. A local perturbation of the potential at the working electrode (WE) falls off in a half-circle into the electrolyte, thereby instantaneously changing the distribution of the electrostatic potential in the entire box (electrochemical cell). The changes will be large, close to the perturbation and quite small far away from it. Hence, the spatial coupling is nonlocal, and it is mediated through the bulk of the electrolyte. Since the electrolyte is an electroneutral medium, the electrostatic potential φ

November 21, 2012

240

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

inside our box (the electrolyte) is given (to a very good approximation) by Laplace’s equation:2,14 ∇2 φ(r, t) = 0,

(1)

which can be solved once the boundary conditions are specified. The most crucial boundary condition is the electrostatic potential at the border to the WE since it is time-dependent. However, at any given time t, a certain potential distribution exists, which mathematically amounts to Dirichlet boundary conditions: ∂φ(r, t)/∂z|z=WE = f (r, t). Assuming a large (rough) counter electrode (CE) and a reversible reaction proceeding with a low overpotential at the CE, we can assume that the potential value is always constant, and we can use this value as the origin of our potential scale: φ(r, t)|z=CE = const = 0. Furthermore, perpendicular to the cell walls, the current flux is zero. To proceed, we recall the structure of the electrochemical interface. It comprises the so-called double layer, which consists of excess charge on the metal electrode surface and an equal amount of counter charge which is carried by ions of the electrolyte that accumulate in front of the surface. Hence, there is a potential drop across the double layer, which we call the double layer potential, φDL and which is given by the difference between the electrostatic potentials of the metal and of the electrolyte at z = WE , φDL = φmetal − φ(r, t)|z=WE . The value of φDL is decisive for the rate at which a certain electrochemical reaction takes place, and thus how much current flows through the cell. Its dynamics is dictated by the differential charge conservation law at the interface2
∂φ(r, t)

∂φDL (r, t) + iF (r, t), −σ =C (2) ∂z
z=WE ∂t where C is the capacitance of the double layer, σ is the conductivity and iF is the Faradaic current, which couples the dynamics of φDL to the electrochemical kinetics at the interface. The differential term on the l.h.s. of Eq. (2) is the local electric current density reaching the electrode from the electrolyte. Now we can grasp how the nonlocal coupling, also referred to as migration coupling, influences the dynamics. Let us assume, because of some fluctuation, φDL (r) changes locally. Since the potential inside the metal is always constant, this leads to a change of φ(r)|z=WE . As a consequence, the entire potential distribution in the electrolyte adjusts to this changed boundary conditions, which also affects the normal component of the electric field at the entire working electrode, and, therefore, also the current flux into the WE is changed at all positions, though to a different amount (l.h.s. of Eq. 2).

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

241

For a given non-uniform potential distribution at the WE, the way in which the non-uniformity decays in the electrolyte to the uniform value at the CE depends on the distance between the WE and the CE. If the CE is sufficiently far away from the WE, any local disturbance of φDL (r) decays into the electrolyte in a characteristic three-dimensional manner. If, however, the CE is close to the WE, a disturbance of φDL (r) has no space to fall off parallel to the electrode and the potential will essentially drop linearly to the potential value at the CE. In this case, the potential profile within the electrolyte is affected essentially only at locations r that are close to the disturbance. Therefore, also the migration coupling remains local. At intermediate distance, the characteristic range over which a disturbance decay is intermediate. Hence, in an electrochemical experiment, the range of the migration coupling depends on the distance between the WE and the CE, which also offers the experimentalist the exceptional opportunity to tune the coupling range experimentally. The two limits are a local (diffusion-like) coupling for vanishing distance between the two electrodes, and a coupling range with some global contribution that adjusts for distances w ≥ 4L, where w is the distance between the WE and the CE and L the extension of the WE in one dimension. There has been some effort in expressing mathematically the migration coupling in a more transparent way than in our original, physically motivated Eq. (2). First of all, it is desirable to separate the terms contributing to the uniform dynamics from those that are only effective in a non-uniform situation, and thus constitute pure spatial coupling terms. When the double layer potential is uniform, the term on the l.h.s. of Eq. (2) takes the form
σ ∂φ(r, t)

= φ|zWE . (3) −σ
∂z w zWE In case of an inhomogeneous potential distribution, the deviation from homogeneity can be expressed as  
∂φ φ

imig. coupling = −σ + , (4) ∂z w
zWE which provides the spatial coupling through the electric field in electrochemical systems. Using a Green function,14 a term for the nonlocal coupling can be derived that does not depend on φ(r, z) anymore, rather, Eq. (2) is mathematically closed in terms of φDL . Most experiments and simulations have been done with ring electrodes. It could be shown that these electrodes could be well described with a simplified cell geometry where the electrolyte is modeled two-dimensional and confined to a cylindrical

November 21, 2012

242

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

surface. The WE and the CE lie at the two ends of the cylinder. For this geometry, the Green function formalism gives the following expression for the migration coupling:  
∂φ φ

+ imig. coupling = −σ ∂z β
zWE  +L/2 =σ Hβ (|x − x |) [φDL (x ) − φDL (x)] dx , (5) −L/2

with Hβ (|x − x |) given by Hβ (|x − x |) =

4β 2 sinh2

π 

π(x−x
) 2β

+

δ(|x − x |) . β

(6)

Here, β = w/L denotes the aspect ratio of the cell and controls the range of the migration coupling. The latter equation is the nonlocal kernel for 1D electrochemical systems. A crucial quantity is the Fourier transform of Eq. (6) given by (q)

Hβ = −q coth(qβ) +

1 , β

(7)

where q ≡ 2πn for periodic boundary conditions. In the limit β → 0 L the NLC becomes local (diffusion-like, ∼ q 2 ) with “diffusion coefficient” σβ/3.17 2.2.

Electrochemical oscillations and the Nonlocal Complex Ginzburg–Landau Equation (NCGLE)

In the following, we focus on oscillating electrochemical systems. In general, besides the double layer potential, also some concentrations of reacting species or surface coverages change with time. Most of the oscillatory systems are linked to an N-shaped polarization curve, and it is the middle branch with a negative differential resistance that renders the stationary dynamics unstable and therefore is the heart of the oscillation mechanism. This type of oscillators has been termed N-NDR oscillator. In this review, we do not consider the reaction mechanism of specific electrochemical N-NDR oscillators. Rather, we consider a generic model which is described by the evolution equations of the double layer potential, the activator in the model, and the concentration of a chemical species, c, the inhibitor, and we first focus on the spatio-temporal dynamics brought about by migration coupling and then by an additional global coupling. A pattern at the electrode is accompanied by both, a spatially varying double layer

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

243

potential and varying concentrations. However, since diffusion of chemical species is slow compared to migration coupling, and a slow diffusion of the inhibitor variable is known not to affect pattern formation qualitatively, only the homogeneous reaction kinetics has to be considered. Our general model for N-NDR oscillators thus reads:  
∂φ φ

, (8) ∂t φDL = f (φDL , c) − σ + ∂z β
zWE

∂t c = g(φDL , c),

(9)

where f (φDL , c) and g(φDL , c) are functions specifying the homogeneous dynamics, and all quantities are assumed dimensionless. Suitable transformations are, e.g. given in Ref. 30. Recently, we have derived a nonlocal complex Ginzburg–Landau equation (NCGLE)16 as a general model for electrochemical oscillators under NLC close to a supercritical Hopf bifurcation. The NCGLE is a partial integro-differential equation since the spatial coupling can no longer be specified by a Laplacian operator as in the CGLE but rather by an integral operator containing a nonlocal kernel. The NCGLE reads16 ∂t W = W − (1 + ic2 )|W |2 W  +(1 + ic1 ) Hβ (|x − x |)[W (x ) − W (x)]dx ,

(10)

WE

where W is the (complex) amplitude, c1 and c2 are dimensionless parameters that can be calculated from the homogeneous dynamics accompanying the nonlocal and nonlinear terms, respectively, and β is the coupling range, the new essential parameter of the NCGLE when compared to the CGLE. β controls both the range and the normalization of the nonlocal kernel Hβ (|x − x |), as described above. The kernel is given by Eq. (6) for 1D ring electrodes. Despite the nonlocality the NCGLE describes, it was derived by means of a center manifold reduction.16 The CGLE is regained from the NCGLE when β → 0, i.e., for vanishing coupling range (local coupling limit) with an extra “diffusion factor” β/3. The factor β/3 is non-essential in this limit (i.e., β and x are no longer independent) and can  be absorbed in the spatial scaling by making the transformation x → β/3 x. Since β → 0, this is equivalent to consider an infinitely large system. We obtain in this case the CGLE. 2.3.

Electrochemical turbulence and the NCGLE

The first manifestation of the nonlocal coupling on traveling waves was observed in a bistable electrochemical reaction.31 During the reduction of

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

H(x−x’)

244

0 −L/2

0 x−x’

L/2

Fig. 1. Coupling functions Hβ for ring working electrodes with circumference L for two different values of β. Solid line: large aspect ratio β resulting in nonlocal spatial coupling. Dashed line: very small aspect ratio β resulting in local (diffusive) spatial coupling.

persulfate on a Ag-ring electrode the velocity with which the interface between a high-current state and a low-current state propagated on the electrode was found not to be constant in time, as it is the case in a diffusively coupled bistable system. Rather, the trigger fronts accelerated during the transition. When we consider the coupling function14 depicted in Fig. 1 for the nonlocal case, the accelerated front propagation becomes comprehensible: The coupling range extends over the entire electrode. Therefore, the effect of the migration coupling on a location in the interfacial region increases as the portion of the electrode that has undergone already the transition to the globally stable state increases. In the oscillatory region, migration coupling can destabilize the uniform oscillation giving rise to electrochemical turbulence. This scenario was studied during the electro-oxidation of hydrogen on a 1D Pt ring-electrode in the presence of Cu2+ and Cl− ions.15 Case 1 in Fig. 2 depicts experiments for a large value of the nonlocal coupling range β. Global current time series are shown together with the temporal evolution of the profile of the double layer potential for three different values of the external applied voltage U . As U is increased, the initially uniform, simple periodic oscillations develop strong spatial modulations before they fall apart into smaller oscillating regions. Hence, a transition from limit cycle behavior to spatiotemporal turbulence is observed upon increasing the overpotential for the hydrogen oxidation and thus the dissipation in the system. This is analogous to

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

245

Fig. 2. Global current time series and spatiotemporal evolution of the interfacial potential, φDL , as a function of position on the ring and time for different values of the applied voltage U and electrode placement corresponding to large β (case 1) [plates (a), (b), and (c)] and low β (case 2) [(d), (e), and (f)]. U: (a) 1.06 V, (b) 1.19 V, (c) 2.14 V, (d) 0.82 V, (e) 0.89 V, and (f) 1.54 V. Electrolyte: H2 -saturated, aqueous 0.5 mM H2 SO4 solution containing 0.1 mM HCl and 0.01 mM CuSO4 . A continuous flow of H2 was maintained throughout the experiments.15

the Rayleigh–B´enard convection where the increase of the temperature gradient drives the system from conductive heat transport over regular convection rolls to more incoherent states. A first impression of the impact of the nonlocality of the spatial coupling on this transition scenario can be seen when comparing these experiments with others done under local coupling (case 2, Fig. 2). We observe that the larger the coupling range the larger are the characteristic lengths of the non-connected domains. More quantitative aspects can be extracted from the Hilbert transform of the experimental data. Examples for the experimental spatiotemporal evolution of the amplitude and the phase in a situation of electrochemical turbulence are plotted in Fig. 3. Clearly, there are locations at which the amplitude vanishes and the phase changes abruptly, see the white circles in Figs. 3(a) and 3(b). This indicates the presence of space-time defects.32 From the phase representations the density of defects for the two transitions

November 21, 2012

246

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

Fig. 3. Experimental spatiotemporal data obtained for phase (a) and amplitude (b) in a situation of electrochemical turbulence. Circles in (a) and (b) exemplify the locations of phase defects. The displayed time interval is 22.5 s. (c) Defect density vs. the applied voltage for local coupling, i.e., β small (solid circles) and nonlocal coupling (i.e., β large) (open circles).15 (c) Spatiotemporal evolution of the modulus of the amplitude |W | obtained from Eq. 10 for c1 = −1.03, c2 = 2.68, L = 100 and β = 1 (c) and 50 (d). Low |W |: dark.16

were determined. The results are summarized in Fig. 3(c) where the defect density is plotted versus the applied voltage for case 1 (solid circles) and case 2 (open circles). In both data sets, the defect density increases with increasing voltage starting from zero, revealing that both series exhibit a transition into a defect turbulent regime. This experimental finding could be elucidated through analysis and simulations employing the NCGLE16 [see Fig. 3, where the spatiotemporal evolution of the modulus of the amplitude is plotted for local coupling (d) and large nonlocal coupling (e)]. The NCGLE does not depend on any free adjustable parameter and was satisfactorily mapped to the dynamics governing an experimental electrochemical oscillator.16 c1 and c2 in the NCGLE, Eq. (10) were explicitly calculated for a model of the reduction of IO− 4 on an Au electrode which is known to be a N-type negative differential resistance (N-NDR) electrochemical oscillator.2,30 The two-parameter bifurcation diagram is plotted for this system in Fig. 4(a). A Hopf line appears, separating a wide region where there is only a single stable stationary state, from an oscillatory regime. The Hopf line has a wide regime where the bifurcation is supercritical. In the vicinity of the supercritical Hopf bifurcation, c1 and c2 were calculated from the steady state, and thus, the dynamics were mapped onto the NCGLE. The resulting

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

247

Fig. 4. (a) Hopf curve for the N-NDR oscillator in the σc vs. U plane (continuous line: Supercritical Hopf bifurcation (SHB); dotted line: subcritical Hopf bifurcation; DH: degenerate Hopf point). Uniform oscillations are unstable between the α = 0 and the DH points. Beyond the DH, the Hopf bifurcation is subcritical and the analysis is not applicable.16 (b) Values for c1 and c2 for the N-NDR dynamics on the SHB shown in Fig. 4. Inset: c1 , c2 and α vs. the applied potential U for the region within the shaded box. Arrows indicate the direction of increasing U.16

curve is shown in grey in Fig. 4(b). The BF curve denotes the instability line α = 1 + c1 c2 = 0, which coincides with the Benjamin–Feir line in the CGLE. A broad parameter range in which the electrochemical oscillator is in the turbulent regime was found,16 confirming our above interpretation that electrochemical oscillators can undergo generically a transition to electrochemical turbulence. With increasing U, the system is driven deeper into the turbulent regime, in agreement with the experimental observations. As already discussed above, the NLC cannot stabilize the uniform oscillation in the BF unstable regime.16 However, there exist a wide variety of coherent structures as standing waves, heteroclinic connections between fixed points and between limit cycles and some other patterns having different translation symmetries17 for a large coupling range that are absent in the CGLE. 3.

3.1.

Electrochemical Oscillators Under Global Coupling (GC) Physical origin of the global coupling

The electric control of an electrochemical cell can easily introduce a global coupling (GC) into the system, and in fact, a GC is present in most electrochemical experiments. Usually, in an electrochemical experiment either the potential of the WE is controlled (potentiostatic or potentiodynamic condition) or the current flowing through the cell is given from

November 21, 2012

248

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

outside (galvonostatic or galvanodynamic control). In the latter case the origin of the GC is apparent: A local fluctuation in the current density (due to a local fluctuation in a state variable, i.e., a concentration or the electrode potential) affects the total current flowing through the system. As a consequence, the galvanostat will drive the potential of the WE to larger or smaller values thereby compensating the deviation in local current density. The change of the WE potential, however, is felt by all positions of the electrode and alters the double layer potential everywhere. Thus, a local change in a state variable affects the dynamics of all other positions of the electrode with the same weight. This is the main feature of a global coupling. All the arguments above hold also for an external ohmic series resistor Re between the WE and the reference electrode (RE) under potentiostatic conditions. A small change in the current changes the voltage drop across the resistor, and thus also the one across the double layer at any position of the electrode. In fact, the galvanostatic mode can be considered as a potentiostatic mode in the limit of U → ∞, and Re → ∞, whereby the set current I is equal to the finite ratio I = U/Re . As we will discuss below, the GC brought about by an external resistor acts always synchronizing and is said to be positive.34 The other important experimental situation which introduces GC in the dynamics of an electrochemical system is any compensation of an IR drop through the electrolyte. This is easily seen when employing an electronic compensation built in most potentiostats. This electronic compensation does nothing else than emulating a negative ohmic impedance in the external circuit. Often, the IR drop compensation is accomplished by using a Haber–Luggin capillary which is placed close to the WE. Also in this configuration the partial compensation of the cell resistance affects the double layer dynamics. If the Haber–Luggin capillary (or equivalently the RE) is placed on the axis of a ring electrode, its effect is again to introduce a negative global coupling into the system.23 If the distance between the RE and different locations of the WE is different, then the GC is weighted accordingly. The strength of the global coupling can be expressed through a parameter γ ≡ 1 − RΩ /(Re + Ru ), with γ ∈ (−∞, 1],23 where RΩ and Ru are the total cell resistance and the uncompensated cell resistance, respectively. Mathematically, the GC relates the value of the double layer potential φDL at one  point x to the spatial average . . . taken over the entire WE φDL ≡ WE φDL (x )dx /L, where L is the length of the WE. An additional term14,23 modifies Eqs. (8) and (9)  ∂t φDL = f (φDL , c, γ) − σ ∂t c = g(φDL , c).


∂φ φ

γσ + (φDL − φDL ), (11) + ∂z β
zWE β (12)

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

249

Note that also the homogenous dynamics depends on γ such that experimentally it is not easy to vary the strength of the global coupling without altering the homogeneous dynamics. From the last term of Eq. (11) it is now easy to see that a positive value of γ tends to smooth any inhomogeneity (synchronizing coupling) and when γ < 0 the GC tends to disfavor the homogeneous state (desynchronizing coupling). 3.2.

Global coupling induced pattern formation in experiments

Most of the patterns observed in electrochemical systems were in fact induced by a GC. Very roughly, we can discriminate three situations: (a) The global coupling destabilizes the uniform stationary state. In this case, the destabilization can lead to a stationary, domain-type scale free pattern, as observed for the persulfate reduction on Ag ring-electrodes or hydrogen oxidation on Pt35,36 as well as the electrocatalytic oxidation of CO on Pt,37 or to standing or traveling waves which emerge in a non-trivial Hopf bifurcation and have been studied in a variety of electrochemical systems.30,38 –41 Furthermore, close to these bifurcation, many variants of partially complex, traveling wave type patterns were observed. (b) Depending on its sign, the global coupling destabilizes the uniform oscillation, giving rise to oscillating cluster patterns,24,26 or suppresses turbulent states, and thus stabilizes the uniform limit cycle. Below, we will discuss this situation in more detail since here it is again possible to derive an extended NCGLE, which captures many of the observed patterns and thus yields insight into the generic dynamics of GC electrochemical oscillators. (c) Special patterns or behaviors that require the presence of both, a nonlocal and a global coupling. Among them are asymmetric target patterns42 or remotely triggered waves.43 Besides traveling waves, cluster patterns are the most robust patterns forming in oscillating electrochemical reactions under negative GC. Figure 5 depicts experimentally observed cluster patterns during the electrooxidation of hydrogen in the presence of Cu2+ and Cl− ions.24 Figure 5(a) shows a typical 2-phase cluster pattern being composed of two domains that oscillate in antiphase with a constant phase shift of π. These patterns have been coined type I cluster pattern to discriminate them from the ones shown in Fig. 5(b). Here, again, clearly, the electrode splits into two domains, each of which oscillates uniformly and with a certain phase shift towards the other one, which here, however, changes with time. Performing a Karhunen– Loeve decomposition reveals that the clustering occurs exclusively in the subharmonic mode onto which the uniform oscillation is superimposed.24 Therefore, these cluster patterns were also termed subharmonic cluster patterns. Another type of peculiar subharmonic cluster patterns were

November 21, 2012

250

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

Fig. 5. Time series of the global current (upper plates), spatiotemporal evolution of the interfacial potential drop along a ring electrode (middle plates), and spatiotemporal evolution of the inhomogeneous part of the interfacial potential drop (lower plates) for (a) Type I clusters and (b) Type II clusters, found at weak and strong global coupling, respectively.24

reported for the anodic dissolution of n-Si(111) in fluoride containing electrolytes.26 In this case, a rectangular 2D electrode was employed and the data were recovered by electrochemical ellipsometric imaging. As can be seen in Fig. 6, a labyrinthine structure develops on the electrode surface, being composed of oxide arms with different thickness. Further analysis revealed three dominant features of the patterns: (1) The labyrinthine pattern existed only in the subharmonic mode, and in this mode adjacent arms oscillated in antiphase. (2) Superimposed to the subharmonic mode was a uniform oscillation that also lead to a pronounced regular oscillation of the global current. (3) The resulting subharmonic labyrinthine cluster pattern was further modulated by an irregular background such that the local oscillators exhibited chaotic amplitude modulations while all maxima of the local time series were locked. Cluster patterns have also been observed

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

251

Fig. 6. Spatiotemporal data during n-Si electrodissolution. (a) and (b) Ellipsomicroscopic snapshots of the Si electrode taken at subsequent maxima of the average light intensity. (c) Spatiotemporal evolution of the local light intensity for the 1D cut indicated in (a). (d) Local time series of the light intensity for the three points indicated in (c). (e) Time evolution of the average light intensity along the 1D cut. (f) Time evolution of the total current.26

in a variety of other chemical oscillating systems under GC or a global feedback.44,45 Yet, from a theoretical point of view many aspects of their formation are still unclear. Type I phase clusters are, in fact, predicted to exist by an extended version of the NCGLE, where a linear global coupling has been added. This will be discussed in Sec. 3.3. Type II

November 21, 2012

252

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

phase clusters typically occur at higher coupling strength, and we will demonstrate in Sec. 3.4 that when combining the NCGLE with a nonlinear GC subharmonic cluster patterns indeed form. 3.3.

Linear global coupling in electrochemical systems and the normal form approach

Using again a perturbation approach, the physical equations (11) and (12) can be mapped to the following extended NCGLE ∂t W = W − (1 + ic2 )|W |2 W + (1 + ic1 )  × Hβγ (|x − x |)[W (x ) − W (x)]dx ,

(13)

WE

where Hβγ is defined by



γ Hβγ (|x − x |) ≡ K Hβ (|x − x |) + , βL 



(14)

with Hβ (|x − x |) given by Eq. (6) and with K being a complex parameter proportional to the conductivity. This extended coupling kernel comprises both NLC and GC. The kernel of Eq. (14) is represented in Fig. 7 for different values of γ and intermediate coupling range β. The global coupling shifts the curves of the NLC function vertically to more positive values (γ > 0) or negative values (γ < 0). In the first case, the GC enhances

Fig. 7. Hβγ calculated from Eq. (14) for β = 0.1, K = 1 and the values of γ indicated in the figure.17

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

253

the synchronizing effect of the NLC, while for negative GC it introduces longrange inhibition. The linear stability analysis of the uniform oscillation25 yields a stability matrix J whose trace must be negative and whose determinant must be positive in order to have stability. The condition for the trace is obtained as25 1 1 γ > − − 2π coth(2πβ) + , β K β

(15)

which holds automatically for positive GC (i.e., γ > 0). The condition for the determinant reads −2α 1 γ > − 2π coth(2πβ) + , 2 β (1 + c1 )K β

(16)

For γ = 0, i.e., in the NCGLE, the condition for the trace is automatically satisfied and only the condition for the determinant remains, which reduces to the well known Benjamin–Feir stability criterion α ≡ 1+c1 c2 > 0, as also discussed above. Next, let us consider the case of a positive global coupling γ > 0. This clearly helps satisfying both inequalities, and has thus a purely synchronizing effect. As a consequence, when considering a BF stable state at γ = 0, a positive global coupling will drive the system further into the stable region. More importantly, a sufficiently large positive GC will stabilize a turbulent state in the entire c1 –c2 plane and arbitrary system size. In contrast, in case of negative global coupling γ < 0, either one of the inequalities (15) or (16) might be violated individually or both of them might be violated, which leads to a large variety of spatiotemporal patterns. Let us first consider the case when the trace becomes positive and inequality (15) is violated, while the condition for the determinant is still satisfied. This means that sufficiently negative GC can destabilize the homogeneous oscillation making the pair of eigenvalues for a given critical inhomogeneous n to cross the imaginary axis. Usually, such a critical eigenvalue is accompanied by an entire band of unstable eigenvalues, and the resulting pattern is, in general, turbulent. As discussed above, Eq. (16) can be violated in the absense of global coupling, and then electrochemical turbulence emerges. A negative value of γ in such a situation might then cause the emergence of a further maximum in the dispersion relation and modulated waves with a characteristic wavelength might arise.25 For larger β and constant γ/β the unstable band displaces to longer wavelengths as observed above for electrochemical turbulence. When both Eqs. (15) and (16) are violated the pattern is turbulent in general, but since there can be simultaneously two bands of unstable modes, these active modes can resonate leading to the formation of phase clusters.25 By

November 21, 2012

254

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

Fig. 8. Type I clusters: Spatiotemporal evolution of |W | (a) and φ (b) obtained from numerically integrating Eq. (13). Parameter values: K = 0.5, γ = −1.5 and c1 = −0.5, c2 = 2.0, β = 0.37 (a), β = 0.25 (b) and β = 0.2 (c). Right: snapshot of W in the complex plane.25

violating the condition for the trace (i.e., making γ more negative), the uniform oscillation is made unstable already by the wavenumber n = 1, that becomes active in the pattern. In its resonance with active modes of shorter wavelengths, the latter breaks the translation symmetry that leads to the cluster formation.25 The calculated phase clusters possess also all the characteristic properties of the experimentally observed type I phase clusters (cf. Fig. 5(a)). For lower β, the walls oscillate (Fig. 8(c)) and in the local coupling limit (here β < 0.004 approx.) we are left with turbulence (not shown). 3.4.

Nonlinear global coupling (NGC) and its normal form

The extended NCGLE with a linear GC does not predict the existence of subharmonic, type II phase clusters. In the above shown experiments (cf. Figs. 5(b), and 6), such phase clusters emerged for rather high values of the

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

255

global coupling. Therefore, we consider now a situation where a nonlinear GC (NGC) is added, that is strong enough to generate strongly resonant behavior. An external forcing46 is known to describe resonant patterns, but it breaks phase invariance, i.e., the dynamics of the complex amplitude is no longer invariant with respect to a shift in phase W → W eiχ . This situation is not met in the experiments made with electrochemical systems under GC. There, phase invariance is preserved. Since symmetry considerations are closely linked to experimental conditions, it was needed then to explain what can be the mechanism leading to the formation of certain subharmonic resonant patterns that have been observed2,21 in experimental systems under GC without introducing any external forcing. In this section we discuss the appropriate normal form to address this problem. The NGC has been introduced27 so that a further experimental observation, the conservation of the oscillatory state of the homogeneous mode, is explicitly accounted for. Quite generally we can consider a reaction–diffusion system governed by a modified Ginzburg–Landau equation (MCGLE)26,27 ∂t W = W + (1 + ic1 )∂x2 W − (1 + ic2 )|W |2 W + B

(17)

B = −(1 + iν)W + (1 + ic2 )|W |2 W

(18)

where

W is a complex amplitude and . . . denotes again spatial averages. The terms on the r.h.s. of Eq. (18) describe the nonlinear effect of the average amplitude W on the dynamics and can be thought as an expansion of a negative GC to third order. Clearly Eq. 17 is invariant under phase transformation W → W eiχ . In Fig. 9(a), a spatial pattern where the L/2 translation symmetry is broken is observed in |W |: the amplitude shifts L/2 spatially at the base frequency so that the same oscillatory state for each point in space is attained at a period that is twice the one of the base frequency ν. An inspection of the phase, plotted in Fig. 9(b), shows indeed two different 27 analysis spatial domains that oscillate with different phases.  ∞ . A Fourier of the complex amplitude in time W (x, t) = −∞ aω (x)e−iωt dω can be performed.26,27 In Fig. 9(c) the cumulative power spectrum of the time  L/2 series |aω |2 = −L/2 |aω (x)|2 dx shows two peaks, one at the main frequency ν and the other at ν/2 that correspond to the main active modes in the pattern. In Figs. 9(d) and 9(e) we plot, respectively, the arrangement of the phase in the complex plane corresponding to the modes at frequency ν and ν/2. A locking in the phase for all oscillators is observed in the main mode, while in the subharmonic mode, a 2-phase cluster exhibiting an Ising

November 21, 2012

256

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

Fig. 9. Type II clusters: Spatiotemporal evolution of (a) the modulus |W | and (b) the phase of the complex amplitude W obtained from Eq. (17) for c1 = 0.2, c2 = −0.8, ν = 1, L = 50 and η = 0.6. (c) Power spectrum |aω (x)|2 of the time series of the amplitude integrated over space. (d) Projection on the complex plane of the mode aν corresponding to the main peak at ω = ν and (e) of the subharmonic mode at half the frequency aν/2 .27 (f) Experimental evolution of the phase.24

wall is observed. These are exactly the same conclusions we have reached above from the Karhunen–Lo`eve decomposition of the experimental data. Furthermore, these Type-II clusters appear at a higher value of the coupling strength compared to the Type-I clusters24 which, in fact, do not require a NGC but a linear GC, as we have shown in the previous section.16 Type-II clusters were also found in experiments with catalytic CO oxidation on

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

257

Pt(110) electrode under delayed global feedback45 although no attempt was made to connect them to a normal form. The experimental results on the spatiotemporal dynamics during Si electrodissolution exhibited Type II clusters, with a turbulent, irregular background superimposed (cf. Fig. 6). These unusual subharmonic cluster patterns were satisfactorily described with Eq. (17) by adding an additional term, a 1:1 self-induced resonant forcing γf , which might be related to the microscopic interfacial dynamics. A resulting simulation is shown in Fig. 10(a). Despite its rather turbulent appearance, further analysis reveals a high degree of organization. In Fig. 10(b) we see, for example, that any individual oscillators, although having complicated modulations in the amplitude, are phase locked to the average signal Wr = ReW , possessing the same base periodicity, as is the case of the Type II clusters. The time

Fig. 10. (a) Spatiotemporal evolution of the real value Wr of the complex amplitude W obtained from Eq. (17) for c1 = −10, c2 = 1.5, γf = 1.55 and ν = 3.1. (b) Local time series of Wr for two individual oscillators and spatial average Wr . (c) Spatial distribution of the phase for the first subharmonic Fourier mode w1/2 of the time series at frequency ν/2. (d) Spatial distribution of the modulus of the amplitude for the first subharmonic Fourier mode w1/2 of the time series at frequency ν/2. (e) and (f) Representation in the complex plane of w1 and w1/2 , respectively.26

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

258

series of Wr is shown for two individual oscillators and Wr is also shown for comparison. It is observed that maxima and minima between the global average signal and the one of individual oscillators occur simultaneously at a frequency ν. The periodicity of the average signal reflects the one of the pulsation of the individual oscillators although the modulation of the amplitude displays a complicated behavior. By performing a Fourier analysis of the time series for each oscillator, two main peaks in the power spectrum are found, one at frequency ν and the other at the subharmonic ν/2. This is the case for the whole spatial distribution of oscillators. From simulations with our theoretical model it was concluded26 that the nonlinear self-induced global coupling is responsible for the 1:1 entrainment of the system, while the self-induced 1:1 resonance contributes to the observed subharmonic 2-phase clustering superimposed to the turbulent background.

4.

Conclusions

We emphasized in the review origin and role of nonlocal and global coupling for pattern formation in electrochemical systems. Special focus was placed on oscillatory conditions. The most frequently experimentally observed patterns were electrochemical turbulence under NLC and cluster patterns under GC. These dynamics could be captured with the normal form approach, reviewed here, which provides a concise and general way of describing patterns in spatially extended oscillatory electrochemical systems. The normal forms go beyond the CGLE1 formulated for reaction– diffusion systems, since the spatial coupling between oscillators is nonlocal or global. For NLC and linear GC, as found in electrochemistry, normal forms were derived rigorously by means of a center manifold reduction,16,25 leading to expressions without freely adjustable parameters. Strongly resonant subharmonic patterns were found in experiments as well24,26 and a normal form with a nonlinear GC was formulated to account for these situations.27 Normal forms are advantageous compared to the specific dynamics of each electrochemical system since they describe the same patterns with only the essential number of dynamical parameters. Normal forms allow for straightforward symmetry considerations of general validity and thus provide deep insight in spatiotemporal pattern formation.

Acknowledgments Financial support from the cluster of excellence Nanosystems Initiative Munich (NIM) is gratefully acknowledged. V. G.-M. acknowledges also

November 21, 2012

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators

259

financial support from the Technische Universit¨ at M¨ unchen — Institute for Advanced Study, funded by the German Excellence Initiative. References 1. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics (Springer, Berlin, 1984). 2. K. Krischer, in Advances in Electrochemical Sciences and Engineering, eds. D. M. Kolb and R. C. Alkire (Wiley-VCH, Weinheim, 2003), p. 89. 3. H. L. Heathcote, Z. Phys. Chem. 37, 368 (1901). 4. W. Ostwald, Z. Phys. Chem. 35, 33 (1900). 5. W. Ostwald, Z. Phys. Chem. 35, 204 (1900). 6. R. S. Lillie, Science, XLVIII, 51 (1918). 7. R. S. Lillie, J. Gen. Phys. 7, 473 (1925). 8. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993). 9. A. S Mikhailov and K. Showalter, Phys. Rep. 425, 79 (2006). 10. Y. Kuramoto, Prog. Theor. Phys. 94, 321 (1995). 11. D. Tanaka and Y. Kuramoto, Phys. Rev. E 68, 026219 (2003). 12. Y. Kuramoto, D. Battogtokh and H. Nakao, Phys. Rev. Lett. 81, 3543 (1998). 13. D. M. Abrams and S. H. Strogatz, Phys. Rev. Lett. 93, 174102 (2004). 14. J. Christoph, PhD. thesis, Free University, Berlin (1999); J. Christoph et al., J. Chem. Phys. 110, 8614 (1999). 15. H. Varela, C. Beta, A. Bonnefont and K. Krischer, Phys. Rev. Lett. 94, 174104 (2005). 16. V. Garcia-Morales and K. Krischer, Phys. Rev. Lett. 100, 054101 (2008). 17. V. Garcia-Morales, R. W. H¨ olzel and K. Krischer, Phys. Rev. E 78, 026215 (2008). 18. R. Imbihl. Prog. Surf. Sci. 44, 185 (1993). 19. F. Mertens, R. Imbihl and A. Mikhailov, J. Chem. Phys. 99, 8668 (1993). 20. F. Mertens, R. Imbihl and A. Mikhailov, J. Chem. Phys. 101, 9903 (1994). 21. H. Levine and X. Zou, Phys. Rev. Lett. 69, 204 (1992); G. Veser, F. Mertens, A. Mikhailov and R. Imbihl, Phys. Rev. Lett. 77, 975 (1993); M. Falcke and H. Engel, J. Chem. Phys. 101, 6255 (1994); M. Falcke, H. Engel and M. Neufeld, Phys. Rev. E 52, 763 (1995); D. Battogtokh and A. Mikhailov, Physica D 90, 84 (1996). M. Kim et al., Science 292, 1357 (2001); R. Imbihl. Prog. Surf. Sci. 44, 185 (1993). 22. R. D. Otterstedt et al., J. Chem. Soc., Faraday Trans. 92, 2933 (1996); J. Lee et al., J. Chem. Phys. 115, 1485 (2001); I. Z. Kiss, W. Wang and J. L. Hudson, Chaos 12, 252 (2002); Y. M. Zhai, I. Z. Kiss and J. L. Hudson, Ind. Eng. Chem. Res., 43, 315 (2004); S. Fukushima et al. Chem. Phys. Lett. 453, 35 (2008). 23. K. Krischer, H. Varela, A. Birzu, F. Plenge and A. Bonnefont, Electrochim. Acta 49, 103 (2003). 24. H. Varela, C. Beta, A. Bonnefont and K. Krischer, Phys. Chem. Chem. Phys. 7, 2429 (2005).

.

November 21, 2012

260

13:2

9in x 6in

b1449-ch12

Engineering of Chemical Complexity

V. Garcia-Morales & K. Krischer

25. V. Garcia-Morales and K. Krischer, Phys. Rev. E 78, 057201 (2008). 26. I. Miethe, V. Garcia-Morales and K. Krischer, Phys. Rev. Lett. 102, 194101 (2009). 27. V. Garcia-Morales, A. Orlov and K. Krischer, Phys. Rev. E 82, 065202(R) (2010). 28. V. Garcia-Morales and K. Krischer, Contemp. Phys. 35, 79 (2012). 29. I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002). 30. F. Plenge, Y.-J Li and K. Krischer, J. Phys. Chem. B 108, 14255 (2004). 31. G. Fl¨ atgen and K. Krischer, Phys. Rev. E 51, 3997 (1995). 32. B. I. Shraiman et al., Physica (Amsterdam) 57D, 241 (1992). 33. K. Krischer, N. Mazouz and G. Fl¨ atgen, J. Phys. Chem. B 104, 7545 (2000). 34. N. Mazouz, G. Fl¨ atgen, K. Krischer and I. G. Kevrekidis, J. Electrochem. Soc. 145, 2404 (1998). 35. P. Grauel, J. Christoph, G. Fl¨ atgen and K. Krischer, J. Phys. Chem. B 102, 10264 (1998). 36. P. Grauel and K. Krischer, Phys. Chem. Chem. Phys. B 3, 2497 (2001). 37. R. Morschl, J. Bolten, A. Bonnefont and K. Krischer, J. Phys. Chem. C 112, 9548 (2008). 38. P. Grauel, H. Varela and K. Krischer, Farad. Disc. 120, 165 (2001). 39. S. Fukushima, S. Nakanishi, Y. Nakato and T. Ogawa, J. Chem. Phys. 128, 014714 (2008). 40. J. Christoph, R. D. Otterstedt, M. Eiswirth, N. Jaeger and J. L. Hudson, J. Chem. Phys. 110, 8614 (1999). 41. P. Strasser, J. Christoph, W. F. Lin, M. Eiswirth and J. L. Hudson, J. Phys. Chem. A 104, 1854 (2000). 42. F. Plenge, H. Varela and K. Krischer, Phys. Rev. Lett. 94, 198301 (2005). 43. J. Christoph, P. Strasser, M. Eiswirth and G. Ertl, Science 284, 291 (1999). 44. V. K. Vanag, A. M. Zhabotinsky and I. R. Epstein, J. Phys. Chem. A 104, 11566 (2000); V. K. Vanag, L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Nature 406, 389 (2000); L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Phys. Rev. E 62, 6414 (2000); M. Bertram and A. S. Mikhailov, Phys. Rev. E 63, 66102 (2001); M. Pollmann, M. Bertram and H. H. Rotermund, Chem. Phys. Lett. 346, 123 (2001). 45. M. Bertram et al., J. Phys. Chem. B 107, 9610 (2003). 46. P. Coullet et al., Phys. Rev. Lett. 65, 1352 (1990); P. Coullet and K. Emilsson, Physica D 61, 119 (1992); P. Coullet, J. Lega, B. Houchmanzadeh and J. Lajzerowicz, Phys. Rev. Lett. 65, 1352 (1990); P. Kaira et al., Phys. Rev. E 77, 046106 (2008); A. L. Lin et al., Phys. Rev. E 69, 066217 (2004); A. L. Lin et al., Phys. Rev. Lett. 84, 4240 (2000); C. Elphick, A. Hagberg and E. Meron, Phys. Rev. Lett. 80, 5007 (1998).

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Chapter 13 QUORUM SENSING AND SYNCHRONIZATION IN POPULATIONS OF COUPLED CHEMICAL OSCILLATORS Annette F. Taylor,∗ Mark R. Tinsley† and Kenneth Showalter† ∗

School of Chemistry, University of Leeds, UK C. Eugene Bennett Department of Chemistry West Virginia University, US

†

Experiments and simulations of populations of coupled chemical oscillators, consisting of catalytic particles suspended in solution, provide insights into density-dependent dynamics displayed by many cellular organisms. Gradual synchronization transitions, the “switching on” of activity above a threshold number of oscillators (quorum sensing) and the formation of synchronized groups (clusters) of oscillators have been characterized. Collective behavior is driven by the response of the oscillators to chemicals emitted into the surrounding solution.

Contents 1. Introduction . . . . . . . . . . . . . 2. Model . . . . . . . . . . . . . . . . 3. Synchronization . . . . . . . . . . . 4. Quorum Sensing . . . . . . . . . . . 5. Spatially-Distributed Particles . . . 6. Clusters . . . . . . . . . . . . . . . 7. Oscillator Death and Multistability 8. Conclusions and Outlook . . . . . . References . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

261

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

262 263 265 267 267 270 273 275 276

November 21, 2012

262

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

Fig. 1. A schematic showing how “cell state”, as measured by the concentration of some species, may depend upon the cell density in a population of cells; for cell density ε > ε3 ; the population is terminally differentiated. Adapted from Ref. 1.

1.

Introduction

The transition from individual to collective behavior in biology is often driven by the production and detection of chemical signals. Connections typically arise directly through specialized channels between cells (gap junctions), or via chemicals emitted into the extracellular solution, in which the dynamics of the organism may be controlled by the cell density (Fig. 1).1 Many cells show density-dependent regulation of proliferation, and bacteria are known to exhibit population-wide activity, such as bioluminescence, following emission of a chemical species (autoinducer) into the surrounding solution. Since group activity requires a threshold number or density of cells, the phenomenon is known as quorum sensing.2 One of the simplest and most fundamental examples of collective behavior is that of synchronization.3 –5 Yeast cells and amoebae of the slime mould D. discoideum display chemical oscillations, and in well-stirred suspensions, populations may develop a collective rhythm. Synchronization of cellular oscillations is believed to be driven by exchange of chemical species with the extracellular solution; oscillatory dynamics depend on the cell density, and the oscillations disappear at low and at high cell densities. Modeling studies1 suggested that the dilution of the chemical signal in the surrounding solution may be responsible for the cessation of oscillations at low cell densities; this was only recently confirmed in experiments.6 The “switching on” of oscillations with increasing cell density was referred to as dynamical quorum sensing.

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

263

Cells, such as yeast, connected via an external solution may be considered as globally coupled oscillators. There are many theoretical studies on the behavior of coupled oscillators,7–9 but comparison with experiments on suspensions of cells has been hindered by difficulties in characterizing the activity of individuals. In order to better understand the collective behavior of populations of coupled oscillators, simpler systems of chemical, electrical or optical oscillators with well-defined kinetics have been exploited.10 –13 In this chapter, we discuss the density-dependent behavior of catalytic micro-particles suspended in catalyst-free Belousov–Zhabotinsky reaction solutions.14,15 Each particle is capable of displaying chemical oscillations and exchanges chemical species with the surrounding solution.16 The concentration of chemical species in the surrounding solution affects the individual dynamics; hence, the number or density of particles is used as a bifurcation parameter. Collective behaviors include gradual synchronization, the sudden appearance of oscillations,17,18 and more complex behavior such as the formation of phase-separated synchronized groups, or clusters,19 of chemical oscillators.

2.

Model

Simulations of populations of catalytic particles were performed using the three-variable ZBKE model20 of the Belousov–Zhabotinsky (BZ) reaction, where X = HBrO2 (activator), Y = Br− (inhibitor) and Z = the oxidized form of the metal ion catalyst. In the well-stirred experiments, each particle undergoes reaction with kinetic terms f (X, Y, Z ) and exchanges X and Y with the surrounding solution, with rate constant k ex .16 Thus for i . . . N particles, the rate of change of X is given by: dXi = −kex (Xi − Xs ) + f (Xi , Yi , Zi ), dt and the concentration in the surrounding solution, Xs , is given by: N V¯
dXs = kex (Xi − Xs ) + g(Xs , Ys ), dt Vs i

where g(Xs , Ys ) contains reaction terms for the catalyst-free solution and V¯ /Vs is the dilution factor: the ratio of the average particle volume to the volume of the solution. The number density of oscillators is given by n = N/V s . Simulations typically involve 1000 oscillators and V s is varied

November 21, 2012

264

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

to alter the number density; however, qualitatively the same results are obtained with changing the number density and keeping V s constant, in both simulations and experiments. The heterogeneity of the population is determined by a Gaussian distribution in one of the parameters: the stoichiometric factor q or the total catalyst concentration C, resulting in a range of oscillatory periods when k ex = 0. For more details on the model, see Ref. 17 (Supplementary Information). Simulations of groups of spatially distributed catalytic particles18 exploited a 3D array of cells diffusively coupled by a six-point Laplacian, with some cells occupied by particles (containing catalyst C ) and the rest with surrounding solution dynamics (C = 0). The profiles for the activator HBrO2 and inhibitor Br− depend on the nature of the catalyst (Fig. 2); the iron complex Fe(phen)2+ (ferroin) or ruthenium complex Ru(byp)2+ 3 3 were both used in experiments and the model parameters were altered to match the catalyst used. Insight into the dynamics can be obtained

Fig. 2. Simulated profiles of activator HBrO2 and inhibitor Br− and phase response curves for a catalytic particle in the BZ reaction with (a) an iron-based catalyst and (b) ruthenium-based catalyst. Adapted from Ref. 19.

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

265

through computation of the phase response curves:21 the phase shift when the oscillator is perturbed at a phase φ in its cycle by a pulse of activator or inhibitor (obtained here using XPPAUT22 ). Phase can be determined from peak to peak, either scaled linearly with time, or using the Hilbert transform (scaled with amplitude); in Fig. 2 the phase is scaled linearly in time. The oscillators have a relatively long refractory period, particularly for the ruthenium catalyst, as they are unresponsive to perturbations in X or Y up to φ ≈ 3π/2. In both cases, pulses of activator mainly result in a phase advance of the oscillation, while pulses of inhibitor mainly delay the next oscillation. In a population of oscillators, the coherence of the oscillations can be measured using the Shinomoto–Kuromoto23 order parameter K :     N N  
  −1
iφj e − N −1 eiφj  K = N (1)   j

j

where   indicates time average and  indicates average over the population. This parameter is 0 for non-oscillating (defining phase as 0 in this case) or not synchronized and 1 for fully synchronized oscillators.

3.

Synchronization

The catalytic microparticles have a range of oscillatory periods that depends on the chemical reactant concentrations in the reaction solution, as well as the particle size24 and catalyst loading. Individual periods can be monitored as color changes during the course of an oscillation in the unstirred solution (Fig. 3(a)). Typically, periods range between 20 and 100 s. However, the period also depends on the exchange rate of chemical species with the

Fig. 3. (a) Spatially distributed catalytic microparticles undergoing chemical oscillations. The white particle is in the oxidized state; trace shows oscillation in particle intensity in time. (b) Experimental set-up for well-stirred experiments. From Ref. 19.

November 21, 2012

266

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

surrounding solution. For experiments involving particles suspended in a stirred solution (Fig. 3(b)), the exchange rate can be controlled by the stirring rate: the higher the stirring rate, the faster chemical species are swept from the particles into the solution. Concentrations in the surrounding solution are monitored using a platinum electrode. For low exchange rates in the well-stirred system, the amplitude of the oscillations in the surrounding solution gradually increased with increasing number density: the experiments involved of the order of 105 particles (104 particles cm−3 ). Images of the catalytic particles taken during the experiment combined with the simulations revealed that the growth in amplitude involved gradual synchronization of the chemical oscillators (Fig. 4(i)).17 This transition can be understood within the framework of the Kuramoto model for synchronization of a population of globally coupled oscillators.25 In such a model, all oscillators are connected through the mean field, the average behavior of the whole population, and, as the coupling strength is increased, there is a growth in the amplitude of the mean field corresponding to the synchronization of the oscillators. In experiments with catalytic particles, the concentration of chemical species in the surrounding solution plays an equivalent role to the mean field, and the coupling strength increases with increasing density of particles. The oscillators mainly respond to changes in the concentration of the activator:

Fig. 4. Simulation results showing two different types of transition to synchronized oscillations in the well-stirred system with increasing density of the catalytic particles: (i) gradual synchronization of oscillators at low exchange rate, and (ii) the sudden switching on of synchronized oscillations at high exchange rate. From Ref. 17.

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

267

when a number of oscillators fire together, a pulse of activator is released into the surrounding solution, which phase advances other (responsive) oscillators. As n is increased, the magnitude of the Xs signal increases, and more of the oscillators join the collective rhythm.

4.

Quorum Sensing

Starting at low number density, increasing the exchange rate results in a growth in the amplitude of the signal in the surrounding solution, corresponding to synchronization of the oscillators; the signal then disappears and images of the reaction vessel show quiescent (red) particles.17 Increasing the exchange rate has a similar role to increasing the coupling strength; however, when oscillators are coupled via the surrounding solution, the activator is diluted and may also decay due to reaction. In this case, the oscillations are no longer supported above a critical exchange rate. However, if the density of particles is then increased for fixed kex , large amplitude, fully synchronized, oscillations suddenly appear (Fig. 4(ii)). For comparison with experiments involving yeast cells, the transition involving “switching on” of oscillatory activity with increasing particle density is referred to as dynamical quorum sensing. The collective nature of the transition can be understood through the order parameter K (Eq. 3) as a function of kex and n. As the exchange rate is increased, the particles synchronize and the coherence K of the population increases to 1 before collapsing to 0, corresponding to the cessation of oscillations on all particles (Fig. 5(a)). For the homogeneous population, the transition occurs for a value of kex that depends on the value of q, and hence the natural period of the population (Fig. 5, colored lines). With increasing number density, the concentration of activator in the surrounding solution increases until activity is suddenly switched on, corresponding to an abrupt transition in K from 0 to 1 (Fig. 5(b)); the value of n for which the transition occurs again depends on the natural period of the homogeneous population. In both cases, the heterogeneous group transition occurs for all particles simultaneously. Both the collapse with k ex and appearance of oscillations with n are therefore collective transitions that overcome the population heterogeneity.

5.

Spatially-Distributed Particles

The globally coupled experiments described above requires that concentrations are homogeneous in the surrounding solution; in cellular systems, such

November 21, 2012

268

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

Fig. 5. Order parameter K in simulations for identical oscillators (colored lines) and for heterogeneous population of oscillators (black line) with (a) increasing kex (n = 1.8 × 104 cm−3 ) and (b) increasing n (kex = 3.0 s−1 ). Histograms of oscillator periods (unstirred conditions: kex = 0.03 s−1 ) with corresponding values of the stoichiometric coefficient q are shown on the right. From Ref. 17.

as bacteria or slime mould, this requires fast transfer to the extracellular solution and slow rates of change in solution. Alternatively, chemical signals may diffusively propagate through a spatially distributed population. Nevertheless, group activity may still emerge above a critical number of cells or group size (rather than density); in the biological literature, this is sometimes referred to as “diffusion sensing”26 rather than quorum sensing. Experiments involving groups of spatially-distributed catalytic particles immersed in catalyst-free BZ solution demonstrate the collective nature of the transition to synchronized activity.18 Quorum sensing transitions are observed in experiments in which the concentrations are tuned such that individual particles do not oscillate. When particles are brought together in a group, the emergence of target waves or spiral waves is exhibited above a critical group size (Fig. 6(b) and (c)). The probability of wave activity is found to increase with increasing group size, and above a critical number of particles, activity is always observed. The particles are locally coupled via the concentration of chemical species in the surrounding solution. In agreement with the experimental

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

269

Fig. 6. Wave activity in spatially distributed groups of catalytic particles; scale bar corresponds to 1 mm. (a) Collective activity in a group of oscillatory particles; (b) and (c) emergence of wave activity in groups of non-oscillatory particles at spiral and target organizing centers, respectively. From Ref. 27.

behavior, above a critical number of particles, wave activity is always observed in simulations (Fig. 7(a)). As the concentration of one of the reactants, bromate, is decreased, the probability of activity decreases, resulting in a less sharp transition for lower concentrations. The emergence of wave activity is found to be associated with a decrease in loss rate of the activator, averaged over the whole group. The degree of heterogeneity also affects the probability of activity — with an increase in the standard deviation of catalyst concentrations, the probability of wave activity decreases (Fig. 7(b)). Thus, heterogeneity suppresses activity in this case. The transition to wave activity is a collective behavior, with the activity dependent on the entire group of particles, rather than the appearance of a pacemaker on a single particle. When groups of oscillatory particles are brought together; more complex behavior than simple target or spiral waves may be observed

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

270

Fig. 7. Emergence of wave activity in 3D simulations of spatially distributed particles. (a) Probability of activity as a function of number of catalytic particles with [BrO− 3 ] = 0.2430 M (solid), 0.2325 M (dashed), and 0.2420 M (dot-dash). (b) Probability of activity for a group of 49 particles with increasing standard deviation of catalyst concentrations on particles (i.e., increased heterogeneity of population). From Ref. 18.

(Fig. 6(a)).27 In these cases, frequency synchronization is observed across the population; however, the phase relationships between particles is dependent on the initial conditions. Despite the patterns appearing more complex, analysis shows the behavior originates from a few organizing centers and the order of the particle activity repeats in time.28

6.

Clusters

Simulations of globally coupled oscillators demonstrate that groups of oscillators separated by a phase difference, known as clusters, may form.8,9,29 Although clusters have been observed in spatially-distributed oscillating reactions and surface reactions (clusters manifest as spatial regions oscillating out of phase with each other),30 –32 there are few experimental examples of cluster behavior in systems consisting of large groups of discrete oscillators. Cluster behavior has been characterized in experiments involving 64 electrochemical oscillators.33 Examples of clusters in large populations of chemical oscillators coupled via the surrounding solution have been observed in suspensions of ruthenium-loaded catalytic particles.19 In these experiments, the electrochemical signal in the surrounding solution is observed to split into two components (Fig. 8). With increasing density, the average amplitude of the oscillations increases, and it takes longer for the signal to split.

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

271

Fig. 8. Electrochemical signal in experiments consisting of well-stirred suspensions of ruthenium-loaded catalyst particles with increasing particle density. From Ref. 19.

Analysis of the images of the particles in these experiments shows that the two components in the global signal correspond to two separate groups of oscillators, with a phase difference between them. The experimental results can be reproduced in simulations with the modified ZBKE model (Sec. 2) using parameters for the ruthenium catalyst. In the experiments discussed in Sec. 3, with ferroin-loaded catalytic particles, the synchronizing force of the activator is dominant. The rutheniumcatalyzed oscillations give larger pulses of the inhibitor Br− compared to the ferroin-catalyzed system, as well as a longer refractory period (Fig. 2). Thus it is not surprising that in the ruthenium-catalyzed system the coupling via the inhibitor tends to dominate rather than the activator, particularly at low densities. The formation of two clusters as the reaction progresses is shown in simulations in Fig. 9(b). Initially the oscillators start with the same (low activator) state. The oscillators fire essentially together, but the oscillators with shorter natural periods lead the group. At the next peak, when the oscillators with shorter natural periods fire, those with longer natural periods are repulsed by the bromide emitted into the surrounding solution and fire at a slightly later time. Thus a small group of particles with short periods is formed in addition to the larger group with period 52 s. The particles with the shortest natural periods fire again, and some more of the oscillators are phase repulsed to join the 2nd group. Following the initial splitting, either a stable configuration of the oscillators is formed, with oscillators in the two groups always firing in the same order, or a stable configuration is not obtained within the timescale of the experiment. Two stable groups give rise to a global signal with constant amplitude in time (Fig. 10). However if the amplitude of the global signal is varying in time, then some of the oscillators are jumping between groups, by a process of phase repulsion or phase attraction arising from the global signal. In Fig. 10, one of the oscillators (blue) at the leading edge of a group is phase advanced when the second group fires and joins this group.

November 21, 2012

272

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

Fig. 9. Formation of clusters in simulations of populations of ruthenium-catalyzed particles coupled via the surrounding solution. (a) Concentration of inhibitor in the surrounding solution. (b) Individual oscillator signals in time and their phase representation (where top of circle corresponds to oscillation peak), their natural periods and the peak-peak period distribution of the first oscillation when the global signal splits. From Ref. 19.

These phase jumps or “switchers” tend to involve particles at the edges of the population distribution of natural frequencies — i.e., those with high or low natural periods and so the appearance of these unstable states largely depends on the exact nature of the period distribution. The number of clusters observed depends on the exchange rate and number density. When either of these parameters is increased, and so the coupling strength is increased, the number of clusters decreases (Fig. 11(a)). The formation of the fully synchronized state requires coupling by the activator: if the exchange of HBrO2 with the surrounding solution is switched off in simulations, then two or more clusters are always observed. At the boundaries between cluster states, the global signal shows intermittency in time, with high frequency oscillations of irregular amplitude followed by lower frequency oscillations (Fig. 11(b)). Analysis shows that the clusters form and disband in time: for the example shown here, the system oscillates between a two cluster and fully synchronized state.

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

273

Fig. 10. Simulations showing (i) stable global signal in time, corresponding oscillators in time and their phase representation (circle), and (ii) unstable global signal in time, and one individual oscillator (blue) jumping to join the other group. From Ref. 19.

Fig. 11. (a) Number of clusters as a function of exchange rate and number density of particles in simulations. (b) Bromide ion concentration in the surrounding solution showing intermittent formation of clusters.

7.

Oscillator Death and Multistability

The transition from activity to quiescence with decreasing number or density of catalytic particles arises because of the loss of activator to

November 21, 2012

274

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

the surrounding solution: individual particles undergo such a transition with increasing dilution factor or exchange rate. This transition therefore does not arise solely as a result of the coupling. All oscillators collapse to a low activator steady state. The phenomenon of oscillator death, or amplitude death, is associated with the collapse of oscillatory behavior arising from the coupling between oscillators.34,35 In general this requires some heterogeneity36 and results in a transition to an inhomogeneous steady state, whereby the group splits and oscillators take either a low or a high activator state. This transition is of interest as it provides a mechanism for terminal differentiation of a population of cells (Fig. 1). Oscillator death was first observed in two continuous flow stirred tank reactors (CSTRs) coupled by mass transfer.37,38 In these experiments, oscillator death was accessed via the anti-phase oscillatory state. Typically, as the coupling strength is increased, the frequency of this state decreases and eventually the oscillators attain the two opposing steady states. This has not been observed in experiments with populations of chemical oscillators. Simulations suggest that exchange of the slow variable is required, and this involves the catalyst Z in the case of the BZ reaction.38 Experiments involving catalyst-loaded particles are therefore not expected to show such a transition. A phenomenon related to oscillator death is that of Turing patterns, which involves coupling via fast inhibitor diffusion and results in the appearance of stationary concentration patterns of a wavelength dictated by reaction parameters. Such a scenario is not possible in the catalytic particles discussed here, as the diffusion coefficients of the activator and inhibitor are approximately equal. However, these patterns have been observed in locally coupled BZ droplets suspended in oil in which fast inhibitor diffusion is possible through the oil phase.12 Theoretical studies suggest that populations of globally coupled oscillators may show multi-stability between cluster states such that any number of clusters may coexist for the same parameter.9,29,39 The coexistence of clusters has been observed in experiments involving globally coupled electrochemical oscillators40 and with photosensitive catalytic particles globally coupled by light.41,42 In the latter experiments, groups of up to 30 particles were found to fully synchronize or form two groups, depending on the initial conditions (Fig. 12(a)). The driving force behind the coexistence of states lies in the existence of both phase delay and phase attraction in the phase response of the photochemical oscillators (Fig. 12(b)). The phase response curve is used to obtain the rate of change of phase difference of two identical oscillators, d∆φ/dt, as a function of their phase difference.25,40,43 Intersection of this curve with 0 gives the steady states and a negative slope gives stable states: if the phase difference is increased, the rate of

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

275

Fig. 12. (a) Coexistence of synchronized state and 2 clusters in a group of globally coupled photosensitive oscillators showing different states emerge when the global coupling is switched on. (b) The phase response curve for 1 photochemical oscillator and (c) the rate of change of phase difference (∆φ = φ2 − φ1 ) of two identical photochemical oscillators. Adapted from Ref. 44.

change of phase difference is negative. Thus the oscillators have stable phase differences, 0 and π. Such an analysis can be extended to a population of oscillators.44,45 In the experiments described here involving catalytic particles globally coupled via chemical species emitted to the external solution, there is no evidence of multistability: the initial conditions do not affect the final number of clusters. With the ruthenium catalyst, the dominant force is that of phase repulsion; under these circumstances the in-phase state is not stable. However, it may be possible to tune the conditions such that the phase-repulsive as well as the phase-attractive part of the activator/inhibitor coupling both play a role. It should then be possible to observe multi-stability in chemical oscillators coupled via an external solution.

8.

Conclusions and Outlook

In this chapter we have described how a population of catalytic particles coupled via chemical exchange with the surrounding solution can be used to investigate transitions to collective behavior. The particles can be

November 21, 2012

13:2

276

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

considered as globally coupled oscillators, where the global coupling is intrinsic to the system rather than an externally imposed feature. Global coupling has been employed in experiments through chemical, electrical and optical means.11 –13 It is of interest to determine which of the observed behaviors are universal and how a system may be driven to a desired state.46 The transition to synchronization in oscillators connected via an external solution requires the release of a chemical species that advances the oscillatory cycle of other oscillators present. Oscillations cease if this species is removed too quickly from the oscillators. In this case, an increase of particle density allows a build up of this species and the sudden appearance of oscillations in a quorum sensing transition. The behavior is collective and overcomes the heterogeneity of the population.17,18 Remarkably, coupling via a single chemical species in an external solution may also lead to the formation of phase-separated groups of oscillators.19 Our experiments were inspired by cellular biological systems such as yeast cells and they may shed light on density-dependent transitions in biology47 . Simulations suggest that genetic oscillators may synchronize their activity when connected via an extracellular solution,48,49 and recent advances in systems biology show that E. coli may be manipulated to display oscillations and synchronize by such a mechanism.50 Bacteria use signalling molecules to trigger certain behaviors, including a concerted attack on a host organism. Synthetic quorum sensing systems may prove useful in understanding how infection arises or even create the possibility to fight it with new methods of drug delivery that mimic biology.

Acknowledgments AFT would like to thank Rita Toth for preliminary experimental results. This work was supported by the US National Science Foundation grant CHE-1212558 (K.S.) and the UK Engineering and Physical Sciences Research Council grant GR/T11036/01 (A.F.T.).

References 1. H. G. Othmer and J. A. Aldridge, J. Math. Biol. 5(2), 169, (1978), ISSN 0303-6812. 2. C. M. Waters and B. L. Bassler, Quorum sensing: Cell-to-cell communication in bacteria, in Annual Review of Cell and Developmental Biology, Vol. 21, pp. 319–346; Ann. Rev. (2005). 3. S. Strogatz, Sync: The Emerging Science of Spontaneous Order (Hyperion, 2003).

November 21, 2012

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

Quorum Sensing and Synchronization

277

4. S. C. Manrubia, A. S. Mikhailov and D. H. Zannette, Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems, World Scientific Lecture Notes in Complex Systems (World Scientific Publishing Company, 2004). 5. A. S. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series (Cambridge University Press, 2001). 6. S. De Monte, F. d’Ovidio, S. Danø and P. G. Sørensen, Proc. Nat. Acad. Sci. USA 104(47), 18377 (2007), ISSN 0027-8424. 7. H. Daido, Phys. Rev. Lett. 73(5), 760 (1994), ISSN 0031-9007. 8. D. Golomb, D. Hansel, B. Shraiman and H. Sompolinsky, Phys. Rev. A 45 (6), 3516 (1992), ISSN 1050-2947. 9. K. Okuda, Physica D 63(3–4), 424 (1993), ISSN 0167-2789. 10. H. Fukuda, H. Morimura and S. Kai, Physica D 205(1–4), 80 (2005). 11. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Science 296(5573), 1676 (2002), ISSN 0036-8075. 12. M. Toiya, V. K. Vanag and I. R. Epstein, Angew. Chem., Int. Ed. 47, 7753 (2008). 13. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo and R. Roy, Phys. Rev. Lett. 105(26) (2010), ISSN 0031-9007. 14. J. Maselko, J. S. Reckley and K. Showalter, J. Phys. Chem. 93(7), 2774 (1989). 15. J. Maselko and K. Showalter, Nature 339, 609, (1989). 16. R. Toth, A. F. Taylor and M. R. Tinsley, J. Phys. Chem. B. 110(20), 10170 (2006), ISSN 1520-6106. 17. A. Taylor, M. Tinsley, F. Wang, Z. Huang and K. Showalter, Science 323, 614 (2009). 18. M. R. Tinsley, A. F. Taylor, Z. Y. Huang and K. Showalter, Phys. Rev. Lett. 102(15), 4 (2009), ISSN 0031-9007. 19. A. F. Taylor, M. R. Tinsley, F. Wang and K. Showalter, Angew. Chem.-Int. Ed. 50(43), 10161 (2011), ISSN 1433-7851. 20. A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein, J. Phys. Chem. 97(29), 7578 (1993). 21. A. T. Winfree, The Geometry of Biological Time (Springer, 1980). 22. G. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students (SIAM, 2002). 23. S. Shinomoto and Y. Kuramoto, Prog. Theor. Phys. 75(5), 1105 (1986), ISSN 0033-068X. 24. K. Yoshikawa, R. Aihara and K. Agladze, J. Phys. Chem. A 102(39), 7649 (1998), ISSN 1089-5639. 25. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, 1984). 26. B. A. Hense, C. Kuttler, J. Mueller, M. Rothballer, A. Hartmann and J.-U. Kreft, Nature Reviews Microbiology. 5(3), 230 (2007), ISSN 1740-1526, doi: 10.1038/nrmicro1600. 27. M. R. Tinsley, A. F. Taylor, Z. Huang, F. Wang and K. Showalter, Physica D, Nonlinear Phenomena. 239(11), 785 (2008), ISSN 0167-2789.

November 21, 2012

278

13:2

9in x 6in

b1449-ch13

Engineering of Chemical Complexity

A. F. Taylor, M. R. Tinsley and K. Showalter

28. M. R. Tinsley, A. F. Taylor, Z. Huang and K. Showalter, Phys. Chem. Chem. Phys. 13(39), 17802 (2011), ISSN 1463-9076. 29. D. Hansel, G. Mato and C. Meunier, Phys. Rev. E 48(5), 3470 (1993), ISSN 1063-651X. 30. V. K. Vanag, L. F. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Nature 406(6794), 389 (2000). 31. M. Kim, M. Bertram, M. Pollmann, A. von Oertzen, A. S. Mikhailov, H. H. Rotermund and G. Ertl, Science 292(5520), 1357 (2001). 32. H. Varela, C. Beta, A. Bonnefont and K. Krischer, Phys. Chem. Chem. Phys. 7(12), 2429 (2005), ISSN 1463-9076. 33. I. Kiss, W. Wang and J. Hudson, J. Phys. Chem. B 103(51), 11433 (1999), ISSN 1089-5647, doi:{10.1021/jp992471h}. 34. D. G. Aronson, G. B. Ermentrout and N. Kopell, Physica D-Nonlinear Phenomena. 41(3), 403 (1990), ISSN 0167-2789. 35. K. Bareli, Physica D-Nonlinear Phenomena. 14(2), 242 (1985), ISSN 01672789. 36. A. Koseska, E. Volkov and J. Kurths, Chaos 20(2), 9 (2010), ISSN 1054-1500. 37. M. Dolnik and M. Marek, J. Phys. Chem. 92(9), 2452 (1988), ISSN 0022-3654. 38. M. F. Crowley and I. R. Epstein, J. Phys. Chem. 93, 2496 (1989). 39. H. Daido, J. Phys. A, Math. Gen. 28, L151 (1995). 40. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Phys. Rev. Lett. 94(24), 248301, (2005), ISSN 0031-9007. 41. R. Toth and A. F. Taylor, The tris (2,2 ’-bipyridyl) ruthenium-catalysed Belousov-Zhabotinsky reaction, Prog. React. Kinet. Mech. 31(2), 59 (2006), ISSN 1468-6783. 42. A. F. Taylor, P. Kapetanopoulos, B. J. Whitaker, R. Toth, L. Bull and M. R. Tinsley, Phys. Rev. Lett. 100(21), (2008), ISSN 0031-9007. 43. G. B. Ermentrout and N. Kopell, J. Math. Bio. 29(3), 195 (1991), ISSN 0303-6812. 44. I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Prog. Theor. Phys. Suppl. (161), 99 (2006), ISSN 0375-9687. 45. A. F. Taylor, P. Kapetanopoulos, B. J. Whitaker, R. Toth, L. Bull and R. Tinsley, Euro. Phy. J. (Special Topics). 165, 137 (2008), ISSN 1951-6355. 46. I. Z. Kiss, C. G. Rusin, H. Kori and J. L. Hudson, Science 316(5833), 1886 (2007), ISSN 0036-8075. 47. T. Gregor, K. Fujimoto, N. Masaki and S. Sawai, Science 328(5981), 1021 (2010), ISSN 0036-8075. 48. D. McMillen, N. Kopell, J. Hasty and J. J. Collins, Proc. Nat. Acad. Sci. USA 99 (2), 679 (2002), ISSN 0027-8424. 49. J. Garcia-Ojalvo, M. B. Elowitz and S. H. Strogatz, Proc. Nat. Acad. Sci. USA 101(30), 10955 (2004), ISSN 0027-8424. 50. T. Danino, O. Mondragon-Palomino, L. Tsimring and J. Hasty, Nature 463 (7279), 326 (2010), ISSN 0028-0836.

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Chapter 14 COLLECTIVE DECISION-MAKING AND OSCILLATORY BEHAVIORS IN CELL POPULATIONS Koichi Fujimoto∗ and Satoshi Sawai† ∗

Graduate School of Science, Osaka University, Machikaneyama-cho Toyonaka, Osaka 560-0043, Japan † Graduate School of Arts and Sciences, University of Tokyo, 3-7-8 Komaba, Meguro-ku, Tokyo 153-8902, Japan Many examples of oscillations are known in multicellular dynamics, however how properties of individual cells can account for the collective rhythmic behaviors at the tissue level remain elusive. Recently, studies in chemical reactions, synthetic gene circuits, yeast and social amoeba Dictyostelium have greatly enhanced our understanding of collective oscillations in cell populations. From these relatively simple systems, a unified view of how excitable and oscillatory regulations could be tuned and coupled to give rise to tissue-level oscillations is emerging. This chapter reviews recent progress in these and other experimental systems and highlight similarities and differences. We will show how group-level information can be encoded in the oscillations depending on degree of autonomy of single cells and discuss some of their possible biological roles.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collective Oscillations in Microbial Populations . . . . . . . . . . . . 2.1. Oscillatory and excitatory response in transcriptional regulation 2.2. Coupling synthetic circuits . . . . . . . . . . . . . . . . . . . . . 2.3. Comparison with BZ reaction . . . . . . . . . . . . . . . . . . . . 2.4. Collective oscillations in budding yeast and Dictyostelium . . . .

279

. . . . . .

. . . . . .

. . . . . .

. . . . . .

280 281 281 284 285 288

November 21, 2012

13:2

9in x 6in

b1449-ch14

K. Fujimoto & S. Sawai

280 3.

Collective Decision Making . . . . . . . . . . . . 3.1. Generalization of the scheme and examples 3.2. Synchrony in decision making . . . . . . . . 3.3. Encoding cell density information . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

1.

Engineering of Chemical Complexity

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

291 291 292 293 294 295

Introduction

Collective rhythmic behaviors are known in a wide spectrum of systems ranging from populations of fireflies1 and bees2 to physical systems such as Josephson junctions3 and electrochemical reactions.4 Mathematical frameworks5 that have developed over the years could guide us to extract the essential ingredients and properties common to these phenomena. When considering the oscillations in cell-cell signaling and gene regulation, it is important to understand how intracellular signaling networks are coupled between the cells to give rise to population-level behavior. Furthermore, by experimentally addressing their dynamics more or less quantitatively in the light of the basic theory, one could begin to understand their roles. As we shall see below, collective oscillations can store group-level information in distinct ways — either in the amplitude or the frequency depending on how they are realized. With only a few exceptions from physico-chemical systems,4,6 this however has remained a theoretical prediction waiting to be tested experimentally. Collective rhythmic behaviors at the organismal level pose a great challenge for those who want to understand how they emerge from dynamics of cellular and subcellular levels. Experimentally, it is often difficult or almost impossible to bridge the large gap between multiple levels of organization; i.e., molecules to cells, cells to tissues. However, recent quantitative studies on glycolytic oscillations in budding yeast and cAMP oscillations in social amoeba Dictyostelium are beginning to fill this gap. In addition, a particle-based Belouzov–Zhabotinsky reaction and artificial gene circuits in bacteria are providing bottom-up approaches to study collective oscillations. These experimental systems take advantage of quantitative measurements that can be made both at the single-cell and population level and molecular, cell and genetic manipulation that are relatively easy. Thus by choosing the appropriate system, it is becoming increasingly tractable to rigorously test how oscillations emerge and how density and other extracellular information are stored in the oscillations. In this chapter, we will first briefly describe the essential features required for oscillatory signaling by taking examples from synthetic approaches. We will then describe recent progress in yeast and Dictyostelium. From studies

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

281

of these relatively simple systems, a unified view of how excitable and oscillatory regulations could be tuned and coupled to give rise to tissuelevel oscillations is emerging. We highlight their similarities and differences and describe the distinct ways by which collective oscillations are realized and how they store group-level information. 2. 2.1.

Collective Oscillations in Microbial Populations Oscillatory and excitatory response in transcriptional regulation

Cell signaling is mediated by biochemical reaction networks full of negative feedback loops. Suppose mRNA (X ) encodes a kinase (Y ) and that Y activates a transcription factor (Z ) that inhibits transcription of X (Fig. 1(a)). Since many of such reactions follow Michaelis–Menten type kinetics with high Hill coefficients, the level of X increases drastically before Z starts to inhibit X. This creates a sufficient overshoot that causes a delay between these two processes — one that acts to increase X and the other that work in the opposite. Due to the all-or-none response in the reaction rate, activation of Z does not begin in until Y reaches a sufficient level. Likewise, Z will not be suppressed unless the level of Y is lowered. These types of nonlinearity in reaction rates realize a sufficient time delay required for persistent oscillations. A simpler reaction with only two variables can give rise to persistent oscillations (Fig. 1(c)) provided that there is an autoregulation. Figure 1(b) illustrates a case where X activates itself and Y, and Y negatively regulates X. In this case, inhibition of X by Y does not take place unless the level of activated Y becomes sufficiently high. Even when the level of activated Y is elevated, the level of activated X does not immediately come down due to presence of auto-regulation (Fig. 1(d)). An illuminating demonstration of the overall logics described above is provided by the synthetic transcription circuit “repressilator”.7 Here, transcription factors LacI, TetR and λCI cyclically repress expression of the other so that they form a relation akin to rock-paper-scissors (Fig. 2(a)). Increase in X inhibits Y, and inhibition of Y acts to decrease X through Z. This forms a negative feedback loop necessary for oscillations. The time delay can be brought about by nonlinearity in the transcriptional dynamics and time required for proteins to be translated. When the circuit is embedded in E. coli together with a GFP-reporter that monitors activity of the tetracyclin repressor, fluorescence of individual bacterium oscillates at about a few hours cycle.7 A more robust oscillator has been developed9 that incorporates a circuit shown in Fig. 2(b). The system consists of the ntrC gene that encodes nitrogen regulator protein NRI that acts as

November 21, 2012

13:2

9in x 6in

(a)

(b)

X

(c)

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

282

Z

b1449-ch14

X Y Y (d)

Fig. 1. A negative feedback regulation combined with strong nonlinearity in cell signaling networks give rise to temporal oscillations. (a) A cyclic circuit composed of three nodes; X activates Y, Y activates Z, and Z represses X. (b) A simpler two-variable circuit with a negative feedback loop and auto-regulation. Autonomous oscillations in the circuit shown in (b); (c) time series of X and Y and (d) the trajectory of a time course in the X -Y plane. The ordinary differential equations dX/dt = −cX X+VX (X 2 +S)/(X 2 +bY +KX ), dY /dt = −cY Y + VY X 2 /(X 2 + KY ) are numerically solved. The first and the second terms of the equations represent degradation of the gene products and transcriptional activity, respectively. X (black solid line), Y (grey dashed line): concentrations of proteins. S: an inducer concentration (cx = 20, VX = 25, KX = 0.04, a = 0.4, cY = 2, VY = 3, KY = 0.1, and S = 4.3 · 10−3 ). Dotted and dashed lines are nullclines,5 dX/dt = 0 and dY /dt = 0, respectively. Following a vector field indicated by black arrows in (d), trajectories from any initial conditions converge to a closed cycle, i.e., limit cycle (black thick line) when the steady state represented by the intersection (grey filled circle) of the nullclines is unstable.

a transcriptional activator. The expression of ntrC and lacI which encodes the Lac repressor are driven by the ntrC promoter. The expression of ntrC is also inhibited by the Lac repressor thus forming a negative feedback loop. The benefit of this construct is that the level of repression by LacI could be tuned by adding non-metabolizable analogue of lactose IPTG (isopropylbeta-D-thiogalactopyranoside) in the medium. A very similar synthetic oscillatory circuit8 makes use of the plac/ara−1 promoter element10 to implement the necessary positive and negative regulation by transcription

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

(a)

(b)

TetR LacI

(d)

(c)

+Arabinose

araC

ntrC λ cI

+Arabinose

-IPTG

-IPTG

lacI

283

lacI -IPTG

(e)

Fig. 2. Synthetic circuits that generate oscillatory gene expression follow a common scheme described in Fig. 1. (a) A cyclic circuit “repressilator”7 composed of three promoter-gene pairs; TetR represses λcI, λcI represses lacI, and LacI represses tetR. A cascade of repression provides necessary nonlinearity and a negative feedback loop required periodic gene expression. (b) A circuit composed of a pair of promoter–gene set.8 LacI represses ntrC and the product of ntrC induces lacI expression. Product of ntrC also induces itself and adds the essential nonlinearity to the reaction kinetics. (c) Transcription of araC and lacI is driven by a hybrid promoter plac/ara−1 that is activated by AraC and repressed by LacI. (d) Time series and (e) a phase-plane trajectory of an excitable behavior exhibited by the equations in Fig. 1 (S = 4.3 · 10−4 ). A perturbation (light grey arrow in e) that exceeds a threshold transiently elevates X and Y following the vector field (grey arrows in e) before returning to the stable steady state (grey filled circle in e).

factors AraC and LacI, respectively (Fig. 2(c)). When AraC binds to arabinose, it becomes a transcriptional activator. By altering concentrations of arabinose and IPTG, strength of both of the feedback loops can be tuned. When repression is sufficiently weak, the system no longer oscillates autonomously, but it is still able to exhibit excitability (Figs. 2(d) and 2(e)).11,12 Upon induction by IPTG, X rises rapidly and then falls to a steady state. This is followed by slow decrease in Y which keeps X inactive for some time. This time window is called a refractory period.13 Due to refractoriness, X is re-activated only when external stimuli are applied at intervals longer than a certain period. A large deviating trajectory upon

November 21, 2012

284

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

suprathreshold perturbation and the refractory period are the two main feature of excitable systems. Excitability is common in systems close oscillatory instability and cells are no exception. Action potential in neurons and cardiomyocytes are well known examples of excitability, however these are just the tip of the iceberg. For example, an excitable gene circuit drives transient differentiation of Bacillus subtilis to the state of competence.14 Here, expression of transcription factor ComK is positively regulated by itself and another transcription factor ComS. Because ComK negatively regulates ComS, the circuit contains both positive and negative feedback loops. Due to stochasticity in gene expression, excitability of the circuit gives rise to sporadic and transient differentiation. An illuminating aspect of Bacillus differentiation is that it can be made to switch between excitable and oscillatory states by changing the level of ComS by an additional copy of the gene under an inducible promoter.15 Excitability in gene expression can also be seen in eukaryotes. In mammalian NF-κB (Nuclear Factor κB) signaling, transient nuclear translocation of the transcription factor RelA is observed only when the cytokine TNFα (Tumor Necrosis Factor α) is repetitively applied at intervals longer than 200 minutes period.16 These real case examples together with the studies of synthetic circuits demonstrate how a negative feedback loop in gene regulation gives rise to excitable and oscillatory response and how they are put to use in the cells. 2.2.

Coupling synthetic circuits

It is important to note that due to cell–cell variability, differences in the phase of the oscillations will increase in time so that after a few cycles, periodic gene expression in individual bacterium will completely be out of sync.7 Therefore the oscillations realized by the simple circuits (Figs. 2(a)–(c)) do not persist at the population level. This is also true in mammalian cells for the oscillatory response in NF-κB described above17 as well as for the periodic synthesis of a tumor suppressor p53 in response to stress signals.18 Asynchronous oscillations of individual cells will cancel out by averaging thus providing a false picture that cells responded transiently with a single peak in the amplitude. Unless the frequency is finely tuned so that they are equal in all cells, some form of cell–cell communication is required for persistent oscillations at the population level. Cell–cell signaling in Vibrio fischeri; a symbiotic marine bacterium that emits bioluminescence in the light organ of a squid19 is well studied and one that has been utilized in synthetic circuits.20,21 Regulation of bioluminescence is mediated by small molecule N-Acyl homoserine lactone (AHL) that induces itself (therefore called an “autoinducer”) and a set

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

285

of enzymes and proteins such as luciferase to produce luminescence. AHL diffuses relatively freely between the cells and penetrate cell membrane. As bacteria will only sense AHL above a certain threshold concentration, the signaling mechanism ensures that bioluminescence is produced only when bacteria are closely packed together (Fig. 3(a)). The positive feedback mediates self-amplification of AHL production and forms the basis of density sensing in an all-or-none fashion. This type of cell–cell communication is referred to as “quorum sensing” (QS) and is prevalent in bacterial populations.19 If one could couple cells harboring oscillatory circuits (Figs. 2(a) and 2(b)) by a QS mechanism, we may be able to see artificial synchronized oscillations in bacteria. Population-level oscillations have been predicted from simulations of coupled circuits similar to the repressilator (Fig. 2(a)). Here, expression of lacI is dependent on AHL encoded by the luxI gene, and the luxI gene is negatively regulated by LacI.22 The first successful demonstration reported recently21 instead based its design on the other oscillatory circuit in E. coli. (Fig. 2(b)). Here, luxI replaces ntrC as the autoregulatory activator gene, and lacI is replaced by aiiA which encodes a degrading enzyme of AHL (Fig. 3(c)). Because AHL freely diffuses between the cells, the positive feedback depends not only on the amount of AHL molecules synthesized within a cell but also on those secreted by others. Above a critical cell density, synchronized oscillations of gene expression emerge at the colony level (Fig. 3(b)). 2.3.

Comparison with BZ reaction

As we have seen, synthetic gene-circuits have provided us with insights into how oscillatory dynamics could take place in a cell and in a group of cells. Similar dynamics have recently been addressed in purely chemical reactions. The well-known Belouzov–Zhabotinsky (BZ) reaction23 involves autocatalytic bromination of malonic acid and a negative feedback by bromide ion that brings the bromoderivatives of malonic acid back to the original reduced form. Depending on HBrO3− concentration, BZ reaction can be either oscillatory or excitable.24 A modified BZ reaction that takes into account the effect of compartmentalization has been studied by Showalter and his colleagues.6 Here, reaction takes place in the catalyst-loaded particles. By suspending the particles in a solution that includes everything required for the BZ reaction except the catalyst ferroin, one can limit the crucial reaction to take place only in the particles while allowing exchange of reactants from the solution — a situation analogous to exchange of autoinducer between intracellular and extracellular environment. Because ferroin only exists in the particle, HBrO 2 is formed and the particles change

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

286

(a)

(b)

(c) Fig. 3. Cell-density dependent transition to collective oscillations.12 (a) Schematics of quorum sensing (QS) and (b) dynamical quorum sensing (DQS). (c) A synthetic system that couples the circuit design in Fig. 1. The luxI gene encodes secreting molecule AHL that can diffuse freely through the membrane to neighboring cells. LuxR is constitutively expressed.21 AHL is an auto-inducing molecule that binds to LuxR and promotes expression of luxI itself. The AHL-LuxR complex also induces expression of aiiA which in turn inhibits luxI expression forming a negative feedback loop. The level of AHL-LuxR is probed by expression of fluorescent protein. Two types of transition to group-level oscillations; (d) Kuramoto-type and (e) dynamical quorum sensing (DQS)-type. In the Kuramoto-type transition, autonomously oscillating cells are desynchronized at low cell density (d, upper panel). Group-level behavior appears first as partially synchronized oscillations of the autoinducer concentration at intermediate densities (e, middle panel) that become fully synchronized at high densities (d, lower panel). In DQS, cells are quiescent at low cell density (e, upper panel). Due to excitability, oscillations emerge by mutual induction above a critical cell density (e, middle and bottom panels). Computer simulations of 1000 cells with a simple circuit similar to (c) are shown. Time course of intracellular autoinducer concentrations for 10 representative cells (thin lines) and population average (thick blue line).

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

(d)

287

(e) Fig. 3.

(Continued )

color from red to blue. This increases auto-catalytically until increased Br− eventually drives the reaction back to the original reduced state. The particle-based BZ reaction has elegantly demonstrated that there could be two ways by which synchronized oscillations could emerge from reactions of many-body. The first type is the so-called Kuramototype transition.5,25 The hallmark of Kuramoto-type transition is that the population-average of the amplitude, i.e., degree of synchronization, increases continuously (thick blue lines in Fig. 3(d) panels) with the coupling strength, e.g. number of particles, diffusion constant, stirring rate, etc. The gradual increase is due to existence of an intermediary state where only a fraction of the elements are synchronized in phase and frequency (thin lines in Fig. 3(d) middle panel). Raising the coupling strength further promotes complete entrainment of all elements (Fig. 3(d) bottom panel). This is in fact what is observed in the particle-based BZ reaction when the number of particle is increased at a low stirring rate.6 A distinctly different type of transition is observed at high stirring rates. Here, the amplitude increases discontinuously in a switch-like manner from zero to a finite value above a critical number of cells (thick blue lines in

November 21, 2012

288

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

Fig. 3(e)). When the number of particles is small, particles are no longer in the oxidized state implying that the individual particles are not oscillating (Fig. 3(e) upper panel). As the number of particle is increased, oscillations appear at some critical density (Fig. 3(e) middle panel). This type of transition is lately referred to as “dynamical quorum-sensing” (DQS),12,26 because there is a density-dependent qualitative change in the behavior at the single-cell level; cells switch from a quiescent to an oscillatory state, and that the change is mediated by an autoinducing signal of some form. The large amplitude at the onset of the DQS-type transition means that the cells are already highly synchronized (thin lines in Fig. 3(e) middle panel) when the group-level behavior first appears. This is in marked contrast to the Kuramoto-type transition where cells oscillate even before the transition and the average amplitude grows continuously (thick blue lines in Fig. 3(d)). From a biological context, distinction between the Kuramoto and the DQS–type transitions is that isolated cells are autonomously oscillatory in the Kuramoto-type whereas in DQS they are merely excitable (Figs. 2(d) and 2(e)).27 Below a threshold of the autoinducer concentration, some cells are randomly excited due to stochasticity of the signaling reaction. This transiently raises the concentration of the autoinducer. Above a critical number of cells, transient pulses of the autoinducer can accumulate and become large enough to mutually excite all cells at once (Fig. 3(e) middle panel). As density is further increased, the frequency of the oscillations increases, because occurrence of the excitatory events increases (Fig. 3(e) bottom panel). In the engineered E. coli (Fig. 3(c)), both the amplitude and the frequency of the collective oscillations increase by increasing the decay rate of extracellular AHL but remain relatively unaffected by the changes in cell density.21 Model simulations21 suggest that these cells are autonomously oscillatory, however further investigations are required to clarify their exact nature. 2.4.

Collective oscillations in budding yeast and Dictyostelium

There are some naturally occurring biological events that are useful to address collective oscillations. When cells of yeast Saccharomyces cerevisiae are starved under limited glucose supply followed by cyanide treatment, the redox state of NADH begins to oscillate at a period of about 30 seconds.28 –30 The oscillations in yeast could be made to persist indefinitely by continuous feeding of the substrate and removal of secreted metabolites.31,32 Since the oscillations persist at the population level, some secreted molecules must be mediating cell–cell communication. A candidate is acetaldehyde which causes a phase shift in the oscillations when applied

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

289

exogenously.33 There are several proposed mechanisms for glycolytic oscillations.34 It is generally believed that inhibition of phosphofructokinase (PFK) by ATP is the source of a negative feedback.35 Although the concept of DQS was firstly put forward in yeast glycolysis,26 the exact nature of its oscillatory transition is still unclear. It has been noted that glycolytic oscillations in Saccharomyces carlsbergensis disappear when cell density is lowered.36 Studies suggest that individual yeast cells are still able to oscillate even when group-level oscillations are absent.30,37 More recent observation also indicates that under well controlled environment, single yeast cells in isolation are unable to oscillate.38 We should note however that the amplitude of yeast oscillations increases continuously with the square root of cell density. This is a characteristic feature of a Kuramoto-type transition. These conflicting observations can be resolved if one assumes that oscillatory instability at the individual cell level increases with cell density and that this plays a key role at the onset of collective oscillations.26 According to the definition of DQS-type transition based on the BZ reaction (see Sec. 2.3), it appears that the glycolytic oscillation is an intermediate of the two forms of transition. A clear demonstration of DQS-type transition came from another model system; social amoebae Dictyostelium discoideum or more commonly known as cellular slime mold. Under starved conditions, cells aggregate synchronously to form a fruiting body.39 The chemoattractant is cAMP that is synthesized and secreted by the cells that propagate as waves at a period of approximately five minutes.12 Despite many earlier efforts, questions regarding the origin of the cAMP oscillations remained unanswered.40 –43 A breakthrough came recently from a live-cell imaging approach that made it possible to study cAMP relay response in single cells to well-defined concentrations and duration of extracellular cAMP stimuli.44 When isolated cells are continuously exposed to nanomolar level of extracellular cAMP, cytosolic cAMP rises transiently within one minute after addition of cAMP. The response attenuates during the next 15 minutes as it continues to oscillate at about 3–6 minutes periodicity. The cells therefore exhibit excitability to a supra-threshold level of extracellular cAMP and that the response is not a single transient peak but of multiple peaks. When well-isolated cells are continuously flushed with phosphate buffer, one sees that cytosolic cAMP remains at the basal level at almost all time. Occurrence of a transient pulse under such a condition is rare and random.44 These observations strongly suggest that cAMP oscillations are property that cannot be accounted for simply by oscillations of single cells. Single cells in isolation washed free of extracellular cAMP do not exhibit cAMP oscillations. How collective oscillations appear in the population can also be studied under perfused conditions. Instead of applying cAMP exogenously, one

November 21, 2012

290

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

Fig. 4. Transition to collective oscillations.12 (a) Dictyostelium cell populations in a perfusion chamber. Starved cells expressing the FRET (fluorescence resonance energy transfer) probe for cAMP epac1camps44 are placed in a chamber (volume ≈ 0.25 ml) and fluorescence is observed under a microscope; buffer is well-mixed in the chamber and exchanged at a fixed flow rate of 2 ml/min. (b) Population averaged changes in cytosolic cAMP estimated from the ratio between fluorescence emission at 485 nm and 540 nm (y-axis). Cell densities are 1/4 monolayer (top panel) and 1/64 monolayer (bottom panel); the flow rate is 2 ml/min. (c),(d) Depending on synchrony at the onset of collective oscillations, the frequency (c) or the amplitude (d) may encode and cell density. Cells are quiescent below a critical density.

could ask what happens when the number of cells is increased under constant flow of buffer (Fig. 4(a)). It is known from earlier studies that the cells secrete cAMP constitutively at a low rate.45,46 We can thus predict that, by adding sufficient number of cells in the flow chamber, the level of extracellular cAMP will be elevated so that cells will react to it just as they did when stimulated in isolated conditions. It was shown that, in fact, cells in a perfusion chamber exhibit synchronized pulsing of cytosolic cAMP above a threshold density of cells (Fig. 4(b)).44 A qualitative change from a quiescent to oscillating state is observed at the level of individual

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

291

cell level when collective oscillations emerge. This indicates that collective oscillations in Dictyostelium appear via DQS (Fig. 3(e)). Synchronized pulsing of cAMP is sporadic when the ratio of cell density and flow rate is small. Its average frequency depends on this ratio until it plateaus around five to six minutes as pulsing becomes more regular in timing (Fig. 4(c)).44 The extracellular cAMP level in the perfusion chamber is expected to increase as the number of cells is raised. Conversely, extracellular cAMP is diluted according to the flow rate. As we saw from the properties of individual cells, sub-threshold level of extracellular cAMP elevates the chance of cells to pulse. These random excitations could begin to coincide because cAMP is secreted. Such stochastically driven selfexcitation events could be examined in details by a mathematical model of coupled excitable elements.44,47 When there is a sufficient number of cells in a chamber, cells that are excited together in this way can elevate extracellular cAMP to a nanomolar range, which will evoke a global pulsing, i.e., almost all cells in the chamber fires. Therefore, collective oscillations in Dictyostelium appear via DQS.

3. 3.1.

Collective Decision Making Generalization of the scheme and examples

Decision making in cell communities often occurs in an all-or-none fashion depending on the cell density. The quorum-sensing (QS) involves a collective switch from one stationary state to another typically in the patterns of gene expression (Fig. 3(a)).19 DQS on the other hand is a collective switch from a quiescent state to an oscillatory state (Figs. 3(b) and 3(e)).6,44 A common theme underlying both QS and DQS is that all cells switch their state simultaneously according to changes in the extracellular environment, and that this decision making is a result of cell–cell communication. Autoinduction is required for the collective switch of DQS and this is also true for QS. As described in coupled synthetic cuiruits, bacterial quorum sensing factor AHL is auto-inducing, and a positive feedback loop mediates synthesis of AHL only when the extracellular concentration of AHL itself reaches a certain level.19 A QS-type scheme is not limited to bacteria but could also be found in animal cells. Secreted Fibroblast Growth Factors (FGF) induces expression of FGF itself via a receptor mediated intracellular signaling cascades.48 There, the so-called autocrine signaling occurs in a switch-like manner and a certain fraction of cell population differentiates above a certain threshold number of cells. This is known as the “community effect”49 in the field of developmental biology. Although, quantitative characterization is still

November 21, 2012

292

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

lacking, such density dependent decision-making appears to be a common theme. Examples include stem cell fate decision,50,51 tumor progression52 and cancer metastasis.53 The main difference between QS and DQS in terms of the underlying kinetics is that in DQS, a strongly nonlinear negative feedback is required in addition to auto-induction. As we saw, cAMP oscillations in Dictyostelium and particle-based BZ reaction are clear examples of DQS. Due to negative feedback regulation, the response becomes transient in time (Figs. 2(d) and 2(e)). DQS-type schemes are becoming evident as well in animal cells. There are countless examples of synchronized rhythmic activities exhibited by populations of cells. Among them, two systems deserve particular attention in the current context. Electrophysiological activities of mammalian suprachiasmatic nuclei (SCN) that support circadian rhythms54 and oscillatory secretion of insulin by pancreatic β cells55 that enhances its hormonal action.56 In both SCN57,58 and β cells,59 isolated cells are merely excitable but populations at high cell density exhibit synchronous oscillations. These observations suggest that many of the oscillatory behaviors in cell populations appear by DQS. 3.2.

Synchrony in decision making

The inspiring aspect of QS and DQS transition is that change in the cellular state could happen simultaneously in a cell population. For such behaviors to take place, an all-or-none response to a common level of signaling molecule (i.e., a threshold response) by individual cells is necessary but not sufficient. When cells are isolated, the timing and the magnitude of the response to exogenous signaling molecules often vary due to cell– cell heterogeneity.44,60 In general, the discrepancy between the precise sharp response at the population level and a more variable responses of isolated cells is circumvented by the presence of auto-induction which provides sufficient entrainment of the response. Here, entrainment refers to synchrony in the timing of the response. Because auto-induction rapidly increases the concentration of the inducing molecule, once a few sensitive cells start to respond, there would be a rapid increase in the concentration with little delay so that other less insensitive cells would respond at almost the same timing. Thus all-or-none response to a common level of signaling molecule and auto-induction are the two key properties for synchrony in cellular decision making. In DQS, the abrupt onset indicates that there are almost no cells that oscillate below the threshold (Fig. 3(e)). At the onset of DQS, autoinduction supports a chain reaction of pulsing that mutually enhances its own rate of occurrence.27,44 In the particle-based BZ reaction, a high

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

293

stirring rate means that there is more shredding of substances from the particles which is equivalent to increasing the secretion rate or coupling between the cells. For an abrupt onset to take place, the timing of the initial response and the subsequent pulses need to be entrained so that there is discontinuous increase in the oscillation amplitude at the population-level (Fig. 3(e)). Accordingly, during Dictyostelium aggregation, even the first pulse is highly entrained.44 This is in striking contrast to the Kuramoto-type transition and yeast glycolytic oscillations where oscillations are poorly synchronized at the onset (Fig. 3(d)). What other properties besides auto-induction are required for the synchronized onset of DQS? As described above, a negative feedback loop is required for DQS. When this feedback constitutes fast activation and slow inhibition, the resulting dynamics exhibit the so-called relaxation oscillations, as observed in a mathematical model of autoinducer regulation coupled with luxI dependent feedbacks61 whose topology is close to Fig. 3(c). Due to separation in timescale, relaxation oscillations exhibit nearly discontinuous pulsatile waveform characterized by fast excitation followed by slow decay to the basal state. The switch between the fast and the slow time evolution supports more rapid synchronization than sinusoidal oscillations.61,62 This is because the time scale separation gives rise to discontinuity in phase response curve. The discontinuity leads to phase synchronization in a finite oscillation cycles whereas, in theory, this takes infinite cycles for sinusoidal oscillations.63 The mechanism can potentially raise synchrony in pulse timing at the onset of DQS. 3.3.

Encoding cell density information

In QS, cell density is encoded digitally in the collective switch, i.e., a population-level all-or-none response. On the other hand, in both Kuramoto (Fig. 3(d)) and DQS (Fig. 3(e)) transitions, i.e., when there is collective switch from quiescence to oscillations, density information may be encoded as analog information. For Kuramoto-type as well as for weakly synchronous DQS, the population-level amplitude reflects degree of synchrony in the phase of individual oscillations (or degree of participation in the excitatory response), thus the amplitude could encode density and other information regarding the extracellular environment (Fig. 4(d)). This is clearly shown for yeasts where the amplitude continuously increases with cell density.26 In contrast, when synchrony at the onset of DQS is high, occurrence of the synchronized pulses increases as the concentration of signaling molecules increases (Fig. 4(c)). Here, it is not the amplitude but the frequency that encodes group-level information. The frequency encoding of population density is clearly shown for Dictyostelium cells in a perfusion chamber44 and

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

294

BZ particles.6 Similarly, β cells dissociated from a pancreatic islet exhibit calcium oscillations whose frequency increases with the number of cells.64 It is important to keep in mind that perfusion chambers and continuous stirred reactors are helping synchronization by mixing the extracellular medium. In many real situations, mixing is limited by diffusion and hence collective events become more localized in space. Propagating waves are common in spatially extended excitable and oscillatory elements, and in case of Dictyostelium the waves of cAMP provide directional cue that guides cell aggregation. Extent of starvation, nutrient availability, cell density, geometry of the microenvironment could all potentially influence frequency of the cAMP oscillations. When cell density or other extracellular factors is spatially heterogeneous, high frequency oscillations from a specific location can propagate as waves to distant regions that otherwise can only give rise to lower frequency pulses. Because in DQS, timing and phase of the pulses are highly synchronous, such a scheme helps the whole population to become entrained to the highest oscillation frequency, thus in Dictyostelium, it may be of benefit for long range spatiotemporal organization of spiral and target waves of cAMP.65 This essentially guides cells to follow those in the most appropriate direction to aggregate and sporulate. In the future, it would be interesting to see whether expression of genes that are frequency dependent16 is decoding66 extracellular and group-level information vital for their survival.

4.

Conclusions

As we saw in the particle-based BZ reaction, the Kuramoto-type and DQS-type transitions can be realized by the same reaction depending on the autonomy of individual cells oscillation and the strength of coupling. It is tempting to speculate that cells make selective use of the transitions depending on the roles transient pulses and oscillations play in the group. For yeast glycolytic oscillations, the amplitude depends on cell density which implies that oscillations are either autonomous or coupling between excitable cells are weak. Although the role of glycolytic oscillations is unknown, appearance or loss of group-level behavior in systems where autonomy of oscillations has a significant role will likely be following a similar scheme. Relation between asynchrony and cell-fate determination may be of relevance in this regard. Expression of bHLH (basic-helix-loophelix)-type transcription factor Her1/7 is highly synchronized in presomitic mesoderm during somite segmentation67,68 and appear as periodic traveling waves that determine precise timing and spacing of cell differentiation.69 These periodic behaviors form the basis of coordinated cellular organization

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

295

that requires precise repetition and ordering of events in a tissue. In embryonic stem cells, however, with exception of neighboring daughter cells that are synchronized, another bHLH-type factor Hes1 level fluctuates asynchronously.70 This gives rise to a broad distribution in the level of Hes1 that potentially acts as a source of divergent response to a common differentiation signal.70,71 In Dictyostelium, it is mainly the frequency of the oscillations that depend on cell density. This arises from the fact that cells are excitable and that coupling between them is directly mediated by the central player of the oscillations — cAMP. Oscillations that arise from DQS may be advantageous for decision-making at the group-level, since it provides an all-or-non switch for a collective behavior. It is often assumed that collective switch in cell population is of QS kind; i.e., the level of an inducing signal is steady in time. However it is equally plausible that the signaling molecules are oscillating in many cases. The two essential features — nonlinear kinetics and negative feedback loops — are prevalent in cell signaling. Live cell imaging and quantitative analyzes in both time and space shall further clarify dynamic and cooperative properties of multicellular collective phenomena.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

J. Buck and E. Buck, Science 159, 1319 (1968). G. Kastberger, E. Schmelzer and I. Kranner, PLoS One. 3, e3141 (2008). K. Wiesenfeld, P. Colet and S. H. Strogatz, Phys. Rev. Lett. 76, 404 (1996). I. Z. Kiss, Y. M. Zhai and J. L. Hudson, Science 296, 1676 (2002). A. T. Winfree, The Geometry of Biological Time (Biomathematics series) (Springer-Verlag, New York, 1980). A. Taylor, M. Tinsley, F. Wang, Z. Huang and K. Showalter, Science 323, 614 (2009). M. Elowitz and S. Leibler, Nature 403, 335 (2000). J. Stricker et al., Nature 456, 516 (2008). M. Atkinson, M. Savageau, J. Myers and A. Ninfa, Cell 113, 597 (2003). R. Lutz and H. Bujard, Nucleic Acids Res. 25, 1203 (1997). P. Ru´e and J. Garcia-Ojalvo, Math. Biosci. 231, 90 (2011). K. Kamino, K. Fujimoto and S. Sawai, Dev. Growth Differ. 53, 503 (2011). J. Keener and J. Sneyd, Mathematical Physiology (Springer, 1998). G. M. S¨ uel, J. Garcia-Ojalvo, L. M. Liberman and M. B. Elowitz, Nature 440, 545 (2006). G. M. S¨ uel, R. P. Kulkarni, J. Dworkin, J. Garcia-Ojalvo and M. B. Elowitz, Science 315, 1716 (2007). L. Ashall et al., Science 324, 242 (2009). P. Paszek et al., Proc. Natl. Acad. Sci. USA. 107, 11644 (2010).

November 21, 2012

296

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

K. Fujimoto & S. Sawai

18. G. Lahav et al., Nat. Genet. 36, 147 (2004). 19. C. Waters and B. Bassler, Annu. Rev. Cell. Dev. Biol. 21, 319 (2005). 20. S. Basu, Y. Gerchman, C. Collins, F. Arnold and R. Weiss, Nature 434, 1130 (2005). 21. T. Danino, O. Mondrag´ on-Palomino, L. Tsimring and J. Hasty, Nature 463, 326 (2010). 22. J. Garcia-Ojalvo, M. Elowitz and S. Strogatz, Proc. Natl. Acad. Sci. USA 101, 10955 (2004). 23. A. Zhabotinsky, Chaos 1, 379 (1991). 24. R. Field and R. Noyes, Farad. Symposia Chem. Soc. 9, 21 (1974). 25. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, 1984). 26. S. De Monte, F. d’Ovidio, S. Danø and P. Sørensen, Proc. Natl. Acad. Sci. USA. 104, 18377 (2007). 27. M. Tinsley, A. Taylor, Z. Huang, F. Wang and K. Showalter, Physica D 239, 785 (2010). 28. B. Hess and A. Boiteux, Annu. Rev. Biochem. 40, 237 (1971). 29. P. Richard, B. Teusink, H. Westerhoff and K. van Dam, FEBS Lett. 318, 80 (1993). 30. B. Chance, et al., Synchronization phenomena in oscillations of yeast cells and isolated mitochondria, in Biological and Biochemical Oscillators, ed. B. Chance (Academic Press, 1973). 31. H. Jacobsen, H. Busse and B. Havsteen, J. Cell. Sci. 43, 367 (1980). 32. S. Danø, P. Sørensen and F. Hynne, Nature 402, 320 (1999). 33. P. Richard, B. Bakker, B. Teusink, K. Van Dam and H. Westerhoff, Eur. J. Biochem. 235, 238 (1996). 34. M. Madsen, S. Danø and P. Sørensen, FEBS J. 272, 2648 (2005). 35. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour (Cambridge University Press, Cambridge, 1996). 36. J. Aldridge and E. Pye, Nature 259, 670 (1976). 37. M. Aon, S. Cortassa, K. Lemar, A. Hayes and D. Lloyd, FEBS Lett. 581, 8 (2007). 38. A. Poulsen, M. Petersen and L. Olsen, Biophys. Chem. 125, 275 (2007). 39. R. Kessin, Dictyostelium: Evolution, Cell Biology, and the Development of Multicellularity (Cambridge University Press, Cambridge, 2001). 40. V. Nanjundiah, Biophys. Chem. 72, 1 (1998). 41. M. Laub and W. Loomis, Mol. Biol. Cell. 9, 3521 (1998). 42. A. Goldbeter, Bull. Math. Biol. 68, 1095 (2006). 43. V. McMains, X. Liao and A. Kimmel, Ageing Res. Rev. 7, 234 (2008). 44. T. Gregor, K. Fujimoto, N. Masaki and S. Sawai, Science 328, 1021 (2010). 45. P. van Haastert, J. Gen. Microbiol. 130, 2559 (1984). 46. M. Dinauer, S. MacKay and P. Devreotes, J. Cell. Biol. 86, 537 (1980). 47. B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Phys. Rep. 392, 321 (2004).

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

Collective Decision-Making and Oscillatory Behaviors in Cell Populations

48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71.

297

H. J. Standley, A. M. Zorn and J. Gurdon, Development 128, 1347 (2001). J. B. Gurdon, E. Tiller, J. Roberts and K. Kato, Curr. Biol. 3, 1 (1993). C. L. Bauwens et al., Stem. Cells. 26, 2300 (2008). Y. S. Hwang et al., Proc. Natl. Acad. Sci. USA 106, 16978 (2009). J. Jouanneau, G. Moens, Y. Bourgeois, M. F. Poupon and J. P. Thiery, Proc. Natl. Acad. Sci. USA 91, 286 (1994). J. Hickson et al., Clin. Exp. Metastasis. 26, 67 (2009). S. Yamaguchi et al., Science 302, 1408 (2003). P. MacDonald and P. Rorsman, PLoS Biol. 4, e49 (2006). O. Schmitz, J. Rungby, L. Edge and C. B. Juhl, Ageing Res. Rev. 7, 301 (2008). A. Webb, N. Angelo, J. Huettner and E. Herzog, Proc. Natl. Acad. Sci. USA 106, 16493 (2009). C. H. Ko et al., PLoS Biol. 8, e1000513 (2010). J. H. Cartwright, Phys. Rev. E. 62, 1149 (2000). P. D. P´erez and S. J. Hagen, PLoS One 5, e15473 (2010). D. McMillen, N. Kopell, J. Hasty and J. Collins, Proc. Natl. Acad. Sci. USA 99, 679 (2002). D. Somers and N. Kopell, Biol. Cybern. 68, 393 (1993). E. M. Izhikevich, SIAM J. Appl. Math. 60, 1789 (2000). F. C. Jonkers, J. C. Jonas, P. Gilon and J. C. Henquin, J. Physiol. 520 Pt 3, 839 (1999). S. Sawai, P. A. Thomason and E. C. Cox, Nature 433, 323 (2005). G. Dupont and A. Goldbeter, Bioessays 20, 607 (1998). I. Riedel-Kruse, C. M¨ uller and A. Oates, Science 317, 1911 (2007). K. Horikawa, K. Ishimatsu, E. Yoshimoto, S. Kondo and H. Takeda, Nature 441, 719 (2006). O. Pourqui´e, Science 301, 328 (2003). T. Kobayashi et al., Genes Dev. 23, 1870 (2009). T. Kobayashi and R. Kageyama, Cell Cycle. 9, 207 (2010).

November 21, 2012

13:2

9in x 6in

b1449-ch14

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Chapter 15 SYNCHRONIZATION VIA HYDRODYNAMIC INTERACTIONS Franziska Kendelbacher and Holger Stark∗ Technische Universit¨ at Berlin, Institut f¨ ur Theoretische Physik Hardenbergstr. 36, 10623 Berlin, Germany ∗ [email protected] An object moving in a viscous fluid creates a flow field that influences the motion of neighboring objects. We review examples from nature in the microscopic world where such hydrodynamic interactions synchronize beating or rotating filaments. Bacteria propel themselves using a bundle of rotating helical filaments called flagella which have to be synchronized in phase. Other micro-organisms are covered with a carpet of smaller filaments called cilia on their surfaces. They beat highly synchronized so that metachronal waves propagate along the cell surfaces. We explore both examples with the help of simple model systems and identify generic properties for observing synchronization by hydrodynamic interactions.

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Synchronization via hydrodynamic interactions in nature 1.2. Theoretical and experimental studies on model systems . 2. Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . . . 3. Synchronizing Helical Filaments . . . . . . . . . . . . . . . . . 4. Metachronal Waves in a Chain of Driven Oscillators . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

300 301 302 304 307 311 316 317

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

300

1.

b1449-ch15

Introduction

Christiaan Huygens, in 1665, discovered that pendulum clocks synchronize the oscillations of their pendulums when they are weakly coupled to each other by a wooden beam to which they are attached.1,2 Since then, synchronization of a collection of interacting dynamical elements has fascinated researchers.3 It is observable in many physical, chemical, biological, as well as social systems. Examples range from the synchronized flashing of fireflies,4,5 the synchronization of the applause by an audience after the last bar of a classic concert6,7 to micromechanic resonators used to construct highly sensitive mass-, spin-, and charge-measuring devices.8 –10 A unifying description for all these phenomena was formulated by Kuramoto.11 Sperm cells have long whip-like tails called flagella along which a bending wave propagates that pushes the sperm cell forward [Fig. 1(a)]. In 1928, James Gray reported the observation from numerous authors that the tails of several sperm cells beat synchronously when they are close to each other.12 –14 The beating flagella initiate flow fields in the surrounding viscous fluid. Taylor, in 1951, was the first to argue that these flow fields

(a)

(b)

Fig. 1. (a) Snapshots of a swimming sperm cell. A wave travels along the flagellum from the left to the right and pushes the sperm cell forward. (b) Three-dimensional stroke of a cilium consisting of a transport stroke (1 → 2) and a recovery stroke (2 → 3 → 4 → 5 → 1). Two-dimensional strokes also do occur.

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

301

mediate so-called hydrodynamic interactions between the flagella which synchronize the flagellar beating.15 This all happens in the microscopic world on the micron scale and at low Reynolds number where inertial forces are negligible against frictional forces. Nature provides numerous examples for synchronization with the help of hydrodynamic interactions which we review in Sec. 1.1. We summarize theoretical and experimental work on model systems in Sec. 1.2. A recent review on the whole subject has appeared recently.16 1.1.

Synchronization via hydrodynamic interactions in nature

The single celled green alga Chlamydomonas reinhardtii moves with the help of two flagella attached to its cell body. They perform a synchronized “breast stroke” interrupted by short intervals of asynchronous beating.17,18 The synchronization of the beating flagella is well described by the Adler equation3 and an estimate for the coupling parameter suggests a hydrodynamic origin for synchronization. The bending wave propagating along a single flagellum is itself the result of a self-organized process. Many dynein motors within the flagellum act synchronously coordinated by the curvature of the flagellum.19 –21 Shorter flagella are called cilia and give rise to a most fascinating phenomenon. Unicellular microorganisms such as paramecium or opalina are covered on their cell surfaces by a carpet of cilia. In general they perfom three-dimensional strokes (Fig. 1(b)). They are synchronized with a small phase difference between neighboring cilia so that a so-called metachronal wave runs along the cell surface, which the microorganism uses for locomotion.22 –27 In Sec. 4. we will indroduce a simple model system where we demonstrate how hydrodynamic interactions lead to metachronal waves.28 Also multicellular organisms such as the Volvox algae29 or comb jellies also called ctenophora30 –32 use ciliary metachronism for locomotion. In addition, two Volvox algae form bounded states close to surfaces with an appealing dynamics.29 Recent experiments on the ciliary carpet on the surface of the flatworm Schmidtea mediterranea may indicate, however, that hydrodynamic interactions are not alone responsible for the formation of stable metachronal waves in biological systems.33 Metachronal waves resulting from ciliary activity are also used to transport liquid such as mucus in the nose or trachea as experiments on the tracheal epithelium of rabbits and other animals demonstrate.34 –38 A single ciliary stroke is devided into a transport stroke with the straight cilium whereas the recovery stroke close to the surface brings the cilium back to its initial configuration (Fig. 1(b)). Metachronal waves come in various

November 21, 2012

302

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

forms depending on how the propagation direction and the power stroke are oriented relative to each other.39,40 A final example refers to bacterial locomotion. Bacteria such as E. coli move forward with the help of a bundle of several rotating helical flagella each driven by a rotary motor embedded in the cell wall.41 Rotating helical filaments can only form a bundle if they all rotate in phase. In their seminal paper, Berg and Anderson already hypothesized that hydrodynamic interactions between the flagella induce a synchronized motion.42 We have confirmed this assumption in Ref. 43 for a simplified model system which we review in Sec. 3. More sophisticated modeling confirms our findings.44,45

1.2.

Theoretical and experimental studies on model systems

Minimal model systems for studying synchronization with hydrodynamic interactions use point-like particles. When they move around in a viscous fluid, they create a flow field that decays as 1/r in a bulk fluid, where r is the distance from the particle. The flow field influences the motion of other particles. Close to a surface it decays faster with distance r. We will comment on details of this picture in Sec. 2. The farfield of a beating cilium coincides with the flow field of a point-like particle. Based on this fact Vilfan and J¨ ulicher represent the three-dimensional ciliary stroke by a point-like particle which is driven by a constant force along an elliptical trajectory tilted against a bounding surface. Coupling two of these phase oscillators hydrodynamically, they observe in-phase or anti-phase synchronization depending on the orientation of the elliptical trajectories.46 To observe synchronization, it is important that the trajectories are tilted so that the particles are slowed down when moving closer to the surface. On the other hand, Lenz et al. considered spherical trajectories parallel to the surface.47,48 When driving the particles with a constant force, synchronization can only occur when the particles are allowed to deviate from the exact circular trajectories by radial excursions. This is a means to slow down or accelerate the angular velocity. The authors reason that interactions decay faster close to a surface hydrodynamic, than in the bulk. So, they restrict them to nearest neighbors and observe the formation of metachronal waves.48 Finally, Uchida and Golestanian study the conditions under which point particles or rotors synchronize when they move on fixed trajectories of arbitrary shape under driving forces that are arbitrary functions of the phase angle.49 They also investigate the collective dynamics in a carpet of more general rotors and identify phase ordering and spiral waves.50,51 Earlier studies on microfluidic rotors or rotational

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

303

molecular motors embedded in cell membranes address positional ordering into hexagonal lattices.52,53 Already in 2002 Cosentino Lagomarsino et al. introduced the rower model which consists of a chain of beads or point particles each periodically driven by an external force on a straight line segment. A geometric switch where the force is reversed when the beads or rowers reach a fixed displacement creates a driven oscillator.54,55 We will review this model in more detail in Sec. 4. based on our previous work.28 Its properties are the following. Two rowers can only synchronize when the driving force is not constant during one half-cycle. When the force decreases they synchronize in antiphase, when the force increases they synchronize in phase. Metachronal waves are possible in both cases but only when we restrict the range of hydrodynamic interactions either artificially to nearest neighbors or by the presence of a bounding surface as in any relevant biological system. Colloidal oscillators realized with the help of switching optical tweezers confirm the antiphase synchronization when they slow down during one half-cycle.56 In a next step towards self-organized ciliary motion one considers semiflexible filaments made as bead-spring chains with bending elasticity. They are attached to a wall and are driven externally. Kim and Netz actuate them at the base with a torque which switches its direction at a fixed angular displacement. Two filaments become phase-locked at a non-zero phase which enhances the pumping efficiency.57 We recently looked at a two-dimensional regular array of mainly stiff rods attached to a surface. The rods perform strokes where they move on a cone tilted with respect to the normal of the surface.58 When we divide the beating cycle into a fast power and a slow recovery stroke, the latter acting when the rod is close to the surface, metachronal waves emerge. Superparamagnetic semiflexible filaments are driven from outside by oscillating external fields where two- and three-dimensional stroke patterns are realizable.58,59 In this system metachronal waves can be imposed artificially by choosing phase shifts between the magnetic field vectors acting on neighboring filaments. We could demonstrate that metachronal waves at a fixed non-zero phase shift strongly enhance the pumping efficiency of a chain of filaments.59 Recently, Osterman and Vilfan showed that metachronal waves emerge in a densely ciliated surface when one searches for beating patterns with optimal energetic efficiency.60 A more accurate modeling of the beating cilia was introduced in a series of papers by Gueron et al.61 –66 and the formation of metachronal waves was demonstrated in Refs. 63 and 64. Guirao and Joanny base their modeling of the cilium on the work of Camalet and J¨ ulicher, where the coordinated motion of the dynein motors and thereby the ciliary

November 21, 2012

304

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

stroke starts with a Hopf bifurcation via a dynamic instability.21 They then observe the spontaneous creation of metachronal waves and an average global flow field when the ciliary stroke becomes asymmetric due to a symmetry breaking transition.67 Quite recent experiments work with active microtubule bundles self-assembled from microtubules and molecular motors. The bundles show cilia-like beating and at high densities metachronal waves occur.68 Taylor treated the flagellar synchronization of several sperm cells by investigating two parallel undulating sheets.15 He found that the dissipated energy of the hydrodynamically interacting sheets has a minimum when they are in phase which he viewed as a strong hint that they will synchronize starting from an arbitrary phase difference. The recent analysis of Elfring and Lauga confirms this behavior. They show that the phase locking has its origin in the front-back asymmetry of the geometry of the flagellar waveform.69,70 An earlier numerical study in two dimensions based on the boundary element method that approximates the flagellum by a sequence of vertices linked by springs is in agreement with Elfring and Lauga’s study.71 Finally, Yang et al. model the sperm cell similarly as semiflexible filament but use a particle-based mesoscopic simulation method, called multiparticle collision dynamics, to calculate the flow field around the sperm cells. They show that many sperm cells form clusters and exhibit swarming behavior.72 Phase locking for simple three-sphere model swimmers was explicitely demonstrated by Putz and Yeomans.73 In the end, we come back to bundling of bacterial flagella. Kim and Powers showed that two rotating rigid helices with parallel orientation do not synchronize.74 One needs some flexibility as Reichert and Stark demonstrated in Ref. 43 for a simplified model system which we review in Sec. 3. Macroscopic-scale models help to study flagellar bundling in more detail.75,76 A combined experimental and theoretical study on torque driven rotating paddles confirms that some flexibility is needed to observe synchronization.77 In the remainder of this chapter, we introduce some basic facts of low Reynolds number hydrodynamics and hydrodynamic interactions in Sec. 2. We then review our work on synchronizing helical filaments with the help of a simplified model system in Sec. 3 and address metachronal waves in a chain of driven oscillators or rowers in Sec. 4. We conclude with final remarks in Sec. 5. 2.

Hydrodynamic Interactions

The flow of a viscous or Newtonian fluid is determined by the Reynolds number Re = ρva/η, the ratio of inertial to visous forces, where ρ is

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

305

mass density and η the shear viscosity. The parameters v and a denote, respectively, a characteristic velocity and length scale. On the micron scale, the Reynolds number is small, Re
1. One can neglect all inertial forces and the velocity field u(r , t) of fluid flow obeys the Stokes equations and the incompressibility condition: 0 = −∇p + η∇2 u

and divu = 0,

(1)

where p is pressure. In adddition, on every bounding surface the no-slip boundary condition applies. A point force F 0 acting at location r  on the fluid initiates the flow field u(r ) = O(r − r  )F 0 , where the Oseen tensor 1 O(r ) = 8πηr


r ⊗r , 1+ r2

(2)

(3)

is the Green function of the Stokes equations in an unbounded fluid. The flow field of a point force is also called stokeslet. Colloidal beads that move under applied forces F j create flow fields that influence the motion of the other beads, so they interact with each other hydrodynamically. Since the Stokes equations are linear, the velocities v i of the beads are proportional to the forces F j ,78  vi = µij F j . (4) j

The important quantities are the mobilities µij . In general, they depend on all the coordinates r i of the beads. If the mobilities are known, equations (4) can numerically be integrated, for example, by the simple Euler method. In leading order the self-mobilities of beads with radius a are given by the inverse of the Stokes friction coefficient, µii = µ0 1 with µ0 = 1/(6πηa).

(5)

Point-like particle means that one observes the flow field at distances much larger than the particle radius a. If one acts with a force F j on a bead, this force also acts on the fluid and creates a stokeslet in the farfield of the bead, u j (r ) = O(r − r j )F j . Another force-free point particle i at position r i takes over the velocity u j (r i ) and the cross mobility for point particles becomes µij = O(r i − r j ).

(6)

November 21, 2012

306

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

Hydrodynamic interactions are long-ranged and decay as 1/r like Coulomb forces, hence difficult to treat. The finite extent of particles changes the argument in two ways. First, the flow field of an isolated particle at position r p becomes u(r ) = (1 + 1 2 2 6 a ∇p )O(r − r p )F 0 , where ∇p is the nabla operator with respect to r p . Second, a particle i placed into a flow field u(r ) moves with a velocity (1 + 16 a2 ∇2i )u(r i ) (Fax´en’s theorem).78 This leads to the Rotne–Prager mobilities which contain terms up to third order in 1/rij ,

 1 1 µij = 1 + a2 ∇2i 1 + a2 ∇2j O(r i − r j ) 6 6  
3 1 a 3 a = µ0 (1 + rˆ ij ⊗ rˆ ij ) + (1 − 3ˆrij ⊗ rˆ ij ) , i = j, 4 rij 2 rij (7) where r ij = r i − r j and rˆ ij = r ij /rij . Near to a planar surface with no-slip boundary condition, the bulk mobilities can no longer be employed. The velocity and pressure fields of a point force for such conditions were first derived by Lorentz more than 100 years ago.79 Blake put these results into a modern form replacing the Oseen tensor by the appropriate Green function, now called Blake tensor.80 The condition of a vanishing fluid velocity field on an infinitely extended plane is satisfied with the help of appropriate mirror images, similar to the image charge approach used in electrostatics. However, in contrast to electrostatics, where it suffices to simply mirror the charge distribution, the hydrodynamic image system is more complicated due to the vectorial argument of the Stokes equations and the incompressibility condition when compared to the Poisson equation. Therefore, so-called stresslet and source-dipole contributions are needed in addition to the stokeslet of the mirrored point disturbance (also called anti-stokeslet). Blake’s tensor reads G Blake (r , r  ) = O(r − r  ) + G im (r , r  ) = O(r − r  ) − O(r − r  ) + δG im (r , r  ),

(8)

where r  is the position of the anti-stokeslet source, i.e., the stokeslet source at r  mirrored at the bounding plane, and δG im (r , r  ) denotes the sourcedipole and stresslet contributions. To make formula (8) more concrete, consider a bounding surface with surface normal oriented along the z axis. Suppose a unit force acts along the y direction at a distance h from the surface. At distances x  h and y  h from the unit force the flow field

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

307

components decay as  1 6h4 GBlake (x) y=0 = yy z=h 8πη |x|5 2   x=0 = 1 12h . GBlake (y) yy z=h 8πη |y|3

and (9)

So, bounding surfaces lead to a faster decay of the stokeslet flow field. Rotne–Prager mobilities close to a surface can also be formulated (see, for example, Ref. 59).

3.

Synchronizing Helical Filaments

Motivated by the bundling of motor-driven helical flagella in E. coli bacteria, we studied in Ref. 43, if hydrodynamic interactions are able to synchronize two helical filaments driven by the same torque D. For simplicity, we considered two stiff helices thus neglecting any effects of elastic deformations within the filament. Figure 2(a) illustrates how the two identical helices are built from equal-sized beads that are connected with each other by (virtual) rigid bonds. The centers of the beads are aligned along the backbone of the helix with equal distances between successive beads. We “fix” the helices in space by anchoring their terminal beads in harmonic traps. This allows for slight shifts and tilts of the helices and thus implies some kind of flexibility, which is important for hydrodynamic synchronization. The flexibility might mimic the fact that one end of the bacterial flagellum is coupled to the rotary motor by a flexible short filament called hook whereas the other end is free. So the whole filament, although pretty rigid, can change its orientation in space. Figure 2(b) shows a picture where the beads are smeared out along the two helices. Each helix is driven by a constant torque Dαi , where αi is a unit vector pointing along the helix axis, and the terminal beads are anchored in harmonic traps with strength K. Ultimately, one has to determine the coupled center-of-mass translations and rotations of the two helices under the applied torques and forces, when they are allowed to interact hydrodynamically. Using Eq. (4), one can calculate translational and also rotational self and cross mobilities for rigid bodies made from beads using the mobility tensors for spherical particles.81 In our simulations, we used the numerical library hydrolib which offers such a routine.82 Integrating the relevant dynamic equations in time (see Ref. 43), we determined the coupled motion of the helices.

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

308

(a)

(b)

Fig. 2. Helix geometry for studying synchronization (here with a phase difference of π/2). (a) All beads of one helix are connected rigidly with each other. (b) For the sake of clarity, the beads are “smeared” out along the helix. The top and bottom beads are anchored in harmonic traps with strength K. The illustrated helices are in their equilibrium positions (i.e., in the absence of driving torques). (Reprinted figure with permission from Ref. 43, copyright (2005) by EDP Sciences.)

The relevant variables are the phase angles φi for the rotational degrees of freedom about the helical axes αi or more concrete the phase difference χ = φ2 − φ1 . For perfectly parallel helices the relative angular velocity χ˙ is proportional to the applied torque D, χ˙ = µ(φ1 , φ2 )D, where the relevant mobility only depends on the two phase angles. Using symmetry operations that map the two-helix system onto itself, one can show that the mobility µ(φ1 , φ2 ) vanishes for infinitely long helices and, therefore, perfectly parallel helices do not synchronize.43 Anchoring the terminal beads of the helices in harmonic traps of finite strength, allows the helical axis to shift, tilt, and undergo a precession-like motion while each helix itself rotates about its respective axis. This enables the synchronization of the helices. Figure 3 plots the phase difference χ = φ2 − φ1 for two trap stiffnesses K as a function of a reduced time τ (K). Since the speed of synchronization

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

309

Fig. 3. Synchronization of rotating helices. The phase difference χ of the two helices tends towards zero during reduced time τ , starting at any value χ < π. The symbols indicate simulation data at two different trap strengths K in reduced units for the terminal beads. The solid line shows the master curve of Eq. (11). (Reprinted figure with permission from Ref. 43, copyright (2005) by EDP Sciences.)

depends on the trap stiffness K, we introduce the reduced time as    dχ  2 , τ (K) = (t − tπ/2 )   π dt t=tπ/2

(10)

where tπ/2 denotes the time where χ = π/2. Starting with χ slightly smaller than π, the phase difference decreases continuously (with steepest slope at χ = π/2) and finally approaches zero, i.e., the two helices do indeed synchronize their phases. Small oscillations due to the explicit dependence on φ1 and φ2 are too small to be resolved here. Introducing the reduced time τ with the help of the maximum synchronization speed, |dχ/dt|t=tπ/2 , obviously maps the curves onto each other, which is also valid for other values of K. We find that the resulting master curve for the phase difference χ obeys the empirical law χ(τ ) =

π (1 − tanh τ ), 2

(11)

as Fig. 3 strikingly reveals. Since the dynamics at low Reynolds numbers is completely overdamped, we expect this law to follow from a differential equation which is of first order in time. Indeed χ(τ ) obeys the nonlinear equation χ(τ ˙ ) = (2/π)χ(τ )[π − χ(τ )], known as the Verhulst equation and originally proposed to model the development of a breeding population.83

November 21, 2012

310

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

Fig. 4. Synchronization speed (taken at a relative phase χ = π/2, as illustrated in the inset) plotted versus the inverse trap strength K −1 in units of a/D, where a is the bead radius. The frequency scale ω0 is the angular velocity of an isolated helix. The symbols indicate values extracted from simulations at different K. The dashed line is an empirical fit to c1 tanh c2 K −1 with c1 = 3.67 and c2 = 0.0685. (Reprinted figure with permission from Ref. 43, copyright (2005) by EDP Sciences.)

However, it is not clear how to derive this equation from first principles in our case. An important result of the investigation is illustrated in Fig. 4, which shows the maximum value of the synchronization speed, |dχ/dt|t=tπ/2 , as a function of K −1 . Clearly, the speed for synchronizing rotating helices decreases with increasing trap stiffness K and tends to zero for K −1 → 0. Phase synchronization does not occur when the helices are strictly parallel to each other as already discussed. We also studied the angular velocity of the rotating helices and find that it becomes maximal at χ = 0. So in the synchronized state the dissipated energy is maximized. Finally, it makes sense to assume that the motor torques driving bacterial flagella are not exactly the same. When the motor torques differ by a value ∆D, the two helices synchronize with a non-zero χ. Increasing ∆D, the phase difference tends towards π/2. For larger ∆D, synchronization is no longer possible. The important result from our simple model system is that helical filaments can indeed synchronize to form bundles but that they need some flexibility. Theoretical and experimental investigations of elastic filaments confirm this observation.44,45,75,76

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

4.

311

Metachronal Waves in a Chain of Driven Oscillators

Using the minimal model of Cosentino Lagomarsino et al.55 that consists of a linear chain of driven oscillators, so-called rowers, we have identified in Ref. 28 generic conditions under which metachronal waves are able to form. We review the main results here. Figure 5(a) illustrates the geometry of our system. Driven by forces, beads move back and forth on line segments with length 2s and distance c. The tilt angle breaks the left-right symmetry of the chain for β = π/2. The force on each bead is directed along the segment and constant in its simplest version (see Fig. 6(a)). The force reverses its direction when the bead reaches a maximum displacement y = 1 or y = −1. Due to this

Fig. 5. (a) Linear chain of rowers or driven oscillators: Beads move on line segments with length 2s back and forth along the y direction. The segments are tilted with respect to the horizontal by an angle β. The distance between the segments is c. (b) The chain of rowers at a distance h above a bounding surface.

Fig. 6. (a) A constant driving force moves the bead from y = −1 to y = 1 (in unit of s) along the segment. When the bead reaches the displacement y = 1, the force direction is reversed (geometric switch) and the bead moves from y = 1 to y = −1, where the force direction is reversed again. The non-zero parameter α = 0.2 in Eq. (12) introduces a fast transport stroke (large force) and a slow recovery force (small force). (b) Harmonic contribution  < 0: Magnitude of force increases during the stroke. (c) Harmonic contribution  > 0: Magnitude of force decreases during the stroke.

November 21, 2012

312

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

geometric switch rule the bead or rower becomes a driven oscillator to which we assign a phase variable ϕ. The phase oscillator is able to slow down or accelerate in response to hydrodynamic interactions with neighboring beads. In addition, we allow a harmonic contribution of the force which either increases the magnitude of the force during the stroke (Fig. 6(b)) or decreases the force value (Fig. 6(c)). The force Fm in reduced units acting on each bead m becomes Fm = (1 + ασm )(σm − ym ),

(12)

where the geometric switch variable σm is positive when the force points into the y direction and negative when it acts against the y direction. A nonzero α allows to distinguish between a transport stroke (large force value |Fm |) and a recovery stroke (small force value |Fm |). Together with β = π/2 it helps to break the left-right symmetry in the rower chain which enables metachronal waves that travel in one direction only. The dynamics of the rower chain is determined by integrating Eq. (4) in time for point particles. We restrict the motion of the beads to the y direction and thereby reduce the simulation time considerably. The chain is either placed in a bulk fluid where the cross mobilities are given by the Oseen tensor. Or we consider the chain at a distance h above a bounding surface where the rowers move parallel to the wall (Fig. 5(b)). Then the Blake tensor of Eq. (8) determines the cross mobilities and the self mobilities also depend on the distance h from the wall (see Ref. 28). We first show the behavior of two phase oscillators moving with a tilt angle β = π/4. Figure 7 summarizes the dynamics. The time

Fig. 7. The synchronization of two rowers depends on the harmonic parameter  of the driving force. Time evolution of the relative phase ∆ϕ = ϕ2 − ϕ1 plotted for two initial values, ∆ϕ = π/2 and −3π/4. Power and recovery stroke are identical, α = 0, the tilt angle β = π/4, and the distance of the rowers in units of s is c = 0.28.

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

313

evolution of the relative phase ∆ϕ = ϕ2 − ϕ1 crucially depends on the harmonic parameter of the driving force. While positive leads to antiphase synchronization regardless of the initial phase difference, the rowers synchronize in phase when is negative. For a constant force ( = 0), ∆ϕ keeps its initial value and does not change in time. The latter is clear since for constant force the mutual hydrodynamic interactions are always identical and thus the phase difference between both rowers does not change. This is even true for α = 0 after averaging over one beat cycle. For = 0, the driving force becomes position dependent and the hydrodynamic interactions are no longer symmetric. In Ref. 28, we explain why two rowers synchronize in phase ( < 0) or in antiphase ( > 0). Recent experiments using a pair of driven colloids demonstrated antiphase synchronization when the stroke slows down during one half cycle.56 To identify metachronal waves in a chain of N oscillators, we introduce N − 1 phase differences ∆ϕn = ϕn+1 − ϕn and define the complex order parameter Z = AeiΦ :=

N −1 1  i∆ϕn e . N − 1 n=1

(13)

The magnitude A and polar angle Φ lie in the respective ranges A ∈ [0, 1] and Φ ∈ [−π, π). For randomly distributed phase differences the complex order parameter Z is close to zero, with A ≈ 0. Stable metachronism means constant ∆ϕn = 0 along the chain. So A = 1 indicates perfect metachronal waves and Φ gives the phase difference between neighboring oscillators. We extensively studied the dynamics of an open rower chain in an unbounded fluid. Figure 8 summarizes some results. With long-range hydrodynamic interactions we could never identify metachronal waves regardless the value of (green and golden curve). For negative , for example, we observe transient patches of in-phase synchronized oscillators that vanish again since the colloids at the boundary of the patches experience a larger friction and lack behind the colloids in the center of the patches. Only when we restrict hydrodynamic interactions artificially to nearest neighbors do metachronal waves appear as the blue and magenta curves in Fig. 8 demonstrate. As discussed in Sec. 2, close to a surface hydrodynamic interactions decay much faster than in a bulk fluid. To investigate its influence on synchronization, we took an open rower chain with < 0 and moved it from larger distances h towards the surface. Figure 9 plots the time-averaged amplitude A¯ of the complex order parameter versus h. At h ≈ 0.35 it indicates a sharp transition from transient in-phase synchronization in the bulk fluid to the formation of metachronal waves close to the bounding

November 21, 2012

314

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

Fig. 8. Absolute value A(t) and polar angle Φ(t) of the order parameter Z(t) as a function of time for a chain of 200 rowers for different parameter sets. The global parameters are c = 0.28 and β = π/4 and for all graphs the rowers start with the same random distribution of phases. Golden line: α = 0.2,  = 0, and long-range hydrodynamic interactions (l.r. HI) are used. Green line: α = 0.2 and  = −0.3 with l.r. HI. Magenta line: α = 0.2 and  = −0.3 but hydrodynamic interactions are artificially restricted to nearest neighbors (n.n. HI). A metachronal wave occurs that travels in one direction. Blue ¯ is reversed and the metachronal line: same situation but α = −0.2. The mean phase Φ wave travels in the opposite direction. (Reprinted figures with permission from Ref. 28, copyright (2011) by EDP Sciences.)

Fig. 9. Average magnitude of the order parameter Z for a rower chain situated at a height h in units of s above a bounding wall. Inset: Average polar angle. The parameters of the open chain with N = 200 rowers are β = π/4, c = 0.28, α = −0.2, and  = −0.4. (Reprinted figures with permission from Ref. 28, copyright (2011) by EDP Sciences.)

surface. The wavelengths of the waves are about 9 to 10 rower distances. Figure 10(left) shows the color-coded phase differences of neighboring rowers as a function of time in a chain of 400 rowers for different heights h. At h = 0.20 a metachronal wave develops from a disordered system and finally stretches over the entire chain. At h = 0.39 the metachronal wave starts to break apart and transient synchronization with larger patches

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

315

Fig. 10. (left) Color-coded phase differences of neighboring rowers as a function of time in a chain of 400 rowers. The simulation starts at a height of h = 0.20 (a), which is then increased to h = 0.39 at times t = 500 (b) and then to h = 1.20 at t = 1000 (c). (right) Snapshots of the displacement variables for a chain segment of 50 rowers labeled by n taken from the phase plots on the left at specific times t. The red arrow gives the direction of a traveling metachronal wave and the red brackets indicate segments of rowers transiently synchronized in phase.

of correlated rowers are visible. Finally, at h = 1.20 the patches become smaller. The right column of Fig. 10 shows snapshots of chain segments which confirm the dynamics. So far, we have no clear understanding why the abrupt transition in Fig. 9 happens at h ≈ 0.35. It certainly depends on the distance of the rowers which in the special case was c = 0.28. This needs further investigations. Our study clearly shows that the long-range nature of

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

316

hydrodynamic interactions impede the formation of metachronal waves. Only when they become short-ranged close to bounding surfaces do metachronal waves occur, which is the biologically relevant case. A stroke that becomes faster during one half cycle ( < 0) leads to in-phase synchronization of a pair of oscillators and thereby to metachronal waves in a linear chain of rowers with wavelengths of 7–10 rower distances. In the opposite case, > 0, two rowers synchronize in antiphase and the wavelengths of the metachronal waves are shorter, around four rower distances. Our future work will address the influence of thermal noise and introduces a distribution in the rower period. 5.

Conclusions

Hydrodynamic interactions are able to synchronize dynamical elements. In particular, the microscopic world in nature provides examples for hydrodynamic synchronization such as beating flagella in sperm cells or in the alga Chlamydomonas reinhardtii, rotating bacterial flagella, and metachronal waves propagating in arrays of beating cilia on tracheal tissue or on the surface of micro-organisms. We have discussed in detail two model systems that help to study synchronization of rotating helical flagella and the emergence of metachronal waves in a chain of driven phase oscillators. These and other studies reveal some generic properties necessary for and associated with synchronization by hydrodynamic interactions. First, the dynamical elements need some flexibility to approach a synchronized state by exploring a larger configurational space. Perfectly aligned helical filaments, for example, do not synchronize but need to change their relative orientations either by elastic distortions of the filaments or through the flexible hook which allows bacterial flagella to change their orientations in space. The driven phase oscillators in our second example need to change their phase velocities along the stroke. Second, long-range hydrodynamic interactions preclude the formation of metachronal waves. Only if we restrict their range, for example, close to bounding surfaces as in any biologically relevant case, do metachronal waves appear in our model system. Third, synchronization is accompanied by an extremum in the dissipated energy. We are currently exploring how the formation of metachronal waves in the rower model is influenced when the phase oscillators are not all identical but display some distribution in their “eigenfrequencies” and when thermal noise is included. These are conditions which occur in nature on the microscopic scale. Finally, we also study the emergence of metachronal waves in a two-dimensional array of model cilia.

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

317

References 1. C. Huygens, Die Pendeluhr, Horologium Oscillatorium, ed. A. Heckscher and A. V. Oettigen, W. Engelmann, (Leipzig, 1913). 2. U. Parlitz, A. Pikovsky, M. Rosenblum and J. Kurths, Phys. J. 5(10), 33 (2006). 3. A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001). 4. J. Buck and E. Buck, Science 159(3821), 1319 (1968). 5. A. Pikovsky, M. Rosenblum and J. Kurths, Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology, ed. L. SchimanskyGeier, B. Fiedler, J. Kurths and E. Sch¨ oll (World Scientific, Singapore, 2007). 6. Z. N´eda, E. Ravasz, Y. Brechet, T. Vicsek and A.-L. Barab´ asi, Nature 403, 849 (2000). 7. J. A. Acebr´ on, L. L. Bonilla, C. J. P´erez Vicente, F. Ritort and R. Spigler, Rev. Mod. Phys. 77, 137 (2005). 8. Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci and M. L. Roukes, Nano Lett. 6(4), 583 (2006). 9. D. Rugar, R. Budakian, H. J. Mamin and B. W. Chui, Nature 430, 329 (2004). 10. A. N. Cleland and M. L. Roukes, Nature 392, 160 (1998). 11. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (SpringerVerlag, Berlin, 1984). 12. J. Gray, Ciliary Movement (Cambridge University Press, Cambridge, 1928). 13. F. Hayashi, Funct. Ecol. 12(3), 347 (1998). 14. D. M. Woolley, R. F. Crockett, W. D. I. Groom and S. G. Revell, J. Exp. Biol. 212, 2215 (2009). 15. G. I. Taylor, Proc. R. Soc. Lond. A 209, 447 (1951). 16. R. Golestanian, J. M. Yeomans and N. Uchida, Soft Matter 7, 3074 (2011). 17. R. E. Goldstein, M. Polin and I. Tuval, Phys. Rev. Lett. 103, 168103 (2009). 18. M. Polin, I. Tuval, K. Drescher, J. P. Gollub and R. E. Goldstein, Science 325, 487 (2009). 19. I. H. Riedel-Kruse, A. Hilfinger, J. Howard and F. J¨ ulicher, Front. Sci. Prog. J. 1, 192 (2007). 20. S. Camalet, F. J¨ ulicher and J. Prost, Phys. Rev. Lett. 82, 1590 (1999). 21. S. Camalet and F. J¨ ulicher, New J. Phys. 2, 24.1 (2000). 22. S. L. Tamm, J. Cell Biol. 55(1), 250 (1972). 23. K.-I. Okamoto and Y. Nakaoka, J. Exp. Biol. 192, 61 (1994). 24. H. Machemer, J. Exp. Biol. 57, 239 (1972). 25. J. Blake, J. Theor. Biol. 45, 183 (1974). 26. J. Blake, J. Theor. Biol. 52, 67 (1975). 27. M. A. Sleigh, J. Exp. Biol. 37(1), 1 (1960). 28. C. Wollin and H. Stark, Eur. Phys. J. E 34(42), 1 (2011), http://epje. edpsciences.org/.

November 21, 2012

318

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

F. Kendelbacher & H. Stark

29. K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley and R. E. Goldstein, Phys. Rev. Lett. 102, 168101 (2009). 30. M. A. Sleigh, J. Exp. Biol. 48, 111 (1968). 31. S. L. Tamm, J. Exp. Biol. 113, 401 (1984). 32. S. L. Tamm and A. G. Moss, J. Exp. Biol. 114, 443 (1985). 33. P. Rompolas, R. S. Patel-King and S. M. King, Mol. Biol. Cell 21, 3669 (2010). 34. M. J. Sanderson and M. A. Sleigh, J. Cell Sci. 47, 331 (1981). 35. D. Eshel and Z. Priel, J. Physiol. 388, 1 (1987). 36. M. A. Sleigh, Cell Motility 2, 19 (1982). 37. P. Satir and M. A. Sleigh, Annu. Rev. Physiol. 52, 137 (1990). 38. L. Gheber and Z. Priel, Cell Motility Cytoskeleton 28, 333 (1994). 39. E. W. Knight-Jones, Q. J. Microsc. Sci. 95, 503 (1954). 40. C. Brennen and H. Winet, Ann. Rev. Fluid Mech. 9, 339 (1977). 41. H. C. Berg, E. coli in Motion, (Springer-Verlag, New York, 2004). 42. H. C. Berg and R. A. Anderson, Nature 245, 380 (1973). 43. M. Reichert and H. Stark, Eur. Phys. J. E 17, 493 (2005), http://epje.edpsciences.org/. 44. P. J. A. Janssen and M. D. Graham, Phys. Rev. E 84, 011910 (2011). 45. S. Y. Reigh, R. Winkler and G. Gompper, Soft Matt. 8, 4363 (2012). 46. A. Vilfan and F. J¨ ulicher, Phys. Rev. Lett. 96, 058102 (2006). 47. P. Lenz and A. Ryskin, Phys. Biol. 3, 285 (2006). 48. T. Niedermayer, B. Eckhardt and P. Lenz, Chaos 18, 037128 (2008). 49. N. Uchida and R. Golestanian, Phys. Rev. Lett. 106, 058104 (2011). 50. N. Uchida and R. Golestanian, Phys. Rev. Lett. 104, 178103 (2010). 51. N. Uchida and R. Golestanian, Europhys. Lett. 89, 50011 (2010). 52. B. A. Grzybowski, H. A. Stone and G. M. Whitesides, Nature 405, 1033 (2000). 53. P. Lenz, J.-F. Joanny, F. J¨ ulicher and J. Prost, Phys. Rev. Lett. 91, 108104 (2003). 54. M. Cosentino Lagomarsino, B. Bassetti and P. Jona, Eur. Phys. J. B 26, 81 (2002). 55. M. C. Lagomarsino, P. Jona and B. Bassetti, Phys. Rev. E 68, 021908 (2003). 56. J. Kotar, M. Leoni, B. Bassetti, M. C. Lagomarsino and P. Cicuta, Proc. Natl. Acad. Sci. USA 107, 7669 (2010). 57. Y. W. Kim and R. R. Netz, Phys. Rev. Lett. 96, 158101 (2006). 58. M. T. Downton and H. Stark, unpublished results. 59. E. M. Gauger, M. T. Downton and H. Stark, Eur. Phys. J. E 28, 231 (2009). 60. N. Osterman and A. Vilfan, Proc. Natl. Acad. Sci. USA 108(38), 5727 (2011). 61. S. Gueron and N. Liron, Biophys. J. 63, 1045 (1992). 62. S. Gueron and N. Liron, Biophys. J. 65, 499 (1993). 63. S. Gueron, K. Levit-Gurevich, N. Liron and J. J. Blum, Proc. Natl. Acad. Sci. USA 94, 6001 (1997). 64. S. Gueron and K. Levit-Gurevich, Biophys. J. 74, 1658 (1998). 65. S. Gueron and K. Levit-Gurevich, Proc. Natl. Acad. Sci. USA 96, 12240 (1999).

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

Synchronization via Hydrodynamic Interactions

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83.

319

S. Gueron and K. Levit-Gurevich, Proc. R. Soc. Lond. B 268, 599 (2000). B. Guirao and J.-F. Joanny, Biophys. J. 92, 1900 (2007). T. Sanchez, D. Welch, D. Nicastro and Z. Dogic, Science 333, 456 (2011). G. J. Elfring and E. Lauga, Phys. Rev. Lett. 103, 088101 (2009). G. J. Elfring and E. Lauga, Phys. Fluids 23, 011902 (2011). L. J. Fauci and A. McDonald, Bull. Math. Biol. 57, 679 (1995). Y. Yang, J. Elgeti and G. Gompper, Phys. Rev. E 78, 061903 (2008). V. B. Putz and J. M. Yeomans, J. Stat. Phys. 137, 1001 (2009). M. J. Kim and T. R. Powers, Phys. Rev. E 69, 061910 (2004). M. J. Kim, J. C. Bird, A. J. Van Parys, K. S. Breuer and T. R. Powers, Proc. Natl. Acad. Sci. USA 100(26), 15481 (2003). M. J. Kim, M. J. Kim, J. C. Bird, J. Park, T. R. Powers and K. S. Breuer, Exp. Fluids 37, 782 (2004). B. Qian, H. Jiang, D. A. Gagnon, K. S. Breuer and T. R. Powers, Phys. Rev. E 80, 061919 (2009). J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996). H. A. Lorentz, Zittingsverlag Akad. Wet. Amsterdam 5, 168 (1896). J. R. Blake, Proc. Camb. Philos. Soc. 70, 303 (1971). M. Reichert, Hydrodynamic Interactions in Colloidal and Biological Systems, Ph.D. thesis, University of Konstanz, Germany (2006). K. Hinsen, Comput. Phys. Commun. 88, 327 (1995). H.-J. Weber and G. B. Arfken, Essential Mathematical Methods for Physicists (Academic Press, San Diego, 1995).

November 21, 2012

13:2

9in x 6in

b1449-ch15

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Part V EVOLUTION, SYNTHETIC BIOLOGY, AND PROTOCELLS

321

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

This page intentionally left blank

322

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Chapter 16 EMERGENCE AND SELECTION OF BIOMODULES: STEPS IN THE ASSEMBLY OF A PROTOCELL

Susanna C. Manrubia and Carlos Briones Centro de Astrobiolog´ıa (INTA-CSIC) Ctra. de Ajalvir km. 4, 28850 Torrej´ on de Ardoz, Madrid, Spain [email protected] Life was probably ubiquitous on Earth some 3,500 million years ago. In a period that could have been as short as 100 million years, cells very similar in structure and metabolism to extant ones had developed from abiotic matter. The pathway from inorganic chemistry to the first self-replicating molecule and the first functional metabolism, both enclosed in a membrane-based compartment, is marked by major difficulties such as a limited availability of simple biomodules, a too slow reaction rate between molecules, the inefficient formation of short homochiral polymers, an unfaithful template copy, or the fragility of the initial self-sustained catalytic cycles. Miller and Urey’s experiment in 1953 moved the problem of the origin of life to the experimental sciences. Since then, significant advances have been accomplished, among them the identification of several RNA molecules which were able to catalyze essential biochemical reactions or the design of protocells able to grow and divide. Experimental achievements are intimately linked to technological advances and to the development of increasingly more realistic theories and models addressing the different stages involved in the origin and evolution of complex chemistry and early life.

Contents 1. 2.

3.

Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The early Earth . . . . . . . . . . . . . . . . . . . . 2.2. Approaches to early life . . . . . . . . . . . . . . . . Synthesis and Accumulation of Biomodules . . . . . . . . 3.1. Life’s origin in the lab: Miller and Urey’s experiment 323

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

324 324 324 325 327 328

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

324

3.2. Sugars and nucleotides . . . . . . . . . . . . . . . . 3.3. Chirality . . . . . . . . . . . . . . . . . . . . . . . 3.4. Prebiotic polymerization of biomodules . . . . . . 3.5. Chemical selection and compartmentalization . . . 4. Template Replication and the RNA World . . . . . . . 5. Theoretical Approaches to Replication and Metabolism 5.1. Template replication and Darwinian evolution . . 5.2. Catalytic cycles . . . . . . . . . . . . . . . . . . . 5.3. Compositional ensembles: the lipid world . . . . . 6. Paths Towards Protocells . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

329 330 331 332 333 334 335 337 338 339 341

Preamble

This chapter contains a non-exhaustive revision of our current knowledge on prebiotic chemistry as well as on the combination of genetic molecules, protometabolic cycles and membrane-based compartments in the pathway from simple building blocks to the first protocells. The subject is so vast that, from the onset, we renounced any attempt to be historically complete or to cover all of the many chemical aspects involved. Instead, we present the basic scenario, the motivation to pursue some key experiments, and some of the conceptual problems that research on prebiotic chemistry and life’s origins has uncovered. The development of a physical environment which is able to sustain complex chemistry is prior to the emergence of life as we know it: the early Earth had to be transformed for life to emerge. We discuss some of the difficulties that, in a hypothetical pathway from inorganic to organic chemistry, should be overcome to generate self-replicating and self-sustained chemical systems. As an illustration of the many efforts made over the last 60 years, we will highlight recent experimental developments and new scenarios that are broadening the way we think about prebiotic chemistry and chemical organization. Admittedly, it is unlikely that we will ever discover how life actually originated. We would be satisfied to find out, maybe in a not so further future, a single continuous pathway leading to the emergence of a protocell of the simplest kind.

2. 2.1.

Introduction The early Earth

Building a habitable planet takes a long time. Our Earth-based knowledge suggests that it takes significantly longer than it is later needed for

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

325

the planet to become inhabited by microorganisms. The transition from inorganic matter to life on Earth could have taken place quite rapidly in a geological scale once our planet cooled down sufficiently for liquid water to be stable on its surface. Four and a half “giga-annum” (Ga) or billion years ago, the Earth was a hot conglomerate of melted rocks, gas and dust. Before life could emerge and develop, the physical and geological features of our planet had to be severely modified. The crust of the Earth differentiated from the core relatively early, some 4.4 Ga ago. A magnetic field developed afterwards, and, in a multistage process that witnessed the addition of different volatiles, a primitive atmosphere composed mainly of nitrogen, carbon dioxide and water vapor, eventually set in. The apperance of oceans was strongly dependent on the delivery of water from extra-terrestrial sources such as comets and meteorites, and on the presence of an atmosphere thick enough to prevent excessive escape of water vapor. There is evidence that proto-oceans might have been present as early as the first crust appeared. At that time, the atmospheric pressure could have reached 273 bars, maintaining oceans at a temperature higher than 230◦C.1 The terrestrial crust was strongly rebuilt for about 500 million years. In fact, although the oldest terrestrial material is a zircon mineral dated 4.4 Ga ago, the oldest rock we know is 4.1 Ga ago. The last major event probably preventing (or hindering) the appearance of life was the Late Heavy Bombardment, which took place between 3.95 and 3.8 Ga ago. Therefore, between the formation of Earth (4.55 Ga ago) and the generation of a habitable surface, some 700 million years elapsed. The precise timing of the events that took place before life appeared is impossible to tell. Indeed, it is likely that life originated (and went extinct) several times during that period, being current organisms the outcome of the only successful trial. There is some consensus in that a prebiotic chemistry stage could have started some 4.4 to 4.2 billion years ago. At that point, the terrestrial environment was ready for chemical reactions able to produce significant yields of relatively complex molecules that, we believe, were subsequently used in the emergence of self-sustained, self-replicating systems. Indirect evidence of life is found some 3.8 Ga ago, when a fractionation of carbon isotopes compatible with biological activity is observed. However, the first incontrovertible fossils are dated to 3.5 Ga ago.56 2.2.

Approaches to early life

Looking at the analogies among extant organisms, Darwin inferred, a century and a half ago, the likely existence of a common ancestor from which we all, from tiny bugs to primates, derive through modification and

November 21, 2012

326

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

selection. That morphological analogy turned into genetic relatedness once DNA was identified as the material carrying the inheritable information. The last universal common ancestor (LUCA) of the three phylogenetic domains (Archaea, Bacteria and Eukarya) lived between 3.8 and 3.5 Ga ago, as the genes of ribosomal RNA and many other of our shared genes demonstrate.57 However, chemistry was probably tinkering with prebiotic molecules as far as 4.4 Ga ago. Little to none evidence of the steps accomplished before LUCA arrived is left. The fossil record does not preserve molecular aggregates or evolving entities smaller than cells. Also, geochemical biomarkers do not inform about past life longer than some 2.5 Ga ago. In turn, relevant clues can be derived from the current biochemical functions: some of the so-called ‘molecular fossils’, such as ribozymes, may tell us about the features of molecular evolution before LUCA. But the fact is that we are trying to reconstruct the pathway from inorganic chemistry to the first protocells mostly in the absence of empirical evidence. Approaches to study prebiotic chemistry and the path to life either try to find out how the building blocks of biochemistry could have been generated from inorganic compounds (starting 4.4–4.2 Ga ago) or look at extant cells and reduce them in order to disentangle how a minimal cell might have looked like (going back to 3.8 Ga ago, at best). There is a large gap in time and in chemical complexity between a bottom-up approach (the first case) and top-down research (the latter). And both ways of looking at the origins of life are strongly conditioned by life as we know it. In 1952 Stanley Miller performed an experiment in Harold Urey’s laboratory that inaugurated the empirical age of prebiotic chemistry. They demonstrated that several proteinogenic amino acids and other essential biomolecules could be abiotically produced. It seemed that the path to synthesizing life in the lab was paved, that the big question of life’s origin would be answered shortly after. Unfortunately, there were many major issues to be sorted out, among them the separation of racemic mixtures of products into homochiral subsets, the synthesis of some compounds essential in modern cells but of low stability (as certain sugars), the concatenation of monomers (e.g. amino acids and nucleotides) into polymers (proteins and nucleic acids), the emergence of a replication machinery or the generation of appropriate cell-like compartments, not to mention catalytic networks, required to establish a primitive metabolism. More than half a century after Miller–Urey’s experiment, we are still far from completely solving the problem. But we certainly have moved forward. Regarding the top-down approach, it is evident that modern cells are extremely complex molecular factories able to perform a huge number of

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

327

different functions. The number of genes that a hypothetical minimal cell requires to complete its basic functions is estimated between 50 and 400.2,58 Lower estimates only include synthesis of DNA, RNA and proteins, and assume that membrane formation and cell division could occur thanks to the action of physico-chemical processes not coded in the cellular genome. Further, this line of research often seeks to understand the performance of small gene circuits and how they can be integrated into functional primitive metabolisms, in a systems biology-like approach. This view is closer to the design of an artificial cell than to the characterization of a true minimal cell. Upper estimates (from 300 to 400 genes) begin with extant cells and try to reduce the number of genes by dispensing apparently non-essential functions. This reduction is qualitative and retrieves minimal cells far too complex to be understood from first constructive principles.

3.

Synthesis and Accumulation of Biomodules

Before the appearance of life, our planet was a very complex chemical reactor. Plate tectonics appeared early in Earth’s history, so one could move from oceans to the summit of mountains through often tempestuous coasts and experience broad variations in temperature. The action of tides created cycles of wetting and desiccation, igneous rocks were present from the very beginning, and sedimentary rocks were forming at least 4.1 Ga ago. The volcanic activity at the bottom of oceans formed submarine vents that created strong temperature and solvent concentration gradients. Such a broad spectrum of environments is impossible to recreate in the laboratory, not to mention the hundreds of millions of years during which the huge terrestrial laboratory was playing with chemistry. The logical inference is that a bunch of different prebiotic experiments were simultaneously taking place, thus transforming the available substrates into complex mixtures of products. Many of those experiments probably were dead ends, others maybe led to self-sustained, replicating systems that were later outcompeted by more efficient solutions. It cannot be discarded that different pieces of what eventually was the first protocell were generated in different environments and later combined. Current attempts to recreate the origins of life should therefore consider this complex (and more realistic) scenario and follow a “systems chemistry”-based approach.59 In any case, some 300 million years before the first protocells, stable and active enough biomodules had to be produced, physico-chemically separated from accompanying compounds, and accumulated in a reusable form.

November 21, 2012

328

3.1.

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

Life’s origin in the lab: Miller and Urey’s experiment

In the decade of the 1920’s, Alexander Oparin and John B. S. Haldane had hypothesized that conditions on the primitive Earth should have permitted the synthesis of organic compounds from inorganic precursors. Three decades later, Miller and Urey decided to test this hypothesis. At their time, the early terrestrial atmosphere was believed to be highly reductive, so they took a mixture of water, methane, ammonia, and hydrogen to recreate it. The activation energy required for chemical reactions to occur was delivered by two electrodes that mimicked lightning through the atmosphere. The reactions took place inside a sealed flask containing the four gases, and the condensed products were collected in a second flask containing liquid water. After a few days, a significant fraction of the carbon in the system was forming organic compounds, with about 2% of it contained in amino acids, glycine being the most abundant.3 Miller and Urey’s spark discharge experiment has been repeated and re-examined several times to confirm and enlarge the repertoire of organic compounds produced: more than 20 different amino acids and a number of other biomolecules have been abiotically synthesized.4 Afterwards, several alternative environments have been tested in the laboratory. Subsequent evidence indicated that early Earth’s atmosphere was not as reducing as previously believed, and due to volcanic activity it likely contained significant amounts of nitrogen, carbon dioxide, hydrogen sulfide, and sulfur dioxide. Under these conditions different repertoires of molecules, including amino acids and nitrites, are produced. One of the effects of the latter is, unfortunately, to rapidly destroy amino acids. However, the addition of iron and carbonate minerals, likely abundant in the early Earth, reverses that effect, and proteinogenic amino acids are again obtained in significant yields.5 Other experiments have explored atmospheres abundant in carbon monoxide and molecular hydrogen, the effect of UV light as a source of energy, and the role that low temperature could have played with regard to the stability of the reaction products. The production of amino acids and other biomodules in significant amounts seems to be unavoidable in any mixture of simple (atmospherically plausible) volatiles exposed to an energy source. Most of the natural amino acids, purines, pyrimidines, and sugars appeared in different variants of the original Miller–Urey experiment. The first step towards the abiotic generation of life has been firmly advanced. Still, the mixtures so produced are highly heterogeneous and racemic. As we will discuss in Sec 3.3, the selection of the chirality of the biomodules is prior to the polymerization of biological macromolecules.

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

3.2.

329

Sugars and nucleotides

Some molecules essential in extant biochemistry, sugars and nucleotides, are not that easily obtainable by abiotic means. Monosaccharides, the simplest sugar molecules, are relevant biomodules in living systems, playing key roles in metabolism and rendering structural or energy rich polysaccharides upon polymerization. Among them, ribose is an important component of many coenzymes and constitutes, together with phosphate, the molecular backbone of RNA. In turn, DNA is characterized by its deoxyribosephosphate backbone. Therefore, the prebiotic synthesis of ribose (and, to a lesser extent, deoxyribose) was soon explored due to their key role in the polymeric genetic macromolecules. Formaldehyde was found to oligomerize in the presence of mineral catalysts to form sugars (in the classical formose reaction, discovered by Butlerow in 1861), although complex, tar-like mixtures of tetroses, pentoses and hexoses are obtained, being ribose a relatively minor product.60 The presence of borate minerals stabilizes ribose in the mixture of sugars, what suggests a plausible mechanism for the accumulation of the precursor of ribonucleotides.61 The synthesis of ribonucleotides had been pursued for over 40 years under the assumption that they should assemble from their three molecular components: ribose, a nucleobase and phosphate. In this scenario, ribose and nucleobases would have been produced independently and then combined. The first synthesis of a purine nucleobase (adenine) was achieved in 1960 through the polymerization of HCN.62 Since then, the formation of purines (and, less efficiently, pyrimidines) has been achieved under different conditions including eutectic phases, ice matrices and drying/ wetting cycles. Mineral and metal surfaces also enhance these processes by concentrating reagents and preventing products from degradation.63 Nevertheless, two very unfavorable reactions are required in order to produce a ribonucleotide: the formation of a glicosidic bond between ribose and the nucleobase, and the phosphorilation of the resulting nucleoside. In particular, no way of joining ribose and canonical pyrimidines has been ever found. Recently, an alternative solution to the pyrimidine+ribose equation leading to the corresponding ribonucleosides has been found. The way out of that conundrum required to escape the old assumption of independent synthesis of nucleobases and ribose and, following a novel “systems chemistry” approach, to look for a path with a common precursor.11 This likely chemical pathway leading to ribonucleosides starts, as many other prebiotic reactions, with very simple building blocks including glycolaldehyde and cyanamide. Interestingly, when inorganic phosphate is added to the mixture, most of the unwanted reactions are eliminated

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

330

and the key intermediate, 2-aminooxazole, is efficiently synthesized. This compound, which contributes both to the sugar and the nucleobase motifs of the ribonucleotide, is volatile enough to be purified by sublimation– condensation cycles. These cycles are by no means complex: day-night variations could suffice to accumulate 2-aminooxazole, a first step in RNA synthesis. The further phosphorilation of the ribonucleoside is facilitated by the presence of urea, which comes from the phosphate-catalyzed hydrolysis of cyanamide. This leads to the final production of pyrimidine ribonucleosides. 3.3.

Chirality

Abiotic chemical reactions where chiral molecules are produced usually yield levorotatory or “left-handed” (L) and dextrorotatory or “righthanded” (D) enantiomers with equal probability. But long polymers, believed to be at the basis of most genetic systems, are only possible if formed exclusively by one of the two enantiomeric types. Life uses L-amino acids and D-sugars: the mechanisms that broke the symmetry of racemic mixtures and yielded enantiomerically enriched products have puzzled researchers for decades.64 As with other accomplishments in prebiotic chemistry, we cannot be certain that the solutions found in the laboratory are those that Nature used. But some plausible, and not necessarily complex, scenarios have been devised: Mixtures with an enantiomeric excess over 99% can be achieved by simply stirring a racemic solution.6 At the root of the abiotic generation of homochirality lie two processes that were probably essential also at other stages: the (autocatalytic) chemical selection of molecules with a certain property (handedness in this case) and their accumulation for possible later use. The theoretical prediction that mixtures with a large enantiomeric excess should result from a process where each enantiomer would catalyse its own production was contemporary to Miller–Urey’s experiment,7 though its experimental demonstration had to wait for half a century.8 In those experiments, the system could not achieve a 100% enantiomeric excess. A theoretical way out could be provided by recycling the less abundant enantiomer to the most abundant type, which would accumulate in crystal form until chiral purity is achieved.9 Shortly after, it was experimentally demonstrated with initially racemic mixtures of sodium chlorate that this process is possible, efficient, and leads indeed to the accumulation of chiraly pure compounds.10 In any case, a growing number of alternative phenomena have been theoretically postulated or experimentally tested to have generated chiral biomodules (and their polymers) from non-chiral matter.

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

3.4.

331

Prebiotic polymerization of biomodules

The polymerization of biomodules (mainly, amino acids and nucleotides) into polymeric macromolecules (polypeptides and nucleic acids) in the absence of enzymes had to deal with the thermodynamically uphill process of water removal, required for condensation reactions. To overcome this limitation, different prebiotic scenarios have been considered, including hydration–dehydration cycles and melting processes, as well as the presence of heterogeneous systems containing either mineral surfaces or lipid domains. Also, these systems should provide a favorable environment for the stabilization of the polymer against hydrolysis once the biomodules have been condensed. The synthesis of polypeptides from amino acids required the conversion of peptide bond formation into a thermodynamically favorable process. Different mechanisms have been proposed, and some have been experimentally tested. The most successful experimental settings involve fluctuating heating cycles in the presence of mineral surfaces (e.g. silica, alumina and the montmorillonite clay),65 the presence of small organic activating molecules (such as imidazol or carbodiimides) in combination with mineral surfaces at low temperature,66 and wetting/drying cycles in the presence of concentrated NaCl solutions and Cu2+ as a metal catalyst.67 Additionally, the use of activated amino acids instead of their natural forms enhances the polymerization rate and the polypeptide length.68 Regarding the non-templated polymerization of ribonucleotides, the activation of the phosphate with different leaving groups (mainly nitrogencontaining heterocycles, such as imidazole, pyridine or purine derivatives) has been assayed. The longest RNA polymers (up to 50-mers) have been obtained using imidazole- and 1-methyladenine-activated ribonucleotides — thanks to the concentration and catalytic properties of montmorillonite interlayers.69 In turn, non-activated ribonucleotides can polymerize up to 25- to 100-mers at high temperature, in a dehydration/rehydration system containing fluid lipid matrices composed of amphiphilic molecules.70 Therefore, a plausible scenario might have involved the dynamic interaction of different biomodules with the montmorillonite clay: the phyllosilicate surfaces or interlayers could have promoted, in contact with the bulk aqueous medium, the synthesis of nucleobases, the polymerization of (activated) ribonucleotides and that of amino acids. The cooperation of this system with the micro-environments provided by amphiphilic-based vesicles is also favored by the experimental evidence. This fact highlights the relevance of heterogeneous catalysis in the origins of life.

November 21, 2012

332

3.5.

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

Chemical selection and compartmentalization

The rate of chemical reactions occurring among the different small biomodules generated through abiotic reactions requires that they were present in a sufficiently high concentration and, preferably, in the vicinity of other chemical species that could act as catalysts. Montmorillonite and other mineral surfaces could have been essential in the production of polymers: they may act by selecting and, in a sense, compartmentalizing the universe of possible reactions among the building blocks of macromolecules. The many different environments simultaneously present on the early Earth surely provided opportunities for synthesis, differential selection, compartmentalization and, occasionally, catalysis of simple biomodules. An interesting setting occurs at the bottom of oceans, in volcanically active regions where hydrothermal vents form. High pressures maintain water in liquid state occasionally above 400◦ C. Surrounding waters rapidly cool down in a gradient that sustains a rich chemical activity12 and a remarkable biodiversity. Since the discovery of black smokers in the late ’70s of the past century, the interest in those submarine formations has steadily increased, to the point that a hydrothermal origin of life has been proposed.13 The abundance of reduced organic compounds, the complex biogeochemistry of those areas, a continuous and concentrated source of energy, the presence of active hydrothermal systems 4.2 Ga ago, or the many lithotrophic microorganisms described these environments as attractive and challenging regions of study in relation to life’s origin.14 Hydrothermal vents have additional advantages to foster the selection and compartmentalization of complex chemical compounds. Thermal gradients inside vent channels promote the separation and differential accumulation of biomodules, both through passive thermal diffusion and enhanced by convection. Interestingly, convection also creates thermal cycles that could be vital to promote complex biochemical reactions, in the same way that polymerase chain reaction (PCR) techniques lead to DNA amplification in current laboratories.15 Further, rocks of volcanic origin are highly porous: the abundance of natural compartments in probably stable physico-chemical conditions could have fuelled further reactions among the chemical species accumulated in such micro-environments. Nevertheless, hypotheses favoring a “hot origin” of life have to face important problems related to the reduced stability of most biomolecules at high temperatures.71 In other environments, still, amphiphilic molecules (e.g. single-chained fatty acids) plausibly synthesized in certain prebiotic scenarios72 may have spontaneously self-assembled into membrane-based vesicles (later called liposomes when they were formed by complex lipids) which in turn could have compartmentalized the first replicating molecules and/or

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

333

protometabolic cycles. Vesicles are relatively stable at a wide range of sizes, being able to grow by the slow addition of fatty acids in the form of micelles. Nevertheless, when a maximum volume-to-surface ratio is reached (or if physico-chemical perturbations affect the stability of the system) the vesicle may divide into two daughter vesicles73 and distribute its internal content between the offspring. This process would have constituted a primordial, very simplified and unregulated version of proto-cellular division. Most scientists would accept that a rich repertoire of molecular compounds and membrane-forming amphiphiles can be produced abiotically. However, the subsequent steps, which are thought to be the advent of template replication, the emergence of autocatalytic reaction networks and the coupling between proto-genome replication and compartment reproduction are major transitions in evolution16 that demand a conceptual shift.

4.

Template Replication and the RNA World

In the context of the origin of life, replication can be defined as any reliable copying process of a polymeric template whose outcome is a new molecule which preserves the specific sequence of the template.88 The first true nonenzymatic self-replicating experimental system used a palindromic DNA hexamer which assisted the ligation of two DNA trimers — each of them complementary to one half of the template.23 Another relevant example of self-replication used 15-mer and 17-mer oligopeptides that covalently bind to each other through the interaction with a 32-mer template.74 The autocatalytic template replication of simpler organic polymeric compounds has been also achieved,75 although the evolutionary connection of this alternative system and the biochemistry operating in current organisms cannot be postulated. A further insight into the replication processes leading to LUCA comes from the top-down approach. The presence of DNA (carrying the genetic information) and proteins (performing metabolic functions) in all extant cells poses a catch-22 like paradox: DNA is required to produce proteins, while DNA replication cannot occur in the absence of proteins. At present, it is known that RNA can perform both functions: it still acts as genomic material in some viruses and all viroids, and, beginning in the early 1980s of the last century,76 it has become clear that some RNA molecules (termed ribozymes) are able to perform catalytic functions in current organisms. Seven classes of natural ribozymes catalyze the cleavage or ligation of RNA: group I and group II autocatalytic introns, RNase P, hairpin, hammerhead, hepatitis delta virus (HDV) ribozyme and Varkud satellite ribozyme.77

November 21, 2012

334

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

The eighth class is the peptidyl transferase center of the ribosome, which catalyzes the formation of a peptide bond between two amino acids during translation.78 The evidence at hand led to the proposal that (as already suggested by Woese, Crick and Orgel in the 1960s) there might have been an RNA world prior to the establishment of the current DNA/RNA/protein world.79 During the RNA world, genotype and phenotype (mainly represented by DNA and proteins, respectively, in extant organisms) should have been combined in a single type of macromolecule: RNA. The development of in vitro selection techniques in 1990 by means of the so-called SELEX method80,81 has increased the repertoire of ribozymes, thus unveiling the functional plasticity of nucleic acids and supporting the plausibility of an RNA world. Nevertheless, it is currently impossible to postulate complex metabolisms based exclusively on ribozymes. Moreover, although template-dependent RNA polymerase ribozymes have been evolved in vitro with progressively better performance (and steadily decreasing the high mutation rate that affects replication in those systems82,83 ), we are still far from envisaging a ribozyme which is able to catalyze its own replication, an essential feature for the evolvability of RNA-based protocells. Additional physico-chemical constraints (as the limited prebiotic abundance of ribonucleotides and the low stability of RNA in solution) might have hindered the de novo establishment of an RNA world. Therefore, different polymers analogous to nucleic acids have been postulated to have preceded RNA at the early stages of the evolution of genetic molecules, thus constituting putative “pre-RNA worlds”.84 These artificial analogues include glycerol-derived nucleic acid (GNA), threose nucleic acid (TNA), locked nucleic acid (LNA) and pyranosyl-RNA (pRNA), as well as a molecule with peptidomimetic backbone: peptide nucleic acid (PNA).

5.

Theoretical Approaches to Replication and Metabolism

Solving the puzzle of the origins of life requires explaining the advent of template replication, metabolism and membrane-based compartmentalization. Replicating systems and metabolic networks might have appeared independent to each other. However, at a certain point they must have combined in the way to a protocell. Before the first protocell appeared as a unit of selection, it is almost certain that replicating molecules were loosely bound to their local system, such that the exchange of chemical information was frequent. LUCA was a late product, highly complex and evolved, of molecular evolution. At an intermediate stage, protocells likely originated in an age of promiscuous mix among genes, where the horizontal

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

335

exchange of hereditary information was the rule. Regarding metabolism, Darwin’s seminal ideas, Oparin’s and Haldane’s hypotheses, and Miller– Urey’s experiment favored a heterotrophic origin of life that has been further pursued by most theoretical models. Simple biomodules are just the starting point of this precellular world. 5.1.

Template replication and Darwinian evolution

Experimental evidence at hand supports that short RNA (or RNA-like) polymers of random or quasi-random sequence could have been produced and accumulated abiotically. In these pools of polymers some biochemical functions (including the activity of simple ribozymes) could have been present. Therefore, basic RNA functionality could have preceded RNAcatalyzed template replication. At the base of this possibility lies the peculiar sequence-structure-function relationship in RNA. An important property of polymeric biomolecules is that they fold in tridimensional structures following thermodynamic rules. The native structure of biopolymers (in particular, RNA and proteins) is critical in the definition of their biochemical function. A key point often disregarded in models of evolution of replicators is the redundancy of the sequence-structure map. A clear example is provided by RNA. In short RNA oligomers (up to 40 nucleotides) of random sequence, the most abundant secondary structures are of the hairpin and stem-loop types.17 Interestingly, it is known that certain RNA molecules with hairpin structure can display RNA ligase activity85 and, thus, they could promote the concatenation of random polymers and the generation of modular, larger molecules even in the absence of template replication. When the huge number of different RNA sequences folding into the same structure is taken into account, the appearance of catalytic function is no longer a formidable obstacle.18,19 The existence of many different genotypes (RNA sequences) causing the same phenotype (their structure and, eventually, function) is supported by a series of more or less direct evidences. A number of experiments with functional RNA molecules have been devised to prove that neutral networks of phenotypes (consisting of all genotypes with the same phenotype) are not only ubiquitous, but in close contact in the space of genotypes. That is, almost any pair of different phenotypes can be retrieved by means of one or a few nucleotidic changes in an appropriate sequence. Schultes and Bartel20 experimentally proved this fact by selecting two evolutionary unrelated ribozymes of almost equal length (89 nucleotides): one was a (synthetic) class III ligase ribozyme that catalyzes RNA ligation; the other was a (natural) HDV ribozyme that catalyzes cleavage and assists in the replication of the viral genomic RNA. It was necessary to change

November 21, 2012

336

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

about 40 nucleotides in each of the original sequences to generate an intersection sequence able to fold into each of the original structures and perform with notable competence the catalytic activity of both molecules under either fold. All intermediate steps between the original sequences and the intersection were catalytically active at levels mostly comparable to the initial ribozymes, and sometimes even better. This was a solid demonstration of the sequence-function redundancy and a clear evidence that dramatic alterations in structure and function are a few mutations away in sequence space.21 Actually, a low fidelity of replication could have been advantageous to maintain variability (even within the same phenotype) and to enhance the adaptation of early molecular populations.22 Thus, we must conclude that the naif relationship one sequence-one function is over-demanding and should not be indiscriminately applied, especially to short RNA sequences. In particular when modelling molecular quasispecies (as discussed below), attention should be focused on the phenotype, and models should systematically assume that a given function, in general, can be retrieved from a huge number of different sequences (genotypes) without a priori homology. A second important aspect of molecular evolution before enzymatic template replication set in is the functional form of the growth rate of the abundance of chemical species (e.g. RNA oligomers). If the limiting step in the production of a certain molecular type is random polymerization, it will accumulate at most linearly in time. Experiments of non-enzymatic template replication yield faster growth rates, but they are still sub-exponential.23 Any sub-exponential increase in the abundance of a population permits coexistence of species.24 Differences in growth rates in linear or parabolic growth, for example, are not sufficient to outcompete other species present. In those plausible early scenarios, thus, survival of the fittest was not yet possible. Instead, different species could have accumulated in the environment and a high chemical diversity was maintained. That molecular heterogeneity was probably advantageous to foster the emergence of an eventually complex biochemistry, and the encapsulation of genetic molecules together with low molecular weight chemical species in reproducing vesicles could have aided in the selection process.25 Most models dealing with replicator dynamics assume that chemical species grow exponentially (although parabolic and hyperbolic growths may have been common in some scenarios25). A classical model is Eigen’s quasispecies,26 where the effect of frequent mutations was first analyzed. Manfred Eigen considered a number of chemical species (polymers), each characterized by a specific sequence and a particular replication rate, and affected by the same error rate of replication. One of the key results of the

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

337

model was that the concentration of the molecular species replicating at the fastest rate vanishes when the error rate of replication approaches a critical (finite) value which is of the order of the inverse of the polymer length. Beyond that threshold, genetic information cannot be maintained and the quasispecies enters into the so-called “error catastrophe”. The limitations derived from Eigen’s model are substantially alleviated when phenotypes, instead of genotypes, are the target of selection. Phenotypes can be maintained with remarkably higher error rates, as has been demonstrated with ribozymes.27 Further, the precise value at which a given phenotype disappears depends on the characteristics of the phenotype itself,28 particularly on the number of sequences representing it (i.e., on the size of its neutral network).29 At present, combined theoretical and empirical evidence suggest that a replication fidelity of 10−3 mutations per nucleotide and round of copy could be enough for RNA molecules of size 7000–8000 nucleotides to be maintained. This value is close to the typical genome length of most RNA viruses, and at the verge of the amount of genetic information hypothetically required to sustain a minimal cell.25 Still, the obstacles found in the abiotic appearance of replicating molecules, including the problem of insufficient replication fidelity, has led to the proposal of alternative systems, among them hypercycles and compositional ensembles. 5.2.

Catalytic cycles

A hypercycle is a population of molecules that interact by aiding in each other’s replication.30 For a hypercycle to be viable, each species has to aid replicating the following one, closing in circle. Competition between replicators is substituted in this scenario by cooperation between species, eventually maintaining a higher amount of genetic information distributed among shorter molecules. The hypercycle is a particular case of catalytic cycle where the growth of the species occurs at a hyperbolic rate. The self-organization of simple chemical species into catalytic cycles, however, also meets profound difficulties, and has been considered highly implausible.31 Autocatalytic networks of interacting proteins were proposed long ago.32 Assuming a finite probability for a polypeptide to catalyze a chemical reaction involving other proteins in the ensemble, it was suggested that a self-sustained autocatalytic network would unavoidably emerge in a diverse enough population of proteins. An obstacle to the feasibility of such networks is however their evolvability.35,37 On the experimental side, no large catalytic cycle has been produced so far, though small cycles have been chemically engineered. Current chemical reaction networks, embedded

November 21, 2012

338

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

in larger systems, cannot work properly without enzymes due to the high specificity of reactions and to energetic balance requirements.33 A stepby-step construction of a large cycle, with selection of viable sub-networks at each stage draws a more plausible path than the spontaneous emergence of a catalytic cycle in full.34 New species and reactions would be slowly incorporated, and the addition of small cofactors first, and efficient catalysts later, could perhaps prevent the proliferation of side reactions. Theoretical advances have arrived by establishing the formal conditions that give plausibility to the appearance of a especially relevant type of catalytic cycles: reflexively autocatalytic and F-generated (RAF) sets. RAF sets capture the idea of catalytic closure, that is of a self-sustaining set supported by a steady supply of (simple) molecules from some reservoir.36 It has been demonstrated that RAF cycles have indeed a high probability of appearance, even if the involved species display modest catalytic activity. RAF sets can be divided into a number of connected autocatalytic cores which can function as units of heritable adaptations in reaction networks.37 This requires that more than one chemical reaction network be encapsulated into a compartment to allow competition and selection of networks.38 The two elements together (plausibility of spontaneous appearance and limited evolvability) bring catalytic cycles back to the stage of important elements in the long way leading to the first protocell. An often discussed problem to obtain stable catalytic cycles is the appearance and taking over of parasitic species. Several mechanisms have been proposed to counteract the deleterious effect of parasites, among them the evolution of catalytic cycles on a surface.39 This spatial restriction could have helped as well in the selection of increased replication accuracy.40 There is another possibility to limit the action of parasites that relies on group selection, as proposed in the stochastic corrector model.16 5.3.

Compositional ensembles: the lipid world

The ease with which simple amphiphilic molecules and lipids of various kinds spontaneously self-assemble to form micelles and vesicles has led to the proposal of a “lipid world”.41 If some particular lipid composition could enhance the incorporation of further molecules, vesicles would grow autocatalytically and eventually divide due to simple physical forces, as discussed above. This would lead to the selection of autocatalytic, fast growing vesicles, in front of other possible compositional ensembles. A major criticism to the lipid world is that it lacks a sufficient capacity for evolvability. The simplicity of the underlying chemistry strongly limits the number of different phenotypes, understood as possible different combinations of lipid species, and in consequence hereditary variation is

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

339

severely limited.42 In addition, it was shown that the replication of those ensembles is so inaccurate, that the “fittest” variant cannot be maintained in the population.43

6.

Paths Towards Protocells

Most authors would agree that an evolutionarily relevant protocell should contain a metabolic subsystem, replicating molecules carrying the heritable information, and a semi-permeable boundary which is able to keep those components together.44 Further, the protocell should be able to divide and distribute the chemical species between daughter protocells. As minimal living organisms, they have to behave as autopoietic systems.45 In the sections above we have discussed different systems that consider only part of those essential elements. The conceptual integration of all of them is feasible,86 and the experimental combination of every pair of elements leading to binary subsystems (template-boundary, metabolism-template and boundary-metabolism) has been tackled over the last decade, as a step forward towards the bottom-up construction of ternary, full-fledged biological systems. In particular, the combination of replicating nucleic acid polymers and reproducing compartments has been partially accomplished.46 Among the various efforts aimed at combining metabolic networks and membrane compartments, a successful approach has allowed the encapsulation of the sugar synthesizing formose reaction into lipid vesicles.87 Current efforts towards the direct construction of ternary systems try to develop artificial cells formed by a lipidic membrane and an informational polymer able to replicate within.47 –49 Major issues are the exchange of nutrients and waste products with the extracellular environment as well as the growth, division and evolvability of such artificial cells,50 before more ambitious goals can be attempted. Additionally, most of these approaches, though highly valuable from a biotechnological viewpoint, rely on the use of complex mixtures of compounds derived from extant organisms (typically, either E. coli extracts or large collections of gene products). Those semi-synthetic approaches cannot be strictly considered as relevant evolutionary paths towards protocells, and their usefulness in the field of the origins of life seems limited.59 In the development of an artificial cell, the information derived from systems biology approaches could be also of relevance.51 Theoretical integrative scenarios, though by no means trivial, are ahead of experiments for obvious reasons. An early proposal of protocell is Ganti’s Chemoton,52 a closed system which grows as a consequence of its internal metabolism (represented by an autocatalytic network), has a bilayer

November 21, 2012

13:2

9in x 6in

340

b1449-ch16

Engineering of Chemical Complexity

S. C. Manrubia & C. Briones

membrane formed by a molecule produced by the metabolism, and contains a replicating molecule carrying genetic information. Life is characterized by the fact that metabolism operates out of thermodynamic equilibrium, taking energy from its environment and using it to generate biomolecules. Thus, any realistic model of a protocell should take energetic balance into account.53 Using knowledge on extant cells, additional important details have been added to the models, as a molecule that could act as energetic currency or particular membrane components able to generate chemiosmotic gradients.54 However, the large formal complexity of models designed under this approach, which usually need a large number of kinetic equations for all chemicals involved, forbids to obtain general principles. The complex dynamics of the many coupled equations make those systems, in general, sensitive to endogenous and environmental fluctuations. Eventually, it is difficult to assess their robustness and evolvability. This nonetheless, these complex models contribute to the identification of the conditions that a viable protocell should verify, and to the understanding of the causal construction of extant-like cells from minimal metabolisms. Simpler formal models have highlighted specific limitations or requirements that protocells have to fulfill to be self-sustainable, for instance, how minority molecules limit the overall growth rate of protocells, requiring that the reproduction of the cell and the replication of the genetic molecule be synchronized.55 Research on protocells and minimal cells, provided that it takes into account evolutionary constraints, nowadays plays a pivotal role in the study of the origins of life. As Luisi and coworkers have stated: “(. . .) there has been an abrupt rise of interest in the minimal cell. It appears that one additional reason for this rise of interest lies in a diffused sense of confidence that the minimal cell is indeed an experimentally accessible target.” 50 This is the feeling that all researchers on the origin of life probably share. The quest for the principles and mechanisms allowing the transition from inorganic to living matter would yield a scientific reward. Answering that question could deeply change our understanding of ourselves, our relationship with any other living being, and, foreseeably, our view of all the inanimate matter that conforms our universe: it may be just life-to-be.

Acknowledgments The authors acknowledge the support of Spanish MICINN through projects EUI2008-00158, BIO2010-20696 and FIS2011-27569, as well as of Comunidad de Madrid through project MODELICO (S2009/ESP-1691).

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

341

References 1. M. Gargaud, Ph. Claeys, P. L´ opez-Garc´ıa, H. Martin, Th. Montmerle, R. Pascal and J. Reisse (eds)., From Suns to Life (Springer, Netherlands, 2006). 2. A. C. Forster and G. M. Church, Mol. Syst. Biol. 2, 45 (2006). 3. S. L. Miller, Science 117, 528 (1953). 4. A. P. Johnson, H. J. Cleaves, J. P. Dworkin, D. P. Glavin, A. Lazcano and J. L. Bada, Science 322, 404 (2008). 5. D. Fox, Sci. Am. (28 March, 2007). 6. D. K. Kondepudi, R. J. Kaurman and N. Singh, Science 250, 975 (1990). 7. F. C. Frank, Biochim. Biophys. Acta 11, 459 (1953). 8. K. Soai, T. Shibata, H. Morioka and K. Choji, Nature 378, 767 (1995). 9. Y. Saito and H. Hyuga, J. Phys. Soc. Jpn. 73, 1685 (2004). 10. C. Viedma, Phys. Rev. Lett. 94, 065504 (2005). 11. M. W. Powner, G. Gerland and J. D. Sutherland, Nature 459, 239 (2009). 12. J. M. Edmond, K. L. von Damm, R. E. McDuff and I. Measures, Nature 297, 187 (1982). 13. J. B. Corliss, J. Baross and S. E. Hoffman, Oceanol. Acta 4, 59 (1981). 14. W. Martin, J. Baross, D. Kelley and M. J. Russell, Nat. Rev. Microbiol. 6, 805 (2008). 15. D. Braun and A. Libchaber, Phys. Biol. 1, 1 (2004). 16. J. M. Smith and E. Szathm´ ary, The Major Transitions in Evolution (Oxford University Press, New York, 1997). 17. M. Stich, C. Briones and S. C. Manrubia, J. Theor. Biol. 252, 750 (2008). 18. S. C. Manrubia and C. Briones, RNA 13, 97 (2007). 19. C. Briones, M. Stich and S. C. Manrubia, RNA 15, 743 (2009). 20. E. A. Schultes and D. P. Bartel, Science 289, 448 (2000). 21. G. F. Joyce, Science 289, 401 (2000). 22. E. J. Hayden, E. Ferrada and A. Wagner, Nature 474, 92 (2011). 23. G. von Kiedrowski, Angew. Chem. Int. Ed. 25, 932 (1986). 24. E. Szathm´ ary and I. Gladkih, J. Theor. Biol. 138, 55 (1989). 25. E. Szathm´ ary, Philos. Trans. R. Soc. B 361, 1761 (2006). 26. M. Eigen, Naturwissenschaften 58, 465 (1971). ´ Kun, M. Santos and E. Szathm´ 27. A. ary, Nat. Genet. 37, 1008 (2005). 28. N. Takeuchi, P. H. Poorthuis and P. Hogeweg, BMC Evol. Biol. 5, 9 (2005). 29. M. Stich and S. C. Manrubia, J. Theor. Biol. 280, 117 (2011). 30. M. Eigen and P. Shuster, Naturwisseschaften 64, 541 (1977). 31. L. Orgel, Proc. Natl. Acad. Sci. USA 97, 12503 (2000). 32. S. Kauffman, J. Theor. Biol. 22, 437 (1969). 33. P. Schuster, Orig. Life Evol. Biosph. 40, 407 (2010). 34. S. Jain and S. Krishna, Proc. Natl. Acad. Sci. USA 98, 543 (2001). 35. S. Lifson, J. Mol. Evol. 44, 1 (1997). 36. W. Hordijk, J. Hein and M. Steel, Entropy 12, 1733 (2010). 37. V. Vasas, Ch. Fernando, M. Santos, S. Kauffman and E. Szathm´ ary, Biol. Direct (2011).

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

342

S. C. Manrubia & C. Briones

38. 39. 40. 41.

G. Wachtershauser, Microbiol. Revs. 52, 452 (1988). S. C. Manrubia and J. F. Poyatos, Europhys. Lett. 64, 557 (2003). P. Szabo, I. Scheuring, T. Czaran and E. Szathm´ ary, Nature 420, 340 (2002). D. Segr´e, D. Ben-Eli, D. W. Deamer and D. Lancet, Orig. Life Evol. Biosph. 31, 119–145 (2001). E. Szathm´ ary, Philos Trans. R. Soc. London B 361, 1669 (2000). V. Vasas, E. Szathm´ ary and M. Santos, Proc. Natl. Acad. Sci. USA 107, 1470 (2010). T. Ganti, The principles of life (Oxford University Press, Oxford, 2003). F. J. Varela, H. R. Maturana and R. Uribe, Biosystems 5, 187 (1974). J. P. Schrum, T. F. Zhu and J. W. Szostak, Cold Spring Harb. Perspect. Biol. 2, a002212 (2010). S. Rasmussen, L. Chen, D. Deamer, D. C. Krakauer, N. H. Packard, P. F. Stadler and M. A. Bedau, Science 303, 963 (2004). S. S. Mansy, J. P. Schrum, M. Krishnamurthy, S. Tob´e, D. A. Treco and J. W. Szostak, Nature 454, 122 (2008). V. Noireaux, J. T. Maeda and A. Libchaber, Proc. Natl. Acad. Sci. USA 108, 3473 (2011). P. L. Luisi, F. Ferri and P. Stano, Naturwissenschaften 93, 1 (2006). V. De Lorenzo and A. Danchin, EMBO Reps. 9, 822 (2008). T. Ganti, BioSystems 7, 15 (1975). K. Ruiz-Mirazo and A. Moreno, Artif. Life 10, 235 (2004). F. Olasagasti, A. Moreno, J. Peret´ o and F. Mor´ an, Bull. Math. Biol. 69, 1423 (2007). A. Kamimura and K. Kaneko, Phys. Rev. Lett. 105, 268103 (2010). W. Schopf and B.M. Packer, Science 237, 70 (1987). C. R. Woese, O. Kandler and M. L. Wheelis, Proc. Natl. Acad. Sci. USA 87, 4576 (1990). A. Moya, R. Gil, A. Latorre, J. Peret´ o, M. P. Garcill´ an-Barcia and F. de la Cruz, FEMS Microbiol. Rev. 33, 225 (2009). K. Ruiz-Mirazo, C. Briones and A. de la Escosura, Chem. Rev., in press. H. J. Kim, A. Ricardo, H. I. Illangkoon, M. J. Kim, M. A. Carrigan, F. Frye and S. A. Benner, J. Am. Chem. Soc. 133, 9457 (2011). A. Ricardo, M. A. Carrigan, A. N. Olcott and S. A. Benner, Science 303, 196 (2004). J. Or´ o, Biochem. Biophys. Res. Commun. 2, 407 (1960). R. M. Hazen and D. A. Sverjensky, Cold Spring Harb. Perspect. Biol. 2, a002162 (2010). D. G. Blackmond, Phil. Trans. R. Soc. London B 366, 2878 (2011). J. Bujd´ ak and B. M. Rode, Orig. Life Evol. Biosph. 29, 451 (1999). R. Liu and L. E. Orgel, Orig. Life Evol. Biosph. 28, 47–60 (1998). B. M. Rode, D. Fitz and T. Jakschitz, Chem. Biodivers. 4, 2674 (2007). H. R. Kricheldorf, Angew. Chem., Int. Ed. 45, 5752 (2006). J. P. Ferris, Phil. Trans. R. Soc. London B 361, 1777 (2006). S. Rajamani, A. Vlassov, S. Benner, A. Coombs, F. Olasagasti and D. Deamer, Orig. Life Evol. Biosph. 38, 57 (2008).

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70.

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

Emergence and Selection of Biomodules

343

71. J. L. Bada and A. Lazcano, Science 296, 1982 (2002). 72. S. E. Maurer, D. W. Deamer, J. M. Boncella and P. A. Monnard, Astrobiol. 9, 979 (2009). 73. P. Stano and P. L. Luisi, Chem. Commun. (Cambridge) 46, 3639 (2010). 74. D. H. Lee, J. R. Granja, J. A. Martinez, K. Severin and M. R. Ghadiri, Nature 382, 525 (1996). 75. T. Tjivikua, P. Ballester and J. Rebek, Jr., J. Am. Chem. Soc. 112, 1249 (1990). 76. T. R. Cech, A. J. Zaug and P. J. Grabowsky, Cell 27, 487 (1981). 77. E. Puerta-Fern´ andez, C. Romero-L´ opez, A. Barroso-delJesus and A. BerzalHerranz, FEMS Microbiol. Rev. 27, 75 (2003). 78. P. Nissen, J. Hansen, N. Ban, P. B. Moore and T. A. Steitz, Science 289, 920 (2000). 79. W. Gilbert, Nature 319, 618 (1986). 80. C. Tuerk and L. Gold, Science 249, 505 (1990). 81. A. D. Ellington and J. W. Szostak, Nature 346, 818 (1990). 82. W. K. Johnston, P. J. Unrau, M. S. Lawrence, M. E. Glasner and D. P. Bartel, Science 292, 1319 (2001). 83. A. Wochner, J. Attwater, A. Coulson and P. Holliger, Science 332, 209 (2011). 84. G. F. Joyce, Nature 418, 214 (2002). 85. J. M. Buzayan, W. L. Gerlach and G. Bruening, Nature 323, 349 (1986). 86. E. Szathm´ ary, M. Santos and C. Fernando, Top. Curr. Chem. 259, 167 (2005). 87. P. M. Gardner, K. Winzer and B. G. Davis, Nat. Chem. 1, 377 (2009). 88. F. J. Dyson, Origins of Life (Cambridge University Press, Cambridge, 1985).

November 21, 2012

13:2

9in x 6in

b1449-ch16

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

Chapter 17 FROM CATALYTIC REACTION NETWORKS TO PROTOCELLS

Kunihiko Kaneko Research Center for Complex Systems Biology, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan [email protected] In spite of recent advances, there still remains a large gape between a set of chemical reactions and a biological cell. Here we discuss several theoretical efforts to fill in the gap. The topics cover (i) slow relaxation to equilibrium due to glassy behavior in catalytic reaction networks (ii) consistency between molecule replication and cell growth, as well as energy metabolism (iii) control of a system by minority molecules in mutually catalytic system, which work as a carrier of genetic information, and leading to evolvability (iv) generation of a compartmentalized structure as a cluster of molecules centered around the minority molecule, and division of the cluster accompanied by the replication of minority molecule (v) sequential, logical process over several states from concurrent reaction dynamics, by taking advantage of discreteness in molecule number.

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-term Sustainment of Nonequilibrium State . . . . . . . . . . . . . . . . Consistency Between Cell Reproduction and Molecule Replication . . . . . . Thermodynamics of a Reproducing Cell . . . . . . . . . . . . . . . . . . . . From Catalytic Reaction Networks to Genetic Information: Kinetic Origin of Genetic Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Minority control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Genetic takeover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Evolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Origin of Compartmentalization and Cell Division . . . . . . . . . . . . . . . 7. Logical Process from Concurrent Reaction Processes . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

346 348 350 351 352 352 355 355 356 356 357

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

K. Kaneko

346

1.

b1449-ch17

Introduction

There still remains a large gap between just a set of chemical reactions and a cell, which autonomously reproduces itself maintaining its state, while it adapts to new environment and evolves. Filling the gap is essential to understand what life is as well as how the life was originated. It is quite difficult, however, to uncover the event of the origin of life from available data. Thus, another promising approach is “constructive” in nature.1,2 Indeed, there have been extensive efforts to construct a reproducing cell by combining several reaction processes. Steps to be taken for the constructive approach of a cell are: (1) A system consisting of (a variety) chemicals (polymers) with some catalytic activity, which reproduces itself as a set, even though the reproduction may not necessarily be precise. (2) A compartment structure that separates the inside from the outside. This is typically achieved by a membrane, such as vesicle (liposome). This membrane also grows by reactions catalyzed by enzymes within. As the size of this “protocell” surrounded by the membrane increases and when it is sufficiently large, it divides into two. (3) Within the protocell surrounded by the membrane, the reaction system (1) works, and the catalysts are synthesized within, so that the synthesis of membrane is coupled to this internal reaction system. (4) The intra-cellular reactions include replication of molecules that carry the information for heredity. The molecule (nucieic acid in the present cell) keeps information on the reaction processes within. (5) The internal reaction process and the synthesis of membrane work in some synchrony, so that a system with the membrane and internal chemicals are reproduced recursively. In fact, extensive attempts have been made to realize these steps. In a so-called PURE system that is extracted from E. coli,3 in-vitro replication is achieved with more than 5000 reaction steps run with 144 species of biomolecules.4 Division process of liposome5 –7 has been realized through synthesis of membrane by precursor molecules, catalyzed by internal enzymes, while protein synthesis from DNA,8,9 as well as replication of DNA10 was achieved. In spite of progresses in synthesis of a reproducing “protocell”, there still remains a large gap between just a set of catalytic reactions and an autonomously reproducing cell.11 Reactions are often suppressed at some stage, due to “jamming” in molecules. It is still difficult for many reaction processes to continue by synchronizing with the synthesis/division process of membrane.

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

347

Why is the synthesis of protocell so difficult? Is it just due to difficulty in finding a suitable set of finely-tuned parameter values for a variety of chemical processess? Or are there basic theoretical issues to fill in the gap? Indeed, several theoretical studies so far proposed important concepts. They include error catastrophe, error threshold, hypercycle, and quasispecies by Eigen,12 parasite problems in catalytic reactions and importance of compartmentalization to resolve the parasite,13 –17 a loose reproduction set by Dyson,18 autocatalytic-set by Kauffman,19 and genetic takeover.20 Some are mathematically formulated, and several models have been proposed,21 –24 while others remain unresolved. In addition, we have more questions to be solved. In this chapter, we discuss the following questions briefly, by proposing some concepts such as chemical-net glass, minoritycontrol, consistency between cell growth and molecular replication, and discreteness-induced transition. (1) Sustainment of nonequilibrium condition: How are nonequilibrium states are sustained endogenously in a set of chemical reactions? As a tentative answer, we show hindrance of relaxation to equilibrium due to negative correlation between abundances of resource chemicals and of catalysts, which we call chemical-net glass.25 (2) Synchrony of several processes: How is recursive production of a set of chemicals satisfying consistency between the cell growth and molecule replication possible? Recursive production of a set of chemicals satisfying consistency between cell growth and molecule replication is shown to be achieved at a critical state.26,27 (3) Adaptation to environment: How can a cell adapt to a huge variety of environmental conditions, autonomously by taking advantage of intracellular processes? Consistency between molecule synthesis and dilution by cell growth under stochastic reaction process leads to generic adaptation.28 (4) Autonomous replication cycle with appropriate energetic transduction: How is energy from nutrients successively and efficiently transformed for cell growth? Metabolic process is regulated through “energy currency” molecules autonomously so that it is consistent with the cell growth.29 (5) Problems of parasites in hypercycle, which replicate themselve without helping the synthesis of others: How can a system of catalytic reaction network continue to grow resisting against possible parasitic molecules that do not help the synthesis of others? First, compartmentalization of a set of chemicals is required so that a protocell can grow and reproduce. Second, an autocatalytic set consisting of a large number of catalysts

November 21, 2012

348

(6)

(7)

(8)

(9)

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

K. Kaneko

with intermingled network structure suppresses the invasion of parasitic molecules.24,30 Genetic take-over, i.e., emergence of specific molecules that carry the genetic information: How did metabolism and genetic replication get married? If genetic take-over was the answer for it, how was it achieved? Minority molecule species in the catalytic reaction network starts to control cell’s reproduction while it is preserved well.31 Origin of compartmentalization: How does a cluster of molecules reproduce itself, synchronized with replications of molecules? The “minority” molecule which is essential within mutually catalytic reaction network, which replicates slowly in a crowded environment, forms a cluster of molecules around it, and when this minority molecule replicates the cluster divides ito two.32 Evolvability: How does evolution of a cell consisting of many molecules progress? Genotype (with a slower change) and phenotype (with a faster timescale) are separated as a result of the minority molecule with slower replication speed.24,31 Sequential (logical) processes from concurrent reactions: How does a “logical” process emerge from concurrent reaction dynamics? Discreteness-induced transition within “jammed” reaction process as well as reaction-net glass by limitation of enzymes leads to successive changes over several quasi-states.33

Of course, the present cells might adopt advanced solutions to the above questions, by using a finely-tuned mechanism through evolution. Still, for evolution to progress, it is expected that there exist some primitive solutions to the above problems in the beginning, and from such generic primitive processes, the current sophisticated mechanism will evolve later.

2.

Long-term Sustainment of Nonequilibrium State

The first question we address is the maintenance of “nonequilibrium” condition. Did a cell utilize some nonequilibrium condition for sustaining metabolism and reproduction? How are exogeneic nonequilibrium conditions embedded endogenously within a set of chemical reactions? How is reproduction of such sustained nonequilibrium states achieved by means of catalytic reaction? To answer these questions, we first discuss how such nonequilibrium state is sustained long.25,34 Let us consider a catalytic reaction network in which an energy is assigned to each chemical species, and the rate for each reaction is determined by the energy difference to satisfy the detailed condition. In a

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

349

closed thermodynamic system consisting of chemical reactions, equilibrium is ultimately attained after a certain relaxation time. The relaxation process is exponential with the time scale given by the reaction kinetic coefficients, as long as we start from an initial state close to equilibrium. This is also true for any initial condition for linear reaction kinetics, i.e., reactions without catalysts or catalytic reactions with fixed concentrations of catalysts. In contrast, the reaction kinetics whose catalysts are synthesized by themselves involve nonlinear terms, because the rate of such catalytic reaction is given by the product of the concentrations of the substrate and the catalyst. In such catalytic reaction networks, we have recently found a mechanism to slow down the relaxation to equilibrium, even in a well-mixed condition assuring spatially homogeneous concentrations. We consider a catalytic reaction network model consisting of k molecule species Xi + Xc ↔ Xj + Xc . Here Xc is a catalyst of the reaction which also belongs to the above set of chemicals (c ∈ {1, 2, . . . , k}). We assign energy Ei to each molecule.25 The ratio of the forward to backward reactions is given by exp(−β(Ej − Ei )) to satisfy the detailed balance condition, where β is the inverse temperature 1/kT . Now, let ε be the variance of energy of each chemical. When the temperature of the system is sufficiently lower than ε (i.e., βε > 1), overall log(t) relaxation appears. The deviation from the equilibrium decreases with log(t), whereas several plateaus appear successively through the course of relaxation. We have studied a variety of reaction networks to confirm that these two characteristics are universal. How many and which type of plateaus appear depend on the network and initial conditions; however, the existence of several plateaus itself is universal. Then, we have revealed a general mechanism for the emergence of plateaus. The plateaus are not metastable states in the energy landscape; rather, they are a result of kinetic constraints due to a reaction bottleneck, originating in the formation of local-equilibrium clusters and suppression of equilibration by the negative correlation between an excess chemical and its catalyst. The existence of such negative correlation depends both on the initial concentrations of chemicals and the network structures; however, even in randomly chosen networks, there exist several sets of chemicals that satisfy the negative correlation. In biochemical reaction processes, the energy variance is rather large, and therefore, the above slow-relaxation is observed even if the temperature is not so low. Hence, the reluctance in relaxation to equilibrium is a rather common feature of catalytic reaction networks. Here it is important to note that many biochemical reactions are facilitated by catalysts, and without

November 21, 2012

350

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

K. Kaneko

them the reactions take enormously long time. As the reaction rate there depends on the abundances of catalysts, cells can autonomously change their timescale through the concentration of enzymes. The above slow relaxation process with bottlenecks is a manifestation of such change in the time scale by the change in concentrations of catalysts. Of course, nonequilibrium condition has to be provided externally in the beginning. Still, in a geophysical or astronomical system, there generally exist a nonequilibrium condition with flow of matter and energy. Then, such nonequilibrium condition supplied exogenously is embedded internally into a cluster of molecules, so that the relaxation is hindered and activity is kept endogenously. Furthermore, we may expect mutual reinforcement of sustainment of nonequilibrium conditions, spatial structure with compartmentalization, and reproduction. By taking advantage of nonequilibrium reaction processes, structure is organized in network and in space, as was also discussed in dissipative structure. Then, compartmentalization by a cluster of molecules is possible, and chemicals form an inhomogeneous spatial structure. This inhomogeneity will further suppress relaxation to equilibrium.

3.

Consistency Between Cell Reproduction and Molecule Replication

Reproduction of a cell consists of a huge number of reactions for membrane synthesis, metabolic processes and replication of genetic information, while all the enzymes needed for such processes are also synthesized within. All components have to be replicated for a cell reproduction, keeping some degree of synchronization with the replication of other intracellular chemicals. How is such recursive production maintained while keeping diversity of chemicals? To discuss this problem, Furusawa and the author26 studied several protocell models consisting of a number of catalytic molecules, which form intracellular catalytic reaction networks. All the catalysts are synthesized as a result of such reactions, starting from nutrient chemicals supplied from the outside. Chemicals are successively transformed from the nutrients through these catalytic reactions. If the reactions progress well, the number of total molecules within this cluster increases, and it is assumed that when the number goes beyond a certain threshold, the cell is assumed to divide into two. We studied a class of models satisfying the above conditions. If the transport of nutrient is at an appropriate level, we have discovered that the cell continues reproduction, approximately maintaining the compositions of chemicals, where the growth is optimized. Reproduction

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

351

of a cell with diversity in chemicals is generally possible, even in this simple setup with mutual catalytic reactions. To study characteristic nature of such reproduction state, we studied statistics of abundances of molecules, for a protocell that keeps reproduction and chemical compositions. We measured the rank-ordered number distributions of chemical species by plotting the number of molecules as a function of its rank determined by the number. The distribution displays a power-law with an exponent of -1. In our model, this power-law of gene expression is maintained by a hierarchical organization of catalytic reactions. Major chemical species are synthesized, catalyzed by chemicals with a slightly smaller amount of abundances. The latter chemicals are synthesized by chemicals with much less abundance, and so forth. This hierarchy of catalytic reactions continues until it reaches the chemical species with minority in number. This hierarachy is not a nature of specific reaction networks, but is commonly observed when the protocell can grow keeping its composition. Furthermore, Furusawa and the author recently found that the flow rate for transport of nutrients is organized autonomously so that this critical state is sustained, if the transport is mediated by one (or few) catalyst(s) within the reaction network.35 Interestingly, this power-law is also confirmed by measuring the abundances of a huge variety of mRNAs, over more than a hundred cell types, by using microarray analysis. Hence, the statistical law as a result of recursive production of a protocell is valid also at the present cell. Adaptation In a class of catalytic reaction networks with growth in a cell volume, a state with a higher growth speed is achieved so that consistency between molecule reproduction and the cell growth is achieved, as a result of dilution by cell growth under stochastic reaction process leads to generic adaptation. Adaptation to a variety of environmental conditions is possible.28,36

4.

Thermodynamics of a Reproducing Cell

A cell receives an influx of some nutrient chemicals; and then effectively uses the energy obtained from the decomposition of these nutrient chemicals to develop useful catalysts and membranes. The chemicals and energy obtained from the nutrients through catalytic reactions are used for cell growth and cell replication. Then, what thermodynamic constraint is imposed for an autonomously reproducing protocell? Can thermodynamic efficiency of cells be understood in the sense of Carnot, or do we need some other scheme to discuss efficiency of reproduction of protocell? Is there an

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

K. Kaneko

352

optimal growth rate for a cell for higher efficiency? These are still totally open questions. To study the cell-state dependence of growth efficiency, Kondo and the author have extended the above mentioned protocell model29 to include energy metabolism, by introducing specific chemicals for “energy currency molecules” such as ATP, which is assumed to be used for cell growth and is synthesized by other catalytic reactions. From simulations of the model, as well as the analysis of the corresponding mean-field theory, three distinct phases with qualitatively different cellular states exhibiting steady growth are found to exist (besides the “death” phase), depending on model parameters. The three phases are characterized by; (i) high cell growth rate and high influx of nutrients, (ii) low growth rate and high influx of nutrients, and (iii) low growth rate and low influx. Thus, the efficiency of conversion of nutrients into the cell growth depends on each phase, and indeed, it is lowest in the phase (ii). Along this line, how cells have achieved thermodynamically efficient growth should be considered, while it is interesting that the existence of the three phases is reminiscent of log, stationary, dormant phases ubiquitous in bacteria and other unicellular organisms. 5.

5.1.

From Catalytic Reaction Networks to Genetic Information: Kinetic Origin of Genetic Information Minority control

In the discussion so far, protocells loosely reproduce themselves by a set of chemicals through catalytic reactions, where “genetic information” may not be necessary, as has been discussed by Dyson,18 Kauffman,19 Lancet,23 and others. However, all the present cells have specific molecules that carry genetic information. Separation of metabolism and genes takes place, or in other words, phenotype and gene that controls the phenotype are separated. This separation is the base of evolution, but how did it take place? Dyson,18 following the idea of Cairns–Smith,20 postulated that genetic takeover took place which transformed loose reproduction into faithful replication. In spite of several discussions, however, theory of this takeover has not been established as yet. To answer the question on the origin of genetic information, we have recently proposed the following hypothesis31 : Consider a reproducing system consisting of mutually catalytic molecules that are encapsulated in a membrane which itself reproduces. Then, molecule species that are minority in number plays the role of heredity-carrier, in the sense that it is preserved well and controls the behavior of this protocell relatively strongly.

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

353

As a first step in an investigation of the origin of genetic information, we study how some molecule species are preserved over cell generations and play an important role in controlling the growth of a cell. Here we consider the following simple model: (i) There are two species of molecules, X and Y , which are mutually catalyzing (minimal hypercycle). (ii) For each of the species X and Y , there are active and inactive types. The active type has the ability to catalyze the replication of both active and inactive types of the other of molecule speciess. Considering that the active molecule type is rather rare, we assume that only one type is active, while there are many inactive types. For example, X and Y are different kinds of polymers, and each type consists of a different sequence of monomers, and only a polymer of a specific sequence has catalytic activity and those of different sequences do not. (iii) The rates of synthesis of the molecules X and Y differ. We stipulate that the rate of the above replication process for Y is much smaller than that for X. This difference in the rates may also be caused by a difference in catalytic activities between the two molecule species. (iv) In the replication process, there may occur structural changes that alter the activity of molecules. Therefore the type (active or inactive) of a replicated molecule can differ from that of the mother. Hence, the probability for the loss of activity is much larger than for its gain. This loss of activity is pointed out by Eigen as error catastrophe.12 (v) When the total number of molecules in a protocell exceeds a given value 2N , it divides into two, and the chemicals therein are distributed into the two daughter cells randomly, with N molecules going to each. Subsequently, the total number of molecules in each daughter cell increases from N to 2N , at which point these divide. We have carried out stochastic simulations by randomly picking up a pair of molecules and making reactions if they can with a given rate. When N is large, the behavior is nothing but that expected from the rate equation of chemical reactions. However, when N is small, under repeated division process of cells with selection, there appears a significant deviation from it. When the total number of molecules N is small, there appears a state of a few active Y molecules and almost zero inactive Y molecules (while the number of X molecules is much larger and there are more inactive X molecules as expected). This state with few active Y molecules and almost zero inactive Y molecules is initially reached by a rare event as a result of fluctuations in molecule numbers, but once such rare fluctuations occurred, they are preserved, since the cell with such compositions can continue reproduction.

November 21, 2012

354

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

K. Kaneko

Note that the continuous rate equation predicts a state with almost zero active Y molecule and several inactive Y molecules. A protocell with such composition cannot continue reproduction, because no X molecules are synthesized, and ultimately there will be no active X molecules either. Here the importance of the smallness in the number of Y molecules is two-fold. First, with such smallness large fluctuation in the number of Y molecules is resulted, which is essential to reach the above rare state. On the other hand, once such a rare state is reached, it is preserved, because the probability to produce inactive Y molecules (that is already extinct) is rather low, since the number of active Y molecules is very small. Then, since such protocell containing active Y molecules can reproduce, such rare state that suppresses inactive Y molecules, once established, is preserved. Note that in this state the active Y molecule is a carrier for heredity, in the following sense.31 Preservation property: The active Y molecules are preserved well over generations. The realization of such state is very rare from the calculation of probability, but, once reached, it is preserved over generations. Also, the number fluctuation of Y molecules is much smaller than that of X. Control property: Consider a structural change in Y molecule, that may occur as a replication error and causes a change of catalytic activity. Since the number of active Y molecules is few, and all the X molecules are catalyzed by them, this influence is enormous. The synthesis speed of a protocell should change drastically. On the other hand, a change to X molecules has a weaker influence, since there are many active X molecules, and influence of change in each molecule is averaged out. Summing up the above argument, the molecule species with slower replication speed and (accordingly) with minor population, comes to possess the properties for the heredity. The state controlled by minority molecule species is termed as minority controlled state (MCS). Note that the compartment is essential to the establishment of this MCS. The necessity of compartmentalization to eliminate inactive states is already discussed in the “stochastic corrector model” by Szathmary.17 So far we have shown the origin of “minimal” genetic information, i.e., just one bit information on the existence/absence of active Y molecule. The minority control, however, gives a basis for genetic information with more bits. First, the minority controlled state gives rise to a selection pressure for mechanisms that ensure the transmission of the minority molecule. Since the active (“minority”) Y molecule is transmitted faithfully, more chemicals will be synthesized with this minority molecule.

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

355

Then, life-critical information is packaged into this minority molecule. Once the minority control mechanism is in place, the minority molecule becomes the ideal storage device for information to be transmitted across generations. 5.2.

Genetic takeover

This MCS may provide a theoretical basis on this genetic takeover. Assume that some molecule species first emerged as a parasitic molecule in a catalytic reaction network for loose reproduction. This molecule then forms a “symbiotic” relationship by achieving mutual catalytic reaction. It now turns to be essential for the reproduction of this chemical reaction network system. When the synthesis speed of this molecule is slower than others, this molecule is minority in number, so that the minority control mechanism would work. Then, genetic take-over by the minority molecule will be completed. Recall the scenario on packaging life critical information into minority molecules. To transfer more information with this minority molecule, it is relevant to embed information into a polymer, since all monomers in the polymer are united so that they are transferred together through replication. Hence, as a minority molecule, stable polymer is a good candidate. 5.3.

Evolvability

As mentioned in Sec. 1, how a cell has acquired evolvability is an important question to be answered. An important consequence of minority control is evolvability (see also Koch37 ). Since only a few molecules of the Y species exist in the minority controlled state, a structural change to them strongly influences the overall catalytic activity of the protocell. On the other hand, a change to X molecules has a weaker influence, on the average, since the variation of the average catalytic activity caused by such a change is smaller, as can be deduced from the law of large numbers. Hence the minority controlled state is important for a protocell to realize evolvability. In fact, Matsuura et al. experimentally designed a reproducing catalyticreaction system,38 where the number of DNA molecules within can be controlled, and showed that when the number is one or few, the systems has evolvability. It is expected that with the generation of minority-controlled state, minority molecules play the role of genetic-information-carrier, thus achieving separation between genotype and phenotype, and leading to evolution.

November 21, 2012

356

6.

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

K. Kaneko

Origin of Compartmentalization and Cell Division

How does a compartment of chemicals that reproduces itself emerge? How is the replication of molecule synchronized with cellular reproduction? To answer this question, it is essential to explain the development of a compartmentalized structure, which undergoes growth and division, from a set of chemical reactions. Kamimura and the author32 considered the mutually catalytic reaction model between X and Y , in Sec. 5. Here we took three more points into account: (i) molecules diffuse in space, and their spatial location (and random walk) are taken into account, instead of the well-stirred model in Sec. 5. (ii) The molecules have a sufficient size, so that they are crowded when they are replicated. (iii) Molecules decompose with a finite rate, which is larger for X (that is synthesized faster) and smaller for Y (synthesized also more slowly). From extensive stochastic simulations, we found that a cluster of cells is formed of a certain size, centered around a single Y molecule. In this MCS, there is a single Y molecule, due to its slower synthesis speed, while since X molecules are synthesized by this Y molecule, Y exists at the center of the cluster of X molecules, whose size is finite due to the decomposition. Then, with a slower rate the Y molecule is replicated to produce a pair of Y molecules, which start to depart by the Brownian motion, while X molecules are synthesized at around each of them. Thus, the cluster starts to form a dumpbell-like structure, and finally when the distance of the Y molecules is larger than the cluster size, two clusters are separated, i.e., divsion of the “protocell” is completed. Thus, the reproduction of a protocell with a growth-division process naturally occurs when the replication speed of one chemical is considerably slower than that of the other chemical. Results of this study show that the division of a protocell is synchronized with the replication of a minority molecule. It is also demonstrated that this growth-division process is robust to errors in molecular replications.

7.

Logical Process from Concurrent Reaction Processes

A salient behavior in an intracellular reaction process is a sequential operation as in a computer program. Only after some process is completed and a certain condition is met, the next step of reactions starts. The next step waits until the previous process is over. Indeed, in the textbook in biology, intracellular processes are often described as “if the condition A is met then the process B progresses”. On the other hand, however, chemical reaction processes are parallel in nature, and a variety of reaction processes work concurrently. Hence, in dynamical-systems description for chemical

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

From Catalytic Reaction Networks to Protocells

357

reactions, all the processes progress in parallel. This point is sometimes criticized as a limitation in dynamical-systems approach to biology. Still, according to chemistry, all the processes have potentiality to progress concurrently. Awazu and the author pointed out two possibilities for the emergence of a sequential process from concurrent catalytic reaction dynamics. The first one is the “chemical-net glass” as discussed in Sec. 2. Here, due to lack of enzyme, progress of certain reactions is hindered, and chemical relaxation process is trapped at a kinetically constrained state. In a complex catalytic reaction network, successive changes over such states are possible. The other possibility is due to discreteness in the molecule number33,39 (see also Refs. 40 and 41). When the number of some catalysts goes to zero, reactions catalyzed by them are stopped over some time span, until they are synthesized from some other reaction paths. In a catalytic reaction network, such bottlenecks to stop reactions appear in time. Indeed, in several simulations, intermittent reaction processes through bottlenecks take place. Residence time for reaction bottlenecks follow a power-law distribution. This kind of discreteness-induced critical behavior may provide a basis for successive switching behaviors in biochemical reaction dynamics. Acknowledgment I would like to thank C. Furusawa, T. Yomo, A. Awazu, A. Kamimura, Y. Kondo, and S. Ishihara for continual discussions. This work was partially supported by a Grant-in-Aid for Scientific Research (No. 21120004) on Innovative Areas. “The study on the neural dynamics for understanding communication in terms of complex hetero systems” (No. 4103), and by ERATO Dynamical Micro-scale Reaction Environment Project.

References 1. K. Kaneko, Life: An Introduction to Complex Systems Biology (Springer, Heidelberg, 2006). 2. K. Kaneko, Complexity 3, 53 (1998). 3. Y. Shimizu, A. Inoue, Y. Tomari, T. Suzuki, T. Yokogawa, K. Nishikawa and T. Ueda, Nat. Biotechnol. 19, 751 (2001). 4. T. Sunami et al., Analyt. Biochem. 357, 128 (2006). 5. P. A. Bachmann, P. L. Luisi and J. Lang, Nature 357, 57 (1992). 6. M. Hanczyc, S. M. Fujikawa and J. W. Szostak, Science 302, 618 (2003). 7. T. Toyota et al., Langmuir 24, 3037 (2008). 8. W. Yu, K. Sato, M. Wakabayashi, T. Nakaishi, E. P. Ko-Mitamura, Y. Shima, I. Urabe and T. Yomo, J. Biosci. Bioeng. 92(6), 590 (2001).

November 21, 2012

13:2

9in x 6in

b1449-ch17

Engineering of Chemical Complexity

358

K. Kaneko

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

V. Noireaux and A. Libchaber, Proc. Nat. Acad. Sci. USA 101, 17669 (2004). K. Kurihara et al., Nature Chem. 3, 775 (2011). J. W. Szostak, D. P. Bartel and P. Luigi Luisi, Nature 409, 387 (2001). M. Eigen and P. Schuster, The Hypercycle (Springer, 1979). J. Maynard-Smith, Nature 280, 445 (1979). M. Eigen, Steps Towards Life (Oxford University Press, 1992). M. Boerlijst and P. Hogeweg, Physica D 48, 17 (1991). S. Altmeyer and J. S. McCaskill, Phys. Rev. Lett. 86, 5819 (2001). E. Szathmary, J. Theor. Biol. 157, 383 (1992). F. Dyson, Origins of Life (Cambridge University Press, 1985). S. A. Kauffman, The Origin of Order (Oxford University Press, 1993). A. G. Cairns-Smith, Clay Minerals and the Origin of Life (Cambridge University Press, 1982). T. Ganti, Biosystems 7, 189 (1975). S. Jain and S. Krishna, Proc. Nat. Acad. Sci USA 99, 2055 (2002). D. Segre, D. Ben-Eli and D. Lancet, Proc. Natl. Acad. Sci. USA 97, 4112 (2000). K. Kaneko, Adv. Chem. Phys. 130, 543 (2005). A. Awazu and K. Kaneko, Phys. Rev. E 80, 041931 (2009). C. Furusawa and K. Kaneko, Phys. Rev. Lett. 90, 088102 (2003). K. Kaneko and C. Furusawa, Biosci. 127, 195 (2008). A. Kashiwagi, I. Urabe, K. Kaneko and T. Yomo, PLoS ONE 1, e49 (2006). Y. Kondo and K. Kaneko, Phys. Rev. E 84, 011927 (2011). K. Kaneko, Phys. Rev. E. 68, 031909 (2003a). K. Kaneko and T. Yomo, J. Theor. Biol. 214, 563 (2002). A. Kamimura and K. Kaneko, Phys. Rev. Lett. 105, 268103 (2010). A. Awazu and K. Kaneko, Phys. Rev. E. 80, 010902 (R) (2009) (Rapid Communication). A. Awazu and K. Kaneko, Phys. Rev. Lett. 92, 258302 (2004). C. Furusawa and K. Kaneko, Phys. Rev. Lett. 108, 208103 (2012). C. Furusawa and K. Kaneko, PLoS Comput. Biol. 4, e3 (2008). A. L. Koch, J. Mol. Evol. 20, 71 (1984). T. Matsuura, T. Yomo, M. Yamaguchi, N. Shibuya, E. P. Ko-Mitamura, Y. Shima and I. Urabe, Proc. Nat. Acad. Sci. USA 99, 7514 (2002). A. Awazu and K. Kaneko, Phys. Rev. E 81, 051920 (2010). Y. Togashi and K. Kaneko, Phys. Rev. Lett. 86, 2459 (2001). N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon, Proc. Nat. Acad. Sci. USA 97, 10322 (2000).

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Chapter 18 CONSTRUCTIVE APPROACH TOWARDS PROTOCELLS Tadashi Sugawara∗,†,‡ , Kensuke Kurihara∗ and Kentaro Suzuki† ∗

Department of Basic Science, Graduate School of Arts and Sciences The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan † Research Center for Complex Systems Biology The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan ‡ [email protected] In this chapter, we describe the construction of a self-reproducing giant vesicle (GV) and the replication of an informational substance (DNA) in the GV. The linkage between these two amplification dynamics has led to generation of a model protocell.

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Protocell . . . . . . . . . . . . . . . . . . . . . . . . . Self-Reproduction of Giant Vesicle . . . . . . . . . . . . . . . 3.1. Robust self-reproducing System . . . . . . . . . . . . . 4. Replication of Informational Substance in GV . . . . . . . . 4.1. Tuning of replication of DNA in GV . . . . . . . . . . . 4.2. GV-size effect on PCR performance . . . . . . . . . . . 5. GV-Based Artificial Cell . . . . . . . . . . . . . . . . . . . . 5.1. Precedent experiments . . . . . . . . . . . . . . . . . . . 5.2. Design of self-reproducing hybrid GV . . . . . . . . . . 5.3. Self-reproduction of GV containing replicated DNA . . 5.4. PCR cycle dependence of frequency of GV division . . . 5.5. Mechanism of linkage between two amplifying dynamics 6. Summary and Prospects . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

360 360 361 362 365 365 366 367 367 367 368 369 371 372 373

November 21, 2012

13:2

9in x 6in

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

360

1.

b1449-ch18

Introduction

Even though the Human Genome Project ended successfully in 2003, such questions as “Where did life originate?” and “What is the boundary between animate and inanimate objects?” have not been answered yet. There are unsettled arguments about the origin of life; whether life originates from the RNA World 1 or the Protein World.2 Many researchers consider that RNA originated first because RNA is an informational substance and it also catalyzes the formation of proteins.3 On the other hand, some researchers insist that protein originated first because it can be synthesized more easily from amino acids. However, from the chemical viewpoint, it is difficult to accept that RNA or proteins could be formed spontaneously in the early stage of the prebiotic era because these macromolecules seem too complex to be generated without biological environments.4 Compared with these materials, the molecular structures in the Lipid World 5 are simple. They, however, are capable of forming such self-assembled structures as micelles and vesicles in water. In this chapter, we intend to answer these profound questions by the experimental results obtained by a constructive approach.6 An example is the construction of a protocell7 using well-defined organic molecules and biopolymers, focusing on the events that could have occurred in the prebiotic era.8 –10

2.

Model Protocell

Protocells are generally defined as hypothetical precursor structures of primitive cells, which are assumed to have been formed at the emergence of life.7,11,12 A protocell constructed using a vesicle may play a significant role in the elucidation of the origin of a living cell1.13 –15 Szostak et al.16 in their paper entitled “Synthesizing life”, pointed out three crucial elements of a protocell: a compartment, a catalyst, and an informational substance. The protocell should show two indispensable dynamics: self-reproduction of the compartment and self-replication of the informational substance. This model implies that the encounter between DNA (RNA World) and an enzyme (Protein World) in a vesicle (Lipid World) is essential to the generation of a model protocell. Because hierarchical dynamics must emerge in such a protocell, this concept is relevant to G´anti’s chemoton model.17 Reproductiveness and recursiveness of protocells have been the focus of interest of theoretical physicists as well.18,19 In this chapter, we describe three important stages for the emergence of a model protocell: first, the self-production of a cell membrane, which

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

361

belongs to the Lipid World (Sec. 3); second, the self-replication of the informational substance, that is, DNA (Sec. 4); and third, linkage between these two self-amplification dynamics leading to the generation of a model protocell (Sec. 5). 3.

Self-Reproduction of Giant Vesicle

Before describing concrete examples, it would be useful to define the selfreproducing system according to Luisis proposal 8 (Fig. 1). Consider a vesicle consisting of the membrane molecule S. If this vesicle incorporates the membrane precursor A and converts it into S within itself, it enlarges and becomes destabilized owing to the surface tension, and then it selfdivides. Such dynamics can be regarded as self-reproduction (Fig. 1(a)). In contrast, if a vesicle incorporates S in its membrane or in a water pool, the enlarged vesicle self-divides. However, such a system cannot be designated as a self-reproducing system because of the lack of metabolism (Fig. 1(b)). Walde et al.20 reported a pioneering system that the number of giant vesicles (GVs, φ > 1 µm) composed of oleic acid and oleate increases in

Fig. 1.

Definition of self-reproduction of compartments.

November 21, 2012

362

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

alkaline water with dispersed oil droplets of anhydride as a precursor of oleic acid. In their model, oleic anhydride is transferred to the vesicular membrane by a surfactant (oleate) and it is hydrolyzed to oleic acid, resulting in the increase in the number of vesicles. They claimed that this system is a self-reproducing system. However, because hydrolysis may occur anywhere in an alkaline solution, the reaction field cannot be specified to the vesicular membrane. Theoretical and experimental investigations on remarkable dynamics such as peeling, budding, division, and birthing have been reported.21 –25 3.1.

Robust self-reproducing system

On the basis of Luisis definition of self-reproduction, a self-reproducing giant multi-lamellar vesicle (GMV) was constructed by Takakura and Sugawara.26 A scenario of the self-reproduction of GMV is as follows (Fig. 2). The vesicle is formed by two-legged artificial amphiphile V with a head group of a trimethylammonium salt. The precursor of the membrane molecule V∗ is a bolaamphiphile that does not form a membrane but dissolves in water, forming small aggregates. Then V∗ is converted to membrane molecule V and electrolyte E in the presence of the amphiphilic

Fig. 2. Dynamics and components of self-reproducing giant multi-lamella vesicle of robust type. (a) Microscopy images of robust self-reproducing GMV. From a mother vesicle (S), a daughter vesicle (T) is formed, and a grand daughter vesicle (U) is formed from T. (b) Mechanism of self-reproduction. Bolaamphiphile (V∗ ), which is a precursor of the membrane molecule V, is hydrolyzed to V and electrolyte E in the vesicular membrane containing a catalyst. The hydrolyzed membrane molecule dissolves in GMV and electrolyte E is released into a thin water layer between vesicular membranes.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

363

Fig. 3. Population change in a 2D diagram of GVs containing fluorescent catalyst before and after self-reproduction. (Insets) Fluorescence microscopy images before and after self-reproduction of GMV containing a fluorescent-probe-tagged catalyst.

acidic catalyst C. The generated V dissolves in the membrane and induces GMV growth. On the other hand, the released E functions as a weak surfactant and contributes to the fission of the squeezed GMV resulting in the generation of two GMVs of similar sizes. The process of the self-reproduction of GMV in a mass scale was analyzed by flow cytometry (FCM).27 The two-dimensional plots of FCM data of GMV containing a fluorescent catalyst obtained before and after the addition of the membrane precursor V∗ is shown in Fig. 3. Each dot represents the amount of catalyst C contained in each vesicle along the vertical axis and the size of each vesicle along the horizontal axis. The distribution shifted downward along the vertical axis by about one order of magnitude. This means that the GMV divided three or four times on average. Note that the size distribution remains almost the same except for the contribution of the very large GMV during the repeated selfreproduction processes. The origin of this interesting maintenance of the size distribution was elucidated by a precise measurement.28 Addition of the amphiphilic catalyst C to the outer aqueous phase and allowing the mixture to stand for four hours restored the amount of catalyst C and then the GMV exhibited division dynamics again. It turned out that the number of GMVs increased by 100 times. This division dynamics drew the interest of theoretical physicists. Recently, Umeda29 has proposed a mechanism of the division of the GMV under the following assumptions (Fig. 4). (i) The GMV is an elastic body and its elastic energy is expressed by Eq. 1. (ii) The added amphiphiles dissolve and pass through the surface of the GMV at a constant rate per area, and diffuse into the inner lamellae. (iii) The distance between

November 21, 2012

364

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

Fig. 4. Simulation of the division dynamics of GMV with 100 lamellae shows the following dynamics. The outer membrane suffers from the positive stress as it expands because the inner membranes pull the outer membrane back, while the core membrane suffers from the negative stress because there are no or only few inner membranes that pull the outer membrane back. As a result, the inner core is divided into two small GMVs, which become nuclei (or seeds) for the reorganization of outer membranes. Because this simulation reproduces the observed dynamics well, it may be a reasonable mechanism for this GMV division. It is speculated that the role of the electrolyte E, which is separated from V∗ , functions as a weak surfactant and mediates the complete division of the pinched GMV (see, text).

two adjacent lamellae is fixed even when the shape of the GMV changes. (iv) Energy minimization determines the mode of deformation of the GMV. E=

i k
(Ai − aNi )2 , 2 m=1 aNi

(1)

where m is the number of lamellae, Ai is the area of the ith lamella, Ni is the number of molecules in the ith lamella, k is the elastic constant, and a is the area per molecule.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

4. 4.1.

365

Replication of Informational Substance in GV Tuning of replication of DNA in GV

Excellent examples of enzymatic reactions proceeding in vesicles have hitherto been reported: synthesis of nucleotides,30 polypeptides,31 and expression of proteins.32,33 However, as far as DNA replication by polymerase chain reaction (PCR) is concerned, only one example has been reported.34 Presumably owing to the low encapsulation efficiency of vesicles with diameters smaller than 1µm, the percentage of the number of the PCR-performed vesicles with respect to the total number of vesicles is less than 0.1%. Shohda et al.35 tuned the conditions of PCR in a robust nonself-reproducing GV. First, poly(ethyleneglycol)-grafted phospholipid [distearoylphosphatidylethanolamine (DSPE-PEG 5000)] was added to a lipid mixture (palmitoyloleoylphosphatidylcholine (POPC) and cholesterol) to enable the GV to tolerate a highly ionic solution. Otherwise, lipids formed amorphous aggregates. Second, GVs with PCR reagents were formed by adding an aqueous mixture of PCR reagents to freeze-dried GVs to increase the encapsulation efficiency of template DNA and other reagents. Although a relatively concentrated solution of the template (0.2 nM) was used, the probability of encapsulation was about 30 templates for a 10 µm diameter GV and was only one template for a 3 µm diameter GV. Third, reasonably long template DNA consisting of 1229 base pairs, which can express the green fluorescent protein, was used. Fourth, the condition of the thermal cycling was optimized carefully. It was found that a twostep cycle, not the three-step cycle that is usually used in PCR was more appropriate for the successful replication of DNA in GV (Fig. 5).

Fig. 5. Optical and fluorescence microscopy images of (a) before and (b) after PCR in GV. Before the thermal cycling, all the vesicles scarcely emit fluorescence. After the thermal cycling, most of the vesicles emit intense fluorescence.

November 21, 2012

366

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

Such large thermal oscillations could be encountered with primitive cells around hydrothermal vents in a deep sea, which is a plausible birth place of these cells.

4.2.

GV-size effect on PCR performance

The performance of PCR was also analyzed by FCM using a fluorescent probe for DNA (Fig. 6). The percentage of DNA-replicated GVs with respect to the total number of GVs was estimated to be approximately 10% (on average). This average value was appreciably larger than that of Oberholzers report described previously.34 Note that only few dots with increased FL intensities were detected for vesicles smaller than 1 µm. This is because the entrapment of important components, such as template DNA and polymerase, is crucial for the PCR performance. Unless the GV contains these two components, the replication of DNA cannot proceed. PCR performance may be affected not only by the entrapment efficiency of the components but also by the lamellarity of GVs; DNA replication hardly proceeds in myelin- or nested type GVs owing to the insufficient volume for enzymatic reactions. Incidentally, recent experimental results have shown that the entrapment efficiency of substrates in a GV is not necessarily the statistical event but depends heavily on the interaction between the components and the vesicular membrane.36 The above FCM analysis suggests that the competition between primitive cells regarding the production of the informational substance had already started when the enzymatic reaction proceeded in a compartment.

Fig. 6. Fluorescence intensities derived from GVs before PCR were lower than 10 along the vertical axis (arbitrary scale), but a group of vesicles with a fluorescence intensity that was larger than 100 appeared after 20 PCR cycles.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

5. 5.1.

367

GV-Based Artificial Cell Precedent experiments

In pioneering investigations on a protocell, Oberholzer et al.37 constructed a model protocell consisting of RNA, Qβ-replicase, and oleate/oleic acid small vesicles (SVs, φ < 100 nm). However, there was no obvious interaction between the vesicular membrane and the replicated RNA; namely, these processes occurred independently. Mansy et al.38 prepared SVs that are semipermeable to nucleotides, and ribose, among others, and demonstrated that short DNA (≈ 30 bp) is synthesized in this model of a hypothetical prebiotic cell by taking nucleotides up from the exterior water phase. Although RNA/DNA synthesis in a vesicle has been extensively studied, the division of such vesicles is mainly performed by a physical procedure, e.g. filtration of a vesicular dispersion. Hence, the intimate interplay between the self-reproduction of a GV and the self-replication of an informational substance in a GV has scarcely been explored yet.

5.2.

Design of self-reproducing hybrid GV

The achievements in research on the self-reproduction of GV and the selfreplication of DNA in GV prompted researchers to study the construction of a model protocell model in which these two dynamics are chemically linked. This is interesting from the viewpoint of simulating the encounter between our primitive protocell and DNA. One of the clues to the self-replication of DNA on the inner surface of the vesicular membrane was obtained using a conjugated amphiphile carrying oligonucleotides and cholesterol at both terminals.39 The primitive linkage between the self-production of a GV and the replication of the encapsulated template DNA in a GV would be achieved if a cationic membrane molecule was blended with a phospholipid in order for the molecule to interact with the replicated polyanionic DNA. Because the positively charged membrane molecule was used for the robust self-reproduction of a GV, as described in the previous section, the membrane molecule V may interact with polyanionic DNA,40 which affects the vesicular division. For this purpose, the composition of a hybrid GV was tuned carefully to satisfy the requirements for the replication of encapsulated template DNA (Table 1). The current self-reproducing GV was made of V, which was of the multi-lamellar type. However, the addition of more than 50% zwitterionic phospholipid (POPC) transformed GVs with a sufficient inner volume for the enzymatic reaction to proceed. Second, an anionic phospholipid (POPG) was added to neutralize the positive charge of the cationic membrane. This treatment is necessary to prevent the adhesion

November 21, 2012

368

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki Table 1. Membrane compositions and list of PCR reagents. Contents

Ratio

POPC (zwitterionic phospholipid) POPG (anionic phospholipid) V (cationic membrane molecule) C (amphiphilic catalyst)

60 mol% 20 mol% 20 mol% 10 mol%

of polyanionic DNA to the cationic vesicular membrane in the initial stage of DNA replication. Third, GVs should be able to tolerate high temperatures and a highly ionic medium. This requirement, indeed, was satisfied by the addition of anionic POPG. The resulting hybrid GV was expected to replicate the encapsulated DNA in GVs by PCR. 5.3.

Self-reproduction of GV containing replicated DNA

A film composed of these lipids POPC, POPG, V which is induced to a molar ratio of 6:2:2, together with the acidic amphiphilic catalyst C, was swell using a buffered solution containing 1229-bp template DNA that expresses GFP, DNA polymerase, two types of primer, deoxyribonucleoside triphosphate (dNTP), and Mg2+ . SYBR Green I (SG) was also added to enable the detection of replicated double-stranded DNA.41 GVs with a diameter of approximately 10 µm were observed frequently in the dispersion under a microscope. The two-step thermal cycling was carried out as described previously.35 Serial divisions of GVs containing replicated DNA produced multiple GVs within seven minutes following the addition of V∗ . Micrographs of the self-reproducing GVs containing replicated DNA were obtained by differential interference contrast microscopy and fluorescence microscopy (Fig. 7(a)). Moreover, as shown in the fluorescence micrographs, the efficient partitioning of the replicated DNA to the newly formed GVs was confirmed.42 The fluorescence intensity due to the dsDNA-SG complex in GVs with the cationic membrane V was analyzed precisely. The analysis indicates that the replicated DNA exists not only inside but also in the periphery of GVs. On the other hand, the distribution of the fluorescence intensity of the replicated-DNA-containing GV without V shows the maximum in the middle of the interior water pool. This finding indicated that DNA was localized on the membrane by V. Control experiments are important to clarify the mechanism understanding the observed dynamics. First, when the thermal cycling was

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

Fig. 7.

369

Chemical linkage between DNA replication and self-reproduction of GV.

conducted on GVs with PCR reagents, GVs did not divide in the absence of V∗ . Second, when no thermal cycling was conducted, less than 10% of GVs (total numbers, 200 GVs) underwent division, and these GVs divided only once within 2 hrs after the addition of the membrane precursor, even though the GVs contained PCR reagents. Third, When GVs without the template DNA were subjected to thermal cycling, GVs divided very rarely even after the addition of the membrane precursor. These control experiments strongly suggest that replicated DNA in GVs and the addition of the membrane precursor are necessary for observing the sequential growth and division of GVs and that the replicated DNA assists the amphiphilic acidic catalyst in converting the precursor to the membrane molecule. 5.4.

PCR cycle dependence of frequency of GV division

From the two control experiments, it is evident that GV division is driven by membrane-adhering DNA in the GV. To confirm the dependence of GV division frequency on the amount of replicated DNA, the PCR cycle dependence on GV division was examined. The procedure was as follows. First, membrane-stained GVs were prepared by mixing them with 0.01 mol% of a phospholipid tagged with a fluorescent probe (RhodDOPE). If the division of the membrane-stained GVs occurs, the amount of fluorescent probe per GV decreases depending on the number of divisions.

November 21, 2012

370

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

Fig. 8. Decay kinetics for the number of GVs with a fluorescence intensity of 1 × 103 (arbitrary unit) after DNA amplification following the addition of V∗ . Plots A, B, and C represent the numbers of remaining GVs with a fluorescence intensity of 1 × 103 after 20, 15, and 10 thermal cycles, respectively. GVs with a fluorescence intensity of 1 × 103 were chosen because of the negligible contribution of divided GVs with a fluorescence intensity higher than 1 × 103 . The pseudo-first-order decay rate constants were evaluated to be k0 = 5 × 10−5 min−1 , k10 = 2 × 10−2 min−1 , k15 = 3 × 10−2 min−1 , and k20 = 3 × 10−1 min−1 for plots A, B, and C, respectively, where ki denotes the initial rate constant for GVs with i thermal cycles.

Because only large GVs were detected under an optical microscope, the decrease in the number of larger GVs (approximately 10 µm diameter) was examined. The FL intensity of such large GVs was estimated to be 103 (arbitrary unit) on the basis of calibration by filtering experiments. The time course of the decrease in the number of GVs with FL = 103 after the addition of V∗ is plotted in Fig. 8. Most of the GVs subjected to 20 thermal cycles, as shown by plot A, decayed rapidly after the addition of V∗ . On the other hand, only 20% of GVs subjected to 10 thermal cycles decayed rapidly and the remaining GVs were nearly unchanged as shown in plot C. The time course of plot B was intermediate between those of the other two plots. This experiment clearly shows that rapid GV division depends on the number of thermal cycles and that the frequency of the division is related to the amount of replicated DNA. Moreover, this finding suggests that if sufficient amounts of PCR reagents are encapsulated in its interior water pool, PCR would produce a reasonable amount of DNA even after 10 thermal cycles. 5.5.

Mechanism of linkage between two amplifying dynamics

The mechanism of self-replication of GVs driven by replicated DNA is discussed here on the basis of the above-mentioned experimental findings.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

371

The morphological change may occur in two steps. The first step is a sort of pre-organization for the production of the membrane molecule and the morphological change of GVs (Fig. 7(b)). The adhesion of DNA to the inner surface of the vesicular outer membrane occurs accompanied by the replication of DNA. Evidence for this is described in Sec. 5.4, when DNA adheres to the inner surface of the outer membranes, only the inner membrane covers the adhering DNA, resulting in an imbalance in the number of membrane molecules between the inner and outer leaflets. Thus, the inner and outer leaflets are ready to deform in the buddingtype manner. However, no deformation or division occurs unless V∗ is added at this stage. The second step starts after the addition of V∗ to the dispersion of replicated-DNA-containing GVs. When V∗ with two cationic head groups is dissolved into the vesicular membrane, V∗ would be captured by polyanionic DNA (Fig. 7(b)). The captured V∗ is likely hydrolyzed with the assistance of the amphiphilic catalyst C, resulting in the increase in the number of membrane molecules in both the outer and inner leaflets around DNA owing to the presence of hydrophilic groups at both ends of V∗ , which allows V∗ to dissolve either from the top or the bottom side. The balanced production of membrane molecule V in outer and inner leaflets induces rapid sequential growth and division dynamics. Angelova and Tsoneva43 reported that the addition of a DNA dispersion to cationic GV membranes caused a rapid division in a concave manner owing to the strong interaction between the oppositely charged substances. The division in the current experiment is similar to the morphological change reported by Angelova et al. However, the deformation occurred in a convex manner because DNA interacts with vesicular membrane from the inner side. Even though this primitive cell does not contain a specific protein for partitioning the replicated DNA, the cooperative interactions between the cationic membrane molecule V, adhering DNA, the cationic amphiphilic catalyst C, and the cationic membrane precursor V∗ make such amplification and partition dynamics possible. Presently, DNA replication in GVs and GV self-reproduction in the presence of replicated DNA are closely linked. The current system satisfies the requirements for an artificial cell proposed by Szostak et al.16

6.

Summary and Prospects

The dynamic GV system in which the membrane-formation reaction is compatible with the replication of DNA was realized. The replicated DNA, which is the product of an enzymatic reaction in the GVs, accelerated the

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

372

division of DNA-rich GVs induced by the addition of V∗ to the exterior water phase. It is interesting to compare the mechanisms of GV division and partition of replicated DNA in this model protocell with those of a prokaryotic cell. In a prokaryotic cell such as E. coli, specific nucleotide sequences (oriCs) on the chromosome bind to the cell membrane early in the replication of the chromosome as mediated by a protein complex.44 Thereafter, the cell membrane between two connected points elongates and becomes pinched, and the genomic DNA is equally distributed between the two daughter cells. Thus, in a prokaryotic cell, specific proteins mediate the separation of DNA and the division of the cellular membrane. By comparison, in our protocell, the replicated DNA is partitioned by the cooperative interactions between DNA, the cationic membrane molecules V, and the bolaamphiphilic membrane precursor V∗ , not by specific proteins. The generation of self-reproducing GVs in which DNA can be replicated is a milestone toward the development of a model of primitive cells that joins the lipid and RNA Worlds. Once the protocell incorporates a molecular transportation system, e.g. vesicular fusion methods for transferring deficient components to the protocell, the linked self-production will occur repeatedly. Moreover, if a genotype of the replicated DNA affects a phenotype of GV, the resulting protocell could be regarded as an evolving protocell to which natural selection is applicable. As for the origin of life, there is an unsettled argument on the precedence of appearance between RNA and proteins as mentioned in Sec. 1. If the current experimental findings could be expatiated, the scenario of the emergence of the first cell driven by the lipid world would be as follows. Once a vesicle acquires the capability of self-reproduction, it could incorporate pre-DNA and pre-protein. Then, these materials, belonging to the RNA, protein, and lipid worlds, could coevolve with each other and eventually convert themselves to a primitive living cell (Fig. 9).

Fig. 9.

Scenario of appearance of first cell from three worlds.

November 21, 2012

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

Constructive Approach Towards Protocells

373

References 1. W. Gilbert, Nature 319, 618 (1986). 2. D. H. Lee, J. R. Granja, J. A. Martinez, K. Severin and M. R. Ghadiri, Nature 382, 525 (1996). 3. R. F. Gesteland, T. R. Cech and J. F. Atkins, (eds.), The RNA World, 3rd edn. (Cold Spring Harbor Laboratory Press, New York, 2005). 4. K. Ikehara, Chem. Rec. 5, 107 (2005). 5. D. Segr´e, D. Ben-Eli, D. Deamer and D. Lancet, Orig. Life Evol. Biosph. 1–2, 119 (2001). 6. H. J. Morowitz, Beginnings of Cellular Life: Metabolism Recapitulates Biogenesis (Yale University Press, Connecticut, 2004). 7. S. Rasmussen, M. A. Bedau, L. Chen, D. Deamer, D. C. Krakauer, N. S. Packard and P. F. Stadler (eds.), Protocells: Bridging Nonliving and Living Matter (The MIT Press, Massachusetts, 2008). 8. P. L. Luisi, The Emergence of Life: From Chemical Origins to Synthetic Biology (Cambridge University Press, UK, 2006). 9. P. Walde, Prebiotic Chemistry: From Simple Amphiphiles to Protocell Models (Springer-Verlag, New York, 2010). 10. P. Pontarotti, (ed.), Evolutionary Biology: Concept, Modeling, and Application (Springer-Verlag, New York, 2009). 11. P. Walde, Bioessays. 32, 296 (2010). 12. P. Stano and P. L. Luisi, Chem. Commun. 46, 3639 (2010). 13. G. R. Fleischaker, S. Colonna and P. L. Luisi, (eds.), Self-production of Supramolecular Structures: From Synthetic Structures to Models of Minimal Living Systems, Vol. 446, NATO Science Series, Ser. C, Mathematical and Physical Sciences (Kluwer Academic Publications, Massachusetts, 1994). 14. T. Sugawara, Evolutionary Biology: Concept, Modeling, and Application (Springer-Verlag, New York, 2009), p. 23. 15. M. M. Hanczyc, S. M. Fujikawa and J. W. Szostak, Science 302, 618 (2003). 16. J. W. Szostak, D. P. Bartel and P. L. Luisi, Nature 409, 387 (2001). 17. T. G´ anti, The Principles of Life (Oxford University Press, UK, 2003). 18. K. Kaneko, Life: An Introduction to Complex Systems Biology (Understanding Complex Systems) (Springer-Verlag, New York, 2006). 19. E. Szathmary, M. Santos and C. Fernando, Evolutionary Potential and Requirements for Minimal Protocells (Springer-Verlag, New York, 2005). 20. P. Walde, R. Wick, M. Fresta, A. Mangone and P. L. Luisi, J. Amer. Chem. Soc. 116, 11649 (1994). 21. B. Boˇziˇc and S. Svetina, Eur. Phys. J. E. 24, 79 (2007). 22. F. M. Menger and S. J. Lee, Langmuir 11, 3685 (1995). 23. P. L. Luisi, P. Walde and T. Oberholzer, Curr. Opin. Coll. Int. Sci. 4, 33 (1999). 24. Y. Sakuma and M. Imai, Phys. Rev. Lett. 107, 198101 (2011). 25. K. Suzuki, T. Toyota, K. Takakura and T. Sugawara, Chem. Lett. 38, 1010 (2009). 26. K. Takakura and T. Sugawara, Langmuir 20, 3832 (2004).

November 21, 2012

374

13:2

9in x 6in

b1449-ch18

Engineering of Chemical Complexity

T. Sugawara, K. Kurihara & K. Suzuki

27. T. Toyota, K. Takakura, Y. Kageyama, K. Kurihara, N. Maru, K. Ohnuma, K. Kaneko and T. Sugawara, Langmuir 24, 3037 (2008). 28. K. Kurihara, K. Takakura, K. Suzuki, T. Toyota and T. Sugawara, Soft Matter 6, 1888 (2010). 29. T. Umeda, Kobe University, Japan, private communication. 30. A. C. Chakrabarti, R. R. Breakera, G. F. Joyce and D. W. Deamer, J. Mol. Evol. 39, 555 (1994). 31. T. Oberholzer, K. H. Nierhaus and P. L. Luisi, Biochem. Biophys. Res. Comm. 261, 238 (1999). 32. T. Oberholzer and P. L. Luisi, J. Biol. Phys. 28, 733 (2002). 33. S. M. Nomura, K. Tsumoto, T. Hamada, K. Akiyoshi, Y. Nakatani and K. Yoshikawa, ChemBioChem. 4, 1172 (2003). 34. T. Oberholzer, M. Albrizio and P. L. Luisi, Chem. Biol. 2, 677 (1995). 35. K. Shohda, M. Tamura, Y. Kageyama, K. Suzuki, A. Suyama and T. Sugawara, Soft Matter 7, 3750, (2011). 36. P. S. T. P. de Souza and P. L. Luisi, ChemBioChem. 11, 1056 (2010). 37. T. Oberholzer, P. L. L. R. Wick and C. K. Biebricher, Biochem. Biophys. Res. Comm. 207, 250 (1995). 38. S. S. Mansy, J. P. Schrum, M. Krishnamurthy, S. Tob, D. A. Treco and J. W. Szostak, Nature 454, 122 (2008). 39. K. Shohda and T. Sugawara, Soft Matter 2, 402 (2006). 40. J. O. Rdler, I. Koltover, T. Salditt and C. R. Safinya, Science 275, 810, (1997). 41. H. Zipper, H. Brunner, J. Bernhagen and F. Vitzthum, Nucleic Acids Res. 32, e103 (2004). 42. K. Kurihara, M. Tamura, K. Shohda, T. Toyota, K. Suzuki and T. Sugawara, Nature Chem. 3, 3 (2011). 43. M. I. Angelova and I. Tsoneva, Chem. Phys. Lipids 101, 123 (1999). 44. G. B. Ogden, M. J. Pratt, and M. Schaechter, Cell 54, 127 (1988).

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Chapter 19 NETWORK REVERSE ENGINEERING APPROACH IN SYNTHETIC BIOLOGY Haoqian Zhang∗,†,‡ , Ao Liu∗ , Yuheng Lu∗ , Ying Sheng∗ , Qianzhu Wu∗ , Zhenzhen Yin∗ , Yiwei Chen∗ , Zairan Liu∗ , Heng Pan∗ and Qi Ouyang∗,†,‡,§,¶ ∗

2010 Peking University Team for The International Genetic Engineering Machine Competition (iGEM), Peking University, Beijng, 100871, China † Center for Quantitative Biology, Peking University Beijng, 100871, China ‡ Peking-Tsinghua Center for Life Sciences at School of physics Peking University, Beijing, 100871, China § The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics Peking University, Beijing, 100871 China ¶ [email protected] Synthetic biology is a new branch of interdisciplinary science that has been developed in recent years. The main purpose of synthetic biology is to apply successful principles that have been developed in electronic and chemical engineering to develop basic biological functional modules, and through rational design, develop man-made biological systems that have predicted useful functions. Here, we discuss an important principle in rational design of functional biological circuits: the reverse engineering design. We will use a research project that was conducted at Peking University for the International Genetic Engineering Machine Competition (iGEM) to illustrate the principle: synthesis a cell which has a semi-log dose-response to the environment. Through this work we try to demonstrate the potential application of network engineering in synthetic biology.

¶ Corresponding

author 375

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

376

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Reverse Engineering Analysis . . . . . . . . . . . . . . 2.1. Quantitative definition of object function . . . . . 2.2. Network definition, dynamics model derivation and calculation . . . . . . . . . . . . . . . . . . . . 2.3. Analysis and results . . . . . . . . . . . . . . . . . 3. Biological Implementation of a SLDRC Network Motif 4. Summary and Discussion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

376 377 377

. . . . .

378 382 384 388 390

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Introduction

Significant advances have been made in synthetic biology researches. For instance, genetic toggle switch,1 genetic oscillator,2,3 push-on push-off genetic switch,4 and genetic program performing edge detection5 have been experimentally synthesized in different laboratories. These achievements demonstrated that artificially designed biological circuits can perform predicted reliable functions and are very useful for biological engineering. However, most of these circuits have been built in a fairly ad hoc manner, in which iterative design and following troubleshooting are often required to exhibit desired functions because of the lack of effective rational design approach.6,7 Therefore, how to develop rational approaches that reliably support the design and construction of genetic circuit is becoming a critical problem in the field of synthetic biology.8,9 Systems biology aims to quantitatively study biological regulatory network, focusing on the interactions between biological components and under what design principles those interactions happen.10,11 Obviously, investigation, understanding and designing are integrally linked. Therefore, it is reasonable to assume that exploiting natural design principles depicted by systems biology for rationally designed artificial biological functional network could be a potential solution to the problem mentioned above.8,9,11 It has been recently revealed in system biology that network topology plays a critical role in robustly producing specific functions — The functional repertoire of a network is limited by its network topology.12 –14 Although the implementation of biological networks dramatically diversifies, there are only a limited number of network topologies that permit a particular biological function, among which some may be more favored for their fewer parameter constraints.12,13 Here we apply the idea that was developed by Ma et al.13 to design a network reverse engineering approach for synthetic biology. The procedure is outlined as the following: for a specific object

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

377

functionality, we enumerate all possible tri-node network motifs and ask which networks can perform the given function? And how robust they are against internal and external fluctuations? We select “positive” network topology motifs, namely, the ones with the highest robustness for desired function, and exclude all other unrelated ones. If the network motifs with highest robustness can be biologically implemented, the desired function will be achieved with less laborious trial-and-error fine-tuning or iterative design. As a proof of concept, we applied this framework to theoretically design and experimentally construct a genetic regulatory circuit that enables bacteria E. coli. to report the presence of mercury (II) ions in water with a semi-log dose response curve (SLDRC)(Fig. 1(a)), in which the output value of the cell (the level of green Florence protein, GFP) is proportional to the log value of the input mercury (II) concentration in water). The major advantage of SLDRC is that the detection error can be reduced, especially near the high and low input concentration value, and working range can be expanded compared with conventional bi-node bioreporter whose dose response curve is typically a Hill Function; the sensitivity range of this type of dose-response curve is very limited.

2.

Reverse Engineering Analysis

The procedure of the biological network reverse engineering design is presented in Fig. 1(b). It contains the following major processes: defining the objective function; network definition and enumeration, computer simulation and analysis. In the following we discuss each process in detail. 2.1.

Quantitative definition of object function

We first determine the object function in quantitative characters according to our expectations. Figure 1(a) provides the main feature of our objection function, which is a semi-log dose response function. The horizontal axis in Fig. 1(a) denotes input, the value of logarithm concentration of divalent mercury ion, and vertical axis stands for output, the value of the expression level of reporter gene, which increases with input in a linear relationship. The output range was defined as HIGHLEVEL minus LOWLEVEL. In addition, we also expected a wide input range; it was predetermined to be wide enough, in our case 10−9 M to 10−5 M. In order to define SLDRC quantitatively for network topology searching, a numerical standard, Pearson Correlation Coefficient r, was also set to represent Input–Output linear relationship in the overall searching work

November 21, 2012

378

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

Fig. 1. (a) The object function for designed networks. r is Pearson correlation coefficient and output range was defined as HIGHLEVEL minus LOWLEVEL. (b) Schematic description of the network reverse engineering approach.

(Fig. 1(a)). If r is larger than a critical value rc (we set rc = 0.98 in our study) we regarded the network topology motif as “positive”, which means it indeed can exhibit SLDRC, else as “negative”, this network motif will be excluded. 2.2.

Network definition, dynamics model derivation and calculation

In this process, we first need to determine the scale of the networks (the number of nodes) and the interaction function among different nodes. Same as in the Ma’s work,13 here we consider only a minimal framework of tri-node network: one node for receiving inputs (Node A in Fig. 2(a)), one node for transmitting output (Node C in Fig. 2(a)), and a third node that

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

379

Fig. 2. Framework of tri-node transcriptional network. (a) All possible tri-node network. Node A receives input signal, Node C transmits output, and Node B plays regulatory roles. The links between nodes stand for TF interaction through transcription and translation with three possible types: positive, negative, or no regulation. There are 16038 possible network motifs in total. (b) For each topology motif, 10,000 sets of network parameters were sampled using LHS method. When sampling a function of N variables, the range of each variable was divided into M equally probable intervals, and M sample points were then placed to meet the Latin Hypercube requirements. In each axis-aligned hyperplane only one cube was filled with sample point. (c) An example of network motif. There is only one link from node A to node C. Whether this link is positive, negative or no regulation was determined by λ, a constant described in context.

can play diverse regulatory roles (node B in Fig. 2(a)). There are nine direct links between nodes, each with three possible types: positive, negative, or no regulation. Therefore, there are altogether 39 = 19,683 tri-node topology motifs because among them there are 3,645 topology motifs that have no direct or indirect links from the input to the output (occlusive Node C), there remain a total of 16,038 possible three-node network motifs that contain at least one direct or indirect causal link from the input node to the output node. We need to evaluate all these possible networks to get the ensemble of positive networks.

November 21, 2012

380

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

The next step is to translate each network into a set dynamics equations that can quantitatively describe its dynamics behavior. For this purpose we need to determine the interactions among nodes in a given network. In this work, derivation of the equation was mainly based on the following assumptions: (1) All three nodes represent transcriptional factors (TF) or reporter gene, such as green fluorescence protein (GFP). The links between nodes stand for gene–gene interaction through transcription and translation process. The transcription was quantified by the equilibrium binding probability P for binding of TF to DNA operator and the maximum transcription rate is constant β. Then we adopted a constant λ to adjust P to make different TFs in equal status. For interaction of multiple TF, we used the multiplication of their λP or 1 − λP to indicate it.12 (2) Only transcription and translation process were considered because other reactions such as mercury (II)-protein binding process reasonably happens much faster and can be regarded to be approximately at steady state compared with the long timescale of transcription-translation process.12,15 –18 Therefore, the equations derived mainly considered protein (TF or GFP) production (transcription and translation), degradation and dilution along with cell growth. (3) It is well known that gene expression has basal level even without TF regulation.19 A repressor will down-regulate the initial expression level and an activator will up-regulate it. Accordingly, we assume each gene have a basal level expression β0 . Furthermore, we assume that every TF species contributes to the final expression level of protein. As an example, we translate the regulatory network presented in Fig. 2(c) as the following: It is widely accepted that the possibility of TF binding to DNA operator in promoter is: P =

(X ∗ /Kd )n , 1 + (X ∗ /Kd)n

(1)

where X ∗ and Kd represents, respectively, the effective concentration of specific TF and the dissociation constant. According to the assumptions (1) and (3), the link from Node A to Node C can be derived as: dXC = β0 + (βA − β0 )λA PAC − αC XC , dt ∗ (XA /Kd )n = β0 + (βA − β0 )λA ∗ /K )n − αC XC , 1 + (XA d

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology


= β0 1 − λA +βA λA

∗ (XA /Kd )n ∗ /K )n 1 + (XA d

381



∗ (XA /Kd )n ∗ /K )n − αC XC , 1 + (XA d

(2)

where β0 is the basal transcription rate, βA the effective transcription rate by TF regulation (A), and αC is the decay rate of C. For the mercury ion binding process, the transcription require A to form a dimer in order to be active: the reaction process are described as the following: A + A
AA, AA + I  AAI. With the assumption (2), we can deduct the effective concentration of ∗ specific TF (XA ) and the concentration of the TF (XA ): ∗ XA = KIX 2A ,

(3)

where I is the concentration of Hg2+ . Substituting the above in Eq. (2), we get the dynamics equation for the network of Fig. 2(a):
 (KIX 2A /Kd )n dXC = β0 1 − λA dt 1 + (KIX 2A /Kd )n +βA λA

(KIX 2A /Kd )n − αC XC . 1 + (KIX 2A /Kd )n

(4)

The first component in the right side of the equation is the basal expression level of the circuit in which 1 − λP is the possibility that specific TF is off the DNA operator; the second component is the effect of TF on protein expression level, in which λP is the possibility that TF is on DNA operator, the third part is the decay rate of the output node. The above translation process can be generalized to get the dynamics equation for any three-node networks. The mathematics formulas are presented in the following: P1i =

(KIX12 /Kd)n ; 1 + (KIX12 /Kd )n

Pji =

Xjn ; j = 2, 3; 1 + Xjn 3 

X1 = [A](i = 1), [B](i = 2), [C](i = 3);



3 



dXi = β0 (1 − λji Pji ) + βm 1 − (1 − λji Pji ) − αi Xi . dt j=1 j=1

(5)

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

382

For each network, we sampled 10,000 sets of network parameters using latin hypercube sampling (LHS) method (Fig. 2(b)). In order to compare r of each circuit (with randomly selected parameters) with estimated standard (r > 0.98), the derived ODE equations was numerically integrated to acquire steady-state expression level of output node C under various input concentration, and then a linear fit of input and logarithm of output was conducted. As our input range was 10−9 to 10−5 M, we selected points in identical logarithmic distance intervals within this range, and then simulated output evolution curve to acquire the steady concentration. We exploited fourth-order Runge–Kutta method to solve ODE equations, in order to reduce expected calculation time. Implicit Runge–Kutta algorithm was adopted to acquire output steady state concentration when Input reached 10−9 M and this concentration was set as initial value for Newton–Raphson method in the later calculation of different input concentrations. Besides, considering the possibility of bistablity resulted from certain network topology motif, we conducted calculation in two directions of input strength varying (from high level to low level and the reverse), respectively. In this way, we exhaustively analyzed a total of 16, 038 × 10, 000 different ODEs. The value of their character (Pearson Coefficient “r”) were thus obtained, of which all were compared with numerical standard (r > 0.98). 2.3.

Analysis and results

As shown in Fig. 3(a), for each network motif we defined a Q value as the number of functional parameter sets that the network satisfies our functional criteria. It stands for the robustness of a network topology motif for SLDRC function: the higher Q value is, the more robust the network motif is. All the network topology motifs were sorted and ranked according to their Q values (Fig. 3(b)). We found out that most network topology motifs have 0 or low Q values while there is only a small fraction of the network motifs with high Q values. Listed in Fig. 3(c) are all of the simplest topology motifs with four or less direct links between the three nodes, whose Q values are all above 100. Their ranks were also indicated in Fig. 3(c). Notably, there is only one 3-link topology motif out of the entire seven simplest topology motifs, and it seems to be necessary among all: two positive links from Node A to Node B and Node C, respectively, and one positive link from Node B to Node C (Fig. 3(c)). We named this network motif as transcriptional coherent feed-forward loop (TCFL).

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

383

Fig. 3. Calculation, analysis and results. (a) Schematic of calculating process. For each network motif, we sampled 10,000 sets of parameters, calculated characters of response curve under each set of parameters to assess whether it fits the standard. Q value was defined as the number of functional parameter sets for each network motif. (b) Racking of Q value of all tri-node networks. Top 74 topology motifs were extracted according to Q values. (c) The simplest network motif is a transcriptional coherent feed-forward loop (TCFL). Top seven simplest topology motifs among them were listed here. Their ranks were also indicated in brackets. All of them contain TCFL implying that it is a common feature.

In order to explore whether TCFL motif is necessary for SLDRC, we analyzed the entire top 74 SLDRC topology motifs (Q > 100). Results shown in Fig. 4(a) demonstrate that TCFL is indeed necessary for SLDRC functionality. Furthermore, to investigate what additional features contribute to SLDRC except minimal TCFL network motif, we clustered top 74 SLDRC network motifs. The results clearly indicate that apart from TCFL and self-activation at Node C, there should be no positive link anywhere else (Fig. 4(b)). The final step in the reverse engineering analysis is to conduct parameter sensitivity analysis. This process is to find sensitive parameters so that to provide theoretical guidance for gene circuit construction. Parameters that significantly contribute to SLDRC functionality are: λ13 , λ12 , λ23 , α1 , α2 , α3 , K13 , K12 , K23. The theoretical anaylsis also suggest in practical gene circuit construction, smaller α3 , larger λ12 and larger λ23 should be adopted.

November 21, 2012

384

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

Fig. 4. TCFL is necessary for SLDRC functionality. (a) Analysis of top 74 networks. We counted all of the TCFL and top 74 SLDRC motifs, and showed that all of the SLDRC network motifs contain TCFL. (b) Clustergram of top 74 SLDRC network motifs. Nine vertical rectangle bars stand for nine links in Fig. 2(a), which are A to A, A to B, A to C, B to A, B to B, B to C, C to A, C to B, C to C, respectively. Red color stands for activation, green for repression and black for no regulation. The network topology motifs on the right are corresponding minimal motifs.

3.

Biological Implementation of a SLDRC Network Motif

After identifying the networks that can perform the objective function, we biologically implemented the simplest Semi-Log Dose Response Curve (SLDRC) network motif, TCFL, into gene circuit. We chose mercury binding transcription activator, MerR, as the input receiving node.20,21 In this work, MerR gene comes from Tn21 from Shigella flexneri R100 plasmid.22,23 MerR forms dimer and tightly sequesters mercury (II) at 10−8 M concentration even in the presence of mM concentration of small molecular thiol competing ligands.16 In the absence of Mercury (II), MerR protein dimer slightly binds to its DNA operator at promoter pmerT, resulting in slight repression of the basal transcription.16,17,21 In the presence of mercury (II), tight binding of mercury (II) causes a conformational change to activate MerR dimer,

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

385

leading into tight binding of MerR dimer to pmerT promoter and RNA polymerase recruiting.16,17 As a consequence, transcription is initiated. Notably, as the binding of MerR dimer to pmerT promoter in the absence of mercury (II) (inactivated form) is weak compared with the case of mercury (II)’s presence (activated form), we reasonably neglected the competitive binding of inactivated MerR dimer in modeling. To characterize MerR-pmerT pair, a conventional bi-node bioreporter circuit only consisting of Node A (MerR) and Node C (green fluorescence protein, GFP) was constructed at first (Fig. 5(a)). Input is mercury (II), and Output is expression level (fluorescence intensity) of green fluorescence protein. In detail, pmerT promoter carrying a sticky end of EcoRI and SpeI restriction sites was cloned upstream of BBa E0840, a GFP generator from Registry of Standard Biological Parts (partsregistry.org).24,25 The expression of MerR was driven by a constitutive promoter, BBa J23103, from Registry of Standard Biological Parts as well. The pmerT-E0840 construct was then cloned into pSB3K3 backbone and BBa J23103-merR into pSB1A3. All the constructions here and after were performed via standard assembly.24,25 When induced with gradient concentrations of mercury (II), bacteria carrying two plasmids pmerT-E0840-pSB3K3 and BBa J23103-merRpSB1A3 were grown in LB broth with ampicillin and kanamycin at 37◦ C and later was reactivated by diluting the overnight culture in a ratio of 1:100 with fresh LB. When OD600 (observing density at 600 nm of wavelength) reached 0.4−0.6, the bacteria was disposed to 96-well plate wells, each owning 500 uL, and different dose of mercuric chloride solution, three duplicates for each concentration. The final concentration varied from 0 to 10−6 M. In-plate culture fluorescence and OD600 was recorded at 20 mins intervals from 0 to 275 mins. Temperature was constant at 37◦ C. The protocol was used throughout this work. As shown in Fig. 5(b), the output, GFP intensity, fit well to hill function. As discussed before, only a narrow range of hill function can be used for accurate measurement. We aslo tested the effect of mercury (II) on cell growth and found that mercury (II) concentration within 0 to 10−6 M does not exhibit significant growth inhibition (Fig. 5(d)). According to the results depicted by our in silico computation, we next implemented tri-node TCFL, the simplest robust network motif for SLDRC function, by adding Node B, which theoretically was expected to transform the dose response curve from Hill function to SLDRC. As suggested in theoretical anaylsis, smaller α3 (degradation rate of GFP), larger λ12 (transcription activation of Node B by MerR-Hg (II)) and larger λ23 (transcription activation of GFP by Node B) should be adopted. Therefore, for Node B, we select a strong transcription activator

November 21, 2012

386

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

Fig. 5. Biological implementation of bi-node bioreporter gene circuit. (a) We chose mercury binding transcription activator MerR as Node A. Input is mercury (II), since it can bind to the merR dimer and activate transcription at pmerT promoter. Output is GFP expression level. Activation of MerR dimer by mercury (II) binding will initiate transcription of GFP at pmerT promoter. (b) The dose response curve exhibits Hill function. Fluorescence increases along with incubating time.

“ogr” obtained from P2 phage.26,27 Once expressed, it can recruit RNA polymerase constitutively and activate transcription at its cognate psid promoter.27 When mercury (II) activates Node A, MerR dimer, ogr will be expressed and activate the expression of GFP at Node C together with activated Node A, MerR (Fig. 6(a)). Additionally, the Ribosome Binding

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

387

Fig. 6. Biological Implementation of tri-node TCLP with SLDRC functionality. (a) Based on TCLP, the simplest SLDRC network topology motif, a specific genetic circuit was constructed. Node A is MerR. For Node B, the gene is transcription activator “ogr” which can activate transcription at psid promoter. Node C is GFP reporter gene whose expression was driven by psid (activated by ogr activator) and PmerT (activated by MerR and mercury (II)). (b) Without further fine-tuning, the dose response curve exhibits SLDRC functionality as dark line shows, while gray line denotes the response curve of bi-node gene circuit shown in Fig. 5. The valid working range for mercury concentration detection was significantly extended from seven fold to nearly 70 fold.

November 21, 2012

388

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

Site (RBS, major determinant of translation rate) of GFP is BBa B0034, the most consensus RBS in Standard Parts Registry and GFP protein has no degradation tag to guarantee low degradation rate. Reassuringly, without laborious parameter fine-tuning in bench work, the dose response curve exhibits SLDRC functionality. The reliable working range for mercury concentration detection was significantly expanded from seven fold to nearly 70 fold (Fig. 6(b)). Following the theoretical analysis, it is possible to fine-tune the parameters to expand reliable working range. Our analysis indicates that any change of K12 and K13 (corresponding to the binding constant of MerR to Node B and Node C pmerT promoter, respectively) will attenuate SLDRC character. Therefore, fine-tuning of the binding constants of MerR to pmerT promoter could probably improve SLDRC functionality. Previous studies had suggested that mutations at semi-conserved region of MerR binding site would change the binding constant of MerR-DNA interactions.28 To construct a spectrum of such parameter, we conducted mutagenesis at semi-conserved region of MerR binding site. A saturated mutagenesis library at semi-conserved region was built, followed by measurement of 100 mutants. If this parameter spectrum was applied to fine-tune TCFL motif gene circuit, it is probable that SLDRC character will be improved.

4.

Summary and Discussion

In summary, we applied biological network reverse engineering approach to synthesize a gene circuit that enables E. coli to report the presence of mercury (II) ions in water with a SLDRC. A minimal framework of tri-node network was first considered: one node for receiving inputs, one node for transmitting output, and one regulatory node. The object functionality was quantified — the input logarithm show a linear relationship with output, in which input is mercury (II) concentration and output is the expression level of reporter gene at output-transmitting node. All three nodes were defined as transcription factors or proteincoding genes and the links between nodes as gene–gene interaction through transcription and translation process. The possibilities of positive, negative or no regulation between any two nodes were considered, resulting in a total about 16,000 different possible topology motifs. Then, for each topology motif, we randomly selected 10,000 sets of network parameters, analyzed their characters, especially Pearson Correlation Coefficient, and compared them with pre-estimated standard. Finally, Q value was defined as the number of parameter sets who enable network to exhibit SLDRC functionality.

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

389

74 robust topology motifs whose Pearson Correlation Coefficient is above 0.998 were investigated, all with Q value higher than 100. Following cluster analysis indicated that, a TCFL is necessary for SLDRC functionality and there’s no any other necessary. Then we conducted parameter sensitivity analysis to find sensitive parameters. All of these efforts provided theoretical guidance for gene circuit construction later then. A gene circuit bearing TCFL topology motif was implemented in E. coli. — According to guidance (lower degradation rate of GFP, stronger transcription activation of Node B by MerR-Hg (II) and stronger transcription activation of GFP by Node B) provided by in silco computation, regulatory Node B was added to interact with input-receiving node and output-transmitting node. This gene circuit was verified to indeed exhibit semi-log dose response curve with Pearson Correlation Coefficient around 0.9. Additionally, a spectrum of binding constant of MerR to Node B and Node C pmerT promoter was built, expected to improve SLDRC character, especially to expand reliable working range of bioreporter. With continuous endeavor in system biology research, more and more advances in our understanding of design principles of natural biological network have been achieved. The advances in systems biology form the fundamental basis for designing synthetic biological systems in synthetic biology area.7 –9 In this context, biological network reverse engineering is an very useful approach, in which design principles of how biological components interact depicted by system biology along with developed computation method is exploited to rationally design synthetic gene circuit with desired functionality. It has been recently revealed in system biology that network topology plays a critical role in robustly producing specific functionality — The functionality of a network can be determined by its network topology.12,13 We speculated that by defining network topology motifs that can achieve desired functionality, the number of possible network motifs can be systematically and rationally limited and the topology motifs with highest robustness can be defined. If the network motif with highest robustness is implemented into biological gene circuit, the desired functionality can be achieved more easily. However, there are still some problems hidden behind. The most vulnerable facette of this biological network reverse engineering approach is the limited complexity of network for exhaustive topology motif searching. The topology motifs of tri-node network can be exhausted, but those of quarto-node network cannot because of our currently limited computation ability. It is currently very difficult to exhaust Quarto-node network motifs and investigate smaller “positive” fraction out of 316 motifs. New mathematical tools are needed to push forward this line of research.

November 21, 2012

13:2

9in x 6in

390

b1449-ch19

Engineering of Chemical Complexity

H. Q. Zhang et al.

Acknowledgments This work is a part of the project for the 2010 team of Peking University in the international genetically engineered machine (iGEM) competition. The team received the First Running UP Prize in 2010. We thank Peking University for its support. This work is also partially supported by the NSF of China (11021463, 11074009, 11174012), the MOST of China (2009CB918500), and the NFFTBS of China (J0630311).

References 1. T. S. Gardner, C. R. Cantor and J. J. Collins, Nature 403, 329 (2000). 2. M. B. Elowitz and S. Leibler, Nature 403, 335 (2000). 3. T. Danino, O. Mondragon-Palomino, L. Tsimring and J. Hasty, Nature 467, 326 (2010). 4. C. Lou, X. Liu, et al. Mol. Syst. Biol. 6, 350 (2010). 5. J. J. Tabor, H. M. Salis, Z. B. Simpson, A. A. Chevalier, A. Levskaya, E. M. Marcotte, C. A. Voigt and A. D. Ellington, Cell 137, 1272 (2009). 6. R. Kwok, Nature 463, 288 (2010). 7. T. K. Lu, A. S. Khalil and J. J. Collins, Nat Biotechnol 27, 1139 (2009). 8. N. Nandagopal and M. B. Elowitz, Science 333, 1244 (2011). 9. A. Randall, P. Guye, S. Gupta, X. Duportet and R. Weiss, Methods Enzymol 497, 159 (2011). 10. F. J. Bruggeman, J. J. Hornberg, F. C. Boogerd and H. V. Westerhoff, EXS (Experientia Supplementum) 97, 1 (2007). 11. C. D. Smolke and P. A. Silver, Cell 144, 855 (2011). 12. U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman & Hall/CRC, 2007). 13. W. Ma, A. Trusina, H. El-Samad, W. A. Lim and C. Tang, Cell 138, 760 (2009). 14. A. B. Artyukhin, L. F. Wu and S. J. Altschuler, Cell 138, 619 (2009). 15. N. E. Buchler, U. Gerland and T. Hwa, Proc. Natl. Acad. Sci. 102, 9559 (2005). 16. J. D. Helmann, B. T. Ballard and C. T. Walsh, Science 247, 946 (1990). 17. D. M. Ralston and T. V. O’Halloran, Proc. Natl. Acad. Sci. 87, 3846 (1990). 18. S. Wickner, M. R. Maurizi and S. Gottesman, Science 286, 1888 (1999). 19. J. Yu, J. Xiao, X. Ren, K. Lao and X. S. Xie, Science 311, 1600 (2006). 20. J. L. Hobman, J. Wilkie and N. L. Brown, Biometals. 18, 429 (2005). 21. A. M. Nascimento and E. Chartone-Souza, Genet. Mol. Res. 2, 92 (2003). 22. C. A. Liebert, R. M. Hall and A. O. Summers, Microbiol. Mol. Biol. Rev. 63, 507 (1999). 23. R. Nakaya, A. Nakamura and Y. Murata, Biochem. Biophys. Res. Commun. 3, 654 (1960).

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

Reverse Engineering Approach in Synthetic Biology

24. 25. 26. 27. 28.

391

A. G. Bower, M. K. McClintock and S. S. Fong, Bioeng. Bugs. 1, 309 (2010). B. Canton, A. Labno and D. Endy, Nat. Biotechnol. 26, 787 (2008). G. E. Christie and R. Calendar, J. Mol. Biol. 181, 373 (1985). B. Julien and R. Calendar, J. Bacteriol. 178, 5668 (1996). S. J. Park, J. Wireman and A. O. Summers, J. Bacteriol. 174, 2160 (1992).

November 21, 2012

13:2

9in x 6in

b1449-ch19

Engineering of Chemical Complexity

This page intentionally left blank

November 21, 2012

13:3

9in x 6in

b1449-index

Engineering of Chemical Complexity

INDEX

ATP (adenosine triphosphate), 80 ATP hydrolysis cycle, 86–87, 90 autocatalytic network, 339 autocatalytic set, 349 autoinducer, 286, 287, 289, 290, 295 auto-induction, 293, 294, 295 automata, 7 autoregulation, 283, 284 azobenzene, 42

accuracy of gradient sensing, 191, 193 acetylene, 33, 35 actin filament, 80 activation energy, 59, 61 activator, 6 active colloidal particles, 120 active fluids, 18 active motion, 17 active polar gel, 204, 205 active self-assembly, 122 actomyosin, 203, 204, 205 adsorption, 52, 60, 61, 63, 66 affinity, 53, 55, 56, 57, 62, 69, 70, 71, 73 AHL, 286, 287, 289, 290, 293 alexander Oparin, 331 all-to-all coupling, 228 amino acid, 328, 331 amphiphilic molecule, 334 amplitude, 216, 219, 224, 225 Forcing amplitude, 221, 222 anomalous dispersion, 147, 150, 151, 152, 159, 164, 165 anomalous phase synchronization, 215, 230, 231 anti phase synchronization, 216, 223, 224 Arnold tongue, 221, 222 Arrhenius plot, 36, 37 artificial cells, 21, 22 asymmetric coupling, 231 asymmetric target patterns in globally coupled electrochemical systems, 249 asymmetrical conformational changes, 102

bacteria, 304, 306, 309, 312, 318 bacterial chemotaxis, 188 ballistic motion, 141 Barkley model, 155, 156, 163 Belousov–Zhabotinsky (BZ), 216, 265, 270, 276, 282, 287 Belousov–Zhabotinsky (BZ) reaction, 9, 12, 16, 18, 148–153, 155, 157, 158, 161, 162, 164, 169, 170, 179, 180, 181, 182, 184, 287, 289, 291, 294, 295, 296 bifurcation, 221 Binnig G., 28, 31 biology, 53, 67, 74 bioluminescence, 286, 287 biomarker, 328 biomodule, 330, 333, 334 bionics, 21 biphenyl, 38, 41 bistability, 52, 60, 63, 64, 66, 73 bi-stable energy, 97 bivariant synchronization index, 230, 231 black smoker, 334 Blake-Tensor, 308, 314 Brownian diffusion, 127 393

November 21, 2012

13:3

9in x 6in

b1449-index

394

Brownian motion, 18, 126 Brownian particles, 108 Brownian rotation, 126 Brownian search-and-forward catch, 87–88 brusselator, 56 bubble propulsion, 132 building blocks, 326 Burgers equation, 162 bursting oscillations, 220, 225 Caenorhabditis elegans (C. elegans), 202, 203, 206, 209, 211 calmodulin, 81 cAMP, 188, 189, 190, 191, 193, 194, 195, 199, 282, 291, 292, 293, 296, 297 catalysis, 334 catalyst, 12 catalytic cycle, 339 catalytic particles, 263, 265, 266, 267, 268, 271, 272, 275, 276, 277 cell, 202, 203, 204, 207, 209 cell density, 287, 289, 290, 291, 292, 293, 294, 295, 296, 297 chain configuration, 229 chaotic current oscillations, 216, 224, 225, 227 chaotic dynamical system, 54, 57 charge transfer, 216, 232 Charles Darwin, 327 chemical contrast, 27, 30, 32 chemical energy, 125 chemical oscillator, 263, 265, 268, 272, 276, 277 chemical patterns, 117, 122 chemical potential, 53, 55, 63 chemical turbulence, 14 chemical wave, 172, 175, 176, 177, 183 chemical, 131, 135, 136, 138 chemically active media, 114 chemically-powered nanomotors, 101, 106 chemically-powered self-propulsion, 105, 109

Engineering of Chemical Complexity

Index

chemical-net glass, 349 350 359 chemical-resistive device, 229 chemoattractant, 188, 189, 190, 191, 192, 193, 194, 199, 291 chemomechanical coupling, 85 chemophoretic, 130, 131 chemotaxis, 133, 134, 142, 187, 188, 189, 192, 193, 194 chirality, 332 cilia, 302, 303, 305, 306, 318 cluster patterns in globally coupled electrochemical systems, 249–258 labyrinthine, 251–252 subharmonic, 250–252, 254–258 type I, 250, 254, 255 type II, 250, 254, 256–258 cluster, 215, 216, 221, 226–229, 232 CO molecule, 32 CO oxidation on platinum, 12, 17 CO oxidation reaction, 13, 14 coarse-grain mesoscopic methods, 106 coarse-grained model (of protein), 92 cohesion, 16 collective, 125, 127, 134, 135, 136, 138, 140, 142, 281, 282, 283, 289, 290, 291, 292, 293, 295, 297 collective behavior, 94, 125, 127, 134 collective dynamics, 119, 120, 121, 122 collective resistance, 223 community effect, 293 compartment, 326, 334 compartmentalization, 356 358 complex Ginzburg–Landau equation, 12, 14, 239 modified with nonlinear global coupling, 254–258 nonlocal, 242-247 nonlocal (with linear global coupling), 252–254 complexity, 52 compositional ensemble, 340 concentration patterns, 16

November 21, 2012

13:3

9in x 6in

b1449-index

Index

conductance, 27, 31, 46 connection topology, 230 constitutive promoter, 387 copolymerization, 51, 53, 67, 69, 71, 72, 74 copper surface, 31, 32, 33, 34, 35, 38, 45 corrosion, 19 cortex, 203, 204, 205, 206, 209, 211 counter electrode, 217, 218, 224 coupled electrochemical oscillators, 215, 216, 222, 230 coupling strength, 222–226, 229–231 critical coupling strength, 225, 230, 231 critical forcing amplitude, 221 cross bridge cycle, 97 cross resistor, 229 cubic autocatalytic reaction, 115, 117 current oscillations, 216, 223–225, 233 curvature flow, 160, 163 Cy3-ATP, 82, 84 cybernetics, 6 dark-field imaging, 87 Debey length, 130 decision Making, 293, 294, 297 delayed phase synchronization, 231, 232 Derjaguin length, 104, 105 desorption, 42, 52, 60, 61, 63, 66 detailed balance, 116, 117 detailed balancing, 53, 54, 56, 57, 68, 73 deuterium, 33, 35 dictyostelium discoidium, 188 dictyostelium, 281, 282, 290, 291, 292, 293, 294, 295, 296, 297 diffusion, 27, 29, 31, 35, 36, 37, 38, 54, 59, 60, 62, 73, 105, 106, 109, 111, 112, 115, 117, 118, 119 diffusion coefficient, 126 diffusion equation, 105, 106, 109, 111, 112, 115, 117 diffusion sensing, 270

Engineering of Chemical Complexity

395

diffusional model, 97 diffusiophoresis, 103, 128, 129, 130, 131, 136, 138 diffusiophoretic, 130, 136, 138 diffusiophoretic interactions, 136, 137 Dimer velocity, 113, 121 direct connection, 230 directionality, 51, 53, 54, 56, 73 discreteness-induced transition, 349 disorder, 57, 68, 69, 70, 71 dissipative structure, 52 DNA, 332, 367, 370, 371, 372 DNA replication, 53, 71, 72 dose response, 377, 379, 386, 387, 388, 389, 390, 391 DQS, 289, 290, 291, 292, 293, 294, 295, 296, 297 dual-electrode flow cell, 224 dynamical differentiation, 226, 227 dynamical order, 51, 52, 53, 56, 58, 67, 73, 74 dynamical quorum sensing, 289 dynein, 80 dynein motors, 303 early atmosphere, 327, 331 early Earth, 326 E. coli, 283, 287, 290 Eigen’s model, 338 elastic network model, 92 electric field, 58, 59, 60, 61, 63, 73 electric field-induced manipulation, 42, 43 electrical coupling, 222–225, 227 electrochemical networks, 217 electrochemical oscillator, 15, 18, 215, 228 electrode array, 215–218, 221, 225–228 electron microscope, 28, 31 electron-induced manipulation, 40, 41 electro-osmosis, 136, 137, 138 electrophoresis, 130, 131, 136, 137 electrophoretic, 130, 138 emergent systems, 138

November 21, 2012

13:3

9in x 6in

b1449-index

396

energetic balance, 342 entropy, 5 entropy production, 53, 54, 56, 57, 69, 70, 71, 73 enzyme, 18, 20, 53, 54 error catastrophe, 339, 349 355 error rate, 339 eukaryotic chemotaxis, 187, 188, 189 evolovability, 357 excitability, 285, 286, 289, 291 excitable, 281, 283, 285, 286, 287, 290, 293, 294, 296, 297 excitable medium, 7 excitation waves, 7 export of entropy, 5 extent of identical synchronization, 229 extrinsic noise, 193 fast scanning tunneling microscopy (STM), 35 fatty acid, 334 Faxen-Theorem, 308 feedback, 12, 13, 215–217, 227, 228, 232 feedback loop, 283, 284, 285, 286, 289, 293, 295, 297 fidelity, 72, 74 field emission microscopy, 51, 58, 60, 65, 67, 73 field ion microscope (FIM), 28 filament tension, 160, 165 fine-tuning, 379, 389, 390 FIONA (Fluorescence Imaging with One Nanometer Accuracy), 84, 86–87 FitzHugh-Nagumo, 151, 152, 155, 157 flagella, 302, 303, 304, 306, 309, 312, 318 flashing ratchet, 97 flow cytometry (FCM), 365, 368 flow field, 302, 304, 306, 307, 308, 309 fluctuations, 54, 56, 57, 73, 74 fluctuation theorem, 54, 55, 56, 73 force generating states in muscle, 96

Engineering of Chemical Complexity

Index

force velocity curve, 95 force-field, 91 forcing frequency, 221, 222 formic acid electrooxidation, 223, 224 fossil, 327 free energy, 52, 53, 54, 55, 68, 69, 71, 72, 73, 74 frequency encoding, 295 frequency synchronization, 272 frequency difference, 230–232 difference enhancement, 231, 232 distribution, 226 FRET, 292 galvanostatic control, 248 Ganti’s chemoton, 341 gel, 17 gel conveyer, 174, 176 generalized synchronization, 220, 230 generic adaptation, 349 353 354 357 genetic circuit, 378, 389 genetic molecules, 326 genetic take-over, 350 geometric switch, 305, 313, 314 giant vesicle (GV), 363, 367, 368, 369, 370, 371 Gillespie’s algorithm, 70, 72 global coupling, 215, 225, 228 in electrochemical systems, 248–258 linear, 252–254 nonlinear, 254–258 global phase synchronization, 221 globally coupled, 265, 268, 269, 272, 276, 277, 278 glycolytic oscillation, 282, 290, 291, 295, 296 gold surface, 31, 42, 46 G¯ o-model, 92 green fluorescence protein, 379, 382, 387 Green-Kubo formula, 56, 73 growth rate, 338

November 21, 2012

13:3

9in x 6in

b1449-index

Engineering of Chemical Complexity

397

Index

Haber-Luggin capillary, 239, 248 hairpin structure, 337 hand-over-hand mechanism, 86–88 harmonic entrainment, 221 forcing, 221 Harold Urey, 328 head, 81 heart, 7 heterogeneous catalysis, 12, 18, 51, 52, 53, 58, 73 hexagon configuration, 229 Hilbert transform, 219, 245 Hill function, 379, 387, 388 homochirality, 332 homoclinic Bifurcation, 220, 224 Hopf Bifurcation, 220, 221 Hopf bifurcation in N-NDR oscillator, 247 Huxley and Simmons’ model, 95 Huxley’s model, 94 hybrid chemical-resistive device, 229 hydrodynamic interactions, 303, 304, 306, 307, 308, 309, 314, 315, 318 hydrolysis, 54 hydrothermal vent, 334 hypercycle, 339, 349 355 hypercycle theory, 74 hysteresis, 63, 64, 147, 151, 152, 155, 156, 157, 165

irreversible, 56, 62 itinerant cluster dynamics, 226, 228

identical synchronization, 215, 219, 220, 223–230 individual Electrode, 217, 223, 227 Resistances, 229 information, 53, 58, 67, 70, 71, 72, 74 inhibitor, 6 in-phase synchronization, 216, 223, 224, 229 intelligent systems, 142 intramolecular strain, 86–87 intrinsic noise, 193 iron dissolution, 220, 221, 224–226, 229, 232

Manfred Eigen, 338 manipulation of single molecules, 27, 28, 29, 31, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47 mass action law, 54 master equation, 67, 73 mean square displacement (MSD), 118, 119, 127, 141 mechanochemical coupling, 89 mechanochemical, 202, 209, 211 MerR, 386, 387, 388, 389, 390, 391 metabolism, 335 metachronal waves, 303, 304, 305, 306, 313, 314, 315, 316, 317, 318

Janus particle, 102, 105, 120 John B. S. Haldane, 331 Karhunen-Lo`eve decomposition, 250, 256 Kinesin, 80 kinetics, 53, 54 kubic harmonics, 59, 65 Kuramoto, 289, 290, 291, 295, 296 Kuramoto model, 268 Kuramoto order, 225, 226, 228 Kuramoto transition, 225, 226 lag synchronization, 220, 225 Lander molecules, 39, 40 Langevin equations, 97, 108 large deviation, 54, 56 last universal common ancestor (LUCA), 328, 336 late heavy bombardment, 327 Latin hypercube sampling, 384 LCST, 170, 178, 183, 184 lever arm domain, 80–81 Levy-walk, 127 lipid, 334 lipid world, 340, 362, 363, 374 live-cell imaging, 291, 297

November 21, 2012

13:3

9in x 6in

b1449-index

398

metallic nanorods, 102 microbial, 283 microbots, 125, 135, 142 microfluidic dual electrode, 218 microfluidic flow cell, 218 microgels, 174, 178, 179, 180 micromotor, 135 microreversibility, 55, 56 micro-swimmers, 17 Miller and Urey’s experiment, 328, 331 Miller index, 59 mineral surface, 334 minimal cell, 329 minority control (minority controlled state, MCS), 349 356 357 358 mobilities, 307, 308, 309 molecular dynamics, 91 molecular dynamics simulations, 106 molecular genotype, 337 molecular machine, 20, 53, 101, 102, 189, 190 molecular motor, 53, 54, 55 molecular phenotype, 337 molecular quasispecies, 338 monomer, 67, 68, 69, 70, 71, 72, 74 montmorillonite clay, 333 morphogenesis, 6, 202, 211 motor, 126, 127, 128, 129, 131, 132, 134 motor domain, 80–81, See also Head motor efficiency, 113 motor protein, 80 multicellular, 281, 289, 297 multi-electrode array, 215–218, 232 multiparticle collision dynamics (MPC), 108, 109, 112, 114 multi-stability, 276, 277 muscle, 80–81, 88, 90, See also Myosin-II mutation, 72, 74 myosin, 80 myosin filament, 90 myosin-II, 81, 85

Engineering of Chemical Complexity

Index

myosin-V, 80, 81, 84–86 myosin-VI, 87–88 NaCl film, 33, 34 NADH, 290 nanobots, 125, 127, 134, 135, 142 nano-car, 44 nanoclock, 67, 73 nanoelectrode, 53 nanomotor, 135 nanoparticles, 128 nanopattern, 52, 59, 60, 63, 66, 73 nanoreactor, 58 nanoscale, 51, 52, 53, 58, 65, 67, 73 nanosystem, 51, 53, 54, 55 nanotechnology, 21, 31, 37 natural frequency, 221, 223, 226, 230 network motif, 379, 380, 381, 384, 385, 386, 387, 391 networks, 18 network topology, 378, 379, 380, 384, 386, 389, 391 neutral network, 337, 339 NF-κB, 286 nickel electrodissolution, 220, 223–231 N-isopropylacrylamide, 169, 170 nonequilibrium, 51, 52, 53, 54, 55, 57, 58, 63, 67, 73, 128 non-identical chaotic oscillators, 225 non-isochronous, 231 nonlinear dynamics, 63, 65 nonlocal coupling in electrochemical systems, 239–242 non-phase coherent chaos, 220, 225, 230 non-reciprocal swimmers, 132 normalized coupling strength, 231 nucleic acid, 328 nucleotide, 328, 332 OD600, 387 Onsager reciprocity relation, 56, 73 open systems, 5, 16 optical tweezers, 82–83

November 21, 2012

13:3

9in x 6in

b1449-index

Index

optimal waveform, 221, 222 orbit flip, 154 order parameter, 229, 230, 267, 269, 270 oregonator, 149, 151, 155, 157, 158 oscillating system, 138 oscillation, 52, 58, 60, 63, 64, 66, 67, 73, 281, 282, 283, 284, 286, 287, 289, 290, 291, 292, 293, 294, 295, 296, 297 oscillator N-NDR, 242, 243, 247 oscillator death, 276 oscillatory behaviour of the actomyosin complex, 97 oscillatory, 281, 282, 283, 284, 285, 286, 287, 290, 291, 293, 294, 296 Oseen-Tensor, 307, 314 osmophoretic, 130 osmotic, 128, 129, 136 out-of-phase synchronization, 229 oxygen, 36, 37 P2 phage, 388 PAR, 202, 203, 204, 207, 208, 209, 211 parasite, 340, 349 partial phase synchronization index, 231 Pauli repulsion, 38, 43 PCR, 370, 371, 372 Pearson Correlation Coefficient, 379, 380, 390, 391 Peclet number, 105, 112 periodic forcing, 221, 222 periodic oscillations, 216, 220, 224, peristaltic motion, 172, 174, 175, 176 phase coherent chaos, 220, 225 definition, 219 difference, 219, 225, 228 transition, 225–226 phase clusters, 272 phase oscillators, 15 phase repulsion, 272, 273, 277

Engineering of Chemical Complexity

399

phase response curve, 221, 266, 267, 277 phase separation, 17 phase synchronization, 215, 218, 221, 224–227, 230, 231 phase synchronized chaos, 227 phase transition, 12, 16, 17 phenotype, 339 phoretic mechanisms, 101, 103 phosphatidylinositol (PtdIns) lipid, 188, 195, 196, 198 photo-emission electron microscopy (PEEM) 12 phthalocyanine, 31, 32 Pi (Inorganic phosphate), 87 PI3K, 188, 189, 197, 198 pinning (of scroll waves), 147, 161, 162, 163, 164, 165 pinning (of spiral waves), 147, 155, 156, 157, 158, 164, 165 PmerT promoter, 386, 387, 388, 390, 391 polarization, 202, 203, 208, 209, 211 polymer, 333, 337 polymer gels, 169, 170, 172, 177, 178, 184 polymerase, 53, 72 polymerization, 129, 134, 333 polymerization motor, 129 population, 263, 264, 265, 266, 267, 268, 269, 270, 272, 274, 276, 277, 278 potentiostatic control, 248 power law, 353 power transduction, 113 prebiotic chemistry, 74, 326 predator-prey, 138, 139 protein, 53, 67, 328 protein machines, 17, 20 protein world, 362 protocell, 326, 341, 347, 348, 351, 353, 358, 362, 363, 369, 374 protometabolic cycle, 326 PtdIns(3,4,5)P3, 188, 189, 190, 195, 197, 198, 199

November 21, 2012

13:3

9in x 6in

b1449-index

400

PtdIns(4,5)P2, 188, 189, 197, 198 PTEN, 188, 189, 190, 195, 196, 197, 198 pump-probe, 35 QM/MM, 93 quasiperiodic oscillator, 216, 221 quasispecies, 74, 338 quorum sensing, 263, 264, 269, 270, 272, 278, 287, 289, 293 rational design, 377, 378 Rayleigh-Benard convection, 245 reaction, 52, 54, 55, 56, 58, 59, 60, 63, 67, 73 reaction-diffusion system, 147, 148, 149, 162 reaction-diffusion turbulence, 12 reference electrode, 217, 218, 223, 224 reflexive autocatalytic and F-generated (RAF) set, 340 refractory period, 285, 286 Registry of Standard Biological Parts, 377 relaxation oscillations, 220, 221, 223–225 replication error rate, 339 repressilator, 283, 285, 287 resonant tunneling, 38, 41 response property, 56, 73 reverse engineering, 377, 378, 379, 380, 385, 390, 391 Reynold’s number (Re), 126, 127 Reynolds-number, 303, 306, 307, 311 rhodium, 52, 59, 60, 61, 62, 64, 66, 67, 73 ribonucleotide, 332 ribose, 331, 332 ribosome, 53 ribosome binding site (RBS), 389 ribozyme, 328, 335, 339 RNA, 332, 336 RNA virus, 339 RNA world, 335, 336, 362, 374 robustness, 379, 384, 391

Engineering of Chemical Complexity

Index

Rohrer H., 28, 31 rolling, 43, 44 saccharomyces, 290 Scallop theorem, 132, 133 scanning tunneling microscopy (STM), 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46 scanning tunneling spectroscopy (STS), 27, 31, 32, 33, 34, 37, 46 schooling, 138 scroll wave, 10, 11, 147, 159, 160, 161, 162, 163, 164, 165 second law of thermodynamics, 56, 57, 58, 67, 69, 73, 74 self-assembly, 120, 122 self-diffusiophoresis, 136 self-organization, 51, 65, 73, 74 self-oscillating gel, 172, 174, 175, 176, 177, 178, 181, 184 self-propelled motors, 102, 118 self-propulsion, 17, 18, 133 self-replication, 367, 370, 372 self-reproduction, 363, 370, 372 semi-log, 377, 379, 386, 391 sequence, 67, 68, 69, 74 sequencing, 72 signal to noise ratio (SNR), 192, 193, 194 Silica-Pt motors, 102 silicon surface, 41, 44 single molecule detection (SMD), 81 single molecule imaging, 81–82, See also TIRFM single-molecule chemistry, 31 slip velocity, 104, 105, 110 small networks, 229, 230 smooth oscillators, 220, 223 soft matter, 16, 17 spatial systems, 271 spatio-temporal dynamics, 53, 54 spatiotemporal pattern, 229

November 21, 2012

13:3

9in x 6in

b1449-index

Engineering of Chemical Complexity

401

Index

sphere dimer, 102, 109, 110, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122 sphere dimer motor, 102, 109, 114, 115, 119, 121 spinodal decomposition, 17 spiral, 13 spiral wave, 8, 9, 10, 12 splitted photosensor, 84 stainless steel, 19 standing waves in globally coupled electrochemical systems, 249 Stanley Miller, 328 star configuration, 229, 230 state transition dynamics, 97 stochastic fluctuations, 187, 190, 191, 199 stochastic process, 53, 56, 57, 73 Stokes efficiency, 114 Stokes equation, 104, 307, 308 Stokeslet, 307, 308, 309 strong binding, 88–89 subdiffusive, 127, 141 sublimation, 37 sugar, 332 sulfuric acid, 222, 223, 226–228 super harmonic entrainment, 221 superdiffusive, 127, 141 supernormal excitability, 154 surface chemical reaction, 12 surface oxide, 60, 61, 64, 66 surfactant, 17 swarming, 120, 122 swarms, 18 switchers, 274 synchronization, 13, 15, 18, 267, 272 synchronization matrix, 230 synchronized, 287, 289, 290, 292, 294, 295, 296 synchrony, 292, 294, 295, 296 synthesizing life, 362 synthetic, 281, 282, 283, 284, 285, 286, 287, 289, 293 synthetic biology, 21, 377, 378, 391

synthetic motors, 102, 114, 120, 122 system biology, 378, 391 system’s chemistry, 331 target pattern, 9 template, 69, 70, 72, 74 template replication, 335, 337 temporal ordering, 58, 67 tension transients, 97 terpyridine, 181 the origin of Life, 362, 374 thermal fluctuation, 94 thermal gradient, 334 thermal noises, 189 thermodynamics, 51, 53, 67, 69, 71, 72, 73, 74 thermophoresis, 103 thin film, 204 time asymmetry, 57, 73 time series, 216, 218, 220, 228 TIRFM (Total Internal Reflection Fluorescence microscopy) 81–82 transcription, 53, 74 transcriptional coherent feed-forward loop (TCFL), 384, 385, 386, 387, 390, 391 translation, 53, 74 transport, 52, 54, 59, 60 traveling waves and nonlocality, 244 in globally coupled electrochemical systems, 249 tunneling current, 28, 29, 30, 33, 35, 43 turbulence, 12, 13 electrochemical, 244–247, 254 Turing instability, 6, 18 Turing patterns, 6 Ullmann reaction, 38, 39 uncoupled frequency, 230 unidirectional coupling, 221 Van der Waals, 38, 43 vesicle, 334, 340

November 21, 2012

13:3

9in x 6in

b1449-index

402

Engineering of Chemical Complexity

Index

vibrations, 33, 35, 38, 40 vibrio fischeri, 286 viscosity, 307 viscosity oscillation, 179, 180, 181, 182

weak binding, 86, 88–89 working electrode, 217, 223, 224, 229

water formation, 52, 62, 63, 64, 67, 73 wave instability, 6

zeta potential, 130

yeast, 264, 265, 269, 278, 281, 282, 290, 291, 295, 296