Principles of Brownian and Molecular Motors (Springer Series in Biophysics, 21) 3030649563, 9783030649562

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Springer Series in Biophysics 21

José Antonio Fornés

Principles of Brownian and Molecular Motors

Springer Series in Biophysics Volume 21

Series Editor Boris Martinac, Molecular Cardiology & Biophysics, Victor Chang Cardiac Research Institute, Darlinghurst, NSW, Australia

The “Springer Series in Biophysics” spans all areas of modern biophysics, such as molecular, membrane, cellular or single molecule biophysics. More than that, it is one of the few series of its kind to present biophysical research material from a biological perspective. All postgraduates, researchers and scientists working in biophysical research will benefit from the comprehensive and timely volumes of this well-structured series.

More information about this series at http://www.springer.com/series/835

José Antonio Fornés

Principles of Brownian and Molecular Motors

José Antonio Fornés Federal University of Goiás Goiânia, Goiás, Brazil

ISSN 0932-2353 ISSN 1868-2561 (electronic) Springer Series in Biophysics ISBN 978-3-030-64956-2 ISBN 978-3-030-64957-9 (eBook) https://doi.org/10.1007/978-3-030-64957-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Nélida, who was always a source of love and inspiration. In memory “Science is an evolution of ideas and approximations.” José A. Fornés, 1998.

Preface

In the last decade with the development of nanotechnology and consequently the emerging of nanomedicine, with the discovery of biological machines, it became necessary the formation of researchers dedicated to this area, able to apply the principles that rule the movement of these nanomachines. It is precisely the subject of this book. This book presents my work on this subject while I was a professor at the Institute of Physics at Goiás University. I am grateful to my many friends and colleagues. I would like to especially acknowledge the help I received from Amando S. Ito and Joaquim Procopio, who influenced very much my scientific career. I am grateful to José Nicodemos T. Rabelo (disciple, at Moscow State University, of the great Russian physicist ˘ who provided the precursor to the Quantum Lanvevin Equation) V .B.Magalinski i, for critical discussion on the subject, and I acknowledge the help I received from Salviano de Araújo Leão, who was always ready to help me with computational software. Also, I would like to express my appreciation to Gonzalo Cordova, my Springer Nature editor in the Netherlands, whose perfect orientation and cordial treatment made this a book reality. It has been a considerable pleasure to work with him. Finally, I want also to express my recognition to Ms Ramabrabha Selvaraj, Project Manager and her team, in Publishing Spi Global for Springer Nature in India, for such a careful and perfect job in the production of the book. Goiânia, Goiás, Brazil September 2020

José Antonio Fornés

vii

Contents

1

Brownian Ratchets and Molecular Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Force-Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Maximum Driving Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Stall Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Smoluchowski-Feynman’s Ratchet as a Heat Engine. . . . . . . . . . . . . . . . . 1.2.1 Parrondo Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Ratchet Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Ratchet Coherency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 First Passage Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Power Stroke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 4 5 5 8 8 10 12

2

The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Discretization of the Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Forward Time Central Space (FTCS) Method . . . . . . . . . . . . . . . 2.3.2 Crank-Nicholson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Program 2.1, F-P Equation, Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 18 18 19 20 21 23

3

Biased Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Parametrization of the Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Building the Fokker-Plank’s Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Periodic Potential Slightly Tilted . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dichotomous Markov Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Generating Dichotomous Markov Noise . . . . . . . . . . . . . . . . . . . . 3.5 Fluctuating Potential, or “Flashing” Ratchet . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Fluctuating Force, or “Rocking” Ratchet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 27 29 30 33 35 36

ix

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Contents

3.7

Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Program 3.1, Euler Equation, Matlab Code . . . . . . . . . . . . . . . . . . 3.7.2 Program 3.2, F-P Equation, Matlab Code . . . . . . . . . . . . . . . . . . . . 3.7.3 Program 3.3, Dichotomous Noise, Matlab Code . . . . . . . . . . . . . 3.7.4 Program 3.4, Flashing Ratchet, Matlab Code . . . . . . . . . . . . . . . . 3.7.5 Program 3.5, Rocking Ratchet, Matlab Code . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 42 42 44 44 46 47

4

The Smoluchowski Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Absolute Reaction Rate Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Mechanochemical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Numerical Computation of Mechanochemical Coupling . . . . 4.4 Program 4.1, Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 51 53 56 58 60 63

5

Rotation of a Dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Langevin Equation for the Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Dipole in a Ratchet Electrical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Program 5.1, Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 69 77 79

6

Ratchet Dimer Brownian Motor with Hydrodynamic Interactions . . . . 81 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.4 Brownian Dynamics with Hydrodynamic Interactions . . . . . . . . . . . . . . . 87 6.4.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.5 Program 6.1, Fortran Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7

Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Fluctuations of the Proton-Electromotive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Parameter Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Calculation of Buffer Equivalent Electrical Capacitance . . . . . . . . . . . . . 7.6 Calculation of IMM Electrical Resistance Rm . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Relaxation Times of the Electrical and Buffer Reservoirs. . . . . . . . . . . . 7.8 Program 7.1, Fluctuations of the PMF, Matlab Code . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 113 114 115 115 116 120 121

Contents

8

Quantum Ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Quantum Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Correlation Quantum Function . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Quantum Overdamped Langevin Equation with Colored Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 The Quantum Underdamped Langevin Equation . . . . . . . . . . . . 8.1.4 The Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Program 8.1a, Moderate Damping, Matlab Code . . . . . . . . . . . . 8.2.2 Program 8.1b, Complete Langevin Equation, Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Transition Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Probability Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.4 Poisson’s Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.5 Detailed Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

123 123 127 129 130 133 140 140 144 148 149 149 149 149 149 150 151

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 B.1 Information Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 C.1 Endoreversible Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 First Passage Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.1 Properties of First Passage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.2 Application to Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 161 162 163

Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 Stochastic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.1 White Noise and Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.2 Spectral Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.3 Properties of Wiener’s Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.4 Stochastic Process Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.5 SDE with Aditive Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.6 SDE with Multiplicative Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1.7 Itô’s and Stratonovich’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 165 165 166 167 167 168 169 174

xii

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Appendix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Stochastic Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1.1 Sekimoto View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1.2 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1.3 Stochastic Energetics: Useful Relations . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 177 178 178 179

Appendix G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.1 Solution of Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Damped Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2.1 Clasification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 182 183 183

Appendix H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 H.1 Electrical and Mechanical Systems Analogies . . . . . . . . . . . . . . . . . . . . . . . . 185 Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 I.1 The Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Appendix J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 J.1 Integral Algorithm for Colored Noise Simulation . . . . . . . . . . . . . . . . . . . . 191 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Chapter 1

Brownian Ratchets and Molecular Motors

Molecular motors are biological molecular machines that are the essential agents of movement in living organisms. In general terms, a motor is a device that consumes energy in one form and converts it into motion or mechanical work; for example, many protein-based molecular motors harness the chemical free energy released by the hydrolysis of ATP in order to perform mechanical work. One important difference between molecular motors and macroscopic motors is that molecular motors operate in the thermal bath, an environment in which the fluctuations due to thermal noise are significant. Wikipedia [1], their action are often described in terms of Brownian Ratchet (BR) and Power Stroke (PS), in other words Brownian ratchet and Power Stroke are models of molecular motors [2]. For an excellent synthesis on molecular motor see [3].

1.1 The Force-Generation 1.1.1 Maximum Driving Force The hydrolysis of one ATP molecule releases free energy G0 of about 0.50×10−19 J [corresponding to 7.3 Kcal/M or 12kB T at typical in vitro temperatures, T [4]]. If all of this free energy could be converted into mechanical energy and move the motor protein through a distance x = d, the step size, the force exerted would be fmax = G0 /d

(1.1)

Considering that one molecule of ATP is sufficient to translocate the motor protein by one step [6], this expression clearly represents the maximal driving force that can be exerted. For a kinesin moving on a microtubule with d = 8.2 nm [5–7] it yields

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_1

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1 Brownian Ratchets and Molecular Motors

Fig. 1.1 Kinesin mediated vesicle movement along microtubule

fmax = 6.2pN . If f is the driving force actually realized, the efficiency of a motor protein may sensibly be defined by  = f/fmax Typical motor speeds v ∼ 10−2 − 1 μm/s [8]. If γ = 6π ηR = 6π 10−3 Kg.m−1 s−1 μm then the Einstein force or viscous drag FE =

kB T v = γ v ∼ 10−2 pN D

(1.2)

Viscous drag is negligible [10] (Fig. 1.1). An excellent paper on the characteristics of Brownian motors was given by Linke et al. [9]. A typical pattern for the stepping of a single kinesin molecule along a microtubule is seen at Fig. 1.2.

1.1.2 Stall Force Motor action is defined as work W being performed against a conservative load force F and the time average of the particle velocity v continue being positive in the interval [Fstall , 0] see Fig. 1.3, the motor does work at the rate P = dW/dt = F v per particle. The stall force is the (negative) force at which the motor has zero velocity on average, see e.g. Fig. 6.4 on Chap. 6.

1.1 The Force-Generation

3

Fig. 1.2 Operation of the force clamp. (a), Sample record from the force clamp, showing kinesindriven bead movement and corresponding optical trap displacement (2mM ATP). Discrete steps of 8 nm are readily apparent. Inset, schematic representation of the motility assay used, showing the experimental geometry (not to scale). The separation between bead and trap was nominally xed at x =175 nm. Bead position was sampled at 20 kHz, filtered with a 12-ms boxcar window for the feedback on the trap deflection, and saved unfiltered at 2.0 kHz. (b), The measured beadtrap separation, x, for the record in (a). (c), Histogram of the displacements in (b), converted to force by multiplying by the trap stiffness (0.037 pN nm−1 ). Solid red line is gaussian fit to these data, yielding a load of 6.5 s´ 0.1 pN (mean s´ s.d:). (Reprinted figure with permission from [10]. Copyright by Springer Nature)

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1 Brownian Ratchets and Molecular Motors

1.2 Smoluchowski-Feynman’s Ratchet as a Heat Engine Feynman famous lectures, [11, 13] include an imaginary microscopic ratchet device to illustrate the second law of thermodynamics. The basic idea belongs to Smoluchowski who discussed it during a conference talk in Munster in 1912 (published as proceedings article in Ref. [12]). As seen in Fig. 1.3, it consists of a ratchet, a pawl and a spring, vanes, two thermal baths at temperatures T1 > T2 , an axle and wheel, and a load. The ratchet is free to rotate in one direction, but rotation in the opposite direction is prevented by the pawl. The system is assumed small so that molecules of the gas at temperature T1 that collide with the vanes produce large fluctuations in the rotation of the axle. Fluctuations are rectified by the pawl. The net effect is a continuous rotation of the axle that can be used to produce work by, for example, lifting a weight against gravity. If Lθ is the torque or the potential energy the weight gains when the ratchet performs a clockwise jump. Then  + Lθ is the energy needed for such a jump, so the rate of clockwise jump is proportional to exp-(Lθ + )/kB T1 (Arrhenius factor). For a counterclockwise jump the energy required is , so the corresponding rate is exp-()/kB T2 , Feynman assumes that this energy is taken from the ratchet bath, There is a weight L0 for which both rate are equal: T1 L0 θ +  =  T2

(1.3)

Fig. 1.3 Smoluchowski and Feynman’s ratchet and pawl system. (Figure from [13] under Licence: Creative Commons Attribution 3.0)

1.3 Ratchet Efficiency

5

If L is chosen to be smaller but close to L0 , then the wheel will move forward very slowly, lifting the weight. Let us calculate the efficiency of the engine. If the ratchet performs N + clockwise jumps and N − counterclockwise jumps, the total work done is (N + − N − )Lθ and the amount of heat taken from bath 1 is (N + − N − )(Lθ + ). Therefore, the efficiency is η=

Lθ Lθ + 

(1.4)

and, in the limit L → L0 (or zero power), the efficiency converges to that of a Carnot cycle, (1.3): η → ηC = 1 −

T2 . T1

(1.5)

1.2.1 Parrondo Criticism Parrondo in his paper together with Español, [14] showed that Feynman’s analysis contains some misguided aspects: Since the engine is simultaneously in contact with reservoirs at different temperatures, it can never work in a reversible way. As a consequence, the engine can never achieve the efficiency of a Carnot cycle, not even in the limit of zero power (infinitely slow motion), in contradiction with the conclusion reached in the Lectures. In spite of this criticism the Feynman analysis, besides its pedagogical interest, has been the inspiration of a currently very active research field on transport induced by Brownian motion in asymmetric potentials [15, 16].

1.3 Ratchet Efficiency The existence of reversible ratchets is of extreme importance for designing Brownian motors with high efficiency. We saw in the previous section that Feynman calculated under very simple assumptions the efficiency of a ratchet, finding that it is equal to Carnot efficiency in the quasistatic limit, and Parrondo revealed the inconsistency of this arguments by proving the intrinsic irreversibility of the system under consideration. Most of the ratchets proposed in the literature are also intrinsically irreversible, [17] and their efficiency turns out to be very low, whereas reversible ratchets posses a comparatively high efficiency. Parrondo et al. [18] calculated analytically and numerically the efficiency of different types of Brownian motors. They found that motors based on flashing ratchets present a low efficiency and an unavoidable entropy production. On the other hand, a certain class

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1 Brownian Ratchets and Molecular Motors

of motors based on adiabatically changing potentials, named reversible ratchets, exhibit a higher efficiency and the entropy production can be arbitrarily reduced. Efficiency is defined as η=

W Ein

(1.6)

where W is the output work and Ein is the input energy. For flashing ratchet the input energy is given by 

Ton

Ein = 0

 dV (x(t)) dx(t) dx

(1.7)

where Ton is the time during which the potential is on and the average is over many ratchet cycles. Another definition of η was given by [19] and [20], namely η=

F v E˙ in

(1.8)

where the numerator is the power delivered by the motor against an external load force, F . Derényi, Bier and Astumian proposed the generalized efficiency [21]. For molecular motors, the task is not only to translocate the motor a distance L, but also to do this during a given time τ, i.e. with a given average velocity  L τ v = L/τ . Since the dissipation via friction 0 γ xdx ˙ = γ 0 x˙ 2 dt is minimal when the motor is moving uniformly (x(t) ˙ = v), the power output, i.e. the minimum necessary power to maintain a motion with an average velocity v against an opposing external force Fext is min Pout ≡ Pin = Fext v + γ v

ηgen =

Pout Pin

(1.9) (1.10)

An efficiency definition that is equivalent to that in Eq. 1.10 was obtained by Suzuki and Munakata [22]. These authors arrived at the following expression called rectification efficiency expressing the fact that this definition can be also used in the absence of a bias force. That is, when the motor rectifies thermal fluctuations without doing work. Namely ηrec =

Fext v + γ v2 Pin

(1.11)

1.3 Ratchet Efficiency

7

where the numerator is the sum o the power the motor needs to generate to perform work against an external force, Fext , plus the power necessary to move in the longtime average at a speed v against a drag force γ v. The denominator is the average input power which can be written as Pin =

1 τ



τ 0

dV dx. dx

(1.12)

for long times τ . Some-what similar efficiency to ηrec was proposed by Wang and Oster, [23, 24], who introduced the so-called Stokes efficiency, ηStokes ≡

γ v2 −Gcycle r + Fext v

(1.13)

A ≡ −Gcycle > 0 is the chemical free energy consumed in one reaction cycle, Fext the conservative force acting on the motor by an external agent (e.g., a laser trap), r the rate of the chemical reaction cycle and v the average velocity of the motor. Machura et al. [25] investigate an often neglected aspect of Brownian motor transport, namely, the role of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise for rocking ratchets in the absence of a load force. They follow the reasoning of Suzuki and Munakata [22], which yields a nonvanishing rectification efficiency also in the absence of an external bias, namely, v2 ηrect =  2 | v − D0 |

(1.14)

BT is the noise intensity and V is the barrier height of the ratchet where D0 = kV potential V (x). They found typically, the current values and the corresponding velocity fluctuations are such that no appreciable rectification emerges in these inertial, rocked Brownian motors. There exist, however, tailored regimes of ratchet profiles and driving parameters for which an enhancement of rectification and optimal transport does occur. For a broader discussion of the energetics of Brownian motors, see the comprehensive review by Parrondo and de Cisneros in a special issue on Brownian motors [26].

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1.4 Ratchet Coherency Another quantity of central interest will be the effective diffusion coefficient 

Deff

x 2 (t) − x (t)2 σ2 ≡ lim = lim t→∞ t→∞ 2t 2t

(1.15)

The means are over the realizations of the stochastic process. The competition between the drift v and diffusivity Deff in advection-diffusion problems is often expressed by a dimensionless number, the Péclet number, P e, [27], Pe =

|v| L Deff

(1.16)

Here L is a typical length scale, in our case the length of a single ratchet element, v is the average stationary velocity of the particle, in our case we used |vcx |. The larger the Péclet number, the more net drift predominates over diffusion.

1.5 First Passage Time How long does it take a diffusing particle to travel from a starting point to a target? This first-passage, or hitting time, (MPFT) [28] 1993, [29] 1967, is a fundamental characteristic of diffusion and has a myriad of applications–to chemical kinetics [29, 30], and neuronal dynamics [31–33] to name a few examples. If a diffusing particle or protein is released at position x (0 ≤ x ≤ L) can diffuse either to the right or to the left. After a time τ , it covers an average distance dx, so that it is located at x = ±dx with equal probability 1/2. The MPFT to first reach an absorbing boundary located at x = 0, T (x, L) is given by (Fig. 1.4)

T (x, L) = τ +

1 [T (x + dx, L) + T (x − dx, L)] 2

(1.17)

We know from the definition of second order partial derivative, ∂ 2T T (x + dx, L) + T (x − dx, L) − 2T (x, L) = ∂x 2 (dx)2 Fig. 1.4 Illustration of the system in one dimension

(1.18)

1.5 First Passage Time

9

From Eq. 1.17 we have T (x + dx, L) + T (x − dx, L) = 2T (x, L) − 2τ

(1.19)

Replacing into Eq. 1.18 and having in account that the diffusion coefficient D = (dx)2 /2τ , we obtain, D

∂ 2T = −1 ∂x 2

(MF P T equation)

(1.20)

The boundary conditions for solving this equation are: At an absorbing boundary, (T = 0, it take no time to get there). At a reflecting boundary, T is constant, ∂T ∂x = 0. In our case we consider the reflecting at x = L. As we are considering only variation of x we can write Eq. 1.20 as d dx



dT (x, L) dx

 =−

1 D

(1.21)

Integrating once no interval [x, L] and considering [dT (x, L)/dx]L = 0, we obtain dT (x, L) L−x = dx D

(1.22)

Integrating once in the interval [0, x], we obtain T (x, L) =

 1  2Lx − x 2 2D

(1.23)

Howard in his excellent book [34] gives an equation to calculate MFPT for the case of a potential barrier, namely 1 T (L) = D



L 0



U (x) exp − kB T

 x

L

 U (y) dy dx exp kB T

(1.24)

In case of a constant force, F = − dUdx(x) or U (x) = −F x, from Eq. 1.24 we obtain

L2 T (L) = 2 2D



kB T FL

2    FL FL exp − −1+ kB T kB T

(1.25)

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1 Brownian Ratchets and Molecular Motors

When U (x) = 12 κx 2 the integral is not algebraically tractable, and it is assumed that the height of the energy barrier is high, UL ≡ U (L) kB T , then Eq. 1.24, gives  tK = τ

 π 4



kB T UL exp UL kB T

(1.26)

where τ = γ /κ is the drag coefficient divided by the spring constant, tK is called the Kramers time, after Kramers, who first derived it in 1940, [35]. For properties of First Passage Phenomena see Appendix D.1.

1.6 Power Stroke We saw that a ratchet is a motor in which the motion is driven directly by thermal fluctuations and rectified, or biased, by chemical reactions. In contrast, a powerstroke motor directly drives the motion. As an example we consider the rotation experiments of the F1 ATPase, the load is the long actin filament attached to the γ subunit, see Fig. 1.5a, [24, 36, 37]. Similarly, in kinesin experiments the motor tows a large bead (the load). Compare this case with the ratchet one at Fig. B.1, Appendix B in Stochastic Energetics. In both cases the ratchet and the power-stroke the energy to drive the motion ultimately comes from chemical reactions taking place in the catalytic site(s) of the motor. For Numerical computation of mechanochemical coupling see Chap. 4. Indeed, a power stroke is large, rapid structural change in a protein that can be used to do mechanical work, see Fig. 1.6. A power stroke has a size on the order of the dimension of the protein itself (several nanometers); this distinguishes it from the much smaller localized structural changes that occur when a protein binds to a ligand or catalyzes a chemical reaction (the size of a chemical bond or a few Ångstroms) [38]. The concept was first formulated for the protein myosin II which drives the contraction of muscle [39]. The γ -subunit acts as a crankshaft that converts conformational changes in the α3β3 ring to a unidirectional rotation and it also transmits conformational changes in an αβ pair to another. Literature treating the rotation of the γ -subunit and the effects that produces see [40–45].

1.6 Power Stroke

11

a Q˙ out = ζ

ω2

Work by Drag Torque

Ẇτ = τ

ω

Work by External Torque

Motlon of the Load

ẆM =

Reactants

-ɸʹsω

Work by Motor Torque

Work by Brownian Torque

b

ChemIcal ReactIon Cycle

r

Products

κBT ζ Q˙ in= I

Fig. 1.5 (a) A long actin filament (the load) is attached to the rotating shaft. There are three catalytic sites in the motor that alternate in sequence to hydrolyze ATP to ADP and phosphate, supplying the energy to turn the shaft. The motor rotates 360 in three steps, each step consuming one ATP. In principle (although not yet in the experiments), the load can be forced by an external conservative torque, such as a laser trap. (b) Four torques acting on the load of a protein motor: the Brownian torque, viscous drag torque, motor torque and the external torque. At equilibrium, in the ∂Qin 2 in absence of chemical reactions and external forcing, ∂Q dt = dt , or kBT = I (ω) (equipartition). (Reprinted figure with permission from [24]. Copyright by Springer Nature)

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Fig. 1.6 Proposals for motility generation mechanisms of motor proteins. (a and b) Power stroke and (c and d) Brownian ratchet. (a) Elastic relaxation. A fuel processing event (e.g., binding of an ATP or release of a hydrolysis product, denoted by a lightning symbol) leads to release of elastic energy. This is an unlikely scenario given the size and mechanical properties of typical motor proteins. (b) Conformational transition. A fuel processing event causes conformational change of the motor head, which changes the equilibrium position of the mechanical element (denoted by swinging rod). Before and after the stroke, the motor is not strained. (c) Flashing ratchet model. Top: The potential V(x) (x: position of the motor) is asymmetric, and the probability distribution P(x) is localized at a minimum. Middle: When V(x) is off, the motor performs free diffusion. Bottom: After V(x) switches back, the motor to the right of the barrier (vertical dashed line) undergoes biased diffusion to the minimum on the right side. This generates to a net current. Switching of V(x) on and off is mediated by a fuel. (d) Rectified diffusion model. A fuel processing event releases the motor from its initial position (top to middle), and the motor diffuses. Conformational change in the motor makes its affinity to the binding sites asymmetric, which results in preferential binding to the forward site (bottom). (Reprinted figure after [40], permission granted by PNAS)

Bibliography 1. https://en.wikipedia.org/wiki/Molecular_motor. Accessed 15 Sept 2020 2. Ait-Haddou, R., Herzog, W.: Brownian ratchet models of molecular motors. Cell Biochem. Biophys. 38, 191–213 (2003) 3. Bustamante, C., Chemla, Y.R., Forde, N.R., Izhaky, D.: Mechanical processes in biochemistry. Annu. Rev. Biochem. 73, 705–748 (2004) 4. Darnell, J., Lodish, H., Baltimore, D.: Molecular Cell Biology, 2nd edn., pp. 832–835. Scientific American Books, New York (1990) 5. Svoboda, K., Block, S.M.: Force and velocity measured for single kinesin molecules. Cell 77, 773–784 (1994) 6. Svoboda, K., Mitra, P.P., Block, S.M.: Fluctuation analysis of motor protein movement and single enzyme kinetics. Proc. Natl. Acad. U. S. A. 91, 11782–11786 (1994) 7. Higuchi, H., Muto, E., Inoue, Y., Yanagida, T.: Kinetics of force generation by single kinesin molecules activated by laser photolysis of caged ATP. Proc. Natl. Acad. U. S. A. 94, 4395–4400 (1997)

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8. Kolomeisky, A.B.: Motor proteins and molecular motors: how to operate machines at the nanoscale. J. Phys.: Condens. Matter 25, 463101 (2013) 9. Linke, H., Downton, M.T., Zuckermann, M.J.: Performance characteristics of Brownian motors. Chaos 15, 026111 (2005) 10. Visscher, K., Schnitzer, J.M., Block, S.M.: Single kinesin molecules studied with a molecular force clamp. Nature 400, 184–189 (1999) 11. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. AddisonWesley, Reading (1966) 12. von Smoluchowski, M.R.: Experimentell nachweisbare derublichen Thermodynamik widersprechende Molekularphanomene. Physik. Zeitschr. 13, 1069 (1912) 13. Suarez, ´ G.P., Hoyuelos, M., Chialvo, D.R.: Fluctuation-induced transport. From the very small to the very large scales. Pap. Phys. 8, art. 080004 (2016) 14. Parrondo, J.M.R., Espanol, ˜ P.: Criticism of Feynman’s analysis of the ratchet as an engine. Am. J. Phys. 64, 1125 (1996) 15. Magnasco, M.O.: Forced thermal ratchets. Phys. Rev. Lett. 71(10), 1477–1481 (1993) 16. Ajdari, A., Prost, J.: Mouvelllent induit par un potentiel periodique de basse symetrie: ´ dieIectrophorese pulsee. C.R. Acad. Sci. Paris, t. 315, Série II, 1635–1639 (1992) 17. Parrondo, J.M.R.: Reversible ratchets as Brownian particles in an adiabatically changing periodic potential. Phys. Rev. E 57(6), 7297 (1998) 18. Parrondo, J.M.R., Blanco, J.M., Cao, F., Brito, R.: Efficiency of Brownian motors. Europhys. Lett. 43(3), 248–254 (1998) 19. Zhou, H.-X., Chen, Y.: Chemically driven motility of Brownian particles. Phys. Rev. Lett. 77, 194 (1996) 20. Sekimoto, K.: Kinetic characterization of heat bath and the energetics of thermal ratchet models. J. Phys. Soc. Jpn. 66, 1234 (1997) 21. Derényi, I., Bier, M., Astumian, R.D.: Generalized efficiency and its application to microscopic engines. Phys. Rev. Lett. 83, 903 (1999) 22. Suzuki, D., Munakata, T.: Rectification efficiency of a Brownian motor. Phys. Rev. E 68, 021906 (2003) 23. Wang, H., Oster, G.: The Stokes efficiency for molecular motors and its applications. Europhys. Lett. 57(1), 134–140 (2002) 24. Wang, H., Oster, G.: Ratchets, power strokes, and molecular motors. Appl. Phys. A 75, 315–323 (2002) 25. Machura, L., Kostur, M., Talkner, P., Łuczka, J., Marchesoni, F., Hänggi, P.: Brownian motors: current fluctuations and rectification efficiency. Phys. Rev. E 70, 061105 (2004) 26. Parrondo, J.M.R., de Cisneros, B.J.: Energetics of Brownian motors: a review. Appl. Phys. A 75, 179–191 (2002) 27. Freund, J.A., Schimansky-Geier, L.: Diffusion in discrete ratchets. Phys. Rev. E 60(2), 1304 (1999) 28. Berg, H.C.: Random Walks in Biology, expanded edition. Princeton University Press, Princeton (1993) 29. Weiss, G.H.: First passage time problems in chemical physics. In: Prigogine, I. (eds.) Advances in Chemical Physics, vol. 13, pp. 1–18. Wiley, New York (1967) 30. Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943) 31. Gerstein, G.L., Mandelbrot, B.: Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41–68 (1964) 32. Bulsara, A.R., Elston, T.C., Doering, C.R., Lowen, S.B., Lindenberg, K.: Cooperafive behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys. Rev. E 53, 3958– 3969 (1996) 33. Burkitt, A.N.: A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol. Cybern. 95, 1–19 (2006) 34. Howard, J.: Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates, Inc. Publishers, Sunderland (2001)

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35. Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940) 36. Yasuda, R., Noji, H., Kinosita, K., Yoshida, M.: F1-ATPase is a highly efficient molecular motor that rotates with discrete 120 ◦ steps. Cell 93, 1117 (1998) 37. Noji, H., Yasuda, R., Yoshida, M., Kinosita, K.: Direct observation of the rotation of F1ATPase. Nature 386, 299 (1997) 38. Howard, J.: Protein power strokes. Curr. Biol. 16(14), R517–R519 (2006) 39. Cooke, R.: The mechanism of muscle contraction. CRC Crit. Rev. Biochem. 21, 53–118 (1986) 40. Hwang, W., Karplus, M.: Structural basis for power stroke vs. Brownian ratchet mechanisms of motor proteins. Proc. Natl. Acad. Sci. U. S. A. 116(40), 19777–19785 (2019) 41. Yang, W., Gao, Y.Q., Cui, Q., Ma, J., Karplus, M.: The missing link between thermodynamics and structure in F1-ATPase. Proc. Natl. Acad. Sci. U. S. A. 100, 874–879 (2003) 42. Gao, Y.Q., Yang, W., Karplus, M.: A structure-based model for the synthesis and hydrolysis of ATP by F1-ATPase. Cell 123, 195–205 (2005) 43. Pu, J., Karplus, M.: How subunit coupling produces the γ -subunit rotary motion in F1-ATPase. Proc. Natl. Acad. Sci. U. S. A. 105, 1192–1197 (2008) 44. Nam, K., Pu, J., Karplus, M.: Trapping the ATP binding state leads to a detailed understanding of the F1-ATPase mechanism. Proc. Natl. Acad. Sci. U. S. A. 111, 17851–17856 (2014) 45. Bason, J.V., Montgomery, M.G., Leslie, A.G.W., Walker, J.E.: How release of phosphate from mammalian F1-ATPase generates a rotary substep. Proc. Natl. Acad. Sci. U. S. A. 112, 6009– 6014 (2015)

Chapter 2

The Fokker-Planck Equation

During the last years with the studies of stochastic processes: neurons networks, molecular motors, dynamics models, anomalous diffusion, disordered media, etc, several methods have evolved to apply the Focker-Planck equation (FPE) to these phenomena. We present here the solution of the Fokker-Planck equation by the Crank-Nicholson formalism. The von Neumann amplification factor, ξ (k), is independent of dt, so the method is stable for any size dt. The method is suitable for modeling molecular motors because the great amount of interactions in these systems, vectors and matrices oriented methods are needed, suited to work with Matlab. In the Appendix of this chapter is given some notions of Stochastic Dynamics

2.1 The Methods The method of Fractional Focker-Planck equation (FFPE) [1] was derived from a generalized continuous time random walk, which includes space dependent jump probabilities which are the result of an external field. In [2] was presented the solution of the FFPE in terms of an integral transformation. In [3] a half order FFPE was derived from the generalized scheme of random walks on the comlike structure. In [4, 5] the FFPE was used to study and describe also the anomalous diffusion in external fields. In [6] the FFPE was used to study ultraslow kinetics. In [7] was introduced a heterogeneous FFPE involving external force fields describing systems on heterogeneous fractal structure medium. In [8] was studied the dynamical properties of bistable systems described by the one-dimensional sub-diffusive FFPE, for the natural boundary conditions as well as the absorbing boundary conditions. In [9] was shown that the subordinated Brownian process is a stochastic solution of the FFPE. In [10] was proven that the asymptotic shape of the solution of the FFPE is a stretched Gaussian and that its solution can be expressed in the form of a function of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_2

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2 The Fokker-Planck Equation

a dimensionless similarity variable for generic potentials. In [11] was performed a numerical solution of the space FFPE. In [12], the globally connected active rotators with excitatory and inhibitory connections were analyzed using the FPE. In [13] a FPE with velocity-dependent coefficients was considered for various isotropic systems on the basis of probability transition approach. In [14] was introduced a new method for solving the FPE arising in the simulation of dilute polymeric solutions. In [15] using the Lie algebraic approach was derived the exact diffusion propagator of the class of FPEs with logarithmic factors in diffusion and drifts terms. In [16] based on the FPE, a formula for the effective diffusion coefficient was derived for a Brownian particle undergoing one dimensional motion in a two-state Brownian motor (ratchet) in which a spatially periodic, asymmetric potential acting on the particle switches between two states stochastically. In [17] was presented an analytical solution of the FPE of non-degenerate parametric amplification system for generation of squeezed light. In [18] was studied the entropy decay of discretized FPEs. In [19] was studied the influence of the periodic potential structure on the diffusion mechanism of a Brownian particle using the FPE. In [20] employing the FPE, was discussed the stability of the self-similar solution on network models. In [21] was presented a procedure for deriving general nonlinear FPEs directly from the master equation. In [22] based on the canonical formalism, the dilatation symmetry is implemented to the FPE for the Wigner distribution that describes atomic motion in an optical lattice. In [23] was performed a multiple scales analysis of the FPE for simple shear flow. In [24] using the Hanggi ansatz to truncate the evolution equation for probability density, an approximate FPE was derived. In [25] was analyze the long-time behaviour of transport equations for a class of dissipative quantum systems with Fokker-Planck type diffusion operator, subject to confining potentials of harmonic oscillator type. One way cells accomplish biomolecular transport is through the use of motor proteins, such as cytoplasmic kinesins and dyneins. Using laser trapping techniques, the biophysical properties of single motor proteins can now be measured. This, coupled with increased biochemical and structural data for molecular motors, has sparked a renewed interest in the mathematical modeling of motor protein function. In general, mechanistic models of energy transduction in motor proteins must be studied numerically, because analytic solutions to model equations exist only for very simple systems [26]. Because the great amount of interactions in these systems, vectors and matrixes oriented methods are needed, suited to work with Matlab/Octave. We present here the solution of the FPE [27] by the Crank-Nicholson formalism [28]. The von Neumann amplification factor, ξ (k) [29], is independent of dt, so the method is stable for any size dt.

2.2 The Fokker-Planck Equation

17

2.2 The Fokker-Planck Equation Let’s consider a Brownian particle in the presence of a potential V (x) capable to localize it in a space region, in such a way to establish, after a sufficient time, an equilibrium distribution, P (x, t) = P (x). The corresponding Langevin equation is m

dx d 2x dV (x) + =− + (2kB T )1/2 ξ(t) 2 dt dx dt

(2.1)

where m is the mass of the particle,  is the frictional coefficient of the particle (Kg/s units), for a spherical particle  = 6π ηa,where η is the viscocity of the medium and a the particle radius; kB is the Boltzmann constant, T is the absolute temperature and ξ(t) is the so called white noise, defined by its statistical properties: ξ(t) = 0 (mean)

(2.2)

ξ(t2 )ξ(t1 ) = δ (t2 − t1 ) (correlation f unction)

(2.3)

The motive to call white noise is simple: the spectral intensity of a stochastic process is the Fourier transform of the correlation function. Now, the Fourier transform of the Dirac delta, δ (t2 − t1 ), is a constant, what means, all the frequencies are present with the same intensity, what characterize the white light. In Appendix E on Stochastic Dynamics we give more details of this noise. If we neglect the inertial term we obtain: 1 dV (x) dt + dx (t) = −  dx



2kB T 

1/2 ξ(t)dt

(2.4)

The corresponding Fokker-Planck equation for the probability distribution P = P (x, t) is  ∂P 1∂ = ∂t 

∂V (x) ∂x P



∂x

+D

∂ 2P , ∂x 2

(2.5)

or D ∂P = ∂t kB T



2 ∂ V (x) ∂V (x) ∂P ∂ 2P P + kB T + ∂x ∂x ∂x 2 ∂x 2

(2.6)

where D = kBT is the Diffusion coefficient. When the equilibrium is stablished, P becomes time independent and Eq. 2.5 changes into  1∂ 

∂V (x) ∂x P

∂x

 (x)

+D

∂ 2 P (x) = 0. ∂x 2

18

2 The Fokker-Planck Equation

Integrating in x follows 1 ∂V (x) ∂P (x) P (x) + D = constant.  ∂x ∂x As P is the equilibrium distribution, its dependency in x is given by Boltzmann distribution, i.e., P (x) ∝ exp −

V (x) kB T

2.3 Discretization of the Fokker-Planck Equation 2.3.1 Forward Time Central Space (FTCS) Method in in in dt

t : t1 = 0, t2 = dt, . . . tn+1 = ndt, . . . x : x1 = 0, x2 = dx, . . . xj +1 = j dx, . . .  P : Pjn = P xj , tn and dx are finite increments. It follows: ∂P (x, t) P (x, t + dt) − P (x, t) = ∂t dt =

Pjn+1 − Pjn dt

,

∂P (x, t) P (x + dx, t) − P (x − dx, t) = ∂x dx n n Pj +1 − Pj −1 , = dx ∂ 2 P (x, t) P (x + dx, t) + P (x − dx, t) − 2P (x, t) = 2 ∂x (dx)2 =

Pjn+1 + Pjn−1 − 2Pjn (dx)2

,

2.3 Discretization of the Fokker-Planck Equation

19

In discrete notation, Eq. 2.6 is: Pjn+1



 (dx)2 ∂ 2 V Ddt = + − 2 Pjn kB T ∂x 2 (dx)2 

dx ∂V + 1 Pjn+1 + 2kB T ∂x



dx ∂V n Pj −1 + 1− 2kB T ∂x Pjn

(2.7)

2.3.2 Crank-Nicholson Method To discretize former equation in the Crank-Nicholson schemeis just to substitute in  the terms affected by squared brackets, P n by 12 P n+1 + P n : n+1 n n n aPjn+1 − cPjn+1 +1 − dPj −1 = bPj + cPj +1 + dPj −1

(2.8)

with a= b= c= d=

 (dx)2 ∂ 2 V 1− −2 2 (dx)2 kB T ∂x 2   Ddt (dx)2 ∂ 2 V 1+ −2 2 (dx)2 kB T ∂x 2

 dx ∂V Ddt 1+ 2kB T ∂x 2 (dx)2

 Ddt dx ∂V 1− 2kB T ∂x 2 (dx)2 Ddt



We define now the matrixes and vectors ⎛

a ⎜ −d ⎜ ⎜ ⎜ 0 E=⎜ ⎜ ⎜ ⎝ 0

−c 0 0 a −c 0 −d a −c ... ... 0 0 ...

⎞ ... 0 ... 0⎟ ⎟ ⎟ ... 0⎟ ⎟, ⎟ ⎟ ⎠ −d a

(2.9)

(2.10) (2.11) (2.12)

20

2 The Fokker-Planck Equation

⎛

b ⎜d ⎜ ⎜ ⎜0 M=⎜ ⎜ ⎜ ⎝ 0

c b c d b c ... ... 0 0 ...

⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ ⎠ d b

... ... ...

⎞ P1n ⎜Pn ⎟ ⎜ 2 ⎟ ⎜ n⎟ ⎜P ⎟ n P =⎜ 3 ⎟ ⎜ . ⎟ ⎟ ⎜ ⎝ . ⎠ PNn ⎛

Using the boundary conditions P0n = PNn +1 = 0, the Eq. 2.8 can be written in matrix form as: E.Pn+1 = M.Pn ,

(2.13)

  Pn+1 = E−1 . M.Pn

(2.14)

2.3.3 Stability Analysis The von Neumann amplification factor, ξ (k), [29], is obtained substituting the independent solutions, or eigenmodes, of the difference equations, namely Pjn = ξ n eikj dx

(2.15)

in Eq. 2.8, giving ξ (k) =

b + ceikdx + de−ikdx a − ceikdx − de−ikdx

(2.16)

independent of dt, so the method is stable for any size dt. As an example of the method’s application we consider a system of Brownian particles (of negligible mass), each particle being acted upon by a linear (elastic) restoring force, Fx = −λx, and having a frictional coefficient  in the surrounding medium. We calculate now the elements a, b, c, d, of the matrixes: Fx = −λx ⇒

∂ 2V ∂V = λx ⇒ =λ ∂x ∂x 2 2

(2.17)

We define k = D2 dt 2 , and k1 = (dx) kB T λ. These parameters values have to conserve (dx) the area under the P (x, t) curves; considering x = j dx, we have, a = 1 − k (k1 − 2) b = 2−a c = k (1 + k1 j )

2.4 Program 2.1, F-P Equation, Matlab Code

Fig. 2.1 Probability distribution at different times: P (x, 0) = (2π 1/2 σ )−1 exp

21



(x−x0 ) 21/2 σ

2

d = 2k − c Fig. 2.1 shows the manner in which an ensemble of Brownian particles approaches a state of equilibrium under the combined influence of the restoring force and the molecular bombardment. The CPU time, in the former example, of this method is 18 times lesser than processing with the Forward Time Central Space (FTCS) method and also this method is useful in describing biological molecular motors where coupled differential equations are involved.

2.4 Program 2.1, F-P Equation, Matlab Code %Solve the Fokkuer-Planck Equation %help FokkerPlanck; clear; x0=80.; L=100; tmax=1000; dt=1.; sigma=1.5; k=.025; k1=.015;

22

2 The Fokker-Planck Equation

tic; % k1 string coefficient lamda; s2=1/(2*sigma2 ); T=floor(tmax/dt)+1; % T is the number of temporal points. x=(1:L)’; % spatial grade, colum vector. p=zeros(L,1); % Initial Probability. %p0=sqrt(s2)/sqrt(2*pi); p0=.3; p=p0*exp(-((x-x0).2 )*s2); p1=p; P=zeros(L,8); dx=1.; kT=1.; E=zeros(L); M=zeros(L); a=1-k*(k1-2.); b=2-a; for j=1:L E(j,j)=a; M(j,j)=b; end for j=2:L c=k*(1.+k1*j); d=-c+2.*k; E(j,j-1)=-d; E(j-1,j)=-c; M(j,j-1)=d; M(j-1,j)=c; end IE=inv(E); pt=ceil(T/8); disp(pt); for tp=1:8 for it=1:pt p=IE*(M*p); end P(:,tp)=p; % Distribution at tp end %plot(x,p1,’.’,x,P(:,1),x,P(:,2),x,P(:,3),x,P(:,4), . . . plot(x,P(:,1),x,P(:,2),x,P(:,3),x,P(:,4), . . . x,P(:,5),x,P(:,6),x,P(:,7),x,P(:,8)) toc F1(:)=P(:,1); F2(:)=P(:,2); F3(:)=P(:,3); F4(:)=P(:,4); F5(:)=P(:,5);

Bibliography

23

F6(:)=P(:,6); F7(:)=P(:,7); F8(:)=P(:,8); F9(:)=p1’; save F1.ext F1 -ascii; save F2.ext F2 -ascii; save F3.ext F3 -ascii; save F4.ext F4 -ascii; save F5.ext F5 -ascii; save F6.ext F6 -ascii; save F7.ext F7 -ascii; save F8.ext F8 -ascii; save F9.ext F9 -ascii; save x.ext x -ascii;

Bibliography 1. Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132 (2000) 2. Barkai, E.: Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E 63, 046118 (2001) 3. Zahran, M.A.: 1/2-order fractional Fokker–Planck equation on comblike model. J. Stat. Phys. 109, 1005 (2002) 4. Lenzi, E.K., Mendes, R.S., Fa, K.S., Malacarne, L.C., da Silva, L.R.: Anomalous diffusion: fractional Fokker–Planck equation and its solutions. J. Math. Phys. 44, 2179 (2003) 5. Zahran, M.A., Abulwafa, E.M., Elwakil, S.A.: The fractional Fokker–Planck equation on comb-like model. Physica A 323, 237 (2003) 6. Chechkin, A.V., Klafter, J., Sokolov, I.M.: Fractional Fokker-Planck equation for ultraslow kinetics. Europhys. Lett. 63, 326 (2003) 7. Ren, F.Y., Liang, J.R., Qiu, W.Y., Xu, Y.: Fractional Fokker–Planck equation on heterogeneous fractal structures in external force fields and its solutions. Physica A 326, 430 (2003) 8. So, F., Liu, K.L.: A study of the subdiffusive fractional Fokker–Planck equation of bistable systems. Physica A 331, 378 (2004) 9. Stanislavsky, A.A.: Subordinated Brownian motion and its fractional Fokker–Planck equation. Phys. Scr. 67, 265 (2003) 10. Xu, Y., Ren, F.Y., Liang, J.R., Qiu, W.Y.: Stretched Gaussian asymptotic behavior for fractional Fokker–Planck equation on fractal structure in external force fields. Chaos Solitons Fractals 20, 581 (2004) 11. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166, 209 (2004) 12. Kanamaru, T., Sekine, M.: Analysis of globally connected active rotators with excitatory and inhibitory connections using the Fokker-Planck equation. Phys. Rev. E. 67, 031916 Part 1 (2003) 13. Trigger, S.A.: Fokker-Planck equation for Boltzmann-type and active particles: transfer probability approach. Phys. Rev. E 67, 046403 Part 2 (2003) 14. Lozinski, A., Chauviere, U.: A fast solver for Fokker–Planck equation applied to viscoelastic flows calculations: 2D FENE model. J. Comput. Phys. 189, 607 (2003)

24

2 The Fokker-Planck Equation

15. Lo, C.F.: Exact propagator of the Fokker–Planck equation with logarithmic factors in diffusion and drift terms. Phys. Lett. A 319, 110 (2003) 16. Sasaki, K.: Diffusion coefficients for two-state Brownian motors. J. Phys. Soc. Jpn. 72, 2497 (2003) 17. Zhao, C.Y., Tan, W.H., Guo, Q.Z.: The solution of the Fokker-Planck equation of nondegenerate parametric amplific ation system for generation of squeezed light. Acta Phys. Sin. 52, 2694 (2003) 18. Arnold, A., Unterreiter, A.: Entropy decay of discretized fokker-planck equations I–Temporal semidiscretization. Comput. Math. Appl. 46, 1683 (2003) 19. Chhib, M., El Arroum, L., Mazroui, M., Boughaleb, Y., Ferrando, R.: Influence of the periodic potential shape on the Fokker–Planck dynamics. Physica A 331, 365 (2004) 20. Kamitani, Y., Matsuba, I.: Self-similar characteristics of neural networks based on Fokker– Planck equation. Chaos Solitons Fractals 20, 329 (2004) 21. Nobre, F.D., Curado, E.M.F., Rowlands, G.: A procedure for obtaining general nonlinear Fokker–Planck equations. Physica A 334, 109 (2004) 22. Abe, S.: Dilatation symmetry of the Fokker-Planck equation and anomalous diffusion. Phys. Rev. E 69, 016102, Part 2 (2004) 23. Subramanian, G., Brady, J.F.: Multiple scales analysis of the Fokker–Planck equation for simple shear flow. Physica A 334, 343 (2004) 24. Liang, G.Y., Cao, L., Wu, D.J.: Approximate Fokker–Planck equation of system driven by multiplicative colored noises with colored cross-correlation. Physica A 335, 371 (2004) 25. Sparber, C., Carrillo, J.A., Dolbeault, J., Markowich, P.A.: On the long-time behavior of the quantum Fokker-Planck equation. Monatchefte fur Mathematik 141, 237 (2004) 26. Oster, G., Hongyun, W., Grabe, M.: How Fo–ATPase generates rotary torque. Phil. Trans. R. Soc. Lond. B 355, 523 (2000) 27. Risken, H.: The Fokker-Planck Equation: Methods of Solution and Applications. Springer, Berlin (1984) 28. Crank, C., Nicolson, N.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Philos. Soc. 43(50), 50 (1947) 29. Press, W.H., Teukolsky, S.A., Vettering, W.T., Flannery, B.P.: Numerical Recipes, The Art of Scientific Computing, p. 625. Cambridge University Press, New York (1987)

Chapter 3

Biased Brownian Motion

We use the definition given by Derényi and Astumian [1] for Brownian ratchets, namely: “Non-equilibrium fluctuations, whether generated externally or by a chemical reaction far-from-equilibrium, can drive directed motion along an anisotropic structure without thermal gradients or net macroscopic forces, simply by biasing Brownian motion. Systems operating on this principle are often referred to as Brownian ratchets, and the transport in such systems is called fluctuation driven transport”. For the historical evolution of the fundamental models of ratchet devices see Chapters 1 and 2 of Cubero and Renzoni [2]. Several references are involved in this context: Bug and Berne [3], Ajdari and Prost [4], Magnasco [5], Astumian and Bier [6], Astumian [7], Astumian and Derényi [8]. First we parameterize the Langevin equation, second in order to perform the numerical simulations we discretize the time, t˜ and we introduce the “Wiener´s  increment” dW t˜ , see also Appendix E, then we build the Fokker-Plank’s matrices. Considering a periodic potential slightly tilded we perform simulations using the Euler’s and Fokker-Plank’s equations. Later we introduce Dichotomous Markov noise, DMN , and we teach how to generate it. Then we analyze the “Flashing” Ratchet, by introducing the DMN into the Euler’s equation. Then several simulations are performed with this equation. Finally we introduce the “Rocking” Ratchet into the Euler’s equation and perform simulations with it.

3.1 Parametrization of the Langevin Equation We consider a system of Brownian particles of negligible mass, each particle being acted upon by a force and velocity-dependent friction, i.e. damped motion in the potential V (x), and also acted by an external force F , in accordance to the Langevin Eq. (2.2), we have: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_3

25

26

3 Biased Brownian Motion

dx (t) = −

DF D dV (x) dt + dt + (2D)1/2 ξ(t)dt kB T dx kB T

(3.1)

where we have used the Einstein’s relation γ = kBDT . In order of working with dimensionless parameters and variables, we perform the following transformations:  F = F0 F , D = D0 D,  ξ(t) = ξ0 ξ(t),  t = τD t˜, x = Lx, ˜ V = V0 V (3.2) The tilded variables are the dimensionless ones. To complement the parameters: τD =

L2 1 V0 kB T , ξ0 = , V0 = , F0 = 1/2 D0 kB T L τD

(3.3)

Correspondingly Eq. (3.1) becomes: ˜  1/2  t˜ F d t˜ + 2D  V0 d V d t˜ + D ξ(t)d d x˜ = −D d x˜

(3.4)

Considering V0 = 0, at the long time limit, after initial transients decay, the particle achieves a steady-state motion with the average velocity 

d x˜ d t˜



F  =D

(3.5)

lim

t˜→∞

If we have n non-interacting particles per unit length, the corresponding steady current is   d x˜  F  J = l n˜ = l n˜ D (3.6) d t˜ lim t˜→∞

 (Fig. 3.1). in the direction of the applied force F

3.2 Numerical Simulations For numerical simulation of Eq. (3.4) we discretize time t. Then when the particle advances from x(tn ) to x(tn+1 ) we obtain         1/2   V0 V x˜  F t˜ + 2D  x˜ t W t˜n tn − D tn t˜ + D n+1 = x˜ 

(3.7)

 W t˜ = dW (t˜). We have to remind that the “Wiener’s increment” dW (t˜) is a Gaussian stochastic process, of width σ = (d t˜)1/2 . Then, at each pass of

3.3 Building the Fokker-Plank’s Matrices

27

Fig.3.1  (a) Linear potential: V (x) = −F x (b) Periodic Potential: V (x) =  V0 2π x 1 4π x  x˜ sin − − F x. In dimensionless units: (a) Linear potential: V ( x) ˜ = − F sin 2π L1 4 L1   1  (x) x˜ (b) Periodic Potential: V ˜ = 2π ˜ −F sin (2π x) ˜ − 14 sin (4π x)

the integration we have to draw dW (t˜) and normalize the result properly. Let us call RG an aleatory number, with Gaussian distribution, centered in RG = 0 and width 1. In MATLAB/OCTAVE RG = rand n, consequently we can write dW (t˜) = (d t˜)1/2 RG . Finally Eq. (3.7) is transformed in the corresponding Euler’s equation of this process, namely:          V0 V x˜ tn t˜ + D F t˜ + 2D  t˜ 1/2 rand n x˜ t tn − D n+1 = x˜ 

(3.8)

In Fig. 3.2 is shown results for Eqs. (3.4) and (3.5) for a given simulation. We can observe that the ! linear Fit at long times limit, coincides with the value given by d x˜ F  = 1 × 0.1 =D Eq. (3.5), d t˜ lim

t→∞

3.3 Building the Fokker-Plank’s Matrices The elements of the matrices E and M are given by Eqs. (2.7)–(2.10), namely:  (dx)2 d 2 V a = 1− −2 2 (dx)2 kB T dx 2   Ddt (dx)2 d 2 V b = 1+ −2 (dx)2 kB T dx 2

 dx dV Ddt 1+ c= 2kB T dx 2 (dx)2 Ddt



(3.9)

(3.10) (3.11)

28

3 Biased Brownian Motion

Fig. 3.2 Results for a simulation corresponding the potential of Fig. 3.1a, see Program 3.1 (Matlab code) at the end of the chapter, with the parameter: N = 20,000; NR = 1000; dt = 0.01; D = 1.; Vo = 0.; F = 0.1; r = 1.; x00 = 0

d=

 dx dV 1 − 2kB T dx 2 (dx)2 Ddt

(3.12)

The parameters as functions of the dimensionless variables and constants are:   t˜ d 2 V  Dd 2 0 − 2 x) ˜ V (d d x˜ 2 2 (d x) ˜ 2

   ˜ d 2V 2 ˜b = 1 + Dd t  ˜ −2 V0 2 (d x) d x˜ 2 (d x) ˜ 2  t˜

 Dd d x˜  d V c˜ = 1+ V0 2 d x˜ 2 (d x) ˜ 2

  t˜ d x˜  d V Dd 1 − d˜ = V 0 2 d x˜ 2 (d x) ˜ 2 a˜ = 1 −

(3.13) (3.14) (3.15) (3.16)

3.3 Building the Fokker-Plank’s Matrices

Fig. 3.3 Periodic Potential: (a) V (x) =      Vo 2π x 1 4π x sin − sin 2π L 4 L

29

Vo 2π

   − sin 2πLx +

1 4

 sin

4π x L

 . (b) V (x) =

3.3.1 Periodic Potential Slightly Tilted The sign + corresponds tilted to the right, Fig. 3.3b, correspondingly the sign − the potential is tilted to the left, Fig. 3.3a.



 2π x 1 4π x Vo sin − sin V (x) = ± 2π L 4 L

(3.17)





 dV (x) Vo 2π x 1 4π x =± cos − cos dx L L 2 L

(3.18)





 2π x d 2 V (x) 4π x 2π − sin + sin = ±V 0 2 L L dx 2 L

(3.19)

The corresponding dimensionless equations are

1 1  (x) V ˜ =± sin (2π x) ˜ − sin (4π x) ˜ 2π 4

(3.20)

 (x) dV ˜ 1 = ± cos (2π x) ˜ − cos (4π x) ˜ d x˜ 2

(3.21)

 (x) " # d 2V ˜ = ±2π − sin (2π x) ˜ + sin (4π x) ˜ 2 d x˜

(3.22)

The corresponding factor V0 = V0 /KB T in the former dimensionless equations is omitted because is considered explicitly in the Euler’s equations. From now on, we

30

3 Biased Brownian Motion

Fig. 3.4 Simulation performed with Program 3.1, with the parameters included:    1  (x) ˜ = cos (2π x) ˜ − 12 cos (4π x) ˜ , V (x) ˜ . Most of the sin (2π x) ˜ − 14 sin (4π x) V ˜ = 2π −4  stochastics realizations, < x >, are to the left < x > < 0. D = 5.10 is in accordance of a 3 nm radius monomeric protein, which the Diffusion coefficient in water is D0 = KB T /(6π ηa) =  = 4.08×10−14 m2 /s, 8.17×10−11 m2 /s, given for the bound state diffusion coefficient D = D0 D consistent with the value D = 4.4 × 10−14 m2 /s for monomeric kinesin

will use the dimensionless variables and constants in the programs and figures and omit the tilde symbol (Figs. 3.4 and 3.5).

3.4

Dichotomous Markov Noise

We consider a stochastic variable η (t) which switches between two values a and b with constant transition rates, DMN. The rate of switching from a to b is μa and from b to a is μb . This two-step process can be described by a probability loss-gain equation or master equation. d P (a, t|x, t0 ) = −μa P (a, t|x, t0 ) + μb P (b, t|x, t0 ) dt d P (b, t|x, t0 ) = +μa P (a, t|x, t0 ) − μb P (a, t|x, t0 ) dt

(3.23) (3.24)

Fig. 3.5 Evolution of the Probability density distribution at different times for different values of  2 0) is the external Force F: The initial Gaussian distribution, P (x, 0) = (2π 1/2 σ )−1 exp (x−x 1/2 2 σ plotted. We can observe even with F = 0. The probability density evolves to the left, meaning that the current is to the left because the potential is slightly tilded to the right, in accordance to Fig. 3.4. Simulation performed with Program 3.2, with the parameters: D = 5 × 10−4 , V0 = 1.0, V (x) ˜ =   1 1 sin x) ˜ − sin x) ˜ (2π (4π 2π 4

32

3 Biased Brownian Motion

where P (a, t|x, t0 ) is the conditional probability that the variable η (t) will assume the value a at some time t given that it was x at earlier time t0 . P (b, t|x, t0 ) can be defined similarly. The condition for the conservation of the total probability is P (a, t|x, t0 ) + P (b, t|x, t0 ) = 1

(3.25)

The initial condition for Eqs. (3.13) and (3.14) are given by: P (xt’t|x, t0 ) = δxt’x at t = t0

(3.26)

From Eqs. (3.13) and (3.14) we obtain the steady state solutions μb μa + μb μa P s (b) = P (b, ∞|x, t0 ) = μa + μb

P s (a) = P (a, ∞|x, t0 ) =

(3.27)

The conditional probabilities P (a, t1 |x, t) and P (b, t1 |x, t) with x = a, b can be obtained by solving the Eqs. (3.13) and (3.14) subject to the conditions (3.15), (3.16) and (3.17):

P=

P (a, t1 |a, t) P (b, t1 |a, t) P (a, t1 |b, t) P (b, t1 |b, t)



 = τc

μb + μa e−t/τc μb (1 − e−t/τc ) μa (1 − e−t/τc ) μa + μb e−t/τc



(3.28) where τc =

1 μa + μb

(3.29)

is the characteristic relaxation time to the stationary state of the DMN . In the foregoing we consider stationary DMN, for which the stationary probabilities of the two states are P robability(η = b) = μa τc , P robability(η = a) = μb τc

(3.30)

We will use the symmetric DMN, for which a = 1 and b = −1, and μa = μb = μ, the corresponding mean value η(t) = μa τc × 1 + μb τc × −1 = (μa − μb )τc = 0.

(3.31)

It does not exist a dichotomous noise with η(t) = 0 if one of the states is zero. Generally is considered η(t) = 0, in order to avoid any systematic bias introduced by the noise in the dynamics of the driven system. The stationary temporal autocorrelation function of DMN is exponentially-decaying:

3.4 Dichotomous Markov Noise

33

 |t − t’| D η(t)η(t’) = exp − , τc τc

(3.32)

where D = μb μa τc 3 (a + |b|)2 = μ2 τc 3 (2)2 =

1 2μ

(3.33)

where we have used τc = (2μ)−1 . Then Eq. (3.18) transforms in,

P=

3.4.1

P (a, t1 |a, t) P (b, t1 |a, t) P (a, t1 |b, t) P (b, t1 |b, t)

 =

1

−2μt ) 2 (1 + e 1 −2μt ) 2 (1 − e

1 −2μt ) 2 (1 − e 1 −2μt ) 2 (1 + e

 (3.34)

Generating Dichotomous Markov Noise

The possible transitions probabilities are schematized in Fig. 3.6. In generating the DMN, we will use only three of them, namely: P (a, t1 |a, t), P (a, t2 |b, t1 ), P (a, t2 |a, t1). We follow the logic of Barik [9], namely: Let the particle is located initially (t) at xn = a. To determine whether the particle moves at time t1 = t + t to another site

Fig. 3.6 Transitions Probabilities

34

3 Biased Brownian Motion

xn+1 = b or remain at the same site xn = a, we consider the conditional probability as given by Eq. (3.24), namely, P (a, t1 |a, t) =

 1 1 + e−2μt 2

(3.35)

An uniformly distributed random number R between [0, 1] is now generated by the computer. This number is then compared against the conditional probability Eq. (3.25). If P (a, t1 |a, t) > R, then we accept the value of the noise a, i.e., xn = a, else we accept the value b or xn+1 = b. If the value of noise is b at t1 (= t + t) then we calculate the conditional probability of jumping to another site xn+2 = a at t2 (= t1 + t) as P (a, t2 |b, t1 ) =

 1 1 − e−2μt 2

(3.36)

On the other hand if the value of the noise is xn+1 = a at t1 (= t + t) we calculate the conditional probability P (a, t2 |a, t1) using Eq. (3.25). We then compare P (a, t2 |b, t1) or P (a, t2 |a, t1 ) against another uniformly distributed random number R1 between [0, 1]. If P (a, t2 |b, t1) or P (a, t2 |a, t1) > R1 then the value of the noise at t2 is xn+2 = a else we accept xn+2 = b. By repeating the procedure we can generate a sequence of random number η(t) switching between two values a and b. In our case between −1 and 1. It is important to note that the time interval between the two steps is always fixed and is equal to t which is much smaller than the correlation time τc of the noise (t  τc ). The Program 3.3 generates the DMN in accordance to the previous procedure of Ref(Debashis). In Fig. 3.7 is shown a diagram of dichotomous Markov noise obtained from Program 3.3. Fig. 3.7 Dichotomous Markov Noise, τ = 0.25

3.5 Fluctuating Potential, or “Flashing” Ratchet

3.5

35

Fluctuating Potential, or “Flashing” Ratchet

The flashing ratchet model was introduced by Ajdari and Prost [4]. The corresponding Euler equation for the fluctuating potential is given by (Figs. 3.8 and 3.9),      1       $ t˜ 1/2 rand n V0 V x˜  F t˜ + 2D x˜ t tn D tn − 1 + η  tn t˜ + D n+1 = x˜  2 (3.37) with

         1 V x˜ t˜n = cos 2π x˜ t˜n − cos 4π x˜ t˜n (3.38) 2 Fig. 3.8 Representation of a Flashing Ratchet

ON

OFF

ON

Fig. 3.9 Fluctuating Potential:The potential switches between the linear, −F.x and the tilted periodic potential: −F.x + V (x) where V (x) is given by any of the curves Fig. 3.3a or b

36

3 Biased Brownian Motion

Fig. 3.10 Flashing ratchet: The simulation was performed with the Program 3.4, following Eqs.  (3.37)  and  (3.38). The thermal fluctuations, D, and the dichotomous Markov noise, DMN = 1  with τ = 0.4, which fluctuates between 0 and 1, both are responsible for the 2 1 + η tn transport. For values F = 0 and D > 4 we observe an inversion of the movement. This is because D activate the particle over the periodic barriers and consequently producing current in the opposite direction

An excellent reasoning explaining the mechanism of “Flashing” ratchet and estimation the particle current in a flashing ratchet was given by [4] and also [11]. Also Doering [12], Astumian [7], and Mogilner et al. [14] treated this problem. The corresponding Fokker-Planck probability density distribution pattern is similar to Fig. 3.6a with the particles moving to the left. Otherwise if the potential is tilted to the left, the particles move to the right (Figs. 3.10, 3.11, 3.12, 3.13, 3.14, and 3.15).

3.6 Fluctuating Force, or “Rocking” Ratchet It was soon realized by Magnasco [5] that other general schemes, like this here, generate directed motion. The fluctuating potential is given by U (x, t) = V (x) − η (t) ± F x,

(3.39)

where η (t) ± F is the fluctuating force process As a consequence, from Fig. 3.16, even though the applied forces ±F are equal, there will be much more drift to the right than to the left and a current is expected. The corresponding Euler equation for the fluctuating potential is given by

3.6 Fluctuating Force, or “Rocking” Ratchet

37

Fig. 3.11 Flashing ratchet with τ = 0.4: The simulation was performed with the Program 3.4, also following Eqs. (3.37) and (3.38). In this case, when is not present an external force, F = 0, we observe that without DMN , non flashing ratchet, the movement is more efficient than with DMN in both directions

Fig. 3.12 Flashing ratchet with τ = 0.4: The simulation was performed with the Program 3.4, also following Eqs. (3.34) and (3.35). In this case, when is present an external force, F , we observe that with DMN , for low values of D, the movement is more efficient than without DMN . At higher values of D without DMN is more efficient

38

3 Biased Brownian Motion

Fig. 3.13 Flashing ratchet with τ = 0.4 : v vs load Force, Fload . We can observe the motor effect: For values negatives of the load Force [−0.225, 0], the corresponding velocity is positive, meaning the motor obtain the energy from its internal mechanism, in our case the dichotomous noise generation source. This computation was performed with Program 3.4

Fig. 3.14 Flashing ratchet with τ = 0.4 : v vs D, with F = 0 and V0 fixed. We observe the curve has a maximum for positive values and a minimum for negative values of < v >, meaning a constant competition of thermal fluctuation and potential values. This computation was performed with Program 3.4, with the potential tilted to the right

3.6 Fluctuating Force, or “Rocking” Ratchet

39

Fig. 3.15 Flashing ratchet with τ = 0.4: v vs V0 , with F = 0 and D fixed. We observe the curve has a minimum, product of a competition between the barrier height, V0 , and thermal fluctuations, D, with the influence of the dichotomous noise. This computation was performed with Program 3.4. with the potential tilted to the right

Fig. 3.16 The tilting ratchet: the activation energy toward the right when the ratchet is tilted to the right is lower than the activation energy toward the left when the ratchet is tilted to the left

40

3 Biased Brownian Motion

  D D D x tn+1 = x (tn ) − V (x (tn )) t + η (tn ) FDN t + Fload t + (2Dt)1/2 rand n kB T kB T 2kB T (3.40)

with V (x (tn )) = ±





 2π x (tn ) 1 4π x (tn ) Vo cos − cos L L 2 L

(3.41)

where Fload is the load force and FDN = |±F | is the dichotomous noise force. In dimensionless units the former equations transform (Figs. 3.17, 3.18, and 3.19):            F  t˜ 1/2 rand n V0 V x˜ tn t˜ + η tn D F x˜ t n+1 = x˜ t n −D DN t˜ + D load t˜ + 2D

(3.42)

with,

         1 V x˜  tn = ± cos 2π x˜  tn − cos 4π x˜ t˜n 2

(3.43)

Fig. 3.17 Rocking Ratchet with τ = 0.4: v vs Fload . We can observe the motor effect: For values negatives of the load Force [−0.775, 0], the corresponding velocity is positive, meaning the motor obtain the energy from its internal mechanism. This computation was performed with Program 3.5. We have used Eqs. 3.42  and 3.43 with the potential tilted to  the right, namely: 1

(x) = cos (2π x) sin (2π x) ˜ − 14 sin (4π x) V ˜ − 12 cos (4π x) ˜ , V ˜ (x) = 2π

3.6 Fluctuating Force, or “Rocking” Ratchet

41

Fig. 3.18 Rocking Ratchet with τ = 0.4: Analogously to the Flashing ratchet, for v vs D, with F = 0 and V0 fixed, the curve has a maximum and a minimum. This computation was performed with Program 3.5, We have used Eqs. 3.42 and 3.43 with the potential tilted to the right

Fig. 3.19 Rocking Ratchet with τ = 0.4: Analogously to the Flashing ratchet, for D constant e F = 0. It exists a minimum in the curve v vs V0 . This figure was generated by Program 3.5. We have used Eqs. 3.42 and 3.43 with the potential tilted to the right

42

3 Biased Brownian Motion

3.7 Programs 3.7.1 Program 3.1, Euler Equation, Matlab Code %Solve the Euler Equation N=20000; NR=1000; dt=0.01; D=0.0005; Vo=1.; F=0.; x00=0.; raizdt=sqrt(2*D*dt); dw=raizdt*randn(N,NR); x=zeros(N-1,NR); V=zeros(N-1,NR); dV=zeros(N-1,NR); for j=1:NR x(1,j)=x00; for i=1:N-1 xij=x(i,j); %V(i,j)=(Vo/(2*pi))*(sin(2*pi*xij/L)- 0.25*sin(4*pi*xij/L)); %dV(i,j)=(Vo/(L))*(cos(2*pi*xij/L)- 0.5*cos(4*pi*xij/L)); %dV is the derivative of V, “i” is the time and “j” is the stochastic realization dV(i,j)=(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij)); %dimensionless x(i+1,j)=xij - Vo*dV(i,j)*dt + D*F*dt + dw(i,j);%dimensionless end end MEANXR=(sum(x,2))/NR;%Mean over the stochastic realizations. t=((1:N)*dt)’; figure(1); plot(t,MEANXR); grid; save t.dat t -ascii; save MEANXR.DAT MEANXR -ascii;

3.7.2 Program 3.2, F-P Equation, Matlab Code %Solution of the Dimensionless Fokker-Planck Equation by Crank-Nicholson scheme %V(x)=(1/2pi)[sin(2pix) - 0.25sin(4pix)], Potential slightly tilted to the right %Intensity s0, Central Position x0, Standard deviation sigma, %Diffusion coefficient D %V (x) = d1, V "(x) = d2, x0=0.5; N=100; tmax=100; dt=0.01; dx=0.01; D=0.0005; F=6.; Vo=1; sigma=0.1 k=0.5*D*dt/dx 2 ; s2=1/(2*(sigma 2 )); s0=1./(sigma*sqrt(2*pi)); T=floor(tmax/dt)+1; %T is the number of temporal points x=((1:N)*dx)’; % spatial grade, column vector p=s0*exp(-((x − x0).2 )*s2);%Initial Probability p1=p; E=zeros(N); M=zeros(N); pt=ceil(T/8);P=zeros(N,8); for j=1:N;

3.7 Programs

d2=-Vo*(2*pi)*(-sin(2*pi*j*dx) + sin(4*pi*j*dx));%dimensionless derivative a=1-k*(d2*(dx 2 )-2.); b=2-a; E(j,j)=a; M(j,j)=b; end for j=2:L; d1=-Vo*(cos(2.*pi*j*dx) - 0.5*cos(4.*pi*j*dx))-F;%dimensionless derivative c=k*(1.+ 0.5*dx*d1); d=2.*k - c; E(j,j-1)=-d; E(j-1,j)=-c; M(j,j-1)=d; M(j-1,j)=c; end IE=inv(E); for tp=1:8 for it=1:pt p=IE*(M*p); P(:,tp)=p; %distribution at times tp end end figure(1); plot(x,P(:,1),x,P(:,2),x,P(:,3),x,P(:,4), . . . x,P(:,5),x,P(:,6),x,P(:,7),x,P(:,8)) F1=P(:,1) F2=P(:,2) F3=P(:,3) F4=P(:,4) F5=P(:,5) F6=P(:,6) F7=P(:,7) F8=P(:,8) F9=p1’ save F1.dat F1 -ascii; save F2.dat F2 -ascii; save F3.dat F3 -ascii; save F4.dat F4 -ascii; save F5.dat F5 -ascii; save F6.dat F6 -ascii; save F7.dat F7 -ascii; save F8.dat F8 -ascii; save F9.dat F9 -ascii; save x.dat x -ascii;

43

44

3 Biased Brownian Motion

3.7.3 Program 3.3, Dichotomous Noise, Matlab Code % Dichotomous noise: To visualize the pattern make N=100, NR=1, to verify = 0,leave as it is. N=100; NR=1; dt=0.1; tau=0.5, a=-1; b=1. %tau=1/(2*mu); t=(1:N)*dt; eta=zeros(N,NR); for j=1:NR; for i=1:N-1; Pat1at = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (-1) Pat2at1 = Pat1at; Pat1bt = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (1) to (-1) Pat2bt1 = Pat1bt; Pbt2bt1 = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (1) Pbt1bt = Pbt2bt1; Pbt1at = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (-1) to (1) Pbt2at1 = Pbt1at; R=unifrnd(0,1); if(Pat1at  R) eta(i+1,j)=-1.; else eta(i+1,j)=1.; end if(Pat1at ≺ R || Pat1at  R) R1=unifrnd(0,1); elseif(Pat2bt1  R1 || Pat2at1  R1) eta(i+2,j)=-1.; else eta(i+2,j)=1.; end end end save t.DAT t -ascii; MEANeta=(sum(eta,2))/NR; % stochastic mean, over NR realizations. figure(1); plot(MEANeta); grid; figure(2); plot(t,eta); grid; ylim([-1.1 1.1]); grid %save MEANeta.dat Meaneta -ascii;

3.7.4 Program 3.4, Flashing Ratchet, Matlab Code % Flashing ratchet N=10000; NR=1000; dt=0.01; F=0.; Vo=5.; D=5.; tau=0.4; eta=zeros(N,NR);

3.7 Programs

45

DMN=zeros(N,NR); raizdt=sqrt(2*D*dt); x=zeros(N-1,NR); V=zeros(N-1,NR); dV=zeros(N-1,NR); dw=raizdt*randn(N,NR); for j=1:NR for i=1:N-1 Pat1at = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (-1) Pat2at1 = Pat1at; Pat1bt = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (1) to (-1) Pat2bt1 = Pat1bt; Pbt2bt1 = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (1) Pbt1bt = Pbt2bt1; Pbt1at = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (-1) to (1) Pbt2at1 = Pbt1at; R=unifrnd(0,1); if(Pat1at  R) eta(i+1,j)=-1.; else eta(i+1,j)=1.; end if(Pat1at ≺ R || Pat1at  R) R1=unifrnd(0,1); elseif(Pat2bt1  R1 || Pat2at1  R1) eta(i+2,j)=-1.; else eta(i+2,j)=1.; end end end for j=1:NR; for i=1:N-1; xij=x(i,j); dV(i,j)=(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij)); %DMN(i,j) = 0.5*(eta(i,j)+1.); x(i+1,j)=xij - 0.5*(eta(i,j)+1.)*D*Vo*dV(i,j)*dt + D*F*dt+ dw(i,j); %x(i+1,j)=xij - D*Vo*dV(i,j)*dt + D*F*dt+ dw(i,j);% DMN=1. end end MEANXR=(sum(x,2))/NR; %MEANdw=(sum(dw,2))/NR; %MEANDMN=(sum(DMN,2))/NR; %MEANeta=(sum(eta,2))/NR; t=(1:N)*dt; figure(1); plot(t,MEANXR); grid; save MEANXR.DAT MEANXR -ascii;

46

3 Biased Brownian Motion

%figure(2); plot(t,MEANeta); grid; %figure(3); plot(t,MEANDMN); grid; %figure(4); plot(t,MEANdw); grid;

3.7.5 Program 3.5, Rocking Ratchet, Matlab Code %Rocking ratchets. N=10000; NR=1000; dt=0.01; Fload=0.0; Vo=8.; tau=0.4; D=5.; FDN=1.; eta=zeros(N,NR); raizdt=sqrt(2*D*dt); x=zeros(N-1,NR); V=zeros(N-1,NR); dV=zeros(N-1,NR); dw=raizdt*randn(N,NR); for j=1:NR for i=1:N-1 Pat1at = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (-1) Pat2at1 = Pat1at; Pat1bt = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (1) to (-1) Pat2bt1 = Pat1bt; Pbt2bt1 = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in (1) Pbt1bt = Pbt2bt1; Pbt1at = 0.5*(1. - exp(-i*dt/tau)); % Transition Probability (-1) to (1) Pbt2at1 = Pbt1at; R=unifrnd(0,1); if(Pat1at  R) eta(i+1,j)=-1.; else eta(i+1,j)=1.; end if(Pat1at ≺ R || Pat1at  R ) R1=unifrnd(0,1); elseif(Pat2bt1  R1 || Pat2at1  R1) eta(i+2,j)=-1.; else eta(i+2,j)=1.; end end end for j=1:NR; for i=1:N-1; xij=x(i,j); dV(i,j)=(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij));%tilted to the right

Bibliography

47

x(i+1,j)=xij - Vo*dV(i,j)*dt+ eta(i,j)*D*FDN*dt + . . . D*Fload*dt + dw(i,j); end end MEANeta=(sum(eta,2))/NR; MEANXR=(sum(x,2))/NR; t=(1:N)*dt; figure(1); plot(t,MEANXR); grid; %figure(2); plot(t,MEANeta); grid; save MEANXR.DAT MEANXR -ascii;

Bibliography 1. Derényi, I., Astumian, R.D.: Brownian ratchets and their application to biological transport processes and macromolecular separation. In: Kasianowicz, J.J., Kellermayer, M.S.Z., Deamer, D.W. (eds.) Structure and Dynamics of Confined Polymers. NATO Science Series. Series 3: High Technology, vol. 87. Springer, Dordrecht (2002). https://doi.org/10.1007/978-94-0100401-5-17 2. Cubero, D., Renzoni, F.: Brownian Ratchets: From Statistical Physics to Bio and Nano-motors. Cambridge University Press. ISBN 978-1-107-06352-5 (2016) 3. Bug, A.L.R., Berne, B.J.: Shaking-induced transition to a nonequilibrium state. Phys. Rev. Lett. 59, 948 (1987) 4. Ajdari, A.J., Prost, J.: Mouvement induit par un potentiel periodique de basse symetrie: Dielectrophorese pulsee. J. Comptes Rendus de l’Académie des Sciences, Paris 315, 1635–1639 (1992) 5. Magnasco, M.O.: Forced thermal ratchets. Phys. Rev. Lett. 71(10), 1477–1481. https://doi.org/ 10.1103/PhysRevLett.71.1477 (1993) 6. Astumian, R.D., Bier, M.: Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett. 72(11), 1766–1769 (1994) 7. Astumian, R.D.: Thermodynamics and kinetics of a Brownian motor. Science 276, 917–922 (1997) 8. Astumian, R.D., Derényi, I.: Fluctuation driven transport and models of molecular motors and pumps. Eur. Biophys. J. 27, 474–489 (1998) 9. Barik, D., Ghosh, P.K., Ray, D.S.: Langevin dynamics with dichotomous noise; direct simulation and applications. J. Stat. Mech. (2006). Online at stacks.iop.org/JSTAT/2006/P03010 10. Bena, I.: Dichotomous Markov noise: exact results for out-of-equilibrium system. Int. J. Mod. Phys. B 20, 2825 (2006) 11. Bader, J.S., Hammond, R.W., Henck, S.A., Deem, M.W., McDermott, G.A., Bustillo, J.M., Simpson, J.W., Mulhern, G.T., Rothberg, J.M.: DNA transport by a micromachined Brownian ratchet device. Proc. Natl. Acad. Sci. U. S. A. 96, 13165 (1999) 12. Doering, C.R.: Randomly rattled ratchets. IL Nuovo Cimento D. 17(7–8), 685–697 (1995) 13. Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361(2–4), 57–265 (2002) 14. Mogilner, A., Elston, T., Wang, H., Oster, G.: Molecular motors: examples. In: Fall, C., Marland, E., Tyson, J., Wagner, J. (eds.) Joel Keizer’s Computational Cell Biology, Chapter 13. A motor driven by a “flashing potential”, vol. 13.2, p. 361 (2002). https://doi.org/10.1016/ S0370-1573(01)00081-3

Chapter 4

The Smoluchowski Model

4.1 Diffusion If a given sample consists of particles or molecules in random motion, then one would expect a net flux from a region of higher concentration to one of lower concentration. The question has been approached from the empirical side leading to a relation known as Fick’s first law. Which states that the flow per unit area J [#/(nm2 .s)], across a surface yz plane is given by1 J = −D

∂C(r, t) ∂x

(4.1)

ˆ + jˆy + kz, ˆ the quantity ∂C(r,t) is the concentration gradient in the x where r = ix ∂x direction and the flow is in fact a vector quantity. Thus Fick’s law when extended to three dimensions becomes 

∂C(r, t) ˆ ∂C(r, t) ∂C(r, t) + jˆ +k (4.2) J = −D iˆ ∂x ∂y ∂z ˆ jˆ, and kˆ are unit vector along the x, y, and z axes. The constant D is known where i, as the diffusion coefficient. The minus sign expresses the fact that flow is from a higher to lower concentration. From now on we consider flow in one dimension, so we omit the vectorial notation. We will use Eq. 4.3 another form of the diffusion equation more useful for solving certain problems. Consider a slab of thickness dx perpendicular to the x axis (see Fig. 4.1). Consider an area A of this slab. The diffusion flow into the slab is given from Eq. 4.3 as −DA ∂C ∂x . The diffusion flow across the corresponding face on the other side of the slab is given by

1 Where

# means the number of particles.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_4

49

50

4 The Smoluchowski Model

Fig. 4.1 Diffusion through a thin slab of thickness dx

Y dx

–D

∂c ∂x

–D

∂c ∂x

+

∂2c dx ∂x2

X

Z

f low = −DA

∂C(x + dx, t) ∂C(x, t) ∂ 2 C (x) dx = −DA + ∂x ∂x ∂x 2

(4.3)

The quantity we wish to determine  net rate of change of the number of  is the Adx, which is given by moles of the diffusing component. ∂C(x,t) ∂t −DA



∂C(x, t) ∂ 2 C (x) ∂C(x + dx, t) ∂ 2 C (x) +DA + dx = DA dx ∂x ∂x ∂x 2 ∂x 2

(4.4)

Consequently

 ∂C(x, t) ∂ 2 C (x) Adx = DA dx ∂t ∂x 2

(4.5)

Dividing through by Adx yields

∂C(x, t) ∂t

 =D

∂ 2 C (x) ∂x 2

(4.6)

The Eq. 4.6 can be also written

∂C(x, t) ∂t





∂Jx =− ∂x

 .

[conservation of particles]

(4.7)

The former equation means the conservation of the number of particles. On the other hand the external field exerts a force on each particle, F = − ∂V∂x(x) , which would produce a drift velocity proportional to the field: v = Fξ . Thus the net motion will be the sum of the Brownian diffusion and the field-driven drift: where ξ = namely

kB T D

,

4.2 Chemical Kinetics

51 Drif t velocity

Jx =

∂C −D ∂x %&'( Diff usion f lux

'(%&  



∂ (V /kB T ) D ∂V ∂C C = −D + C − kB T ∂x ∂x ∂x %&'(

(4.8)

Drif t f lux

At equilibrium Jx = 0, then ∂Ceq D ∂V =− Ceq ∂x kB T ∂x

(4.9)

which leads to the Boltzmann distribution Ceq (x) = C0 exp (−V (x)/kB T ) .

[Boltzmann distribution]

(4.10)

Rather of considering a cloud of particles, we can think in terms of the probability of finding a single particle at (x, t). In doing this we have to normalize the concentraL tion in Eq. 4.9 by dividing by the total population p(x, t) ≡ c(x, t)/( 0 c(x, t)dx). Inserting Eq. 4.9 expressed in terms of p(x, t) into the conservation law Eq. 4.7 yields the Smoluchowski equation ⎡

⎤

⎢ ⎥

 ⎢ ∂ ∂ (V /kB T ) ∂p(x, t) ∂ 2 p (x, t) ⎥ ⎥. = D⎢ p + ⎢ ∂x ⎥ 2 ∂t ∂x ∂x ⎣%&'( %&'(⎦ Drif t

[Smoluchowski Equation]

Diff usion

(4.11)

Considering the interval 0 ≤ x ≤ L. The variables x, t are normalized in accordance to Eqs. (3.2) and (3.3), Then Eq. 4.11 can be written as ∂p(x, ˜ t˜) ∂ =2 ∂ x˜ ∂ t˜

 

 ∂ 2 p x, ˜ t˜ ∂V p + ∂ x˜ ∂ x˜ 2

(4.12)

where t˜ and x˜ are now dimensionless, and the potential V , is measured in units of kB T . Depending of the system, in equation 4.12 has to specify the boundary conditions, namely, the value of p(x˜ = 0, t˜), p(x˜ = 1, t˜), and p(x, ˜ t˜ = 0), where p(x, ˜ t˜) is defined on the x interval [0, 1].

4.2 Chemical Kinetics To understand Brownian motors, we have to consider chemical reactions, which supply the energy to drive Brownian motors. We will here take up the problem of the rates at which chemical reactions take place. We start considering the simplest reaction

52

4 The Smoluchowski Model

AB A+C  D

(4.13)

A+C  E+F The rates of the reactions can be written, as d[A] = −k1 [A] + k2 [B] dt d[A] = −k3 [A][C] + k4 [D] dt d[A] = −k5 [A][C] + k6 [E][F ] dt

(4.14)

The brackets indicate concentrations in the classical version of the theory, although the problem may be recast into activities. The k s are called rate parameters or rate constants; that is, they do not depend explicitly on the concentrations, and have the dimensions of (time)−1 . If in the second of Eqs. 3.13, we make A− ≡ A, H + ≡ C, D ≡ [H.A] ≡ A0 then we get A− + H +  A0

(4.15)

The former Eq. 4.15 resembles nucleotide hydrolisis which is one of the most common energy supply to drive molecular motors. If we center only in the amino acid (nucleotide), we see it exists in two states: charged (A− ) and neutral (H.A ≡ A0 ). Then Eq. 4.15 from the view point of the amino acid is simply kf∗ A−  A0

(4.16)

kr Accordingly the second Eq. 4.15 with k3 = kf and k4 = kr1 , “f ” of forward and “r” of reverse. d[A− ] = −kf [H + ][A− ] + kr [A0 ] = dt − kf∗ [A− ] + kr [A0 ]

(4.17)

where kf∗ = kf [H + ] is called a pseudo-first order rate constant. Returning to the rate constants, k’s, in general they are functions of the temperature and other environmental variables. Such purely experimental variables usually have an exponential dependence on temperature. Arrhenius in 1889 formulated the following equation (empirical generalization).

4.2 Chemical Kinetics

53

k = Ae−E/RT

(4.18)

A is the frequency factor and E is known as the energy of activation, meaning that there is an activation barrier to the reaction, A measures the collision rate and e−E/RT is the probability that a collision will be sufficiently energetic to get over the barrier. The ideas considered in Eq. 4.18 have been extended in the theory of absolute reaction rates.

4.2.1 Absolute Reaction Rate Theory The first concept that must be considered is that of the reaction coordinate, we will denote by ξ . For the reaction Eq. 4.15, ξ(t) is the distance between the ion (H + ) and the amino acid charge, A− . The spatial scale of this coordinate is much smaller (i.e. angstroms) than the spatial scale of the motor’s motion. In Fig. 4.3 is plotted the energy diagram corresponding to Eq. 4.16, at the top of the potential barrier along this coordinate is the activated complex. We represent this complex by a shallow potential. The width of the well is δ and its height may be represented as about 12 kB T , the mean thermal energy per degree of freedom. The complex may be visualized as a harmonic oscillator with the reaction coordinate being a degree of vibrational freedom. The complex can be assumed to have an effective reduced mass m with respect to the vibrational and a force constant of κ (Fig. 4.2). Following with the amino acid hydrolis we write the reaction as Reactants  Activated Complex → Products A− + H +  M ‡ → P roducts

(4.19)

The rate of the forward reaction will be given by Rate = (Concentration of activated complexes) × (Rate of crossing the barrier)

(4.20)

= C ‡ × (Rate of crossing) We use the double dagger notation (‡) to indicate parameters associated with the activated complex. We assume a partial equilibrium; the reactants are in equilibrium with the activated complex, namely   G‡1 C‡ ‡ = K = exp − [A− ][H + ] RT

(4.21)

54

4 The Smoluchowski Model

Fig. 4.2 Model of a chemical reaction indicating that the activated complex is a very shallow energy well ∼ 12 kB T at the top of the barrier

where G‡f is the Gibbs free-energy difference between the activated complex and the reactants. To obtain the rate of crossing we divide the mean velocity of crossing by the distance to be traversed. We assume a crossing will involve a single oscillation so that the distance involved is the order of 2δ, the length transversed in the time of a single vibrational period T , Rate of barrier crossing =

v 2δ

(4.22)

The period of a classical harmonic oscillator is given by  T = 2π

m κ

(4.23)

The ground state energy of an oscillator from quantum theory is h 0 = 4π



h κ = m 2T

(4.24)

where h is the Planck’s constant. The right-hand equality comes from introducing Eq. 4.23 into Eq. 4.24. Noting our assumption that the depth of the well is ∼ 12 kB T we get

4.2 Chemical Kinetics

55

2( 1 kB T ) 20 kB T 1 = = 2 = T h h h

(4.25)

Comparison with Eq. 4.22 shows that there is a barrier crossing rate independent of δ, κ, and m, and as is seen from Eq. 4.25 is dependent only on the barrier height. Since we assume the same barrier height ( 12 kB T ) for the shallow well at the top of barriers for all reactions. the crossing rate is generalized to a universal function of temperature alone. If we now go back to Eq. 4.21 and substitute Eqs. 4.21 and 4.25 we get

 kB T G‡ Rate = exp − [A− ][H + ] h RT

(4.26)

From a consideration of Eq. 4.17, we get Rate = kphenomenological [A− ][H + ]

(4.27)

Hence kphenomenological =

 G‡ kB T exp − h RT

(4.28)

This result is the fundamental expression of the theory of absolute reaction rates. Returning to Eq. 4.17, at equilibrium the forward and reverse reactions occur at equal rates, principle of detailed balance, as a consequence kf∗ [A− ]eq = kr [A0 ]eq K=

(4.29)

kf∗ (kB T / h)exp(−G‡f /RT ) [A0 ]eq = = [A− ]eq kr (kB T / h)exp(−G‡r /RT )

(4.30)

The first equality in Eq. 4.30 comes from the definition of the equilibrium constant, the second comes from Eq. 4.29 (detailed balance), and the third from Eq. 4.17. By reference to Fig. 4.3, we note that we can rewrite Eq. 4.30 as  K = exp

−G‡f + G‡r RT



= exp −

Gproducts − Greactants RT

 (4.31)

This result, is identical to the thermodynamics, which shows the consistency of this kinetic approach. Following Mogilner et al. [1], we treat reactions by the 2-state model, whose equations of motion are:

56

4 The Smoluchowski Model

d[A− ] d[A0 ] =− = net flow over the energy barrier = Jξ = kf∗ [A− ] − kr [A0 ] dt dt (4.32) where we have used Eq. 4.17. The former Eq. 4.32 in the vector form: d P = Jξ = K.P, dt

P=

p− p0



 K=

,

kf∗ −kr −kf∗ kr

 (4.33)

where p− and p0 are the probabilities to have a negatively charged and neutral amino acid, respectively.

4.3 A Mechanochemical Model For a Brownian motor the potential V (x, t) in Eq. 4.11 has to be broken into two parts: V (x, t) =

VI (x, t) %&'( Internally generated forces

+

VL (x, t) %&'(

(4.34)

External load forces

Each chemical state is characterized by its own probability distribution pk (x, t), where k ranges over all the chemical states, and each chemical state is characterized by a separate driving potential, Vk (x, t) Thus there will be a Smoluchowski equation 4.11 for each chemical state, and these equations must be solved simultaneously to obtain the motor’s motion. For the amino acid hydrolisis Eq. 4.16, the total change in probability, p(x, ξ, t), is given by ∂ ∂t



p1 p2

 = Net flow in x + Net flow in ξ '(%& '(%&



 Jξ1 (∂/∂x1 ) Jx1 =− + Jξ2 (∂/∂x2 ) Jx2 



−k12 p1 +k21 p2 ∂/∂x1 [p1 ∂ (V1 /kB T )/∂x1 + ∂p1 /∂x1 ] + =D ∂/∂x2 [p2 ∂ (V2 /kB T )/∂x2 + ∂p2 /∂x2 ] −k21 p2 +k12 p1 (4.35)

where p− ≡ p1 and p0 ≡ p2 . Also k12 ≡ kf∗ and k21 ≡ kr∗ . Observe that Jξ1 = −Jξ2 . We can visualize the mechanochemical coupling in Fig. 4.3

4.3 A Mechanochemical Model

57

Fig. 4.3 The mechanochemical phase plane. A point is defined by its spatial and reaction coordinates (x(t), ξ(t)). (Reprinted Figure with permission, [1], pag.339, Copyright by Elsevier (2005))

At steady state, the total flux in the spatial dimension J = −D (p1 ∂ (V1 /kB T )/∂x1 + ∂p1 /∂x1 ) − D (p2 ∂ (V2 /kB T )/∂x2 + ∂p2 /∂x2 ) = j1 + j2

(4.36)

is a constant independent of x. Equilibrium requires that j1 and j2 are identically zero and that k12 p1 − k21 p2 is identically zero. In this case, the probability densities are proportional to the Boltzmann distributions pi (x, t) ∝ exp−

Vi (x) kB T

(4.37)

which forces the following constraint on the rates k12 (x) V1 (x) − V2 (x) ∝ exp k21 (x) kB T

(4.38)

If Eq. 4.38 is not obeyed, then in general the system will experience a net flux (i.e., J  0). These types of systems have been referred to generically as “flashing” ratchets. As an example of a model for motor protein moving along a polymer we consider the two states of the nucleotide-binding site being occupied or empty. The mechanical forces that the motor experiences are assumed to arise from the potentials V1 and V2 . This example was already treated, using another formalism,

58

4 The Smoluchowski Model

by Wang et al. [2]. Following this reference we consider the potentials: V1 (x) =







 2π 1 4π 1 6π 2A sin x − sin x + sin x π L 2 L 3 L

V2 (x) = 0

(4.39) (4.40)

This model resembles kinesin movement along the microtubule, see Chap. 1. State 2 corresponds (V2 = 0), meaning that dynein is able to diffuse along the microtubule. State “1” corresponds with dynein heads being empty or occupied with ATP. The potential V1 is periodic but spatially asymmetric. L = 8 nm correspond with the repeat length of the microtubule. we approximate at t = 10−4 s the system has reached the steady state. Because Eq. 4.38 is not satisfied the steady state deviates from equilibrium and produces a steady state flux.

4.3.1 Numerical Computation of Mechanochemical Coupling In order to solve Eq. 4.35 we follow the Crank-Nicolson method for solving the Fokker-Planck of Chap. 2. Then applying to the system Eq. 4.35 n+1 n n n n a1 Pjn+1 (1) − c1 Pjn+1 +1 (1) − d1 Pj −1 (1) = (b1 − k12 dt)Pj (1) + c1 Pj +1 (1) + d1 Pj −1 (1) + k21 dtPj (2) n+1 n n n n a2 Pjn+1 (2) − c2 Pjn+1 +1 (2) − d2 Pj −1 (2) = (b2 − k12 dt)Pj (2) + c2 Pj +1 (2) + d2 Pj −1 (2) + k21 dtPj (2)

(4.41) Expanding the system, a1 P2n+1 (1) − c1 P3n+1 (1) − d1 P1n+1 (1)

=

(b1 − k12 dt)P2n (1) + c1 P3n (1) + d1 P1n (1) + k21 dtP2n (2)

a1 P3n+1 (1) − c1 P4n+1 (1) − d1 P2n+1 (1)

=

(b1 − k12 dt)P3n (1) + c1 P4n (1) + d1 P2n (1) + k21 dtP3n (2)

a1 P4n+1 (1) − c1 P5n+1 (1) − d1 P3n+1 (1)

=

(b1 − k12 dt)P4n (1) + c1 P5n (1) + d1 P3n (1) + k21 dtP4n (2)

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

(4.42)

a2 P2n+1 (2) − c2 P3n+1 (2) − d2 P1n+1 (2)

=

(b2 − k21 dt)P2n (2) + c2 P3n (2) + d2 P1n (2) + k12 dtP2n (1)

a2 P3n+1 (2) − c2 P4n+1 (2) − d2 P2n+1 (2)

=

(b2 − k21 dt)P3n (2) + c2 P4n (1) + d2 P2n (2) + k12 dtP3n (1)

=

(b2 − k21 dt)P4n (2) + c2 P5n (2) + d2 P3n (2) + k12 dtP4n (1)

a2 P4n+1 (2) − c2 P5n+1 (2) − d2 P3n+1 (2)

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

In order to solve this system of equations, following the method of Chap. 2, we have to define the following matrixes:

4.3 A Mechanochemical Model ⎛

−c1 0 0 a1 −c1 0 −d1 a1 −c1 ... ... 0 0 0 ...

59 ⎞ ... 0 ... 0 ⎟ ⎟ ... 0 ⎟ ⎟, ⎟ ⎟ ⎠

a1 ⎜ −d1 ⎜ ⎜ 0 E1 = ⎜ ⎜ ⎜ ⎝

⎛

c1 b1 c1 d1 b1 c1 ... ... 0 0 0 ...

b1 ⎜ d1 ⎜ ⎜0 L1 = ⎜ ⎜ ⎜ ⎝

⎛

K12

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝



−c2 a2 −c2 −d2 a2 −c2 ... ... 0 0 0 ...

a2 ⎜ −d2 ⎜ ⎜ 0 E2 = ⎜ ⎜ ⎜ ⎝

−d1 a1

⎞ ... 0 ... 0 ⎟ ⎟ ... 0 ⎟ ⎟, ⎟ ⎟ ⎠

⎛

0 0 0

... ... ...

⎛

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

K21

0 k12 dt

E1 0 0 E2



,

c2 b2 c2 d2 b2 c2 ... ... 0 0 0 ...

b2 ⎜ d2 ⎜ ⎜0 L2 = ⎜ ⎜ ⎜ ⎝

d1 b1

k12 dt 0 0 k12 dt 0 0 0 k12 dt 0 ... ... 0 0 0 ...

E=

⎛

M=

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

⎞ ... 0 ... 0 ⎟ ⎟ ... 0 ⎟ ⎟ ⎟ ⎟ ⎠ −d2 a2

⎞ ... 0 ... 0 ⎟ ⎟ ... 0 ⎟ ⎟, ⎟ ⎟ ⎠ d2 b2

k21 dt 0 0 k21 dt 0 0 0 k21 dt 0 ... ... 0 0 0 ...

L1 − K12 K21 K12 L2 − K21

... ... ...

0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 k21 dt

 (4.43)

n = 0, the system Eq. 4.41 can be written Using the boundary conditions P0n = P2N in matrix form as (Fig. 4.4):

E.Pn+1 = M.Pn ,

  Pn+1 = E−1 . M.Pn

⎛

⎞ P1n (1) ⎜ P n (1) ⎟ ⎜ 2 ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ n ⎟ (1) P ⎜ ⎟ Pn = ⎜ Nn ⎟ ⎜ P1 (2) ⎟ ⎜ n ⎟ ⎜ P2 (2) ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ . ⎠ n (2) P2N

(4.44)

60

4 The Smoluchowski Model

Fig. 4.4 Mechanochemical coupling: The diffusion coefficient of the dynein along the micros ∗ tubule and in solution are Ddynein = 9 × 103 nm2 /s, and Ddynein ≈ 107 nm2 /s respectively k12 = k21 = 8, 400 s−1 . Details in Program 4.1

4.4 Program 4.1, Matlab Code %Solution of the system of Fokker-Planck Equations with mechanochemical %chemical coupling by the Crank-Nicholson scheme. %V1(x)=(2A/pi)[sin(2pi*x/L)- 0.5sin(4pi*x/L) + (1/3)sin(6pi*x/L). %V2(x)= 0. %d21 = 2nd derivative of the potential at the state one. %d22 = 2nd derivative of the potential at the state two. %Intensity s0, Central Position x0, Standard deviation sigma, %Thermal energy kBT, A=10kBT, (1 ATP 20kBT per molecule of 2 heads) %Vo = A/kBT dimensionless Vo. %L is the motor step per reaction cycle, 8nm. %D∗ = 9x103 nm2 /s dynein diffusion coefficient along the microtubule. %Do 107 nm2 /s dynein diffusion coefficient in solution. %D = D ∗ /Do = 9x10− 4 dimensionless diffusion coefficient. %tauD = L2 /Do = 6.4x10− 6s. %k12∗ = k21∗ = 84001/s transition rates. %k12 = k21 = 5.376x10− 2 dimensionless transitions rates. %dt∗ = 7.1x10− 7s from Wang et al. (2003). %dt = dt ∗ /tauD = 0.111 dimensionless dt.

4.4 Program 4.1, Matlab Code

61

%dx∗ = 8nm/(2N ) = 0.04nm. %dx = dx ∗ /L = 0.04nm/8nm = 5x10− 3 dimensionless dx. x0=20.; N=100; tmax=100; dt=0.111; dx=5.e-3; D=9.e-4; Force=0.; Vo=10.; sigma=4.; ka = (1/2) ∗ D ∗ (dt/dx 2 ); ka1 = V o ∗ dx 2 ; ka2=Vo*dx/2; k12dt=6.e-3; k21dt=6.e-3; k12=5.376e-2; k21=5.376e-2; s2 = 1/(2 ∗ (sigma 2 )); s0=1./(sigma*sqrt(2*pi)); T=floor(tmax/dt)+1; % T é o num. de pontos temporais x=((1:2*N)’)*dx*8; f = s0 ∗ exp(−((x − x0).2 ) ∗ s2); p1=f; F=zeros(2*N,9); E=zeros(2*N);E1=zeros(2*N); E2=zeros(2*N); M=zeros(2*N);M1=zeros(2*N); M1=zeros(2*N);K12=zeros(2*N); K21=zeros(2*N); L1=zeros(2*N); L2=zeros(2*N); KR21=zeros(2*N);KL12=zeros(2*N); %STATE 1 for j=1:N d21=(8*pi)*(-sin(2*pi*j*dx)+2*sin(4*pi*j*dx)-. . . 3*sin(6*pi*j*dx)); a1=1-ka*(-2 + ka1*d21); b1=2-a1; K12(j,j)=k12*dt; E1(j,j)=a1; L1(j,j)=b1; end %STATE 2 for j=N+1:2*N d22= 0.; a2=1-ka*(-2 + ka1*d22); b2=2-a2; K21(j,j)=k21*dt; E2(j,j)=a2; L2(j,j)=b2; end %STATE 1, DIAGONAL NEIGBOURGH for j=2:N d11=4.*(cos(2.*pi*j*dx)-cos(4.*pi*j*dx)+cos(6.*pi*j*dx))-Force; c1=ka*(1.+ka2*d11); d1=ka*(1.-ka2*d11);

62

4 The Smoluchowski Model

E1(j,j-1)=-d1; E1(j-1,j)=-c1; L1(j,j-1)=d1; L1(j-1,j)=c1; end %STATE 2, DIAGONAL NEIGBOURGH for j=N+2:2*N d12=0.; c2=ka*(1.+ka2*d12); d2=ka*(1.-ka2*d12); E2(j,j-1)=-d2; E2(j-1,j)=-c2; L2(j,j-1)=d2; L2(j-1,j)=c2; end for j=N+1:2*N KR21(j-N,j)=k21*dt; KL12(j,j-N)=k12*dt; end E=E+E1+E2; M = M + L1-K12 + L2-K21 + KR21 +KL12; IE=inv(E); pt=ceil(T/8); for tp=5:5:25 for it=1:pt f=(IE)*(M*f); end F(:,tp)=f; % distribution at times tp end figure(1); plot(x,F(:,5),x,F(:,10),x,F(:,15),x,F(:,20)) F1=F(:,5); F2=F(:,10); F3=F(:,15); F4=F(:,20); save F1.dat F1 -ascii; save F2.dat F2 -ascii; save F3.dat F3 -ascii; save F4.dat F4 -ascii; save x.dat x -ascii; %============================================================= %Flux1 IN STATE 1 %============================================================= G4=zeros(2*N,1); Flux12=zeros(2*N-1,1); dV1=4.*(cos(2.*pi*x)-cos(4.*pi*x)+cos(6.*pi*x));

Bibliography

63

for i=1:N G4(i) = F4(i); Flux12(i)=-D*dV1(i)*G4(i); end Y4 = diff(G4)/dx; Flux11=-D*Y4; Flux1=(Flux11+Flux12); %figure(2); plot(x,dV1); figure(3); plot(x,G4); %figure(4); plot(x1,Y4); save F1.dat F1 -ascii; save F2.dat F3 -ascii; save F3.dat F4 -ascii; save F4.dat F4 -ascii; save F5.dat F5 -ascii; save x.dat x -ascii; %============================================================ %Flux2 IN STATE 2 %============================================================ H4=zeros(2*N,1); for i=N+1:2*N H4(i)=F4(i); end Z4 = diff(H4)/dx; dV2=0.; Flux21=-D*Z4; %Flux22=-D*(dV2*H6’); Flux22=0. Flux2 = Flux21; %============================================================ Totalflow = Flux1 + Flux2; TotalflowMean = sum(Totalflow)*dx; disp(TotalflowMean) %Flux mean %============================================================ %figure(5); plot(x,H4) %figure(6); plot(x1,Z4) figure(7); plot(x1,Totalflow)

Bibliography 1. Mogilner, A., Elston, T., Wang, H., Oster, G.: Molecular motors: examples. In: Fall, C., Marland, E., Tyson, J., Wagner, J. (eds.) Joel Keizer’s Computational Cell Biology, Chapter 12. Modeling Chemical Reactions vol. 12.3, p. 336 (2002). https://doi.org/10.1016/S03701573(01)00081-3 2. Wang, H., Peskin, C.S., Elston, T.C.: A robust numerical algorithm for studying biomolecular transport processes. J. Theor. Biol. 221, 491–511 (2003)

Chapter 5

Rotation of a Dipole

Here we consider a dipole in a viscous medium under the influence of an oscillating electric field and thermal noise, then we add an extra ratchet field. We analyze the properties of rotation of both cases. Because of the very low Reynolds numbers involved with molecular processes, we considered overdamped Langevin dynamics, as a consequence the inertia term becomes negligible. We observed rotation for some values of the parameters.

5.1 Introduction In recent years, with the advance of the nanotechnology, new techniques have evolve to visualize single-molecules, as scanning tunneling [1] and atomic force microscopies [2]. In Ref. [3] was observed rotation of a single molecule within a supramolecular bearing. After the first examples of unidirectional rotating molecular motors based on simple organic molecules [4, 5], several works were reported on this subject as [6–14]. Also great theoretical efforts have been devoted to understand and to propose nanoscale rotors and stators as molecular motors capable to transform efficiently a driven random rotation into a directed translational motion [15–17]. As a biological example of a molecular motor consisting of rotor and stator is the rotatory motor of bacterial flagella which is driven by a transmembrane electrochemical gradient of protons, [18], also is the ATP synthase which has been very well studied: [19–21]. Most of these motors are called Brownian motors, for an excellent review on this subject see Ref. [22] and references within. Although ‘brownian motors’ are known to permit thermally activated motion in one direction only, the concept of channelling random thermal energy into controlled motion has not yet been very extended to the molecular level and the basic principles are not very well understood. Experiments suggest that dynamics is inescapable and may play a decisive role in the evolution of nanotechnology, [23]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_5

65

66

5 Rotation of a Dipole

In Ref. [17], was shown that colloidal suspension of ferromagnetic nanoparticles, so-called ferrofluids, are ideal systems to test theoretical predictions on fluctuation driven transport experimentally. Here we consider one of the simplest rotors in nature: a dipole in a viscous medium under the influence of an oscillating electric field and thermal noise. By applying the Langevin equation we observe rotation for some values of the parameters.

5.2 Langevin Equation for the Rotor The energy of a dipole in an electric field is given by  = −pE(t) cos θ  E(t) WpE = −p.

(5.1)

where |p|  = q × d, q being the positive charge of the dipole and d is the distance between both charges, the vector p goes from the negative to the positive charge. In our case the field varies as E(t) = E0 sin (ωE t), with τ%E = 2π ωE−1 , then (Fig. 5.1) WpE = −pE0 sin (ωE t) cos θ

(5.2)

The corresponding torque on the dipole is E pE (θ, t) = −∂θ W% p = −pE(t) sin θ

(5.3)

pE (θ, t) = −pE0 sin (ωE t) sin θ

(5.4)

or

Fig. 5.1 Schematic of the system

5.2 Langevin Equation for the Rotor

67

Then the corresponding Langevin equation for the system is I

dθ d 2θ = pE (θ, t) − frot + B dt dt 2

(5.5)

As we are in the regime of null inertia (I = 0), we have: frot

dθ = pE (θ, t) + B dt

(5.6)

where frot [J.s] is the rotational frictional coefficient, and is related to the diffusion rotational constant, Drot [s−1 ] by the Einstein relation: Drot =

kB T frot

(5.7)

where kB is Boltzmann’s constant and B is the Brownian torque, given by, B = (2frot kB T )1/2 ξ (t)

(5.8)

where ξ (t) is the thermal noise, defined by its statistical properties, namely: ξ (t) = 0.

(5.9)

ξ (t2 ) ξ (t1 ) = δ (t2 − t1 )

(5.10)

i.e. the correlation time of the noise is zero. The corresponding discretization of Eq. 8.46 is performed by multiplying both members by dt, dividing by frot and performing the integration in the interval (t, t + t), namely θ =

1 frot



t+t t

 E 1/2 % p (θ, t) dt + (2Drot )

t+t

ξ (t) dt

(5.11)

t

or θ =

1 frot

pE (θ, t)t + (2Drot )1/2 W (t)

(5.12)

where pE (θ, t) is the mean value of pE (θ, t) in the considered interval, and the last term, we use the definition of the Wiener’s process, [24]. At the limit t → dt the mean value pE (θ, t) %  pE (θ, t), and W (t) = dW (t). We have to remind that the “Wiener’s increment ” %dW (t) is a Gaussian stochastic process, of width σ = (dt)1/2 . Then, at each pass of the integration we have to draw dW (t) and normalize the result properly. Let us call RG an aleatory number, with Gaussian distribution, centered in RG = 0 and width 1. In MATLAB/OCTAVE

68

5 Rotation of a Dipole

%RG = rand n, consequently we can write dW (t) = (dt)1/2 RG . Finally Eq. 5.12 is transformed in the corresponding Euler’s equation of this process, namely θj +1 = θj +

Drot E  (θj , t)t + (2Drot t)1/2 RG kB T p

(5.13)

Or using Eq. 5.3 θj +1

Drot = θj − kB T

/

∂WpE (θ, t) ∂θ

0 t + (2Drot t)1/2 RG

(5.14)

θj

Then in dimensionless units:  1/2   θ˜j +1 = θ˜j + D pE (0) pE (θ˜j , t˜)t˜ + 2D RG rot  rot t˜

(5.15)

where we have used, τB =

θ02 0 2Drot

θ = θ0 θ˜ ,

,

0  Drot = Drot Drot

t = τB t˜,

(5.16)

The use of the parameter θ0 will become evident in the next section. Also we have defined:  pE (0) =

pE0 kB T

(5.17)

       pE θ˜j , t˜ = sin ωE τB t˜ sin θ0 θ˜j

(5.18)

and $t, t = τB 

ωE t = ωE τB t˜,

θ = θ0 θ˜

(5.19)

A first basic quantity of interest is the average angular velocity in the long-time limit, i.e., after transients due to initial conditions have died out, namely  θ (t) ω∞  = θ˙ % ∞ ≡ lim t→∞ t

(5.20)

Another quantity of central interest will be the effective diffusion coefficient 

Deff

θ 2 (t) − θ (t)2 σ2 = lim ≡ lim t→∞ t→∞ 2t 2t

The means are over the realizations of the stochastic process.

(5.21)

5.3 Dipole in a Ratchet Electrical Potential

69

5.3 Dipole in a Ratchet Electrical Potential Consider the following potential acting on the dipole (Fig. 5.2):  



0 1 2π θ Vrat 4π θ − sin sin Vrat (θ ) = 2π θ0 4 θ0

(5.22)

The corresponding electric field modulus at the position of the positive charge is: Erat (θ ) = −



 

2π θ 1 4π θ dVrat (θ ) 0 cos − cos = Erat dθ θ0 2 θ0

(5.23)

V0

0 = − rat . The corresponding electric field modulus at the position of the where Erat θ0 negative charge is:

Erat (θ + π ) = −Erat (θ )

(5.24)

Then the corresponding force modulus at the position of the positive charge is: Frat (θ ) = qErat (θ )

(5.25)

and the corresponding force modulus at the position of the negative charge is: Frat (θ + π ) = −qErat (θ + π ) = qE(θ )

(5.26)

We observe they are the same. Then the total ratchet torque acting on the dipole is: pE(rat) (θ ) = 2qErat (θ ) namely: Fig. 5.2 Dipole under an electric and ratchet field

d = pErat (θ ) 2

(5.27)

70

5 Rotation of a Dipole

pE(rat) (θ )

= pE(rat)0



  θ θ 1 cos 2π − cos 4π θ0 2 θ0

(5.28)

The corresponding Eqs. 5.13 and 5.15 are:  Drot E  Drot E(rat)   θj t + p p θj , t t + (2Drot t)1/2 RG kB T kB T (5.29) Or in dimensionless units (Fig. 5.3): θj +1 = θj +

     1/2   θ˜j +1 =θ˜j +D pE(rat) (0) pE (0) pE(rat) θ˜j , t˜ t˜+D pE θ˜j , t˜ t˜+ t˜ RG rot  rot  (5.30) with pE(rat)  p (0) =

pE(rat)0 , kB T

 pE (0) =

pE0 kB T

(5.31)

    1  pE(rat) = cos 2π θ˜ − cos 4π θ˜ 2

(5.32)

 pE = − sin(ωE τB t) sin(θ0 θ˜ )

(5.33)

and

In Fig. 5.4, (a), (b), (c) were performed from data obtained from Eq. 5.29 with E(rat) p (θ ) = 0. and Eqs. 5.20, 5.21, which describe a dipole in a fluctuating electric field with thermal noise. Figures (d), (e), (f) were obtained from the solution of the Eq. 5.29 which contains the two terms pE (θj , t) and pE(rat) (θj ) different from zero. We observe through the plotted variables that the introduction of the ratchet field changes drastically the behaviour of the system. Being the most relevant the invertion of rotation. In Fig. 5.4 is shown the noise and symmetry effect in the movement produced by the ratchet. We observe that noise stabilizes the movement and the necessity of symmetry breaking in order to have movement [25]. In Fig. 5.5 we observe the influence of the ratchet field parameter θ0 , Eq. 5.23, (θ0 = π/2, θ0 = 2π ) in the mean rotation of the system. The results were E(rat)  0., pE = 0.; obtained from Eq. 5.29 for different system torques, namely: p E(rat)

E(rat)

 0., pE  0.; and p = 0., pE (symmetric ratchet)  0., ΓpE = 0.; Γp  0.0. In Fig. 5.6 we observe the influence of the ratchet field parameter θ0 , Eq. 5.23, (θ0 = π/2, θ0 = 2π ) in the results of Eq. 5.29 for the mean rotation of the system E(rat) p  0.0, pE  0.0. We can see the case θ0 = 2π duplicate the mean number of rotations the case θ0 = π/2. Intuitively, the likelihood that the particle exceeds a

5.3 Dipole in a Ratchet Electrical Potential

71

Fig. 5.3 (a) Mean number of rotations, < n(t) > (b) Mean rotations per seconds, < ω(t) >, (c) Mean squared angular displacement, < σ (t)2 >: ωE = 4.0 × 105 rad/s, Drot = 2.0 × 106 rad2 /s, τB = 2.5 × 10−7 s, E(rat)0 = E0 = 4.0 × 109 V/m, p = e0 × 5 nm = 240.Debye, B (max) = 4.14 × 10−3 pN.nm, θ0 = π/2, dt = 0.5τB . Temporal steps, i, N = 4 × 104 , Number of realizations, j , NR = 103 , Figures (d), (e), (f) same as former but E(rat) (θ) with the add of ratchet field torque, p

72

5 Rotation of a Dipole

Fig. 5.4 ωE = 4.00 × 107 rad.s−1 , τB = 2.5 × 10−9 s, E0 = 4 × 109 V.m−1 , p = 5e0 .nm = 240 Debye, Drot = 2 × 108 rad2 .s−1 , dt = 0.1 ∗ τB , Temporal steps, i, N = 2 × 104 , Number of realizations, j , NR = 103

maximum of the ratchet potential is greater than the likelihood of overpassing four maximum. In Fig. 5.7 we observe the influence of the ratchet field parameter θ0 , Eq. 5.23, (θ0 = π/2, θ0 = 2π ) in the results of Eq. 5.29 for the mean angular velocity, E(rat) < ω >, of the system p (θ )  0.0, pE  0.0. We can see the case θ0 = 2π duplicate the angular velocity, < ω >, the case θ0 = π/2. In Fig. 5.8 we observe the influence of the ratchet field parameter θ0 , Eq. 5.23, (θ0 = π/2, θ0 = 2π ) in the results of Eq. 5.29 for the Mean squared angular E(rat)  0.0, ΓpE  0.0. We observe displacement, < σ (t)2 >, of the system p an increase of the effective diffusion coefficient for the case θ0 = 2π compared to the case θ0 = π/2. In Fig. 5.9 are shown typical load-angular velocity at long times, ωstat of the Brownian ratchet. Both figures were obtained from simulations performed with E(rat) Eq. 5.29. Figure 5.9a shows a system with p  0., ΓpE  0., for a ratchet E(rat)

 0., pE tilded to the right, and Fig. 5.9b is shown a pure ratchet system, p = 0., with the ratchet tilded to the left. We observe in this last system that the range in which the system behaves as a motor is greater than the former in Fig. 5.9a. From the Program at the end of the chapter more simulations can be performed for better understanding of the system.

5.3 Dipole in a Ratchet Electrical Potential

73

Fig. 5.5 The influence of the ratchet field parameter θ0 , Eq. 5.23, in the mean dipole rotation. We used the same parameters as Fig. 5.4

74

5 Rotation of a Dipole

Fig. 5.6 The influence of the ratchet field parameter θ0 , Eq. 5.23, in the mean dipole rotation. We used the same parameters as Fig. 5.4 (green curve of Fig. 5.5)

5.3 Dipole in a Ratchet Electrical Potential

75

Fig. 5.7 The influence of the ratchet field parameter θ0 , Eq. 5.23, in the mean angular velocity, < ω >. We used the same parameters as Fig. 5.4

76

5 Rotation of a Dipole

Fig. 5.8 The influence of the ratchet field parameter θ0 , Eq. 5.23, in mean squared angular displacement, < σ (t)2 >, of the system. We used the same parameters as Fig. 5.4

5.4 Program 5.1, Matlab Code

77

Fig. 5.9 Typical ωstat vs Fload curves, We used the same parameters as Fig. 5.4

5.4 Program 5.1, Matlab Code Solve the Eq. for the rotation of a dipole in an Electric Field E %help rotor; E(rat) % T dip ≡ pE , T rat ≡ p E(rat)

% T dipo ≡ pE (0), T rato ≡ p (0) clear; N=20000; NR=1000; Drot=2.e8; omega=4.e7; d=50.e-10; eo=1.6e-19; kT=4.14e-21; E=4.e9; Eo=4.e9; pi=3.14159; tB=1/(2.*Drot); tE=2*pi/omega; dt=0.1*tB; x=zeros(N-1,NR); w=zeros(N-1,NR);

78

5 Rotation of a Dipole

sumx2=zeros(N-1,NR); sumx=zeros(N-1,NR); sigma2=zeros(N-1,NR); raizdt=sqrt(dt/tB); dw=raizdt*randn(N-1,NR); Trat=zeros(N-1,NR); Tdip=zeros(N,NR); frot=kT/Drot; TBmax=sqrt(2.*frot*kT); q=1*eo; p=q*d; %Tdipo=p*E; Tdipo=0.; Trato =p*Eo; %Trato =0.; %Tload=(k.e-3)*Trato; % k = 1,2,3,. . . . xo=2*pi; %xo=pi/2; for j=1:NR t=0.; x(1,j)=0.; sumx2(1,j)=0.; sumx(1,j)=0.; for i=1:N-1 xij=x(i,j); %============================================ % DIMENSION %============================================= t=t+dt; phase=omega*t; Tdip(i,j)=-Tdipo*sin(xij)*sin(phase); Trat(i,j)= Trato*(cos(2.*pi*xij/xo) - 0.5*cos(4*pi*xij/xo)); %x(i+1,j) = xij+(dt/frot)*Tdip(i,j) + dw(i,j); %x(i+1,j)=xij + (dt/frot)*Trat(i,j) + dw(i,j);% - (dt/frot)*Tload; %ratio = dw(i,j)/((dt/frot)*Trat(i,j)); %ratio1 = dw(i,j)/((dt/frot)*Tdip(i,j)); %ratio2 = Trat(i,j)/Tdip(i,j); x(i+1,j)=xij+(dt/frot)*Tdip(i,j)+(dt/frot)*Trat(i,j)+dw(i,j); % -Tload*(dt/frot); w(i+1,j) = x(i+1,j)/t; %w(i+1,j) = (x(i+1,j)-x(i,j))/dt; %============================================= % DIMENSIONLESS %============================================= %raizdt=sqrt(dt); %ddw=raizdt*randn(N-1,NR);

Bibliography

79

%t = t + dt; %phase = omega*t; %Tddip(i,j)=-Tddipo*sin(2*pi*xij)*sin(phase); %Tdrat(i,j)= Tdrato*(cos(xij) - 0.5*cos(2*xij)); %x(i+1,j)=xij+Drot*Tddip(i,j)*dt + Drot*Tdrat(i,j)*dt + ddw(i,j); %−T load ∗ Drot ∗ dt/(kT )2 ; %======================================================== sumx2(i + 1, j ) = sumx2(i, j ) + x(i + 1, j )2 ; sumx(i + 1, j ) = sumx(i, j ) + x(i + 1, j ); sigma2(i, j ) = (sumx2(i, j ) − (sumx(i, j )2 )/(i + 1))/(i + 1); end end mtheta=sum(x,2)/NR/6.2832; mw = sum(w,2)/NR/6.2832; msigma2=sum(sigma2,2)/NR/6.2832; t=((1:N)*dt)’; figure(1); plot(t,mtheta); grid; figure(2); plot(t,mw); grid; t=((1:N-1)*dt)’; figure(3); plot(t,msigma2); grid; %disp(mw/1.E8); %======================================================== save t.dat t -ascii; save msigma2.dat msigma2 -ascii; save mtheta.dat mtheta -ascii; save mw.dat mw -ascii;

Bibliography 1. Binning, G., Quate, C.F., Gerber, C.: Atomic force microscope. Phys. Rev. Lett. 56, 930 (1986) 2. Binning, G., Rohrer, H.: Scanning tunneling microscopy–from birth to adolescence. Rev. Mod. Phys. 59, 615 (1987) 3. Gimzewski, J.K., Joachim, C., Schlittler, R.R., Langlais, V., Tang, H., Johannsen, I.: Rotation of a single molecule within a supramolecular bearing. Science 281, 531 (1998) 4. Kelly, T.R., De Silva, H., Silva, R.A.: Unidirectional rotary motion in a molecular system. Nature 401, 150 (1999) 5. Koumura, N., Zijistra, R.W.J., Van Delden, R.A., Harada, N., Feringa, B.L.: Light-driven monodirectional molecular rotor. Nature 401, 152 (1999) 6. Feringa, B.L., Koumura, N., Van Delden, R.A., Ter Wiel, M.K.J.: Light-driven molecular switches and motors. Appl. Phys. A 75, 301 (2002) 7. Palffy-Muhoray, P., Kosa, T., Weinan, E.: Brownian motors in the photoalignment of liquid crystals. Appl. Phys. A 75, 293 (2002)

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5 Rotation of a Dipole

8. Van Delden, R.A., Koumura, N.A., Schoevaars, A., Meetsma, A., Feringa, B.L.: A donor– acceptor substituted molecular motor: unidirectional rotation driven by visible light. Org. Biomol. Chem. 1, 33 (2003) 9. Galajda, P., Ormos, P.: Complex micromachines produced and driven by light. Appl. Phys. Lett. 78, 249 (2001) 10. Galajda, P., Ormos, P.: Rotors produced and driven in laser tweezers with reversed direction of rotation. Appl. Phys. Lett. 80, 4653 (2002) 11. Sestelo, J.P., Kelly, T.R.: A prototype of a rationally designed chemically powered Brownian motor. Appl. Phys. A 75, 337 (2002) 12. Vacek, J., Michl, J.: Molecular dynamics of a grid-mounted molecular dipolar rotor in a rotating electric field. Proc. Natl. Acad. Sci. U. S. A. 98, 5481 (2001) 13. Horinek, D., Michl, J.: Molecular dynamics simulation of an electric field driven dipolar molecular rotor attached to a quartz glass surface. J. Am. Chem. Soc. 125, 11900 (2003) 14. Leigh, D.A., Wong, J.K.Y., Dehez, F., Zerbetto, F.: Miniaturized gas ionization sensors using carbon nanotubes. Nature 424, 174 (2003) 15. Zolotaryuk, A.V., Christiansen, P.L., Nordén, B., Savin, A.V., Zolotaryuk, Y.: Pendulum as a model system for driven rotation in molecular nanoscale machines. Phys. Rev. E 61, 3256 (2000) 16. Porto, M.: Molecular motor based entirely on the Coulomb interaction. Phys. Rev. E 63, 030102(R) (2001) 17. Engel, A., Müller, H.W., Reimann, P., Jung, A.: Ferrofluids as thermal ratchets. Phys. Rev. Lett. 91, 060602 (2003) 18. Kleutsch, B., Läuger, P.: Coupling of proton flow and rotation in the bacterial flagellar motor: stochastic simulation of a microscopic model. Eur. Biophys. J. 18, 175, 191 (1990) 19. Elston, T., Wang, H., Oster, G.: Energy transduction in ATP synthase. Nature 391, 510 (1998) 20. Oster, G., Wang, H., Grabe, M.: How Fo–ATPase generates rotary torque. Phil. Trans. R. Soc. Lond. B 355, 523 (2000) 21. Bustamante, C., Keller, D., Oster, G.: The Physics of Molecular Motors. Acc. Chem. Res. 34, 412 (2001) 22. Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57–265 (2002) 23. Gimzewski, J.K., Joachim, C.: Nanoscale science of single molecules using local probes. Science 283, 1683 (1999) 24. Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1985) 25. Reimann, P., Hängii, P.: Introduction to the physics of Brownian motors. Appl. Phys. A 75, 169–178 (2002) 26. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

Chapter 6

Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Keywords Hydrodynamic interactions · Motor proteins · Molecular dynamics · Brownian dynamics · Fluctuation phenomena · Random processes · Noise · Brownian motion

We use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of an elastically coupled dimer Brownian motor in a ratchet potential. The only external forces considered in our system were the load, the random thermal noise and an unbiased thermal fluctuation. We observe differences in the dynamic behaviour if hydrodynamic interactions are considered as compared to the case without them.

6.1 Introduction Brownian motors are small physical micro- or even nano-machines that operate far from thermal equilibrium by extracting the energy from both, thermal and non-equilibrium fluctuations in order to generate work against external loads. They present the physical analogue of bio-molecular motors that also work out of equilibrium to direct intracellular transport and to control motion in cells. In such bio-molecular motors, proteins such as kinesins, myosins and dyneins, move unidirectionally on one-dimensional “tracks” while hydrolysing adenosine triphosphate (ATP). These molecular motors are powered by a ratchet mechanism, [1], they convert the nonequilibrium fluctuation into directed flow of Brownian particles in an asymmetrical periodic potential (ratchet) without any net external force or bias. Several authors have studied theoretically the transport of two coupled particles modeling the two heads of a motor protein [2–13]. Nonequilibrium fluctuations, whether generated or by a chemical reaction far from equilibrium, can bias the Brownian motion of a particle in an anisotropic medium without thermal gradients, a net force such as gravity, or a macroscopic electric field. Fluctuationdriven transport is one mechanism by which chemical energy can directly drive the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_6

81

82

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

motion of particles and macromolecules and may find application in a wide variety of fields, including particle separation and the design of molecular motors and pumps. Zimmermann and Seifert [14] studied the efficiencies of a molecular motor for a generic hybrid model applied to the F1-ATPase they obtained good quantitative agreement with the experimental data. Pinkoviezky and Gov [15] motivated by the observed pulses of backward-moving myosin-X in the filopodia structure, they model interacting molecular motors with an internal degree of freedom, introducing a novel modification to the approximation scheme. In the present work we use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of a elastically coupled dimer Brownian motor in a ratchet potential. In section entitled “The Model” we describe the forces acting on an oscillating dimer in a ratchet potential with a load force and an external unbiased fluctuation, which acts simultaneously on two particles. In the section entitled “Brownian dynamics with hydrodynamic interactions” we describe the formalism given by Ermak and McCammon (1978) [16], which couples the forces described in “The Model” section and thermal noise with the diffusion tensor. A striking feature of fluid mechanics in the viscously dominated regime, or equivalently, at low Reynolds number, is the long range of hydrodynamic interactions. For example, the Stokeslet, or flow field induced by a point force, falls off inversely with distance. Hydrodynamic interactions have been considered by several authors to explain different phenomena: Kemps and Bhattacharjee, [17], used a particle tracking model for colloid transport near planar surfaces covered with spherical asperities This model provides a preliminary step in investigating how geometrically tractable asperities alter the undisturbed flow field and hydrodynamic interactions between the particle and the substrate. Hydrodynamic interactions allow in average for directed motion of a three-sphere system, the spheres are connected by two identical active linker arms. Each linker arm contains molecular motors and elastic elements and can oscillate spontaneously, see [18]. Microorganisms are often subjected to swimming in close proximity to each other as well as other boundaries. The resulting hydrodynamicn interactions may have puzzling effects on their swimming speed, trajectory, and power dissipation. These effects were investigated by Ramia et al. [19], each microorganism consisted of a sphere propelled by a rotating helix. It was found that only a small increase (less than 10%) results in the mean swimming speed of an organism swimming near and parallel to another identical organism. In a later paper, Kim and Powers [20], focus on hydrodynamic interactions by considering two rotating rigid helices. They suppose the helices were driven by stationary motors, and obtained complementary results to those of Ramia et al., since the hydrodynamic interactions between their helices were stronger. Fornés [21], showed that hydrodynamic interactions induce movement against an external load in a ratchet dimer Brownian motor. In the present paper we show that hydrodynamic interactions introduce differences in the behaviour of a ratchet dimer Brownian motor as compared without them. We report differences in the effective diffusion coefficient, Deff , mean vcx component of the mass center velocity and spatial cross correlations in x direction.

6.2 The Model

83

6.2 The Model We consider an elastically coupled dimer in 3 dimensions in an asymmetrical potential tilted to the left (ratchet) in the x direction, see Fig. 6.1. We considered a linear superposition of three spatial harmonics,



 2π xi 1 4π xi V0 sin + sin Urat (xi ) = − 2π L 4 L

(6.1)

The corresponding force on the particles produced by the ratchet potential is given by: Frat (xi ) = −





 2π xi 1 4π xi V0 ∂Urat (xi ) cos − cos = ∂xi L L 2 L

(6.2)

In the former equations xi is the x coordinate of particle i, i = 1 and 2, to distinguish the dimer particles. The total force on the particles will be: Ftotal (xi ) = Frat (xi ) + Fload + Acos(ωt) + X(t)

(6.3)

where X(t) is a fluctuating force produced by the thermal noise. The effect of this force is manifested in the positions of the particles, in the program of brownian dynamics through the subroutine Gauss. We define rij the vector from the center of particle i to the center of particle j , for particles i = 1, j = 2, we have rij = x12 i + y12 j + z12 k Fig. 6.1 Dimer in a ratchet potential

(6.4)

84

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

i, j, k are unit vectors in the direction of the cartesian axis, with x12 = x1 − x2 y12 = y1 − y2

(6.5)

z12 = z1 − z2 1/2  2 2 2 r12 = x12 + y12 + z12 The pair potential V12 between the two subunit spheres is V12 =

kT (l0 − r12 )2 2δ 2

Then the modulus of the harmonic force is, 1 1 1 kT 1 1 F12 = 1 (l0 − r12 )11 δ

(6.6)

(6.7)

where κ = δ −2 is the strength of the harmonic potential and l0 is the equilibrium position (Fig. 6.2). Then the components of the harmonic force are,

Fig. 6.2 The pair potential V12 versus (l0 − r12 ) is ploted. Observe that r12 are the distances obtained from the realizations of the stochastic process, l0 = 8nm, κ = 6.43pN/nm

6.3 Hydrodynamic Interactions

85

x12 r12 y12 = F12 r12 z12 = F12 r12

Fx12 = F12 Fy12 Fz12

(6.8)

The corresponding components of the harmonic force on each dimer particle are, x12 = −Fhar (x2 ) r12 y12 Fhar (y1 ) = F12 = −Fhar (y2 ) r12 z12 Fhar (z1 ) = F12 = −Fhar (z2 ) r12

Fhar (x1 ) = F12

(6.9)

Then the forces acting on the dimer particles are: Fx1 = Fhar (x1 ) + Frat (x1 ) − Fload + εx (t) Fy1 = Fhar (y1 ) Fz1 = Fhar (z1 )

(6.10)

Fx2 = −Fhar (x1 ) + Frat (x2 ) − Fload + εx (t) Fy2 = −Fhar (y1 ) Fz2 = −Fhar (z1 ) The load force, Fload , acts to oppose the motor’s forward progress, εx (t) = A cos (ωt) is an external unbiased fluctuation, which acts simultaneously on two particles.

6.3 Hydrodynamic Interactions Consider a particle embedded in a viscous liquid, the movement of the surrounding incompressible fluid at the regime of Stokes-flow (low Reynolds number Re = dρ η v) is governed by ∇p − η∇ 2 v = f (r) ∇v = 0

(6.11) (6.12)

86

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Fig. 6.3 The particle at r exerts a force f(r) onto the liquid which affects the velocity v(r)

where d is the size of the system ρ and η are the density and the viscosity of the fluid respectively, v is the velocity field, see Fig. 6.3 p (r) is the local pressure and f (r) is the force density. The general solution of these inhomogeneous linear equations is given by the super-position of a solution of the homogeneous equation, vext (r) , which is the externally imposed flow field, and a special solution  vind =

    dr O r, r f r

(6.13)

of the inhomogeneous   equation. The Green’s function of Eq. 6.13 is called the Oseen-tensor O r, r , its Cartesian matrix elements are given by [25, 26] ⎤  

r r − r − r i j i j 1 ⎦ ⎣δij + Oij (r, r ) ≡

2

8π η |r − r | |r − r | ⎡

(6.14)

1 1  Thus, the hydrodynamics mediates a long-range interactions  1/ 1r − r 1 between the force f acting at r and the velocity v induced at r. The total velocity field becomes v (r) = vext (r) + vind (r)

(6.15)

The relation between the Diffusion and Oseen tensor is given by   Dij = D0 δij I + 1 − δij kB T Oij

(6.16)

D0 = kB T /6π ηa is the diffusion coefficient of a single subunit sphere, δij is the Kronecker delta, I is the unit tensor and a is the particle radius. Equation 6.16 can be split in two, namely

6.4 Brownian Dynamics with Hydrodynamic Interactions

87

Dij = D0 δij , i, j on the same particle,   − → → rij ⊗ − rij a 3 Dij = D0 I+ , i, j on different particles 4 rij rij2

(6.17)

→ − → rij is the dyadic product, for particles i = 1, j = 2, we have rij ⊗− ⎡

⎤ x12 " # − − → r→ 12 ⊗ r12 = ⎣ y12 ⎦ x12 y12 z12 = z12 ⎤ ⎡ 2 x12 x12 y12 x12 z12 ⎣ y12 x12 y 2 y12 z12 ⎦ 12 2 z12 x12 z12 y12 z12

(6.18)

As an example we show the Diffusion tensor for two particles in a two dimension system, ⎡ 1 ⎢ ⎢ ⎢ ⎢ 0 ⎢

 D = D0 ⎢ 2 ⎢3 a x12 ⎢ 2 ⎢ 4 r12 1 + r12 ⎢ ⎣ 3 a x12 y12 4 r12

2 r12

3 a 4 r12

0

3 a x12 y12 4 r12 r 2 12 3 a 4 r12

2 x12 2 r12

3 a x12 y12 4 r12 r 2 12

1

1+

1+

2 y12 2 r12





⎤ 3 a x12 y12 4 r12 r 2 12 3 a 4 r12

1+

1

0

0

1

2 y12 2 r12

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.19)

6.4 Brownian Dynamics with Hydrodynamic Interactions Consider a system of N spherical interacting Brownian particles suspended in a hydrodynamic medium, the displacement of particle i during t is given by Ermak and McCammon (1978), [16], namely ri = ri0 +

2 Dij0 Fj0 j

kB T

t + Ri (t)

(6.20)

where the superscript “0” indicates that the variable is to be evaluated at the beginning of the time step. Fj0 is the force acting on particle j . Ri (t) is a random displacement with a Gaussian distribution function whose average value is zero and the correlation is Ri (t) Rj (t) = 2Dij0 t.

88

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Dij0 = D0 δij , i, j on the same particle,   − → − → ⊗ r r a 3 ij ij Dij0 = D0 I+ , i, j on different particles 4 rij rij2

(6.21)

D0 = kT /6π ηa is the diffusion coefficient of a single subunit sphere of a radius, − → → rij ⊗− rij is the dyadic product, for particles i = 1, j = 2, we have ⎡

⎤ x12 " # − − → r→ 12 ⊗ r12 = ⎣ y12 ⎦ x12 y12 z12 = z12 ⎤ ⎡ 2 x12 x12 y12 x12 z12 ⎣ y12 x12 y 2 y12 z12 ⎦ 12 2 z12 x12 z12 y12 z12

(6.22)

In our case of a dimer in three dimensions the tensor Dij0 is a 6 × 6 matrix. For details on hydrodynamic interactions and they applications, see our article [21], and Refs. [30], [22–26]. A first basic quantity of interest, in our case, is the average center of mass velocity in the x direction, vcx , in the long-time limit, i.e., after transients due to initial conditions have died out, is given by 

rcx (t) − rcx (t0 ) vcx  = lim t→∞ t − t0

 (6.23)

where rcx (t) = [x1 (t) + x2 (t)] /2. In the program dt = t − t0 is a constant. The means are over the realizations of the stochastic process. A quantity of central interest will be the effective diffusion coefficient, easy to compute by the equation given by [27]: Deff

1 = lim 6 t→∞

3" #2 4 rc (t) − rc (t0 ) t − t0

(6.24)

where "

" #2 #2 rc (t) −rc (t0 ) = [rcx (t) −rcx (t0 )]2 + rcy (t) −rcy (t0 ) + [rcz (t) −rcz (t0 )]2 (6.25)

The competition between the drift v and diffusivity Deff in advection-diffusion problems is often expressed by a dimensionless number, the Péclet number, P e, [28], Pe =

|v| L Deff

(6.26)

6.4 Brownian Dynamics with Hydrodynamic Interactions

89

Here L is a typical length scale, in our case the length of a single ratchet element, v is the average stationary velocity of the particle, in our case we used |vcx |. The larger the Péclet number, the more net drift predominates over diffusion. We performed the simulation in dimensionless units. Energy is in units of kT . Distance is in units of the separation distance l0 and time is in units of l02 /D0 , We used in the simulations the following parameters: l0 = 1, L = 1, κ = δ −2 = 100, radius = 0.25, D0 = 1., A = 7.0, ω = 5., F0 = V0 /L = 4.0, t = 0.00125, the simulation time was t = 1250, which corresponds to 1 × 106 steps. The corresponding dimension units are: l0 = 8.nm, L = 8.nm, κ = 6.43pN/nm, radius = 2.0nm, D0 = kT /(6π ηa) = 1.226 × 10−9 m2 /s, A = 3.60pN , t = 52.2ns, t = 65.2ps, ω = 0.6GH z. F0 = 2.057pN . In Brownian dynamics simulations with hydrodynamic interactions the size of the physically meaningful time step is restricted to values which are sufficiently long t mi D0 /kT . In our case mi D0 /kT = 3.4fs where we have used mi = 138 × 10−24 Kg for the protein dimer unit of 2.5Å of radius. The average velocity of a molecular motor is a function of the load force resisting the motor’s advancement. One of the characteristic of a molecular motor is the load force-velocity curve. In Fig. 6.4 we show vcx  as a function of the load Force Fload . At the stationary state, the ratio SE (vcx ) / vcx  ≤ 10−6 , where SE (vcx ) is the standard error of the mean velocity vcx  . In the range −4 ≤ Fload ≤ 0 we observe, the motor continue with a positive velocity in spite of the negative load force (motor effect). We also observe a substantially increase of the motor velocity in the case with hydrodynamic interaction as compared without it.

6.4.1 Efficiency The motor efficiency is defined as the ratio of the output work to input energy η=

W˙ E˙ in 

(6.27)

where W˙ = |Fload vcx | is the average work done against the load per unit of time and E˙ in is the average input power, both quantities averaged with respect to all random processes and time, see Chap. 6, 100. of Reference [29]. We can estimate the efficiency in our case the periodic force Fp = A|sin(ωt)|, then the average input power per cycle, E˙ in , will be E˙ in  = Fp vcx (t) + |Fload v|

(6.28)

90

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Fig. 6.4 Mean x component of the mass center velocity versus the load Force, vcx  vs Fload

as in the long times limit vcx is constant, the former equation can be approximated as  1 τ Fp vcx (t) ∼ Avcx  sin(ωt)dt (6.29) τ 0 Then 2Avcx  + |Fload vvcx | E˙ in  = π

(6.30)

where we have used τ = π/ω. Then the efficiency is, η=

1 2A π |Fload |

+1

(6.31)

independent of vcx . For the case Fload = −1 and A = 7, η = 0.18. The ratchet works as a motor only in the interval Fstall < Fload < 0, since only in this case, the ratchet is performing work against the external load. In Fig. 6.5 we observe  the behaviour of the mean squared displacement of the mass center position, [rc (t) − rc (t0 )]2 as a function of time in the long time limit. The effective duffusion coefficient is giving by the slope of the linear fitting,

6.4 Brownian Dynamics with Hydrodynamic Interactions

91

 Fig. 6.5 Mean squared displacement of the mass center position, [rc (t) − rc (t0 )]2 as a function of time in the long time limit. The effective diffusion coefficient is giving by the slope of the linear fitting, Deff = B/6, in accordance to Eq. 6.24. Fload = −1

Deff = B/6, in accordance to Eq. 6.24. We observe greater slope for the case with hydrodynamic interactions. In Fig. 6.6 is shown the effective diffusion coefficient as a function of the load Force, Deff vs Fload , we observe that hydrodynamic interactions increase the effective diffusion. In both cases with and without hydrodynamic interactions Deff remains almost constant. In Fig. 6.7 is shown the Péclet number as a function of the load force, P e vs Fload , we did not find differences between the Péclet numbers with hydrodynamic interactions and without them. In Fig. 6.8 is shown the mean x component of the mass center position as a function of time, rcx  versus T ime, in the long-time limit. for a given load Force, Fload = −1, We observe a linear relation, with rcx  in the case with hydrodynamic interactions greater than the case without them. In Fig. 6.9 we observe the behaviour of the average center of mass velocity in the x direction, vcx  versus T ime, in the long-time limit for a given load force, Fload = −1. The velocity is substantially greater for the case with hydrodynamic interactions. In Fig. 6.10 is shown the spatial cross correlations in x direction as a function of Lag time, Corr[rx (1).rx (2)] versus Lag time. We observe that the correlation is higher in the case with hydrodynamic interactions. A similar result was found

92

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Fig. 6.6 Effective diffusion coefficient as a function of the load Force, Deff vs Fload

Fig. 6.7 Péclet number as a function of the load force, P e versus Fload

6.4 Brownian Dynamics with Hydrodynamic Interactions

93

Fig. 6.8 Mean x component of the mass center position as a function of time, rcx  versus T ime. In the long time limit. Fload = −1

Fig. 6.9 Mean x component of the mass center velocity as a function of time, vcx  versus T ime. In the long time limit. Fload = −1

94

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Fig. 6.10 Spatial cross correlations in x direction as a function of lag time, Corr[rx (1).rx (2)] N5 −1 versus Lag time. Corr [rx (1).rx (2)]j = rx (1)j +k rx (2)k , Fload = −1 k=0

by Houtman et. al. [30] who developed a simple 2D-lattice model in order to test the influence of hydrodynamic interactions on the collective transport of molecular motors, which is important for the understanding of cell growth and development. Houtman et. al. showed that long range collective hydrodynamic interactions lead to a substantial increase in the effective velocity of motors attached to a filament. Their results were also supported by experiments. In conclusion hydrodynamic interactions influence substantially the dynamics of a ratchet dimer Brownian motor, consequently they have to be considered in any theory where the molecular motors are in a liquid medium.

6.5 Program 6.1, Fortran Code THE FOLLOWING FILE “motor.in”, HAS TO BE ADDED TO THE DIRECTORY IN ORDER TO RUN THE NEXT PROGRAM: FILE motor.in: ================================================= initposi*tape2 *tape3 *output00*tape5 *tape6 *tape7 *tape8 *

6.5 Program 6.1, Fortran Code

tape9 *lastposi*tape11 *tape12 *tape13 *tape14 *tape15 *tape16 * nstep :0001000000 isave :0000000001 iprint :0000010000 itape :0000000001 dt :000.001250 =================================================== c program motor c this program performs brownian dynamics with hydrodynamic c interactions (Ermak and Mccammon, J. Chem. Phys. 69, 1352, 1982) c for a harmonic dimer) implicit none common / block1 / rx, ry, rz common / block2 / d, xic common / block3 / fx, fy, fz common / block4 / A, Fload, gamma, w, Fo integer itape, step, i, nstep, isave, iprint, step1 integer n, n3 parameter ( n = 2, n3 = 3 * n ) real rx(n), ry(n), rz(n), fx(n), fy(n), fz(n) real rx0(n), ry0(n), rz0(n) real d(n3,n3), xic(n3) real drxsq, drysq, drzsq,sigma,StDev real pi, dt,ts real consii, consij real acv, acvsq real avv, flv real v, radius, vn, r real acr,acrsq,avr, avrsq, flr real rcx0, rcy0, rcz0 real rx210, ry210, rz210, r21sq0, r210 real rcx, rcy, rcz, rc2, rc, rsq real rx21, ry21, rz21, r21sq, r21 real acdn,acdiff, acrotn, acrot, deff, peclet, cos1 real acd, vcx, vcy, vcz, sumvcx, sumvcy, sumvcz, vc, t real sumr, sumr2,dlamda,a,w, Ein, t1, Fload, DPI real rmean, rmeansq, deltar, ranf, dummy,work,efficiency real Fo , sumFrat, kF, dtW, vcx2, sumvcx2,tau,mass real gamma, STOKESEFF, ENERGCONVEFF, KINETICEFF, VCX2EQ real CUBEROEFF, Frat, SUZUKIEFF real mvcx, mvcx2, sumvc,mvcy, mvcz real vcy2, vcz2, vcx1, t2, acdnx,DEFFX,acdiffx parameter (pi = 3.1415927, DPI=pi*2) character*8 fn(16), tape tape = ’tape****’

95

96

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

c *************************************************************** 101 format(8(a8,1x)) 102 format(10x,i10) 103 format(10x,e10.6) c *************************************************************** open(unit=5,file=’motor.in’,form=’formatted’) open(unit=4,file=’motor.out’,form=’formatted’) open(unit=9,file= ’energy.dat’,status=’unknown’) open(unit=10,file= ’mvcx.dat’,status=’unknown’) c ** read input data ** read(5,101)fn read(5,102)nstep read(5,102)isave read(5,102)iprint read(5,102)itape read(5,103)dt c write(4,’(” **** program motor **** ”)’) write(4,’(” brownian dynamics simulation ”)’) write(4,’(” with hydrodynamic interactions ”)’) write(4,’(” harmonic dimer ”)’) c ** write input data ** c write(4,’( //1x ,a )’) title write(4,’(” number of atoms ”,i10 )’) n write(4,’(” number of steps ”,i10 )’) nstep write(4,’(” save frequency ”,i10 )’) isave write(4,’(” output frequency ”,i10 )’) iprint c write(4,’(” configuration file name ”,a )’) cnfile write(4,’(” timestep ”,f10.4 )’) dt c ** read in initial configuration ** if (itape.eq.1) then call setup (fn(1), 1) else call readcn (fn(10),10) end if c ==================================== radius = 0.25 consii =1. c ==================================== c with Hydrodynamic Interactions, consij: consij = 0.75*radius*consii c ==================================== c without Hydrodynamic Interactions, consij: c consij = 0. c ==================================== c ** zero accumulators ** acv = 0.0

6.5 Program 6.1, Fortran Code

acvsq = 0.0 flv = 0.0 acr = 0.0 acrsq = 0.0 flr = 0.0 acdiff = 0.0 acdiffx =0.0 acrot = 0.0 vcx = 0.0 SUMR = 0.0 SUMR2 = 0.0 Ein=0. work=0. sumFrat =0. sumvcx=0. sumvcx2=0. sumvcy=0. sumvcz = 0. sumvc = 0. c ** write out some useful information ** write(4,’(” dt = ”,f10.5)’) dt write(4,’(/” ** brownian dynamics begins ** ”/ )’) c write(4,’(” step v/n ”/ )’) c ********************************************************* c ** main loop begins ** c ********************************************************* c ** calculate the diffusion tensor and systematic ** c ** forces at the beginning of the step ** do i = 1, n rx0(i) = rx(i) ry0(i) = ry(i) rz0(i) = rz(i) end do rcx0 = (rx0(1)+rx0(2))*0.5 rcy0 = (ry0(1)+ry0(2))*0.5 rcz0 = (rz0(1)+rz0(2))*0.5 rx210 = rx0(2)-rx0(1) ry210 = ry0(2)-ry0(1) rz210 = rz0(2)-rz0(1) r21sq0 = rx210*rx210 + ry210*ry210 + rz210*rz210 r210 = sqrt(r21sq0) A = 7. Fload = -1 gamma = 0.1 mass=2. w = 5.

97

98

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

Fo = 4. do 100 step = 1, nstep t = step*DT c ** define first position for calculating transl. diffusion ** c ** and rotation rate ** call force ( consii, consij, v, r, t) c ** calculate the correlated normal variates ** call covar ( dt ) c ** move the atoms ** call move ( dt ) c ** calculate instantaneous values for previous step ** vn = v / real ( n ) rcx = (rx(1)+rx(2))*0.5 rcy = (ry(1)+ry(2))*0.5 rcz = (rz(1)+rz(2))*0.5 rc = sqrt(rcx**2 + rcx**2 + rcx**2) rx21 = rx(2)-rx(1) ry21 = ry(2)-ry(1) rz21 = rz(2)-rz(1) r21sq = rx21*rx21 + ry21*ry21 + rz21*rz21 r21 = sqrt(r21sq) acdn = (rcx-rcx0)*(rcx-rcx0)+(rcy-rcy0)*(rcy-rcy0)+ * (rcz-rcz0)*(rcz-rcz0) acdnx= (rcx-rcx0)*(rcx-rcx0) acrotn = (rx21*rx210 + ry21*ry210 + rz21*rz210)/ * r21/r210 acv = acv + vn acvsq = acvsq + vn * vn acr = acr + r acrsq = acrsq + r * r acdiff = acdiff + acdn acdiffx = acdiffx + acdnx acrot = acrot + acrotn cos1 = acrot/step vcx =(rcx - rcx0) /dt sumvcx = sumvcx + vcx mvcx=sumvcx/step sumvcx2 = sumvcx2 + (vcx**2) mvcx2 = sumvcx2/step vcy =(rcy - rcy0) /dt sumvcy = sumvcy + vcy mvcy=sumvcy/step vcz =(rcz - rcz0) /dt sumvcz = sumvcz + vcz mvcz=sumvcz/step

6.5 Program 6.1, Fortran Code

99

vc = sqrt( mvcx**2 + mvcy**2 + mvcz**2) t = dt*step if(step.GT.7.E5) then step1 = step - 7.E5 t2 = dt*step1 write(10,*) t2, mvcx endif c mvcx = sumvcx/step1 c mvcx2 =sumvcx2 /step1 c———————-FORNES—————DRXSQ = (rx(1)-rx(2))**2 DRYSQ = (ry(1)-ry(2))**2 DRZSQ = (rz(1)-rz(2))**2 R = SQRT(DRXSQ + DRYSQ + DRZSQ) RSQ = R*R SUMR = SUMR + R SUMR2 = SUMR2 + RSQ do i = 1, n rx0(i) = rx(i) ry0(i) = ry(i) rz0(i) = rz(i) end do rcx0 = (rx0(1)+rx0(2))*0.5 rcy0 = (ry0(1)+ry0(2))*0.5 rcz0 = (rz0(1)+rz0(2))*0.5 rx210 = rx0(2)-rx0(1) ry210 = ry0(2)-ry0(1) rz210 = rz0(2)-rz0(1) c SUZUKI EFFICIENCY if(step.GT.7.E5.AND.step.LT.700503.) then step1 = step - 7.E5 t1 = dt*step1 Frat = Fo*(cos(DPI*rcx) -0.5*cos(2*DPI*rcx)) sumFrat = sumFrat + Frat*vcx endif 100 continue RMEAN = SUMR/real(NSTEP) RMEANSQ = SUMR2/real(NSTEP) DELTAR = SQRT(abs(RMEANSQ - RMEAN**2)) DEFF = acdiff/real(nstep)/6/DT DEFFX = acdiffx/real(nstep)/2/DT PECLET = ABS(VCX)/DEFFX VCX2EQ = DEFFX*gamma/2. c write(*,*) VCX2EQ, mvcx2 c ========================================================

100

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

C EFFICIENCY: STOKESEFF, ENERGCONVEFF, KINETICEFF c ======================================================== c Spiechowicz, Hänggi, Luczka, 2015-Int. Conf. on Noise c DT = eq. =kT/m = gamma*DEFF/mass C ======================================================== c VCX2EQ = 19.56142 c VCX2EQ = gamma*DEFFX/mass STOKESEFF = mvcx**2/(mvcx2 - VCX2EQ) ENERGCONVEFF = abs(Fload*mvcx)/(gamma*(mvcx2 - VCX2EQ)) KINETICEFF = w*mvcx**2/(4*pi*gamma*(mvcx2 - VCX2EQ)) CUBEROEFF = abs(Fload*mvcx)/(abs(Fload) + gamma*(mvcx2 - VCX2EQ)) SUZUKIEFF = (abs(Fload)*mvcx + gamma*mvcx2)/(-sumFrat) c ======================================================== WRITE(*,’(” Fload =”,f10.7)’) Fload WRITE(*,’(” gamma =”,f10.7)’) gamma WRITE(*,’(” mass =”,f10.7)’) mass WRITE(*,’(” Fo =”,f10.7)’) Fo WRITE(*,’(” w =”,f10.7)’) w WRITE(*,’(” =”,f10.7)’) MVCX c WRITE(*,’(” VC =”,f10.7)’) VC c WRITE(*,’(” VCX/VC =”,f10.7)’) VCX/VC WRITE(*,’(” =”,f10.2)’) MVCX2 WRITE(*,’(”eq=”,E12.5)’) VCX2EQ WRITE(*,’(” - eq =”,f10.7)’) (mvcx2 - vcx2eq) WRITE(*,’(” sumFrat =”,f10.2)’) sumFrat WRITE(*,’(” !Fload!* =”,f10.7)’) abs(Fload)*mvcx WRITE(*,’(” gamma* =”,f10.7)’) gamma*mvcx2 WRITE(*,’(”RMEAN =”,f10.5)’) RMEAN WRITE(*,’(”RMEANSQ =”,f10.5)’) RMEANSQ WRITE(*,’(”DELTAR =”,f10.5)’) DELTAR WRITE(*,’(” DEFF =”,f10.5)’) DEFF WRITE(*,’(” DEFFX =”,f10.5)’) DEFFX WRITE(*,’(” PECLET =”,f10.5)’) PECLET WRITE(*,’(” STOKESEFF =”,f10.5)’) STOKESEFF WRITE(*,’(” ENERGCONVEFF =”,f10.5)’) ENERGCONVEFF WRITE(*,’(” KINETICEFF =”,f10.5)’) KINETICEFF WRITE(*,’(” SUZUKIEFF =”,f10.5)’) SUZUKIEFF WRITE(*,’(” CUBEROEFF =”,f10.5)’) CUBEROEFF WRITE(4,’(/” ** brownian dynamics ends ** ”///)’) c ** calculate and write out running averages ** nstep = nstep - 7e5 avv = acv / real ( nstep ) acvsq = ( acvsq / real ( nstep ) ) - avv ** 2 avr = acr / real ( nstep ) avrsq = acrsq / real (nstep )

6.5 Program 6.1, Fortran Code

101

Deff = acdiff / real ( nstep )/6/DT acrot = acrot / real ( nstep ) c ** calculate fluctuations ** if ( acvsq .gt. 0.0 ) flv = sqrt( acvsq ) if ( acrsq .gt. 0.0 ) flr = sqrt( avrsq - avr * avr) write(4,’(/” averages ”/ )’) write(4,’(” = ”,f10.6)’) avv write(4,’(” = ”,f10.6)’) avr write(4,’(” = ”,f10.6)’) avrsq write(4,’(” translational diffusion = ”, f10.6)’) Deff write(4,’(” rotation rate = ”, f10.6)’) acrot write(4,’(/” fluctuations ”/)’) write(4,’(” fluctuation in = ”,f10.6)’) flv write(4,’(” fluctuation in = ”,f10.6)’) flr write(4,’(/” end of simulation ”)’) c *************************************************************** c ** main loop ends ** c *************************************************************** c ** write out the final configuration from the run ** call writcn(fn(10),10) stop end subroutine readcn(cnfile,itape) common / block1 / rx, ry, rz c *************************************************************** c ** subroutine to read in the configuration from unit 10 ** c *************************************************************** integer n parameter ( n = 2 ) character cnfile*(*) real rx(n), ry(n), rz(n) integer itape c *************************************************************** open(unit=itape,file=cnfile,form=’formatted’,status=’unknown’) rewind(itape) do i = 1,n read(itape,*)rx(i),ry(i),rz(i) end do return end subroutine writcn(cnfile,itape) common / block1 / rx, ry, rz c *************************************************************** c ** subroutine to write out the configuration to unit 10 ** c ***************************************************************

102

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

integer n parameter ( n = 2 ) character cnfile*(*) integer itape real rx(n), ry(n), rz(n) c *************************************************************** open(unit=itape,file=cnfile,form=’formatted’,status=’unknown’) rewind(itape) do i = 1,n end do return end subroutine force ( consii, consij, v, r, t) common / block1 / rx, ry, rz common / block2 / d, xic common / block3 / fx, fy, fz common / block4 / A, Fload, gamma, w, Fo c *************************************************************** c ** routine to compute systematic forces and the diffusion tensor ** c ** ** c ** principal variables: ** c ** ** c ** integer n number of atoms ** c ** integer n3 number of degrees of freedom ** c ** real rx(n),ry(n),rz(n) positions ** c ** real fx(n),fy(n),fz(n) forces ** c ** real d(n3,n3) the diffusion tensor ** c ** real xic(n3) correlated random normal deviates ** c ** real consii constant in the diffusion tensor ** c ** real consij constant in the diffusion tensor ** c ** real v the potential energy ** c ** ** c ** usage: ** c ** ** c ** force is called in a brownian dynamics program to calculate ** c ** the systematic force on each atom and the elements of the ** c ** diffusion tensor. ** c *************************************************************** integer n, n3 parameter ( n = 2, n3 = n * 3 ) real consii, consij, v, DPI real rx(n), ry(n), rz(n), fx(n), fy(n), fz(n) real d(n3,n3), xic(n3) integer ic, jc, i, j real rxi, ryi, rzi, fxij, fyij, fzij, fij

6.5 Program 6.1, Fortran Code

103

real fxi, fyi, fzi, rij, rrijsq real rijsq ,rxij, ryij, rzij, vij, wij, oij, rpij c *************************************************************** c ** zero forces and potential ** c *************************************************************** do 10 i = 1, n fx(i) = 0.0 fy(i) = 0.0 fz(i) = 0.0 10 continue v = 0.0 r = 0.0 c ** loop over all pairs of atoms ** do 100 i = 1, n - 1 rxi = rx(i) ryi = ry(i) rzi = rz(i) fxi = fx(i) fyi = fy(i) fzi = fz(i) ic = 3 * ( i - 1) + 1 do 99 j = i + 1, n rxij = rxi - rx(j) ryij = ryi - ry(j) rzij = rzi - rz(j) rijsq = rxij * rxij + ryij * ryij + rzij * rzij c ** calculate off-diagonal blocks of diffusion tensor ** c ** here we assume the rotne-prager tensor form ** c ** take rpij = 0 instead below for oseen tensor ** jc = ( j - 1 ) * 3 + 1 rpij = 0. rij = sqrt( rijsq ) rrijsq = 1.0 / rijsq oij = consij / rij d( ic , jc ) = oij + rpij + * ( oij - 3.0 * rpij ) * rxij * rxij * rrijsq d( ic+1, jc+1 ) = oij + rpij + * ( oij - 3.0 * rpij ) * ryij * ryij * rrijsq d( ic+2, jc+2 ) = oij + rpij + * ( oij - 3.0 * rpij ) * rzij * rzij * rrijsq d( ic , jc+1 ) = * ( oij - 3.0 * rpij ) * rxij * ryij * rrijsq d( ic , jc+2 ) = * ( oij - 3.0 * rpij ) * rxij * rzij * rrijsq d( ic+1, jc+2 ) =

104

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

* ( oij - 3.0 * rpij ) * ryij * rzij * rrijsq d( ic+1, jc ) = d( ic , jc+1 ) d( ic+2, jc ) = d( ic , jc+2 ) d( ic+2, jc+1 ) = d( ic+1, jc+2 ) c ** calculate systematic forces ** delta = 0.1 deltasc = delta**2 vij = 0.5*(1./deltasc)*(1.-rij)**2 wij = (1./deltasc)*(1.-rij)*rij fij = wij * rrijsq fxij = fij * rxij fyij = fij * ryij fzij = fij * rzij v = v + vij fxi = fxi + fxij fyi = fyi + fyij fzi = fzi + fzij fx(j) = fx(j) - fxij fy(j) = fy(j) - fyij fz(j) = fz(j) - fzij write(9,*) 1. - RIJ, VIJ 99 continue fx(i) = fxi fy(i) = fyi fz(i) = fzi 100 continue DPI=6.2832 FAC = A*abs(sin(w*t)) DO I = 1,2 FX(I)= FX(I)+ Fo*(cos(DPI*RX(I)) - 0.5*cos(2*DPI*RX(I))) + * Fload + FAC END DO do 50 i = 1, n c ** calculate on-diagonal blocks of diffusion tensor ** ic = 3 * ( i - 1 ) + 1 d( ic , ic ) = consii d( ic+1, ic+1 ) = consii d( ic+2, ic+2 ) = consii d( ic , ic+1 ) = 0.0 d( ic , ic+2 ) = 0.0 d( ic+1, ic+2 ) = 0.0 50 continue c ** fill the lower triangle of the diffusion tensor ** do 70 ic = 1, n3 - 1 do 60 jc = ic + 1, n3

6.5 Program 6.1, Fortran Code

105

d( jc, ic ) = d( ic, jc ) 60 continue 70 continue return end subroutine covar ( dt ) common / block2 / d, xic c *************************************************************** c ** routine to compute 3n correlated random normal deviates. ** c ** ** c ** principal variables: ** c ** ** c ** integer n number of atoms ** c ** integer n3 number of degrees of freedom ** c ** real d(n3,n3) the diffusion tensor ** c ** real xic(n3) correlated random normal deviates ** c ** real xi(n3) uncorrelated random normal deviates ** c ** real l(n3,n3) a lower triangular matrix ** c ** real dt reduced timestep ** c ** ** c ** usage: ** c ** ** c ** covar is called in a brownian dynamics simulation after the ** c ** the diffusion tensor has been constructed in force. on exit ** c ** the array xic contains the correlated gaussian displacements. ** c ** ** c ** ********************************************************** ** c ** ** warning ** ** c ** ** l * l(transpose), where l is a lower triangular ** ** c ** ** matrix. this is expensive for a large matrix ** ** c ** ** and you may find a more efficient or accurate ** ** c ** ** machine code routine in the common scientific ** ** c ** ** libraries such as nag or imsl. if the matrix ** ** c ** ** is not positive definite the method will fail. ** ** c ** ********************************************************** ** c ** ** c *************************************************************** integer n, n3 parameter ( n = 2, n3 = n * 3 ) real d(n3,n3), xic(n3) real dt integer i, j, k real gauss, dummy, l(n3,n3), sum, xi(n3) c *************************************************************** c ** calculate the lower triangular matrix l **

106

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

c *************************************************************** l(1, 1) = sqrt(d(1, 1)) 9 l(2, 1) = d(2, 1) / l(1, 1) l(2, 2) = sqrt(d(2, 2) - l(2, 1) * l(2, 1)) 10 do 60 i = 3, n3 l(i, 1) = d(i, 1) / l(1, 1) do 40 j = 2, i - 1 sum = 0.0 do 30 k = 1, j - 1 sum = sum + l(i, k) * l(j, k) 30 continue l(i, j) = ( d(i, j) - sum ) / l(j, j) 40 continue sum = 0.0 do 50 k = 1, i - 1 sum = sum + l(i, k) * l(i, k) 50 continue l(i, i) = sqrt( d(i, i) - sum) 60 continue c ** calculate correlated random displacements ** do 80 i = 1, n3 c ** calculate uncorrelated random normal deviates ** c ** with zero mean and variance 2.0 * dt ** xi(i) = gauss ( dummy ) * sqrt( 2.0 * dt ) sum = 0.0 do 70 j = 1, i sum = sum + l(i, j) * xi(j) 70 continue xic(i) = sum 80 continue return end subroutine move ( dt ) common / block1 / rx, ry, rz common / block2 / d, xic common / block3 / fx, fy, fz c *************************************************************** c ** routine to move the atoms in a brownian dynamics simulation ** c ** ** c ** principal variables: ** c ** ** c ** integer n number of atoms ** c ** integer n3 number of degrees of freedom ** c ** real rx(n),ry(n),rz(n) positions ** c ** real fx(n),fy(n),fz(n) forces **

6.5 Program 6.1, Fortran Code

107

c ** real d(n3,n3) the diffusion tensor ** c ** real xic(n3) correlated random normal deviates ** c ** real dt reduced timestep ** c ** ** c ** usage: ** c ** ** c ** move is called after force and covar to move the atoms. ** c *************************************************************** integer n, n3 parameter ( n = 2, n3 = n * 3 ) real rx(n), ry(n), rz(n), fx(n), fy(n), fz(n) real d(n3,n3), xic(n3) real dt real f(n3), sumx, sumy, sumz integer i, ic, jc c *************************************************************** c ** place forces in a temporary array of size 3n ** c *************************************************************** do 10 i = 1, n ic = ( i - 1 ) * 3 + 1 f(ic) = fx(i) f(ic+1) = fy(i) f(ic+2) = fz(i) 10 continue c ** move the atoms ** do 30 i = 1, n ic = ( i - 1 ) * 3 + 1 sumx = 0.0 sumy = 0.0 sumz = 0.0 do 20 jc = 1, n3 sumx = sumx + d( ic , jc ) * f(jc) sumy = sumy + d( ic+1, jc ) * f(jc) sumz = sumz + d( ic+2, jc ) * f(jc) 20 continue rx(i) = rx(i) + sumx * dt + xic( ic ) ry(i) = ry(i) + sumy * dt + xic( ic + 1 ) rz(i) = rz(i) + sumz * dt + xic( ic + 2 ) 30 continue return end real function ranf ( dummy ) c *************************************************************** c ** function ranf returns a uniform random variate between 0 and 1** c ** **

108

6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions

c ** *************** ** c ** ** warning ** ** c ** *************** ** c ** ** c ** good random number generators are machine specific. ** c ** please use the one recommended for your machine. ** c *************************************************************** integer l, c, m parameter ( l = 1029, c = 221591, m = 1048576 ) integer seed real dummy save seed data seed / 0 / c *************************************************************** seed = mod ( seed * l + c, m ) ranf = real ( seed ) / m end real function gauss ( dummy ) c *************************************************************** c ** function gauss returns a uniform random normal variate from ** c ** a distribution with zero mean and unit variance. ** c ** ** c ** reference: ** c ** knuth d, the art of computer programming, (2nd edition ** c ** addison-wesley), 1978. ** c *************************************************************** real a1, a3, a5, a7, a9 parameter ( a1 = 3.949846138, a3 = 0.252408784 ) parameter ( a5 = 0.076542912, a7 = 0.008355968 ) parameter ( a9 = 0.029899776 ) real sum, r, r2 integer i c *************************************************************** sum = 0.0 do 10 i = 1, 12 sum = sum + ranf ( dummy ) 10 continue r = ( sum - 6.0 ) / 4.0 r2 = r * r gauss = (((( a9 * r2 + a7 ) * r2 + a5 ) * r2 + a3 ) * r2 + a1 ) **r end subroutine setup(cnfile,itape) common / block1 / rx, ry, rz integer i, n

Bibliography

109

parameter(n = 2) real rx(n),ry(n),rz(n) character cnfile*(*) open(unit=itape,file=cnfile,form=’formatted’,status=’unknown’) rewind(itape) do i=1,n rx(i)=ranf(dummy)-0.5 ry(i)=ranf(dummy)-0.5 rz(i)=ranf(dummy)-0.5 write( itape,*) rx(i),ry(i),rz(i) end do return end subroutine wheader common / block1 / rx, ry, rz common / block4 / step integer n,step parameter ( n = 2 ) real rx(n), ry(n), rz(n) real xi,yi,zi 10 format(f8.6,1x,f10.6,1x,f8.6) if( mod(step,10).eq.0) then do i=1,n xi = rx(i) yi = ry(i) zi = rz(i) write(step,10)xi,yi,zi end do end if return end

Bibliography 1. Reimann, P., Hänggi, P.: Materials science and processing. Introduction to the physics of Brownian motors. Appl. Phys. A 75(2), 169–178 (2002) 2. Wang, H.Y., Bao, J.D.: The roles of ratchet in transport of two coupled particles. Phys. A 337, 13–26 (2004) 3. Wang, H.Y., Bao, J.D.: Cooperation behavior in transport process of coupled Brownian motors. Phys. A 357, 373–382 (2005) 4. Wang, H.Y., Bao, J.D.: Transport coherence in coupled Brownian ratchet. Phys. A 374, 33–40 (2007) 5. Fornés, J.A.: An oscillating electric field with thermal noise increases the rotational diffusion and drives rotation in a dipole. J. Colloid Interface Sci. 281, 236–239 (2005)

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6. von Gehlen, S., Evstigneev, M., Reimann, P.: Dynamics of a dimer in a symmetric potential: Ratchet effect generated by an internal degree of freedom. Phys. Rev. E 77, 031136 (2008) 7. Lipowsky, R., Chai, Y., Klumpp, S., Liepelt, S., Müller, M.J.I.: Molecular motor traffic: from biological nanomachines to macroscopic transport. Phys. A 372, 34–51 (2006) 8. Tao, Y.G., Kapralb, R.: Design of chemically propelled nanodimer motors. J. Chem. Phys. 128, 164518 (2008) 9. Howard, J.: Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates, Sunderland (2001) 10. Block, S.M.: Nanometres and piconewtons: the macromolecular mechanics of kinesinTrends. Cell Biol. 5, 169–175 (1995) 11. Visscher, K., Schnitzer, M.J., Block, S.M.: Single kinesin molecules studied with a molecular force clamp. Nature 400, 184–189 (1999) 12. Schnitzer, M.J., Visscher, K., Block, S.M.: Force production by single kinesin motors. Nat. Cell Biol. 2, 718–723 (2000) 13. Speer, D., Eichhorn, R., Evstigneev, M., Reimann, P.: Dimer motion on a periodic substrate: spontaneous symmetry breaking and absolute negative mobility. Phys. Rev. E 85, 061132 (2012) 14. Zimmermann, E., Seifert, U.: Efficiencies of a molecular motor: a generic hybrid model applied to the F1-ATPase. New J. Phys. 14, 103023 (2012) 15. Pinkoviezky, I., Gov, N.S.: Modelling interacting molecular motors with an internal degree of freedom. New J. Phys. 15, 025009 (2013) 16. Ermak, D.L., McCammon, J.A.: Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69(4), 1352–1360 (1978) 17. Kemps, J.A.L., Bhattacharjee, S.: Particle tracking model for colloid transport near planar surfaces covered with spherical asperities. Langmuir 25(12), 6887–6897 (2009) 18. Günther, S., Kruse, K.: A simple self-organized swimmer driven by molecular motors. Eur. Phys. Lett. 84, 68002 (2008) 19. Ramia, M., Tullock, D.L., Phan-Thien, N.: The role of hydrodynamic interaction in the locomotion of microorganisms. Biophys. J. 65, 755–778 (1993) 20. MunJu, K., Powers, T.R.: Hydrodynamic interactions between rotating helices. Phys. Rev. E 69, 061910 (2004) 21. Fornés, J.A.: Hydrodynamic interactions induce movement against an external load in a ratchet dimer Brownian motor. J. Colloid Interface Sci. 341, 376–379 (2010) 22. Grimm, A., Stark, H.: Hydrodynamic interactions enhance the performance of Brownian ratchets. SoftMatter 7, 3219 (2011) 23. Polson, J.M., Bylhouwer, B., Zuckermann, M.J., Horton, A.J., Scott, W.M.: Dynamics of a polymer in a Brownian ratchet. Phys. Rev. E 82, 051931 (2010) 24. Dickinson, E.: Brownian dynamic with hydrodynamic interactions: the application to protein diffusional problems. Chem. Soc. Rev. 14, 421–455 (1985) 25. Oseen, C.W.: Hydrodynamik, Akademische Verlag Leipzig (1927) 26. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Claredon Press, Oxford (1986) 27. https://ocw.mit.edu/courses/nuclear-engineering/22-103-microscopic-theory-of-transportfall-2003/lecture-notes/lec3.pdf 28. Freund, J.A., Schimansky-Geier, L.: Diffusion in discrete ratchets. Phys. Rev. E 60(2), 1304 (1999) 29. Cubero, D., Rensoni, J.: Brownian Ratchets. Cambridge University Press, Cambridge (2016) 30. Houtman, D., Pagonabarraga, I., Lowe, C.P., Esseling-Ozdoba, A., Emons, A.M.C., Eiser, E.: Hydrodynamic flow caused by active transport along cytoskeletal elements. Europhys. Lett. 78, 18001 (2007)

Chapter 7

Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial Membrane

The intermembrane mitochondrial space (I MMS) is delimited by the inner and outer mitochondrial membranes and defines a region of molecular dimension where fluctuations of the number of free protons and of trans-membrane voltage can give rise to fluctuations in the proton-electromotive force (P MF ) across the inner mitochondrial membrane (I MM). We have applied the fluctuation-dissipation theorem (F DT ) to an electrical equivalent circuit consisting of a resistor (Rm ) in parallel with a capacitor (Cm ) representing the passive electrical properties of the I MM, in series with another capacitor (Cb ) representing the proton buffering power of the I MMS fluid. An access resistance, Ra , was defined as a link between the capacitor Cb and the membrane. Average P MF fluctuations across the I MM were calculated for different assumptions concerning the inter-membrane space dimensions. The calculated average P MF fluctuations were in the vicinity of 100 mV for relaxation times in the few microseconds range. The corresponding fluctuational protonic free energy is about 10 kJ/mole what compares with the binding energy for protons in different transporters. This suggests that fluctuations in PMF can be of relevance in the universe of forces influencing the molecular machinery embedded in the I MM.1

7.1 Introduction The synthesis of ATP in eukariotic cells occurs at the expense of proton flow driven by the proton-motive force (P MF ) across ATP synthase molecules inserted in the inner mitochondrial membrane (I MM). Proton gradients as high as 1 pH unit can

1 Reprinted from [1], Copyright DOI:https://doi.org/10.1103/PhysRevE.55.6285 by the American Physical Society.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_7

111

112

7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .

occur at that location [2] and superposition with electrical potential difference can lead to P MF s attaining the 200 mV mark. Since the intermembrane space in most of its traject has a width of only about 10–20 nanometers this dimension limits a region of molecular proportions where fluctuations in thermodynamic parameters may become an important part of the forces acting upon the molecular machines inserted alongside the inner mitochondrial membrane. This paper analyses some consequences of the particular geometry of the intermembrane space (I MS) in mitochondria, which may result in relatively important fluctuations of proton-motive force across the I MM.

7.2 Theory The free energy change, G, for the creation of an electrochemical gradient by an ion pump is, [2] (the SI system of units is employed throughout): G = RT ln

c2 + zF  c1

(7.1)

where c2 c1 −1 is the concentration ratio for the ion that moves, z is the ion valence, F is Faraday’s constant, R is the gas constant, T is the absolute temperature and  is the transmembrane difference in electrical potential measured in Volts. For the case of protons: ln

c2 = 2.3(log[H + ]out − log[H + ]in ) = 2.3pH c1

(7.2)

and Eq. (7.1) with z=1 reduces to: G = 2.3RT pH + F 

(7.3)

The protonmotive force (P MF ) is defined by: P MF = 2.3

RT pH +  F

(7.4)

When G = 0 (zero chemical driving force, P MF = 0), Eq. (7.3) can be used to relate the variations of pH across the membrane with voltage changes: pH = −

F  e =− 2.3RT 2.3kT

where e is the electronic charge and k the Boltzmann constant.

(7.5)

7.3 Fluctuations of the Proton-Electromotive

113

7.3 Fluctuations of the Proton-Electromotive In order to determine the fluctuations of the P MF across the I MM we need to know the susceptibility, α(ω), of the system comprised by the following elements (see Fig. 7.1): (1) The inner mithocondrial membrane with its associated resistance and capacitance. (2) The proton buffer compartment associated with the intermembrane fluid. (3) The access resistance between the two first elements. In doing the above associations we can resemble the system to a set of electrical resistors and capacitors. The I MM and its surrounding solutions is represented by the membrane capacitor, Cm . The conductive pathways across the I MM which include the pump channel and other leaks (including decouplers) are collectively represented by the term Rm . The buffering capacity of the intermembrane fluid is represented by an electrical equivalent (calculated below) defined as Cbuff er = Cb . This compartment is electrically connected to the I MM by an access resistance Ra . In this way, the relation between the Fourier components of the fluctuational charge, qω , and the corresponding components of the proton-electromotive force at that frequency, (P MF )ω , is (see for instance Procopio and Fornés [3]):

Fig. 7.1 Diagram showing the mitochondrial membrane and electrical equivalent

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7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .

qω = α(ω)(P MF )ω

(7.6)

The impedances of the membrane, Zm , and of the buffer system ,Zb , can be written respectively as: 1 1 = + iωCm , Zm Rm

1 1 = Ra + Zb iωCb

(7.7)

where i is the imaginary unit. The total impedance ZT is given by: ZT = Zm + Zb . From the relation between the susceptibility and the impedance, α(ω) = ωZi ω , we get for the real, α (ω), and imaginary, α

(ω), parts of the susceptibility:

α (ω) = −

ωRm τm 1+(ωτm )2

+

Ra ωτb

(7.8)

ωD(ω)

α

(ω) = −

Rm 1+(ωτm )2

+ Ra

(7.9)

ωD(ω)

where D(ω) is given by: D(ω) =

ωRm τm Ra + ωτb 1 + (ωτm )2

2

+

Rm + Ra 1 + (ωτm )2

2 (7.10)

where τm = Rm Cm and τb = Ra Cb are the corresponding relaxation times of both systems. Then the corresponding spectral density of the mean square of the fluctuational proton-motive force, [(P MF )2 ]ω will be given by [3]: [(P MF )2 ]ω =

α

(ω) 2kT | α(ω) |2 ω

(7.11)

where | α(ω) |2 = [α (ω)]2 + [α

(ω)]2 . The mean square of the fluctuating protonmotive force acting across an I MM patch is given by the integral: < (P MF )2 >=

1 π



∞

[(P MF )2 ]ω dω

(7.12)

0

7.4 Parameter Definitions In order to estimate the I MM patch capacitance we consider an idealized mitochondrion with an inter-membrane space about 15–20 nm wide. This minimum

7.6 Calculation of IMM Electrical Resistance Rm

115

dimension limits a cube with 15–20 nm side that defines a fluctuational domain (see Fig. 7.1). The inter-membrane space (space between the 2 mithocondrial membranes) can then be partitioned into a great number of parallel volumes that are considered to function independently, i.e. are uncorrelated. The I MM patch lining each of these elementary volumes has then an area of (225–400) nm2 . The patch electrical capacity is given by: Cm = 0

A d

(7.13)

where A is the area of the small I MM patch lining each elementary volume,  and 0 are, respectively, the relative dielectric constant of the membrane ( = 2) and permittivity of vacuum (0 = 8.85 × 10 −12 F m −1 ) and d is the I MM membrane thickness.

7.5 Calculation of Buffer Equivalent Electrical Capacitance The change of electrical charge, Q, associated with a given change, pH , of the pH of an elementary volume, V , of the intermembrane fluid is given by: Qb = F V [H + ] = F V βpH

(7.14)

where β is the specific buffering capacitance of the intermembrane fluid equal to 10 mM of acid added per pH unit change, β = [H + ]/pH , see Lauger [4]. This charge, stored in buffer capacitance, corresponds to a voltage change given by: b =

2.3RT pH F

(7.15)

The electrical equivalent of the buffer capacitance will then be: Cb =

Qb βF 2 V = b 2.3RT

(7.16)

7.6 Calculation of IMM Electrical Resistance Rm We assume a density of AT P synthase molecules such that there is an average of 1 pumping unit facing each cubic fluctuational domain whose lateral area is about 400 nm2 . The single-channel conductance of the AT P synthase is taken as

116

7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .

1 pSiemens at pH approximately 7.4 (Lauger [4], Lill et al. [5]), what corresponds to a resistance Rm = 1 × 1012 .

7.7 Relaxation Times of the Electrical and Buffer Reservoirs Coupling between the buffer and the membrane systems is expected to be of importance when the characteristic relaxation times (as defined by their corresponding RC) of both systems, are of the same order. We use this condition to calculate the access resistance Ra of the electrical link between the I MM and the buffer compartment, namely, τ = τm = τb , assuming that proton movement between the membrane and buffer compartments is fast relative to the proton translocation across the I MM (see Discussion). From this condition we obtain: Ra =

τ Cb

(7.17)

Figures 7.2 and 7.3 summarize our results. The coupling condition of the relaxation times of the membrane and buffer systems (Eq. (7.17)) makes the “spectral density” function (Eq. (7.11)) to decrease assimptotically with increasing radial frequency (see Fig. 6.2). This creates a convergence condition for the integral of Eq. (7.13), allowing for the determination of the mean square P MF fluctuation for a range of fluctuational domain sizes (Fig. 7.3). Figure 7.3 spans a relatively wide range of possible sizes of fluctuational domains and displays the corresponding mean P MF fluctuation and associated relaxation times. The factors limiting the size of a fluctuational domain were defined by its physical barriers (delimiting membranes, etc.) and the lack of correlation with neighbouring domains (distance factor). In the present case the physical limits are defined by the 2 mitochondrial membranes, limiting the extension of the intermembrane space. We chose this as a maximum size for a fluctuational domain, defining thus the value of 20–30 nm as an upper size limit for a fluctuational domain, with 10 nm being a definite size possibility in many types of mitochondria. Taking 15 nm as “typical” width for the I MS, Fig. 7.3 indicates a mean P MF fluctuation of 100 mV at pH 7.4 with a corresponding relaxation time of 1 μs. On another extreme, for a fluctuational domain size of 70 nm we obtain a mean P MF fluctuation of about 40 mV with a corresponding relaxation time of 18 μs. Both the amplitude of the mean P MF fluctuation and its characteristic relaxation time are relevant in any attempt at estimating the effects of a fluctuating P MF on the molecular machinery resident in the I MM. In order to be of relevance such fluctuational force is required to have a minimum amplitude and duration which are to be compared with the dynamic characteristics of the specific system upon which it acts, in this case, protons interacting with ATP synthase molecules. We shall consider two main effects of the fluctuating

7.7 Relaxation Times of the Electrical and Buffer Reservoirs

117

Fig. 7.2 Spectral density of the mean square of the fluctuational P MF (Eq. (7.9))

field upon the proton dynamics in the above system. The first effect concerns free proton translocation and the second is related to proton association/dissociation with binding sites along the transporter molecule. The rate of proton translocation associated with AT P synthesis is about 1200 s−1 , giving a translocation time of 8×10−4 per proton. Assuming that proton translocation across the F1 sector in either direction is a 3 step process (Stein and Läuger [6]) this gives 2.6×10−4 s as the mean time associated with each step. Proton translocation across the F0 sector is a substantially faster process with typical turnover time of 6×105 protons×s−1 per channel (for a conductance of 1 pSiemens and driving force of 100 mV). This defines a transit time for protons across the channel/pump synthase molecule of the order of 1–2×10−6 s (see Läuger [4] and Lill et al. [5]). Since complete proton translocation across the transporter involves many protonation-deprotonation reactions with specific sites alongside the channel region it is expected that the individual steps last only a fraction of the total translocation time, i.e. a fraction of 1 μs. This is to be contrasted with the dwelling time for protons in sites, estimated by Kazianowicz and Bezrukov [7] as ranging around the 100 μs mark.

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7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .

Fig. 7.3 Representation of < (P MF )2 >1/2 and the associated relaxation times (τ ) as a function of the possible sizes of fluctuational domains

The binding/debinding proton kynetic constants can, in principle, be modified by fluctuational electrochemical forces in the few μs characteristic times, provided the fluctuating energy has both amplitude and characteristic time of the same order as the energies involved in proton binding to and debinding from sites. The product of the fluctuational P MF by the proton charge, gives the fluctuation in proton energy. Taking a fluctuational P MF of about 100 mV and proton charge equal to 1.6×10−19 C, a fluctuational energy of about 10 kJoule×mole−1 is obtained. This is to be compared with the energy barrier for a water pore formation in a bilayer which is about 100 kJoule×mole−1 (Marrink et al. [8]) and 63 kJoule×mole−1 for proton dissociation from water (Deamer and Nichols [9]). In biological systems Woodhull [10] finds a pKa = 5.3 for the proton binding into nerve Na+ channels, what gives G = 30 kJoule×mole−1 . More recent studies of Kasianowicz and Bezrukov [7] in the α- toxin channel point to an effective pKa = 5.5 for proton binding with resulting G = 31.5 kJoule×mole−1 . These values of binding energy compare with the estimated fluctuations in proton energy due to the fluctuating P MF . Therefore, the fluctuational proton energy is of the same order as its binding energy what implicates the above described fluctuations as influencing proton binding/debinding. Concerning the times involved Gutman et al. [11] have reported protonation/depro-

7.7 Relaxation Times of the Electrical and Buffer Reservoirs

119

tonation times of about 2–3 μs in I MM preparation, what is of the same order as the relaxation time of the P MF fluctuations we describe above. The access resistance best estimated to match the common relaxation time of the membrane and buffer “compartments” was found to be 9 × 109 , what is about 100 times smaller than the proton channel resistance. This value is high as compared with proton resistance of an aqueous pathway having equivalent geometry. We took as an assumption that the limiting factor in proton translocation was the channel crossing step. In effect, DeCoursey and Cherny, 1996 [12] concluded that H+ diffusion is not a rate limiting step in the overall proton translocation across channels. Additional reason for adjusting the access resistance value is that the mechanisms of proton translocation near or at biological interface are far from being adequately understood. Values of reported, calculated or predicted proton conductance vary considerably (Nagle and Morowitz [13], Heberle et al. [14]). The general consensus, however, is that proton translocation across microdomains is significantly faster as compared to bulk water. Since proton conductivities in or near surfaces are subject to great controversy (see Kasianowicz and Bezrukov [7]) it is reasonable to make the access resistance an adjustable parameter, rather than estimate its value from the proton mobility in bulk water and geometric parameters. Considering that typical time averaged P MF s across I MM are about 140 mV (Läuger [4]), the values of mean P MF fluctuation here reported can be considered as relevant to the proton dynamics in the μs time scale. The effects of this P MF fluctuations should be interpreted along the following lines: (a) Coupling of P MF fluctuations with pump/synthase conformational states, different pump states and their corresponding dwell times (Stein and Läuger [6]). (b) Coupling of the fluctuations with protonation/deprotonation of sites: part of the influence of the fluctuating P MF could be felt at the protonation/deprotonation dynamics and part involved in the modulation of the channel conductance for other ions (see Kazianowicz and Bezrukov [7]). The above calculated fluctuations of P MF across the I MM should be interpreted in connection with the molecular machines under their influence. Two general assumptions about the pump/synthase reacting time serve as the basis of our analysis: (1) The assumption of a “slow” pump mechanism translate into a system reacting to only time averaged P MF , calculated from the known values of pH and voltage across the I MM. (2) If the reacting time of the synthase machine is “fast”, say in the 10 μs range, then the fluctuational changes of P MF may importantly interfere with its function. We conclude that the 100 mV fluctuations in P MF having characteristic time of 1 μs may be relevant for the processes of proton translocation inside the F0 sector of the AT P ase. Lesser amplitude P MF fluctuations with corresponding longer relaxation times might influence conformational changes in the AT P ase molecule.

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7 Fluctuations of the Proton Electromotive Force Across Inner Mitochondrial. . .

7.8 Program 7.1, Fluctuations of the PMF, Matlab Code % Fluctuational Domain size = 30 (nm) %—–Assumes that TauBuffer = TauMembrane and calculates Racces Nsteps = 700; N=Nsteps; w=(1:N)*0; PMFsqw=(1:N)*0; RT = 2493; kT = 4.143E-21; avogadro = 6.02E+23; faraday = 96500; pi = 3.1416; electron = 1.602E-19; epszero = 8.85E-12; eps = 2; integral=0.; diff = 1.E-07; % Diffusion coefficient of proton in m2/sec = 1 x 10-8 d = 7E-09; %Membrane width in meters cubeside = 1.9E-08; % = Cube side in meters = intermembrane space memarea = cubeside2 %Membrane Area patch in m2 cubevolume = cubeside3 ; %Cube volume in m3 Beta = 10; %Buffering capacitance – 10 Moles of acid added %in 1 m3 = 1 unit change in pH %From Lauger (Electrogenic ion pumps) pH = 7.4; molarm3 = (10( − pH )) * 1000; %Concentration of protons in moles/m3 protonsm3 = molarm3 * avogadro; %Num. of protons/m3 numproton = cubevolume * protonsm3 ; %Number of protons in the cube SCC = (molarm3 / (.000063)) * (1E-12); %Lauger, at pH aprox. = 7.2 %—-SCC is the single-channel cond. of the pump normalized for the given pH Rm = 1 / SCC %Resistance of membrane patch in ohms Cm = eps * epszero * (memarea / d) Taum = Rm * Cm % seconds Taubuff = Taum %Assumes that Taubuff = Taum Cbuffer = (Beta * cubevolume * faraday * faraday) / (2.3 * RT) Raccess = Taubuff / Cbuffer %————-Frequencias inicial e final em Hz —– freqinicial = .001; freqfinal = 1.E+06; freqmedia = (freqfinal - freqinicial) / 2; omegainicial = freqinicial * 2 * pi; omegafinal = freqfinal * 2 * pi; %Nsteps Number of steps in integration deltaOmega = (omegafinal - omegainicial) / Nsteps %Pass of integration di=1; i=1; for omega = omegainicial : deltaOmega : omegafinal i=i+di; frequency = omega / (2 * pi); Domega1 = ((omega * Rm * Taum) / (1 + (omega * Taum)2 )) + (Raccess / (omega * Taubuff)); Domega2 = (Rm / (1 + (omega * Taum)2 )) + Raccess; domega = (Domega12 ) + (Domega22 ); %Eq. (7.8) alfa1A = (omega * Rm * Taum) / (1 + (omega * Taum)2 );

Bibliography

121

alfa1B = Raccess / (omega * Taubuff); alfa1 = -(alfa1A + alfa1B) / (omega * domega); % Eq.(7.6) alfa2A = Rm / (1 + (omega * Taum)2 ); alfa2B = Raccess; alfa2 = (alfa2A + alfa2B) / (omega * domega); % Eq.(7.7) alfasquare = (alfa12 ) + (alfa22 ); PMFsqOmega = (2 * kT * alfa2) / (alfasquare * omega);% Eq.(7.9) integral = integral + deltaOmega * PMFsqOmega; % Eq. (7.10)*pi w(i)=omega; PMFsqw(i)=PMFsqOmega; end figure(1); plot(w, PMFsqw);grid; PMFsq = integral / pi % Eq(10) MPMF = 1000*sqrt(PMFsq) %Mean PMF fluctuation “mV”

Bibliography 1. Procopio, J., Fornés, J.A.: Fluctuations of the proton-electromotive force across the inner mitochondrial membrane. Phys. Rev. E 55(5), 6285–6288 (1997) 2. Lehninger, A.L., Nelson, D.L., Cox, M.M.: Principles of Biochemistry. Worth Publishers, Inc., New York (1993) 3. Procopio, J., Fornés, J.A.: Fluctuation-dissipation theorem imposes high-voltage fluctuations in biological ionic channels. Phys. Rev. E. 51, 829 (1995) 4. Läuger, P.: Electrogenic Ion Pumps. Sinauer Associates, Inc. Publishers, Sunderland (1990) 5. Lill, H., Althoff, G., Junge, W.: Analysis of ionic channels by a flash spectrophotometric technique applicable to thylakoid membranes: CF0, the proton channel of the chloroplast ATP synthase, and, for comparison, Gramicidin. J. Membrane Biol. 98, 69 (1987) 6. Stein, W.D., Läuger, P.: Kinetic properties of F0F1-ATPases. Theoretical predictions from alternating-site models. Biophys. J. 57, 255 (1990) 7. Kasianowicz, J.J., Bezrukov, S.: Protonation dynamics of the alpha-toxin ion channel from spectral analysis of pH-dependent current fluctuations. Biophys. J. 69, 94 (1995) 8. Marrink, S.J., Jahnig, F., Berendsen, J.C.: Proton transport across transient single-file water pores in a lipid membrane studied by molecular dynamics simulations. Biophys. J. 71, 632 (1996) 9. Deame, D.W., Nichols, J.W.: Proton flux mechanisms in model and biological membranes. J. Membrane Biol. 107, 91 (1989) 10. Woodhull, A.M.: Ionic blockage of sodium channels in nerve. J. Gen. Physiol. 61, 687 (1973) 11. Gutman, M., Kotlyar, A.B., Borovok, N., Nachliel, E.: Reaction of bulk protons with a mitochondrial inner membrane preparation: time-resolved measurements and their analysis. Biochemistry 32, 2942 (1993) 12. DeCoursey, E., Cherny, V.V.: Effects of buffer concentration on voltage-gated H+ currents: does diffusion limit the conductance? Biophys. J. 71, 182 (1996) 13. Nagle, J.F., Morowitz, H.J.: Molecular mechanisms for proton transport in membranes. Proc. Natl. Acad. Sci. USA 75, 298 (1978) 14. Heberle, J., Riesle, J., Thiedemann, G., Oesterhelt, D.N.A., Dencher, N.A.: Proton migration along the membrane surface and retarded surface to bulk transfer. Nature 370, 379 (1994)

Chapter 8

Quantum Ratchets

In statistical mechanics, there exists a commonly accepted postulate according to which a “small” system interacting with a “large” system (a heat bath), which stays in the state of statistical equilibrium, eventually approaches the state of statistical equilibrium itself. Bogolyubov, [1, 2] rigorously proved the validity of this postulate for the first time. As the system and the heat bath, he considered an oscillator linearly coupled to a collection of harmonic oscillators, simulating the bath in which the single oscillator is immersed. However, without explicitly working out the statistical properties of the fluctuations. These properties, together with a quantum mechanical transcription of the model, has been accomplished by Magalinski˘i [3]. Is this model we will unwrap in this chapter and we apply it to understand a quantum ratchet in the quantum range. For an excellent review of the evolution of the quantum Langevin equation, see P. Reimann [4].

8.1 The Quantum Langevin Equation Consider an oscillator with mass m and frequency ω0 , linearly coupled with a set of a large number of independent harmonic oscillators with frequencies ωk (k = 1, 2, . . . , N; N 1), simulating the bath in which the single oscillator is immersed. The Hamiltonian of the system is given by H =

 2 1 1 2 2 p2 + mω02 x 2 + pk + ωk2 xk2 + gx βk xk 2m 2 2 k

(8.1)

k

where x and p are the corrdinate and momentum of the single oscillator, xk and pk are those of the k − th oscillator of the medium, and gβk is the coefficient of the coupling with the k − th oscillator. To obtain the equation of the motion for the single oscillator we have to solve de system of Hamilton’s equations corresponding © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9_8

123

124

8 Quantum Ratchets

to the Hamiltonian of Eq. 8.1 namely, x˙ = q˙k =

∂H ∂p

∂H ∂pk

p˙ = −

∂H ∂x

(8.2)

p˙k = −

∂H ∂qk

(8.3)

From Eqs. 8.1, 8.2, 8.3, we obtain, x¨ + ω02 x = −

g 2 βk qk m

p m

(8.4)

q˙ = pk

(8.5)

x˙ =

k

q¨k + ωk2 qk = −gβk x

In accordance of Appendix, the solution of Eq. 8.5 is given by, 

I m(ξk βk = −g ωk ωk

t

pk0 sin(ωk t) + qk0 cos(ωk t) ωk 0 (8.6) Replacing in the former Eq. 8.6 sin(ωk (t − τ ))dτ = ω1k d[cos(ωk (t − τ ))], we get qk =

I m(ξk βk qk = = −g ωk ωk



t

x(τ )sin(ωk (t − τ ))dτ +

x(τ )d[cos(ωk (t − τ ))] +

0

pk0 sin(ωk t) + qk0 cos(ωk t) ωk (8.7)

Performing the integration by parts, we can write, g

2 k



t

βk qk =

K (t − τ ) x(τ ˙ )dτ − K (0) x(t) + x(0)K (t) − f (t)

(8.8)

2 β2 cos(ωk (t)) ωk2 k

(8.9)

0

where K (t) = g 2 and f (t) = −g

2 k

βk

pk0 qk0 cos(ωk (t)) + sin(ωk (t)) ωk

 (8.10)

and 5 qk0 and pk0 are the initial values of the canonical variables. Replacing g k βk qk given by the former equation in Eq. 8.8, we obtain,

8.1 The Quantum Langevin Equation

 mx¨

+ mω02 x

+

t

125

K (t − τ ) x˙ (τ ) dτ − xK (0) + q0 K (t) = f (t)

(8.11)

0

We can observe that the influence of the bath is through the random force f (t) and a dissipative force, third term of the equation 8.11. We derive now the correlation quantum function c(t − τ )f = f (t)f (τ ), where the average is over all the microscopic states of the oscillators of the medium. If the medium is a thermostat, i.e., if these states are canonically distributed, we have pi0 pk0  = δij E(ωk , kT ), qi0 qk0  = δij E(ωk , kT )ωk−2 ,

(8.12)

qi0 pk0  = 0. where E (ωk , kB T ) =



1 h¯ ωk hω ¯ k coth 2 2kB T

(8.13)

is the average energy of the k-th oscillator at temperature T , and kB is the Boltzmann constant. From Eqs. 8.10, 8.13, 8.13 we obtain, cf (t − τ ) = f (t)f (τ ) = g 2

2

βk2 ωk−2 E(ωk , kT ) cos[ωk (t − τ )]

(8.14)

k

Assuming the frequency spectrum of the oscillators of the bath sufficiently dense and replacing the sums by integrals by the rule, 2



∞

Fk = A

F (ω)ω2 dω

0

k

we get  K(t) = Ag

∞

2



β 2 (ω) cos(ωt)dω,

(8.15)

β 2 (ω)E (ωk , kB T ) cos(ωt)dω

(8.16)

0 ∞

cf (t) = Ag 2 0

In order to right the Magalinskii˘ equation 8.11, in the usual Langevin’s equation, we have to make the following definitions, K(t) = 2δ(t),

x(0) = 0.

(8.17)

126

8 Quantum Ratchets

then mx¨ +  x˙ + mω02 x = f (t)

(8.18)

From Eq. H.6 Z(ω) = (ω) + i

 m 2 ω − ω02 ω

(8.19)

where ω0 = (k/m)1/2 is the resonance frequency and k is the force constant the oscillator. The force f(t), 

t

f (t) =

K (t − τ ) x˙ (τ ) dτ

(8.20)

0

From Appendix C, we can write, the force in the frequency, ω, domain, f (ω) = Z(ω)x(ω) ˙

(8.21)

where f (ω) and x(ω) ˙ are the Fourier components of the dissipative force  f (t) =

∞

 f (ω)exp(−iωt)dω =

0

∞

Z(ω)x(ω)exp(−iωt)dω ˙

(8.22)

0

Then the real part of f(t) is,  Re[f (t)] =

∞

Re[Z(ω)]x(ω) ˙ cos(ωt)dω

(8.23)

0

namely, 

∞

Re[f (t)] =

(ω)x(ω) ˙ cos(ωt)dω

(8.24)

0

Comparing with Eq. 8.15,we have, (ω) = Ag 2 β 2 (ω)

(8.25)

then  K(t) =

∞

(ω) cos(ωt)dω

(8.26)

0

If there are an infinite number of oscillators in the bath and their frequencies are continuously distributed. We can identify (ω) = N(ω)k(ω), where N(ω)dω is the number of oscillators whose natural frequencies are between ω and ω + dω and

8.1 The Quantum Langevin Equation

127

k(ω) is the average force constant of the oscillators whose frequency is ω. Finally, from Eq. 8.16, we obtain the general formula that connects the correlation of the fluctuation force with the dissipative properties of the system: 

∞

cf (t) =

N(ω)k(ω)E (ωk , kB T ) cos(ωt)dω

(8.27)

0

8.1.1 The Correlation Quantum Function  The correlation function cf (t − t ) = f (t)f (t ) S , where f (t) is the c-number quantum noise for the force, in the continuum limit is [5] cf (t − t ) =

1 2



∞ 0

N(ω)k(ω)hω. ¯ coth

hω ¯ 2kB T



cos[ω(t − t )]dω

(8.28)

where kB denotes the Boltzmann constant and T the absolute temperature. For a Lorentzian distribution of bath modes characterized by a density function N(ω), (ω) =

0 , 1 + (ωτc )2

(8.29)

where 0 is the dissipation constant and τc refers to the correlation time and τs is the system characteristic time, namely: τc =

h¯ , 2kB T

τ=

τc , τs

y = ωτc τs ,

$t = (t − t )/τ τs 

(8.30)

Replacing Eq. 8.29 in Eq. 8.28 and using the Einstein relation 0 .D0 = kB T , we get, $t) = 02 cf (

D0 τc



∞

dy 0

y $t) coth(y). cos(y  1 + y2

(8.31)

 the velocity quantum noise correlation function (t)(t ) is, $t) = cv (

D0 τc



∞

dy 0

y $t) coth(y). cos(y  1 + y2

(8.32)

From now on, we will use the effective diffusion coefficient instead D0 , in the former and next equations, [7, 8],

128

8 Quantum Ratchets

Deff (x) =

#−1 1 " 1 − λβU

(x) β0

(8.33)

where the prime denotes the derivative with respect to the coordinate x. The quantum correction parameter λ describes quantum fluctuations in position given by λ=



 τc 0 h¯ γ + 1+ , π 0 Mπ

β=

1 kB T

(8.34)

Here, (z) is the digamma function, γ  0.5772 the Euler-Mascheroni constant. Note that for kB T  h¯ 0 /M, λ becomes

 h¯ τc 0 γ + ln , (8.35) λ= π 0 Mπ where, for a moderate damping, [9],

0 = M0

with

0 = 2π

2

U0 k B T L2 M

1/2 (8.36)

where the dimensionless U0 is the amplitude of the ratchet potential and M is the mass of the particle.

8.1.1.1

Dimensionless Parameters

Defining (x) ˜ = U (Lx)/U ˜ , x = Lx, ˜ λ = L2 λ˜ , β˜ = U /kB T , U , τs = L2 /U F = (U/L) F

(8.37)

we obtain the rescaled dimensionless diffusion coefficient  −1  

(x) ˜ = 1 − λ˜ β˜ U ˜ D eff (x)

(8.38)

we have used in 8.33: U

(x) =

U $

x) U ( L2

(8.39)

The barrier height U is the difference between the maximal and minimal values of the unbiased potential.

8.1 The Quantum Langevin Equation

129

8.1.2 The Quantum Overdamped Langevin Equation with Colored Noise We follow the formalism, except for the noise, given by [10] and used by [11], namely x˙ = −

F λ U (x) + − 0.5 U

(x) +  (t) 0 0 0

(8.40)

U (x) is the internal energy generated by the motor, F is an external force. See bellow for the noise  (t). Then from the fitting of Fig. 8.1, we can write, I ntegral =

 1 1 1 t 1 exp − τk τk

3 2 Dk k=1

(8.41)

or,  1 1 3 1  ts 1 2  ( x) ˜ D D eff k ˜ (t˜)˜ (t˜ ) = exp − τ τk τk τ





(8.42)

k=1

where we have considered that the system itself is which perceives the time scale of the noise, consequently, we must measure time in units of τs , Equation 8.42, was equivalently written by [6],

Fig. 8.1 Integral from Eq. 8.32 fitted with 3 exponentials

130

8 Quantum Ratchets

 1 1  2 3 1  ts 1 τs  Dk τs

 ˜ (t˜)˜ (t˜ ) = Deff (x) ˜ exp − τc τk τk τc

(8.43)

k=1

8.1.2.1

The Dimensionless Quantum Overdamped Langevin Equation

From Eqs. 8.37, 8.38, 8.46, we obtain  d x

(   − 0.5 = −$ U ( x) + F λU x ) + ˜ t˜ d t

(8.44)

where we have used U

(x) =

U  U

( x) L3

(8.45)

In Fig. 8.2 are represented the contributions, (blue curves) of the first term, (Fig. 8.2a), and third term, (Fig. 8.2b). we can observe that this third term, quantum correction, dominates the equation in this simulation. In Fig. 8.3 we can see that the system behaves as a brownian motor, for the given parameters.

8.1.3 The Quantum Underdamped Langevin Equation In reference [12] Denisov et al. investigated the quantum ratchet effect under the influence of weak dissipation. They predicted a ratchet current when all relevant symmetries are violated (Fig. 8.4). in obtained these results they used a FloquetMarkov master equation approach. In this section we will use the quantum Langevin equation to treat the same problem, namely (Fig. 8.5), M

dv = −Mγ v − U (x) − 0.5λU

(x) + F +  (x, t) dt

(8.46)

We follow the parametrization of [12], namely,  1 M L 1 1 1 , x= = x, ˜ λ = 2 λ˜ , τs = , kL 2π kL k L U0 kL  dv U0 U0 d v˜ 1 6 v, ˜ = kL MU0 h¯˜ , v= , h¯ = ˜ M dt M dt kL Then the complete quantum Langevin equation is given by,

γ =

1 γ˜ , τs

τc $0 (8.47) = 0.5U τs

8.1 The Quantum Langevin Equation

˜ Fig. 8.2 Contributions to =    Eq. 8.44 for Fload U0 2π 1 4π −sin x + cos x , tilted slightly to the left 2π L 4 L

131

0.

and

  d v˜

(x)   +  x, ˜ − 0.5λ˜ U ˜ +F ˜ t˜ = −γ˜ v˜ − $ U (x) d t˜

λ˜ =0.146,

U (x)

=

(8.48)

We observe in Fig. 8.6a λ increases as kB T tends to zero (temperature), which means quantum effects increase. We usually use the following ratchet, lightly tilded to the right.

132

8 Quantum Ratchets

Fig. 8.3 Brownian motor behavior of Eq. 8.44 for FDN = 1., τDN = 0.4, T = 1.o K, β = U = 10, U0 = 5., L = 8.nm, λ˜ = 9.311 nm2 , τs = 2.887 ps, U/L = 17.26 f N Force Unit, L − −31 Kg,  = 0 τs = 2771.0 m/s Velocity Unit, γ = 0 = 6.84 THz, M(e ) = 9.1091 × 10 M.0 = 6.23 Kg.aHz, νc = 0.823THz, δt = 10−12

Fig. 8.4 Linear Fit at the long time limit of the curve x ˜ vs t˜ in order to obtain the mean velocity for a given temperature, same parameters as Fig. 8.3 with Fload = 0

8.1 The Quantum Langevin Equation

133

Fig. 8.5 Contributions to Eq. 8.44 for F˜load = 0. and λ˜ =0.146

U (x) = U0 (+cos(kL x) − 0.25cos(2kL x)) U (x) = U0 kL (−sin(kL x) + 0.5sin(2kL x)) U

(x) = U0 kL 2 (−cos(kL x) + cos(2kL x)) U

(x) = U0 kL 3 (+sin(kL x) − 2sin(2kL x))

(8.49)

 U (x) = U0 U (x) ˜ U (x) = U0 kL $ ˜ U (x) $

(x) U

(x) = U0 kL 2 U ˜

(x)  U

(x) = U0 kL 3 U ˜

(8.50)

8.1.4 The Ranges Indeed the ranges were classified by J. Ankerhold et al [10], which were obtained starting from the classical Smoluchowski equation which is similar to the FokkerPlanck equation, we follow this classification:

8.1.4.1

Classical Range γ  νM

(8.51)

134

8 Quantum Ratchets

Fig. 8.6 (a)λ vs kB T , γ = 0.1, h¯ = 1.; (b) functions and parameters dimensionless

λ

vs

γ , kB T = 1.E − 4, h¯ = 1. The

where νM = kB T /h¯ is the first Matsmubara freqüency. 8.1.4.2

Classical Smoluchowski Range ω0 h¯  1. kB T

1 2 2 Mω0 (q

U

(q0 ) M , − q0 )2

8.1.4.3

Quantum Range

where ω02 =

and

kB T hγ ¯

(8.52)

from the Brownian motion in the harmonic potential U (q) =

γ νM

(8.53)

8.1 The Quantum Langevin Equation

8.1.4.4

Quantum Smoluchowski Range γ h¯ , 2 kB T ω0

8.1.4.5

135

1 γ

and

kB T  h¯ γ

(8.54)

Discretization of the Quantum Underdamped Langevin Equation

In order to build the software, we will discretize Eq. 8.61, transforming it in two equations. We can write the particle velocity as v(i, ˜ j) =

x(i ˜ + 1, j ) − x(i ˜ − 1, j )  2dt

(8.55)

where the index i represents the time and the index j the stochastic representation. Finally the two equations for solving the Eq. 8.61 are given by,    +  x,  x(i ˜ + 1, j ) = x(i ˜ − 1, j ) + 2v(i, ˜ j )dt ˜ t˜ dt

(8.56)



(x(i,  ]dt  v(i ˜ + 1, j ) = v(i ˜ − 1, j ) + 2[−γ˜ v(i, ˜ j) − $ U (x(i, ˜ j )) − 0.5λ˜ U ˜ j )) + F

(8.57) We consider now the model of Ref.[12]. A quantum particle in a time-dependent potential obeys the Schrödinger equation / 0 ∂ h¯ 2 ∂ 2 i h¯ ψ(x, t) = − + U (x, t) ψ(x, t) ∂t 2M ∂x 2

(8.58)

U (x, t) = U0 u(x) − xE(t), where u(x) = u(x + L) with max| u(x) |∼ 1. The driving E(t) is a time-periodic field with zero mean, E(t + T ) = E(t),E(t)T = 0.. By the transformation |ψ → exp − hi¯ xA(t) |ψ, we bring the Schrödinger equation 8.58 to the spatially periodic form [13] ∂ i h¯ ψ(x, t) = ∂t



 #2 1" pˆ − A(t) + u(x) ψ(x, t) 2

(8.59)

t with the vector potential A(t) = − 0 E(t )dt and the momentum operator pˆ = −i h∂/∂x. The quantum Langevin Equation corresponding to the Hamiltonian in ¯ Eq. 8.59 is given for by   d x˜ = v˜ +  x, ˜ t˜ d t˜

(8.60)

136

8 Quantum Ratchets

d v˜

(x)  $ (x) ˜ − 0.5λ˜ U ˜ + E(t˜) + γ˜ A(t˜) = −γ˜ v˜ − U d t˜

(8.61)

with E(t˜) = E1 cos(ω˜ t˜) + E2 cos(2ω˜ t˜ + θ ) A(t˜) = −

# sin(ω˜ t˜) " E1 + E2 cos(ω˜ t˜ + θ ) ω˜

(8.62) (8.63)

In the present model, Ref.[12], with U0 = 1., we have (x) U ˜ = +cos(x) ˜ − xE(t) $ ˜ = −sin(x) ˜ − E(t) U (x)

(x)  ˜ = +sin(x) ˜ U

(8.64)

Then the Eq. 8.61 gives   d x˜ = v˜ +  x, ˜ t˜ d t˜   d v˜ = −γ˜ v˜ + 1. − 0.5λ˜ sin(x) ˜ + E(t˜) + γ˜ A(t˜) d t˜

(8.65) (8.66)

In Fig. 8.7 are shown the diagramm of these functions in order to compare the influence of each term in the final results. In Fig. 8.8 is shown the representation of Fig. 8.7 in three dimensions, namely Figure 8.10a depicts the ratchet velocity (current) as a function of the dissipation strength γ . For γ > 0 we observe a purely dissipation-induced quantum ratchet current. This current is negative for faint dissipation, but crosses zero and becomes positive with increasing dissipation. This current reversal behaviour resembles the one found for the corresponding classical problem [14] but even there has not explained analytically (Fig. 8.9). Figure 8.10b shows the ratchet velocity (current) as a function of the phase lag θ for two different dissipation strength. In the Hamiltonian limit γ → 0, the current vanishes at the symmetry points θ = 0 and θ = π . This effect was very well explained by Denisov et al.[12]. For finite dissipation, the current exhibits multiple current reversal upon changing the phase lag θ .

8.1 The Quantum Langevin Equation Fig. 8.7 Contributions to Eq. 8.61 for λ˜ = 12.36, E1 = 1.6; E2 = 2.; θ = π2 ; ω˜ = 0.8.. The functions and parameters are dimensionless

137

138

Fig. 8.8 Representation of Fig. 8.7 for U(x,t) in 3 dimensions Fig. 8.9 (a) Poincaré map v (velocity) vs x, obtained from Eqs. 8.61, Program8.1b, under quantum range, N=2E5, NR=1 γ νM , γ = 0.1, kB T = 1.E − 4, h¯ = 1., M = 1., dt = 1.E − 4, E1 = 1.6, E2 = 2., ω = 1. θ = π , functions and parameters are dimensionless. (b) Idem. with v and x as a function of time

8 Quantum Ratchets

8.1 The Quantum Langevin Equation

139

T Fig. 8.10 Mean velocity, v = T1 0 v(t)dt, (a) v vs γ , (b) v vs θ for different values of γ . Program8.1b, moderate quantum range γ /νM = 20., N = 2000, N R = 1, kL = 1., F ext = 0.; kB T = 1.E − 4, h¯ = 1., M = 1.,dt = 1.E − 4, E1 = 1.6, E2 = 2., ω = 1. θ = 0., functions and parameters are dimensionless

140

8 Quantum Ratchets

8.2 Programs 8.2.1 Program 8.1a, Moderate Damping, Matlab Code clear; %MODERATE DAMPING N=1000; NR=100; L=8.E-09; Fext=-0.e-1; pi=3.14159; kB=1.38054e-23; hp=1.05457266E-34; T=1.0; M=9.1091E-31; dt=1.e-12; tau=0.4; FDN=1.; qe=1.6E-19; %======================================================== % QUANTUM PARAMETERS and limits %======================================================== kBT=kB*T beta=1/kBT; Dbeta=10.; Uo=Dbeta/2; DU=Dbeta; %================================================ %Parameters from P. Reimann et al. Phys Rev Lett 79,1,10,(1997). and % L. Machura et al arXiv:cond-mat/0611277v2 [cond-mat.soft] 10 Jan 2007. %======================================================= %Wo=(2*pi2 /L)*sqrt(Uo/M) %Gamma=M*Wo; moderate damping % ts = Gamma.L2 /DU %characteristic time. %======================================================= Wo=((2*pi)/L)*sqrt(Uo*kBT/M); Gamma=M*Wo; ts=(Gamma*L2 )/(DU*kBT); Velocityunit=(L/ts); Forceunit=DU*kBT/L; tc=hp/(2*pi*kB*T);%Thermal correlation time tcs=tc/ts; nu=2*pi*kBT/hp %(for T = 1pK, nu=0.2618 1/s MATSUBARA FREQUENCY R1=Wo/nu; y=1+(tc*Gamma/(M*pi)); lambda=(hp/(pi*Gamma))*(-psi(0,1) + psi(0,y)); lambda1=(hp/(pi*Gamma))*log(hp*beta*Gamma/(2.*pi*M)); %Another aproximation of lambda. Dlambda=lambda/L2 ; %======================================================= % CLASSICAL RANGE Wo « nu, %========================================================= % QUANTUM RANGE Wo » nu %========================================================= % DICHOTOMOUS MARKOV NOISE %=========================================================

8.2 Programs

141

eta=zeros(N,NR); for j=1:NR for i=1:N-1 Pat1at = 0.5*(1. + exp(-i*dt/tau)); %Probability to stay in "-1" Pat2at1 = Pat1at; Pat1bt = 0.5*(1. - exp(-i*dt/tau)); %Transition Probability 1->0 Pat2bt1 = Pat1bt; Pbt2bt1 = 0.5*(1. + exp(-i*dt/tau)); %Probability to stay in "1" Pbt1bt = Pbt2bt1; Pbt1at = 0.5*(1. - exp(-i*dt/tau)); %Transition Probability 0->1 Pbt2at1 = Pbt1at; R=unifrnd(0,1); if(Pat1at > R) eta(i+1,j)=-1.; else eta(i+1,j)=1.; end if(Pat1at < R || Pat1at > R) R1=unifrnd(0,1); elseif(Pat2bt1 > R1 || Pat2at1 > R1) % || significa or eta(i+2,j)=-1.; else eta(i+2,j)=1.; end end end %====================================================== tR=tc/ts; INVTR=1/tR; A1=6.50618; A2=3.89653; A3=3.01154; t1=8.33507e-5; D1=A1*t1; t2=0.01425; D2=A2*t2; t3=0.3896; D3=A3*t3; x=zeros(N-1,NR); v=zeros(N-1,NR); Dd1U=zeros(N-1,NR); Dd2U=zeros(N-1,NR); Dd3U=zeros(N-1,NR); DDeff=zeros(N-1,NR); eps=zeros(N-1,NR); h=zeros(N-1,NR); eps1=zeros(N-1,NR); h1=zeros(N-1,NR); eps2=zeros(N-1,NR);

142

8 Quantum Ratchets

h2=zeros(N-1,NR); eps3=zeros(N-1,NR); h3=zeros(N-1,NR); gw=zeros(N-1,NR); a1 = abs(randn(1)); b1 = abs(randn(1)); a2 = abs(randn(1)); b2 = abs(randn(1)); a3 = abs(randn(1)); b3 = abs(randn(1)); E1 = exp(-(dt/(t1*tR))); E2 = exp(-(dt/(t2*tR))); E3 = exp(-(dt/(t3*tR))); xij=x(1,1); %U(i,j)=-(Uo/(2*pi))*(sin(2*pi*xij/L) - 0.25*sin(4*pi*xij/L)); %Potential tilded to the left. %Dd1U(1,1)=(Uo/L)*(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij))*(L/DU); Dd2U(1,1)=-Uo*(2.*pi/L2 )*(-sin(2.*pi*xij) + sin(4.*pi*xij))*(L2 /DU); %Dd3U(1,1)=Uo*((2.*pi2 )/L3 )*(-cos(2.*pi*xij) + 2.*cos(4.*pi*xij))*(L3 /DU); DDeff(1,1)=1./(1-Dlambda*Dbeta*Dd2U(1,1)); %DDeff(1,1)=D; eps1(1,1) = sqrt((-2.*(DDeff(1,1)*D1)/(t1*tR))*(1-E12 )*log(a1))*cos(2*pi*b1); eps2(1,1) = sqrt((-2.*(DDeff(1,1)*D2)/(t2*tR))*(1-E22 )*log(a2))*cos(2*pi*b2); eps3(1,1) = sqrt((-2.*(DDeff(1,1)*D3)/(t3*tR))*(1-E32 )*log(a3))*cos(2*pi*b3); eps(1,1)=eps1(1,1)+eps2(1,1)+eps3(1,1); for j=1:NR for i=1:N-1 xij=x(i,j); Dd1U(i,j)=-(Uo/L)*(cos(2.*pi*xij) - 0.5*cos(4.*pi*xij))*(L/DU); Dd2U(i,j)=-Uo*(2.*pi/L2 )*(-sin(2.*pi*xij) + sin(4.*pi*xij))*(L2 /DU); Dd3U(i,j)=-((Uo*(2.*pi)2 )/L3 )*(-cos(2.*pi*xij) + 2.*cos(4.*pi*xij))*(L3 /DU); DDeff(i,j)=1./(1-Dlambda*Dbeta*Dd2U(i,j)); %======================================================== % QUANTUM LANGEVIN EQUATION %======================================================== x(i+1,j)=xij - Dd1U(i,j)*dt - 0.5*Dlambda*Dd3U(i,j)*dt +eta(i,j)*FDN*dt. . . + Fext*dt + eps(i,j)*dt; v(i+1,j)=(x(i+1,j)-xij)/dt; %======================================================== % NOISE %================================================

8.2 Programs

143

a1 = abs(randn(1)); b1 = abs(randn(1)); a2 = abs(randn(1)); b2 = abs(randn(1)); a3 = abs(randn(1)); b3 = abs(randn(1)); h1(i,j) = sqrt((-2.*(DDeff(i,j)*D1)/(t1*tR))*(1-E12 )*log(a1))*cos(2*pi*b1); h2(i,j) = sqrt((-2.*(DDeff(i,j)*D2)/(t2*tR))*(1-E22 )*log(a2))*cos(2*pi*b2); h3(i,j) = sqrt((-2.*(DDeff(i,j)*D3)/(t3*tR))*(1-E32 )*log(a3))*cos(2*pi*b3); eps1(i+1,j)=eps1(i,j)*E1 + h1(i,j); eps2(i+1,j)=eps2(i,j)*E2 + h2(i,j); eps3(i+1,j)=eps3(i,j)*E3 + h3(i,j); eps(i+1,j)=(eps1(i+1,j)+eps2(i+1,j)+eps3(i+1,j)); %======================================================== end end MEANXR=(sum(x,2))/NR; MEANVR=(sum(v,2))/NR; %MEANEPS=(sum(eps,2))/NR; %MEANDeff=(sum(DDeff,2))/NR; t=(1:N)*dt; t=t’; save t.dat t -ascii; figure(1); plot(t,MEANXR); grid; figure(2); plot(t,MEANVR); grid; figure(3); plot(MEANXR,MEANVR); grid; %disp(MEANVR) %t=(1:N-1)*dt; %t=t’; save MEANXR.DAT MEANXR -ascii; save MEANVR.DAT MEANVR -ascii;

144

8 Quantum Ratchets

8.2.2 Program 8.1b, Complete Langevin Equation, Matlab Code clear; %Dimensionless Complete Langevin Equation with aceleration, N=230000; NR=1; kL=1.; C1=0.; Fext=0.; FDN=0.; dhp=1.; dGamma=0.1; x00=1.e-6; Uo= 1; dkBT=1.; M=1.; dt=1.e-4; E11=1.6; E22=1.98; w=1.; r=0.5; theta=r*2*pi; %========================================================= % QUANTUM PARAMETERS and limits for the noise %========================================================= % Here the unit of energy is Uo. Dbeta=Uo./dkBT; %========================================================= %Parameters from S. Denisov et al. Europhys. Lett. 85, 40003, (2009). %========================================================= qrt(Uo/M) ts=(1/kL)*sqrt(M/Uo); Gamma=dGamma/ts; GAMMA=M*Gamma; kBT=Uo*dkBT; hp=(dhp/kL)*sqrt(M*Uo); tc=hp/(2*pi*kBT);%Thermal correlation time nu=2*pi*kBT/hp; %(for T = 1pK, nu=0.0417/s FIRST MATSUBARA FREQUENCY. y=1+(tc*Gamma/pi); lambda=(hp/(pi*GAMMA))*(-psi(0,1) + psi(0,y)); Dlambda=lambda*kL2 %Dlambda=0.; To use when you want simulate classical regime. %========================================================= %CLASSICAL RANGE gamma « nu, %========================================================= %QUANTUM RANGE gamma » nu %========================================================= F=(1:N)*0; F=F’; F1=(1:N)*0; F1=F1’; x=zeros(N,NR); v=zeros(N,NR); xij=x00; v(1,1)=0.; %========================================================= % DICHOTOMOUS NOISE (iN CASE YOU WANT TO USE) %=========================================================

8.2 Programs

145

%eta=zeros(N,NR); %for j=1:NR %for i=1:N-1 %Pat1at = 0.5*(1. + exp(-i*dt/tau)); % Probability to stay in "-1" %Pat2at1 = Pat1at; %Pat1bt = 0.5*(1. - exp(-i*dt/tau)); %Transition Probability 1->0 %Pat2bt1 = Pat1bt; %Pbt2bt1 = 0.5*(1. + exp(-i*dt/tau)); %Probability to stay in "1" %Pbt1bt = Pbt2bt1; %Pbt1at = 0.5*(1. - exp(-i*dt/tau)); %Transition Probability 0->1 %Pbt2at1 = Pbt1at; % R=unifrnd(0,1); % if(Pat1at > R) % eta(i+1,j)=-1.; % else % eta(i+1,j)=1.; % end %if(Pat1at < R || Pat1at > R) % R1=unifrnd(0,1); % elseif(Pat2bt1 > R1 || Pat2at1 > R1) % || significa or % eta(i+2,j)=-1.; %else % eta(i+2,j)=1.; %end %end %end %========================================================= DU=zeros(N-1,NR); Dd1U=zeros(N-1,NR); Dd2U=zeros(N-1,NR); Dd3U=zeros(N-1,NR); DDeff=zeros(N-1,NR); %Potential slightly tilded to the right.? C1=0. Denisov et al potential DU(1,1)=(cos(kL*xij) - C1*0.25*cos(2*kL*xij)); Dd1U(1,1)=-sin(xij) + C1* 0.5*sin(2.*xij); Dd2U(1,1)=-cos(kL*xij) + C1*cos(2.*kL*xij); Dd3U(1,1)=sin(kL*xij) - C1*2.*sin(2.*kL*xij); DDeff(1,1)=1./(1-Dlambda*Dbeta*Dd2U(1,1)); %========================================================= % THERMAL QUANTUM NOISE %========================================================= tR=tc/ts; INVTR=1/tR; A1=6.50618; A2=3.89653; A3=3.01154; t1=8.33507e-5; D1=A1*t1;

146

8 Quantum Ratchets

t2=0.01425; D2=A2*t2; t3=0.3896; D3=A3*t3; DDeff=zeros(N-1,NR); eps=zeros(N-1,NR); h=zeros(N-1,NR); eps1=zeros(N-1,NR); h1=zeros(N-1,NR); eps2=zeros(N-1,NR); h2=zeros(N-1,NR); eps3=zeros(N-1,NR); h3=zeros(N-1,NR); a1 = abs(randn(1)); b1 = abs(randn(1)); a2 = abs(randn(1)); b2 = abs(randn(1)); a3 = abs(randn(1)); b3 = abs(randn(1)); E1 = exp(-(dt/(t1*tR))); E2 = exp(-(dt/(t2*tR))); E3 = exp(-(dt/(t3*tR))); eps1(1,1) = sqrt((-2.*(DDeff(1,1)*D1)/(t1*tR))*(1-E12 )*log(a1))*cos(2*pi*b1); eps2(1,1) = sqrt((-2.*(DDeff(1,1)*D2)/(t2*tR))*(1-E22 )*log(a2))*cos(2*pi*b2); eps3(1,1) = sqrt((-2.*(DDeff(1,1)*D3)/(t3*tR))*(1-E32 )*log(a3))*cos(2*pi*b3); eps(1,1)=eps1(1,1)+eps2(1,1)+eps3(1,1); for j=1:NR xij=x00; v(1,1)=0.; disp(j) for i=2:N-1 xij=x(i,j); %======================================================== Dd1U(i,j)=-sin(kL*xij) + C1* 0.5*sin(2.*xij); Dd2U(i,j)=-cos(kL*xij) + C1*cos(2.*kL*xij); Dd3U(i,j)=sin(kL*xij) - C1*2.*sin(2.*kL*xij); DDeff(i,j)=1./(1-Dlambda*Dbeta*Dd2U(i,j)); %======================================================== % QUANTUM LANGEVIN EQUATION %======================================================== F(i)= E11*cos(w*(i-1)*dt)+E22*cos(2*w*(i-1)*dt + theta); F1(i)= -(1./w)*sin(w*(i-1)*dt)*(E11 +E22*cos(w*(i-1)*dt + theta)); %======================================================== x(i+1,j) = x(i-1,j) + 2.*v(i,j)*dt + eps(i,j)*dt;

8.2 Programs

147

v(i+1,j) = v(i-1,j) + (-dGamma*v(i,j)- Dd1U(i,j) -0.5*Dlambda*Dd3U(i,j) +. . . F(i) + dGamma*F1(i) + Fext )*2.*dt; % to add in case you want to simulate an external force Fext = eta(i,j)*FDN, tau = ; %======================================================== % NOISE %======================================================== a1 = abs(randn(1)); b1 = abs(randn(1)); a2 = abs(randn(1)); b2 = abs(randn(1)); a3 = abs(randn(1)); b3 = abs(randn(1)); h1(i,j) = sqrt((-2.*(DDeff(i,j)*D1)/(t1*tR))*(1-E12 )*log(a1))*cos(2*pi*b1); h2(i,j) = sqrt((-2.*(DDeff(i,j)*D2)/(t2*tR))*(1-E22 )*log(a2))*cos(2*pi*b2); h3(i,j) = sqrt((-2.*(DDeff(i,j)*D3)/(t3*tR))*(1-E32 )*log(a3))*cos(2*pi*b3); eps1(i+1,j)=eps1(i,j)*E1 + h1(i,j); eps2(i+1,j)=eps2(i,j)*E2 + h2(i,j); eps3(i+1,j)=eps3(i,j)*E3 + h3(i,j); eps(i+1,j)=(eps1(i+1,j)+eps2(i+1,j)+eps3(i+1,j)); %========================================================= end end MEANXR=(sum(x,2))/NR; MEANVR=(sum(v,2))/NR; MEANEPS=(sum(eps,2))/NR; MEANDeff=(sum(DDeff,2))/NR; t=(1:N)*dt; t=t’; save t.dat t -ascii; figure(1); plot(t,MEANXR); grid; figure(2); plot(t,MEANVR); grid; figure(3); plot(MEANXR,MEANVR); grid; save XCg1E-1-thetapi.DAT MEANXR -ascii; save VCg1E-4-thetapi.DAT MEANVR -ascii; %=========================================================

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8 Quantum Ratchets

Bibliography 1. Mitropolskii, Y.A., Petrina, D.Y.: On N. N. Bogolyubov’s works in classical and quantum statistical mechanics.. Ukr. Math. J. 45, 171–214 (1993). https://doi.org/10.1007/BF01060975 2. Bogolyubov, N.N.: Elementary example for establishing statistical equilibrium in a system coupled to a thermostat. On some statistical methods in mathematical physics, Publ. Acad. Sci. Ukr. SSR, Kiev 1945, pp. 115–137 (in Russian) 3. Magalinski˘i, V.B.: Dynamical model in the theory of the Brownian motion. Sov. Phys. JETP 9, 1381 (1959), JETP 36, 1942 (1959) 4. Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57–265, ¯ (2002), 208–209 5. Banerjee, D., Bag, B.C., Banik, S.K., Ray, D.S.: Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects. J. Chem. Phys. 120(19), 8960 (2004) 6. Hänggi, P., Jung, P.: Colored noise in dynamical systems. Adv. Chem. Phys. 89, 239 (1995) 7. Machura, L., Kostur, M., Hänggi, P., Talkner, P., Luczka, J.: Consistent description of quantum Brownian motors operating at strong friction. Phys. Rev. E. 70, 031107 (2004) 8. Luczka, J., Rudnicki, R., Hänggi. P.: The diffusion in the quantum Smoluchowski equation. Phys. A 351, 60 (2005) 9. Reimann, P., Grifoni, M., Hänggi, P.: Quantum ratchets. Phys. Rev. Lett. 79(1), 10–13 (1997) ¯ 10. Ankerhold, J., Pechukas, P., Grabert, H.: Strong friction limit in quantum mechanics: the quantum smoluchowski equation. Phys. Rev. Lett. 87, 086802 (2001) 11. Machura, L., Luczka, J., Talkner, P., Hänggi, P.: Transport of forced quantum motors in the strong friction limit. Acta Phys. Polonica Series B 38, 1855–1863 (2007) 12. Denisov, S., Kohler, S., Hänggi, P.: Underdamped quantum ratchets. Europhys. Lett. 85, 40003 (2009) 13. Denisov, S., Morales-Molina, L., Flach, S., Hänggi, P.: Periodically driven quantum ratchets: symmetries and resonances. Phys. Rev. A 75, 063424 (2007) 14. Yevtushenko, O., Flach, S., Zolotaryuk, Y., Ovchinnikov, A.A.: Rectification of current in acdriven nonlinear systems and symmetry properties of the Boltzmann equation. Europhys. Lett. 54, 141 (2001)

Appendix A

A.1 Master Equation A.1.1 Transition Rate Consider discrete states of a system Sj , where j is an integer. Temporal change among discrete states is called state transition. The so-called transition rate, ωi→j , from the discrete state Si to the discrete state Sj is defined as follows: • Suppose that a system is in the state Si at a time t. • The (conditional) probability that the system makes the transition to a different state Sj during an infinitesimal time lapse, dt, is ωi→j dt.

A.1.2 Probability Flux Consider a system in the state Si with the probability Pi . Between Si and Sj , there is the flow of probability Pj ωi→j from Si to Sj and Pi ωj →i from Sj to Si . We call the net flow to the probability per unit of time the (net) probability flux and denote it by Ji→j (=-Jj →i ): Ji→j = Pi ωi→j − Pj ωj →i = −Jj →i

(A.1)

A.1.3 Master Equation Let us denote by Pi (t) the probability to find the system at a time t in the state Si . In accordance to the former statements, the redistributed probabilities Pi (t + dt) should satisfy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

149

150

Appendix A

Pi (t + dt) = Pi (t) −

2

Ji→j dt

(A.2)

j

Then we have the following evolution equation for Pi called the master equation: 2 dPi =− Ji→j , dt

(A.3)

j

where the sum runs for all states, Sj As an example of application of the Master Equation we will consider the Poisson Process.

A.1.4 Poisson’s Process Let us consider a discrete set of events that successively happen at irregular time intervals (random). For example, the events can be radioactive emitions (α particles, β or γ ) from a source or the colitions suffered for a gas molecule. We also assume taht the probability to happen an event in a infinitesimal time interval, dt is independent of t and equal to λdt with λ constant. We are interested in the random variable M(t) (the number of events happening between 0 and t). We call Pm (t) the probability of happening m events till t. Then we have the conditions to apply the Master equation, with ωi→j = λδm,m +1

(A.4)

One transition can only increase by one the event counting. Then Eq. A.1, Jm−1→m = λ (Pm−1 − Pm ) = −Jm→m−1

(A.5)

dPm = λ (Pm−1 − Pm ) dt

(A.6)

Then the master

The initial conditions for this system of equations are Pm (t = 0) = 1 f or m = 0 Pm (t = 0) = 0 f or m  0

(A.7)

The solution of this system, with this initial conditions, is Pm (t) =

(λt)m exp(−λt) m!

(A.8)

A.1 Master Equation

151

This probability distribution is called Poisson Distribution, with the following properties M(t) = λt,

(A.9)

2 = λt σM

(A.10)

A.1.5 Detailed Balance We consider the steady sate of a master equation that satisfies: Ji→j = 0, ∀i, ∀j (detailed balance)

(A.11)

Pi ωi→j = Pj ωj →i , ∀i, ∀j (detailed balance)

(A.12)

or, equivalently,

We call such steady state the state of detailed balance. Detailed balance allows us to formulate relations among rates parameters. This may be easily seen in the isomerization reactions shown in Fig. A.1. At equilibrium, detailed balance means that k1 [A] = k2 [B] ,

k3 [B] = k4 [C] ,

k5 [C] = k6 [A] ,

(A.13)

from which k1 k3 k5 = k2 k4 k6

(A.14)

More general relations may be developed for complex reaction networks. From Chap. 2, Eq. 2.4 we obtain the overdamped limit of the Langevin equation,  x˙ (t) = −

Fig. A.1 Reaction sequence of three isomeric compounds, A, B, and C

dV (x) +  (2D)1/2 ξ(t) dx

(A.15)

152

Appendix A

where  = Mγ (units ≡ Kg/s) In this system regardless of the potential’s symmetry, detailed balance is valid, that is, the probability for a particle to make a thermally activated transition from x to (x + x) equals the probability for the reverse step from (x + x) to x, for arbitrary x and x. This implies that no exists net transport in thermal equilibrium. To turn the system into a Brownian motor, one needs to break detailed balance by driving the system away from thermal equilibrium, One way of doing so is to introduce time dependent noise in the potential as 0.5 (1 + ηDN (t)) V (x(t)) as the case of flashing ratchet or adding a fluctuational force, ηDN (t)F in the case of rocking ratchet, where ηDN (t) is dichotomous noise. That is why several authors define a Brownian motor as a device that achieves directed motion of particles by rectifying thermal fluctuations. We see that thermal fluctuations are crucial without them does not exist movement of particles.

Appendix B

B.1 Information Flow When a fluctuation occurs, a local order is established and a knowledge (for example number of particles) of this local molecular system can be performed by a device capable to process information faster than the relaxation time of the fluctuation. Our objective is, without entering in the mechanisms of the process, (see Refs. [1] and [2] for possible mechanisms) the “minimum” flow of information (bits/s) a device would have to process in order to dissipate an ionic fluctuation in the solution through a channel within the size of a biological one; and compare this result with some of the already existent man made devices. In doing so we make use of the relation between the entropy production and the information lost (bits/s) when the fluctuation is dissipating in a volume of the size of a biological channel, of course, the device would have to process information in a faster way in order to decide whether this fluctuation would be utilize by de cell or not. Procopio and Fornés [3], analyzed the possibility that nature occuring ionic fluctuation could be utilize by the cell membrane system to lead a transient ionic flow, and to eventually produce ion concentration gradients across the membrane. It was found that there is a broad range of concentrations for which significant fluctuations of them take place in enclosing volumes having diameters comparable to the membrane thickness. In these cases, the dissipation of a fluctuation close to the membrane could result in a transient net flow of the corresponding ionic species through the channel. In order that a channel could correlate its state of aperture to the phase of a nearby ion density fluctuation, a flow of information is required by the second law of thermodynamics Brillouin [4] and Bennett [1]. Molecular entities such as biological channels, “carriers”, and eventually ATPases, may be important molecular devices for an efficient processing of information in the form of fast conformational changes. Such structures are in principle capable of performing the coupling of fluctuating parameters as concentration or density to fast molecular conformational changes. Studies © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

153

154

Appendix B

employing different techniques have shown (5) that channel proteins experience important conformational changes in response to binding or proximity of ions, and also [5], computer simulations of internal motions of proteins can be carried out for time periods of the order of a few hundred picoseconds. Our calculation is a first step in the quantification of the process, showing the possibility of its existence. We consider here an ionic channel placed along the x axis at a distance d from the origin, (see Fig. B.1). The entropy production (J/K.m3 .s) while an ionic species a is being dissipated in the solution at the neighborhood of the channel mouth was given by (Fornés [6]):

(x − d)2 + y 2 + z2 σa = Q[(x − d) + y + z ] exp − 2Da (t + τ0 ) 2

2

2

(B.1)

where Q is given by: Q=

RKa ca νa τ03  2 4Ma Da (t + τ0 )5

(B.2)

with Ka given by: Ka = 1 −

e3



3/2

2m

NA π (kT )3



1/2 za4

2

−1/2 ca za2

(B.3)

a

MEMBRANE . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .R . . . . FLUCTUATION---> . . . . d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.. .. . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . CHANNEL . . > x . . . r . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. B.1 Ionic fluctuation close to a membrane channel. (Reprinted figure with permission from Fornés, [8], copyright by Elevier)

B.1 Information Flow

155

where ca is the concentration in moles/m3 , NA is the Avogadro number, νa is the number of ions of type a in one molecule of the electrolyte, za is its corresponding valence, m is the dielectric constant of the medium, e is the elementary charge (esu), T the absolute temperature and k is the Boltzmann constant, R is the gas constant and Ma is the atomic weight of the ion a, and τ0 is the initial relaxation time of the fluctuation, Da is the diffusion coefficient, and  = δca /ca is the relative amplitude of the fluctuation. For a symmetrical monovalent electrolite of concentration c, Eq. B.3 transforms in: Ka = 1 −

e3 (NA π )1/2 −1/2 c (2m kT )3/2

(B.4)

Then the rate in which the entropy is being created inside the channel (J/K/s) will be given by: 

σa,ch = Q

l+d d



r 0

(x − d)2 + ρ 2 2πρdρdx (B.5) [(x − d) + ρ ] exp − 2Da (t + τ0 )

2

2

where l is the channel length and r the channel radius. After performing the integrations in Eq. B.5 we obtain:

σa,ch =

ca νa fa5 Da4 Ka 2 3 2 √ 1 3 2 π τ0  R[ π erf (fa l)[exp(−(rfa )2 )[ + (rfa )2 ] + ] Ma 2 2 − (lfa ) exp(−(lfa )2 )[1 − exp(−(rfa )2 )]]

(B.6)

[2Da (t + τ0 )]−1/2 and erf is the error function defined as: erf (u) = where √ fa= u (2/ π ) 0 exp(−x 2 )dx. In order to assure the fluctuation will dissipate inside the channel we choosed τ0 = l 2 /(2Da ) in Eq. B.6. The mean rate in which the entropy is being produced inside the channel by the ionic species a will be: σ

a,ch

1 = τ0



τ0 0

σa,ch (l, t)dt

(B.7)

where σa,ch (l, t) denotes the right hand member of Eq. B.6. As a consequence of this dissipation, information was lost at a mean rate (bits/s)

I a,ch =

σ a,ch k ln 2

(B.8)

So, if a system exits in the channel capable of performing the coupling of fluctuations in concentration to fast molecular conformational changes, it will have at least to be able to process the information given by Eq. B.8.

156

Appendix B

Fig. B.2 Information Flow to dissipate an ionic fluctuation ( = 0.1) through a membrane channel. Na+ ion from a symmetrical monovalent electrolyte (DN a + = 1.33 × 10−9 m2 /s) at room temperature (298 K). (a) τ0 = 9.38 × 10−9 s (channel size: r = 5 Å, l = 50 Å), (b) τ0 = 2.34 × 10−9 s (channel size: r = 5 Å, l = 25 Å). (Reprinted figure with permission from Fornés, [8], copyright by Elevier)

In Fig. B.2, I a,ch is shown as a function of ca , for given values of , and τ0 for N a + ion from a symmetrical monovalent electrolite at room temperature (298 K). We can observe that for common values of biological concentrations, if this system exists, it will have to process information about Mbits/s for a fluctuation with  = 0.1. This value is not so large as compared with some of the already existent man made ones which process information at a rate of 104 M bits/s (Peled [7]).

Bibliography 1. Bennett, C.H.: Demons, engines and the second law. Sci. Am. 257(5), 108–116 (1987) 2. McClare, C.W.F.: Chemical machines, Maxwell’s demon and living organisms. J. Theor. Biol. 30, 1–34 (1971) 3. Procopio, J., Fornés, J.A.: Local transient fluctuational density as producing ionic flow through cell membranes. J. Colloid Interface Sci. 134, 279 (1990) 4. Brillouin, L.: Science and Information Theory, 2nd edn. Academic, New York (1962) 5. Lauger, P.: Dynamics of ion transport systems in membranes. Phys. Rev. 67, 1296 (1987) 6. Fornés, J.A.: Ionic fluctuations in solution-entropy production. Phys. Lett. A 175, 14 (1993) 7. Peled, A.: The next computer revolution. Sci. Am. 257(4), 35 (1987) 8. Fornés, J.A.: Information flow to dissipate an ionic fluctuation through a membrane channel. J. Colloid Interface Sci. 177, 411–413 (1990)

Appendix C

C.1 Endoreversible Thermodynamics “Endoreversible thermodynamics is a subset of irreversible thermodynamics aimed at making more realistic assumptions about heat transfer than are typically made in reversible thermodynamics. It gives an upper bound on the energy that can be derived from a real process that is lower than that predicted by Carnot for a Carnot cycle, and accommodates the exergy destruction occurring as heat is transferred irreversibly.” [1]. Endoreversible thermodynamics was discovered in simultaneous work by Novikov [2] and Chambadal, [3] although sometimes mistakenly attributed to Curzon and Ahlborn [4]. Later, Rebhan [6], studied a Carnot engine with thermal losses and friction. He found that the efficiency of these machines was bounded from above by the Curzon-Ahlborn expression. Van den Broeck, [7] has shown, in the framework of non-equilibrium thermodynamics, that the efficiency of any thermal engine is bounded from above by ηCA . Curzon and Ahlborn in the original derivation for the efficiency of a Carnot engine with thermal losses, time appears explicitly and this is disappointing in the framework of classical thermodynamics. Miranda, [8] made a derivation without any explicit reference to time. The efficiency of a heat engine is given by, see Fig. C.1 η=

W QC =1− QH QH

(C.1)

The Carnot’s engine is reversible, see Fig. C.2, this means non-entropy production, Stot = SE = 0. Stot = −

QC QH + + SE = 0.then, TC TH

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

157

158

Appendix C

TH

Fig. C.1 Heat Engine. (After Jae Dong Noh (2015), [5])

QH

E QC

TC

TH

Fig. C.2 Carnot Engine. (After Jae Dong Noh (2015), [5])

W

QH

C QC

TC T1

Fig. C.3 Endoreversible Engine. (After Jae Dong Noh (2015), [5])

Q1 =

TA

W

C

T1 - T A

P =W

TB T2

ηCarnot = 1 −

TC η TH

Q2 =

( TB - T2(

(C.2)

The endoreversible engine see Fig. C.3 was given by [4], The endoreversibility condition is, α (T1 − TA ) β (TB − T2 ) = TA TB

(C.3)

Obtained by maximizing P with respect to TA and TB , then the efficiency at the maximu power is  ηMP = 1 −

T2 T1

(C.4)

Bibliography

159

Table C.1 Comparison of Chambadal-Novikov efficiency with Carnot efficiency and with the observed efficiency for some real plants Power source West Thurrock (UK) coal fired power plant CANDU (Canada) nuclear power Larderello (Italy) geothermal

Carnot ChambadalObserved T2 (◦ C) T1 (◦ C) efficiency Novikov efficiency efficiency 25 565 64.1% 40% 36% 25

300

48%

28%

30%

80

250

32.3%

17.5%

16%

Bibliography 1. https://en.wikipedia.org/wiki/Endoreversible_thermodynamics. Accessed 21 Sept 2020 2. Novikov II: The efficiency of atomic power stations (a review). J. Nuclear Energy 7(1–2): 125– 128 (1958) 3. Chambadal, P.: Les centrales nucléaires, vol. 4, pp. 1–58. Armand Colin, Paris (1957) 4. Curzon, F.L., Ahlborn, B.: Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 43, 22–24 (1975) 5. Noh, J.D.: Efficiency of a Brownian Heat Engine proffered at ICTS Bangalore (2015,10,28) can be seen on YouTube 6. Rebhan, E.: Efficiency of nonideal Carnot engines with friction and heat losses. Am. J. Phys. 70, 1143 (2002) 7. van den Broeck, C.: Thermodynamic efficiency at maximum power. Phys. Rev. Lett. 95, 190602 (2005) 8. Miranda, E.N.: On the maximum efficiency of realistic heat engines. Int. J. Mech. Eng. Edu. (2012)

Appendix D

D.1 First Passage Phenomena D.1.1 Properties of First Passage Time This topic is based on a Lecture proffered by Sidney Redner in YouTube [1] We ask the following questions: (1) What is the probability of eventually hitting the origin starting at x? (2) What is the time to hit the origin.? We will use a continuous method, using the diffusion equation in one dimension, ∂c ∂ 2c =D 2 ∂t ∂x c(x, t = 0) = δ(x − x0 ) initial b.c.

(D.1)

c(x = 0, t) = 0. absorbing b.c. We will use the electrostatic method of images, [2] in order the solution accomplish the absorbing boundary condition, namely



 (x − x0 )2 (x + x0 )2 exp − − exp − c(x, t) = √ 4Dt 4Dt 4π Dt 1

(D.2)

Flow of particles hitting the origin, is equivalent to the probability that a random walker hit the origin for the first time,

∂c j (t) = −D ∂x

x=0



x2 = √ exp − 0 4Dt 4π Dt 3 x0



© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

(D.3)

161

162

Appendix D

The probability to hit the origin at any time, is obtained integrating the former Eq. D.3, namely 

∞

J =

j (t)dt = 1

(D.4)

0

And the mean return time will be ∞ t = 0∞ 0

tj (t)dt j (t)dt

=∞

(D.5)

The Eq. D.4 means that the return is certain and the Eq. D.5 means that the return time is infinity. This is an intriguing property of diffusion in spatial dimensions d 2 is that a diffusing particle is certain to reach any finite-size target, but the average time for this event is infinite. This property of eventually reaching any target is known as recurrence, [3].

D.1.2 Application to Chemical Kinetics This topic is based on a Lecture proffered by Sidney Redner in YouTube [4] We ask the question: How efficiently diffusion reactants actually undergo reaction? We consider a sphere of radius a immersed in a fluid surrounded of particles in Brownian motion. We ask: how quickly are particles absorved by the sphere? see Fig. D.1 The reaction rate K is defined by K=

N particles absorved time c

(D.6)

where c is the concentration of particles. We calculate the flow of particles absorved by the sphere Fig. D.1 Sphere of Diffusion coefficient D immersed in a fluid with particles performing Brownian motion at concentration c

Bibliography

163

∂c = D∇ 2 c ∂t c(r > a, t = 0) = 1 c(r = a, t > 0) = 0

(D.7) (D.8) (D.9)

This is a classical boundary problem, instead, now we will consider the steady state, D∇ 2 c = 0.

(D.10)

c(r = a) = 0

(D.11)

c(r → ∞) = 1

(D.12)

The solution of the Laplace equation is c(r) = 1 −

a ≡ escape probability r

(D.13)

Then K is given by      K= −D ∇c −d S = 4π Da

(D.14)

For nonsteady state the solution is

 a K = 4π Da 1 + √ π Dt

(D.15)

Bibliography 1. Redner, S.: Lecture on YouTube (1919): Random Walks-6-First Passage Phenomena 2. Jackson, J.D.: Classical Electrodynamics, p. 26. Wiley, London (1963) 3. Krapivsky, P.L., Redner, S.: First-passage duality. J. Stat. Mech. (2018) 093208 4. Redner, S.: Lecture on YouTube (1919): Random Walks-7.2-Elementary Applications of First Passage Phenomena

Appendix E

E.1 Stochastic Dynamics The stochastic differential equations, known also as Langevin Equations can be with aditive or multiplicative noise. The stochastic infinitecimal used in their interpretation and solution has different rules than those used in the usual infinitesimal calculus. The Stochastic Dynamics term refers to the temporary evolution of random variables subject to noise. We give here the basic konwledge on this subject. For more knowledge and examples consult the bibliography.

E.1.1 White Noise and Wiener Process A stochastic process (SP) very important for the “Stochastic Calculus” is called “White Noise”, ξ(t), defined by its statistics properties: ξ(t) = 0 ξ(t2 )ξ(t1 ) = δ (t2 − t1 )

E.1.2 Spectral Intensity The electromagnetic signals emited by electronic circuits have stochastic components, called noise. we will consider stationary signals, of mean null, X = 0. The intensity of the frequency component ω of the signal X(t), called spectral intensity, I (ω), is related with the auto-correlation function, X(t + τ )X(t), through the Wiener-Khintchine theorem, namely © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

165

166

Appendix E

1 I (ω) = 2π



∞

−∞

X(t + τ )X(t) exp (−iωτ ) dτ

(E.1)

Then the reason for calling it white noise is because the spectral intensity of a SP is the Fourier’s transform of the auto-correlation function. Then, the Fourier’s transform of the Dirac’s delta is a constant, this means, that all frequencies are present with the same intensity, which characterize the white light. Let’´s notice, the correlation time of white noise is zero. All SP has a finite corelation time. There are however circumstances in which the time of correlation of a SP is so short that treating it as white noise is a good approach.

E.1.3 Properties of Wiener’s Process Related to white noise ξ(t), is the “Wiener Process” W (t), defined as 

t

W (t) =

ξ(t )dt

(E.2)

0

Some properties of W (t) follow from its definition: (1) (2) (3)

W  (t) = 0, follow from ξ(t) = 0, W (t)W (t ) = min(t, t ), W (t) is an SP Markoviano, Gaussiano, because, being an integral of an SP whose correlation time is zero, the probability distribution is

ω2 PW (ω, t) = √ exp − 2t 2π t 1

(E.3)

with σ 2 = t. (4) The conditional probability (or transition probability) is

(ω − ω0 )2 1 exp − T (ω, t|ω0 , t0 ) = √ 2(t − t0 ) 2π(t − t0

(E.4)

√ with width W (t) = σW = t. This fact, matches the Wiener’s process very special features. For example, it is non-differentiable but is continuous namely √ dW W (t) t = lim ∼ →∞ t→0 dt Δt t

(E.5)

From Eq. E.2 we’d hope the derivative of W (t) would be ξ(t). What happens is that white noise is not a SP mathematically well-defined. The transition

E.1 Stochastic Dynamics

167

probability T (ω, t)|ω0 , t0 ), given by Eq. E.4, is a solution of the diffusion equation in ω, with the diffusion coefficient D = 1, ∂T (ω, t|ω0 , t0 ) 1 ∂ 2 T (ω, t|ω0 , t0 ) = ∂t 2 ∂ω2

(E.6)

with the initial condition T (ω, t0 |ω0 , t0 ) = δ(ω − ω0 )

(E.7)

This means the matter which is spread is at the initial time t0 , all concentrated at the origin ω0 .

E.1.4 Stochastic Process Derivative As an stochastic process , X(t), is not a function, in the usual sense, we have to say what we understand for theirderivative, X(t) ˙ X(t) ≡ dt

(E.8)

˙ “X(t) means the stochastic process whose realizations are the derivatives of the realizations of the SP X(t)”.

E.1.5 SDE with Aditive Noise We will integrate the Eq. 2.4 from t to t + t and after we will take the limit t → dt, 

t+t 0

  1 dX t = − γ

 0

t+t

dV (X(t )) dt + dX(t )



2kB T γ

1/2 

t+t

ξ(t )dt

0

(E.9)

or X(t) ≡ X(t + t) − X(t) = A(X)t + B (W (t + Δt) − W (t)) ≡ A(X)t + BW (t), we have used the definition of Wiener’s process, Eq. E.2 and A(X) = − γ1

(E.10) dV (x(t )) dx(t ) ,

this last term means the mean valor in the interval t. In the limit t → dt

168

Appendix E

the mistake we make to replace A(X) by A(X(t)) in Eq. E.10 is negligible for dt infinitesimal, then we can write dX(t) = A(X(t))dt + BdW (t)

(E.11)

√ We saw that the “Wiener’s increment”, dW (t), is a Gaussian SP, with σ = dt. That’s why, an each integration pass we’ll have to draw dW(t) and normalize the result appropriately. Let’s call RG a random number, with Gaussian distribution, centered at RG = 0 and width 1. In MATLAB/OCTAVE RG = randn. With this convention the last term of Eq. E.11 can be written as BdW (t) =

√ dtBRG

In case de m independent noises, dW (t) will be a column vector of m components and we have to replace randn by randn(m, 1). The more used algorithm in the integration of stochastic equations it’s Euler’s. We divide the integration interval [0, tmax ] in n intervals of size dt. We create the “vector” t = (t1 , t2 , . . . tj , . . . , tn+1 ) = (0, dt, 2dt, . . . , tmax ). For efficiency of the procedure, we draw once all the “Wiener’s increments”, creating the dW vector. In MATLAB/OCTAVE we write dW = sqrt (dt) ∗ randn(1, n) The i mth integration pass will be, x(i + 1) = x(i) + A(x(i)) ∗ dt + BdW (i) When we perform several realizations of the stochastic process we have an extra index, j corresponding to the j mth realization, namely x(i + 1, j + 1) = x(i, j ) + A(x(i, j )) ∗ dt + BdW (i, j )

E.1.6 SDE with Multiplicative Noise We write now the system of equations in terms of the derivatives and white noise, letting, however, both A and B depend on X and may also have depending explicit in t, dX = A(X, t) + B(X, t)ξ(t) dt

(E.12)

Eq. E.12 is called stochastic differential equation with multiplicative noise. Tranforming it as we done in the case of aditive noise

E.1 Stochastic Dynamics



t+t

X(t) =

169

  A X(t ), t dt +



0

t+t

  B X(t ), t ξ(t )dt

(E.13)

0

  The first integral can be replaced, in the limit t → dt by A X(t ), t dt. √ For the second, let’s remember, W (t) ∼ t. Then ΔX it can also contain terms O and the mistake that is made to substitute the integral by (X))   B X(t ), t W (t) it can be just as or greater than O(t). So the SDE with multiplicative noise should be treated by an special stochastic calculation, which may be “Itô’s calculus” or “Stratonovich’s calculus”

E.1.7 Itô’s and Stratonovich’s Calculus E.1.7.1

Convergence in Quadratic Mean

Be X1 , X2 , . . . , Xn a sequence of random variables. we say Xn coverges in quadratic mean to X if ! lim (Xn − X)2 = 0.

n→∞

(E.14)

We say, then, that the limitinquadraticmean of Xn is X, what we represent by qm − lim Xn = X n→∞

(E.15)

Let’us consider situations in which the random variables of the sequence are functions of an stochastic process, this is, Xn = Xn (t) and X = X(t). In this case the mean of Eq. E.14 is a mean over the realizations of the SP, N ! 2 1 2 2 = xn,R (t) − xR (t) (t) − X(t)) (Xn N

(E.16)

R=1

where xn,R (t) is the value that acquires the function xn in the realization R of X, at the instant t. Be G (X(t), t) a function of the SP X(t) and possibly, explicit function of t. In turn, X(t) depends on W (t), in the sense for each realization of dW (t) corresponds a realization of X(t). Generally, X(t) is the solution of a SDE that has W (t ) as noise. the value of X(t) only is influenced by the values of W (t ), with t ≤ t, meaning, X(t) and the variations W (t ) in times later to t are independent random variables. This property is expressed in stochastic calculus by the qualification ‘not in advance” for a variable X(t). Itô’s integral for G (X(t), t), from t0 to t is defined as

170

Appendix E



t

N 2   G X(t ), t • dW (t ) = qm − lim G (X(tn ), tn ) W (tn ), t→0

0

(E.17)

n=0

where N = (t − t0 )/t tn = t0 + nt W (tn ) = W (tn+1 ) − W (tn )

(E.18)

The symbol •dW (t ) is used in Eq. E.17 to indicate that is Itô’s integral.

E.1.7.2

Properties of Itô’s Integral

(1) 

t

W (t ) • dW (t ) =

0

 1 W (t)2 − W (t0 )2 − (t − t0 ) 2

(E.19)

(2) 

t

 n 1  W (t)n+1 − W (t0 )n+1 − n+1 2

W (t )n • dW (t ) =

0



t

W (t )n−1 dt

0

(E.20)

where the last integral, on the right, is an usual Riemann’s integral. (3) The mean over the realizations of any Ito’s integral is null. 

t









G X(t ), t • dW (t ) = 0.

(E.21)

0

(4) Itô’s integrals whose integration measures are powers of Wiener’s increment: 

t

N 2   G X(t ), t • (dW (t ))m ≡= qm − lim G (X(tn ), tn ) (W (tn ))m , t→0

0

n=0

(E.22) Also can be shown [1–4].  0

and

t

  G X(t ), t • (dW (t ))2 =

 0

t

  G X(t ), t dt

(E.23)

E.1 Stochastic Dynamics



t

171

  G X(t ), t • (dW (t ))m = 0 f or m > 2.

(E.24)

0

besides 

t

  G X(t ), t • dW (t )dt = 0

(E.25)

0

As (dW (t ))m is only used in integrals, we can write, the following Itô’s differential relations: (dW (t))2 = dt dW (t)dt = 0 (dW (t))m = 0, f or m > 2

E.1.7.3

(E.26)

Itô’s Differential

Consider a continuous function of t and W , F (t) ≡ F (W (t), t), meaning, F depends of t through W (t), also it can also depends explicitly of t. In spite W (t) depends of t in a “non-differentiable”, we guest F depends of W in a differentiable way (for ex. F = W n ). A differential of F is obtained by expansion of Taylor till dt 1 = dt, dF (t) ≡ F (t + dt) − F (t) =

∂F 1 ∂ 2F ∂F dW (t) + dt + (dW (t))2 + . . . ∂W ∂t 2 ∂W 2 (E.27)

or ∂F dW (t) + dF (t) = ∂W



∂F 1 ∂ 2F + ∂t 2 ∂W 2

 dt,

(E.28)

where we used Eqs. E.26.

E.1.7.4

Itô’s Stochastic Differential Equation

We consider again Langevin’s equation in the integral form, Eq. E.13 

t+Δt

X(t) ≡ X(t+t)−X(t) =

 A X(t ), t dt + 





0

t+t

0

  B X(t ), t ξ(t )dt

(E.29) We saw that, in the limit t → dt, A X(t ), t can be replaced by A (X(t), t) in the first former integral. The same procedure cannot be used in the second integral: 



172

Appendix E

  Suppose instead we replace B X(t ), t by B (X(t), t) we perform the replacement   B X(t ), t ⇒ B



 X(t) + X(t + t ,t , 2

(E.30)

In this case 

1 X(t) = A (X(t), t) t + B X(t) + X(t), t W (t) 2

(E.31)

because X(t + t) = X(t) + X and 

t+t

ξ(t )dt = W (t)

0

Iterating the Eq. E.31 in X : replazing X which appears in the argument of B by the value of X given by the proper Eq. E.31, and expanding B in Taylor’s serie, follow X(t) = A (X(t), t) Δt + B (X(t), t) ΔW (t) +

(E.32)

1 ∂B [A (X(t), t) t + B (X(t), t) W (t) + . . .] W (t) + . . . 2 ∂X

(E.33)

Taking the infinitesimal limit, t → dt, X → dX, W → dW and neglecting terms higher than dt 1 , we obtain dX(t) = A (X(t), t) dt + B (X(t), t) • dW (t) + 1 ∂B B (X(t), t) (dW (t))2 2 ∂X

(E.34) (E.35)

or dX(t) = AI (X(t), t) dt + B (X(t), t) • dW (t)

(E.36)

1 ∂B B (X(t), t) 2 ∂X

(E.37)

where we defined AI (X(t), t) = A (X(t), t) +

we have used Eq. E.26 The Eq. E.36 is called “Itô’s Stochastic Differential Equation (Itô-SDE)” It corresponds to the usual form of replacing the Langevin’s equation with white noise, Eq. E.12, not very well mathematically defined, by a differential

E.1 Stochastic Dynamics

173

equation well defined, in Itô’s calculus. That’s why it’s also known as “Itô-Langevin Equation”.

E.1.7.5

Stratonovich’s Calculus

  The Stratonovich’s integral of a function G X(t ), t is defined as 

t







G X(t ), t dW (t ) = qm − lim

t→0

0

N 2 n=0

 X(tn ) + X(tn+1 ) , tn W (tn ), G 2

(E.38) When X(t) can be written explicitly as a function of W (t) the integration rules are the same as the Riemann’s integrals. That’s why we don’t use any symbol to indicate that the integral is Stratonovich’s. For example: 

t

W (t )dW (t ) =

t0

 1 W (t)2 − W (t0 )2 2

(E.39)

and 

t t0

W n (t )dW (t ) =

 1  W (t)n+1 − W (t0 )n+1 n+1

(E.40)

See [2] for demostrations of this and another integrals.

E.1.7.6

Stratonovich’s Stochastic Differential Equation

In the limit t → 0, Making the replacements Δt → dt, X → dX, and W → dW in Eq. E.31, we have an Stratonovich’s SDE, namely dX(t) = A (X(t), t) dt + B (X(t), t) dW (t)

(E.41)

where we have maked dX = 0 in the limit inside B. It´s understood that the integration of this equation has to be done in accordante to Stratonovich’s integral. The Eqs. E.36 and E.41 are equivalent, in the sense the first one is integrated by Itô and the second by Stratonovich have to result in the same solution, in another way, they have to generate the same realizations XR for the same realizations WR (t). Similarly, if we had an Itô’s SDE in the form dX(t) = A (X(t), t) dt + B (X(t), t) • dW (t) Then the following Stratonovich’s equation,

(E.42)

174



1 ∂B B (X(t), t) dt + B (X(t), t) dW (t) dX(t) = A (X(t), t) − 2 ∂X

Appendix E

(E.43)

is equivalent.

Bibliography 1. Itô, K., McKean, H.P. Jr.: Diffusion Processes and Their Sample Paths. Classics in Mathematics. Springer, Berlin/Heidelberg/New York (1996) 2. Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1985) 3. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974) 4. Stratonovich, R.L.: Introduction to the Theory of Random Noise. Gordon and Breach, New York (1963)

Appendix F

F.1 Stochastic Energetics Consider the reduce Langevin Eq. (2.1), m

dx d 2x = − + (2kB T )1/2 ξ(t) 2 dt dt

(F.1)

with  = mγ in units of kg/s. ξ(t) = 0

(F.2)

ξ(t2 )ξ(t1 ) = δ (t2 − t1 )

(F.3)

The Eq. F.1 is an stochastic process, is better written in the way of It´s calculus p dt m  p dp = − dt + (2Γ kB T )1/2 dW m

dx =

(F.4) (F.5)

where dW = ξ(t)dt, follows the relations of Eqs. F.6, namely (dW )2 = dt dW dt = 0 (dW )m = 0, f or m > 2 We want to obtain the equation for the Kinetic Energy Power,

(F.6) dEk dt ,

namely

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

175

176

Appendix F

  1 d p2 dEk = dt 2 dt  2 Then we need to calculate d p , namely   d p2 = (p + dp)2 − p2 = (dp)2 + 2pdp

(F.7)

(F.8)

To finish our calculation, from Eq. F.5, we get (dp)2 (dp)2 = 2kB T

(F.9)

where we have used Eqs. F.6. Finally after adding to Langevin Eq. F.1 the internal and external forces: − dU dx and FL being the load force, respectively, in flashing ratchet an external force is the driving force F (t). Performing the average on the realizations of the stochastic process, we obtain  2     ! p p d p2  p dU = − 2 + kB T − + F (t) dt 2m m m dx m m

(F.10)

− dU dx is the motor force FM , sometimes is FM = G/L, where G is the free energy produced by chemical reactions and L is the step of the molecular motor. then  2   ! p d p2 p !  p = − 2 + kB T + FM − FL (F.11) dt 2m m m m m where we have not considered a drive force F (t). The former Eq. F.11 as a function of de velocity and at th regime of constant velocity is given by, !  0 = − v 2 + kB T + FM v − FL v m

(F.12)

This equation was already given by [1], deduced from Kramers equation. The representation of the motor is given in the next figure: (dQ/dt)out is the power dissipated by the motion of the motor, (dQ/dt)in is the power supplied to the motor by thermal fluctuations of the fluid, −FL v is the rate of work done by the load force FL , FM v is the rate of work done by chemical reaction driving the motor. J-M. Park et al. (2016) performed an excelent work related to the efficiency at maximum power and efficiency fluctuations in a linear heat-engine model [2] (Fig. F.1).

F.1 Stochastic Energetics

177

F.1.1 Sekimoto View Ken Sekimoto [3] faced the question of how Langevin dynamics is related to the first and second law of equilibrium and also non-equilibrium thermodynamics, in doing so he considered U = U (x, a) in Eq. F.10 where a represents the effect of an external agent (or agents), with an external force Fa = −∂U (x, a)/∂a, then dU , dU =

∂U (x, a) ∂U (x, a) dx + da ∂x ∂a

(F.13)

Then the Eq. F.10 transform in, where we have added a load force, FL ,  2         ! p p ∂U p ∂U p ∂U  d p2 p ! p − = − 2 + kB T − − − FL + F (t) dt 2m m ∂a m m ∂x m ∂a m m m (F.14)  p ∂U where we have added − m ∂a on the left side of the former equation in order to equilibrate it. Written in another way, where we have used Eq. F.10, then 

           dEk dWext dQ dU dWL dWdrive + =+ − − + dt dt dt dt dt dt

(F.15)

where 

 2  p dQ  = − 2 + kB T dt m m

(F.16)

is the power dissipated in the thermal bath. Equation F.15 as a function of the velocity at the regime of constant velocity, (dEk /dt) = 0, !  Fa v +  v 2 + FL v = kB T + FM v + F (t) v m Fig. F.1 Energy balance on a protein motor

(F.17)

178

Appendix F

where we have used − dU /dt = FM v, and dWext /dt = Fa v

F.1.2 Entropy Production With respect to the rate of entropy production, excelent demostration, namely kB  dS =− + kB2 T dt m

dS dt ,

3

Cubero and Renzoni [4] gave an

∂lnP ∂p

2 4 (F.18)

where P is the probability finding the particle at (x , p ) at time t and is  density of



given by P (x , p , t) = δ(x(t) − x )δ(p(t) − p ) . The total entropy production is obtained by adding the change of entropy of the ˙ heat bath Q T , namely 1 dQ dS  + = 2 T dt dt m



p T 1/2

+ kB T 1/2 m

∂lnP ∂p

 ≥ 0.

(F.19)

Equation (F.19) is a manifestation of the second law of thermodynamics.

F.1.3 Stochastic Energetics: Useful Relations F.1.3.1

Jarzynski’s Equality

It is known from elementary thermodynamics that when the system is driven from the initial to the final state by a reversible process, a work W is performed so that W = −F (isothermal process). If, in contrast, the process connecting initial and final states is irreversible, the work W differs in general from the free-energy difference because there are potentially infinity irreversible processes connecting the initial and the final states (only one isothermal reversible, though), and in each of these processes the work performed will be different. Therefore, one can treat the ensemble of possible stochastic processes, and perform the statisticall mean W . Now in Stochastic Thermodynamics, we write the second law as W  ≥ F

(F.20)

Indeed, in classical thermodynamics Eq. F.20 is written without the brackets implying that it must hold for all processes. The virtues of the Jarzynski’s equality is that it gives meaning to the average in Eq. F.20, namely

Bibliography

179

! e−W/kB T = e−F /kB T

(F.21)

A rigorous formal derivation can be found in the original Jarzynski’s papers [5, 6]. An excelent article on Jarzynski’s equality illustrated by simple examples was given by, Híjar and Zárate, [7].

F.1.3.2

Crooks Theorem

Many theoretical developments based on the Jarzynski equality have appeared. Many of them represent equivalent formulations adopting different perspectives. Most noted is the so-called Crooks theorem [8], usually formulated as PF (W ) = exp [(W − F )/kB T ] PB (−W )

(F.22)

where the left-hand side is the ratio of the probability of work W when going from the initial to the final state (forward probability) to the probability of work −W when going from the final to the initial state (backward probability). The equivalence of the Jarzynski and Crooks formulations can be readily demonstrated [8]. Among other theoretical developments we mention extensions to cover the Langevin dynamics, when the system is in equilibrium with a thermal bath Imparato and Peliti [9].

Bibliography 1. Mogilner, A., Elston, T., Wang, H., Oster, G.: Molecular motors: examples. In: Fall, C., Marland, E., Tyson, J., Wagner, J. (eds.) Joel Keizer’s Computational Cell Biology, Chapter 12, 12..6.4 p. 348, Eq. (12.71): Modeling Chemical Reactions (2002). https://doi.org/10.1016/ S0370-1573(01)00081-3 2. Park, J.-M., Chun, H.-M., Noh, J.D.: Efficiency at maximum power and efficiency fluctuations in a linear Brownian heat-engine model. Phys. Rev. E 94, 012127 (2016) 3. Kesimoto, K.: Langevin equation and thermodynamics. Prog. Theor. Phys. Suppl. 130, 17 (1998) 4. Cubero, D., Renzoni, F.: Brownian Ratchets, p. 99. Cambridge University Press, Cambridge (2016) 5. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997) 6. Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: a master-equation approach. Phys. Rev. E 56, 5018–5035 (1997) 7. Híjar, H., Ortiz de Zárate, J.M.: ıJarzynski’s equality illustrated by simple examples. Eur. J. Phys. 31, 1097 (2010) 8. Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721 (1999) 9. Imparato, A., Peliti, L.: Work-probability distribution in systems driven out of equilibrium. Phys. Rev. E 72, 046114 (2005)

Appendix G

Forced Oscillations

G.1 Solution of Equation x¨ + ω02 x =

F (t) , m

(G.1)

where ω0 is the free oscillations frequency. Equation G.1 can be integrated in a general form for an arbitrary external force F (t). This is easily done by rewriting the equation as

or where

d 1 (x˙ + iω0 x) − iω0 (x˙ + iω0 x) = F (t) dt m dξ F (t) − iω0 ξ = dt m ξ = x˙ + iω0 x

(G.2) (G.3)

is a complex quantity. Equation G.2 is of the first order. Its solution when the righthand side is replaced by zero is ξ = Aexp(iω0 t) with constant A. We seek a solution of the inhomogeneous equation in the form ξ = A(t)exp(iω0 t), obtaining for the function A(t) the equation A˙ = F (t)exp(−iω0 t)/m. Integration gives the solution of Eq. G.3:  ξ = exp(−iω0 t) 0

t

1 F (t)exp(−iω0 t)dt + ξ0 m

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

(G.4)

181

182

Appendix G

where the constant of integration ξ0 is the value of ξ at the instant t = 0. This is the required general solution; the function x(t) is given by the imaginary part of G.4, divided by ω0 1 .

G.2 Damped Oscillations Consider the equation x¨ = −ω02 x − γ x˙

(G.5)

The solution is x = exp(rt) andobtain r for the characteristic equation r 2 + γ r +    γ 2 2 . The general solution of Eq. G.5 is − ω ω02 = 0. whence r1,2 = − γ2 ± 0 2 x = c1 exp(r1 t + c2 exp(r2 t)

(G.6)

γ 2

Two cases must be distinguished. if < ω0 , we have two complex conjugate values of r. The general solution solution of the equation of motion can then be written as  08 7 /  γ 2 γ ω02 − t (G.7) x = re A exp − t + i 2 2 where A is an arbitrary complex constant, or as  γ  (G.8) x = a exp − t cos (ωt + θ ) 2   2  with ω = ω02 − γ2 and a and θ real constants. If γ2  ω0 , the amplitude of the damped oscillation is almost unchanged during the period 2π ω . Next, let γ2 > ω0 . Then the values of r are both real and negative. The general form of the solution is   7 / 7 /

0 8

0 8  γ 2  γ 2 γ γ 2 2 x=c1 exp − − + −ω0 t +c2 exp − −ω0 t 2 2 2 2 (G.9) In this case when the friction is sufficiently strong, the motion consists of a decrease in |x|, i.e. an asymptotic appoach as (t → ∞) to the equilibrium position.This type of motion is called aperiodic damping. Finally, in the special case γ2 = ω0 , the characteristic equation has the double root r = − γ2 . The general solution of the differential equation is then  γ  (G.10) x = (c1 + c2 t) exp − t 2 This is a special case of aperiodic damping. 1 The

force F (t) must, of course, be written in real form.

G.2 Damped Oscillations

183

G.2.1 Clasification γ ω0 2 γ < ω0 2 γ = ω0 2

(a) overdamped (b) underdamped (c) critical − damping

(G.11)

G.2.2 Summary G.2.2.1

(a) Overdamped Oscillations,

γ 2

 ω0

The equations for the coordenates: position, x, and velocity, v, are given (Fig. G.1)  γ  γ  x = a exp − t cos t +θ (G.12) 2 2  γ   γ  γ  γ t + θ + sin t +θ (G.13) v = −a exp − t cos 2 2 2 2

G.2.2.2

(b) Underdamped Oscillations,

γ 2

< ω0

 γ  x = a exp − t cos (ωt + θ ) 2  γ  γ v = −a exp − t [cos (ωt + θ ) + sin (ωt + θ )] 2 2 with ω =

  2  ω02 − γ2

G.2.2.3

(c) Critical Damping Oscillations,

γ 2

(G.14) (G.15)

= ω0

 γ   γ   x = exp − t x0 + v0 + x0 t 2 2  γ  γ  γ   v0 + x0 t v = exp − t v0 − 2 2 2

(G.16) (G.17)

184

Appendix G

Fig. G.1 v vs x and x, v vs t, Overdamped (a), (b), Eqs. G.12, G.13: N = 20. θ = 0., γ = 5. dt = 0.1, Underdamped (c), (d), Eqs. G.14, G.15: N = 400, θ = 0., γ = 0.06, dt = 0.3, ω0 = 0.6, Criticaldamped (e), (f), Eqs. G.16, G.17: N = 40, θ = 0., γ = 1.2, dt = 0.3, ω0 = 0.6

Appendix H

H.1 Electrical and Mechanical Systems Analogies The establishment of a formal analogy between the differential equations expressing two different types of problems permits a formal transfer of known solutions of problems of one type to those of the other. The method of complex amplitudes developed in connection with electric circuits has a useful application in mechanical problems where generalized definitions of mechanical impedances or susceptibilities are involved. The differential equation of a simple (L,R,C)-circuit acted upon by a sinusoidal electromotive force is  di 1 t L + Ri + i dt = E (H.1) dt C 0 Differentiating Eq. H.1 we obtain L

d 2i i dE di +R + = dt C dt dt 2

(H.2)

Consider, on the other hand, a mechanical system of a damped oscillator excited by an external sinusoidal force. Its equation is m

d 2x dx + kx = F +b dt dt 2

(H.3)

One observes that Equations H.2 and H.3 are of the same form and that the following corresponding quantities indicate the analogy between electrical and mechanical problems: 



dE 1 ,k ; ,F (H.4) (i, x) ; (L, m) ; (R, b) ; C dt © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

185

186

Appendix H

The corresponding electrical, Ze , and mechanical, Zm , impedances for both systems are:  L 2 2 ω − ω0e (H.5) Ze = R + i ω  m 2 2 Zm = b + i ω − ω0m (H.6) ω where i is the imaginary unit, ω0e = (1/LC)1/2 and ω0m = (k/m)1/2 are the corresponding resonance frequencies of both systems. The corresponding susceptibities, α(ω) = i/ωZ(ω), are:   2 2 + iωγ ω − ω0e e  αe (ω) =  (H.7) 2 2 L ω2 − ω0e + (ωγe )2   2 2 + iωγm ω − ω0m  αm (ω) =   2 2 + (ωγ )2 m ω2 − ω0m m

(H.8)

where γe = R/L and γm = b/m are respectively the electrical and mechanical damping factors or resonance widths.

Appendix I

I.1 The Fluctuation-Dissipation Theorem One way of formulating the FDT is by formally regarding the spontaneous fluctuations of a quantity x as due to the action of some random force f , meaning that the environment senses the system through the generalized susceptibility, α(ω), and respond with a fluctuating force. The Fourier components xω and fω are related by: xω = α(ω)fω

(I.1)

The relation between the generalized impedance Z(ω) and α(ω) is: Z(ω) =

i ωα(ω)

(I.2)

dxω dt

(I.3)

As xω = x0ω e−iωt we can write: fω = Z(ω)

The spectral densities of the fluctuation are given by (x 2 )ω =| α(ω) |2 (f 2 )ω

(I.4)

h¯ ω 2kT

(I.5)

The results of the FDT are:

(x 2 )ω = hα ¯ (ω) coth

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

187

188

Appendix I

Correspondingly: (f 2 )ω =

hα hω ¯

(ω) ¯ coth 2kT | α(ω) |2

(I.6)

The mean square of the fluctuating quantity is: 1 < x >= π



2

0

∞

h¯ (x )ω dω = π



∞

2

α

(ω) coth

0

hω ¯ dω 2kT

(I.7)

These formulae constitute the FDT, established by Callen and Welton [1]. They relate the fluctuations of physical quantities to the dissipative properties of the system. At energies kT hω ) ≈ 2kT /hω, ¯ (classical limit) we have coth(hω/2kT ¯ ¯ and | α(ω) |2 ≈ | α (0) |2 . Then Eq. I.7 becomes: < x 2 >=

2kT π

 0

∞

α

(ω) dω ω

(I.8)

Using the Kramers and Kronig’s relations this integral can be written as [2]: < x 2 >= kT | α (0) |

(I.9)

Averaging Eq. I.4 in frequency in the classic region, we have: < x 2 >=< (x 2 )ω >=

(I.10)

and in order for Eqs. I.9 and I.10 to be compatible, we obtain: < f 2 >=

kT | α (0) |

(I.11)

From Eqs. I.9 and I.11 we obtain: 1

1

< x 2 > 2 < f 2 > 2 = kT

(I.12)

This is the classical analogy of the Heisenberg uncertainty principle, [3]. This equation shows a constant equilibrium between the system and the environment, 1 when < f 2 > 2 increases in the ambient, the systems reacts in such a way as to inhibit the fluctuation of the corresponding physical quantity x and vice versa in order to mantain the product constant equal to kT . The FDT can be generalised to the case where several fluctuating quantities xi are considered simultaneously [2]. In this case, Eqs. I.5 and I.6 have to be reemplaced by:

Bibliography

189

  h¯ ω 1  ∗ (xi xk )ω = ih¯ αki − αik coth 2 2kT

(I.13)

  hω 1  −1 ¯ −1∗ coth (fi fk )ω = ih¯ αik − αki 2 2kT

(I.14)

with xiω = αik fkω,

−1 fiω = αik xkω

(I.15)

∗ the corresponding where αki are the matrix elements of the susceptibility and αki complex-conjugate element.

Bibliography 1. Callen, H.B., Welton, T.A.: Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951) 2. Landau, L.D., Lifshitz, E.M.: Statistical Physics, p. 386. Pergamon Press, Oxford (1988) 3. Fornés, J.A.: Fluctuation-dissipation theorem and the polarizability of rodlike polyelectrolytes: an electric circuit view. Phys. Rev. E. 57, 2110 (1998)

Appendix J

J.1 Integral Algorithm for Colored Noise Simulation We follow the formalism given by Fox et al., [1]. Consider the summarized Langevin equation, x˙ = f (x) + gω

(J.1)

where gω is Gaussian white noise, with the properties, gω  = 0,

(J.2)

gω (t)gω (s) = 2Dδ(t − s)

(J.3)

where D is the diffussion coefficient and δ is the Dirac δ-function. In order to obtain exponentially correlated colored noise to drive Eq. J.1 instead of using gω is to replace Eq. J.1 with the pair of equations, x˙ = f (x) + ,

(J.4)

1 ˙ = − ( + gω ) , τ

(J.5)

The driven noise  is now exponentially correlated colored noise (OrnsteinUhlenbeck process, 1930, [2]), see also Gillespie, 1996, [3]. (t) = 0, (t)(s) =

 |t| D exp − , τ τ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9

(J.6) (J.7)

191

192

Appendix J

After this summary of colored noise we can start with the integral algorithm. Integrating Eq. J.5 we obtain

 

 t 1 t t −s (t) = exp − (0) + gω (s) ds exp − τ τ 0 τ  



 1 t+t t + t t + t − s (0) + gω (s) (t + t) = exp − ds exp − τ τ 0 τ

(J.8) (J.9)

Then  

 1 t+t t t + t − s (t) + gω (s) (t + t) = exp − ds exp − τ τ t τ 

t (t) + h(t, t), (J.10) = exp − τ

h(t, t is Gaussian, and with zero mean, because gω also has these properties. Terefore all of its properties are determined by its second moment,

 ! D 2t 1 − exp − , h2 (t, t) = τ τ

(J.11)

For initiating the numerical simulation, we need to know The Box-Mueller algorithm, [4], this algorithm is used to generate Gaussian noise from two random numbers which are uniformly distributed on the unit interval. Thus, to start the simulation, an initial value for  is needed, and is obtained from the Box-Mueller algorithm, namely, m = random number,

(J.12)

n = random number,

(J.13)



1/2 2D = − ln(m) cos(2π n) τ

(J.14)

 t E = exp − τ

(J.15)

Then set

After that, the exponentially correlated, colored noise is obtained by the lines x(t + t) = x(t) + (f (x(t)) + )dt a = random number,

(J.16) (J.17)

J.1 Integral Algorithm for Colored Noise Simulation

193

b = random number,

1/2  2D  1 − E 2 ln(a) h= − cos(2π b), τ (t + t) = E + h

(J.18) (J.19) (J.20)

After Eq. J.20, the algorithm loops back to Eq. J.16 and continues as long as one would like. In our case where we numerically f it the correlation function cv Eq. 8.31, with a superposition of three exponential, Eq. 8.42; the quantum noise i is thus an Ornstein-Ulhenbeck process with properties k (t) = 0,

 1 1 3 1 t 1 2  Dk

k (t)l (t ) = δkl exp − τk τk

(J.21) (J.22)

k=1

The c-number quantum noise (t) due to the heat bath is therefore given by (t) =

3 2

k

(J.23)

k=1

Equation J.23 implies that (t) can be realized  as a sum of several Ornstein5 $t)= 3k=1 k (t)l (t ) Then the simulation Ulhenbeck noises k satisfying cv ( proceeds as follows: ak = random number, k = 1, 3,

(J.24)

bk = random number, k = 1, 3. / 01/2 eff Dk D ˜k = −2 ln(ak ) cos(2π bk ) τ τk 

t˜ Ek = exp − τk τ

(J.25)

x( ˜ t˜ + t˜) = x( ˜ t˜) + (f˜(x( ˜ t˜)) + )d t˜

(J.26)

(J.27) (J.28)

After that, the exponentially correlated, colored noise is obtained by the lines: Generate new random numbers ak = random number, k = 1, 3,

(J.29)

bk = random number, k = 1, 3.

(J.30)

194

Appendix J

/

 eff 2Dk  D 1 − Ek2 ln(ak ) hk = − τ τk ˜k (t˜ + t˜) = ˜k Ek + h˜ k ( ˜ t˜ + t˜) =

3 2

˜k (t˜ + t˜)

01/2 cos(2π bk ),

(J.31) (J.32) (J.33)

i=1

After Eq. J.33, the algorithm loops back to Eq. J.28 and continues as long as one would like. In our case where we numerically f it the correlation function cv Eq. 8.31, with a superposition of three exponential, Eq. J.33. eff , hk , are matrices, A(i, j ), with the first index i being the Indeed k , , D time and the second j being the number of the stochastic process realization. See program at the end of the chapter.

Bibliography 1. Fox, R.F., Gatland, I.R., Roy, R., Vemuri, G.: Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise. Phys. Rev. A 38(11), 5938 (1988) 2. Uhlenbeck, G.E., Ornstein, L.S.: On the theory of the Brownian motion. Phys. Rev. 36, 823 (1930) 3. Gillespie, D.T.: Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E 54(2), 2084 (1996) 4. Box, G.E.P., Muller, M.E.: A note on the generation of random normal deviates. Ann. Math. Stat. 29(2), 610–611 (1958)