Physical Principles of Electro-Mechano-Biology: Multiphysics and Supramolecular Approaches (Studies in Mechanobiology, Tissue Engineering and Biomaterials, 25) 3031379802, 9783031379802

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Studies in Mechanobiology, Tissue Engineering and Biomaterials 25

Christian Brosseau

Physical Principles of Electro-MechanoBiology Multiphysics and Supramolecular Approaches

Studies in Mechanobiology, Tissue Engineering and Biomaterials Volume 25

Series Editor Amit Gefen, Department of Biomedical Engineering, Tel Aviv University, Ramat Aviv, Israel

The series Studies in Mechanobiology, Tissue Engineering and Biomaterials is intended for a general international professional audience with interest in advancing the fields of mechanobiology, tissue engineering and biomaterials, including for example scientists and practitioners in the fields of bioengineering, biomedical engineering, mechanical engineering, materials engineering, biophysics, biology, and medicine. Indexed by SCOPUS.

Christian Brosseau

Physical Principles of Electro-Mechano-Biology Multiphysics and Supramolecular Approaches

Christian Brosseau Department of Physics University of Brest Brest, France

ISSN 1868-2006 ISSN 1868-2014 (electronic) Studies in Mechanobiology, Tissue Engineering and Biomaterials ISBN 978-3-031-37980-2 ISBN 978-3-031-37981-9 (eBook) https://doi.org/10.1007/978-3-031-37981-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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

Contents

1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

1 5

Background

2 Elementary Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Anatomy of a Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electrical Properties of Cell Membranes . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Electrical Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Single-Cell Modelling by a RC Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cellular Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Mechanical Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Single-Cell Deformation Modelling by a Network of Spring and Dashpot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Extension to Multicellular Structures and Biological Tissues . . . . . 2.4.1 Electrical Properties of Tissues and Cell Suspensions . . . . 2.4.2 Equivalent Electrical Circuit of Biological Tissues . . . . . . . 2.4.3 Mechanical Properties of Tissues . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Brief Sketch of the History of EMB: Where Good Ideas Come From . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Early Ideas on EMB: Standing on the Shoulders of Giants . . . . . . . 3.2 Continuum Approaches of the Electrical and Mechanical Properties of Biological Materials: Lessons Learned . . . . . . . . . . . . 3.3 EMB Model Building and Phenomenology: How We Got to Know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Computational EMB: A Multidisciplinary Approach . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 17 17 23 31 36 40 43 45 45 46 48 67 68 70 78 81 86

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Part II

Contents

Calculation Methods

4 Analytical Approaches of EMB at Multiple Scales . . . . . . . . . . . . . . . . . 4.1 How Physics Scales EMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 ED and EP Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 ED and Transmembrane Voltage-Dependence of Membrane Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Macroscopic Models of the Cell EP Energetics and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effective Dielectric Properties of Biological Materials . . . . . . . . . . 4.3.1 Polarization in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Polarization and Permittivity in an Alternating Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dielectropheresis and Electrorotation of Cells . . . . . . . . . . . . . . . . . Appendix 1: Electric Dipole and Polarizability . . . . . . . . . . . . . . . . . . . . . . Appendix 2: ITV and Electrodiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3: Maxwell Stress Tensor and Electrostatic Force Acting on an Isolated Body in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4: Electro-Thermal Effects in Biological Materials . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 102 107

5 Computational Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Some Preliminary Considerations on the Finite Element and Boundary Integral Element Methods . . . . . . . . . . . . . . . . . . . . . . 5.2 Numerical Determination of Cm (Vm ) and Rm (Vm ) and Electrostatic Control for an Isolated Cell Model . . . . . . . . . . . . 5.3 Proximity-Induced ED and Membrane Capacitance Coupling Between Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Modelling Cell Membrane ED by Alternating Electric Fields . . . . 5.5 Effective Dielectric Properties of Cells . . . . . . . . . . . . . . . . . . . . . . . 5.6 Electrostatic Forces Between Biological Cells . . . . . . . . . . . . . . . . . 5.7 Strain Energy in Multicellular Environments . . . . . . . . . . . . . . . . . . 5.8 An Approach to Electrical Modeling of Realistic Multicellular Structures Before EP . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 An Electromechanical Boundary for the Cell Membrane ED . . . . . 5.10 Electrical Coupling of the Cell Membrane and Nucleus Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Computing the ITV for an Isolated Cell Model . . . . . . . . . . . Appendix 2: Two-State Model of a Random Assembly of CS Spherical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3: Thermal Noise in Cell Membrane and Nuclear Envelope Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

108 110 125 125 136 143 147 149 152 156 158

183 189 195 200 207 216 222 229 235 239 256 260 263 265

Contents

vii

Glossary of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Chapter 1

Preface

For centuries physics and biology have intersected. Molecular biological machinery ultimately relies on cells functioning at physiological temperature i.e. 36.5–37.5 °C (97.7–99.5 °F). Over a 1–2 °C window, this single variable will not play a crucial role as in standard physics, where one can change the temperature and scan vastly different energy scales. By contrast, electric fields and mechanical stresses can have huge impact on the electrodeformation (ED) and electroporation (EP) and electropermeabilization of biological cells. Some of the most difficult problems in biophysics today involve understanding how systems order themselves and why they sometimes fail to do so. How do proteins consistently fold into unique structures, given the myriad possible paths available to them? What are the forces between cells, and how do those forces determine the 3D structure of tissues (and living things)? We view these examples as both important and convincing enough to motivate the study of electro-mechano-biology (EMB). EMB has been an active research topic over the past decade and has led to intriguing questions promising exciting developments for clinical applications. EMB has evolved in the past decade through evaluating the consequences of experimental measurements as well as exploiting computational tools that permit exploration of new model building and clinical possibilities. In science it is worth tracking backwards, not only to see where we have come from but also to be humbled by the vastness of what is still unknown. Questions which were posed in the early days of bioelectricity are still inspiring scientists today. The history of EMB may be traced back to the eighteenth century when Galvani and Volta debated whether the twitching of frog legs when contacted with a metal wire was due to “animal electricity (Galvani’s point of view)” or “contact electricity (Volta’s point of view)”. History of science taught us that the concept of “animal electricity” was not needed [1]. The field of developmental bioelectricity was born. In a different perspective, D’Arcy Thompson [2] pointed out correlations between regular patterns in biological systems and mechanical phenomena. D’Arcy Thompson’s expansive treatise relating to the importance of mechanics in biological systems represent one of his finest achievements. In the early 1900s, he proposed that physical methods, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9_1

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such as force balance and energy minimization, are just as applicable to biological systems as inorganic ones. A new scientific field was spawned, called mechanobiology. As a result, the academic community that studies the boundaries between electropermeabilization and mechanobiology has developed the EMB terminology to describe different forms of knowledge creation within and across the disciplines of physics and biology. Since then, EMB is regularly found in applications ranging from delivery of genetic material to food processing and environmental management. Work in this field over the past two decades rapidly advanced, with the last five years seeing significant innovation. Looking at the boundary of physics, mechanics and biology, the problem of cell membrane permeabilization (CMP1 ) presents an excellent opportunity for crossdisciplinary science and engineering and a context in which fundamental physics is called for to answer complex questions. Hence, many topics in biology comprise a growing area of inter-disciplinary science and provide significant opportunities for the application of the first-principles of physics which offer many beautiful examples of the interconversion of force, energy and entropy in matter systems. In parallel with this development, a major feature of biology in the twenty-first century is its transformation from phenomenological and descriptive discipline to quantitative and predictive one. Models and experiments show that rather than being a passive bystander in the function of membrane-bound proteins, the membrane can at times have an essential role in determining the function of these proteins. The data that emerge often reveal functional and quantitative relations between biologically interesting parameters, e.g. the open probability for ion channels as a function of voltage or membrane tension, and carry with them an imperative for models of the underlying phenomena [6]. The underlying goal of the current cell EMB research is to combine theoretical, experimental and computational approaches to construct a realistic description of cell electromechanical behaviors that can be used to provide new perspectives on the role of physics and mechanics in human diseases. Additionally, due to the large separation of length scales2 between the membrane thickness and the size of the cytoplasm, it is challenging to understand quantitatively the electromechanical couplings in these systems. With developing advanced multiphysics and multiscale numerical analysis we expect to learn many cross-properties of biological materials which involve multiple spatial or temporal scales. In that respect, one important issue is to undertsnad how do passive and active microscopic material properties manifest itself on the macroscale and give rise to spatio-temporal structures and symmetry-breaking phenomena. But multiscale modeling also relies on heavily on physics to define and 1

Notice that in this book electropermeabilization and EP are often used interchangeably to describe the permeabilization status of the membrane upon application of a strong electric field. Some investigators have argued for the effectiveness of this terminological distinction, while other investigators provided different perspectives on this subject [3–4]. Different techniques of CMP such as magnetoporation, sonoporation and optoporation have been extensively developed, see e.g. [5]. 2 We refer to length scales as levels of biological organization. In general, they are relatively welldefined. However, it remains often uneasy to give precise quantitative values for the lengths at which we swtich from one level to another one. The major length scales will be described later in Sect. 4.1. In this book, we will focus our attention to the cell and tissue levels.

1 Preface

3

constrain that ways that molecular and cellular processes can scale up to produce tissue and organ-scale physiology. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities: here, most of our simulations were performed using the multi-physics COMSOL® software package [7]. Since EMB involves several decades of spatial scales (from nm (membrane thickness) through µm (cell size) up to cm (tissue sample)) and temporal scales (from ns to h), understanding its first-principles features requires a multiscale modelling approach, ranging from molecular simulations to large-scale continuum models of cells and tissues. An additional level of complexity arises from the intricate coupling between the subcellular components and interactions between cells in tissues. The emergence of complexity in self-organizing biological systems poses fundamental challenges to their quantitative description: multiscale modeling and machine learning (ML) techniques are introduced to reduce the complexity of biomolecular systems while maintaining an efficient description of the observables of interest. These challenges are cross-generational and their considerations are likely to require cross-disciplinary explorations, instead of being the purview of a particular sub-discipline. There are some specific points connected with present volume which deserve further comments. In the first place, we review the recently developed mechanistic understanding of EMB in which the focus is primarily on the couplings between the electric and mechanical fields. Our emphasis is on the analytical and computational aspects of EMB at the cellular level. This book is divided into two parts. In the first part, we start by defining and discussing the relevant basic aspects of the electrical and mechanical properties of cell membranes. We provide an overview of some of the ways analytical modelling of cell membrane ED and EP appears in a variety of contexts as well as a contemporary account of recent developments in computational approaches that can feature in the theory initiative, particularly in its attempt to describe the cohort of activities currently underway. Intended to serve as an introductory text and aiming to facilitate the understanding of the field to non-experts, this part does not dwell on the set of topics, such as cellular mechanosensing and mechanotransduction,3 irreversible EP,4 and atomistic molecular dynamics (MD) modelling of membrane EP. The reader is encouraged to refer to the many references cited in this part of the book to gain a complete mechanistic understanding of these subjects. Complementary to the atomic and molecular-level events that are currently 3

Mechanosensing, which focuses on understanding how force is converted into the biochemistry that regulates cell behavior, remains one of the biggest puzzles plaguing biophysics. In many cases, the existence of forces acting directly on molecules or cells is required in order to trigger the correct biological response. This is in essence what mechanotransduction is: the ability to alter biological outcomes through mechanical forces. In the case of mechanosensitive ion channels, a signal can be produced when forces acting within the PB rise to a level sufficient to produce a conformational change in the protein channel and consequently alter its conductance [8]. 4 Irreversible EP uses pulsed electrical fields to induce an irreversible process of permeabilization of cell membranes, thus triggering a substantial alteration in the physiological equilibria of cells, ultimately leading to cell apoptosis. Irreversible EP is attributed to PB rupture by uncontrolled pore growth. It is an efficient approach to solid tumor ablation [9]. In reversible EP, electric pulses are not aimed at killing cells, but to aid in the uptake of therapeutic drugs.

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addressable only by simulations of MD, what we present in this part is a supramolecular level understanding of EMB. This choice reflects my research interests, and is also the starting point for many other avenues of research on EMB. The second (and larger) part of the book is devoted to a presentation of the necessary analytical and computational tools to illustrate the ideas behind EMB and illuminate physical insights. The scope of this book is limited: other authors would emphasize different problems and examples. Brief notes on the history of EMB and its many applications, describing the variety of ideas and approaches are also included. Through this part, we touch on the first-principles background and practical calculation methods with the intention of highlighting aspects that are not actually found in a single volume. Paradoxically, our main focus is on equilibrium continuum approach ignoring altogether the molecular details of membrane heterogeneities which have been evidenced by the great successes of structural biologists. Continuum methods have proved to be extremely useful quantitative tools, allowing the accurate prediction of the magnitude of electromechanical effects. The great impact of equilibrium continuum methods results from their intuitively yet physically robust model of membranes, requiring only moderate computational expense and the use of a finite element (FE) multiphysics software. One crucial take-home lesson from this approach is that it combines the principles of physics and computer science that aims to develop a quantitative understanding of biological phenomena allowing us to predict and simulate biological behaviors. However, many puzzles remain in both more phenomenologically oriented and more theoretically oriented contexts which form the basis for a rich research subject in the future. This particular field is such a melting pot. You can come from physics and computation. You can come from mechanical engineering. You can come from the biological scientist’s point of view. You can start from almost anywhere and find that your expertise is applicable and at the same time you have a lot of specialties to learn about. This exchange of knowledge and ideas provides excellent opportunities to make innovative advances based on an integrative approach of all these disciplines [10]. Teamwork is key. Thus, we try to use this material in traditionally separate departments and audiences including computer science, biology and physics disciplines. Since there is a great need for a cross-disciplinary analysis of EMB subjects and pooling expertise from many disciplines can be a challenge for a newcomer we attempt to provide a comprehensive list of primary sources which would be beneficial for advanced undergraduate and graduate students in physics and biophysics in order they get a better understanding of what EMB can involve. We hope that this gentle introduction of EMB will inspire readers to undertake a more scholarly investigation of those topics they find especially interesting. The text can also serve biomechanical engineers who seek to broaden their biophysical engineering “toolbox”, to include some of the fundamentals of multiphysics and supramolecular analysis of biological materials. Finally, I have supplied with each chapter an extensive bibliography, not only of papers referred to, but of those bearing on the subject. Too many bibliographies in review articles and other places lose most of their usefulness by not including the titles of references, as well as author and journal references.

References

5

In many ways, this book is autobiographical in nature, drawing on the author’s personal contributions in the subject, interactions with other teams, PhD and postdoc mentoring, and lessons learned. Scientific work hardly seems worth the effort unless at least one other person responds to it. The book had a long gestation period and built on some of the work I’d done with my research group. In this instance, many folks were critically important to the development of the ideas in this book: Melvin Essone Mezeme, Tomo Murovec, Danish Shamoon, and Elias Sabri in particular were all very generous with their expertise but we also benefited tremendously from conversations with numerous members of the biophysics community. Obviously, all shortcomings in this book are due to my failure and are no fault of their own.

References 1. W. Bernardi, in The controversy on animal electricity in eighteenth-century Italy: Galvani, Volta and Others, in Nuova Voltiana, ed. by F. Bevilacqua, L. Fregonese (Hoepli, Milano, 2000); N. Kipnis, Changing a Theory: The case of Volta’s contact electricity, in Volta and the History of Electricity, ed. by F. Bevilacqua, L. Fregonese (Hoepli, Milano, 2000); M. Bresadola, Medicine and science in the life of Luigi Galvani. Brain Res. Bull. 46, 367–380 (1998) 2. D.W. Thompson, On Growth and Form (Cambridge University Press, 1917) 3. T. Kotnik, L. Rems, M. Tarek, D. Miklavˇciˇc, Membrane electroporation and electropermeabilization: mechanisms and models. Annu. Rev. Biophys. 6, 63–91 (2019) 4. V. Vajrala, J. R. Claycomb, H. Sanabria, J. H. Miller Jr., Effects of oscillatory electric fields on internal membranes: an analytical model. Biophys. J. 94, 2043–2052 (2008); U. Zimmerman, G. A. Neil, Electromanipulation of Cells (CRC Press, 1996). 5. X. Du, J. Wang, Q. Zhou, L. Zhang, S. Wang, Z. Zhang, C. Yao, Advanced physical techniques forgene delivery based on membrane perforation. Drug Deliv 25, 1516–1525 (2018) 6. R. Phillips, T. Ursell, P. Wiggins, P. Sens, Emerging roles for lipids in shaping membraneprotein function. Nature 459, 08147 (2009) 7. COMSOL® Multiphysics version 6.0. 8. B.L. Ricca, G. Venugopalan, D.A. Fletcher, To pull or to be pulled: parsing the multiple modes of mechanotrasduction. Curr. Opin. Cell Biol. 25, 558–564 (2013); J.D. Humphrey, M.A. Schwartz, G. Tellides, D.M. Milewicz, Role of mechanotransduction in vascular biology: focus on thoracic aortic aneurysms and dissections. Circ. Res. 116, 1448–1461 (2015); A.- L. Le Roux, X. Quiroga, N. Walani, M. Arroyo, P. Roca-Cusachs, The plasma membrane as a mechanochemical transducer. Phil. Trans. Roc. Soc. B 374, 20180221 (2019); T.J. Kirby, J. Lammerding, Emerging views of the nucleus as a cellular mechanosensor. Nat. Cell Biol. 20, 373–381 (2018) 9. R.V. Davalos, L.M. Mir, B. Rubinsky, Tissue ablation and irreversible electroporation. Ann. Biomed. Eng. 33, 223–231 (2005); M.L. Yarmush, A. Goldberg, G. Sersa, T. Kotnik, D. Miklavˇciˇc, Electroporation-based technologies for medicine: principles, applications, and challenges. Ann. Rev. Biomed. Eng. 16, 295–320 (2014); N. Jourabchi, K. Beroukhim, B.A. Tafti, S.T. Lee, E.W. Lee, Irreversible electroporation (Nanoknife) in cancer treatment. Gastrointest. Interv. 3, 8–18 (2014); K.N. Aycock, R.V. Davalos, Irreversible electroporation: background, theory, and review of recent developments in clinical oncology. Bioelectricity 1, 214–234 (2019)

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10. C. Brosseau, Some reflections on the impact of physics, materials science, and engineering on biology and medicine. Ann. J. Mater. Sci. Eng. 1, 8 (2014); Editorial, Can physics deliver another biological revolution? Nature 397, 89 (1999); R. Phillips, S.R. Quake, The biological frontier of physics. Phys. Today 59, 38–43 (2006); J. Ross, A.P. Arkin, Complex systems: from chemistry to systems biology. Proc. Natl. Acad. Sci. U.S.A. 106, 6433–6434 (2009); P. Schwille, Bottom-up synthetic biology: engineering in a tinkerer’s world. Science 333, 1252–1254 (2011)

Part I

Background

Chapter 2

Elementary Concepts and Definitions

And what’s so small to you Is so large to me If it’s the last thing I do I’ll make you see. Suzanne Vega

In order to provide context and help for newcomers in the area of biological physics, the goal of this chapter is to develop a feeling for the physics of cell membranes. Our aim is to present our interpretation of EMB in a language familiar to physicists. Biological materials are exquisitely structured and depend critically on their dynamic 3D organization to perform their physiological functions. Many parameters are involved that need to be known if function is to be understood and predicted. But all models of cells must lie within the constraints set by physics. Contrastingly with the reductionist view1 which is effective in many areas of physics the whole of a living system cannot be completely understood by the study of its individual parts. That’s because biological properties emerge for reasons that are not at all apparent if one’s analysis begins at the smallest length scale. Implicit in approach of biological physics is an intention to study living organisms as a network of interacting parts and recognition that complex biological systems such as cells and tissues are dynamic and behave differently under varying exogenous stimuli. As the contemporary literature on EMB modelling and physical measurements spans more than three decades and is too vast and scattered to review in depth, we instead highlight a few studies that illustrate important biophysical principles that will be of utmost importance for us. The structures of potential interest include the individual cell and its membrane.

1

In a reductionist research strategy, systems are analysed into their components, their configurations, and their interactions in order to understand the system as a whole. Whether or not this reductionist perspective can always be completely successful is a controversial question. Especially in biophysics, there is a long tradition of using ideas from statistical physics to think about the emergence of collective behavior from the microscopic interactions, with the hope that this functional collective behavior will be robust to our ignorance of many details in these systems. In terms of numerical simulations this hierarchical reductionism is also called coarse-graining, i.e. the resolution is reduced on purpose to lose the details of the lowest levels.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9_2

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2 Elementary Concepts and Definitions

In practice, most of the attention will be focused at the cellular level, focusing on the electrical and mechanical properties of membranes. Thus, we begin with a brief overview of the basic morphological characteristics of the membrane enveloping living cells which separates the cytoplasmic medium (cytosol) from the extracellular medium (ECM). The ECM of cells is a complex 3D fibrous meshwork with a wide distribution of fibres and gaps that provide complex biochemical and physical cues, which are very different from uniformly 2D surfaces [1]. Since we are interested in the ED and EP mechanisms we first place an emphasis on the electrical properties of membranes. We outline the interpretation of typical cell membrane electrical measurements and show how this allows a basic introduction into the mechanisms underlying ED and EP. Since perturbations to the mechanical environment can affect cell behavior and its normal physiology, we next briefly review the current understanding of how the mechanical characteristics of cell relate to underlying architectural changes and describe how these changes evolve with membrane deformability in response to an applied stress. The chapter closes with a discussion of possible generalizations to cell assemblies and simple models of biological tissues.

2.1 Anatomy of a Cell Molecular biological machinery ultimately relies on cells functioning at physiological temperature. Modern biology has coincided with the emergence of new ways to visualize cells, i.e. the cellular interior. A full mechanistic understanding of biological function can emerge only if we are able to integrate all relevant information at multiple levels of organization to recreate dynamic interactions. Multicellular organisms are characterized by enormous diversity in their multitude of different cell types. However, each cell in a living organism is enclosed by a membrane that physically separates the ECM from the intracellular components that are suitable for the survival of the cell [1].2 While the cell membrane is typically two orders of magnitude smaller than the resolution of the optical microscope, one of the first estimations of the membrane thickness was obtained from measurements of the electrical impedance of a suspension of cells [2]. Despite its nanometric thickness, a cell membrane constitutes a formidable barrier for many compounds, so specific proteins reside in the membrane that tightly control and facilitate active and passive transport of ions and small molecules across the membrane. This is mainly conveyed through transmembrane proteins and peripheral membrane proteins that associate with the inner leaflet of the membrane. Since a significant fraction of membrane proteins are associated with the cytoskeleton structure they are essentially immobile within the membrane plane.

2

Many cell models have been proposed over the years, some of them can be accessible online via the CellML repository (http://www.cellml.org).

2.1 Anatomy of a Cell

11

Typical quoted numbers for cells in the human body are in the tens of trillions.3 From the physical standpoint, the basic form of single-shell model of eukaryotic4 cell can be traced back to a profound paper by Schwan in 1957, which suggested that most eukaryotic biological cells in suspension, being highly heterogeneous objects, can be described by the canonical core–shell (CS) structure, i.e. a dielectric nanometric membrane, a phospholipidic bilayer (PB),5 surrounded by conducting cytoplasm and extracellular medium[2, 4–8]. It is worthwhile to note that typical cell membranes may contain hundreds of different lipids, asymmetrically distributed between the two bilayer leaflets and are crowded with proteins covering an estimated membrane area as large as 30%. The proportions of the components vary but lipids dominate, reaching 102 lipid molecules for every protein in some membranes. The exact composition is highly diverse and depends on the cell type, ECM, and environmental factors. It was also reported that membranes of mammalian cells contain an amount of 20– 50% cholesterol, which is an important component of lipid rafts involved in signal transduction and endocytosis. Specific interactions between the different components in the membrane lead to a specific ordering and segregation of these components in the plane of the membrane. This induces a long-range order and cooperativity within segregated domains. Furthermore, instead of being static, cell membranes are highly This on the order of a hundred times the estimated 200 billion or 2 × 1011 stars in our galaxy. There are two main types of cells, prokaryotic and eukaryotic. Prokaryotes are cells that do not have membrane bound nuclei, whereas eukaryotes do. The rest of our discussion in this book will focus only on eukaryotes. Eukaryotes represent only a small portion of the diversity and abundance of life on Earth. The total biomass of bacteria and archae is typically 40 times greater than that of all animals combined (and approximately 1300 times more biomass than humans) [1]. Structurally, the cell boundary of bacteria is more complex than that of mammalian cells and can include one (grampositive) or two cell (ram-negative) membranes composed of PB, and in most bacterial species, a cell wall [1]. 5 Phospholipids have a polar head and two hydrophobic hydrocarbon tails. The cell membrane is composed of two layers of phospholipids with the hydrophilic head groups facing the aqueous intra- and extracellular environments, while the hydrophobic acyl chain aligns laterally forming the hydrophobic core of the bilayer. Due to these properties, when a large number of lipid molecules are placed in a solution they self-assemble, under suitable conditions, into 2D structures consisting of two leaflets (or monolayers). The lipid molecules are oriented so that the tails of the molecules in each leaflet are in contact with each other, while the heads are in contact with the ambient solution [3]. The same forces that drive phospholipids to form bilayers also provide a self-healing property. A small tear in the bilayer creates a free edge with water; because this energetically unfavourable, the lipids spontaneously rearrange to eliminate the free edge. Thus, the prohibition against free edges has the profound consequence: the only way for a bilayer to avoid having edges is by closing in on itself and forming a sealed compartment. The acyl chains of the phospholipids intercat with each other by van der Waals interactions, while the interactions of the polar groups at the interface with the solution are expected to be largely of the Coulombic and hydrogen-bonding types. The polarization of membranes can have various reasons. On average, about 80% of the lipids of biological membranes are zwitterionic. Zwitterionic lipids possess permanent electrical dipole moments. Depending on the specific membrane, 10 to 40% of the lipids carry a net negative charge. Biomembranes also contain integral and peripheral proteins with asymmetric distribution between inside and outside, which carry positive and negative charges. Due to such compositional asymmetries, a spontaneous electrical dipole moment of the membrane can be generated in the absence of an externally applied potential. 3 4

12

2 Elementary Concepts and Definitions

dynamic and characterized by distinct phases and internal fluctuating structures, thus allowing proteins to function under out of thermodynamic equilibrium conditions [1]. How cells establish and maintain a well-defined size is a fundamental issue of cell biology [1]. Bacterial cells are usually not more than 2–3 μm in size while eukaryotic cells are much larger.6 Each eukaryotic cell consists of two parts: the cell body (with typical radis 10–20 μm) and the nucleus (typically 10 μm), with the exception of red blood cells, which have a diameter of 8 μm and no nucleus.7 The interior of living cells is a molecular-crowding environment, where large quantities of various molecules coexist and a fragile soft material in which the effects of biochemistry, molecular crowding and physical forces are complex and inseparable [1]. Investigations into the nature of this environment are essential for an understanding of both the elaborate biochemical reactions and the maintenance of homeostasis occurring therein. Recent developments in various techniques to analyse and visualize the cellular interior, e.g. NMR, optical and electron microscopy, neutron scattering, and large-scale molecular dynamics simulation have provided an unprecedented and detailed microscopic view of cells. The reader is encouraged to refer to Fig. 2.1 to see the basic heterogeneous and complex architecture of a cell membrane is the PB, in which are embedded many proteins that perform a variety of functions including energy transduction, signaling, transport of ions, etc. All these considerations suggest that the cell is a very crowded place [1]. Ion channels are pore-forming transmembrane proteins (Fig. 2.1) that allow ions to pass through the membrane Voltage-gated ion channels are sensitive to mV-scale transmembrane electric potentials and respond to these voltages by undergoing a conformational change that allows selected ions to pass. With an outer diameter of 1.5 nm and an inner pore diameter of 0.3–0.4 nm, voltage-gated ion channels in the membrane allow monovalent cations such as Na+ and K+ to pass.8 Ion channels conduct at rates of up 107 ions per second and are capable of differentiating between ions of similar sizes. K+ channels conduct K+ ions at rates up to 103 times that of the slihghtly smaller Na+ ion. It is worthwhile to note that the right panel of Fig. 2.1 will be at the very basis of the coarse-grained (equilibrium continuum) analysis of EMB throughout this book. Continuum approaches are useful when the length scales of interest are much larger than microscopically relevant structural features that act as small perturbations to global variations in properties. Even if membrane dynamics 6

Although the cell size varies strongly between different cell types, populations of cells of the same type typically display a well-defined uniform size that is well regulated [1]. 7 Eukaryotic cells come in many shapes and sizes, but a common feature is that the nucleus is usually positioned at the center of the cell. The mechanism for this centering is a fundamental question of cellular organization that has long puzzled cell biologists. In this regard, microtubules appear to play a key role because they are one of the few cellular structures whole length scale approaches that of the whole cell, and also becausetheir rapid polymerization dynamics allow them to explore the entire cytoplasmic space. We alos note that off-center locations are common in specific biological circumstance, e.g. asymmetric division during embryogenesis. 8 Ion channels are gated, i.e. they can switch conformation between a closed and an open state, only the former allowing ions to flow. MD simulations on model nanopores reveal that a narrow ( V EP . Experimental estimates for V EP fall in the range of 0.2–1.5 V [25–29], so the huge electric field strength across the membrane is on the order of 108 Vm−1 . Note that this field is an order of magnitude higher than the electric fields associated with electric breakdown in the atmosphere, i.e. Paschen’s law. It is worth noting that electric fields five times as high have been measured in membranes with no evidence of electric breakdown of the PB [30]. As the number of pores increases, the membrane conductivity σm (resp. the membrane resistance Rm ) increases (resp. decreases) until critical pore size and number are attained [31–34] which have for effect to counteract the V m increase. Visualization of isolated pores by fluorescence imaging during the application of potential allows estimating their conductance, typically 400 pS, in support of the hydrophilic pore model [31–36]. From the electrical point of view, we assume that a biological cell contains highly conductive (material parameters are typically for cytoplasm: σc ≈ 0.3 Sm−1 and relative permittivity εc ≈ 60; extracellular medium: σe ≈ 1.1 Sm−1 and relative permittivity εe ≈ 76) aqueous electrolytes separated by a very thin, low-conductivity membrane (σm ≈ 10–5 Sm−1 , relative permittivity εm ≈ 11 and thickness d m = 5 nm). Within a simple electrostatic model, the cell membrane can be considered as a capacitor that can be charged by applying a field across the membrane. Electrically speaking, the membrane can be considered as a capacitor separating two conductors. The capacitive nature of the membrane is essential for cell viability in external fields because it acts as a shield to the cytoplasm. The conventional equations describing electrostatic effects in a biological cell predict proportionality between the total charge Qm accumulated at the inner and outer surfaces of the membrane and Vm , i.e. Qm = Cm Vm . An implicit-yet fundamental- assumption underlying membrane electrical modelling is that it can be described by a parallel-plate capacitor made √ with Am is and thickness d « two identical [plates. The capacitance with plate area A m m )] ( √ log( Am /dm ) ε0 εm Am –12 −1 √ , where ε0 ≈ 8.85 × 10 Fm is the vacuum Cm = dm 1 + O 2 A /d m m permittivity [34]. Thus, its capacitance per unit area of membrane is C˜ m = ε0 εm /dm since the second term in the brackets can be neglected. Given that in its plane the membrane is volumetrically incompressible (Am dm = constant), the capacitance changes proportionally to the square of the changing planar area Am [1]. Typical data obtained on biological cells and artificial PB indicate that the membrane capacitance per unit area is of the order of 10–2 Fm−2 , which represents a very large capacitance per unit area because of the nanometric thickness of the membrane [2, 8, 37]. The larger the area, the larger the capacitance will be. Cell capacitance may change due to local membrane thickness variations at lipid rafts, which are approximately 10% thicker than non-raft membranes[1, 23]. The electrical energy stored in the membrane capacitance at physiological resting potential is typically 2 × 10–5 Jm−2 . It is small compared to the interfacial free energy of a PB membrane (on the order of 10–3 Jm−2 ) and explains why EP is not observed under physiological conditions. Keep also in mind that the membrane has also an associated areal resistance R˜ m = dm /σm of the order of 10–2 Ω m2 . Assuming a /uniform permittivity in the membrane, the field across the/membrane is Em = Vm dm and the force acting on this capacitor is Fm = Cm Vm2 2dm . This results in a pressure on the membrane of 104 Nm−2 at

20

2 Elementary Concepts and Definitions

100 mV (voltages close to 100 mV are typical in the voltage clamp electrophysiology experiments used to measure protein conductance). This pressure has a direction normal to the membrane surface. This implies that increasing the force results in a reduction of thickness and an increase of area. Additionally, the membrane resistance value is in the range 109 –1011 Ω [2, 8]. These values have been repeatedly confirmed experimentally. Consider a spherical cell of radius R = 10 μm, which represents a surface area 4π R2 of typically 10–9 m2 , actual membrane capacitance of the order of 10 pF, and actual resistance 109 Ω at low values of V m .20 The nominal resistance is difficult to measure since electrodes will puncture the membrane and thus decrease its apparent conductivity. If one sets Vm to 5 mV, this would require a charge transfer of 1.6 × 10–14 C across the membrane, i.e. corresponding to 9.8 × 104 univalent cations (each bearing an elementary charge of 1.6 × 10–19 C). If the cytoplasm is modelled as a physiological solution (0.10 N) KCL containing 3 × 1010 univalent K+ , it means that the generation of a voltage difference across the membrane of 5 mV necessitates the change in the number of K+ of 3 ppm. This illustrates the strength of the electric effects, i.e. the membrane has a very large capacity to store charge with little voltage rise. Membrane capacitance may also change due to local membrane thickness variations at lipid rafts, which are 10% thicker than non-raft membranes. While the intracellular and extracellular fluids (e.g., of the resting neuron) share a common ingredient, water, they differ significantly from each other in the relative concentrations of ions they contain (Fig. 2.2). For instance, the concentration of K+ ions is higher in the cytoplasm compared to the extracellular space, whereas Na+ and Cl− ions are found at higher concentrations extracellularly. Note that Na+ and K+ behave differently in the ECM: the electric drift K+ flux has a much smaller magnitude in comparison to diffuse K+ flux, whereas for Na+ the magnitude of electric drift flux is comparable to diffusive flux [38]. Fluctuations in the concentration of these charged species must produce fluctuations in the local electric field. The diffusion produced by these thermal fluctuations can have important effect on several biological functions, e.g. resting potential, and cell motility.21 As Eisenberg [38] puts it, “ion channels are the valves of cells”, or put another way, ion channels and electrostatic interactions play a crucial role in controlling the biological functions. There are many experimental techniques can be used to measure V m , e.g. patchclamp electrodes, potentiometric dyes [39]. Voltage-clamp experiments provide a Corresponding to a gravimetric membrane capacitance of typically 500 Fg−1 , i.e. close to the largest specific gravimetric capacitances composed of reduced graphene oxide films used for supercapacitor electrodes. 21 Thermal fluctuations are noise from the environment which give the charge some random chance to get energy from the background and jump over an energy barrier. To get a feel for the importance of thermal effects, note that the natural energy unit of physical biology is the piconewton-nanometer. The piconewton is the characteristic force generated by molecular machines and the nanometer is the typical length scale [1]. Instruments capable of mechanically probing and manipulating single cells and biomelocules in this range of units have provided opportunities for examining the processes responsible for operation of cellular machinery, the forces arising from molecular motors and the interactions between cells, proteins and nuclear acids. 20

2.2 Electrical Properties of Cell Membranes

21

Fig. 2.2 The extracellular fluid of the resting neuron is similar to seawater, with large concentrations of chloride and sodium ions, but small concentrations of potassium ions. This is in sharp contrast with the intracellular fluid of the cytoplasm [from Laura A. Freberg, Discovering Behavorial Neuroscience: An Introduction to Biological Psychology (Cengage Learning, 2016)]

means to manipulate the ITV while simultaneously monitoring ion channel activity, and current-clamp mode enables measurement of membrane voltage changes that result from ion flux via ion channels. While powerful, such techniques do not directly interrogate the localized charge distributions at the cell membrane nor do they capture the variations in the distribution of ions near the charged surface of the membrane [19]. Conversely, the localized ion flux mediated by channel activity may lead to fluctuations in the charge distribution at the membrane. Stretch-activated may be also activated by electroconformation [19]. Ion channel activity can mediate a spectrum of signaling pathways as distinct families of ion channels are gated, or activated, by different signals. In addition, ion channels can exhibit selectivity for specific species of ions, resulting in either membrane depolarization or hyperpolarization. As an aside, it is also noting that infrared light excites cells through an electrostatic mechanism. Infrared pulses are absorbed by water, producing a rapid local increase in temperature. This heating reversibly alters the electrical capacitance of the membrane, depolarizing the target cell. This mechanism is fully reversible and requires only the most basic properties of cell membranes [36]. As outlined above, the ITV is defined as the the difference in voltage between the interior and exterior of a cell. This results in long-range effects, acting globally on transmembrane proteins, such as ion channels. It has been also suggested that the ITV can be viewed as more locally. That is, each membrane–solution interface has its own surface potential, which is defined by the charged lipids in the membrane and the counterions in solution. This surface potential is often referred to as the zeta potential with a characteristic Debye length, which is the distance at which the

22

2 Elementary Concepts and Definitions

potential decays to 1/e of its maximum [19]. Because the inner and outer leaflets of the cell membrane carry different charged lipids, the zeta potential at the ECM and intracellular side also differ. An alternative definition of the transmembrane potential is the difference in these two surface potentials [19]. In this case, the asymmetrical distribution of charged lipids can affect the ITV in several ways. First, a negative zeta potential can attract positively charged ions to the membrane surface, forming an ionic double layer. As a result, the ionic gradient directly adjacent to the membrane may differ substantially from the gradient measured in the bulk solutions, such as in whole cell patch-clamp experiments. The second contribution of charged lipids to this alternative, locally defined ITV arises from interactions within the bilayer. MD simulations have shown that the ITV can be created solely from the difference in surface potential between the two leaflets, independently from the ionic concentration differences in the bulk solutions on either side of the membrane [19]. Several notes are in order. Firstly, it is worth mentioning that the MC and MR are measurable quantities in the entire biologically relevant frequency range, i.e. from dc to say 100 MHz by impedance spectroscopy techniques [38]. Secondly, Everitt and Haydon [40] proposed an interpretation of the MC of a lipid membrane (uniform thickness, infinite MR, uniform surface charge density) separating two aqueous phases by considering its surface charge density, electrolyte concentration dm + ε08π ,22 and applied potential. The capacitance per unit area was given as C˜1 = 4π ε0 εm εc η m where the first term represents the standard geometric capacitance and the second term corresponds to the planar electrical double layers contribution (η is proportional to the Bjerrum length, which is defined as the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, and to the electrolyte conductivity [7]). From this expression, we observe that the electrical double layers contribute only weakly to the MC since the right term represents only 1% of the total membrane capacitance for Bjerrum lengths within physiological bounds (a few nm) and electrolyte conductivities above 5 mSm−1 (which is around 3 orders of magnitude lower as compared to typical buffer conductivities used in EP experiments). Thirdly, it should be stressed that the relative permittivity of the membrane and the surrounding aqueous electrolyte are not spatially uniform. Namely, in hydrophobic region of the lipid tails the relative permittivity is very small, while in the region of lipid headgroups it is decreasing in direction from the outer solution towards the boundary between the hydrophilic and hydrophobic part of the lipid bilayer [37]. The local fluctuations of permittivity are not considered in the analytical expression of the membrane capacitance. Fourthly, the consideration of the membrane as a capacitor separating two conductors represents a simplification if one would like to describe the membrane electrostatics in details. The differential capacitance of electric double layer on both sides of the membrane may be strongly influenced by different conditions in electrolyte solution, e.g. finite size of ions, asymmetry in the size of the ionic species [37]. Sixthly, 22

Note that the spatial variation of relative permittivity (due to dipole and quadrupole orientational ordering) within the membrane and in close vicinity to both surface membranes is not considered in this formula.

2.2 Electrical Properties of Cell Membranes

23

biological membranes are capacitors that can be charged by applying a field across the membrane. The charges on the capacitor exert a force on the membrane that leds to electrostriction (EL), i.e. a decrease of d m . Since this force is quadratic in voltage, negative and positive voltage have an identical influence on the physics of symmetric membranes. This is not the case for a membrane with an asymmetry leading to a permanent electric polarization, e.g. due to the lipid composition which is different on the two monolayers of the membrane, or from membrane curvature [37]. One crucial take-home lesson is that Table 2.1 lists the input parameters which are commonly used in analytical calculations and simulations [10, 39, 41]. The reader is also referred to other sources which describe physical parameters used in the analysis of biological cells exposed to electric fields [1, 2, 4–6, 39, 42–45].

2.2.2 Single-Cell Modelling by a RC Equivalent Circuit Model Membrane charging is a necessary condition for subsequent membrane permeabilization. In order to predict the behavior of membrane charging and simulate the development of electric potentials across the membrane, physical or distributed circuit approaches can be used. In a physical model, Maxwell’s equations are solved to obtain the electric potential. In an equivalent circuit model, resistances, capacitances, and voltage sources are used to describe charging and discharging processes. The model is built either in the frequency- or time-domain, in which each component has a clear physical meaning. Frequency-domain analysis applies to linear systems. The time-domain analysis is useful to provide assessments of transient electric excitations that are relevant to the timescales of processes inherent in the system under investigation. Figure 2.3a tells us that in presence of an external electric potential, a cell behaves as an array of resistors, R, and capacitors, C. RC are essential stand alone or inherent components of virtually every electronic circuits. This figure illustrates the effect of adding a resistive path for current flow in the ECM, in parallel with with the capacitance of the cell membrane. The electrical analog of the current path for the conducting charges in the cytoplasm is a resistor. The current through the membrane im consists of two components: the component that flows onto the capacitor C m and the component that flows through the resistor Rc . The mathematical formulation of this problem consists in solving Kirchoff current continuity and voltage laws which consider continuous time variations of electric potentials and currents for a step turn-on of a dc homogeneous electric field E pulse in order to get the steady-state value of the ITV [2, 39]. Figure 2.3b, c illustrate another important feature: only the membrane capacitance is considered, and the membrane resistance is neglected [30]. When a voltage, either continuous (dc) or alternating (ac) is applied to the circuit there is some “time-delay” between the input and output terminals. Solving for V m yields an exponential form of Vm (t) = Re I (1 − exp(−t/[(Re + Rc )Cm ])), where I

0.04–1 3.5–4.4 × 10−11 10–10 7 × 10−10 0.2–2 0.1–1 ×

Sm−1 Fm−1 Sm−1 Fm−1 Sm−1

σc ε0 εm σm ε0 εe σe

Cytoplasm conductivity

Cell membrane permittivity (without pores)

Cell membrane conductivity (without pores)

ECM permittivity

ECM conductivity

102 –103

Nm−2

Gc

Cytoplasm shear modulus

[46] 102

s Nm−2

τ rel,c Ye

Cytoplasm viscous relaxation time

10–2 –1

ECM Young modulus

[46]

15–1.5 × 103

Nm−2 s

ηc

Cytoplasm viscosity

[46]

[5]

[46]

102– 103

Nm−2

ηm

[1, 46]

[1, 46]

Yc

10–3 –1

Nm−2

[39, 41]

[39, 41]

[10, 21, 26, 32]

[10, 21, 32]

(continued)

[10, 19, 21, 26, 32, 39]

[10, 21, 32, 38, 39]

Membrane viscosity

103 –106

Nm−2

Ym

Membrane Young modulus

10–6

10−5

[10, 19, 21, 26, 32] [10, 21, 26, 32, 39]

Cytoplasm Young modulus

10–7

s s

τm τMWS

Time constant of membrane charging

to

Maxwell–Wagner-Sillars polarization time s

3–7 × 10−10

Fm−1

ε0 εc

Cytoplasm permittivity

[10, 19, 26, 32, 39, 41] [39]

2.71 × 10–9

m

d me

Membrane dielectric thickness

3–15 × 10–9

m

dm

Membrane thickness

Sources [10, 26, 32, 39, 41]

Value or range 5–50 × 10–6

Units m

Symbol R

Parameters

Cell radius

Table 2.1 Summary of standard parameters used in the quantitative analysis of ITV and pore formation equations for cells exposed to electric fields

24 2 Elementary Concepts and Definitions

10−4 –3 × 10−3 10−11

4.6–9.2 × 0.5–1.35 4.6–7 × 10−10

Sm−1 Fm−1 Sm−1 Fm−1

σ nm ε0 εnm σn ε0 εn

NE conductivity

NE permittivity

Nucleus conductivity

Nucleus permittivity

10–4 -10−5

Nm−1

Γ0

Initial membrane tension (without pores)

1.8–2.4 × 10−11

Jm−1

γ

Membrane pore edge tension

1.5 × 109

m−2

N0

Equlibrium membrane pore density at Vm = 0 V

45 0.8 × 10–9

kT

rm

Equilibrium radius at V m = 0 V

m

U*

Membrane pore energy at transition

[22, 26, 41]

[39, 41, 44]

[26, 41]

[26, 41]

[26, 41]

[26, 39, 41, 44] 0.5–0.8 × 10–9

2.46 m

r*

( / )2 q = rm r ∗

Minimum radius of hydrophilic pores at V m = 0 V

Membrane pore creation constant

[26, 39, 41, 44]

1.4 × 109

J m−2 s−1

β α

Steric repulsion energy

[26, 47]

[26, 47]

[15, 19]

[15, 19]

[15, 19, 44]

Membrane pore creation rate coefficient

10−19

1–10 × 6–20 × 1013

m−2

σ nPC NPC

NPC conductivity

NPC density

10−3

Sm−1

[15, 19, 41, 44]

[15, 19, 41, 44]

[19, 41, 44]

[15, 41, 44] [15, 19, 41, 44]

d nm

NE thickness

7–10 × 10–9

m

Rn

Nucleus radius

[46]

[46]

4–11 × 10–6

s

Gn τ rel,m

Nucleus shear modulus

Membrane viscous relaxation time m

103 –104

Nm−2

Sources

[46]

15–30 × 106

Nm−2

Gm 1–102 × 10–8

Value or range

Units

Symbol

Parameters

Cell membrane shear modulus

Table 2.1 (continued)

(continued)

2.2 Electrical Properties of Cell Membranes 25

0.7 × 10–3 2–5 × 10−2

N Jm−2 Jm−2

F max Γ Γ”

Maximum electric force at Vm = 1 V

Surface tension of membrane (lipid heads/water)

Surface tension of membrane (lipid tails/water)

We also include the references where these numbers were taken from.

0.1–5 × 10−14

m2 s−1

D 10−3

Value or range

Units

Symbol

Parameters

Membrane pore radius diffusion coefficient

Table 2.1 (continued)

[22, 26, 41]

[22, 26, 41]

[26, 41]

[26, 41]

Sources

26 2 Elementary Concepts and Definitions

2.2 Electrical Properties of Cell Membranes

27

Fig. 2.3 a The electrical CS model (here 2D) depicts the cell as a dielectric membrane surrounding a homogeneous cytoplasm that is conducting. The PB membrane is equivalent/to a combination of a capacitor of capacitance C m and a resistor of resistance Rm (in (a), Gmem = 1 Rm ). The capacitor, C m , charges up through the resistance, Rc , when a voltage source is applied to the circuit powered by a voltage or a current source. Since Rm is very large compared to Rc and Re , a convenient simplification occurs and Rm can be ignored (adapted from [30]); b A cross-section schematic diagram illustrating a resistive-capacitive combination of the single CS structure of an idealized spherical biological cell of radius R; c The equivalent lumped circuit model of (b). The membrane, of uniform thickness, is considered as passive (the ideal capacitor presumption leads to a nominally infinite membrane resistance (no leakage currents), i.e. σm can be considered as negligible compared to σc and σe ) and the cytoplasm and extracellular medium are described by resistances (adapted from [36]). The cytoplasm is described by a resistance Rc and the external medium is identified by a resistance Re

is the constant current injected at one end of the cell from time t. The time delay is (Re + Rc )Cm . Thus, the voltage rises exponentially during injection of a constant current. When the current injection ends, the decay of voltage is also exponential, with same time constant. An important problem in using such CS model for describing the ITV is the calculation of the membrane charging time τm as a function of the RC characteristics. One way to formalize that might/be going on here is to take advantage of the quasistatic approximation since ωεc,e ε0 σc,e « 1, where ω and ε0 denote the angular frequency of the (harmonic) electric field and the permittivity of free space, the displacement currents in the cytoplasm and in the ECM are negligible compared to the conduction current (typically below 100 MHz corresponding to the β dispersion), i.e. the electromagnetic field cannot distinguish the details of the / cell which can be described by an effective medium [45]. Additionally, ωεm ε0 σm « 1 means that the the displacement current in the membrane is negligible below 1 kHz (α dispersion). Then, the (spherical) cellular dielectric properties can be described by an equivalent RC network, i.e. coarse-grained modeling of materials. In the following, we will mainly focus on dc stimulation (steady-state or pulse electric field). This approach yields Schwan’s equation [2], i.e. the membrane charges according to.23 ( / ) A more exact expression is Vm (t, θ ) = fER cos θ 1 − exp(−t τm ) , where f is a dimensionless term which[ relating (the electrical properties of each component and the cell geometry [11], f = ] )

23

σe 3dm R2 σc + 3dm2 R−dm3 (σm −σc ) 3 ) ( 2 R3 (σ +2σ ) σ + 1 σ −2(R−d )3 (σ −σ )(σ −σ ) . m e m c m m e m 2 c

function reduces to the constant 3/2.

Under physiological conditions, σm « σe , σc , the

28

2 Elementary Concepts and Definitions

Vm (t, θ ) =

3 ER cos θ (1 − exp(−t/τm )), 2

(2.1)

˜ (σc +2σe ) where τm ∼ denotes the time constant of membrane charging and θ is = RCm2σ c σe the polar angle measured from the center of the cell with respect to the direction of the field.24 This charging time constant is a measure of the time during which the cell interior is exposed to the applied pulsed electric field. This is also equivalent to the statement that the outer membrane becomes increasingly transparent for ac electric fields when the angular frequency of the oscillation exceeds a value given by the inverse of the charging time (see, later the Appendix 2 in Chap. 4). Considering the membrane physical characteristics (Table 2.1), τm is in the order of 0.1–1 μs at physiological conditions for a 10 μm cell, meaning that if τm is small compared with the duration of E, a convenient simplification occurs and steady-state Schwan’s equation is Vm (θ ) = 23 ER cos θ expressing a linear dependence of V m on the cell radius R, thus predicting that cells should porate at threshold electric fields that go as R−1 . Since σm can be considered as negligible compared to σc and σe , the linear dependence of the ITV on R is explained by the fact that most of the potential dropped across the cell is dropped across the membrane. Thus, the cytoplasm of the cell is shielded from the applied field, i.e. in the case of a membrane with zero conductivity, the field in the cell is also zero. As the cell gets larger and larger, the shielded region also increases and thus, in order for the line integral of the electric field between the two electrodes to remain constant, the ITV must increase linearly with R. It is also worth noting that Eq. (2.1) is valid for a nondeformed spherical cell and electric pulse strength well below the EP threshold and pulse duration larger e ∼ than the Maxwell–Wagner-Sillars (interfacial) polarization time τMWS = ε0 σεcc +2ε +2σe = ε0 εe , which represents the time constant for charge accumulation at heterogeneous σe interfaces of dielectric materials with different permittivities and conductivities [15]. The electric field acts on the free charges at the interfaces and gives rise to a force which is tangential to the interface. The characteristic timescale τMWS is on the order of 10−7 s. Considering typical values of the characteristic time τm , measurements on the dynamics of membrane charging require fast diagnostic tools like pulsed laser fluorescence microscopy exhibiting a temporal resolution of 5 ns. A low conductivity ECM has for effect to increase τm . Thus, in order to induce EP with μs pulses, field amplitudes of typically 106 Vm−1 must be applied. If the pulse duration is in the range of 100 μs–1 ms, a much lower pulse amplitude (i.e. 5 × 104 Vm−1 ) can be used [11]. Various authors have also commented that the field strength in the membrane is much higher than the strength of the applied electric field [2, 4, 39]. Other variations of parallel combinations of resistors and capacitors todeal with the presence of ionic channels and pumps in the cell membrane, as well as pores in the NE are considered in the literature [7]. A more useful means of visually representing the vector nature of an electric field is through the use of electric field lines. The reason is that this is a very effective

24

The exact expression reads τm =

[11].

R R˜ m

RC˜ m c σe + σ2σ c +2σe

, but this equation simplifies since

R R˜ m

« σc ≈ σe

2.2 Electrical Properties of Cell Membranes

29

way of presenting the distorsion of these lines in such a way it can be grasped at a glance. Based on the results shown in Fig. 2.4, we can have a good, intuitive basis for understanding the electric field line patterns in the cell and cell deformation. During the charging process, the charge densities on the inner and outer membrane surfaces can become unbalanced depending on the value of the electrical conductivity ratio Λ of the aqueous solutions inside and outside the cell. For biological cells and physiological condition, Λ varies over a range of values 0.1–1 [15, 39]. In this case, when an external dc field is applied, the cell will initially deform into an oblate spheroid (Fig. 2.4a) and eventually adopts a prolate shape (Fig. 2.4b) at long times. In the steady-state, the charges redistribute themselves to produce a field that essentially cancels the applied physics inside the cytoplasm. The resulting change in potential difference must then appear almost entirely across the membrane. The time scale for that process is on the order of τ m . If the rise time of the applied pulsed electric field is faster than the charging time of the membrane, the electric field will pass through the membrane into the cytoplasm and affect internal cell structures. Although sub-μs charging time is enough to temporarily surpass the cell membrane capacitance, the actual pulse duration in standard EP practice is within sub-ms to ms range to allow effective payload delivery while ensure successful recovery of the integrity of cell membrane and the survival of the treated cells. This is also consistent with the fact that a much stronger electric field magnitude has to be applied for ns EP, because in this situation the pulse duration is too small for the membrane charging to complete. Up to this point I have tacitly assumed that the cell has spherical symmetry and is excited by a dc electric field. The ITV for an isolated cell exposed to a dc field can

Fig. 2.4 Schematic diagram of the electric field and induced charge distribution around an initially spherical cell embedded in an electrolyte solution (Λ < 1) and submitted to the electric field E in the absence of any imposed flow. a During the charging phase of the membrane (t < τ m ), the cell is pulled into an oblate ellipsoid with the long axis aligned perpendicular to the direction of the applied electric field. The electric stress tends to compress the cell in the direction of the electric field. The electric field lines penetrate the cell interior and the deformation is assumed to be small, typically a few % of R [10, 39]; b Once the membrane is fully charged (steady state [t ≫ τm or low frequencies)], the accumulated charges on the internal and external sides are equilibrated. An oblate-to-prolate shape transition is predicted. In that case, the field in the cell is zero. The dashed lines indicate the cell deformation (adapted from [39])

30

2 Elementary Concepts and Definitions

be also evaluated analytically for other canonical shapes, e.g. for a prolate spheroid with the axis of rotational symmetry aligned with the direction of the applied electric field [11, 39] Vm (θ, t) = E0

b−

√

( 2 ) ( ) b − a2 a cos(θ ) ) ( √ (1 − exp(−t/τm )), 2 2 a2 a2 cos2 (θ ) + b2 sin2 (θ ) ln b+ ba −a 2 2

b −a

(2.2) where a and b represent respectively the semi-minor axis b to semi-major-axis a of the ellipsoid. If a uniform and spatially homogeneous oscillating electric field is applied to a spherically cell with angular frequency ω [2, 4, 13, 21], the magnitude of the oscillatory ITV is Vm =

1 3 ER cos θ √ . 2 1 + (ωτm )2

(2.3)

From Eq. (2.3), one recovers that the induced steady-state ITV in the low frequency limit, i.e. ωτm « 1, is given by Eq. (2.1). The ITV decreases rapidly (∼ ω−1 ) above the characteristic frequency τm−1 , i.e. typically, 1 MHz for mammalian cells. This implies a decrease of the ITV of several orders of magnitude for a given field amplitude as the frequency is increased from dc to several MHz. As displayed in Fig. 2.4, for applied fields with a frequency of less than about 1 MHz, the molecules in the cytoplasm are not directly affected by the external field. Note that the assumption we require to make use of Eq. (2.3) is that the electromagnetic wavelength of the radiation probing the system is much larger than all inhomogeneity length scales. We remind the reader that a basic consequence of this quasistatic condition is that the size of the system cannot be as large as one wish with respect to the frequency. Indeed, based on quasi-electrostatic and penetration depth criteria, this approximation is valid provided that ω is much smaller than R−1 1015 , with R measured in μm [21]. The parallel RC configuration shown in Fig. 2.3a leads to a Debye’s single relaxation time mechanism and is compatible with Eq. (2.3). It is also worthy to note that biological cells exhibit various frequency-dependent behaviors when they are subjected to an ac field excitation, e.g. dielectrophoresis (DEP), electrotation (ER), see Sect. 4.5 [21]. Previous theoretical and experimental studies [11, 15, 39] show that spherical cells in a uniform ac electric field can deform to prolate and oblate geometries depending on angular frequency ω and electrical conductivity ratio Λ. At low frequency (typically, below 1 kHz), the cell deforms to the prolate shape with the longer axis oriented along the field direction. At intermediate frequency (>1 kHz and 10 MHz), the spherical shape is again observed. For an isolated irregularly shaped cell and dense cell distributions the ITV estimation relies either on experimental means, e.g. using potentiometric fluorescence dyes [11, 26, 28], or numerically [6, 28, 39, 41]. In 3D, the best we can do for irregularly shaped cells is to solve Laplace equation by numerical methods, see, e.g.

2.3 Cellular Mechanical Properties

31

References [11, 39, 41]. This statement raises an important question from another perspective. Clearly if a NE needs to be considered within a double CS representation of the cell, then two time constants of membrane charging need to be calculated [7]. An electrical equivalent circuit different from that shown in Fig. 2.1 should be considered that takes the internal structure into account. The application of an electric field to the cell suspension results in conduction currents in the suspending medium and cytoplasm and a corresponding displacement current through both membranes, the outer one and the membrane surrounding the nuclear structure. Additionally, Schwan’s equation does not hold for short electric pulses, i.e. well smaller than τ m , for which the pulse duration and repetition are parameters of paramount importance for estimating V m [32, 39, 48]. It is also worthy to note that at frequencies below 104 Hz, the interior of the cell is screened from the applied field, while at higher frequencies the electric field is coupled capacitively through the membrane [21]. Most of the potential drop across the cell is effective across the membrane. As a result, the cytoplasm is shielded from the applied field. During electric field stimulation, the electric current(density through ) the cell membrane is obtained by summing the capacitive current C˜ m V˙ m (t) and / the conduction current (σm (Vm + Vrest ) dm ), where V m + V rest is the overall transmembrane voltage. It is worth noting that unlike Vrest which is constantly present in the cell membrane, the ITV varies with position over a cell membrane according a cosine law, i.e. Vm (θ ) ∝ cos θ : a spherical cell will polarize such that the maximum and minimum of the overall V m occur at the poles of the cell, and is equal to V rest at the equator. During the stage of membrane charging, negative and positive charges within the cell accumulate at the regions facing the cathode and anode respectively. Correspondingly, the pole closest to the cathode (θ = π) is depolarized, while the one closest to anode (θ = 0) is hyperpolarized. The ITV at the hyperpolarized pole is higher due to the negative V rest . Therefore, EP initially occurs at the pole of the cell closest to the anode, followed by the pole closest to the cathode. However, this analysis cannot reproduce the flattening of V m observed in electroporated membrane regions, i.e. at the poles [11, 39, 41]. From the above analysis, it is also important to retin that the ITV depends on the spatiotemporal profile of the membrane conductivity. For a large number of pores in an electroporated membrane, the membrane conductivity increases leading to a decrease of V m , thus preventing the creation of additional pores [48]. In closing, we observe that the ITV in spherical shelled cells with a nucleus in its center and interfacial distributions of free charges on the faces of the membrane has been also obtained numerically [20, 21].

2.3 Cellular Mechanical Properties So far emphasis has been on the electrical properties of cells. A cell embedded in a multicellular organism will experience a wide range of mechanical stimuli over the course of its life leading to shape change. Fluid flows and neighboring cells

32

2 Elementary Concepts and Definitions

actively exert stresses on the cell, while the ECM presents a set of passive mechanical properties that constrain its physical behavior.25 Cell mechanical properties are fundamental to the organism but remain poorly understood.26 More specifically, the measurement of the biomechanical properties at a single-cell-scale level remains a difficult and often challenging task. Indeed, a range of subcellular structures mediate different aspects of mechnostransduction, including the ion channels, membrane cytoskeleton, nucleus and cell-adhesion complexes. This originates also from the fact that, during the test, intrinsic changes of the biological structure can interfere with the actual property being measured. Furthermore, at the single-cell-scale, mechanical features can be drastically different from one site to the other. Any alteration of mechanical forces is likely to cause a disruption in cell normal functioning, thereby producing a disease state. From a mechanical standpoint, three generic properties make membranes ideal boundaries for cytoplasm and ECM: softness, impermeability and strength. As noted earlier softness and impermeability allows facile shape change while restricting the passage of macromolecules, polar or charges small molecules and ions, so that biochemical functionality can be efficient in specialized compartments (Fig. 2.1). PB also endows membranes with considerable strength, needed to maintain the substantial osmotic pressures that are related to the differential chemical potentials. Ever since the discovery of the CS structure of biological cells, physicists have puzzled over the modelling of their mechanical properties. Most of the knowledge dealing with the rheological properties of cells originates from earlier studies on soft matter and complex fluids.27 It is nonetheless of interest, both phenomenologically and theoretically, to see if any consistent simple stress–strain relationships of cells and subcellular components exist, and how these relationships are altered as a result of mechanical or electrical stimuli. How cells respond after the mechanical stress has been removed? Some argue that they return to their undeformed states like an elastic solid. Indeed, this is often a good assumption for single cells with conserved volume. Others argue that they behave like a liquid or a material that has plastically and irreversibly yielded under force. There has been a longstanding debate over consistency of the answers to these questions [47, 51–56]. Recent studies have indicated that the rheological behavior of the cytoskeleton follows a power law response to an applied strain or stress, called soft glassy rheology28 [51, 53–56]. For 25

The collagen network is, in vertebrates, the primary component of the ECM, a dense composite of molecules that surrounds cells and tissues and gives them structural support. Depending on the ECM stiffness, the cells may be prompted to change or maintain their behavior. 26 In a reductionist research strategy, understanding the mechanical properties associated with each component and the interconnections among components can contribute to the overall (effective) mechanical properties of the cell. 27 The last statement raises an important question from a biological perspective. Unless standard soft materials, living cells can develop an active response when submitted to stresses generated by a variety of excitations [49, 50]. This response is due to mechanotransduction, which represents the ability of cells to transform mechanical stresses into biochemical signals (and vice versa) in order to achieve a specific function. 28 The power law rheology exhibited by foams and collois is an intrinsic feature of materials composed of numerous discrete elements that experience weak interactions with inherently disordered and metastable microstructural geometry.

2.3 Cellular Mechanical Properties

33

such response the time and frequency data did not conform to conventional spring and dashpot viscoelastic models (see Sect. 2.3.2).29 Electron microscopy images make it clear that cells are able to deal with large mechanical stresses in the ECM even though they are enveloped by a fragile and soft membrane of molecular thickness. There is now growing evidence showing that eukaryotic cells can sense mechanical cues in the surrounding microenvironment to regulate their functions, i.e. proliferation, differentiation, apoptosis, and homeostasis, etc. [46, 57–66]. While the biological effects of forces are eventually most obvious in the context of cell activity, such as breathing, heart pumping, blood flow, compression and tension due to muscle contraction, mechanical forces also regulate cell migration and even cell adhesion to the ECM [57]. As is displayed in Fig. 2.1 living cells are complex materials exhibiting a high degree of structural hierarchy and heterogeneity coupled with biochemical processes that constantly remodel their internal structure. Two important mechanical properties relevant to the cellular scale are its microrheological and adhesion (with respect to the ECM or to other cells) behaviors. Invstigating these behaviors requires a broad range of forces, beginning from single molecule interaction forces in the range of pN to cell stretching forces (100 nN) wher the contact area is large. To understand the cellular response to mechanical stress, a great variety of experiments have been conducted to apply a mechanical stimulus to a single cell, and study its response, e.g. micropipette aspiration, atomic force microscopy (AFM) indentation, particle tracking laser microrheology, magnetic twisting cytometry, and manipulation by optical tweezers30 [55, 67, 68]. Therefore, a rich phenomenology of rheological behaviors and models has been found in cells depending on amplitude, frequency and spatial location of loading [69]. More specifically, a variety of rheological models have been proposed to deal with the interpretation of experimental data [1, 13, 50, 68, 70–72]. Furthermore, from the clinical point of view, many observations reveal strong differences between the mechanical properties of biological samples of physiological and pathological cells, e.g. through genetic mutations, malignant cells have different cellular characteristics than begnin cells such as increased nuclear to cytoplasmic ratios, larger cell volumes and wider EC, cancer tissues can be up to tenfold stiffer than healthy tissues [70]. Recent suggestion was also made that in primary tumors high mechanical tension and matrix stiffening are important in cancer progression, and high fluid/solid pressure in the primary tumor often accompanies tumor growth [70]. Under physiologically relevant timescales cells are intrinsically viscoelastic under mechanical loading, as they display a combination of both elastic and time-dependent responses to deformation. The importance of choosing an appropriate quantitative model to account for deformation imposed during a given stress has long been pursued by cell rheology model builders [47, 52]. Of course, one of the difficulties encountered 29

Elastic materials are characterized by an instantaneous and reversible deformation, like a spring that extends under an applied force and snaps back to its original length when the force is removed. By contrast, viscous materials are liquid-like, exhibit flow and undergo irreversible deformation when subjected to force. 30 Optical tweezers allow for manipulating micron-sized objects using pN level optical forces.

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Fig. 2.5 A sketch of different phenomenological models used in the literature to describe cell mechanics. a The first scenario considers a possible mechanical equivalent circuit that combines springs and dashpots in various arrangements, e.g. Kelvin-Voigt that considers the biological complexity of the cell as a single phase homogeneous continuum material; b The second scenario imagines a tensegrity device, based of cables (actin stress fibers) and struts (microtubules), that subjects the cell to some prestress network; and c The third scenario assumes crowded particles that are trapped in a complex energy landscape with wells of different depth such that spontaneous transitions from one to another well have a very low probability, i.e. soft glassy rheology that describes cells as being akin to soft glassy materials close to a glass transition. In such a system, thermal energy is not sufficient to drive structural rearrangement and, thus out of equilibrium trapping occurs (from [72])

to describe the cell’s reaction to a mechanical perturbation is to simplify the complex features of the cell and to apply suitable coarse-graining. Additionally, geometries are simplified and anisotropies due to the specific position of the nucleus and other organelles are neglected. For instance, phenomenological models with little or even no microscopic details such as a simple viscoelastic continuum model (Fig. 2.5a), a tensegrity model assuming the cell as pre-stressed cable network (Fig. 2.5b), and a soft glassy material model have been studied (Fig. 2.5c). After removal of the mechanical load, the cell shape recovers only incompletely to its original undeformed state due to additive plastic energy dissipation during cell deformation which is intimately related to elastic cytoskeletal stresses [51]. During changes in cell shape the maximal rate at which shape change can occur is dictated by the rate at which the cytoplasm can be deformed because it forms the largest part of the cell [71]. Membrane tension, stress, and strain distributions have been characterized in the vicinity of a microelectrode using FE analysis of a multiscale electro-mechanical model of pipette, media, membrane, actin cortex, and cytoplasm [72]. Ultimately, one important goal is to see how well the mechanical properties of the membrane of eukaryotic cells fit with a flexible PB as we understand it. Do the mechanical properties of cell membranes resemble the properties of neat PB, perhaps with parameters adjusted to account for solutes? Or are there ways in which membrane components, e.g. transmembrane proteins, fundamentally change the mechanics? This subject has motivated a lively collection of models [1, 13]. Detailed efforts have invested in characterizing the membrane mechanics in cellular systems with a range of complex rheological behaviors, i.e. elastic, viscoelastic, poroelastic, nonlinear, etc. While models that treat the cell and its nucleus as a

2.3 Cellular Mechanical Properties

35

homogneous isotropic material have proven successful in many instances, they fail to predict major physiological observations such as the transmission of the forces applied via focal adhesions to long distances in the cytoskeleton [63]. Assuming a cell is a homogeneous and isotropic material, the cell can be characterized by just two parameters, e.g. elastic (Young’s) modulus and the Poisson ratio [34]. Typical values for the cell elastic modulus range from 102 to 105 Pa (5–6 orders of magnitude smaller than that of metals ceramics and polymers), and the cellular viscosity is in the order of a few hundred Pas [55].31 Cell membranes can be modeled with a Poisson’s ratio of 0.49 that approaches incompressibility condition. With the data accessible to us, we have listed in Table 2.1 selected typical physical parameters which can be used for modelling the mechanical properties of an elastic cell, i.e. the cell tends to return to its original geometry after application of the force. However, the complexity of the structural organization of a “real cell” (Fig. 2.1) cannot be simply be described by a linear elastic model, as cells can typically withstand large strains and exhibit strain hardening [67]. Finally, it is worth observing that one crucial take-home lesson from many studies devoted to the analysis of the force balance in cell membranes is that the coupling of the membrane to the cytoplasm is an important ingredient in the recent numerical simulations [72] and observations [73–75]. The cytoplasm is not just a spectator-it influences the physics [74]. In response to the mechanical cues, cells often adjust their cytoskeletal32 tension such that many of the mechanical information are translated into a level of inherent cellular traction forces, and in turn into intracellular signals 31

Cells are rarely subject to inertia. The Reynold’s / number which describes the ratio of inertial to viscous forces in a system is given by Re = ρvl μ, where ρ is density, v is velocity, μ is viscosity, and l is a relevant scale. Since human cells has a typical size of 10 μm and moves at speeds no greater than 10 μms−1 , cells experience a Re of 10–4 , meaning that viscous forces on a cell are 104 times as great as those of inertia [1, 64]. 32 Network of subcellular filaments form meshes and bundles that endow individual cells with their ability to sustain external mechanical forces. Three cytoskeletal filaments are of special interest to cell mechanical properties: actin microfilaments, microtubules, and intermediate filaments. These filamentous biopolymers are much less flexible than synthetic polymers, yet they can still exhibit significant conformational changes driven by thermal Brownian fluctuations. In polymer physics, a measure of filament flexibility is given by the persistence length, L p , which defines a length scale over which a filament remains unbent in a thermal bath. Actin is one of the abundant proteins in eukaryotes that form polarized filaments that interact with an array of ancilliary proteins. From a mechanical standpoint, actin filaments are semiflexible on the length-scale of the cell, are highly dynamic and rapidly re-organize enabling cells to migrate and change shape [57]. Under compression actin filaments can buckle, depolymerize and slide. Filamentous actin is a polymer composed of actin monomers. Its L p is on the order of tens of μm, diameter 5–10 nm, Young’s modulus of elasticity is very high, E ~ 103 MPa, (e.g., comparable to that of nylon and collagen fibrils). In response to stretching of isolated actin filaments, their tension-strain curve exhibits initial strain-hardening at low tensions (0–50 pN) and low strains (0.4–0.6%), which is indicative of filament’s conformational changes. Microtubules are polymers of α- and β-tubulin dimers organized as hollow tubes (25 μm outer diameter and 12 μm inner diameter). They have nearly the same elastic modulus as actin filaments (E ~ 103 MPa), but a much greater L p ~ 103 μm. Intermediate filaments are family of proteins (e.g., vimentin, desmin, keratin, lamin) ~ 10 nm diameter, which are much more flexible (L p ~ a few μm) and much more extensible (E ~ 100 to 101 MPa) than either actin filaments or microtubules.

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2 Elementary Concepts and Definitions

regulating the related functions [50]. Most significantly, the cellular cytoskeleton provides a resistive force which locally opposes the applied tension and prevents the propagation of force through the membrane [72].

2.3.1 The Mechanical Membrane In this section we summarize results concerning the membrane mechanical properties [50, 76, 77]. The membrane is continuously deformed in- and outward in response to a wide range of intracellular and intercellular forces [57]. Mechanostransduction involves stress transfer from the CM through the cytoskeletal network to the nucleus. Since the membrane is the origin of the ECM stimuli, it is important for understanding the mechanics of a cell. There have been several attempts to address this question, particularly in the context of mechanobiology [57]. On the observational side, since the PB is very thin compared to their lateral dimensions, the membrane can be modelled as surface. Consequently, the idea that has received the most attention is the 2D fluid description of the cell membrane for understanding the membrane mechanics [76, 77]. In an aqueous medium, lipids aggregate into 2D bilayer sheets and adopt a configuration that minimizes the exposure of their hydrophobic parts. The in-plane flow of lipids suggest that the membrane resembles a nearly incompressible viscous fluid, while either by out-of-plane bending and in-plane stretching it behaves somewhat like an elastic solid [1, 68]. Membrane structures are often associated with a variety of modes of electromechanical coupling. Hence, mechanosensitive channels can be activated, de-activated, or switched from one of the multiple states to another as a result of mechanical factors, such as stress, or strain in the surrounding membranes. In cellular membranes, the channels can be also governed by the viscoelastic-like interaction between the membrane and underlying cytoskeleton [69]. Other recent studies revealed force-carrying connections from the cell membrane to the nucleus, perhaps playing a significant role in the regulation of specific gene expression [78]. Due to the fluid nature of the PB, membranes are characterized by a low shear modulus, typically in the range 4–10 mNm−1 . Due to the small stretch in PB, membranes have also a high elastic modulus 103 Nm−2 , and a variable viscosity, e.g. 0.4–2.1 10–3 Pas for an erythrocyte, which depends on membrane composition, i.e. viscosity influences the translational and rotational diffusion of proteins and lipids in membranes [1, 46]. More specifically, a model of protein diffusion within membranes was suggested, which describes the PB as a thin layer of viscous fluid, surrounded by a less viscous fluid, where the diffusion coefficients of membrane proteins scale logarithmically with the radiu they occupy in the membrane and decrease as afunction of membrane viscosity [1, 46]. Processes where physical forces applied to membranes create deformations that are translated into classical chemical signal-transduction pathways, are called curvature-dependent mechano-chemical signal translation. A first approach to study the deformation of membranes is phenomenological: these properties at length scales several times molecular lengths can be described by an effective free energy which depends only on the symmetry and conservation laws of

2.3 Cellular Mechanical Properties

37

their equilibrium phases. On length scales larger than the membrane thickness, the bilayer can be modeled as a smooth continuous surface, and its elastic free energy is related to the membrane area and the local curvatures. In the context of a equilibrium continuum model of elasticity theory, changing the shape of materials costs energy [79]. This model relies on the same principle that is the basis for the definition of equilibrium: equilibrium is attained when the free energy of the system is minimum. Lipid membranes can be described by its free energy using just the geometry of the membrane through curvatures. The elastic deformation response of the membranes to mechanical force is primarily due to their resistance to bending and stretching. Ignoring the fluctuations of the lipid molecules in the membrane, the elastic energy cost associated with membrane stretching and bending during deformation can be described by Canham-Evans-Helfrich (CEH) free energy [68, 80–82] that is a function of the local shape of the membrane 1 Kbend 2

∫ membrane

( ) 1 1 2 1 dA + + Kstretch c1 c2 2

∫ dA

(ΔA)2 . A0

(2.4)

membrane

The integration in Eq. (2.4) is carried out over the whole surface of the membrane. The first term describes the bending energy, where Kbend is the membrane bending rigidity, c1 and c2 are the membrane principal radii of curvature which are surface invariants with respect to similarity transformations (translations and rotations), and dA is a surface element (Fig. 2.6). It is worth noting that a Gaussian term is irrelevant here since there is no topological membrane remodeling and the spontaneous curvature is assumed to be zero [81]. Another implication of membrane curvature is that the inner and outer membrane leaflets, which have an equal tension in a planar configuration, exhibit differential tensions on bending. The second term of Eq. (2.4) represents the lowest order of the in-plane tension energy where Kstretch is the membrane area stretch modulus, and A = A0 + ΔA and A0 are, respectively, the perturbed and unperturbed surface areas of the membrane. Characteristic values of Kbend for PB are in the range 5–25 kT (10–19 J) which is strongly influenced by membrane proteins and cytoskeletal elements [1]. Thus, the elastic energy of the membrane can be written solely in terms of its geometric properties. This is a noteworthy aspect of membranes because it indicates that for length scales only a few times bigger than the thickness of the membrane, the behavior of the membrane can be described solely in terms of its geometric properties. References [82, 83] furthermore addressed some of the issues with the measurement of Kstretch which is found in the range 50–70 kT (0.25–0.35 Nm−1 ). Note that the asymmetry in the membrane can lead to spontaneous membrane curvature [82].33 Using Eq. (2.4), one can estimate that the corresponding rupture strain of a typical lipid membrane leads to a 2–5% areal stretch increase, and is estimated to be in the range 8–10 mNm−1 [83]. Thus, in that respect membranes are not 33

The ability to break symmetry in response to external stimuli is an essential property of most eukaryotic cells [83].

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2 Elementary Concepts and Definitions

Fig. 2.6 (top) Illustrating the response of the membrane to lateral stretching: the free energy penalty (area change) due to membrane stretching changes the membrane area and the average distance between the springs (representing the elastic constants) by stretching the membrane from its equilibrium area; (middle) Describing the membrane as a continuous elastic sheet. The idea behind this description is that there is a cost in terms of free energy that must be paid for perturbing the PB away from some underformed reference state. A relaxed membrane lies flat when undisturbed; (bottom) Illustrating the response of the membrane to out-of-plane deformation due to the bending energy penalty (adapted from [1]): a equilibrium state, and b when a bending force is applied to the whole membrane, the structure of the PB is slightly deformed leading to a mechanical restoring force (by the springs). Gradients in curvature can drive molecular movement within the membrane plane

particularly stretchable and are often treated as incompressible. The elegance of the CEH analysis lies in its ability to predict the shape of the membrane under different conditions, e.g. the biconcave shape of the red blood cell [80]. When subjected to either ac or dc electric fields GUVs deform and stretch (Fig. 2.4). Depending on the intensity, pulse duration or frequency of the field, as well as the ratio between the internal and external conductivities of the aqueous solutions, the shape of the GUV membrane can significantly vary [82]. The small deformation regime is governed by the bending rigidity of the membrane, i.e. first term in Eq. (2.4). For stronger deformations, the electric field leads to elastic stretching of the membrane and this regime is now driven by the second term of Eq. (2.4). Moreover, due their small thickness, the resistance to shear in fluid membranes is negligible. External applied voltages induce changes in the mechanical properties of membranes, e.g. flexoelectricity and piezoelectricity [38], which are in turn responsible for protein conformational modulation of ion channels [9]. Surface charge adds an electrostatic term on the membrane free energy of Eq. (2.4). Then, it is expected

2.3 Cellular Mechanical Properties

39

that electrostatics impacts the viscoelastic response of membranes [38]. Since the pH of the aqueous ECM regulates surface charge density by modifying the charge state of ionizable groups in the membrane, it is also expected that the membrane deformability depends on pH [38]. In particular, mechanical, interfacial electrical, and internal electrical measurements combine to support the view that changes in Kbend result from alterations in interfacial electrostatics [38]. The 2D fluid description of the cell membrane is an interesting scenario, but has several questionable aspects, i.e. cell rheology is timescale free, cells are predominantly viscoelastic over timescales up to minutes, and under mechanical force, cells often stiffen and fluidize [47, 51, 52]. While many cellular features such as the cytoskeleton and membrane protein composition are ignored in the 2D fluid model, it is amazing that the shape of a red blood cell can be described with an equation for mechanical equilibrium by minimizing the associated elastic energy. This was the advent of mechanobiology at the cellular level in the use of energy to describe mechanical forces and membrane shapes. Some work has been done on addressing this concern [77, 84–87]. In contrast, alternative models of the 2D fluid picture have been proposed [88]. One such model considers that a membrane has rigidity and resists deformations perpendicular to its plane. Furthermore, cell membranes can exhibit heterogeneities, e.g. rafts which are thought to be dynamic, ordered lateral domains comprised mainly of phosphatidylcholine, cholesterol, and sphingomyelin molecules, with dimensions ranging from tens to hundreds of nm. There is however the possibility of another scenario, namely to consider that the membrane and cytoskeleton should be considered as a composite material-not as two separate structures [67, 72]. The product of Kstretch with the areal stretch yields the tension in the membrane. The quantitative measurement of the location and magnitude of the forces applied by cellular machinery promises to yield important insights into the mechanisms of membrane-remodelling processes. In that respect, spatial variations in the composition of lipids and/or protein concentration in the membrane can cause spatial variations in the membrane tension [83]. Membrane mechanics has been postulated to plays an important role in the aggregation of transmembrane proteins There has also been an ever increasing need to study the pore topology and stability of NE [89]. While the NE possesses unique mechanical properties, which enable it to resist cytoskeletal forces, we have very few insights into their electromechanical properties. Over the last decade significant effort has been dedicated to addressing this question, particularly in the context of the nano-architecture of the NE [89]. How the NE ruptures is not fully understood, but it is clear that the rupture can be caused by electromechanical stresses.

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2.3.2 Single-Cell Deformation Modelling by a Network of Spring and Dashpot Many approaches have been suggested to model a cell’s mechanical properties, especially its elasticity. The mechanical response of a cell as observed in experiments is largely consistent with the assumption of a linear viscoelastic behavior [50, 64, 79, 80]. When the system is elastic, stresses are proportional to deformations, whereas a viscous material will flow under an applied force, with stresses proportional to the rate of deformation. Continuum mechanics provides a well-established foundation for describing the macroscopic behaviors of both fluids and solids. The simplified fluid description has dissipation with no elasticity (dashpot); the simplified solid has elasticity but no dissipation (spring). The former implies an infinitely fast relaxation of the material’s microscopic components, whereas the latter implies no relaxation at all. Current literature is quite extensive regarding rheological models to understand the cellular response to mechanical stress. Among them, those using networks of spring and dashpots have the ability to describe a viscoelastic behavior, KelvinVoigt (viscous and elastic elements are connected in parallel, Fig. 2.5a) and Maxwell (viscous and elastic elements are connected in series) models [90, 91].34 However, the elements in these lumped parameter viscoelastic models cannot be unambiguously associated with cellular components, in sharp contrast with Schwan’s model (Fig. 2.3). Instead, the action of all elastic components of the cell can be represented by a linear spring (or springs) and the action of all viscous components by a daspot (or dashpots). The Kelvin-Voigt model captures the property that cells flow and deform at the scale of minutes and hours but have a solidlike behavior ion the scale of seconds or shorter (compared with characteristic time constant). The creep ( ( its / )) compliance response is J (t) ∝ 1 − exp −Yt η , where Y is the elasticity modulus (which determines the ability to sustain material’s shape under mechanical stress) and η is the viscosity (which characterizes the rate at which a fluid flows under a specific load). Under load, the material relaxes to the new equilibrium conformation of the spring with a time constant given by the ratio of damping to elasticity. In the Maxwell model, the displacement increases with a larger (than linear) dependence on the applied force. To illustrate that idea, it is worthwhile to first investigate the construction of the above-mentioned models from a strictly macroscopic point of view to see where at least some of the challenges might arise. Consider a Maxwell element (a spring of stiffness G in series with a dashpot of coefficient η). Such a model /develops a stress τ related to the deformation γ by λτ˙ + τ = ηγ˙ , where λ = η G denotes the mechanical relaxation time, γ is the strain and τ˙ is the time derivative of the stress. The advantage of this spring-dashpot construction is that it is simple enough to define two kinds of behavior depending on the value of λ: if t « λ,, the material behaves as an elastic solid of stiffness G, with a stress τ = Gγ , and if t ≫ λ,, the material behaves as a fluid of viscosity η and τ = ηγ˙ [50, 55]. 34

Spring-pot-based models have been also introduced to describe the viscoelastic properties for which the constitutive law is defined through fractional derivatives, with applications to cell materials [92].

2.3 Cellular Mechanical Properties

41

When the deformation is small enough and if the imposed strain γ (t) ∝ sin(ωt) varies with same angular frequency, ω, as the stress τ ∝ sin(ωt + φ), the elastic and viscous moduli are described by the frequency dependent response G ' (ω), of the elastic modulus, and G '' (ω), of the frictional loss modulus, where ω is the angular frequency. Figure 2.7 schematically indicates a typical curve for a polymeric biomaterial, such as an actin solution, or a network of actin polymers with crosslinks [49]. At low frequencies, if the material flows and when no interactions between structural components exist that prevent flow, the typical slopes for G ' and G '' are respectively 2 and 1. If interactions are present and strong enough (e.g. crosslinks), then no flow is possible at low rates of deformation leading to constant G ' and G '' as graphically illustrated in Fig. 2.7. At intermediate frequencies, the material is viscoelastic. Both G ' and G '' are of the same order and G ' exhibits a plateau. At larger frequencies, both moduli increase with a slow slope (exponent between 0.1 and 0.6). At very high frequencies, the aterial reaches a glassy behavior charcaterized by a constant G ' and a decreasing G '' . It is also possible to predict a similar behavior using a continuous distribution of relaxation times [49]. There is an important interesting general lesson here. By making use of such spring and dashpot models, we find that it is indeed possible to make quantitative predictions on the constitutive behavior of single cells, e.g. multiaxial stress–strain relations, changes of mechanical properties with time and/or in response to electrical stimulus. While these mechanical equivalent circuits that combine springs and dashpots in various arrangements are of phenomenological nature and contain no microscopic details, the way they are designed is arbitrary. Additionally, cells are multi-component structures and likely to exhibit more than one relaxation time.

Fig. 2.7 A typical viscoelastic spectrum for soft biological material (log–log scale) [49]. The solid (resp. dashed) lines correspond to an uncrosslinked (resp. crosslinked) material. In [49], a generic model of soft matter was introduced, with interactions represented by a mean-field noise temperature x. For 1 < x < 2, both storage and loss modulus vary with frequency as ωx−1

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Moreover, current measurements can take seconds to hours to perform, during which biological processes can modify the cell elasticity. By necessity, these models are oversimplified because proteins and cells present highly anisotropic, heterogeneous, nonlinear mechanical properties that vary widely and depend on the composition, architecture, and environmental conditions, as well as the direction, nature and rate of load application. Thus, things aren’t always that straightforward. At this point it seems appropriate to discuss the following remark. Many problems in physics can be reduced to analysis of systems possessing two distinct time scales, namely fast and slow degrees of freedom. Then, one may estimate the effect of the slow variables on the fast ones in the adiabatic approximation [34]. To illustrate this point in the cell context, we can treat deformation of the cell as slow variable characterized by a mechanical relaxation time τ rel,c which is the range 10–2 –1 s (Table 2.1) compared to the typical value of the time constant (fast variable) of membrane charging τ m and MWS characteristic time τMWS . Rapid oscillations in time accompany such polarization and capacitive charging, and lead to cancellations in processes whose characteristic time scale is much longer-that is, in processes associated with motion of the “slow” variables. For example, if t ≫ τm no internal electric field is actually present in the interior of the cell, as displayed in Fig. 2.4. These considerations suggest that the characterization of electromechanical couplings requires large changes in frequency, and are therefore associated with fast variables. Thus, one cannot put electrical and mechanical relaxations on similar footing. But, as we have seen, the development of deformation in response to cell stress can involve nontrivial structures, e.g. the cytoskeleton is a dynamic structure that is constantly changing its organization and composition. Soft-condensed matter physics has provided, in the past decades, many of the relevant concepts and methods allowing successful description of biological materials. In particular, the transition between solid-like and fluid-like behavior in cells has been well characterized [47, 51, 52]. More specifically, alternative models, such as the soft glassy rheology models that describe cells as being akin to soft materials composed of numerous discrete elements that experience weak interactions with intrinsic disordered and metastable microstructural geometry have been worked out recently [53–55, 92–98]. No characteristic time scales appear that would correspond to a viscoelastic model of one or several springs and dashpots. As shown in Fig. 2.5c, this model considers energy traps (metastable states) [49] related to structural disorder, implying an energy barrier necessary to leave a specific metastable state. These nonthermal interactions are represented by a mean-field noise level x. For 1 < x < 2, both storage and loss modulus vary with frequency as ωx−1 , and hence remain at a constant ratio [47, 51, 52]. Within this picture, the material is elastic when x = 1 because the individual elements remain trapped in their metastable states. When x = 2, the elements are not restricted by energy barriers anymore resulting in purely viscous dynamics. However, one of the critical questions, which is being subjected to systematic experimental scrutiny, is how the variability of the cell mechanical properties can be dealt with since they display a largely heterogeneous, and anisotropic multi-component cytoskeleton architecture [54–56].

2.4 Extension to Multicellular Structures and Biological Tissues

43

Applying or measuring displacements (and inferring forces from these measurements) across the length scales of proteins and cells requires a range of techniques and several biochemical sensors and microfabricated devices have been developed towards this goal. Optical and magnetic tweezers, Förster Resonance Energy Transfer (FRET) molecular tensions sensors, and AFM are widely used to study conformational changes of individual mechanosensitive proteins under mechanical load, while optical stretchers, micropipette aspiration, AFM and microelectromechanical systems (MEMS) enable single cell mechanobiological studies [93]. The determination of the static shear viscosity (10–100 Pas) and elastic modulus (5–20 Pa) confirms the viscoelastic character of the cytoplasm of living cells [50, 99]. A central challenge in dealing with continuum-based computational models of cell mechanics is choosing material laws which are able to represent the strain-stree relationships of cells and subcellular components and their change as a result of mechanical and electrical stimuli. The Neo-Hookean strain energy, Mooney-Rivlin and Ogden hyperelastic constitutive equations have been also used to describe the nonlinear deformation of cells and cell membranes [100]. There have been also several attempts to address mechanical fatigue from cycling straining and the effect of fluctuations in stress or deformation at the single-cell level, see e.g. [101]. This allowed measuring the systematic changes in morphological and biomechanical characteristics of healthy human red blood cells during its 10-day life span, and also their membrane mechanical properties. The erythrocyte’s remarkable mechanical properties originate from the unique architecture of its cell wall, which is the main load bearing component as there are no stress fibers inside the cell. Thus, it is very flexible, and yet resilient enough to recover the biconcave shape whenever the cell is quiescent. Such study can provide insights into the accumulated damage as they squeeze through microvasculature, and eventual catastrophic failure of these cells causing hemolysis in various pathologies.

2.4 Extension to Multicellular Structures and Biological Tissues So far, most treatments of ED and EP of cell membranes have focused on the electrical and mechanical properties of isolated cells based on the paradigmatic CS cell model subject to strong local (electrical and/or mechanical) constraints. Physicists are also trying to understand the behavior of multicellular systems, where collective effects, correlation and mutual interaction dominate, and the dynamics of cellular self-assembly in an effort to improve their ability to accelerate tissue knowledge and fabrication. Although cells are the building blocks of tissues and are investigated largely in terms of their genetic and biological attributes, it is physical laws that they must ultimately follow, irrespective of the underlying biological processes. Therefore, understanding and applying the physical principles of the dynamics of multicellular systems may help control tissue engineering. One important aspect of tissue-like

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environments is that cells are confined. Furthermore, tissues are composed of similar types of cells that work in a coordinated fashion to perform a common task.35 The diversity of function arises from diversity of form, in part via the complexity of cellspecific gene exression, which defines the distinct 3D molecular anatomy and cell properties of each tissue. In vivo, cells are tightly connected to each other via cellcell junctions. The obvious question is how the presence of cell-cell pathways for transmitting force can impact the corresponding the non-uniform displacement field. During tissue morphogenesis,36 individual cells self-assemble into complex tissues and organs with specialized shapes and functions. Understanding the complex tissue organization and properties requires knowledge of the mechanism responsible for coordinating cell behaviors between the different cells [102]. Little is known quantitatively about how cells coordinate forces across a tissue during morphogenetic rearrangements. Additionally, tissues are dynamic multicellular structures that are extremely remodeled, particularly during embryonic development [102]. Tissues show remarkable nonlinear stress–strain responses, anisotropy and poroelasticity. Not only do cell mechanical properties exhibit rate-dependent behaviors such as non-linear viscoelasticity or thermodynamic instabilities, but the ECM itself also changes with time. Cells secrete and remodel the ECM that comprises their tissue microenvironment in a load history-dependent manner. Understanding the material responses is crucial towards understanding the physiological and pathological functionality of tissues in their in vivo context. More specifically, predicting how biological tissues adapt to changes in their environments is extremely valuable for understanding physiological adaptation and disease progression. To better understand the underlying collective effects that govern tissue electromechanics, physical models are needed that can faithfully predict the effective properties from the properties of their constituents and microstructures. Our discussion can be extended to continuum (macroscopic) approaches of biological tissues consisting of heterogeneous cell aggregates, each of them being described by specific properties such as distributions of cell-cell distance and cell size [103].

35

Four basic types of tissues are found in animals. Epithelium is a type of tissue whose main function is to cover and protect body surfaces but can also form ducts and glands or be specialized for secretion, excretion, absorption and lubrication. Connective tissues perform such diverse functions as binding, support, protection, insulation and transport. Despite their diversity, all connective tissues are comprised of living cells embedded in a non-living cellular matrix consisting of extracellular fibers or some type of ground substance. Muscle tissue is specialized for contraction. Nervous tissue is specialized for the reception of stimuli and conduction of nerve impulses. 36 Process in which a collection of interconnected cells, i.e. a tissue, reshapes itself. To do that, the tissue must transform from a strong, rigid system to a flowing, fluidlike system in order to take on a new shape, at which point it becomes rigid again. In the morphogenesis of an embryo, the flowing tissue does not spill and spread randomly but transforms into a particular new geometry, necessary for healthy development.

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2.4.1 Electrical Properties of Tissues and Cell Suspensions It is now well established that some human diseases are related to changes in electrical properties of tissues. For example, malignant cells have a different impedance signature than begnin cells as function of frequency [104]. Simulations suggest also that the unperturbed solid tumour growth is related to changes in the surface charge density over time between the tumour and the surrounding healthy tissue [104]. Overall, there is a close relationship between biological and electrical parameters in tissues. Cancer cells have higher intracellular Na+ , lower intracellular K+ and Ca2+ concentrations, and more negative charges on their cell surfaces [11, 41]. The role of charge as well as spatial changes in the field distributions in cellular regulation in tissues was pionneered by many authors [105]. Contrasting with dilute cell suspensions which possess relaxation processes that can be modelled by Debye equation (see, e.g. [41]), dense tissues and multicellular assemblies have different behaviors due to the irregular shape of cells and their close proximity interactions. Theoretical models and simulations have suggested a number of phenomenological models involving Cole–Cole, Cole-Davidson, Havriliak-Negami, and the use of a continuous distribution of relaxation times [41]. In these models, the tissue is represented in a homogeneous manner: such homogenized models are coarse-grained, and hence relevant for simulating phenomena on the tissue scale (mm). To be realistic, these models need the knowledge of many parameters if physiological function is to be understood and predicted. What is not so well known is how these parameters are determined. However, given the large number of parameters than potentally can affect the electrical behavior of tissues, the theory cannot be well extended from the computation of the electrical properties of a single cell to understand the role of ion migration and diffusion of large cell assemblies, and there is some level of uncertainty that impedes the progress of knowledge of this subject.

2.4.2 Equivalent Electrical Circuit of Biological Tissues Methods based on modeling biological tissues by lumped circuit elements have been also proposed in order to analyze the electrical characteristics of biological materials [6, 48, 95–98, 106, 107]. For example, in [97] the authors describe a transport lattice method to model a multicellular system with irregular shapes and nonlinear interactions. This method allows for a convenient description of electrical behavior in a complex biological geometry that may contain inhomogeneities and anisotropies. Such method consists in solving the Kirchoff current continuity law at each of the nodes of this discretized system. The system geometry, shown in Fig. 2.8, for a single cell is based on a large electric circuit comprising resistors and capacitors for passive charge transport and storage within electrolytes. The electrolyte models are distributed spatially and are connected to their nearest neighbors on a Cartesian lattice. For a full 3D analysis, each such RC combination is connected

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Fig. 2.8 (left) Schematic diagram of 2D portion of the cell modelized by the transport lattice method: 7 × 7 nodes, 84 components, two electrolytes separated by a membrane (distinguished by the bold line) which is not discretized due to its small spatial extent (adapted from [97]). (right) Complex tissue circuit with distributed RC elements. The tissue model is made of a 3D matrix of connected single cell equivalent circuits (from [95])

to its six nearest neighbors. Due to its finite dimension, the local membrane is not discretized. There is no transport in the perpendicular direction to this 2D model. Voltages applied along the top and bottom boundaries of the system model provide the nominal uniform electric field. A set of equations coupling the voltage values with its nearest neighbors is then obtained. The resulting matrix equation is solved to yield the time-evolving ITV. Then, the equivalent circuit is related to the asymptotic EP model, which determines the pore density as function of time [28]. A 3D equivalent tissue circuit [95] is shown in Fig. 2.8. There have been several attempts to analyze the electrical properties of simple models of tissues thanks to an infinite 3D lattice [107], or by considering random polydisperse distributions of spherical cells [48], or by simulating realistic multicellular structures based on segmentation and classification of 3D reconstruction of cell images [41]. But these approaches have questionable aspects. One is that they rely on poorly understood electrostatic coupling at short distances in living cells. Another is that it requires higher densities than those actually simulated. But the most problematic feature of these approaches is that they rely on homogeneous and isotropic conditions, whereas we know that the cell components are heterogeneous and anisotropic, meaning that that the material’s electrical properties depend on the direction of the applied electric field.

2.4.3 Mechanical Properties of Tissues Cells within tissues are exposed to a combination of mechanical and biochemical signals that often cannot be easily disentangled. Additionally, tissues experience combinations of compression, tension, and shear stress along different dimensions, making it difficult to determine what type of stress or strain is sensed and what

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cellular structure are responding. More specifically, spatial aggregation and reorganization of cell assemblies is important in physiological and pathological developmental processes, i.e. tissue wound healing, aging, and disease [102]. Many human diseases are related to changes in mechanical properties of tissues, e.g. for patients suffering from arteriosclerosis, the arteries lose some of their elasticity and become thicker and stiffer, in liver fibrosis, excessive fibrous connective tissue is characterized by a hardening of this organ. These are just a couple of examples that emphasize the importance of cell mechanics in biology. Magnetic tweezers (calibrated forces from 0.05 pN up to 150 pN), optical tweezers (0.1 pN to 100 pN), and AFM (5–104 pN) are the most common tools for investigating the mechanical properties of tissues [58]. Recently Mueller polarimetric imaging has been reported as a promising tool to probe the microstructure of biological tissues, both ex vivo and in vivo, i.e. collagen microstructure [103]. There have been several attempts to model collective cellular processes from the mechanical interactions of individual cells. However, understanding the interplay between subcellular mechanisms and ECM properties that produces movement requires a mathematical model that links molecular-level behavior with macroscopic observations on forces exerted, cell shape and speed, but how to formulate a multiscale model that integrates the microscopic steps into a macroscopic model is still poorly understood [104]. Since cells are filled with water and molecules, they are fairly incompressible and vary in their elasticity and affinity for sharing edges with other cells. If cells prefer contact with other cells, their edges may be long and their shapes oblong, whereas if they prefer little contact, they are more circular. Discrete computational models can be useful to track and predict the behavior of individual cells in a 3D tissue [104, 108–110]. In such models, the individual cells are modeled as viscoelastic shells around a liquid core representing a dense actin cortex surrounding a fluid cytoplasm, cells in contact adhere via intercellular bond forming a flat interface, and the 3D tissue is defined by a random distribution of points representing individual cells. While cells contain a fibrous cytoskeleton embedded with other structures (Fig. 2.1), tissues are composed of cells confined within ECM. Constitutive models to describe soft tissues often exhibit non-linear mechanical responses. One might be tempted to consider theories developed to predict the responses of synthetic soft materials such as elastomers and hydrogels. However, in general, due to the complexity and multiscale nature of biomaterials these approaches fail to correctly describe the strain-driven changes in tissue rheology [111]. More specifically, it was shown that an important characteristic of cytoplasm and ECM biopolymers is their high bending rigidity compared to most synthetic polymers [5, 39]. In addition, the highly nonlinear elastic response of biopolymer networks, in which the shear rigidity can increase by orders of magnitude upon strains of only a few percent [7, 47]. Cell-cell and cell-matrix adhesion mechanisms play a major role in the maintenance and/or reconstruction of multicellulat structures. It has long been recognized that defining how cells detect mechanical properties and how they adapt both acute responses and genetic programs [56, 109]. Quantifying the cell-generated forces in vivo is technically challenging and requires novel strategies that capture mechanical information across molecular, cellular, and tissue length scales, while

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allowing these studies to be performed in physiologically realistic biological models [110]. Within this context, Brodland and coworkers combined a FE analysis with video force microscopy that allows detailed, dynamic force maps to be produced from time-lapse images, in order to estimate forces during Drosophila development [112]. Cell migration is also crucial to the function of numerous cell types [57, 113]. Although necessary for physiological processes such as immune response, wound healing and tissue development, cell migration can also have detrimental effects by enabling metastatic cells to invade new organs. In vivo, migration occurs in complex environments and often requires a high cellular deformability, a property limited by the cell nucleus. Indeed, the deformability of the nucleus can be thought of as a rate limiting step in cell migration and therefore significant forces are required for a cell to pass through spaces that are smaller than the size of the undeformed nucleus. Nucleus can act as an intracellular ruler to measure cellular shape variations when motile cells squeeze through tight spaces or when deform in densely packed tissue microenvironments [114]. What force magnitudes and directions are necessary to achieve this is extremely challenging to evaluate [115]. In closing, we wish also to mention the interesting electrical and mechanical stimulations of biological materials and tissues. There have been several attempts to address this question particularly in the context of tissue engineering for innovative regenerative medicine and therapeutic devices [116]. So far, this chapter has been an ode to electrical and mechanical properties in the context of biological membranes which are considered to be one of the most important functional parts of cellular systems. Since the corresponding physical scales are largely governed by electrostatic interactions and entropy it is high time to provide a flavor of the way physics touches on these important properties. At some level EMB model building can be thought of as tuning “why?” questions into “how?” questions, e.g. how the cell remains in a highly organized state despite its many components and avoids the restriction of thermodynamic equilibrium through permanent dissipation [117]? In reaching the end of formal remarks, it’s worth emphasizing that a substantial list of unsolved problems still faces us. Now, it is worthwhile to ask how multifaceted approaches including physics, mechanics, and computational electromagnetics allow us to address cell biological questions. The focus of the next chapter is on the scope of ideas to illustrate the breadth and depth of EMB research through a diverse et of case studies.

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Chapter 3

A Brief Sketch of the History of EMB: Where Good Ideas Come From

Any living cell carries with it the experiences of a billion years of experimentation by its ancestors. You cannot expect to explain so wise an old bird in a few simple words. Max Delbrück

To make a very long story short, we try to give some perspectives on the key steps in which EMB modelling has developed over the years. The intended scope of the remarks includes both the interpretation of laboratory measurements, as well as the relevance of physical models intended to describe those measurements since models are a fundamental step in the scientific discovery. The basic ideas in this subject were all essentially in place several decades ago. However, the time seems right to call them back to mind. The idea of bioelectricity, first introduced to physics in the late eighteenth century, had a renaissance in the late 1930s when potential phenomenological applications were recognized and the notorious effective medium problem to describe the long-wavelength (quasitatic) properties of heterogeneous biomaterials was introduced. On the other hand, living cells can sense mechanical forces and convert them in biological responses, and similarly biological signals can influence mechanical forces. Therefore, studies in the EMB of cells and tissues are of utmost importance for biotechnology and human health and have rapidly evolved during the past decades. As an outgrowth of that effort, new solutions to understand the changes in cellular structure, its response and function under electromechanical force, were discovered whose implications have had far-reaching consequences well beyond the initial phenomenological domain, i.e. diagnosis and treatment of disease. An important connection and reason for this ubiquity is the relation to ED and EP of cells and tissues, where many new results have been found. Because of the diverse phenomenological applications, the discovery of a truly novel EMB approach and the potential to exploit biomedical applications have had a sustained and consequential impact and are likely to do so in the foreseeable future. Naturally, research in EMB is connected to the study of the behavior of complex systems, but four comments do seem appropriate at the outset. First, while the electrical and mechanical properties of biological materials were characterized and modelled independently by separate

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9_3

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scientific communities the collaborative work at the very basis of EMB across disciplinary boundaries has really taken off only from the mid-1990s [1]. Second, science and innovation at the interfaces can be disruptive but it largely depends on the collaboration between the specialties and sub-specialties. Third, a clear unifying tendency of contemporary science is the internally driven interdisciplinary research that leads to a growing overlap in the fundamental sciences such as biology, physics and chemistry. Fourth, the accuracy with which the relevant physical parameters have been determined experimentally has improved markedly in the last few years, making a much more meaningful comparison between theory and observation possible. The results of this confrontation, as we shall see, are quite encouraging and suggestive. In this chapter, we combine disparate literature discussions of the electromechanical properties of cells and tissues in order to get a timeline and perspectives in this area. The long history of scientific interest in EMB has generated an enormous phenomenology presented in a vast array of scientific publications. For present purposes only, a few of the primary concepts will be examined. To appreciate each example one must understand the context in which it occurs, and so we will outline here the essential physics as it arises. This theorist-guided non-chronological journey might strike readers as too reductive, but I tried to help readers better see where we came from and urges them to think harder about where we are going.

3.1 Early Ideas on EMB: Standing on the Shoulders of Giants The story of EMB begins when experimental studies related to the electrical control of cell physiology were reported a long time ago by Galvani [2] and Volta [2] who deserve to be considered true pioneers for their work on “animal electricity” and bioelectricity. The famous experiments of Galvani [2] dealing with frog nerve-muscle provided evidence for “animal electricity”. Volta’s work has opened a veritable cornucopia of interesting problems in electricity. At that time the ability for electric charge to instigate shape change in a sold dielectric was observed in Leyden jars, the first electrical capacitors, by Volta [2]. Pioneering studies were also done by Matteucci who described frog’s leg contraction in terms of potential differences between the electrically stimulated and the intact nerves. In a similar vein, the experimental physiologist du Bois-Reymond discovered the nerve action potential, that is the difference in the electric potential across the cell membrane caused by the flow of ions through the ionic channels [2]. These important investigations have laid the conceptual groundwork upon which actual studies on mechanosensing and mechanotransduction are based. More than a century ago, it was suggested that mechanical forces could drive tissue formation [3]. In the classic text On Growth and Form, Thompson [4] proposed also that mechanical forces were at the very basis for shaping tissues and organs during embryonic development, including examples from mammals, arthropods, and

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individual eukaryotic cells. From this pioneering work, the concept of emergence became increasingly popular and is now commonly applied in many scientific disciplines since it provides an understanding to the formation of complex structures from simpler elements. Many examples of regular patterns in biological systems from the molecular level up to plants and animals have been discussed in [3]. Ever since the work of D’Arcy Thompson, there is no denying that physical forces and mechanics are of utmost importance in shaping biological entities, and that importance goes beyond the structural role played by mechanics to hold cells together. However, it is only recently that we are beginning to understand how individual cells transduce mechanical force into biochemical signals. Although biomechanics and mechanobiology are well recognized nowadays, only some of the mechanisms of these complex systems have been elucidated. Biology has been a descriptive science for a long time, and has lacked the abstract theoretical framework that allows you to draw complex inferences about phenomena. In our present context, it is convenient to consider that the experimental efforts to look for new physics in this framework, by precision measurements, have been one of the drivers of our understanding of the physics of cell membranes. Early ideas about the cell structure have been the source of a good deal of controversy since it has become well accepted that cells are the most basic units of life and, consequently, that all known living organisms are made up of cells. The development of the optical microscope was a precondition for the discovery of cells.1 Cells were first observed under the microscope by Hooke in 1665. However, critical observations by optical microcopy about cell structure were made by Brown in 1833 as a constant component of plant cells [5]. Next, nuclei were also observed and recognized as such in some animal cells. It soon became apparent that nuclei, both isolated and studied within intact cells, had physical properties different from the cytoplasm [5]. In 1839, Luckey [5], who are considered the founders of the cell theory, recognized the common features of cells to be the membrane, nucleus, and cell body and described them in comparisons of various animal and plant tissues. Later, in the 1850s, Virchow suggested that the cell, as the simplest form of life-manifestation that nevertheless fully represents the idea of life, is the organic unit, the indivisible living one. He first published his idea that all cells arise from other cell other cells meaning that spontaneous generation of living things from innominate matter does not occur. This statement also helped to provide a basis for the idea of evolution as presented by Darwin a few years later [5]. It is of interest to see that, at the end of the nineteenth century, mechanical forces were suggested to shape tissues and organs during embryonic development [3, 4]. However, addressing this concern required unavailable tools to directly test such ideas experimentally. 1

Using single-lensed microscopes of his own design and make, van Leeuwenhoek was the first to observe and to experiment with microbes, which he originally referred to as “animalcules”. In the late 1600s, he was the first to relatively determine their size. Most of the “animalcules” are now referred to as unicellular organisms. He was also the first to document microscopic observations of muscle fibers, bacteria, spermatozoa, red blood cells, and among the first to see blood flow in capillaries.

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To keep their insides in and their outsides out, cells and their substructures are enveloped by PB membranes. Most significantly, as earlier as in 1925, Gorter and Grendel [6] hypothesized that biological membranes of eukaryotic organisms can be characterized by PB long before tools such as X-ray diffraction and NMR had made their way onto the scene of contemporary biological science. Using Langmuir’s through, they measured the area occupied by lipids that were extracted from a known number of erythrocytes. Then they measured the surface area of whole erythrocytes and calculated that the lipids of a single erythrocyte could be accommodated by a PB. Refs. [7, 8] furthermore addressed and documented some of the issues with cell membrane observation from freeze-fracture electron microscopy images, which eventually lead to conclude that all cell-organelle membranes had a common structure. It turns out that this concept was later strengthened by many authors who produced artificial plane PB membrane from lipids extracted from bovine brain [7]. Most significantly, [8, 9] argued for the PB model of membranes displayed in Fig. 2.1. But this last statement raises an important question about the structure of membranes and the molecules that make them up that. This point leading eventually to the fluid mosaic model of membrane, which characterizes it as a two-dimensional continuous fluid bilayer of lipids with freely diffusing embedded proteins, will be discussed next.

3.2 Continuum Approaches of the Electrical and Mechanical Properties of Biological Materials: Lessons Learned A turning point in the story is the experimental evidence that cells possess a resistive dielectric membrane that surrounds a conducting electrolyte. Inspired by the ground-breaking work of Höber in the 1910s, which opened the door to characterizing the frequency dependence of cell suspensions and muscle tissue, investigators have pursued applications of dielectric spectroscopy analysis in addition to exploiting the Maxwell–Wagner-Sillars interfacial polarization (β dispersion) [10]. In subsequent years, Fricke published a series of papers on the mathematical modelling of the electrical conductivity and capacity of disperse systems [11, 12]. These pioneering studies indicated that the membrane is molecular thick [11, 12]. The value of the membrane capacitance per unit area, of the order of 10−2 Fm−2 , likely stems from the first measurements ever made in the famous experiments of Hodgkin and Huxley [13]. On the one hand, Ohm’s law is used to relate the ion channel and intracellular resistances to the membrane voltage. Kirchoff’s current law provides the other key physical principle that Hodgkin and Huxley used to develop a mathematical model to explain the behavior of nerve cells in a squid giant axon, based on nonlinear dynamics of electrically coupled ion channels. Their circuit model, which was developed well before the advent of electron microscopes or computer simulations, was able to give scientists a basic understanding of how nerve cells work without having a detailed understanding of how the membrane of a nerve cell looked. A major driving force for

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many model buildings and phenomenological activities in the electrical properties of biological materials has been dielectric spectroscopy. Cole and coworkers published several papers [12] on the electrical impedance of biological eggs. At the same time, it has been little appreciated that Cole was the first scientist to use cantilever-beam theory to probe the mechanical properties of sea urchin egg cells [12]. Subsequently, Foster and Schwan described how frequency-dependent material parameters (permittivity and conductivity) vary with the frequency of the applied electric field, with four dispersion mechanisms2 (named α, β, δ, and γ ), each of which characterize a different polarization mechanism (Fig. 3.1) [15]. With the seminal work of Schwan, the subject of biomedical engineering almost immediately reached a mature stage. Schwan is best known for many biophysical studies related to electrical properties of cells and tissues, and on non-thermal mechanisms of interaction of fields and forces with biological systems [15]. It is worth noting that the idea of classifying cancers based on their electrical properties was first proposed by Fricke and Morse [11]. Schwan’s CS model of a biological cell possess several important virtues and can be considered as a member of simplified models3 in physics that allowed the exploration of a rich variety of electrical and mechanical properties, leading eventually to a comprehensive set of physical models of EMB [16]. Ease of analytical calculation for an allowed configuration is certainly 2

Dispersions can be differentiated in terms of dipole orientation and charge carrier dynamics. At low frequencies, it is easy for dipoles to orient in response to the change in the applied field, whereas charge carriers travel long distances. Thus, permittivity is high and conductivity is low because carriers can be easily trapped at defects and interfaces. As frequency in increased, the dipoles are less able to follow the field and charge carriers are less likely to be trapped. Consequently, permittivity decreases and conductivity increases. The α-dispersion characterizes the electrode polarization due to diffusion of counter ions through the electrical double layer along cell membranes, typically between 10 and 10 kHz, while the relaxation time is tupically 6 ms. It is notoriously difficult to analyse a-dispersions since many experimental artefacts may influence the measurements such as electrode polarizations. The β-dispersion deals with the capacitive-short circuiting of the poorly conductive cell membrane in the typical range 10–10 MHz, with a relaxation time close to 300 ns. Additionally, in some situations a weak δ dispersion above 100 MHz and below 5 GHz originating from relaxation of proteins can be observed. The γ -dispersion considers the dipolar relaxation of water molecules above 1 GHz [13, 14]. One might be tempted to consider these three dispersions by Debye relaxations. However, considerable empirical evidence suggests that Debye relaxation is inaccurate for interpreting the dispersion widths. Other approaches to deal with the understanding of these apparent widths rely on a RC relaxation time distribution or other phenomenological dielectric responses [12]. 3 Simplified models (such as two-level systems and harmonic oscillators in physics) are well be described by a small number of parameters directly related to experiment. The sensitivity of any newphysics search to a few-parameter simplified model can be studied and presented as a function of these parameters. The primary intended applications for simplified model results are: (i) identify the boundaries of search sensitivity. For example, 2D simplified models can illustrate these boundaries very clearly; (ii) characterize new physics signals. If new physics is observed, it will be important to fully characterize the range of parameters that it may involve; (iii) derive limits on more general models. Constraints on a wide variety of models can be deduced from limits on simplified models. The accuracy of the model is dependent on the validity of the assumptions, and the development of a relevant model is usually an iterative process, with successive generations providing better approximations of real systems.

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Fig. 3.1 a Representations of α, β, δ, and γ dielectric dispersions in biological materials. For the α dispersion, the real part of the permittivity can reach up to 106 –107 below 100 Hz. The imaginary part of the permittivity can be calculated via the Kramers-Krönig relationships. The β dispersion characterizes interfacial polarization at the interface membrane-electrolyte. This dispersion occurs near the cell membrane where the movement of charges particles are limited. The δ dispersion arises from the dipolar moments of proteins and other large molecules such as biopolymers, cell organelles, and protein bound water. The γ dispersion occurs due to the presence of free water content in the cells and tissues. b Showing the path of currents through cells as a function of frequency of the applied electric field. At low frequencies, the current travels around cells in the ECM. At intermediate frequencies, the current continues to pass around cells, but begins to act on the space within the cell membrane. At high frequencies, the current affects the space inside the cells entirely (from [12])

one. Another is that it represents an extreme limit of symmetry for dense forms of cells and can be translated in terms of electrical circuits. Beyond those features, it has an economical mathematical definition, while nevertheless exhibiting complex and fascinating phenomena. Many similarities between analog electrical and biological circuits have been also reported in [17].

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The introduction of the long-wavelength (quasistatic) approximation in the 1940s prompted researchers to turn their attention to determining the polarization mechanisms in heterogeneous dielectric media which highlighted the importance of effective medium approaches, see e.g. [18]. The literature on theories of effective or average response of composite media is voluminous and dispersed among many fields [14, 18–23]. Improving the efficiency of such calculation is important for a large variety of applications, e.g. colloidal and interface science, electrochemistry and biophysics. Biological materials are such materials for which any response to an electric field impacts the charging mechanisms along interfaces of the constituents with different dielectric properties. So, the hunt is on to find methods of modelling phenomena of biological relevance involving cells and tissues without having the price to pay in excessive data of molecular simulations. From a modelling perspective, several analytical methods have been developed to better our understanding of the fundamental physics of cells, particularly those subjected to electric fields [14, 18–26]. However, due to the complex multiscale nature of the problem, the analytical solutions require restrictive assumptions, e.g. canonical cell shapes, equilibrium continuum and quasistatic models, in order to solve the governing equations, but they can be tested. Over the period 1980–2000, Asami and Irimajiri [18], and contemporary extensions of these investigations [19–23] presented detailed analytical effective medium CS analysis on the effects of using an electric field on cells with different shapes and dielectric properties of the cytoplasm and ECM (ion concentration, presence of organelles, etc.) and compared their results with experimental observations. Later, numerical treatments by making use of a variety of well-developed techniques (FE, finite-difference time-domain, multiple multipole, boundary integral equation, etc.) of this problem have opened a wide window of opportunities to consider more realistically shaped biological cells, cell interactions, and cell packing in tissue models [23]. In 1970, in order to explain the biconcave shape of red blood cells, Canham [24] proposed a bending energy density dependent on the squared mean curvature. Three years later Helfrich proposed a definition of the curvature elastic energy per unit area of a closed isotropic PB, followed by an often-cited related paper by Evans [24]. The integral in Eq. (2.4) is often referred as the CEH energy because of their truly farsighted and pioneering contributions which allow determining the equilibrium cell shape by the elastic properties of its membrane. Based on CEH energy, several authors showed that the response of membranes to the application of force is well approximated by the CEH theory and the stability of cell membrane under mechanical forces is enhanced by cross-linking proteins [25]. The diffraction limit prevents the use of optical microscopy for accurate reporting at resolutions below 200 nm, which means that identifying and localizing single biomolecules remains out of reach by using this probe. Beginning in the twentieth century and continuing through today, electron microscopists have focused on the structure of the PB cell membrane. Similarly, confocal and super-resolution fluorescence microscopy have emerged as powerful tool for the capture of 3D cellular structure. Early models of cell membranes simply depicted membrane lipids as fluid

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entities within a homogenous matrix, with their main function being the accommodation of membrane proteins. More recent models include the heterogeneous distribution of lipids both between the two leaflets and laterally within the membrane. Most phospholipids have an asymmetrical distribution between the outer and inner leaflets of the cell membrane which also hosts discrete domains enriched in specific molecules (Fig. 4.1). Biologists have long debated the mechanism of membrane organization and, in particular, whether it has anything to do with thermodynamics phase separation in other liquids. In this context, an important step was the fluid mosaic model of membrane structure which was proposed based on thermodynamic princips of organization of membrane lipids and proteins [26]. The mosaic is made of proteins that are inserted into the fluid, which is the PB (left panel of Fig. 4.1). In this paradigm, the bilayer is considered a uniform semipermeable barrier that serves as a passive matrix for membrane proteins. The proteins are usually arranged so that their hydrophobic surfaces are buried in the lipid. Within this model, the fluid is considered as isotropic: diffusion of the lipids and proteins are random unless it is constrained by the cytoskeleton or by the high concentration of membrane proteins. This model posits a cell bilayer membrane that can flow within its plane, but that has elastic properties when stretched or bent. In the initial version of this model, transmembrane proteins diffused freely within the plane of the PB. In addition to explaining the various hierarchies that appear in the cell structure, CS models can also be used to understand dynamics of the membrane permeabilization. Most significantly, all these directions prompt thinking about ED and EP models of cell membrane considering it a partially permeable boundary surrounding the cytoplasm. In the 1960s Coster and Hope’s pioneered the first descriptions of the membrane EP of the giant cells of Chara coralline, a eukaryotic alga [27]. Artificially created planar lipid bilayers and GUVs have been used in a number of experiments to gain further insight in the cascade of mechanisms that are operative during the electric pulse [28]. These simplified model systems offer several advantages for studying molecular interactions which occur in biological membranes [28, 29]. More specifically, GUVs provide the benefit of isolating the function of the membrane from the complex cytoplasm organization, while resembling the size of the cell and curvature of the cell membrane. Additionally, the composition of the membrane can be controllable, e.g. the GUVs can be prepared from lipids with different phase-states and cholesterol can be also added to form ternary phase systems [28]. Within this context, the ED of vesicles has been theorized by many authors [29–32]. Calculations of the induced potential drop (i.e. ITV) when the cell is subjected to applied fields at low frequencies were reported by many authors for a variety of spheroidal cells [30–32]. Many results dealing with the role of the amplitude and duration of the electric pulses on the EP mechanism have been obtained by several studies [30–38]. Several early experiments showed cell transfection in vitro through EP using strength of several kV/cm with pulse duration of a few μs electric pulses [33]. Many widely disparate EP models have been described in the literature, but the attempts to obtain a unified description of all the physical mechanisms involved in EP have so far been more notable for their ingenuity. Chernomordik et al. [34] were pioneers in

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proposing a single-cell EP treatment containing a thin, rigid membrane in a series of seminal papers. Most significantly, the above studies showed that EP becomes observable when the ITV reaches a certain threshold, typically between 0.2 and 1 V,4 and the percentage of permeabilized cells increases as the electric pulse amplitude is increased while the percentage of cells surviving the application of the electric field decreases [35–46]. On the other hand, many authors addressed some of the issues related to EP of cell suspensions and tissues [45–49]. In these situations, the density of electric field lines at a specific location in space reveals information about the strength of field at that location, i.e. intercellular interactions such as gap junctions. As recalled in Chap. 1, the prevalent theory for membrane permeabilization arises from a minimum energy principle. Within this perspective, several authors have suggested models to explain membrane EP. A prevailing view from pioneering EP and membrane rupture studies is that pore growth and PB rupture are controlled by long-lived metastable defect states that precede fully developed pores. Visual demonstrations of the transversal pores formed in the membranes were not made until the electron microscope studies due to the difficulty of detecting these transient small structures [8]. Researchers have often used calcium concentration as a measure of permeability since this ion has a greater extracellular concentration at physiological conditions. Hence, Sengel and Wallace [42] were able to visualize in real time the presence of individual voltage-induced pores in droplet interface bilayers with total internal reflection fluorescence microscopy by detecting a fluorescent signal proportional to the flux of Ca2+ flowing through a pore. When ITV is over some threshold, the prevalent EP model is that transient hydrophilic pores form in the lipid domains or within some membrane proteins that fill with aqueous media and allow translocation of extracellular or intracellular materials into or out of the cell [31, 32]. Under the influence of an exogenous electric field, the potential drop across the cell is sustained by the cell membrane, which is a poor conductor, while the field in the much more conductive ionic medium of the cytoplasm is small. The model can predict a stable pore if the expansion leads to a reduction in surface tension. These pores are then destroyed randomly with a mean life time orders of magnitude longer than the duration of the pulse, estimated to range from ms to even mn. Once cells detect the breached membrane, they activate the membrane repair processes to restore boundary integrity to again inhibit molecular transport. Moreover, based on multiple CS models of cell there have been several attempts to address the question of distributed equivalent RC circuit representations to obtain the ITV using either a time domain or a spectral analysis [29]. A recent suggestion by [29] was made in this context: the authors pointed out that the cell membrane, when exposed to ac electric fields, acts as a first-order low pass filter while the NE acts as a first-order bandpass filter.

4

It may be worthy of note that several investigators consider that there is no fixed voltage threshold for EP. It was suggested that the ITV value at which pore creation increases depends not only on the rate at which ITV rises but also on a complex kinetic behavior within the energy landscape of pore states leading to hydrophilic pores [35].

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EP appears as an emergent property involving nonlinear local electric and mechanical interactions with membrane. Chizmadzhev et al. [34] suggested that pores are caused by transient local membrane defects. The transient aqueous pore process of EP is based on continuum models of membrane pores, mechanical and electrostatic energy contributions, and thermal fluctuations, usually in the form of a continuous Smoluchowski equation for the growth and decay of pores [32].5 Further refinements were added by Weaver and Mintzer [31] by suggesting that the pore creation energy barrier depends on V m , as well as Neu and Krassowska [37] who used an asymptotic model which approximates the Smoluchowski equation that disregards pore size change. This model was also applied by Smith et al. [38] to study the EP of an isolated spherical cell as a function of the electric pulse magnitude and duration. The important features of this deterministic model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. Vasilkoski and coworkers developed a probabilistic nucleation model and incorporated it into a Smoluchowski equation-based model to solve for an absolute rate equation that describes the formation of pores in the membrane over a wide range of timescales [35]. Later, Krassowska and Filev [36] provided a full model of the whole-cell EP in which the spatial and temporal dependence of the EP statistics (pore radius and number) were given equal consideration. The aforementioned studies, along with many others, addressed some of the issues of the dynamics of electropores such as location on the membrane, creation, destruction and size change, but in all of these models pores exhibit a similar kinetic behavior, although direct comparisons to quantitative experimental data are scarce, e.g. [46]. Gowrishankar and Weaver [47] proposed a transport lattice model with passive and active interactions including more than 105 local transport elements. Since the processes of pore creation and evolution are stochastic in nature, the ITV value at which pore creation increases was found to depend not only on the rate at which ITV rises, but on a complex kinetic behavior within the energy landscape of pore states leading to hydrophilic pores [35]. Morshed et al. [47] described also an electrical circuit equivalent model containing dynamic components for cell membranes and electropores. In this model, the dynamic behavior of the electropores on the cell membrane during EP is modeled by a voltage-controlled resistor. This resistor is activated by a voltage-gate switch at a two-stage cascaded integrator and the model implements the voltage-controlled resistor to predict the breakdown voltage of cell membrane at EP. Shagoshtasbi et al. [47] used a nonlinear size-dependent equivalent circuit model of a single-cell EP to investigate the electromechanical behavior of cells in microfluidic chips during EP. Akimov et al. [43] attempted to model the cell membrane as a continuous liquid crystal medium and developed a model of hydrophilic pore formation induced by an external stress, such as lateral tension or transmembrane electric potential. What determines the size of the pore? When a membrane ruptures, the energy of the pore is determined by membrane tension, line tension, and ITV. The 5

The concept of using a diffusive motion across an energy landscape was originally discussed by Smoluchowski. As a consequence of the fluctuation–dissipation theorem, the evolution of the pore density can be expected to follow a diffusive Brownian motion across the energy landscape.

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proposed model explicitly considers the optimal shape of the pore surface obtained by energy minimization for each state of pore formation and growth. Within this model, and once formed, the membrane pore can expand, relocating lipids from the pore to the rest of the membrane surface. This can lower the areal stretch and hence, reduce the membrane tension. Large stable pores of up to a μm in diameter can potentially be obtained if continual lipid relocation decreases the areal stretch after rupture. However, these models assume neither where EP will locally occur nor the magnitude of the ITV at which significant numbers of pores will be created. Recently, it was suggested that pores can nucleate within some membrane proteins, and that pore formation is affected by the actin cytoskeleton [49]. In the 1990s, irreversible EP was also found to be an effective technique for tumor ablation in multiple types of cancer [50], attributable to its unique ability to induce cell death, even in close proximity to vasculature and nerves. In a related context, the evaluation of local heating during exposures of a single cell to electrical pulse train, high-frequency irreversible EP has been also reported [50]. It is also worth stressing that EP is not solely an electromechanical phenomenon. It can be also accompanied by electrochemical effects involving cell oxidation and the generation of reactive oxygen species. Within this context, it is believed that during and after EP lipid peroxidation plays a significant role in the dynamics of pore creation and resealing in the membrane. Importantly, the electric pulse parameters can significantly impact the extent of oxidative stress [44]. The mechanical properties of biological materials are described by how they respond to an applied force. Most analyses of cell and tissue mechanics relied on continuum theories that describe their effective electrical and mechanical properties. While such phenomenological models quantitatively capture some of the behaviors of cells and tissues, they often contain fitting parameters that lack clear physical interpretations, thus providing reasonable but limited insights into the underlying mechanisms responsible for cell and tissue mechanics. Among the many attempts to address the question of pore formation in the membrane it was suggested that the applied field triggers orientational changes of interfacial water molecules and lipids depending on their dipole moment. The tilting of lipids is eventually followed by spontaneous insertion of water molecules and a few lipid headgroups into the membrane hydrophobic core, followed by the formation of a toroidally shaped hydrophilic pore of size close to that of protein ion channels [48]. The advent of new cryofixation 2D and 3D reconstruction electron microscopy techniques finally supplied the necessary resolution to discern the primary components of the cell membrane. The lipid-raft model proposed by Brown and Rose [51] added another layer of complexity and hierarchy for membrane model. Revisions to the fluid mosaic model gradually added an increasing number of constraints due to specific interactions of membrane proteins with other proteins, with lipid domains, or with cytoskeleton and ECM [52]. More significantly, the importance and roles of specialized membrane domains, such as lipid rafts and protein/glycoprotein complexes, in describing the macrostructure, dynamics and functions of cellular membranes as well as the roles of membrane-associated cytoskeletal fences and

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ECM structures in limiting the lateral diffusion and range of motion of membrane components have been documented [52]. These membrane domains are likely to have an impact on membrane proteins. Additionally, as the NE is not permeable to most of the molecules inside the cell, NPC exist on its surface that allow highly selective translocation over the NE. Since its discovery in 1954, the NPC has generally been accepted as the only means of communication between the cytoplasm and the nucleoplasm [52], i.e. molecules smaller than 30–40 kDa can passively penetrate the NPC from the cytoplasm into the nucleus and vice versa, whereas bigger molecules require interaction with nuclear transport receptors and form importin/exportin complexes that are guided through the pores. Several authors used fractal analysis to capture cell morphological complexity and correlate it to other biophysical properties of the membrane such as its capacitance derived from the electrorotation measurements [53]. In Ref. [53], the scale-invariant property of the membrane of biological cells is examined by applying the MinkowskiBouligand method to digitized scanning electron microscopy images of the cell surface. It is observed that the membrane exhibit fractal behavior, i.e. the structure is close to scale invariance, and the derived fractal dimension gives a good description of its morphological complexity. One way to formalize what might be going on here was to take advantage of the self-similarity property to propose a fractal single-shell model in place of the standard single-shell model described above.

3.3 EMB Model Building and Phenomenology: How We Got to Know Ever since the contemporary development of EMB, physicists have puzzled over the electromechanical couplings in biological materials [54–56]. It is now apparent that mechanical forces can regulate a wide variety of biological processes from cell migration and morphogenesis to cell adhesion to ECM. More specifically, mechanical cues from the ECM trigger signaling cascades that alter gene expression and affect various processes, including cell motility and fate. It has become clear that most eukaryotic cells can generate intracellular forces that cat on the surrounding ECM or neighboring cells. Applying force to the cell-ECM unit leads to structural deformations and rearrangements of the ECM, and deformation of nearly every subcellular structure, including the position of the mitochondria, endoplasmic reticulem, and the nucleus (Fig. 4.1). The next levels in the mechanistic understanding of EMB came from several experimental studies and phenomenological models for developing a mechanistic understanding of the mechanical responses of single cells, dilute suspensions in vitro, and tissues upon application of a strong electric field. The connection between mechanics and cell function has been studied in various contexts. Many biological systems are, amazingly, able to perform a multitude of tasks by dynamically adjusting their mechanical properties without changing their composition. For example, cells undergo mechanical deformation when subjected

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to external forces and geometric constraints, e.g. a red blood cell with a diameter of 7–8.5 μm is subjected to a 100% elastic deformation as blood flows through narrow capillaries with inner diameter smaller than 3 μm. Many normal and diseased conditions of cells are dependent on or are regulated by their mechanical environment, and the deformation characteristics can provide significant information about their biological function. Most of the modeling efforts in EMB have been devoted to mammalian cells. Similarly, growth and elongation of bacteria6 are related to mechanical stress and strain within the envelope associated with osmotic pressure (typically 100 kPa) [54]. Technological advances have also related the mechanical properties of bacteria to cell division and cell envelope remodeling [54]. Although bacteria lack a cytoskeleton, they do have a number of cytoskeletal-like molecules that serve mechanical functions. Mechanobiology describes the relationship between a cell and its environment. The sharp contrast in hardness between a tumour and the surrounding tissue is at the heart of mechanobiology [55]. Researchers have shown that the forces generated by, and acting on, tissues influence the way tumours grow and develop. It was suggested that the chemomechanical response may be transmitted over great distances within the cell by a tensegrity tissue matrix structure. Tensegrity was first described by Fuller in the late 1950s as an architectural principle relying on balancing continuous tension and discontinuous compression forces to provide a stable prestressed structure [56]. Then, this building principle was visualized by the sculptures, made by the artist Snelson [56], that are composed of stainless-steel bars and tension cables. Next, Ingber proposed the model of cellular tensegrity in the early 1980s emphasizing the importance of tension among cytoskeletal structures in the cell’s ability to configure a holistic sense of the forces at play [56]. If there is tension in the cytoskeletal filaments, then a local change in one part of the filament is quickly transmitted to all connected parts. It became apparent that the determination of constitutive behavior of single cells and tissues, i.e. multiaxial stress–strain relations, changes of mechanical properties with time and/or in response to electrical stimulus was one of the two pillars which define EMB. The convergence between mechanobiology and electrostatics of biological materials was part of the continuing effort of merging elasticity theory and electrostatics. This research was initiated by a couple of seminal papers by Toupin [57] in the 1950s. In addition to the Maxwell equations the force balance must be satisfied [58]. The key question is therefore what is the appropriate stress tensor for electroelastic systems and materials? Within this perspective, Maxwell stress tensor, a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum, plays an important role in the dynamics of continuum media interacting with external fields [58]. The electric field at the cell interface is not only the driving force for the polarizability and conduction phenomena described by Maxwell equations but also induces simultaneously a mechanical stress field. The coupling between the electrical and mechanical fields at interfaces which are nanometric in scale underlies EMB, e.g. 6

EP is also performed on bacteria for many important applications such as food sterilization [54].

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mechanoelectrical transduction can activate proteins in the cytoplasm to affect intracellular activities. For example, imaging EP of endothelial cells reveals significant, reversible disruption of interphase microtubules and actin filaments [59]. The unique characteristics of miniaturization and integration enable microfluidics techniques to overcome many of the drawbacks of standard EP as a variety of microelectrodes can be incorporated into the chips to generate the field necessary for EP [60]: (1) The EP microchips can provide uniform electric field distribution; (2) Single cells can be manipulated on chips to probe cell heterogeneity; (3) The utilization of transparent materials for microchips allows in situ observation and real-time monitoring of EP process using fluorescent probes; (4) Most significantly, many authors [60] argued that transient membrane pores are formed by the mechanical squeezing of a cell as it passes through a microfluidic constriction that is smaller than the cell diameter. Experiments on tension propagation in cells have yielded hugely diverse outcomes [60, 61]. Within the context of EMB, a microfluidics-based approach capable of quantifying simultaneously Young’s modulus and membrane capacitance of aspirated single cells through a narrow constriction channel (Fig. 3.2) is a convenient means to monitor the electromechanical couplings along the cell trajectory in a continuous manner [61]. To date several constriction channels that mimic pore/capillaries in vivo have been discussed by a number of authors [61]. This breaking cellular symmetry mechanism requires reorganizing the cytoskeleton, and in many cases, is actually driven by cytoskeleton mechanisms [59]. However, little is known about how local sensing of

Fig. 3.2 A constriction channel device for simultaneous characterization of the mechanical and electrical properties of single cells (from [61]). The height and width of the constriction channel are chosen to match the cell size inn order that the cell can squeeze during the 1D motion along the long axis of the channel if the input pressure is large enough. This microfluidic technique forces cells to deform as they enter and transit through a narrow channel under an applied hydrostatic pressure. Typical measurement data include cell stiffness, C m , entry time, transit time and shape-change parameters. The cell can squeeze through a constriction smaller than its size if the input pressure is large enough

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mechanical force is transduced into biochemical signals that result in cell responses. Alteration of such forces can likely to cause a disruption in their functioning, leading eventually to a disease state. Quantifying cell-generated forces in vivo is technically challenging and requires efficient strategies that capture mechanical information across a variety of length scales, while allowing these studies to be performed in physiologically realistic biological models. Most significantly, several authors [46, 47] argued for the use of continuum approaches, according to which accurate details of stress and strain distributions induced at the cell level by an electric field excitation can assist in the refining of accurate microscale models. The last statement raises an important question from a multiscale perspective. There have been several attempts to address this question, particularly in the context of differential geometry which provides a variational formulation to a number of physical phenomena, including elasticity and fluid dynamics, electrophoresis, electrokinetics and electrohydrodynamics [60]. Although the details are still debated, the concern is that closed-form continuum models for a multiphysics understanding of EP should be inseparable of a multiscale analysis.

3.4 Computational EMB: A Multidisciplinary Approach The last step we mention here is relatively recent; it has to do with the use of computation models to predict the electromechanical behavior of biological systems. The 21th century has seen tremendous progress in computational science and engineering, fast algorithms and new visualization techniques. These include also multifaceted models to study how the behavior and dynamics of PB affect permeability for molecules entering and exiting a cell subjected to an electric field excitation. Computational modeling serves as an important tool to characterize and predict the behavior of these many complex biophysical systems, and predictions from computational studies are developed alongside experimental testing, creating a synergistic feedback loop to accelerate discovery. Using simplified cell and tissue models, one can test hypothesis and understand which specific feature gives rise to particular outcome [60]. Equally important has been the development of methods to map molecular properties onto the language of continuum electrostatics and mechanics and the recognition that many molecular effects can be treated accurately with a model that ignores the microscopic description of molecules in aqueous solution. While the standard FE technique was developed in different fields of engineering and physics from the 1970s, extensive use of FE simulations in biology is more recent and dates back of the early 2000s [61, 62]. Since then, this rapidly changing field evolved and expanded, so do the skill sets necessary to excel in it. The main reasons for its popularity are its efficiency, ease of computation, and speed of implementation. In addition, material as well as geometric nonlinearities can be easily incorporated. Many powerful commercial FE software packages have become available to accommodate biologically-distinctive features such as COMSOL® Multiphysics [63], Ansys Multiphysics [64], and FreeFEM [65], just to mention a few. Alternate computational approaches have been also used

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for particular applications, e.g. the boundary integral model, based also on solving PDEs but which requires discretization on only the domain boundary, thus reducing the computational time [23, 66] A cultural change is in progress since historically, biologists have not been comfortable talking to physicists and computer scientists, who in turn do not really understand a lot of the issues in biology. Collaborations between biologists and computer scientists can be fruitful when they bring synergetically their respective expertise, learn to speak the language of the other and well fitted. We expect that combining capabilities for measuring deformation and immunofluorescence by continuous-flow microfluidic techniques will be of help in this matter. Over the past two decades, an incredibly large body of FE simulations has been developed to model coupled partial differential equations developed in the realm of continuum-based models [61, 62]. Continuum electrostatics and mechanics provide well-established foundations for describing the behaviors of cells and tissues at the supramolecular scale. These continuum-based approaches have also been used to study the properties of subcellular components, such as the properties of the nucleus and its associated structures. There is a long record of mechanical studies of homogenization7 of perforated thin films and perforated layers, including biomaterials, see e.g. [62]. Such approaches could provide valuable information for evaluating the impact of pore content and arrangement, as well as inter-pore distance, on the overall mechanical properties of membranes during EP. Numerical theoretical models have been proposed for analysing how different spatial organization of cell populations can represent simplified models for analysing the EMB of real tissues [69]. The ability to quantify the electrostatics and mechanics of cell-cell interactions by multiscale approaches [70–74] is essential in enabling the progress of numerous applications, particularly in understanding the nature of cellular forces and how they can drive collective phenomena. More specifically, it is of utmost importance to understand the association between internal the individual subcellular structures over a range of spatial and temporal scales on the collective tissue and organ responses to external stimuli. In a very pedagogical way, McCulloch [69] discussed the need for multiscale modelling of organ when we put ourselves in the shoes of the physician by considering the heart diseases for which electrical, mechanical and transport functions all linked together (Fig. 3.3). Such cardiac tissue model can aid our understanding of complex cardiovascular disorders using a reductionist approach. No matter how the scale separation can be chosen, the actual coupling between scales is at the heart of multiscale simulations where microscopic processes (small spatial scales, fast dynamics) are coupled to macroscopic processes (large spatial scales, slow dynamics). Modeling an emergent complex behavior such as EP requires simultaneous dynamical descriptions on both macroscopic and microscopic length

7

Homogenization refers to theoretical methods for which the contribution of smaller scale is simplified by seeking an asymptotic limit for the problem solution assuming scale separation between microscopic and macroscopic domains [67, 68].

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Fig. 3.3 A multiscale and multiphysics approach of cardiac electrophysiology and biomechanics for the purpose of modelling cardiac electromechanical function in heath and disease: from the voltage-dependent kinetics of ion channels and pumps that carry Na+ , K+ , Ca2+ and Cl− ions a to the capacitance of the membranes in a cell model b to the resistive coupling between neighboring muscle cells at the tissue scale c to the 3D anatomy of the cardiac chambers d to the integrated model of whole heart electrical activity e. The approach also considers the pumping mechanics of the cardiac chambers h to explain the filling and contraction of the cardiac chambers and flows in the circulatory system i and down in the scale of the molecular motors f, and can include also the internal structure of the myocytes g (from [69])

scales. That is why besides the continuum FE simulations, MD simulations8 have contributed key insight into the structure and dynamics of pores in the membrane by providing a model of molecular motion with sub-nanometer level detail and femtosecond resolution, which is not possible with continuum-scale simulations. While the concept of MD simulation was originally developed in the early 1950s, the ab-initio MD method, invented by Car and Parrinello in 1985, has been applied with remarkable success to a variety of problems in condensed matter, chemical physics,

8

Many software’s for the interactive visualization and analysis of cell membrane by MD simulations have been considered in the literature, e.g. GridMAT-MD, FATSLiM, APL@Voro [75–77].

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materials science, and biochemistry, and is now a standard tool for molecular simulation. In MD simulations, the starting point is a set of atomic positions in 3D space. The time evolution of these atoms is then simulated by integrating Newton’s second law which requires an empirical potential energy in order to describe the forces acting on the atoms. In the beginning of the 2000s, MD simulations showed that pores providing a high-conductivity state of the membrane arise from structural rearrangement of the phospholipids in the membrane [48, 75, 78–80]. These simulations have provided novel insight into many features of the process of local defect and pore creation and destruction, as well as the transport of ions, molecuales or even DNA through membrane pores. For example, Joshi and Schoenbach [78] found that a MD scheme can be combined with a distributed circuit model for a self-consistent analysis of the transient membrane response for cells subjected to an ultrashort (nanosecond) high-intensity (∼0.01 Vnm−1 spatially averaged field) voltage pulse. The dynamical, stochastic, many-body aspects are treated at the molecular level by resorting to a course-grained representation of the membrane lipid molecules. Coupling the Smoluchowski equation to the distributed electrical model for current flow provides the time-dependent transmembrane fields for the MD simulations. These MD simulations confirmed the presence of hydrophobic and hydrophilic pores during pore nucleation, while the radii of stable open pores are found in good agreement with experimental estimates [75]. Simulation data from very different systems and models show how water penetration and local defect formation can determine the free energies of many membrane processes. Such simulations also showed that PB can deform and allow the spontaneous and homogeneous formation of nanoscale defects and pores in the membrane due to thermal fluctuations of the lipid molecules when exposed to a diverse range of stimuli. Overall, MD simulations provide information that complements experimental studies, allowing microscopic insight into experimental observations and suggesting novel hypotheses and experiments, e.g. the pore formation is predominantly driven by the electric field reorientation of the water dipoles at the water-bilayer interface and not by tensile electric stress [75]. However, this field of multiscale modelling is data-intensive and the simulated systems are restricted to very small sizes and times on the order of nm and ns due to a demanding computational expense. Additionally, the terabyte data engendered by MD simulations represent a staggering quantity of information, even for relatively small systems. For example, a MD calculation on a patch of lipid bilayer containing a single ion channel and surrounded by water ands ions (104 particles) dealing with length scales of the order of 10–30 nm and time scales of hundreds of ns, woefully inadequate for accessing materials processes, generates a terabyte worth of data and requires several days of computational time [76]. There have been also several attempts to address the effects of bacterial physiology on EP by MD simulations of both gram-negative (Escherichia coli) and gram-positive (Staphylococcus aureus) bacteria, see e.g. [77]. Direct communication between continuum, coarse-grained and MD approaches is not straightforward. Exciting developments have seen fundamental multiscale simulations EP phenomena using mixed molecular/continuum or coarse-grained schemes, in which the electropores in the membrane are treated in full molecular details while the overall mechanical properties of the cell are treated using continuum theories,

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i.e. the displacement field is continuous [81]. The multiscale coarse-graining method determines a coarse-grained interaction potential from atomistic force information through a variational minimization procedure. However, the location of subcellular level on the interface between stochastic (e.g. pore nucleation) and deterministic (e.g. membrane elasticity) events makes it difficult to include in a multiscale model since it can be difficult to couple models across this interface. These approaches allow for a significantly expanded range of simulated space and time scales in comparison with atomistic methods, but their upper limit in spatiotemporal scales remains and largely depends on the level of coarse-graining used. A basic message of this brief historical examination is that already in the late eighteenth century, long before tools such as electron microscopes and NMR spectrometers had made their way onto the scene of modern biophysics, scientists had already gleaned a sense of the important ingredients with which the qualitative and quantitative understanding of EMB can be achieved. This essay has also attempted to convey some of the excitement that has arisen because of the advent of the ability to build models of EMB that intrinsically involve multiple scales in either space and time or both. If blind spots are still there, I apologize to the many workers whom I have slighted out of ignorance. We know quite precisely what the equations of the EP and ED theories are.9 Furthermore, we have ample evidence that these modelling approaches are correct. Yet, there are still absolutely striking features we don’t understand. Historical strides in our fundamental understanding of EMB have not typically relied on detailed physical characterization and models of the underlying molecular behavior. However, EMB is a melting pot since it provides a rich playground for novel structure–function relationship of cells and heterogeneous tissues. This is a research area with several open questions in phenomenology and model building, and deep questions in terms of couplings between more complex and realistic cell models. History suggests that the present technical limitations are only temporary. Present numerical simulation techniques will likely be refined or replaced by clever application of physical concepts and with them will come new discoveries about the EMB principles. Life presents many interesting questions for physicists and they have a lot to contribute in the field [82, 83]. With this history as background, we will now return to the contemporary problems posed in the introductory remarks of this book.

9

It is an important discipline to try to visualize the physical behavior the theories predict under different conditions. Perhaps the most profound comment about the right questions to pose was made by Paul Dirac: “I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it”.

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Part II

Calculation Methods

Chapter 4

Analytical Approaches of EMB at Multiple Scales

I didn’t really think of this as moving into biology, but rather as exploring another venue in which to do physics. John Hopfield

Our interests in biophysics are driven by questions about “how things work” and “why things are” which leads naturally to morphology and emergent dynamic behavior. We are basically interested in getting an insight into how quantitative cell biophysics is implemented today and can offer the appropriate methods to deepen our knowledge and insight. In general, the objectives of computational models are to replicate the behavior of the complex system it parallels and to do so are based on actual known properties of the sytem constituents. Achieving these goals require the model to span a range of spatial and temporal scales and incorporate information from multiple disciplines. The challenge of building models that integrate structurally across physical scales of biological organization from individual molecule to cell to organ and organism is a defining problem of contemporary biophysics. The development of very complex computational models motivates closer examination of the basis of the physical models underpinning the simulations. Advances and innovations in microscopy and biological imaging have provided descriptions of unprecedented detail of the constituent parts and basic structures of living organisms. However, experimental complexity usually restricts observations to restricted spatial and/or temporal scales. A particularly fertile example that serves as the backdrop for the present part of this book is given by EMB. In this chapter we summarize some of the ways the EMB appears in a variety of contexts that can feature in the theory initiative, particularly in its attempt to unify the cohort of EMB activities currently underway. The purpose of this chapter is threefold. First, we aim to discuss topics in physics that are particularly relevant for the scientific community in order to delineate and sharpen EMB’s potential. The organization of this chapter reflects this identification of topics, with each section covering one area. Second, within each section we summarize the state of the art of each of these areas, both from a theoretical and an observational point of view, highlighting central analytical methods and their applications as well as general theoretical and experimental results and their practical applications. Third, in order to keep this chapter both manageable and useful, rather than presenting an extensive © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9_4

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detailed review of each topic, we have opted for sufficiently concise reviews, referring to excellent recent articles in the archival literature. We have relegated technical details on several topics to appendixes.

4.1 How Physics Scales EMB From the discussion so far, it should be clear that to accomplish our goals here at the interface between physics and biology, a thorough grounding in the laws of physics is necessary. The reason for this is simply that the laws of physics govern everything in the observable universe at all measurable scales of length, mass, and time. It is, after all, some notion of functional behavior that distinguishes life from inanimate matter, and it is a challenge to quantify this functionality in a language that parallels our characterization of other physical systems. Since biological systems are hierarchically structured the level of biological organization considered in quantitative models is of of utmost importance. One of the key hallmarks of biological function is ordering in space and time. With the emergence of fundamental length and time scales whose influence permeates every aspect of physical behavior, one might have anticipated that the EMB modelling at larger scales (tissues and organs) and of matter at smaller scales (molecules and proteins conformations) would diverge. It is a profound, and at first sight astonishing, fact that this does not happen. One finds, instead, far-reaching resemblances between phenomena at very different scales of time and distance, occurring in systems as different superficially as the electric polarization of cell membranes and those of composite materials, or the mechanical properties of cells and those of viscoelastic gels. So far, this progress reads like a standard reductionist triumph [1]. It is also interesting to observe that the cell remains in a hihgly organized state and despite its large number of interacting components it avoids the restrictions of thermodynamic equilibrium through permanent energy dissipation. Additionally, natural phenomena can be observed at different scales, thus one needs to choose the appropriate scale in order to compare with available experimental data [2]. As already emphasized a comprehensive understanding of biological systems requires a multiscale approach. The distinction between different scales is based on characteristic lengths of objects and times of the phenomena under investigation. First, this requires identify and characterize molecular and cellular components of the considered biological material, then conceptualize the way the components interact with one another. To come to grips with what it really means, we need to get a little bit quantitative about the scales that drive EMB. Figure 4.1a shows the wide range of biological and physical processes at various temporal, spatial, and elastic modulus scales from the subcellular structure to the tissue to the organ. Models developed at the subcellular scale (10–6 –10–9 m) consider the behavior of the physical and biochemical state (i.e. involving genes, cytoskeleton, proteins dynamics and nucleus) of a single cell. This represents a formidable task since the relationships between the morphological details, the biological functions, and the physical modelling are only partially known [3]. At the cell scale (10–4 –10–5 m, one is interested in describing the behavior of

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a large number of interacting cells and molecules. The cell scale is thus strongly connected with the subcellular scale, but modelling at this scale can be performed by largely ignoring the details of single cell models. The macroscopic scale considers tissues (10–3 m) and organs (10–1 m) of the organism. In describing organs, the physical model needs to consider a specific task. Additionally, it is worth observing that these phenomena occur over a broad range of characteristic time scales (Fig. 4.1b). The latter are interlinked to the spatial scales since cellular processes are governed by the dynamics of its various subcellular components, which in turn is affected by the movement of the cells in a tissue. Of specific importance are the experimental tools and methodologies that can provide data to compare with theoretical models. An exemple is provided by the elastic modulus of various biomaterials ranging from cells to organs (Fig. 4.1c). The resistance of single cells to elastic deformation, as quantified by an effective elastic modulus, ranges from from 10 to 103 Pa, orders of magnitude smaller than that of metals and polymers. As described in Chap. 2 the deformablility of cells is largely determined by the cytoskeleton, whose rigidity is influenced by the mechanical and chemical environments including ceel-cell and cell-ECM interactions. The fact that cells make directed decisions regarding how to use energy, i.e., where to direct intracellular components or where to move, suggests that energy is harnessed in specific ways [5]. There are now well identified processes by which thermodynamic energy is converted to biochemical process and mechanical force. A myriad of cellular processes require energy, frequently in the form of ATP [5]. Specific cellular processes carry associated energetic costs which can be quantified by the rate of Gibbs energy change. Cellular energetic costs include biosynthesis, signalling, maintaining chemical gradients, motility, gene regulation and building of cellular structures such as the cytoskeleton [5]. However, a thermodynamic process which occurs in an unstructured environment has no memory of its action. Life must, therefore, have mechanisms to harness the energy of its chemical reactions into functional meaning. These mechanisms are to be found in the structural components of the cell. From a general thermodynamic standpoint, cell membranes are very soft and flexible because their bending elasticity modulus is of the order of kT, where kT is the thermal energy. Hence, membranes exhibit thermal shape fluctuations which can be easily visualized under the optical microscope. There is an ongoing effort suggesting that electromechanical mechanisms like those described below are governed by an intricate interplay between, for example, electrostatic repulsion and attraction forces, mechanical deformations, thermal energy and chemical bonds energies. These mechanisms involve multiple scales in space and in time, but also in energy which reflect the rich interplay between deterministic and stochastic energies that dictate phenomena in nearly all the molecular processes of life. On these scales many fundamental physical concepts have a large impact on biology, among them thermal fluctuations, cooperativity, self-assembly, or elasticity. We observe that membrane strain energy (~5 to 10 kT ) and whole media strain energy (~103 kT ) estimates from our numerical models [6] are well within the range of the physically relevant processes displayed in the energy metrics diagram shown in Fig. 4.2.

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Fig. 4.1 Three panels illustrating the wide range of: a the physical aspects of the cellular environment that are sensed by cells are force and geometry at the nano-to-micrometre level. Higher lengthscales are relevant to tissue shapes and, ultimately, the shape of organs; b A multitude of time scales are emerging to describe the biochemical reponses and are intimately linked to the cellular function; and c Range of values for the elastic modulus of biologica cells and comparisons with those of engineering metals and polymers (adapted from [4])

Fig. 4.2 Illustrating the energy scale in the cellular context [7]. The cell exists on a physical scale that is largely governed by electrostatic interactions, e.g. van der Waal interactions and hydrogen bonding, and entropy

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Thermal fluctuations of the lipid molecules agitate molecules in solution over a broad range of times and distances. Since the population of pores in the membrane is attributed to thermal fluctuations it is also interesting to compare the relevant energies of EMB to kT. The scale kT is important since processes that operate on the scale of kT or below happen spontaneously and reversibly [8]. It is also worthy to note that biological membranes also operate at temperatures very close to some phase transitions in pure lipids. The coexisting phases in PB and nanodomains dramatically alter both the structure and dynamics of the cell PB [9]. From the extensive literature dealing with this subject, reversible EP is an electromechanical phenomenon rather than an electro-thermal phenomenon. Since this phenomenon occurs on a ns to ms time scale in a lipid bilayer cell membrane in the range of 5–10 nm thick, thermal phenomena are negligible, as in 10 ns, heat can diffuse about 120 nm in water (thermal diffusivity about 10–6 m2 s−1 ) and about 40 nm in the membrane (thermal diffusivity of about 2 × 10–7 m2 s−1 ), both of which are large relative to the scale on which heat is being dissipated. Consequently, the heat generated during reversible EP is spread widely, and the temperature rise is negligible (less than 1 K) [10]. This may not be the case for irreversible EP and hyperthermia as ablation technologies which may induce thermal damage to adjacent structures [11]. It has been shown that repeated electric pulse application generates Joule heating leading to a rise of membrane temperature of a few K and proximal temperature gradients in excess of 107 km−1 in the polar membrane region, indicative of the strong possibility of thermo-diffusive intracellular fluid motion and transport, or the enhancement of local electric fields in a strongly polar liquid [12]. Before constructing a computational model, important questions need first to be considered: Do individual cells need to be represented in their complexity, or can their electrical and mechanical behaviors be efficiently captured by a suitable simplified model with minimal mathematical complexity? Do cytoskeletal consituents or other force-generating structures need to be included? Is the behavior fundamentally 3D or is a 2D approximation sufficient? What interactions between cells should be incorporated into the model? Suitable assumptions can bring simplifications but will be quantitatively tested. Broadly speaking, two kinds of approaches that capture and simulate the response of subcellular components, cells and tissues submitted to external stimuli are used. One the one hand, continuum-based models are applicable when the smallest of interest is much larger than the length over which the structure and specific interest of the cell vary (Fig. 4.3). The continuum hypothesis is one of the primary cornerstones of predictive models for a variety of physical applications. Continuum-scale theories rely on conservation principles to provide tools for predicting the spatio-temporal evolution of fields, e.g. electric field and stress, and other salar parameters, e.g. voltage. This approach will be the focus of this chapter. On the other hand, when the length scale of interest is comparable to the structural featues of the system under study, it becomes necessary to explore dynamics at the molecular-scales for which this hypothesis requires refinement. A prime example of this is the modelling of pore dynamics in electropermeabilized membranes. For example, continuum approaches cannot describe

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Fig. 4.3 Schematic diagram of the spectrum of length scales usually encountered in EMB. MD (micro/nanostructured) modelling is used in understanding structure and interaction at very small length scales, whereas continuum modelling is used at larger scales (adapted from [13])

molecular scale microscopic processes, such as the molecular and small ions transport across the pores in membrane, which involve understanding the complex nature of the membrane morphology. In contrast to molecular-level models, the existence of electropores is a basic assumption of continuum-based models, e.g. [14], and not a result of numerical simulation. As a result, the dynamics of the pore should be addressed at the molecular level. MD modelling is a direct solution of Newton’s laws for individual molecules and shows excellent agreement to experiments, reproducing the underlying liquid structure, equilibrium thermodynamic properties, flow dynamics and diffusion; a priori prediction of dynamic coefficients such as viscosity and surface tension, as well as fluid dynamics for canonical flows. Naturally, MD is therefore an ideal tool to study the dynamics of the pores in electropermeabilized membrane. Yet, complex phenomena in this region of a membrane can involve physics at larger length- and time scales than is practicable to model atomistically. Coupling MD and continuum approaches, either directly as part of the same simulation or indirectly by parameterizing equations, represents a great challenge for defining a multiscale analysis of ED and EP. Overall, the fundamental aspects of ED and EP of cell membrane are governed by the coupling phenomena between electrostatic forces, mechanical strain, thermodynamical fluctuations and molecular process energies. Current efforts towards the understanding of these phenomena which are controlled by multiple spatial and temporal scales imply a multiphysics analysis to describe the cooperative and self-assembly aspects.

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4.2 ED and EP Modelling The long history of scientific and technological interest in EP and ED has generated an enormous phenomenology presented partly in the approximate electric pulse strength-duration space diagram shown in Fig. 4.4, which involves at least seven orders of magnitude in duration and three orders of magnitude in strength. This diagram provides a useful guide to experimenters since the electric field strength and the pulse duration are the two most important parameters chosen to elaborate EP protocols.

Fig. 4.4 Schematic diagram of the electric pulse strength-duration space with important examples of electric field effects in biological systems ranging from ED, including reversible or irreversible EP and membrane disruption. Typically, when a biological cell is subjected to an electric pulse of magnitude above a few kVcm−1 and duration in the range of μs-to-ms numerous hydrophilic pores are formed in the membrane which becomes permeable (hatched region). Strong electric pulses of short duration induce electric breakdown in membrane integrity. Breakdown leads to formation of transient pores occurring on the order of ps-to-ns. In the minutes to hours following EP, a porous membrane reseals to again inhibit molecular transport. Pioneering experimental studies with intracellular effects by several tens of kVcm−1 and sub-microsecond pulses are provided in Ref. [15]. For reversible EP, temporary and limited pathways for molecular transport are formed, but at the end of the electric pulse, the transport ceases, and the cell remain viable. At very large pulse duration, the reversible breakdown turned into an irreversible mechanical breakdown, associated with the destruction of the membrane, i.e. the cell cannot regain its homeostasis after EP, leading eventually to cell apoptosis. In that case, pores do not reseal [16]

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Reversible EP is widely applied to delivery of variety of molecules into cells or tissues, such as fluorescent dyes, DNA, RNA, proteins, peptides, drugs, and nanoparticles, in a controlled fashion. The combination of short high-voltage and long lowvoltage pulses can enhance EP efficiency while maintain cell viability [15, 16]. It is suggested that high-voltage pulses contribute to electropermeabilization, while lowvoltage pulses provide electrophoretic force to drag DNA towards the cell membrane and/or insert it into the cells. Electric pulses when sufficient short cause higher power dissipation in the membrane with respect to the ECM [15]. DNA molecules need to overcome multiple obstacles including extracellular matrix, cell membrane, cytoplasm (especially cytoskeletal network), NE and even cell wall (for bacteria, algae, fungi, and plant cells) to achieve ultimate gene expression. Irreversible EP occurs under intensive electric parameters and typically leads to cell death after the procedure. It has been suggested that irreversible EP is likely caused either by generation of permanent pores in a portion of membrane region after extensive electropermeabilization or by chemical stress resulting from molecular transport through transient pores. That’s a fine sketch, as far as it goes. It’s coherent, largely taken from experimental phenomenology. But, a closer look at Fig. 4.4 shows that ED is not explicitly considered in this diagram. It should be emphasized that this phenomenon eventually impacts the threshold values displayed here since it is widely recognized that membrane cells can react and adapt to electromechanical stresses [17]. However, the absence of an in-depth understanding of the physical mechanisms by which the MC and MR vary as V m is increased renders the identification of the physical parameters and behaviors that control the EP of a deformable membrane speculative. To go further, we need to consider the different cell components shown in Fig. 1.1, and their electromechanical couplings, which vary with the type of cell exposed and ECM property. Do they fill in our sketch? In particular, do they bring out remaining gaps? The answers, broadly speaking, are yes. Overall, our sketch will grow much stronger and more coherent. However, resolving it requires considerations of another order. Up to this point, I have been able to be rather vague about what actually happens at the membrane scale. We’ve seen that the … But it has not been necessary to speculate about the dynamics of pore in the cell membranes. Now, we want to go into more details on the calculation which guide us to … The main ideas have been in place—ever promising, but mostly unfulfilled—for 4 decades or more.

4.2.1 ED and Transmembrane Voltage-Dependence of Membrane Capacitance An exposure of a biological cell to an external electric field with sufficient amplitude undergoes ED. Prior work has focused on the ITV dependence of C m where we need to account for the fact that charges on a capacitor generate mechanical forces, i.e. Maxwell stress tensor [18–21]. A nice synopsis of the field of electromechanics is

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discussed in Ref. [22]. By way of consequence, these forces can change the dimensions of the capacitor, e.g. d m decreases, and the capacitance C m increases. Hence, we expect a non-linear behavior of the MC as V m is increased. Classical EL theory allows us to provide a crude estimate of the magnitude of the force on a membrane which is a quadratic function of voltage when the membrane is polarized. Assuming a membrane with constant area and small thickness change, Mosgaard and co-workers [23] suggested that the capacitance is proportional to the force and can be described as a quadratic function of voltage. Based on experimental data on artificial bilayer membrane, Alvarez and Latorre [24] demonstrated that the dependence of MC on V m can be written as Cm = Cm0 + a' (Vm − Vrest )2 ,

(4.1)

( ) where Cm0 = Cm Vm = Vrest and a' denotes a fitting( parameter. ) Heimburg suggested that the change in C m can be expressed as Cm = Cm0 1 + χ Vm2 , where χ is a constant (close to 10–2 V−2 ) and should be attributed to EL, i.e. an increase in Am and a decrease in d m [25]. Multiple RC configurations in the time domain have been also proposed, see e.g. Refs. [26–28], for describing the pre-transitional state of ED of the membrane underlying the EP phenomenon. The majority of the current literature on RC models is focused on characterizing ionic currents through different porosity stages and does not consider relevant spatial scales, e.g. d m . An illustrating example is shown in Fig. 4.5 where the authors developed a model that describes cellular electroviscoelastic membrane based on thermodynamic principles [26]. Within this analysis, it should be realized that the presumption of pure capacitive membrane is relaxed, but the MC and MR are constant. These ideas have motivated a lively collection of RC circuit equivalent models. Further details and a discussion of these efforts can be found in Refs. [25, 26]. Several

Fig. 4.5 RC circuit equivalent model of the ED and the electromechanical coupling in a cell membrane. The left panel represents the electrical circuit where V, R, C, q denote respectively the membrane voltage, MR, MC, and the rate of the transferred charge (current).The right panel represents the mechanical part of the model, where ε˙ p , V mech , N and T stand for the rate of passive strain, mechanical component of the voltage, membrane resultant and the transformer ratio [26]

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authors, e.g. Refs. [29, 30], have also argued that the increase of MC is related to the sphere-to-spheroid transformation, i.e. surface expansion of the membrane of an initially spherical cell exposed to the electric excitation as V m is increased. Notice that this statement is completely consistent with experimental observations that have equilibrium, the degree been made in many places, e.g. Ref. [31]. In mechanical / of stretching can be estimated by the aspect ratio b a, where b and a are the two principal semiaxes along and perpendicular to the electric field direction, respectively. A prolate spheroid (of revolution) has surface area defined 2π a2 + 2π ab sin−1 (e)/e, √ 2 with e = b − a2 /b. The maximum deformation corresponds to about 10% in terms of aspect ratio [17, 29] and is expected to depend on the initial surface tension of the membrane and the excess area of an initially spherical cell of the same volume, / ( / ) / δAm Am0 ∼ = ln Γ Γ0 kT 8π K,

(4.2)

where Am0 is the initial surface area for which the tension is Γ0 , Γ is the membrane tension when the cell is exposed to the electric field, K is the membrane bending stiffness (K ∼ = 102 kT ), and kT is the scale factor for energy. Needham and Hochmuth considered that V m induces a Maxwell (electrocompressive) stress tensor on the membrane which manifests as an additional membrane tension (MT) which is proportional to Vm2 [30]. MT relates to cellular phenomena that induce membrane shape changes at constant volume resulting in a molecular scale shifting of membrane lipid equilibrium configuration giving rise to tension within the membrane that cannot exceed the lysis tension (Fig. 4.1). But although there has been a good deal of attention directed towards probing different RC circuit equivalent models to analyze ED, relatively little exploration has been dedicated to studying EP because of the no-scale structure of these models. Furthermore, the possibility of such electrical circuit equivalent models for describing the membrane ED at large values of V m , the membrane EP and the collective behavior of cell clusters (tissues) raises a group of open questions, e.g. do these models produce representative electrical waveforms that can be tested by comparison with experiments?

4.2.2 Macroscopic Models of the Cell EP Energetics and Dynamics We turn now to presenting the prevalent aqueous pore model for membrane permeabilization of a single cell which arises from a minimum energy principle. Even if this model allows for increased permeability across the membrane, recent studies suggest that chemical and structural changes in the lipids and proteins of the membrane can also contribute to the EP mechanism [31, …]. Beyond V EP , V m cannot be further increased, and even can decrease due to the charge transport into the membrane [68]. When V m exceeds a threshold V EP , nanoscale electropores are created within

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the membrane through the localized rearrangements of the PB, leading to a rapid membrane discharge and a decrease in MR. As noted previously, the V EP value is typically several hundred mV, irrespective of cell types […,31,…]. Broadly speaking, the process of pore formation from the intact membrane [Fig. 4.6a(i)] can be divided into two steps which are schematically illustrated in Fig. 4.6 [32]. This figure illustrates such a model for which the kinetics of pore formation is described by the transition over an energy barrier created by the intersection of two different pore configurations. On the one hand, a hydrophobic pore where the lipid phase is broken up, and, on the other hand, a hydrophilic toroidal pore (Fig. 4.6a). For small radii, the hydrophilic pores are confined within a local energy minimum while for large radii, there exists a local maximum beyond which a pore may grow indefinitely. The free energy of the hydrophilic pore is changed upon application of ITV such that its free energy, along with this local maximum, is decreased (Fig. 4.6b). At some critical potential, the local maximum restraining pore expansion vanishes and the pore grows until the bilayer is destroyed [33].

(a)

(b)

Fig. 4.6 a Sequence of lipid bilayer rearrangements (not to scale) leading from (i) the unperturbed membrane (stable barrier exhibiting permeability only to select certain small uncharged and/or hydrophobic molecules), to (ii) formation of a non-conducting hydrophobic pore, that is characterized by the protusion of a thin water needle in the membrane core, after exposure to an intense electric field, to (iii) stabilization of the membrane after reorientation of lipid head groups and creation of a hydrophilic pore, now allowing for passage of previously impermeable molecules. In that case, the lipid head groups are required to be spaced further apart. b Illustrating the energy landscape of pore states leading to hydrophilic pores within the transient aqueous pore model of EP. The dashed (resp. solid) line represent the hydrophobic (resp. hydrophilic) pore free energy. The star symbol denotes the hydrophobic-hydrophilic transition. The free energy of induced pores decreases with increasing ITV beyond a critical radius. The numbers denote the ITV values. Reproduced with permission from [33]

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In the state (i) the PB serves a s a semipermeable barrier that separates the cytosol from ECM, only allowing diffusion of certain small uncharged and/or hydrophobic molecules. Large or charged molecules are transported through discrete transmembrane proteins organized into channels and pumps. Although structurally stable, the lipid molecules in the PB are held together by van der Waals forces, creating a fluidlike structure in which each molecule is constantly moving within the PB. Several comments are in order. Firstly, hydrophobic pores [Fig. 4.6a(ii)] are formed due to lateral thermal fluctuations of lipid molecules because the pores provide lower energy states than the non-perturbed PB. Exogeneous electric fields lower the activation energy necessary for the stochastic pore formation process. Secondly, as the pores expand to a critical radius, the lipid headgroups reorient to shield the aqueous defect from the hydrophobic membrane core, thereby forming hydrophilic pores lined by the polar heads of phospholipids in order to minimize the energy of the membrane [Fig. 4.6a(iii)]. There have been several attempts to address the question of metastability of the hydrophilic pores. By MD simulations, it was argued that it is due to the compensating of positive and negative curvature effects at the edge of a pore [34]. Likewise, MD simulations predict that hydrophobic pores are less than 1 nm in diameter and reseal within ms, whereas hydrophilic pores are roughly 1–10 nm in diameter and reseal with mn to hours [35, 36]. Thirdly, the PB components of the cell membrane are largely heterogeneous. Thus, it is expected that the nature of the lipids will impact the formation of the pores. For instance, lipids having a smaller volume of nonpolar components are expected to pack more redily into the curved region of the pore [Fig. 4.6a(iii)] since that would reduce the exposure of the hydrophobic region of the structure of the aqueous medium and concomitantly decrease the surface free energy in the curved surface of the pore. A contrario, we observe that cholesterol molecules in the cell membrane, which have small polar heads, exert an inhibiting effect on the formation of pores and will decrease the electrical conductance of the membrane [37]. Fourthly, MD simulations of membranes at various electric field strengths E(have suggested / ) that the pore formation time follows a simple exponential, ∝ exp −ΔμE kT , with the activation dipole moment upon pore formation Δμ [36]. This implies that pores with similar characteristics to electropores appear as well at vanishing electric fields. Now we can ask why EP can be considered as a threshold-like phenomenon [29, 38]. That means that EP takes place when V m exceeds a threshold V EP above which electrically conductive (hydrophilic) nanopores start forming in the membrane. The literature suggests several ways to deal with this issue and also provides estimates of V EP. Such estimates do suggest that dynamically formed pores in membrane could contribute significantly to observable events. The pores are primarily formed where the local electric field is maximal and expand in size if the electric field is sustained until critical pore size and number are attained [39]. Observe that many authors argue there is another threshold value of the electric field that should not be crossed if membrane recovery and resealing should occur; otherwise irreversible EP is operative and the cell loses its viability. Some work has been done on addressing this concern, see in particular Refs. [12, 31, 40].

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Reversible EP occurs when the MT reaches a critical magnitude. To address this issue, Needham and Hochmuth [30] argued on the basis of simple back-onthe-envelope calculation of the storage of elastic energy in a flat incompressible membrane at equilibrium as a result of work done at the boundaries of the system by either tensile and electrocompressive stress, that the tension in the membrane takes the form Γ = AδAm /Am0 − C˜ m Vm2 /2,

(4.3)

where A is a specific constant and δAm /Am0 denotes the relative increase in surface membrane area. Then, a general failure criterion of a membrane is suggested when Γ = 0, i.e. / Vmc =

2AδAm /Am0 C˜ m ,

(4.4)

providing a critical membrane voltage required for breakdown. Needham and Hochmuth [30] conducted a series of electromechanical experiments for three kinds of membrane systems (artificial lipid bilayers) and showed that this relation is verified experimentally for 50-μs pulses. The tricky thing is that we do not know a priori if this analysis gives us the correct EP condition, i.e. VEP = Vmc . An important parameter the researchers can tune in addition to the voltage strength is its pulse duration. Akinlaja and Sachs [40] attempted to determine whether mechanical tension and electric stress couple too cause breakdown in cells. In their experimental work dealing with HEK293 cells voltage pulses of increasing amplitude were applied until the authors observed a simultaneous sudden decrease of Rm and increase of C m . Complicating the interpretation, they found that the mechanisms of high field/short pulse and low field/long pulse breakdown are fundamentally different: for pulses of 50 μs duration, breakdown required > 0.5 V and was dependent of the tension whereas for pulses of 50–100 ms duration, breakdown required 0.2–0.4 V and was independent of tension. Reference [40] also identified that the critical energy (voltage) underlying the EP criterion cannot be compared with experimental data without a model that incorporates time. When Vm = VEP , typically in the range from 0.2 to 0.6 V when V m exists for times of about 100 μs or longer (Fig. 4.4), then electrical breakdown and destructive rupture of the membrane manifest as a very rapid drop in Rm . This can be explained by the large population of transient pores in the membrane. Having identified the critical nature of EP, we now attempt to gain further insight in understanding the physical framework that aims to capture the electromechanical behavior of membranes and link this to creation of pores in the membranes by the transition over an energy barrier due to thermal fluctuations of the lipid molecules as V m is increased [33]. Furthermore, under electric field stimulation, it is reasonable to predict that electropores appear and disappear in the membrane in a stochastic manner and exhibit fluctuating size. The transient nature of the electropores places significant limitations

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on the characterization of their dynamics. Recently Ref. [33] provided an improvement in our experimental understanding of electroporation by imaging individual voltage-induced defects in a bilayer membrane by detecting a fluorescent signal proportional to the flux of Ca2+ flowing through a pore. Using optical single-channel recording, Sengel and Wallace [33] were able to track multiple isolated electropores in real time in planar droplet interface bilayers (formed from the contact between an aqueous droplet and a hydrogel surface immersed in a phospholipid/ oil solution). They observed individual mobile pores that fluctuate in size and with a range of dynamic behaviors revealing complexity in their interaction and energetics. Furthermore, their observation is consistent with the fact that the potential well that supports hydrophilic pores (Fig. 4.6) widens and shifts to larger radii when ITV is increased. Stabilization of hydrophilic pores can be also realized by heterostructures contained in the membrane. We furthermore observe that due to the highly unfavorable line tension of the pore rim, membrane pores are generally not expected to be energetically stabilized. Fošnariˇc and coworkers described a theoretical model that predicts the existence of stable pores in a lipid membrane, induced by the presence of anisotropic inclusions [41]. Furthermore, they argue that the optimal pore size is governed by the shape of the anisotropic inclusions, i.e. saddle-like inclusions favor small pores, whereas more wedgelike inclusions give rise to larger pore sizes. Experimental values of the mean times of pore opening across black lipid membranes of diphytanoylphosphocholine at voltages between 250 and 550 mV have been reported in Ref. [40]. After pore formation, ECM molecules enter the cell electrokinetically and diffusively. Cross membrane transport refers to ion or molecule transport across the electroporated membrane through aqueous pores. Based on minimization of the membrane energy, Barnett and Weaver [42] proposed an asymptotic Smoluchowski model to describe the creation, destruction and evolution of cylindrical pores, which was later modified by Neu and Krassowska [43]. On the basis of coarse-grained descriptions of the membrane, several groups concluded that cell EP can be described by the formation of hydrophilic pores in the membrane, see e.g. Refs. [44, 45], resulting in a significant increase of σm and transmembrane current density, namely jm (t) = σm (Vm + Vrest )/dm + C˜ m V˙ m (t) + jEP (t),

(4.5)

where the first term on the right-hand side of Eq. (4.5) is the leakage current density across the intact membrane, the second term is the displacement current density, and the third term jEP denotes the ionic current density flowing through the pores at a specific location [46]. An expression of jEP based on the Nernst-Planck equation in a geometry encompassing an isolated electropore and for a binary electrolyte solution was suggested as a function of V m [42, 46–49] jEP = N σ˜ Vm

2π r 2 , π r + 2dm

(4.6)

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where N is the local pore density and σ˜ provides an approximate of the effective conductivity of the aqueous solution filling the pores, assuming that pore current is predominantly Ohmic owing to large membrane thickness compared to the pore radius [46, 48, 50]. Within this approach, the electrical conductance of the pores depends on the concentration of ions and dominates the overall conductance of the membrane since σ˜ ≫ σm . It is difficult to calculate the precise form of σ˜ from first principles. However, it has been suggested that σ˜ = (σs (r) − σc )/ ln(σs (r)/σc ), where denotes the soft layer conductivity at the soft-layer lipid bilayer interface (which generally varies with position r on the surface), i.e. σs (r) = z 2 (Ʌ+ c+ (r) + Ʌ− c− (r)), with z being the valence of the positive/negative ions (assuming a symmetric electrolyte), c± is the concentration of positive/negative ions, and Ʌ± is the molar conductivity of positive/negative ions [26, 50, 51]. Based on an estimation of the average conductance of a single pore, the total number of pores required to account for the total measured conductance can be calculated. One finds typically 1010 pores/m2 . Thus, the nanometric pores completely dominate the electrical conductance of the PB membrane but they occupy only 2 × 10–7 of the total area [17, 51]. The value of V m governs the dynamics of the pore density per area as a function of their radius and time t )[ )] ( ( N˙ (t) = αexp (Vm /VEP )2 1 − (N /N0 )exp −q(Vm /VEP )2 ,

(4.7)

where N 0 is the pore density in the unelectroporated membrane (V m = 0 V), and α and q are two parameters encoding the EP process: α is the average pore creation rate (109 m−2 s−1 [14, 31, 47]) and q is an EP constant (=2.46 [14, 31, 47]). Note that at t = 0, N = 0, whereas the first derivative of N is not zero, but α. This equation can be solved at each point on the membrane, and at any time the total number of pores on the cell membrane is given by the surface integral of the pore density, rounded to the nearest integer. Since this model relies on the exponential of the squared ITV it requires small time steps to resolve in numerical simulations. For derivations of this expression and comprehensive discussions of the underlying theory, the reader is referred to Refs. [48, 52]. Within this model, one assumes that all identical noninteracting pores have reached their equilibrium size before the end of the electric pulse for μs EP since the pulse length is much larger than the characteristic timescale of pore evolution [42, 53]. There has been much effort directed to develop stochastic energy-based EP models [54] in order to predict the EP activation energy, V EP , and the equilibrium pore size. Firstly, consider the membrane as an infinitely thin film without internal structure subjected to external lateral tension Γ , i.e. this would correspond to the situation for which V m = V ext . The energy of a cylindrically symmetric pore with radius r can be written using Derjaguin’s equation [55], quantifying how membrane tension Γ promotes rupture by pore formation and expansion while pore edge energy density γ opposes growth. Thus, with reference to Fig. 4.6 (iii), the work required to create a circular pore in the cell membrane with radius r, at V m = 0 and at constant surface tension Γ , is given by [56]

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U (r) = 2π γ r − π Γ r 2 .

(4.8)

The first term in Eq. (4.7) is the energy cost of having an edge of perimeter 2π r (the circumference of the circular pore), exposed to the surrounding medium, and γ is the edge energy density of the pore (γ ∼ = 10−11 Jm−1 [57]). The second term contains the hydrophilic surface tension (lipid heads/water leaflet), typically Γ ∼ = 10−3 Jm−2 [54], and is the gain in energy achieved by reducing the membrane area by an amount π r 2 . Consequently, the pore formation involves a balance between two forces. The system energy has a maximum at ) radius rc = γ /Γ defining ( critical πγ 2 the activation energy barrier to pore formation Γ . Such a thermodynamically ( 2) is accessible circular pore can be created only if the critical energy barrier πγ Γ by thermal fluctuations. According to Eq. (4.8), pores with r < r c are reversible and tend to close, whereas those with r > r c grow indefinitely, and would eventually cause membrane breakdown. This prediction has been found to be in conflict with many experiments because pores larger than r c have been observed to reseal [56]. In addition, assuming a linear relation between the reduction in elastic energy and the area of the pore, as written in the second term in Eq. (4.8), applies to small pores only. In particular, it ignores nucleation of a hydrophilic transmembrane pore that involves a rearrangement of lipid molecules from an orientation parallel to membrane normal to perpendicular. Alternative models can be obtained by assuming other forms of the free energy dependence on the pore area. Consider that a source current proportional to the exogenous electric field drives an increase in V m beyond V EP, thus membrane defects begin to occur. The creation of pores in the membrane then alters the stored energy of the membrane capacitor by modifying the permittivity of portions of the capacitor from that of the lipids to that of water. If the pore, is assumed to be of cylindrical shape with radius r (> r*, the smallest radius of hydrophilic pores, typically 0.5 nm [43, 49, 58, 59]), small compared to the Debye length, the change in free energy energy U(r, V m ) due to thermal fluctuations of the lipid molecules, reads U (r, Vm ) = −

π (εe − εm )Vm2 r 2 + 2π γ r − π Γ r 2 , 2dm

(4.9)

where again the first term accounts for the drop in the electrostatic energy stored in the membrane induced by pore formation; the second, for the line tension acting on the circumference of the pore; and the third, for the surface tension of the membrane. The first term serves to lower the barrier to pore formation significantly under electrical tension. In like fashion as above, this free energy form leads to a critical pore size at which the energy reaches an activation energy [42, 46, 53]. In this case, the system energy has a maximum at critical radius ( ) rc = γ / +C˜ m Vm2 /2 .

(4.10)

4.2 ED and EP Modelling

117

Within this approach, the rate of change of the pore radius is determined by D ∂U (r) , where D is the diffusion coefficient of pore radius [45, 54], i.e. a pore r˙ = − kT ∂r must be thermally activated to overcome the nucleation activation energy. The pores expand in size with time if the electric field is sustained until an unstable pore size is reached, e.g. up to 50 nm [39, 60]. This model predicts that the electric pulse produces more but smaller pores on the hyperpolarized cell hemisphere and fewer but larger pores on the depolarized hemisphere. Two more comments are in order. Firstly, note that the expression of the free energy found in [49] for the case of hydrophobic state (r < r*) reads as ( / ) I1 r l π U (r, Vm ) = 2π Γ dm r ( / ) − (εe − εm )Vm2 r 2 , 2dm I0 r l '

(4.11)

Where ┌ ’ is the hydrophobic surface tension (lipid tails/water leaflet), l is a characteristic length of hydrophobic intercations (l ≈ 10−9 m [49, 61]), and I n denotes the nth order modified Bessel function of the first kind [62] Secondly, Neu and Krassowska suggested a correction ) the pore size is within the subnanometer ( for γ/when range by changing γ → γ 1 + C r 5 , where C = 1.39 × 10–46 Jm−4 [43]. We also note an additional feature of Eqs. (4.9) and (4.11) by plotting in Fig. 4.6b the energy profile of pore states leading to hydrophilic pores, assuming typical values for the physical parameters appearing in Eqs. (4.9) and (4.11) for each value of V m we consider. At this point, it is worth noting that Deng and coworkers [63] considered a different form of the free energy of the cell membrane by introducing a supplementary strain energy term U s (r) due to the large deformation of the membrane during EP (r > r*), U (r, Vm ) = −

π (εe − εm )Vm2 r 2 + 2π γ r − π Γ r 2 + Us (r). 2dm

(4.12)

These authors showed that it is this strain energy term which provides resistance to the pore growth and eventually stabilizes the pore size. Furthermore, the strain ∑ energy form they used is expressed with a six-order polynomial function Us (r) = 6j=0 aj r j , where the coefficients aj s are determined phenomenologically. Tests performed on chicken red blood cells (R = 10 μm and d m = 10 nm) determined EP pore nucleation activation energy of the order of 10 kT, equilibrium size of 3.5 nm and V EP = 0.82 V. Kroeger and coworkers [64] have also investigated a curvature-driven pore dynamics model and showed that the aqueous viscosity of the extracellular medium can impact pore dynamics. Recently, more complicated pore conductance models [51] has been developed based on the electrokinetics-based-Poisson-NernstPlanck formalism considering a toroidal pore shape, pore selectivity for different ionic species and the electric double layer at the membrane-extracellular medium interface. The models presented above allow us to study both temporal and spatial aspects of EP. It is also desirable to have simple circuitry that manages charge, operating in

118

4 Analytical Approaches of EMB at Multiple Scales

synchrony with the mechanical strain that is being applied to the membrane. Analysis based on lumped parameter RC models has been also extensively studied in the literature [18, 65–68]. An illustrating example has been recently reported by Sweeney and coworkers [65] to cope with the molecular transport into a single idealized spherical cell immersed in an aqueous buffer. This model (Fig. 4.7) includes a cell membrane circuit model (reversible primary process) coupled with a phenomenological dualporosity model and simple diffusion (irreversible secondary process). During the former, a source current I s drives an increase of V m (noted U in Fig. 4.7), with formation of pores allowing ions to flow across the membrane. Ionic currents slow the V m increase until a dynamic equilibrium is reached. In the latter, the transport of solute is considered from a high extracellular concentration into a cell initially containing no solute. A nonlinear size-dependent equivalent circuit (Fig. 4.8) model of a single-cell EP was presented by Shagoshtasbi and coworkers [69]. Based on the Kirchhoff laws and continuity equations, this method allowed the authors to get the time response of V m , pore size and pore number at different stages of permeabilization. This analysis relies on an earlier strain energy model that considers the change in membrane strain energy due to pore formation (rather than due to a change in membrane tension). The

Fig. 4.7 a Electrical circuit model of the cell membrane charging: σm is given by the parallel resistance of the naïve membrane (blue, left) and porous membrane weighted by the membrane fraction in each of 2 stages (green and magenta, right), N is the fraction of the membrane area that is conductive of small / ions and M is the fraction of the membrane area permissive of the entry of larger solutes, U = Vm VEP , α, β, δ and η are constants; b The naïve membrane contributes a conductivity and permittivity to the electrical model (blue, left). The N membrane fraction contributes to the permeabilized conductivity of ionic currents but does not permit the transport of larger solutes (green). The M membrane fraction permits diffusive transport of solute (magenta) (Reproduced with permission from [13])

4.2 ED and EP Modelling

119

Fig. 4.8 Equivalent circuit analog describing the nonlinear electromechanical EP model consists of two main electromechanical parts. The dashed-line rectangular box includes the components of cell: C m , series Rs , shunt Rsh , and cytoplasm Rc resistances are functions of the cell size r c , Rm is a function of N and r. The equivalent circuit for the pore dynamics is illustrated on the right-hand side. The two controlled voltage sources are the corresponding energy source for the applied voltage and the surface tension 2π Γ r. The line tension 2π γ is represented by a constant load; the nonlinear spring (k s (r)r) for the membrane elasticity is represented by a nonlinear capacitance. The energy dissipative element (kT /D) is represented by a resistance. The total current passing through the cell membrane I m is part of the total EP current I i (Reproduced with permission from [69])

model was designed under the tacit assumption of a uniformly polarized membrane exposed to a pulse shaped electrical stimulus and includes an assumption of nonlinear elasticity in the membrane. Although the main feature of this model is the coupling of the electrical circuit with the mechanical process of EP, it does have several limitations, e.g. this is partly due to the fact that no direct assessment of V EP can be obtained from this model. Several models describing EP using a continuous Smoluchowski equation, which governs the distribution of pore density as a function of pore radius and dynamics, have been suggested [70, 71] ) ( ∂ ∂n(r, t) n(r, t) ∂n(r, t) =D + W (r, Vm ) , ∂t ∂r ∂r kT

(4.13)

where n(r, t)dr is the number of membrane pores per unit area between r and r + dr at time t [70], D is the diffusion coefficient of the pores in the pore radius space, m) , where again U(r, V m ) represents the shift in and the force W (r, Vm ) = ∂U (r,V ∂r membrane free energy due to the presence of a pore in the membrane, e.g. Equation (4.9). The implementation of the asymptotic EP model in terms of an equivalent circuit is described in [72]. The next aspect of cell EP analysis to be considered is a macroscopic model, introduced in a series of papers by the group of Krassowska [14, 43, 48, 50, 53], which provides a means for investigating the mechanisms and effects of EP. Despite its success, the Smith, Neu and Krassowska (SNK) model can only be considered as an effective theory description of EP which leaves many unanswered questions about the nature of reality at distance scales shorter than ≈ nm, i.e. pore nucleation mechanism [43]. From the perspective of today’s understanding

120

4 Analytical Approaches of EMB at Multiple Scales

of electropermeabilized lipid vesicle membranes the approach of SNK [43] attempts to predict the expansion of macropores based on the integration of the MST over the internal surface delimiting a pore in the membrane. One question these authors addressed was how the process of evolution of macropores proceeds during and after the electric pulse excitation. When V m exceeds the EP threshold, homogeneous nucleation of membrane defects begins to occur by thermal fluctuations of the lipid molecules and energy does redistribute to maintain consistent boundary conditions through a perturbation to the electric field [58]. It is important to notice that Eq. (4.13) has no analytical solution, and therefore the density distribution of pores in the membrane must be solved numerically [43, 54, 57]. Standard assumptions = 0) is negligible for physiologically relevant include that the diffusion terms (i.e. ∂n ∂r conditions and the characteristic timescale of U is longer than 0.1 μs. In addition, the large computational time required for solving Eq. (4.1) on tractable space and time scales constitute serious drawbacks [43]. To tackle these issues, Neu and Krassowska conducted a thorough analysis in [43] and proposed an asymptotic model to the one presented above, which simplifies Eq. (4.13) in Eq. (4.7) and D dr = − W (r, Vm ). dt kT

(4.14)

The expression of the pore energy U (r, Vm ) for conducting pores (r > 0.5 nm) has been extensively debated in the literature, with particular emphasis on the definition of the energy term related to the induced transmembrane potential Vm [43, 57, 58, 70]. From the perspective of today’s understanding, the approach of SNK, which consisted in defining the Vm related energy term from the integration of the electric stresses (MST) over the internal boundaries of a toroidal pore [15], has been by far the most reemployed numerical model for cell membrane EP simulation over the past two decades [17, 47, 50, 51, 73–80]. An important remark here is that imbalances in the conductivities of media surrounding the membrane were neglected, i.e., only the case Λ = 1 was investigated in the SNK model [43], while the vast majority of membrane EP simulation studies have focused their efforts on modeling the response of systems showing conductivity imbalances of one order of magnitude between the media surrounding the membrane [17, 50, 53, 73–80]. In the SNK model, W (r, Vm ) is defined for r* < r < d m , as W (r, Vm ) =

−Vm2 fSNK (r)

( ) 4β r ∗ 4 + 2π γ − 2πeff r, − r r

(4.15)

and for r < r* as, W (r, Vm ) =

−Vm2 fSNK (r)

+U

∗

(

r∗ r

)−2

.

(4.16)

The first term of Eq. (4.15) accounts for the electric force induced by the local value of Vm ; the second represents the steric repulsion between lipid heads lining the

4.2 ED and EP Modelling

121

pore, the third, for the edge tension opposing the expansion of the circumference of the pore, and the fourth introduces the surface tension of the membrane [43, 53]. The second term was introduced to describe the formation of hydrophilic pores beyond a critical radius r* (r* = 1 nm in Fig. 4.6 and Table 4.1). Here,fSNK (r) represents a force per unit square voltage [43], β denotes the steric repulsion energy, and finally, 2σ ' − σ0 Γeff = 2σ ' − ( )2 , 1 − Ap /A

(4.17)

is the effective tension of the porated membrane felt by the pore of radius r[53], A is the surface area of the total membrane and the total pore area Ap = ∫where ∫ ⃝ N π r 2 dS with S denoting the membrane surface, and where Γ0 is the surface S

tension of the membrane without pores, Γ ' is the tension of the hydrocarbon-water interface (Table 4.1). In Eq. (4.16), U * is the critical energy at the hydrophobichydrophilic transition (Table 4.1). Figure 4.6 illustrates the behavior of U (r, Vm ) as a function of r for different values of V m . Substituting Eq. (4.15) into Eq. (4.14) and simplifying with a number of assumptions [43] leads to Eq. (4.7) in which α and q are fitting parameters. In relation with the above equations describing pore dynamics, more sophisticated pore conductance models [51] have been developed by making use of MD simulations or a Poisson-Nernst-Planck equation. In another direction, the Smoluchowski equation, i.e. Equation (4.13), can be supplemented by a Langevin equation describing the curvature-driven growth of pores in charged membranes [54, 64]. Whereas pores grow irreversibly and rupture the membrane for intermediate values of voltage, pores grow and reseal reversibly for low and high voltage values. The dominant term in Eq. (4.15) originates from the electrical force acting on a hydrophilic (conductive) pore which is induced by the local transmembrane potential. This is understood intuitively as the application of V m changes the free energy of the hydrophilic pore such that its free energy is reduced [81]. In the SNK model [43], the first term of Eq. (4.15) takes the form fSNK (r) = Fmax

r + r1 . r + r2

(4.18)

This expression was obtained by SNK as a phenomenological approximation of their numerical computations when Λ is set to 1, F max is the maximal radial force per unit squared volt, and r 1 and r 2 are two constants [43]. In [81], Sabri and coworkers derived a Λ dependent extension of the SNK model, also based on a similar approximation of our numerical results which has a different analytical expression, namely (

) 1 χ , fɅ = Fmax (r + r1 ) − r + r2 r + r3

(4.19)

0

∫∞ [

] exp(−iωt)dt − d φ(t) dt 0

∫∞

−∞

∫ +∞

τ τ0 −τ

, τ < τ0

)

/

1 π

( ) exp − 4τπ0 , for β =

1 2

exp(−x) exp(−u cos(π λ)) sin(u sin(π λ))dx τ 4π τ0

0

∫∞

)(1−β)/ 2

(ωτ0 )1−α cos(π α / 2) 1+(ωτ0 )1−α sin(π α / 2)

1+2(τ / τ0 )1−α sin(π α / 2)+(τ / τ0 )2(1−α)

sin((1−β)ϕ3 )/ π

)1−β

| ( / )| = A |ln τ τ0 | ≤ Ξ | ( / )| = 0 |ln τ τ0 | > Ξ [ [ ( / )]2 ] √B exp −B2 ln τ τ0 π

(

0, τ > τ0

(

g[ln(τ )]d [ln(τ )] = 1, A and B are constants

( ) exp(−x) sin ωτ0 x1/ λ dx

( )(1−β)/ 2 1+2(ωτ0 )1−α sin(π α / 2)+(ωτ0 )2(1−α)

Note that g(τ ) is written in a form that is normalized for the variable ln(τ ), i.e.

Kohlrausch-Williams-Watt (KWW) [102]

Log-normal distribution

Log-rectangular distribution

Havriliak-Negami (HN) [111]

sin((1−β)ϕ2 )

Davidson-Cole (DC) [110]

1 ( )1−β 1+(jωτ0 )1−α

sin βπ π

(cos(ϕ1 ))1−β sin((1 − β)ϕ1 )

1 (1+jωτ0 )1−β

Cole–Cole (CC) [109]

[

sin(απ ) cosh[(1−α) ln(τ / τ0 )]−cos(απ )

1 2π

cos(π α/2) (ωτ0 1+2(ωτ0 )1−α sin(π α/2)+(ωτ )2(1−α)

1 1+(jωτ0 )1−α

Debye (

–

ωτ0 1+(ωτ0 )2

1 1+jωτ0 )1−α

g([ln(τ )])

ε'' /Δε

Model

( ) ' /Δε ε(ω) − ε∞

' , ϕ = tan−1 (ωτ ), ϕ = tan−1 Table 4.1 A summary of different distributions of relaxation time with parameters: Δε = εs − ε∞ 1 0 2 [ ( ) ] [ ] ) ( λ λ 1−α −cos(πα) , φ(t) = exp − τt0 with 0 < λ < 1, and u = xτ ϕ3 = π2 − tan−1 (τ0 /τ )sin(πα) τ0

] ,

122 4 Analytical Approaches of EMB at Multiple Scales

4.2 ED and EP Modelling

123

where F max , r 1 , r 2 , r 3 and χ are constants (values are given in [81]) which depend only on Λ. Current strategies for visualizing the time-dependent behavior of individual electropores in vesicles rely on detecting the dynamics of solute flow between the internal and external volumes and imaging the membrane itself [29], or using optical single-channel recording and fluorescence imaging of bilayers. While there has been significant theoretical treatment of planar membranes (e.g. SNK) there has been little study combining theoretical and experimental approaches of cell and vesicle membranes with other common shapes (spheroids, ellipsoids) during pore expansion [17]. Moreover, the collective nature of SNK’s model, i.e. coupling of individual pores through the membrane lipid-water interfacial tension, is still very much unknown, and a wide variety of direct and indirect detection experiments are actively searching for evidence of post-pulse collective membrane resealing kinetics. Fast digital imaging offered insight into the deformation and permeabilization of giant unilamellar vesicles subjected to electric pulses of varying strength/duration. The aspect ratio (defined as the ratio of semi-major axis b to semi-minor-axis a of the ellipsoid) of the ellipsoidal deformation for a vesicle (initially spherical) represents a reliable metric for the underlying morphology of the vesicle subjected to electric pulses as it allows to define the transmembrane potential from analytical considerations. Recently [7], there has been an increased emphasis on the role anisotropy may play in the broad set of phenomena described above. When b/a is large at elevated V m , EP dominates, and maximum membrane deformation coincides with maximum pore aperture. The transmembrane potential for an ellipsoidal membrane can be evaluated by making use of Eq. (4.85). Unfortunately, SNK’s interpretation also suffers from a number of drawbacks. Several comments dealing on SNK’s model are in order. First, this model predicts that when a cell is exposed to a 1 ms pulse, pore creation occurs on a time scale of μs. After formation is complete, pores continue to expand in size for the duration of the 1 ms pulse, although, by the end of the pulse, additional changes in pore sizes are small [50]. The pore evolution is driven by the cell’s surface tension since the ITV peaks after the membrane capacitor finishes charging (at τm ). As pores grow, the overall surface tension of the cell decreases thus lowering the system energy. Second, since different regions on the cell membrane take different time to achieve the threshold ITV, both the size and density of the pores are spatially heterogeneous at various locations on the cell surface. Based on the pore size, the pores could be divided into two populations: small pores (< 2 nm in diameter) and large pores (>2 nm in diameter) [50, 82]. The highest pore density is created at the poles, but the largest pores are primarily located on the border of the electroporated regions (close to the cell equator). In addition, the hyperpolarized half of the cell has more but smaller pores, whereas depolarized half of the cell has fewer but larger pores. Third, the number of pores increases as the applied electric field increases, but that, except for those fields very close to the EP threshold. Fourth, SNK’s model assumes that all pores in the membrane have the same dynamics. Fifth, the probabilistic nature of nucleation based on the classical nucleation theory [83] suggests that, if the free energy decrease requested for pore nucleation is driven by electric stress, the rate of pore nucleation reads

124

4 Analytical Approaches of EMB at Multiple Scales

) BVm2 (t) − δc , Θ = A exp kT (

(4.20)

where δc is the energy barrier between the hydrophilic and hydrophobic membrane pore states (typically, 45 kT [83–85] with reference to Fig. 4.6) when V m = 0 V, A is a pre-exponential factor ptroportionnal to the number of possible pore nucleation sites and the frequency of lateral lipid fluctuations, and B is a model parameter [49, 84]. Sixth, SNK’s model is deterministic, but the reality of EP must be more stochastic. Nanavati and coworkers [86] analysed the stochastic distributions of pore conductance and suggested that the local lipid composition and the local membrane curvature have significant effects on pore dynamics. Tolpekina and coworkers [87] showed that the cost of pore formation is 15–20 kT. They further used an atomistic dipalmitoylphosphatidylcholine membrane and reported the 85 kJmol−1 minimum cost for the pore nucleation. Seventh, following the pulse, pore size decreases rapidly leading to the elimination of the large pore population within μs. Pore resealing takes longer with a mean lifetime estimated to range to range from milliseconds to even minutes or hours, in agreement with a number of experimental observations reported in the literature [38]. There is no known mechanism for this large range of pore lifetime which remains a puzzle plaguing cell EP. Any explanation for this puzzle must involve physics beyond the SNK’s model. Eight, it should also be noted that although EP occurs mainly in the PB domains of membranes, the membrane proteins and cytoskeleton also contribute to the process by affecting the behavior of PB. Yet, accumulation of hydrophilic pores on cell membrane facilitates the exchange of the water-soluble molecules and ions through the cell membrane. An associated issue is that ED is not considered in SNK’s model. Since the original studies on the EP of single cell mentioned above, the question of the multilayer shell and multicellular system models for understanding tissue-level electric field effects has been an important subject. Evolving subcellular features, e.g. mitochondrial membrane, prominently in models of a biological cell approximated by multilayer spheres with homogeneous, lossy and dispersive dielectric properties for each layer have been presented (see, e.g. Ref. [88]). Cytoskeleton-mediated deformation of the nucleus has long been considered a pathway through which shear stresses applied to the cell are transduced to gene-regulating signals [89]. Multilayered spherical cell models with concentric and non-concentric nucleus were proposed but considering the full morphological information in these models is elusive without precise 3D spatial and dynamical data. Lumped parameter models can connect also cell-level description to tissue-level phenomena [90–92]. A transport lattice model has been used to analyze multicellular 2D structures with nonlinear active sources of V rest , including simple representations of local and passive membrane EP [72]. Kirchoff’s laws provide basic procedures for deriving mathematical models for electrical circuits. Within this approach, a cell assembly can be described by a large electrical system in which local RC components interact through paths that connect nearby cells. But overall, these equivalent circuit models of EP are either too complicated (i.e. for very large electrical circuit) or too simplified (i.e. capacitively coupled

4.3 Effective Dielectric Properties of Biological Materials

125

nearby cells and more distant layers of cells) to be generalizable to realistic tissue models. Another comment is in order. The transient permeabilized state of the membrane allows transport of molecules into the cytoplasm. Granot and Rubinsky [93] first developed a mass transfer model at a single cell level during reversible EP based on Krassowska and Filev [14] and diffusion equation which is solved subject to initial conditions, which are the initial concentration in the ECM, and boundary conditions, which is modeled as no mass flux. Other physical models dealing with EP-mediated mass transport to cells and tissues were also proposed more recently [94].

4.3 Effective Dielectric Properties of Biological Materials Understanding the properties of the dielectric properties, from dc to GHz, and the underlying mechanisms responsible for these properties of biological materials has long been pursued by physicists. The historical examination described in Chap. 3 has demonstrated that many elegant models are constrained, e.g. CS models, requiring some to substantial fine-tuning [95–99]. This demonstrates the remarkable success of the synergy between theory and experiment where ideas are proposed, models are built, and parameters are constrained. From the applications standpoint, the comparison between normal and cancer tissues was also well characterized by many authors, see e.g. [98] showing that the permittivity of cancer tissue is greater than that of normal tissue. The material treated in this section may be divided into three major topics. We begin by summarizing the classical principles behind polarization of biological materials in a static electric field. Special topics include a summary of the main physical concepts such as polarizability and permittivity. The second subject concerns the response of biological material to a harmonic perturbation, dielectric spectroscopy. Thirdly, we examine the basic mechanisms that govern the electrostatic force (EF) acting on a single biological cell (modelled by a spherical CS structure) in an electric field.

4.3.1 Polarization in an Electric Field The dielectric properties of biological materials are a measure of their interaction with electromagnetic fields. The electrodynamics of continuous media has been the guiding principle in the development of the … over the last several decades [20, 99–103]. A fundamental fact of the electrodynamics of continuous media is that the microscopic (or local) field acting upon atoms or molecules is generally different from both the applied and average macroscopic fields, because of interparticle interaction. Its best-known manifestation is the famous Onsager relation for dielectrics

126

4 Analytical Approaches of EMB at Multiple Scales

[100–103]. The simplest solutions of the Maxwell equations are the time independent solutions, especially the point multipole (point charge, dipole, quadrupole, etc.) solutions for electrostatic fields. The interested reader may consult Appendices 1, 2 and 3, respectively for precise definitions of the electric dipole properties, and calculations of the ITV and EF. Let us consider a homogeneous polarizable body occupying a volume Ω in free space and bounded by a surface S. The body is assumed to be isotropic and is immersed in an ambient time-independent spatially uniform electric field E. From a microscopic point of view, the interaction of an electric field with a dielectric medium produces a separation of negative and positive charges; this separation of charges produces the polarization in the medium. In general, the displacement of the charges in the materials is proportional to the applied electric field. The linear dependence is described by P = ε0 (ε − 1)E.

(4.21)

It is necessary in practical applications to be able to determine the sources of polarization within a material [100–103]. Accordingly, it will be helpful to review briefly the basic polarization mechanisms by relating them to the dimension scales over which scales can be separated within the material (Fig. 4.9). At the atomic level, a first contribution is derived from the distorsion of the electron cloud about an atom or molecule, and is termed the electronic polarization P e . At the molecular level, two contributions arise: one comes from the separation of ionic species, or atomic nuclei, this is termed the ionic polarization P i , and the other comes from the re-orientation of molecules possessing permanent dipoles P o . On a larger scale, a contribution can be observed from charges that accumulate at interfaces in heterogeneous materials where the inhomogeneities can be present eithet intentionally or as a result of impurities. This contribution is known as the MWS polarization P MWS . Finally, there may also be a contribution from localized conduction, on a scale which is smaller than the physical bounds of the material under and where the charges cannot be considered as truly free, e.g. this is the case for intrinsically conducting polymers where charges are able to move along the backbone of the polymer chain and have low mobility for inter-chain conduction via tunnelling or hopping mechanisms. This contribution is termed the localized conduction polarization P loc [103, 104]. Hence, we express the total polarization of a material by P = P e + P i + P o + P MWS + P loc .

(4.22)

Next, we give here some basic facts about the case of field distributions which are independent of time, i.e. the distribution of charges must be time independent. We first focus on the classical and important example of the application of Laplace’s equation concerning a spherical cavity of radius a in a dielectric that is subject to a uniform static electric field far from the cavity. Sphere is the simplest canonical shape. However, because most particles of interest are not spherical, it is important

4.3 Effective Dielectric Properties of Biological Materials

127

E

E=0 0

electronic

orientation

ionic

(a)

localized conduction

electronic

Maxwell-WagnerSillars

localized conduction

orientation

ionic

Maxwell-WagnerSillars

(b)

Fig. 4.9 Schematic interpretation of polarization mechanisms: a at zero field, b in a static applied electric field E. Electronic polarization deals with the displacement of nuclei and electrons in the atom under the influence of an external electric field. Orientation polarization orients the permanent dipoles. For ionic polarization positive ions are displaced in the direction of the applied field while the negative ions are displaced in the opposite direction, therefore giving a resultant dipole moment to the material body. MWS polarization deals with space charge

to show how departing from a spherical shape affects the static polarizability. A prototypical nonspherical particle is the spheroid. The exact analytical solution for a single, homogeneous, isotropic spheroid (an ellipsoid of revolution) has been also obtained by using the method of separation of variables. The second purpose of this section is to establish the connection of permittivity to the polarizability of the system. Along the way we pick up notation and acquaint the reader with the kind of thinking that underpins topics in electrostatics. Let us assume that there are no volume charges so the electric potential satisfies the Laplace equation ∇ 2 V (r) = 0,

(4.23)

where V (r) is the electric scalar potential at a point r. There are many physical processes for which the solution of the Laplace equation on the exterior of a body of general shape is central to the theoretical description [100–102]. Of great importance to many electrostatics problems is the polarizability. In this section, we develop some of the basic properties of such quantity in preparation for later consideration of various problems such as the determination of the local field inside a small particle, the depolarization factor of arbitrarily shaped inclusions, and the evaluation of the electric

128

4 Analytical Approaches of EMB at Multiple Scales

field induced transmembrane potential of a CS model of biological cells. To begin with, let us introduce spherical polar coordinates (r, θ , φ) relative to an origin placed at the centre O of a sphere: r is the radial distance from the origin, θ is the polar angle measured with reference from the z-axis and φ is an azimuthal angle measured from the plane x–z. We characterize the field distribution within the system by specifying the potential function V (r, θ, φ) everywhere within it. We will show that in spherical polar coordinates system, the general separable solution V of Eq. (4.23) consists of a sum of Legendre polynomials of cosθ whose coefficients depend on different integer powers of r. For simplicity, we consider a problem which is symmetric around the z-axis so that the potential is independent of the azimuthal angle φ, i.e. V is axisymmetric. Now let us proceed to the demonstration that the static polarizability of a spherical particle is proportional to its volume. We consider a uniformly polarizable homogeneous sphere of radius R, with permittivity ε1 surrounded by an infinite medium (continuum) characterized by permittivityε2 . Without loss of generality, we apply the external uniform electric field E0 along the z-axis, as in Fig. 4.10. Outside the sphere, the potential V satisfies Eq. (4.23) since no free charges are present and because the electric field is supposed to be produced by sources which are very far removed from the position of the sphere. On its surface, Eq. (4.23) is not valid since there is an apparent surface charge according. Inside the sphere, however, Eq. (4.23) can be used again. Far from the sphere the potential must have the form V (r → ∞, θ ) = V0 = −E0 z = −E0 r cos θ.

(4.24)

The potential is even about the x-axis, i.e. invariant under the substitution θ → −θ . This solution suggests that the potential both inside and outside the sphere should

z R

E0

ε1 ε0

ε2

x

Fig. 4.10 A dielectric sphere of radius R and with permittivity ε1 placed at the origin within the surrounding environment of permittivity ε2 . The sphere is subject to a uniform, constant, external field E0 . Our convention is that E0 is directed along the z-axis is applied to the system. Thus, the boundary condition we wish to apply is V (r → ∞, θ ) = V0 = −E0 z = −E0 r cos θ . We also require that V (r) is well behaved ar r = 0 because there is no charge here

4.3 Effective Dielectric Properties of Biological Materials

129

be proportional to the Legendre polynomial of order 1 P1 = cosθ.1 Now, Eq. (2.2.3) in spherical coordinates is written as ( ) ( ) ∂ 1 ∂ 1 ∂ 2 ∂ r V (r, θ ) + 2 sin θ V (r, θ ) = 0. r 2 ∂r ∂r r sin θ ∂θ ∂θ

(4.25)

The general solutions of Eq. (4.25) are given by2 ) ∞ ( ∑ Bn An r n + n+1 Pn (cos θ ), r n=0

(4.26)

) ∞ ( ∑ Dn n Cn r + n+1 Pn (cos θ ). Vin = r n=0

(4.27)

Vout = and

Inside the sphere, Eq. (4.27) can be written in the form Vin =

∞ ∑

Cn r n Pn (cos θ ).

(4.28)

n=0

1

The Legendre[ polynomials are]solutions of Legendre’s equation [62]. The Legendre’s differential ( )d d equation is dx P(x) + n(n + 1)P(x) = 0, where n is an arbitrary complex number 1 − x2 dx introduced for solving the equation. This equation can be solved by power series. The solution is convergent for |x| < 1 and arbitrary degree n, but is divergent at x = ±1. The only solution is to make the degree n a non-negative integer. In this case, the solution is called polynomial, which [( Legendre )n ] dn may be expressed using the Rodrigues formula Pn (x) = (2n n!)−1 dx x2 − 1 . The multiplicative n constant in front of each polynomial ) been chosen so that the polynomials satisfy the orthogonality ( has ∫1 2 δmn in the range |x| ≤ 1, where δmn = 1 if m = n, and zero property −1 Pm (x)Pn (x)dx = 2n+1 ) ( otherwise. The first five Legendre polynomials are: P0 (x) = 1, P1 (x) = x, P2 (x) = 21 3x2 − 1 , ( ( ( ) ) ) P3 (x) = 21 5x3 − 3x , P4 (x) = 18 35x4 − 30x2 + 3 , and P5 (x) = 18 63x5 − 70x3 + 15x [62]. ( ) )' −1 ( 2 Seek solutions of the form V (r, θ ) = f (r)g(θ ). Then, f −1 r 2 f ' ' = − g ' sin θ g sin θ and both )' ( ' sides must be constant, λ say. Consequently, g sin θ = −λg sin θ. If we set ζ = cos θ , and use the / / / / chain rule to replace d d θ by d d ζ , we have d d θ = − sin θ d d ζ . Making use of this change ] [( ) of variable we have ddζ 1 − ζ 2 ddgζ + λg = 0, which is Legendre equation. For well-behaved solutions at ζ = ±1 (i.e. θ = 0, π) we need λ = n(n + 1) for some non-negative integer n, in ( )' which case g ∝ Pn (ζ ) = Pn (cos θ ). Similarly, it is easy to that f must satisfy r 2 f ' = λf to which n the solution is f = An r n + rBn+1 . The general solution to Laplace’s equation for the axisymmetric case is therefore given by Eq. 4.26 or Eq. (4.27). A similar analysis for the non-axisymmetric case, i.e. V depends now of the azimuthal angle φ shows that the(general solution ) to Laplace’s equa∑ ∑n n + Bmn P m (cos θ ) exp(imφ), tion can be written in the form V (r, θ, φ) = ∞ r A mn n+1 n n=0 m=−n r where P[nm (ζ ) are the]associated Legendre functions which satisfy the associated Legendre equa[ ] ( ) m tion ddζ 1 − ζ 2 ddgζ + n(n + 1) + 1−ζ 2 g = 0 when m and n are integers such n ≥ 0 and −n ≤ m ≤ n.

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4 Analytical Approaches of EMB at Multiple Scales

There is no term ∝ r −(n+1) because there is no charge at r = 0 which would cause the potential to become singular at the origin. At the outside Vout =

∞ ∑ Bn Pn (cos θ ) − E0 r cos θ. r n+1 n=0

(4.29)

In writing Eq. (4.29) we have used the fact that the Legendre functions are linearly independent, all coefficients An are equal to zero, except A1 = −E 0 in order that the potential reduces to that corresponding to a uniform field of strength E 0 at distances far from the sphere. On the surface of the sphere the potential must be continuous on passing from the inside to the outside the sphere; this continuity of the potential guarantees that the tangential component of E will be continuous across the surface of the sphere as is required by curl E = 0. Thus, one finds, for r = R and any value n = Cn Rn . On the surface of the sphere the normal component of of n except 1, RBn+1 the electric displacement D must be continuous. Using the orthogonality property of n . From the Legendre polynomials, this condition gives ε2 nCn Rn−1 = −ε1 (n + 1) RBn+2 these two continuity equations it follows that Bn = 0 and C n = 0 for all values of n except n = 1. For n = 1, these two equations can be written as B1 − E0 R = C1 R, R2

(4.30)

and ( ε1 C1 = −ε2

) 2B1 + E0 . R3

(4.31)

Equations (4.30) and (4.31) can be solved in order to find B1 and C 1 . We have −ε2 3 1 C1 = ε13ε E , and B1 = εε11+2ε R E0 . Substituting these results into Eqs. (4.28) and +2ε2 0 2 (4.29), we obtain the following expressions for the potential and field in the region inside the sphere Vin (r, θ ) = − Ein = −

3ε2 E0 r cos θ , ε1 + 2ε2

3ε2 ∂ Vin = E0 , ∂z ε1 + 2ε2

(4.32) (4.33)

and in the region outside the sphere ) ] [ ( ε1 − ε2 R3 , Vout (r, θ ) = −E0 r cos θ 1 − ε1 + 2ε2 r 3 [ ( ) ] ε1 − ε2 R3 ∂ . Eout = − Vout = E0 1 − ∂z ε1 + 2ε2 r 3

(4.34)

(4.35)

4.3 Effective Dielectric Properties of Biological Materials

131

Equation (4.33) implies that the electric field inside the sphere is constant. However, the field outside the host medium is not constant and is equal to the incident field plus an additional field that would be produced by an electric dipole situated at the origin of the sphere. Indeed, it is worth observing that the second term of the right-hand side of Eq. (4.35) corresponds to the potential of a sphere-centered point dipole p = pz directed along the z-axis with dipole moment p = 4π ε2 ε0 R3

ε1 − ε2 E0 = αd ε2 E0 , ε1 + 2ε2

(4.36)

where α d (kg−1 A2 s4 ) denotes the dipole polarizability of the dielectric sphere. When the influence of neighboring cells cannot be ignored, a multipolar approach should be used [1, 13–15, 17]. Considerable progress has been made in recent years in characterizing the static polarizability of inhomogeneous media. Analytical expressions of the static electric polarizability are also available for more complex shapes, i.e. for the cytoplasm (permittivity εc )-membrane (permittivity εm and thickness d m ) spherical particle embedded the ECM of permittivity εe ), it may be expressed in the form α = 3V ε0

(2εm + εc )(εm − εe ) + ϖ (εc − εm )(2εm + εe ) , (2εm + εc )(εm + 2εe ) + 2ϖ (εc − εm )(εm − εe )

(4.37)

/ )3 ( with ϖ = R − dm R and R denotes the outer radius of this 2-shell model of cell. Several tacit assumptions need be considered: (a) a continuum description holds for the system, (b) the electromagnetic wavelength of the radiation probing the system is much larger than all inhomogeneity length scales, (c) the three phases are spatially isotropic, homogeneous, and non-dispersive, (d) the membrane is passive, i.e. the heterogeneous structure of the cytoskeleton (ion channels and ion pumps) is ignored, and (e) no surface charges are present on the faces of the membrane. Whether a continuum description may hold for nanosized membranes only a few molecules thick is still a matter of debate. The quasistatic assumption (b) is essentially required for treating the system as effectively homogeneous. A basic consequence of the effective modelling of heterostructures [28, 31, 32] is that the size of these structures cannot be as large as one wishes with respect to the frequency. Indeed, based on quasielectrostatic and penetration depth criteria, this approximation is valid provided that the frequency F of the electromagnetic wave is much smaller than R−1 1014 μm−1 Hz for the spherical CS structure considered [31, 32]. While (c) is a working hypothesis, it has become clear that the phases 2 and 3, i.e. the intracellular space, are spatially heterogeneous regions [2, 3]. Thus, assumption (c) can be regarded as crude since the real cell is actually a heterogeneous composite of ions, peptides, and metabolites. Concerning assumption (d), this simple model does not include the effect of the counterion layer which may become important under specific conditions, e.g., when cells are surrounded by weak electrolytes. Next, it is worth observing that the latter equation can be written as

132

4 Analytical Approaches of EMB at Multiple Scales

α = 3V ε0 where γ =

(εc −εm ) . (2εm +εc )

(εm − εe ) + ϖ γ (2εm +εe ) , (εm +2εe ) + 2ϖ (εm − εe )

(4.38)

If we expand the terms of the numerator and the denominator, γ εm 1+2ϖ 1−ϖ γ −εe

and then divide them both by 1 − γ , we obtain α = 3V ε0 ε

1+2ϖ γ m 1−ϖ γ

+2εe

. If we compare

this expression with Eq. (4.36), we find that the dipole polarizability of the core– shell particle is equivalent to that of a homogeneous dielectric sphere of radius R, characterized by an equivalent permittivity given by [ εcm = εm

] / ϖ −1 − 2(εm − εc ) (2εm +εc ) / . ϖ −1 + (εm − εc ) (2εm +εc )

(4.39)

As an aside, observe that the corresponding polarizability for the two-dimensional case (concentric circles or cross-cuts of cylinders) is α = 2Aε0

(εm +εc )(εm − εe ) + ϖ (εc − εm )(εm +εe ) , (εm + εc )(εm +εe ) + ϖ (εc − εm )(εm − εe )

(4.40)

where A is the surface of the core–shell particle and ϖ now refers to the fractional area [13, 14]. Extensions of these calculations to non-spherical shelled structures, e.g. cylindrical and toroidal, can be found in Refs. [13, 14]. For a general multi-layered shell, a similar procedure can be generalized within a dipolar analysis. The procedure is to start with the innermost object and the layer enclosing it, and to define an equivalent permittivity unsing Eq. (4.39). Then, the procedure is repeated on the new, now homogeneous object and the layer enclosing it. The same procedure is repeated until the outermost shell has been incorporated into the equivalent permittivity. At the end, the multi-layered object is thus replaced by an equivalent homogeneous field, sphere with a radius equal to that of outermost shell [13, [ 14]. For a harmonic ] we assume sinusoidal variation such as E(r, t) = Re E(r) exp(−jωt) , where E(r) is spatially dependent. Additionally, we can assume that the cytoplasm and ECM are homogeneous dielectric media with Ohmic electrical conductivities σ c and σ e , respectively. The method previously described ( / to )obtain the effective ( dipole /can )be used again by substituting ε0 εc → ε0 εc + σc jω and ε0 εm → ε0 εm + σm jω . The dipole field exhibits a R−3 dependence so its contribution to the net electric field is significant only at distances close to the sphere. In addition, we can show that Eq. (4.36) corresponds to a uniform polarization along the z-axis per unit volume −ε2 E0 . This polarization would produce a depolarizing electric field within 3ε2 εε11+2ε 2 −ε2 the sphere given by − εε11+2ε E0 [20, 100, 101]. As an aside we note that Eq. (4.36) 2 indicates that a polarizable isotropic sphere subject to a constant uniform electric field acquires a dipole moment which is aligned with the field. Consequently, the mechanical torque on the sphere is zero. It is worth noting that this statement holds true for a time-varying electric field if the polarization adjusts instantaneously to the new field direction as the latter changes. If a nonuniform electric field is applied, the

4.3 Effective Dielectric Properties of Biological Materials

εN εi+1 εi ε1 aN

ε3

R

ε2

R ε1

θ

θ

z

a1 E0

133

ai

z E0

ai+1

(a)

a1

a2

(b)

Fig. 4.11 a A N-layer concentric spherical structure is exposed to a uniform electric field, b case of a single CS model (without NE) with inner radius a1 and outer radius a2

sphere will be polarized and experience the dielectrophoretic force F = (p · ∇)E (see Eq. (4.90) of Appendix 3). If we assume that the applied electric field E does not vary significantly over the diameter of the sphere, we can use only the first term of a Taylor expansion of the electric field, when determining the dipole moment and force. Then, by substituting Eq. (4.36) one obtains the lowest-order approximation of the dielectrophoretic force given by Eq. (4.92). Next we focus on a method for solving Laplace’s equation specialized to the CS model of a spherical cell in suspension. Here, our purpose is to extend this solution to the situation of a N-layer concentric spherical structure. Having in mind the evaluation of the electric field ITV of a CS model of biological cells we specialize the solution to this specific already found that the general solution of ( Wen have ) ∑ case. Bn A P r + θ ), where the constants A and B Eq. (4.23) is V (r, θ ) = ∞ (cos n n n=0 r n+1 are coefficients to be determined from boundary conditions. Now, the geometry of interest is illustrated in Fig. 4.11a. For the N-layer concentric spherical structure, the radii are assumed to be a1 ,…, aN , and the permittivity and conductivity are set to ε1 , σ 1 ,…, εN , σ N . In a very similar way that was used to obtain Eqs. (4.26) and (4.27), the potential for each layer can be written as Vin (r, θ ) =

∞ ∑

A1n r n Pn (cos θ ) for r ≤ a1 ,

(4.41)

n=0

V2 (r, θ ) = Vi (r,θ ) = and

) ∞ ( ∑ Bn2 Pn (cos θ ) for a1 ≤ r ≤ a2 , A2n r n + n+1 r n=0

) ∞ ( ∑ Bni Ain r n + n+1 Pn (cos θ ) for ai−1 ≤ r ≤ ai , r n=0

(4.42)

(4.43)

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4 Analytical Approaches of EMB at Multiple Scales

Vout (r, θ ) =

∞ ∑ BnN Pn (cos θ ) for r > aN . rn + 1 n=0

(4.44)

The superscript i in Ain , Bni represents the i-th layer, and the subscript n refers to the n-th term in the series. As was shown above, on the interface between the i-th and i + 1-th spheres the potential must be continuous and the normal component of D must be continuous. These conditions imply that Vi = Vi + 1 for r = ai ,

(4.45)

and εi

∂Vi ∂Vi + 1 = εi + 1 for r = ai , ∂r ∂r

(4.46)

From these two continuity equations it follows that Ain = 0 and Bni = 0 for all values of n except n = 1. Thus, these two coefficients for each layer can be calculated by solving the linear equations system obtained from the boundary conditions. Likewise, the electric field in the given regions can be obtained by taking derivatives of the potentials V i . Hence, the electric fields are Ein = −A11 for r ≤ a1 , Ei (r, θ ) = −Ai1 +

(4.47)

) B1i ( 3 cos2 θ − 1 for ai - 1 ≤ r ≤ ai , r3

(4.48)

) B1N ( 3 cos2 θ − 1 for r > aN , 3 r

(4.49)

Vout (r, θ ) = E0 +

where the uniform electric field E0 directed along the z-axis is applied to the system. Now if we consider the special case of Fig. 4.11bof CS spherical cell we have Vin = A11 r cos θ for r ≤ a1 ,

(4.50)

) ( B2 V2 = A21 r + 21 cos θ for a1 ≤ r ≤ a2 . a1

(4.51)

and Vout = −E0 r +

B13 cos θ for r > a2 . a12

(4.52)

4.3 Effective Dielectric Properties of Biological Materials

135

As an interesting exercise, the reader is asked to prove that using the boundary conditions Eqs. (4.45) and (4.46) four linear equations of the coefficients A11 , A21 , B12 , and B13 are obtained A11 = A21 +

B12 , a13

B2 B13 = A21 + 31 , 3 a2 a2 ) ( B2 ε1 A11 = ε2 A21 − 2 31 , a1 −E0 +

(4.53)

(4.54)

(4.55)

and ( ) ( ) B12 B13 2 ε2 A1 − 2 3 = ε3 −E0 − 2 3 . a2 a2

(4.56)

After lengthy but straightforward algebra, these coefficients can now be expressed as A11 = −

3ε2 ε3 E0 , ε2 (ε1 + 2ε3 ) + 2v(ε1 − ε2 )(ε3 − ε2 )

(4.57)

A21 = −

ε3 (ε1 + 2ε2 )E0 , ε2 (ε1 + 2ε3 ) + 2v(ε1 − ε2 )(ε3 − ε2 )

(4.58)

ε3 (ε1 − ε2 )E0 a13 , ε2 (ε1 + 2ε3 ) + 2v(ε1 − ε2 )(ε3 − ε2 )

(4.59)

ε2 (ε1 − ε3 ) − v(ε1 − ε2 )(ε3 − ε2 ) E0 a13 , ε2 (ε1 + 2ε3 ) + 2v(ε1 − ε2 )(ε3 − ε2 )

(4.60)

B12 = and B13 =

( / )3 / where ν = (1 − φ3 ) 3 and φ3 = a1 a2 is the fraction of the total particle volume occupied by the phase 3. Despite its success, the N-layer CS description, which only considers inclusion of concentric cell organelles, such as the NE, it leaves many unanswered questions about the description of reality of irregularly shaped cells. Among these so-far unanswered questions are the influence of cell morphology on its electrical and mechanical responses, an understanding of the coupling between CS, and why the scales associated with cell and nucleus are so disparate [31, 72, 88].

136

4 Analytical Approaches of EMB at Multiple Scales

4.3.2 Polarization and Permittivity in an Alternating Electric Field In this section we wish to extend the concepts of polarization, polarizability, and permittivity which characterize the response of materials to static electric fields to the case of time-varying fields. Uniquely in comparison with static measurements, relaxation phenomena involve an unparallel wealth of information, since any single datum point of a static measurement corresponding to a given temperature pressure, composition, etc. is represented by tens and in some cases hundreds of points defining the time or frequency spectrum corresponding to this point. The tools available for the measurement of dielectric relaxation, both in the frequency and the time domains, have undergone dramatic improvements in recent years and it is now possible to make measurements over an extremely wide range of frequency covering more than 10 decades from 10–1 Hz to GHz, with extensive automation of the procedures and of the control of measuring conditions [98, 99]. A comparable range is available in the time domain and their respective advantages and drawbacks may be used to optimal advantage [98, 99]. Although dielectric spectroscopy (DS) does not possess the selectivity of the NMR and ESR methods, it can offer important and sometimes unique information on the dynamic and structural properties of almost any kind of materials (Fig. 4.12). DS is especially sensitive to intermolecular interactions and is able to monitor cooperative processes. Via molecular spectroscopy, it provides a link between the

Fig. 4.12 DS offers a valuable opportunity to investigate the electrical behavior of biological materials under the influence of an external electric field. The number indicate the time domain DS and corresponding frequency, f , range (adapted from Raicu and Feldman [99])

4.3 Effective Dielectric Properties of Biological Materials

137

investigation of the properties of the individual constituents of a complex material and the characterization of its bulk properties [98, 99]. In real materials, the complexity of the many-bodied interactions and the heterogeneous environment experienced by the different constituents of these materials is not amenable to exact analytical description. Exceptions are the Debye and related models, which describe the reorientation of molecules in the microwave and far-IR regions of the spectrum. In what follows, we shall consider simple models for the motion of a bound electron in the presence of an applied electric field. As the electric field tries to separate the electron from the positively charged nucleus, it creates an electric dipole moment. Averaging this dipole moment over the volume of the material gives rise to a macroscopic dipole moment per unit volume, i.e. a polarization vector P. A common assumption in local-field theories is that the local field is uniform at distances much shorter than the wavelength of electric field, i.e. amounting to the so-called “mean-field approximation”. When a time-varying electric field is applied to a dielectric material, the polarization response of the material, assumed to be linear, homogeneous and isotropic, cannot be instantaneous. In a similar manner, when the field is removed suddenly, the polarization decay caused by thermal motion follows the same law as the relaxation (or decay function) of polarization. In the simplest case of a medium with a single relaxation mechanism, e.g. Debye type, the system will relax toward a new equilibrium as a first-order process characterized by a relaxation time τ. In that case, the displacement D(r, t) = εε0 E(r, t) would be determined by the electric field at the very same moment t. In the time domain, the most general causal linear relationship between the displacement and electric fields can be described by the convolutional constitutive relationship, D(r, t) =

' ε∞ ε0 E(r, t)+ε0

∫t

) ( ) ( ε t − t ' E r, t ' dt '

−∞ ' = ε∞ ε0 E(r, t)+

∫t

−∞

) d ϕ(t ' ) ( E r, t − t ' dt ' , ' dt

(4.61)

' where ε∞ denotes the high-frequency limit of the permittivity ε(ω). The first term in the right-hand side of Eq. (4.61) represents the instantaneous response (the socalled infinite frequency response) whereas the second term represents the retarded response coming from the energy initially absorbed by the medium and subsequently returned (partially) by it. This second term is delayed in time because of material inertia and is responsible for the energy dispersion. The representation in Eq. (4.61) provides an instructive interpretation in separating the material response of the medium into a contribution from the vacuum and the actual material interaction (expressed in terms of a convolution integral). Indeed, it is motivated in describing the material property in the form of D = εε0 E = ε0 E + P, in terms of an electric polarization vector P(r, t). The dielectric response function Φ(t) is defined as

138

4 Analytical Approaches of EMB at Multiple Scales

Fig. 4.13 Illustrating the CC representation (Argand diagram) of the complex permittivity of a dielectric material which is characterized by a single relaxation time

ωτ = 1

ω

ε" (ε '∞ +ε s ) / 2

ε '∞

Δε / 2

ε'

εs

( ) ' ϕ(t) = ε0 εs − ε∞ [1 − φ(t)], where / εs denotes the static limit of the permittivity ε(ω) and φ(t) = ⟨p(r, t) · p(r, 0)⟩ ⟨p(r, 0) · p(r, 0)⟩ is the normalized autocorrelation function of the macroscopic fluctuating dipole moment of the unit volume sample p(r, t). The connection between the polarization vector of the unit volume sample P(t) and the average of the total dipole moment ⟨p(r, t)⟩ may be written , where V is the volume of the sample, and ⟨...⟩ denotes an as P(r, t) = ⟨p(r,t)⟩ V ensemble average. The ] complex permittivity is related to the relaxation function by [ ' ε−ε∞ d φ(t) = TL − dt , where TL means Laplace transform [99–102]. ' εs −ε∞ In the case of harmonic time dependence, i.e. the frequency regime of the electric field is proportional to exp(−jωt), the dispersion of a considerable number of liquids and dielectrics can be represented by the Debye relation [101, 102] described by a single relaxation time τ 3 ' εs − ε∞ ' ε(ω) = ε' (ω) − jε'' (ω) = ε∞ + 1 + jωτ (( ) ) ( ' ' ) εs − ε∞ ωτ εs − ε∞ ' −j = ε∞ + . 2 2 2 1+ω τ 1 + ω τ2

(4.62)

It is also worth to note that the Debye Eq. (4.62) describes what is also termed anomalous dispersion, which is dispersion in which ε' (ω) falls with frequency.4 This dispersion extends over roughly a decade in frequency from/ one third to three times F c . At the relaxation (or characteristic) frequency Fc = 1 2πτ the permittivity is halfway between its low- and high-frequency values. Observe that if ε'' (ω) is plotted [ / ]2 [ ]2 ' ) 2 + ε'' = versus ε' (ω) a semi circle (Fig. 4.13) is realized ε' − (εs + ε∞ / ]2 [ ' ) 2 . (εs − ε∞

3

In the case of dielectric dispersive biological materials, a generalized n-th order dispersion relation can be used [97–101]. 4 For a Debye relaxation mechanism, the relaxation function is given by an exponential decay ε−ε' 1 φ(t) = exp(−t / τ ) for t ≥ 0, where the time constant τ is the relaxation time. Hence, εs −ε∞' = 1+jωτ ∞ in that case.

4.3 Effective Dielectric Properties of Biological Materials

139

This is the so-called Cole–Cole (CC) representation of the complex permittivity (also termed diagram).5 The center of the semicircle is on the real axis at ( the Argand ' ) ' the point εs + ε∞ /2, and its apex occurs where ω = 2πFc = τ −1 . Δε = εs − ε∞ is the relaxation strength which is related to the amount of dipoles involved in the relaxation process. Other graphical representations that may be useful in showing dielectric relaxations include ε' versus ωε'' which should be a straight line ε' =/ εs − ' + τ −1 ε'' ω. It ωτ ε'' , and ε' versus ε'' /ω which should be also a straight line ε' = ε∞ should be noted that if a distribution of relaxation times is present, the corresponding plots in the complex permittivity no longer is a semicircle centered on the real axis but might resemble semicircular arcs with centers below the real axis, or skewed arcs. For example, it is convenient to write the real and imaginary parts of the permittivity for an ensemble of Debye processes with a∫continuous relaxation time distribution ∞ g(τ), such that the normalization condition 0 g(τ )d τ = 1 is verified, as '

ε (ω) =

' ε∞

) ( ' + εs − ε∞

∫∞ 0

g(τ ) d τ, 1 + ω2 τ 2

(4.63)

and ) ( ' ε (ω) = εs − ε∞ ''

∫∞ 0

g(τ )ωτ d τ. 1 + ω2 τ 2

(4.64)

The general approach, in interpreting experimental data, is to infer the distribution function and identify the physical mechanisms that account for it. As the underlying integrals in Eq. (4.63) and Eq. (4.64) are nonlinear, inverting them is nontrivial, see e.g. [107]. While g(τ ) in principle can be obtained directly from the data without preconceptions about its functional form, this inversion is notoriously susceptible to numerical instabilities, i.e. it belongs to the class of mathematically ill-posed problems [108]. Instead, it is common practice to assume a distribution function and numerically fit the data to it. Observe also that different physical effects can give rise to a distribution of relaxation times. The simplest cause is a distribution in some important parameters throughtout the sample. Alternatively, the relaxation might arise from a single thermally activated process with a distribution of activation energies. Many different functions g(τ ) have been proposed for use in fitting dielectric data. Several of them are related to physical model, while others serve only to parameterize the data without clarifying the underlying mechanisms involved in Although the CC representation is useful to describe the experimental values of ε' (ω) and ε'' (ω) when the considered dielectric material is characterized by a single relaxation time, it is sometimes ' and τ by a different graphical plot. If we plot ωε '' (ω) and ω−1 ε '' (ω) preferable to determine εs ,ε∞ ' versus ε (ω) we obtain two straight lines with respective slopes τ −1 and τ, and ε' intersecting ' , respectively. This arises because the above analysis enables us the horizontal axis at εs and ε∞ ' τ ( ) εs −ε∞ ε −ε' ω2 τ ) = to write Eq. (4.62) as ωε'' (ω) = ( s1+ω∞2)τ 2 = τ −1 εs − ε' (ω) and ω−1 ε'' (ω) = (1+ω 2τ 2 ) ( ' ' . τ ε (ω) − ε∞ 5

140

4 Analytical Approaches of EMB at Multiple Scales

ε 1, σ 1

d1

ε 2, σ 2

d2

(a)

R1

C1

V ~

ε, σ R2

(b)

d1

C2

(c)

Fig. 4.14 Schematics of: a the two-layer capacitor, b its equivalent circuit, and c an equivalent single layer capacitor

relaxtion. Table 4.1 summarizes different (logarithmically symmetric) distribution functions that have been proposed in the literature, see e.g. [109–111]. Why is the log-normal ubiquitous? The simplest answer appeals to unspecified multiplicative processes and the central limit theorem. Next, we evaluate the remaining polarization contribution in Eq. (4.22) which concerns interfacial polarization, or MWS effect.6 For heterogeneous sytems dispersion occurs in the bulk properties from the charging of the interfaces within the material. This phenomenon does not arise from relaxation in the bulk phases of the material, but is a consequence of the boundary conditions on the fields at the interfaces between phases. Interfacial effects can dominate the dielectric properties of colloids and emulsions. For that purpose, we use a RC network analysis modeling two slabs in series. With reference to Fig. 4.14, we consider a simple parallel plate capacitor composed of two distinct dielectric layers. The dielectric layers are characterized by their respective permittivities (ε1 and ε2 ), conductivities (σ 1 and σ 2 ), and thicknesses (d 1 and d 2 ). The permittivities and conductivities are assumed to be independent of frequency. The boundary condition on the electric field component normal to the interface can be written as ε1 E1 = ε2 E2 if the interface is free of charge. The capacitor can be represented by the equivalent circuit of two RC parallel circuits in series, where each layer is represented by a parallel RC combination. A single-time relaxation (Debye type) for the two-layer system can be determined as follows. To find the effect of the interface, we replace the two layer capacitor with an equivalent single layer capacitor using the rule for series addition of two capacitances. The resultant parallel capacitance values converge both at low and high frequencies, and the capacitance at low frequency is higher than that at high frequency. Considering the parallel model of two slabs in series displayed in Fig. 4.14b, we have a Debye dispersion even without any dipole relaxation in the dielectric. The dispersion is due to a conductance in parallel with a capacitance 6

The interfacial polarization theory, was developed first by Maxwell [112], and later extended by Wagner [113] and Sillars [114].

4.3 Effective Dielectric Properties of Biological Materials

141

for each dielectric, so that the interface can be charged by the conductivity. It is convenient to write the permittivity as ε = ε' − jε'' = ε' − jσ/ωε0 , with ε' (ω) = εa +

εb − εa , 1 + ω2 τ 2

(4.65)

and σe (εb − εa )ωτ + , (4.66) ωε0 1 + ω2 τ 2 ( ) ε1 d1 σ22 +ε2 d2 σ12 d ε1 ε2 ε1 d2 +ε2 d1 1 σ2 , σe = σ1 dd2σ+σ , and d = where εa = (d 2 , εb = ε d +ε d , τ = ε0 σ d +σ d 1 2 2 1 1 2 2 1 2 d1 2 σ1 +d1 σ2 ) d 1 + d 2 . Athough Eqs. (4.65) and (4.66) may appear to be complicated expressions at first glance, it can be shown that these expressions look like Eq. (4.62) of the Debye model for orientation polarization, except that the loss term includes a contribution from conductivity. We note in closing that the electric field strength will be smaller on the high-permittivity side. The ratio /of current densities on sides 1 and 2 will / / is equal to j1 j2 = σ1 E1 σ2 E2 = σ1 ε2 σ2 ε1 . In the special case where εa = εb , i.e. when the condition σ1 ε2 = σ2 ε1 , no loss will be observed. The interface has zero density of free charges. However, if σ1 ε2 /= σ2 ε1 , the difference in current densities implies that the interface is actually charged. It will be charges if σ 1 = 0 and σ 2 > 0. Similarly, a three-layer system has two single-time relaxations, but mathematical expressions for them are rather complicated. At this point, an example may be useful to illustrate the aforementioned concept. Consider a lossy dielectric sphere of permittivity ε1 = ε1' − jε1'' = ε1' − jσ1 /ωε0 and radius R embedded in a host medium of permittivity ε2 = ε2' − jε2'' = ε2' − jσ2 /ωε0 and submitted to an ac electric field. From Eq. (4.36), we write the polarizability as ε'' (ω) =

( )( ) σ1 − σ2 1 + jωτ0 ε1 − ε2 3 , αd = 4π R ε0 = 4π R ε1 + 2ε2 σ1 + 2σ2 1 + jωτMWS 3

where two relaxation times have been defined: the first ) ( ε1 + 2ε2 , τMWS = ε0 σ1 + 2σ2

(4.67)

(4.68)

is associated with the MWS surface polarization, and the other ( τ0 = ε0

) ε1 − ε2 , σ1 − σ2

(4.69)

characterizes the accumulation of free charges at the particle-medium interface. There have been several attempts to address the question of modelling the electrical permittivity of a spherical CS structure, particularly in the context of dielectric properties of suspensions of biological cells [88, 100, 116–118]. This subject

142

4 Analytical Approaches of EMB at Multiple Scales

of research is related to our ability of treating a dielectric material with effective parameters [102, 103, 115, 116]. Recall that the effective permittivity of composites makes sense if the inhomogeneities in the material are considerably smaller in size than the spatial variation of the electromagnetic field, i.e. its wavelength. In this low-frequency region, scattering effects from the inhomogeneities can be neglected. The efforts to look for new strategies by considering more irregular shapes of cells and assemblies of cells have been one of the drivers of theoretical programs for understanding the origin of field distorsion [31, 88, 118]. Such modelling remains an extremely active area to address a range of the puzzles of EMB, and continues to push the envelope of new experimental probes to characterize in situ the electric field distribution, e.g. electric impedance tomography (EIT). Experimental measurement of the anisotropy ratio of the conductivity tensor can be performed thanks to magnetic resonance EIT (MREIT). MREIT is based on reconstructing images of true conductivity with high spatial resolution by obtaining current density information using magnetic resonance imaging and measuring surface voltage potential [88]. Theoretical analysis of the estimating the effective permittivity of composites calls for the calculation of polarizability of the inhomogeneities that compose the material. This is an old problem in physics dating back to the very beginning of modern electromagnetism and has been the subject ofsome considerable debate, see e.g. [119]. Many mixing rules have been suggested to describe the permittivity of cells in suspensions [102–103,106,115,Pethig]. In the modern scientific parlance, mixing rules refer to formulae that give the effective permittivity of a mixture as a function of the permittivities of the pure species and their volume fractions only. Thus, they do not distinguish between material types with different microstructure. Unfortunately, with the passing of time, considerable confusion with the term has arisen. As pointed out earlier, the strength of these modeling methods lies in their predictive ability. Performing direct measurements on each material sample, for all possible phase property values and volume fractions, is prohibiting from a time and cost standpoint. Empirical relations are more useful for correlating data than predicting them. These formulas have often been predicated on heuristic arguments, and some of them contain free parameters that need be determined by fitting experimental results. Although these empirical descriptions generally have limited transferability, and their ability to give quantitative results should always be carefully scrutinized, such approaches can be quite useful in exploratory studies. It is hardly surprising, therefore, if, when applied to the same problem, different formulas give badly disparate predictions. Empirical laws also may be manipulated to isolate a certain physical effect, thus providing a degree of freedom that is generally not available to the first principles approaches. The only assumptions in these approaches are that the mixed medium is considered to be linear and time invariant. There exist many empirical laws which can be classified into this category. For example, the Maxwell Garnett equation, which describes the effective permittivity of a suspension of isolated spherical inclusions randomly positioned within a host material, can be derived by solving the electrostatic equations with the corresponding boundary conditions at the interfaces. Another mixture equation [99–102] was derived for the case of a double shell model of cell, i.e. for the cytoplasm (permittivity εc )-membrane (permittivity εm and thickness d m ) spherical

4.4 Dielectropheresis and Electrorotation of Cells

143

particle of outer radius R embedded the ECM (permittivity εe ); the cell cytoplasm embedding a nucleus envelope of thickness d n and permittivity εn , and a nucleoplasm of permittivity εnc . This equation may be expressed in the form ε = εm

2(1 − ϖ1 ) + (1 + 2ϖ1 )E1 , (2 + ϖ1 ) + (1 − ϖ1 )E1

(4.70)

/ )3 ( ( / )3 2 )+(1+2ϖ2 )E2 with ϖ1 = 1 − dm R , E1 = εεmc 2(1−ϖ , ϖ2 = Rn (R − dm ) ,E2 = (2+ϖ2 )+(1−ϖ2 )E2 / )3 ( εn 2(1−ϖ3 )+(1+2ϖ3 )E3 , ϖ3 = 1 − dn Rn , E1 = εεncn . In the general case where one εc (2+ϖ3 )+(1−ϖ3 )E3 considers dielectric losses, every real-valued permittivity in Eq. (4.70) should be changed in ( σx ) . ε0 εx → ε0 εx − j ω

(4.71)

4.4 Dielectropheresis and Electrorotation of Cells The ability to manipulate cells remotely without direct contact is of utmost importance for applications in microfluidics and biotechnology [120–126]. When a cell is exposed to a non-uniform electric field, either dc or ac, electric charges accumulate accumulate in boundaries and the cell experience a DEP translational force or torque [120, 121]. The direction of the DEP force is independent of the direction of the field, but only on the field gradient. Electrical forces act on both the cells and the ECM and have their origin in the charge and electric field distribution in the system. DEP requires only simple instrumentation and can generate positive (Fig. 4.15 (left panel) as well as negative forces Fig. 4.15 (right panel). In a certain frequency range, some neutral cells are more polarizable than the ECM and experience positive DEP forces: they are attracted toward the region where the gradient of electric field is higher [122, 124]. In contrast, the right panel of Fig. 4.15 illustrates the case corresponding to cells that are less polarizable than the ECM: the forces are directed away from from the high electric field region. By applying an appropriate local electric field pattern, any particle with a dielectric constant different to that of the surrounding medium can be manipulated with DEP. Nowadays, DEP and electrorotation (EROT) techniques are widely used to manipulate, transport, separate and sort different types of particles [122, 124, 127, 128]. There are two theoretical approaches that are commonly used to evaluate the DEP force and torque on a spherical polarizable object (of radius R and permittivity εcm ), such as cell, suspended in a homogeneous, isotropic dielectric fluid (permittivity εe ) which is subjected to nonuniform ac (quasistatic) electric fields. The reader is referred to Appendix 3 for the basic definitions of MST and EF. The DEP force can be calculated in two ways. The first, and more fundamental is based on the MST formulation, i.e. Eq. 4.99) [20]. The second approach involves considering effective

144

4 Analytical Approaches of EMB at Multiple Scales

Fig. 4.15 Schematic principle of the DEP mechanism for a spherical object submitted to a nonuniform electric field (from [122]). The figure shows the polarization, induced net dipole and DEP force direction for a particle that is (left, positive DEP) more polarizable than the surrounding medium and (right, negative DEP) less polarizable

dipole or multipole moments. The object’s dipole has two origins [121, 123, 124] Firstly, there is the permanent dipole which is due to the reorientation of the charges on the object’s surface from the action of the electric field. The force on a dipole in an electric field is given by Eq. (4.90) [20]. Several authors have shown that the differences observed in the force calculations between the two different simulations methods arise because the approximations in the two methods are taken to different orders [120–122, 129]. In Eq. (4.90), the induced higher order multipolar moments other than dipole moment are neglected. We further assume that the dipole is small compared to the length scale of the nonuniformity of the imposed field. When the object is suspended in an electrolyte and placed in an electric field, the charges inside the object and inside the medium will be redistributed at the objectmedium interface depending on the polarizability of the object and the medium. The rule of thumb is as follows. If the polarizability of the object is larger than that of the medium, more charges will accumulate at the object’s side. If the polarizability of the medium is larger than that of the object, more charges will accumulate at the medium’s side. This non-uniform distribution of charges induces a dipole across the object aligned with the applied electric field. When the object-medium system experiences a non-uniform electric field, the object feels different forces (Fig. 4.15) for the case of a spherical object. A polarizable sphere of permittivity εcm embedded in a medium of permittivity εe , Eq. (4.39), has for effect of perturbing the electric field leading to an effective moment

4.4 Dielectropheresis and Electrorotation of Cells

( p = 4π εm

145

) εcm − εe R3 E = αE, εcm + 2εe

(4.72)

where E is the quasistatic field imposed by electrodes, a is the polarizability of the object, and the term between brackets is termed the Clausius–Mossotti factor. Because the particle is a lossless spherical object, the moment p is parrallel to E. Evaluating the force on the polarizable object, which is a cell in an ECM, and making use of Eq. (4.92) of Appendix 3 yields ( F = 2π εm

) εcm − εe α R3 ∇E 2 = ∇E 2 . εcm + 2εe 2

(4.73)

According to Eq. (4.73), and in accordance with our earlier remark, an object will be either attracted to or repelled from a region of strong electric field intensity, depending on whether εc > εe , or, εc < εe respectively. The corresponding torque, i.e. Eq. (4.91), is zero because the dipole moment and electric field are always parallel. Jones has calculated the different multipolar force contributions [121]. Now, if the object under consideration is modelled as a spherical CS core (with permittivity εc ) and shell with permittivity εm and thickness d m ), and if we again assume that the nonuniformity of the electric field is weak on the scale of the object’s size, it can be shown that Eq. (4.73) can be changed in Eq. (4.74) by replacing α by Eq. (4.38). From examination of Eq. (4.73), we observe that DEP is a nonlinear phenomenon, is present only when the electric field is non-uniform, does not deend on the polarity of the electric field, is proportional to the object volume, and depends on the electrical properties of the object and the external medium. Now, assuming sinusoidal time variation for the nonuniform and stationary electric field, the time-averaged dipolar DEP force on a spherical object can be written [121, 129] as ( ⟨F⟩ = 2π εm Re

) εcm − εe R3 ∇E2rms , εcm + 2εe

(4.74)

where Re(…) means the real part, Erms is the root-mean square of the applied electric field, and the following substitution permittivity Eq. (4.71). Generally, the object will be driven out a high field region at low frequencies and into a hight field region at mid frequencies and out a high field region again at very high frequencies [130]. The DEP crossover frequency (CF), which is the angular frequency point for which the DEP force switches from positive to negative DEP, reads / ωCF =

(σe − σcm )(σcm + 2σe ) −1/2 = τMWS ε0 (εcm − εe )(εcm + 2εe )

/

(σe − σcm ) , ε0 (εcm − εe )

(4.75)

cm +2εe where τMWS = ε0 σεcm denotes the MWS polarization time characterizing the +2σe β dispersion (Table 4.1). It is the frequency where the conductivity of the object

146

4 Analytical Approaches of EMB at Multiple Scales

is equal to that of the medium. Making use of a RC model of a single cell ˜ m = R˜ −1 C˜ m and conductance G with capacitance m , one arrives also at ωcross = /( )2 2 . If the particle is in a rotating electric field, the ˜m ˜ m − 9R2 G √1 4σe − RG 2 2RC˜ m

dipole continually tries to follow the field, a torque is exerted on the induced dipole which causes the particle to rotate [121, 129]. In that case, the time-averaged torque reads ) ( εcm − εe ⟨N ⟩ = −4π εm Im R3 |E|2 , (4.76) εcm + 2εe where Im(…) indicates the imaginary part. Thus, the torque depends on the imaginary −εe , which is nonzeo only if there is a loss mechanism. part of εεcmcm+2ε e Besides characterization of polarization effects, DEP and EROT of cells have been well studied for distinguishing, trapping and selectively manipulating cells [125, 128, 131]. In closing, it is also interesting to observe that coupled hydrodynamic, electric, and thermal systems have been studied by making use of of the incompressible Navier–Stokes equations and continuity equations [132]. We have presented a brief sampling of some of the important implications of EMB and its underlying resurgence and persistence in recent years. While far from being exhaustive, the issues discussed in this chapter yield tractable testable examples that can be rigorously studied in detail. Taken together, the diversity of subjects presented here provides a valuable and lively snapshot of a field that is in rapid development. Further, the diversity of problems iillustrated here demonstrates how themes such as self-assembly and organization, dynamics of pore formation, and cell interaction in a tissue model, all play out in EMB. Despite its significant successes, the SNK model can only be considered an effective (continuum) description of EP which leaves many unanswered questions, but it continues to inspire hundreds of research papers yearly. Among these so-far unanswered questions are the stochastic origin of the pores and an explanation for the ability of cells to transit constrictions 30–80% smaller than their own diameter [133]. Any explanation for these above mentioned puzzles must involve physics beyond the continuum description. The SNK also does not explain the observed asymmetry of pattern formation at membranes due to ED. Any explanation for these above mentioned puzzles must involve physics beyond the SNK model. Further, many themes that permeate modern topics in EMB such as the importance of cell morphology, the interplay of mechanics and organization at many different scales, all play a role in the following chapter. Thus, EMB has had and will continue to have an enormous impact for the study of biological materials [134]. Given the interesting aspects of EMB that have been uncovered in terms of computational multiphysics models, we next move on to present important recent developments in this direction.

Appendix 1: Electric Dipole and Polarizability

147

Appendix 1: Electric Dipole and Polarizability The electric dipole moment for a pair of opposite point charges ± e relative to an arbitrary origin is defined as er, where r is the position vector from the origin to the charges. Its direction is toward the positive charge. Its field at large distances, i.e. distances large in comparison to the separation of the two charges, depends almost entirely on the dipole moment as defined above. Recall that for a collection of point charges ei and position vectors ri the total dipole moment of can be written as p=

∑

ei ri .

(4.77)

i

Now if a dielectric ∑ material consisting of elementary charges ei contains no net charge, then we have i ei = 0, implying that the electric moment is independent of the choice of the origin. Eq. (4.77) can be also written as p = (rp − rn )Q, ∑

ei ri

∑

(4.78)

ei ri

where rp = i,positive and rn = i,negative represent the position vectors for posiQ Q tive and negative charges, respectively, and Q denotes the total positive charge. By definition, Eq. (4.102) can be also expressed as p = aQ,

(4.79)

where a denotes the distance separating the two point charges e and –e. More precisely, Eq. (4.79) defines the electric dipole moment of a system of charges with zero net charge. From Eq. (4.78), one sees that the dipole moment vector is directed from the negative charge to the positive charge because the position vector of a point is directed outward from the origin to that point. Such system represents a physical (finite) electric dipole. Now, from this definition one can introduce the notion of point (ideal) dipole in the following way: the vector position a is divided by the number of charges, n, and the charge e is replaced by en. Then, the ideal dipole is obtained in the limit n → ∞. More generally, for a continuous distribution of charge confined to a volume ∫ p(r) = ρ(r)rd τ . For a V, the corresponding equation for the dipole moment is V ∑ distribution of point charges, ρ(r) = i ei δ(r − ri ), where δ(r) is the 3D Dirac delta distribution. Substitution of this expression in p(r) is equivalent to Eq. (4.77) when the system of charges has an overall neutral charge. Many natural molecules are examples of finite electric dipole moments (also termed permanent dipole moment). This is because the barycenters of the positive and the negative charge distributions do not coincide. Molecules having such kind of permanent dipole moments are called polar molecules, e.g. water molecule (Fig. 4.16). Otherwise, non-polar dielectrics are materials which do not possess permanent dipole moment, e.g. hydrocarbons.

148

4 Analytical Approaches of EMB at Multiple Scales

H p

+

105°

+

H

Oxygen

−

Fig. 4.16 Dipole moment of the water molecule. The asymmetry of the water molecule leads to a dipole moment in the symmetry plane pointed toward the more positive hydrogen atoms. The measured magnitude of this dipole moment is p = 6.2 × 10–30 Cm. Treating this system like a negative charge of 10 electrons and a positive charge of 10e, the effective separation of the negative / and positive charge centers is a = p 10e = 0.0039 nm. This charge separation is very small compared to an atomic radius, i.e. 0.15 nm for the effective radius of H in liquid form. The polar nature of water molecules allows them to bond to each other in groups and is associated with the high surface tension of water

Apart from these permanent dipole moments, temporary induced dipole moments can arise when a particle is submitted to an external electric field. Under the influence of an electric field, positive and negative charges are moved apart. Thus, the particle is polarized. The values of molecular dipole moments are usually expressed in Debye units (1 D = 3.33 × 10–30 Cm). The permanent dipole moment of non-symmetrical molecules is typically in the range from 0.5 to 5 D, e.g. measurements reveal that water has a dipole moment of 6.2 × 10–30 Cm = 1.85 D. When a polarizable body is placed in a uniform electric field E 0 (caused by a charge distribution) in vacuum its dipole moment will in general change. Let us define the induced dipole moment p as the difference beween the dipole moments before and after application of the electric field. In most cases polarizable bodies are linearly polarized, that is we can write p = αE0 ,

(4.80)

where α is the scalar polarizability of the body. Hence, polarizability is defined as the relation between the dipole moment induced in an object, and the incident electric field which induces the dipole moment. The value of α depends on the shape and the material of the object, and the permittivity contrast between the object and its environment. Three comments are in order. First, much more complicated expressions have been proposed for other canonical shapes, but generally the problem of deriving α must be solved analytically [135, 136]. Second, we note that in the case of a dielectric sphere the entire volume is polarized, while for the conducting sphere the induced dipole moment arises from surface charges alone. Third, in Eq. (4.80) α is defined as a scalar quantity. For a class of simple, symmetric objects, e.g. regular polygons and polyhedra [137], α is a scalar. However, the polarizability can be generalized into a tensorial quantity provided that the polarization remains linear

Appendix 2: ITV and Electrodiffusion

149 ↔

p = α E0 ,

(4.81)

with α ↔ being a second-rank tensor (dyadic) that generally depends on inclusion orientation, shape, size, and ratio of the inclusion permittivity to the matrix permittivity [138]. This tensor has also been called the Pólya-Szeg˝o matrix [139]. In this case, the induced dipole moment will depend on the orientation of the inclusion with respect to the applied electric field, e.g. ellipse [138], and ellipsoid [139]. Although simple in principle, the calculation of the polarizability tensor for inclusions of general shape is a mathematical problem of notorious difficulty. Indeed, the ellipsoid [140] is the only three-dimensional shape for which exact analytical results have been obtained. It is also worth commenting that the Polya-Szegö polarization tensors appear in problems of potential theory related to certain issues arising in hydrodynamics and in electrostatics [141, 142].

Appendix 2: ITV and Electrodiffusion The dual purpose of this appendix is to briefly review several useful expressions for the ITV [143–145] and to clarify operationally what is meant by ionic electrodiffusion. For a constant dc homogeneous electric field, 3D spherical geometry, and polar angle θ measured from the center of the cell with respect to the direction of the applied electric field, the equivalent Eq. (4.1) reads Vm,dc =

3 ER cos θ. 2

(4.82)

For a ac homogeneous electric field, 3D spherical geometry, and polar angle θ measured from the center of the cell with respect to the direction of the applied electric field, the equivalent Eq. (4.1) becomes Vm,ac =

1 3 ER cos θ √ . 2 1 + (2π Fτm )2

(4.83)

Notice that if F is maller than 100 kHz, it is straightforward to verify that Vm,ac (θ ) ≈ Vm,dc (θ ). As a different question, one may ask how Eq. (4.82) is changed when spatial symmetry breaking emerges during slow transitions in ITV. A simple example that will turn out to be highly relevant to our work ahead is provided by a prolate spheroid with the axis of rotational symmetry aligned with the electric field. The answer is [142, 150].

150

4 Analytical Approaches of EMB at Multiple Scales

( 2 ) b − a2 a cos(θ ) ( √ 2 2) √ Vm = E , 2 2 2 a b+ b −a b sin (θ ) + a2 cos2 (θ ) b − √b2 −a2 ln a

(4.84)

where b and a are the radii of the spheroid in the directions parallel and perpendicular to the electric field, respectively. By another calculation for an oblate spheroid with the axis of rotational symmetry aligned with the field [143, 150], we get Vm = E

2

√ a b2 −a2

) ( 2 a − b2 a cos(θ ) ) ( √ . 2 2 a arctan √b2 −a2 − a b sin (θ ) + a2 cos2 (θ )

(4.85)

Three main limitations of analytical models for calculating the ITV are that the Schwan equation calculates V m assuming no pore formation, no electrodeformation, homogenized domains, and spatially constant ionic concentrations. However, there are many situations for which the latter assumption may be questionable. Thus, understanding the relationship between ITV and ion fluxes has become of great interest. Most of the numerical models for analyzing the ITV assume that the ion concentrations are constant in time and space. However, biological sytems require continual inputs of mass and energy to stay alive. They are open systems that require flow of matter and specific chemicals across their boundaries. Over the past several decades, there have been several attempts to describe the effect of ionic gradients across the cell membrane. For example, a series of electroneutral models for ionic electrodiffusion have been developed for homogenized domains [146–149]. These references develop a system of coupled, time-dependent, non-linear PDEs to describe ionic electrodiffusion in a given sample domain configuration. Knowing the ITV, it is of interest to develop an expression for the charge density perturbation on each membrane surface by making use of Poisson equation [146]. Considering again the single-shelled cell model displayed in Fig. 4.16, the electrostatic potential (in the zero flow limit7 ) obeys Poisson’s equation ∇ · (εk ∇)Vk = −

∑ i

ρk,i = −

∑

Fzk,i ck,i ,

(4.86)

i

where ρ k,i is the volume charge density, ck,i is the concentration, zk,i is the charge valence, F is Faraday’s constant (96 485 Cmol−1 ), the subscript i represents the ionic species of classical electrophysiology (i.e. i = Na+ , K+ , or Cl− ), and the subscript k = 1,2,3 corresponds to exterior, membrane, and interior regions, again. When steric effects are negligible and assuming electroneutrality, the Nernst-Planck equation for 7

Numerical models including ionic electrodiffusion (electrodynamics) and osmotic water flow (hydrodynamics) in biological tissues are discussed in [5,7], and generalize earlier models that deal only with electrodynamics. If flow field needs be considered, there is a supplementary contribution due to the advection-driven fluxes of ions in the right-hand side of Eq. (4.87). The transport of electrolytes in a Newtonian solvent through a rigid porous medium is also discussed in [5].

Appendix 2: ITV and Electrodiffusion

151

mass conservation allows us to evaluate the rate of change of the spatial dependence of the electric potential and ion concentrations ( ) ∑ ∂ck,i =∇· Dk,i |zzk,i|FRT ∇ck,i Vk + ∇ck,i , k,i ∂t i

(4.87)

where Dk,i is the effective diffusion coefficient of ion species i in the k domain, R is the gas constant (8.314 JC−1 mol−1 ) and T is the absolute temperature [147, 148]. In the right-hand side of Eq. (4.87), the first and second terms represent the drift of ions induced by the polarization of the cell diffusion, respectively of the i species. Observe that Eq. (4.87) is valid in the cytoplasm and ECM domains. We have also neglected the flow field that can be described by the Stokes equation. In the membrane domain, an effective diffusion coefficient Deff,i = (1 − p)Dm,i + pDi is often employed, where p is the proportion of the area occupied by pores, Dm,i = 0.01Di , and Di is the species diffusion coefficient in the membrane without pores [147]. Since the product of the volume of a pore and the ion density iss mall compared to one, the probability of finding an ion in the pore iss mall and we must interpret ck,i and Vk as ensemble averages. Values of the different parameters used in Eq. (4.87) are listed in Table 4.2 for the most abundant electrolytes in biological systems. In Ref. [150], DeBruin and Krassowska evaluated the ion-specific EP density current jEP , Eq. (4.6), as a function of V m and predicted that an electroporated single Table 4.2 Parameter values of electrodiffusion models for ions typically present in an excitable cell: sodium Na+ , potassium K+ , and choride Cl− (physiological temperature) Symbol

Parameters Electric charge

Na+

zNa

Electric charge Cl− Electric charge

Units

+

zK

Sources

1

zCl −

K+

Value or range -1

+

1

Diffusion coefficient Na+

DNa +

m2 s−1

1.33–1.78 × 10−9

[146–149]

Diffusion coefficient

Cl−

−

m2 s−1

2.03–3.83 × 10−9

[146–149]

Diffusion coefficient

K+

+

m2 s−1

1.96 ×

DCl DK

10−9

[148]

Initial intracellular Na+ concentration

0 ccNa +

mM

12–22

[146–149]

Initial ECM Na+ concentration

0 ceNa +

mM

100–151

[146–149]

Initial intracellular K+ concentration

0 ccK +

mM

125–155

[148–150]

Initial ECM K+ concentration

0 ceK +

mM

4–4.5

[146, 148–150]

Initial intracellular Cl− concentration

0 ccCl −

mM

4–4.5

[146–150]

Initial ECM Cl− concentration

0 ceCl −

mM

104–123

[146–150]

152

4 Analytical Approaches of EMB at Multiple Scales

cell will have a ITV profile that is symmetric about the equator and a pore density distribution that is larger at the hyperpolarized end of the cell. More sophisticated multiphysics models including the effect of electrolyte fluid convective transport, thermal gradient, osmotic pressure, and mechanical forces have been proposed to describe ion channels [146, 151]. For example, Dk,i in Eq. (4.87) needs to a position dependent function in many situations, such as ion channels [152]. In like fashion with a non-vanishing flow velocity, Eq. (4.87) should be modified so that the rate of change of the ion species i is balanced by the convective transport of the incompressible fluid flow, density gradient, electrostatic gradient and the flux of the fluid kinetic energy. Additionally, animal cells avoid osmotic lysis because they can balance osmotic pressure between the two sides of their membrane by actively pumping out Na+ ions with the Na+ -K+ -ATPase enzyme [153]. Because of this pump-leak mechanism, the regulation of cell volume and membrane polarization are coupled by the pump-leak mechanism, which renders the membrane impermeable to Na+ ions. However, superficial charges of the membrane modify the ion distribution near the membrane sides, i.e. electric double layer [146, 153]. Numerical simulations of Eqs. (4.86, 4.87) have been used to study the double layer charging for the scenario of overlapping double layers, i.e., when the double layer thickness is comparable to the pore radius in the membrane [154].

Appendix 3: Maxwell Stress Tensor and Electrostatic Force Acting on an Isolated Body in an Electric Field There have been several attempts to address this question, particularly in the study of the dielectric properties of biological cells under the influence of an external electric field [155–165]. On the other hand, the electrical polarizability of biological cells in the presence of a layer of localized, partially bounded, charges at the two cell membrane interfaces has been also well determined within the dipolar approximation […]. In this we find several interesting implications of the relationships between MST and polarizability with minimal mathematic complexity. The MST is critical for practical force calculations in numerical codes. The purpose of this appendix is to propose an expression for the EF on a homogeneous, spherical-shaped body resulting from the application of an electric field by making use of the MST. Consider first a small dipole of charges + q and –q and a denotes the distance separating the two point charges (Fig. 4.17). By small, we mean that |a| is small compared to the length scale of the imposed electric field E. Defining the net Coulomb force F = qE(r + a) − qE(r), we have F = qa · ∇E,

(4.88)

by developing this expression as Taylor series up to first order. From the very definition of the dipole moment, the EF can be approximated as

Appendix 3: Maxwell Stress Tensor and Electrostatic Force Acting …

153

Fig. 4.17 Definition of parameters used in the calculation of the EF on a small dipole due to an electric field E (adapted from [160])

F = p · ∇E.

(4.89)

Hence, a net force occurs if ∇E /= 0. That is if the electric field is inhomogeneous. Within this approximation, the net electrostatic torque writes N=

−a a × q E+ × (−q)E = p × E. 2 2

(4.90)

The dipole contribution to the total electric field cannot exert a force on itself and therefore is not included in E. Apart from acting on electrostatic charges electric fields exert forces on polarized and polarizable particles. Consider a homogeneous dielectric sphere (relative permittivity ε1 and conductivity σ1 ) of radius a embedded in a host medium (relative permittivity ε2 and conductivity σ2 ) submitted to an electrostatic field E0 . Substitution of Eq. (1.2.16) into (4.89) yields F = 2π ε2 ε0 a3

ε1 − ε2 ∇E02 , ε1 + 2ε2

(4.91)

which is the standard expression for the dielectrophoresis force on a dielectric sphere [158–160]. Observe that the dielectrophoretic force can lead to particle chaining in assemblies of particles. Applications to fluidics were also largely considered in the literature [159]. Now if one considers an electric field with harmonic time dependence E = E0 exp(−jωt), the time-averaged EF and torque on the dielectric particle read [161, 162] respectively ) ε1 (ω) − ε2 (ω) 2 ⟨F⟩ = 2π ε2 ε0 a Re ∇Erms , ε1 (ω) + 2ε2 (ω) (

3

where E rms is the root-mean square magnitude of the ac electric field, and

(4.92)

154

4 Analytical Approaches of EMB at Multiple Scales

) ( ε1 (ω) − ε2 (ω) ⟨N⟩ = −4π ε2 ε0 a3 Im E2. ε1 (ω) + 2ε2 (ω) 0

(4.93)

Thus, from Eq. (4.92), a lossy polarizable particle experiences either a positive or a negative EF depending on the particle’s and medium’s permittivity and of frequency. If the particle is more polarizable than the medium, the particle is attracted to the high intensity electric field regions. Conversely, if the particle is less polarizable than the medium, the particle is repelled from the high intensity field regions. Note that combining Eq. (4.90) and Eq. (1.2.16) gives zero for the torque, because the dipole moment and electric field are always parallel for a spherical particle. To escape from this restriction, the particle must be lossy, nonspherical, or possess a permanent dipole moment. The frequency specta of both the EF and torque can provide important information on the dielectric properties / of polarizable particles. Separating the real and imaginary components of εk − jσ/ k ωε0 in Eq. (4.91) and writing the complex permittivity of each phase as εk − jσk ωε0 gives [ ⟨F⟩ = 2π ε2 ε0 a

3

] 2 σ1 − σ2 ω2 τMWS (ε2 − ε2 ) 2 ( ) ) +( , ∇Erms 2 2 1 + ω2 τMWS 1 + ω2 τMWS (σ1 + 2σ2 ) (ε1 + 2ε2 ) (4.94)

and [ ⟨N⟩ = −4π ε2 ε0 a3

] 3ωτMWS (ε1 σ2 − ε2 σ1 ) ( ) E2, 2 1 + ω2 τMWS (ε1 + 2ε2 )(σ1 + 2σ2 ) 0

(4.95)

2 is the MWS relaxation time. CFs, at which the EF changes where τMWS = ε0 σε11 +2ε +2σ2 ) ( (ω)−ε2 (ω) = 0,8 i.e. from attraction to repulsion or vice versa, correspond to Re εε11(ω)+2ε (ω) 2 for a lossy particle [157, 162, 163]

/ ωCF

1 = ε0

(σ2 − σ1 )(σ1 + 2σ2 ) −1/ 2 = τMWS (ε1 − ε2 )(ε1 + 2ε2 )

/

(σ2 − σ1 ) . ε0 (ε1 − ε2 )

(4.96)

Consequently, the EF exerted by nonuniform ac electric fields can be harnessed to move and manipulate polarizable particles suspended in liquid media, e.g. biological cells (see Chap. 6). Using rotating electric fields, controlled rotation (torque) can be induced in these same particles. It is worth noting that in this approach multilayered particles (such as those considered in Sect. 4.3.1 and later in Chap. 6) have to be replaced by an equivalent homogeneous particle with a radius equal to that of the outermost shell. One may notice that contrary to the expression of the EF, Eq. ( ) For two complex numbers z = x − iy and z ' = x' − iy' , the condition Re z/z ' = 0 implies that ' ' xx = −yy . 8

Appendix 3: Maxwell Stress Tensor and Electrostatic Force Acting …

155

(4.96) does contain any reference to a length scale. Thus, ωCF is expected to be size independent. The above analysis, based on the approximation that only the dipole is induced, fails to predict observed behavior for a particle situated near a null or in a highly nonuniform field because of the neglect of induced higher-order moments. A published equivalent multipolar moment theory [163], while it enables calculation of these higher-order EF force contributions, is not easily interpreted or used because it is expressed in terms of a Legendre function expansion of the field. Another formulation of the multipolar EF theory expresses the net force as the gradient of a series of scalar electromechanical potential functions [160]. Another approach for calculating EFs is based on the MST wherein the stress tensor [Eq. (1.1.51)] ) ( 1 2 ∗ , E · E E − = εε T 0 2 ↔

(4.97)

is integrated over a domain ∂Ω of the particle ∫∫ ∫∫ ↔ ↔ F = ⃝ T · d S = ⃝ n · T dS, ∂Ω

(4.98)

∂Ω

where ε is the permittivity of the medium surrounding the particle, dS is the surface element, ∂Ω denotes the closed surface enclosing the particle, n is the outward unit perpendicular to the object surface, and dS is an infinitesimal surface∫∫ element [159, 160, 162].9 One may easily verify that Eq. (H10) reduces to F = ⃝ εε0 E 2 ndS in ∂Ω

the case of a perfect conductor [168]. The time-averaged force along the i (=x,y,z) 9

For an applied electromagnetic field with a frequency less than 100 MHz, the effects due to the magnetic component of the field can be ignored [1,11]. Hence in view of the quasistatic approximation, the exposure of a cell to an electric field is enough to study the cell response. A note of caution is in order here. It is well known that an alternative approach to EP is to use magnetic fields. Furthermore, applications of magnetic-field driven processes are quite broad because of the robustness of using a magnetic field, which avoids certain specific features often encountered with electric fields, such as electrochemical reactions. This approach has been taken by many research groups reporting contactless treatments of the biological objects using pulsed magnetic fields [12]. Since the initial work on MP in the last decade, which opened up an entirely new way of approaching CMP, the ability to manipulate CMP with magnetic fields has made great advances. This subject is particularly challenging because biological materials are diamagnetic, with a susceptibility very close to that of water. Thus, the differences in the diamagnetic susceptibilities of cellular components are very low, which leads to tiny effects. However, by using high-gradient magnetic field, a number of intriguing effects can be detected. In the case of cells suspended in a weakly diamagnetic medium, the volumetric force is proportional to the square of the field gradient. This gives the approach an enormous potential because significant changes in cell functions, shape and spatial organization can be made possible by using a spatially non-uniform magnetic field with a sufficient gradient (up to 107 Tm−1 ) [13]. The authors argued that the effects of magnetic fields with a gradient of the order of 106 Tm−1 can create magneto-mechanical stress and change the probability of the opening/closing of mechanosensitive ion channels [13].

156

4 Analytical Approaches of EMB at Multiple Scales

Cartesian coordinate axis can be calculated integral )] of Eq. (H10) [ (using the surface ∫∫ ⟨ ⟩ ⟨ ⟩ 1 ∑ 1 ∗ ∗ as ⟨F⟩i = ⃝ Tij nj dS with Tij = 2 Re εε0 Ei Ej − 2 k Ek Ek . Expanding Eq. ∂Ω

(4.97) in matrix format in a Cartesian basis leads to ⎤ Ex2 − Ex2 − Ex2 2Ex Ey 2Ex Ez 1 ⎢ ⎥ 2Ex Ey Ey2 − Ex2 − Ez2 2Ey Ez T = ε0 ε⎣ ⎦. 2 2 2 2 2Ex Ez 2Ey Ez Ez − Ex − Ey ⎡

(4.99)

Two remarks are in order. First, the force is entirely determined by the electric field on the arbitrary surface ∂Ω. Second, it is interesting to oberve that no material properties enter the expression for the force; the entire information is contained in the electric field. If the object deforms when it is subject to an electric field we have to include the electrostrictive forces. Similarly, the torque imparted by the electromagnetic field on a spherical particle of radius a reads [Eq. (1.1.53)] ∫∫ (↔ ) N = a ⃝ n × T · n dS.

(4.100)

∂Ω

This method is regarded as the most rigorous approach to the derivation of EF. Observe that Eqs. (4.98) and (4.100) involve the computation of surface integrals that may be difficult to obtain for certain geometrical shapes without relying to numerical simulations. If one considers dielectric particles the magnetic field should not be included in the stress tensor. Several authors [160–162, 165] showed that the differences observed in the EF calculations between the induced dipole moment and Maxwell stress tensor approaches arise because the approximations in the two methods were taken to different orders. It is also worth noting that Eqs. (4.98) and Eq. (4.100) can be generalized to a body of arbitrary shape by enveloping the particle and a portion of the surrounding medium with a spherical Gaussian surface [163].

Appendix 4: Electro-Thermal Effects in Biological Materials The purpose of this appendix is to provide a brief discussion of how electrical energy is dissipated into heat conduction in the context of biological materials in response to an electric field excitation. At this point it is fair to say that heat transfer and dissipation management are key issues for a continuum thermodynamics description of cell metabolism and functions [169]. What is unique about thermodynamics is its universality describing the properties of macroscopic systems as well as at the microscopic scale. Thus, heat transfer will proceed by thermal conduction and convection. Conduction is heat transfer by thermal fluctuations within a material without any motion of the material as a whole. For heat transfer between two plane surfaces, the rate of conduction heat transfer is described by Fourier law. Convection is heat

Appendix 4: Electro-Thermal Effects in Biological Materials

157

transfer by mass motion of a fluid when the heated fluid is caused to move away from the source of heat, carrying energy with it. Using 3D computational simulations of intracellular fluid motions, Howard and coworkers [170] were able to quantify the convective velocities that could result from the temperature differences observed experimentally. The interested reader in convection transfer is refered to studies in the archival literature, see e.g. [171]. Observe that multicellular organisms depend on convection to bring materials close enough to cells so diffusion to and across cell membranes can provide what the cell needs to live. Below the ionization threshold, interactions between electromagnetic waves and biological materials can be described by either thermal or nonthermal effects [172]. The former are related to the absorption of energy of the wave which is described by the Poynting theorem and induce molecular motion in the material leading to a rise of temperature by the Joule effect [172]. The latter correspond to alterations in the biological material, e.g. changes in protein conformation, free radical formation, which cannot be explained by a temperature increase [172]. What is important to notice is that the imbalance in terms of conductivity between the differents parts of the cell and the ECM may cause small temperature differences but significant spatial gradients which can trigger lipid peroxidation phenomenon [173, 174]. Several methods for sensitively probing temperature variations on nanometre scales have provided a major driving force for many model building and phenomenogical activities in biological, physical and chemical research in the last few decades. A thermometer capable of subdegree temperature resolution over a specific range of temperatures as well as integration within a living system can provide a powerful tool for many issues including temperature-induced control of gene expression, tumour metabolism, and cell-selective treatment of disease. By combining local lightinduced heat sources with sensitive nanoscale thermometry, it can also be possible to engineer biological processes at the subcellular level. Several sensitive nanoscale thermometry studies have been reported recently, e.g. Refs. [174–179]. For example, the local intracellular temperature distribution inside living cells can be characterized by fluorescence lifetime imaging microscopy [180]. Based on such cationic fluorescent polymeric thermometer, a temperature gradient among sub-cellular locations has been monitored with high temperature resolutions (0.01–1.0 °C) [179]. Recently, experimental evidence based on intracellular thermometry has suggested that cell nucleus and mitochondria are hotter than the cytoplasm and cell membrane environment, therefore introducing the possibility of localized temperature gradientdriven convective circulation of a 1 °C temperature differential between the heated nucleus and the outer cell membrane [180, 181]. Intracellular thermometry and the heat transfer properties at the cellular and subcellular scales have been also characterized by transient plasmonic imaging [182]. Thanks to its high temporal-spatial resolution, this plasmonic method allows the measurement of the thermal conductivity of different regions within single cells. Currently, there is interest across a broad range of disciplines in biophysics surrounding the thermodynamic description of biomaterials from both a fundamental and applicative view point, e.g. thermo-electricity, thermo-diffusion, and thermo-acoustics [10]. However, the status of electro-thermal

158

4 Analytical Approaches of EMB at Multiple Scales

effects in biological materials, and their effects in different pathologies, is still far from settled. The temperature distribution in the material is described by the following equation [10, 170] ρcp

∂T (t,r) = ∇(K · ∇T (t,r)) + J (r) · E(r), ∂t

(4.101)

where ρ is the mass density, cp is the specific heat capacity, T is the temperature, t is the time, K is the thermal conductivity, and J · E is the heat generation rate per unit volume, or the Joule heating term [170]. Typical values of the parameters in Eq. (4.102) are: ρ ≈ 103 kgm−3 , cp ≈ 3.5 × 103 Jkg−1 K−1 , and ρcKp ≈ 0.13 mm2 s−1 [170]. In the following we shall consider a homogeneous, linear and isotropic biological material. Taking into account convection transfer should include supplementary heat generation and heat sink terms in the right-hand side of Eq. (4.101) yielding the Pennes bioheat equation [171]. From Ohm’s law Joule heating can be written as J · E = σ E 2 , Eq. (4.101) reads K 2 ∂T (t, r) σ E(r)2 = ∇ T (t, r) + . ∂t ρcp ρcp

(4.102)

Equation (4.102) has no simple general analytical solution. However, a Gaussian profile may be thought to originate from radial cooling of an instantaneous line source of heat [183]. In that case, itegration of Eq. (4.102) can be performed in a 2D model to get temperature distribution between two cylindrical electrodes, ( the ( r )2 ) T (t, r) = A exp − B , where A and B can be determined by fitting the Gaussian profile to the measured temperature distribution. Another kind of solution to Eq. (4.102) is available if radial cooling of the ) is approximated by 1D diffusion (/vessel wall ρc

p x √ , where C denotes a fit parameter in the x-direction, i.e. T (t, x) = Cerf 4K t and erf(x) is the error function [184]. In closing, it is worth noting that several authors discussed the cases of nonlinear and/or anisotropic materials by introducing a sigmoid description of in the former case and a conductivity tensor for the latter situation [171, 185].

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4 Analytical Approaches of EMB at Multiple Scales through state-space modeling. Bioengineering 9, 499 (2022); A. Mohammadi, L. Bianchi, S. Asadi, P. Saccomandi, Measurement of ex vivo liver, brain and pancreas thermal properties as function of temperature. Sensors 21, 4236 (2021); R.V. Davalos, B. Rubinsky, Temperature considerations during irreversible electroporation. Int. J. Heat Mass. Transf. 51, 5617–5622 (2008); Y. Mandel, B. Rubinsky, Treatment of uveal melanoma by non-thermal irreversible electroporation: electrical and bioheat finite element model of the human eye. J. Heat Transf. Trans. ASME 134, 111101 (2012); K. Kurata, S. Nomura, H. Takamatsu, Three-dimensional analysis of irreversible electroporation: estimation of thermal and non-ther-mal damage. Int. J. Heat Mass Transf. 72, 66–74 (2014); P A. Garcia, R.V. Davalos, D. Miklavˇciˇc, A numerical investigation of the electric and thermal cell kill distributions in electroporation-based therapies in tissues. PLoS One 9, e103083 (2014); M.J.C. van Gemert, P.G.K. Wagstaff, D.M. de Bruin, T.G. van Leeuwen, A.C. van der Wal, M. Heger, C.W.M. van der Geld, Irreversible electroporation: just another form of thermal therapy? Prostate 75, 332–335 (2015); M.R. Wang, N. Yang, Z.Y. Guo, Non-Fourier heat conductions in nanomaterials. J. Appl. Phys. 110, 064310 (2011) L. Challis, Mechanisms for interaction between RF fields and biological tissues. Bioelectrochem. 26, S98-S106 (2005); J. Lin, P. Bernardi, Computational methods for predicting field intensity and temperature change, ed. by F. Barnes, B. Greenebaum, in Bioengineering and Biophysical Aspects of Electromagnetic Fields (CRC Press, 2007); R. Funk, T. Monsees, N. Özkucur, Electromagnetic effects: from cell biology to medicine. Prog. Histochem. Cytochem. 43, 177–264 (2009); R. Funk, T. Monsees, Effects of electromagnetic fields on cells: Physiological and therapeutical approaches and molecular mechanisms of interaction: a review. Cells Tissues Organs 182, 59–78 (2004) T. Heimburg, Thermal Biophysics of Membranes (Wiley, Hoboken, 2008); M. Cifra, J. Fields, A. Farhadi, Electromagnetic cellular interactions. Prog. Biophys. Mol. Biol. 105, 223–246 (2011) W. Milestone, Q. Hu, A.M. Loveless, A.L. Garner, R.P. Joshi, Modeling coupled single cell electroporation and thermal effects from nanosecond electric pulse trains. J. Appl. Phys. 132, 094701 (2022) G. Kucsko, P.C. Maurer, N.Y. Yao, M. Kubo, M. Noh, H.J. Noh, P.K. Lo, H. Park, M.D. Lukin, Nanometre-scale thermometry in a living cell. Nature 500, 54–58 (2013) S. Dutz, R. Hergt, Magnetic nanoparticle heating and heat transfer on a microscale: basic principles, realities and physical limitations of hyperthermia for tumour therapy. Int. J. Hyperthermia 29, 790–800 (2013) J.M. Yang, H. Yang, L. Lin, Quantum dot nano thermometers reveal heterogeneous local thermogenesis in living cells. ACS Nano 5, 5067–5071 (2011) M. Nakano, T. Nagai, Thermometers for monitoring cellular temperature. J. Photochem. Photobiol. C 30, 2–9 (2017) G. Baffou, H. Rigneault, D. Marguet, L. Jullien, A critique of methods for temperature imaging in single cells. Nat. Methods 11, 899–901 (2014) K. Okabe, N. Inada, C. Gota, Y. Harada, T. Funatsu, S. Uchiyama, Intracellular temperature mapping with a fluorescent polymeric thermometer and fluorescence lifetime imaging microscopy. Nat. Commun. 3, 705 (2012); S. Uchiyama, T. Tsuji, K. Ikado, A. Yoshida, K. Kawamoto, T. Hayashi, N. Inada, A cationic fluorescent polymeric thermometer for the ratiometric sensing of intracellular temperature. Analyst 140, 4498–4506 (2015); T. Hayashi, N. Fukuda, S. Uchiyama, N. Inada, A cell-permeable fluoresecent polymeric thermometer for intracellular temperature mapping in mammalian cell lines. PLoS One 10, e0117677 (2015); R. Tanimoto, T. Hiraiwa, Y. Nakai, Y. Shindo, K. Oka, N. Hiroi, A. Funahashi, Detection of temperature difference in neuronal cells. Sci. Rep. 6, 22071 (2016) D. Chrétien, P. Bénit, H.H. Ha, S. Keipert, R. El-Khoury, Y.T. Chang, M. Jastroch, H.T. Jacobs, P. Rustin, M. Rak, PLoS Biol. 16, e2003992 (2018); R. Pinol, J. Zeler, C.D.S. Brites, Y. Gu, P. Tellez, A.N. Carneiro Neto, T.E. da Silva, R. Moreno-Loshuertos, P. Fenandez-Silva, A.I. Gallego, L. Martinez-Lostao, A. Martinez, L.D. Carlos, A. Milan, Nano Lett. 20, 6466 (2020);

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Chapter 5

Computational Approaches

If you ask Nature the right question, she will give you the right answer. Richard P. Feynman

As biology advances into this century, new levels of quantitative understanding of biological systems are emerging from the strong partnership between experiment, theory, and simulation, with accurate simulation of these systems being of paramount importance for understanding problems involving many interacting degrees of freedom. Particularly due to insights from numerical algorithms, computational approaches, in particular, have played a big role in recent developments by allowing a study of regimes of parameters that would otherwise be intractable. Computational models cannot replace experiments nor they can prove that particular mechanisms are at work in a specific situation. But they can demonstrate whether or not a given mechanism is sufficient to produce an observed phenomenon. Models that are used to describe the multiscale and multiphysics in biophysics are generally too complex to allow exact solutions to be written down. In generic terms, computational approaches use numerical algorithms to study the physical principles underlying biological phenomena and processes. They provide means of approximating solutions for theoretical biophysical problems lacking closed-form solutions, and simulating systems which can be often checked by experiments. The choice of an appropriate computational approach represents an important part of the modelling process because it will impact the accuracy of the solution and the requested computational time. It is worthwhile at the outset of this chapter to define what it means to model something: recognize the implications of the conceptual, mathematical, and algorithmic steps of model construction, and comprehend what the “big picture” defined in the previous chapters can and cannot do. Living organisms are complex systems. Complexity means that these systems are composed a different interconnected parts and exhibit collective and emergent properties which do not necessarily arise from the properties of the individual components [1]. Modeling the emergent cell membrane

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9_5

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EP behavior requires simultaneous dynamical descriptions at continuum and molecular scales. Within this context, we need new methods and concepts to deal with that complexity, a need we share with many different disciplines, ranging from economy to sociology. Here, most of the space will be devoted to continuum-based approaches relying on FE simulations, some of which connect the cytoskeleton, cell properties, and tissue mechanics. The FE method is a standard technique used to solve continuum-scale constitutive equations arisng in EMB and is based on finding a functional that is minimized or maximized by nature, such as, e.g. the energy in the electric field. This functional is then extremized over a discretized space to determine the approximate field. As will be explained in a moment the FE method has long been used to compute electromagnetic and mechanical field problems in models which are not overloaded by the many details of the topology of cells and tissues [2]. There have been several attempts to address the question of multiscale methods for homogenization problems in the mathematical literature [3]. However, it should be recognized at the outset that it is at the molecule scale that the functions of biological systems emerge, i.e. the “ultimate scale” is the size of the molecule. Nanophysics is thus needed to characterize nanopores that are 1 V [20]. Upon approach to this limit, we observe a significant deviation from prolate cell morphologies as illustrated in Fig. 5.5. Consequently, the cell shape is no longer spheroidal. / A close inspection of the aspect ratio change follows the C m behavior up to b a ∼ = 1.4, beyond which the large strain produces complex axisymmetric cell shapes which can be described using a variety of different parameterized forms, e.g. Cassini curves [11]. This shows an important finding that the higher the ratio, the more dominant the effect of the ITV which increases the MC and decreases the MR. Another fact arising from our simulation is Rm (Vm ) scales as Vm−2 . With an understanding of the ITV sensitive capacitance behavior of a cell membrane under steady-state electric field excitation, we performed additional simulations to investigate the ellipticities and associated 3D cell shapes, and whether they might be prolate or oblate depending on the polarization and magnitude of the electric field excitation [14]. There is no general analytical formula of V m which can be used for arbitrary cell shape, but analytic calculations allow one to compute V m for a prolate spheroid with the axis of rotational symmetry aligned with the electric field, i.e. Equation (4.84). Using the procedure outlined above, we performed a set of calculations for different values of the steady-state

5.2 Numerical Determination of Cm (Vm ) and Rm (Vm ) and Electrostatic …

193

Fig. 5.5 Membrane capacitance C m (blue lines), resistance Rm (green lines), and cell aspect ratio b/a (red lines) as a function of the exogeneous electric field. d m = 5 nm, R = 5 μm and Y m varies from 19 MPa (solid line) to 22 MPa (dashed line) to 25 MPa (dash-dotted line) 28 MPa (dotted line). a and b characterize the spheroidal cell being respectively the long and short semi-axes, θ denotes the angle between the surface normal and the field direction, from [14]

electric field ranging from 1 to 5 kV/cm, i.e. different values of the aspect ratio b/ a ranging from 1.02 to 1.36. We find that Eq. (4.844.84) reproduces the numerical results to better than 5% accuracy over the range of θ explored, where θ refers to the angle measured from the z axis with respect to the applied electric field direction (Fig. 5.3). Since the membrane thickness is not uniform, significant deformations of the membrane can arise (up to 5%). Observe that the effective membrane thickness d m under electric field excitation can be fully parametrized by dm = dm0 −c cos2 (q) + d sin2 (q),

(5.24)

where d m0 is the unperturbed membrane thickness, and c and and d are two constants [14]. Positive values of (d m − d m0 )/d m0 denote tensile strain whereas negative values denote compressive strain [14]. A 500-mV increase causes a 1.5% increase in C m , 0.7% d m decrease, and 0.7% Am increase such that the nominal volume of the membrane remains constant. This is corroborated by experimental data showing that the thickness of the membrane does not change by more than a few percent throughout the EP process [21]. While these simulations present an opportunity to witness complex dielectric phenomena in cell membranes they suffer from a number of limitations. Firstly, our analysis has focused on nonporated membranes. In Ref. [22], Weaver and coworkers estimated the contribution of transient aqueous pores to capacitance and found that the time average property of a large population of transient aqueous pores has similar features with a quadratic dependence of capacitance. Secondly, an important constraint of the continuum model comes from the fact that the MC and MR features are independent of the molecular details of the membrane in which they

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occur but are dominated by the geometry and can be explained with simple mechanical models. Schwan’s equation predicts that cells should porate at threshold electric fields that go as R−1 , however [23] found that the electric field needed to induce poration varied between three types of cell, but all confirmed the lack of size dependence. It is also worth referring to a recent study of Liang and coworkers [24]. In this study, the authors reported on a 3D analysis of the positive and negative optically induced DEP forces on cells stimulated by nonuniform ac bias potential. The MC per unit area and areal MR of four types of cells were derived by characterizing their DEP crossover frequencies using micro-vision techniques and were found of the order of 10–2 Fm−2 and 10–2 Ω m2 , respectively, i.e. remarkably close to our numerical data. Thirdly, we made the assumption about the scalar nature of the membrane permittivity. It can be argued that this assumption is not very physical (i.e. the cell membrane is heterogeneous and the electric-field induced stretching of the membrane exhibits a peculiar anisotropy), but nevertheless our results allow us to shed some light on the mechanical strain involved in the ED. Fourthly, in [14] we provide another figure when the conductivity ratio between the cytoplasm and the extracellular medium varies by two orders of magnitude but keep all other structural and material parameters constant. We find that C m and Rm are only weakly affected in the range of electric field magnitude explored. We further note that charge near the membranes (ionic double-layer) contributes with a capacitance that acts in series with C m . Within a mean field approximation [25], i.e. assuming that the double layer capacitance is equal to the capacitance of a planar capacitor with thickness equal to the Debye length and permittivity ε0 εe , a correction to the C m value is obtained at the percent level in accordance with experimental capacitance values for biomimetic bilayer membranes at different solution conductivities [26]. Fifthly, since we only considered a single cell, a statistical analysis that considers the influence of different variables was not possible. These trends, we believe, are real despite the limitations. Note also that this model is flexible to incorporate a variety of other biological attributes such as internal organelles and nucleus membrane. Here, the cell membrane is assumed to have elastic properties when stretched. An obvious next step is to consider viscoelastic (or even poroelastic [27]) cell materials; however, this is a much more difficult problem, because the mechanism by which the membrane is deformed is crucially affected by the detailed temporal sequence of electric field excitation. In closing this section, we briefly comment on the thermal dependence and electrical stimulation of C m [28]. By making use of rapid temperature pulses, C m was predicted to increase. This temperature dependence was related to an intramembrane thermomechanical effect wherein the phospholipid bilayer undergoes axial narrowing and lateral expansion accurately predicts a potentially universal thermal capacitance increase rate of ∼0.3%°C−1 [29].

5.3 Proximity-Induced ED and Membrane Capacitance Coupling Between …

195

5.3 Proximity-Induced ED and Membrane Capacitance Coupling Between Cells Topic: Research and development in the field of tissue engineering has gained considerable attention of late [11, 12, 30, 31]. While these studies provide important details about the role of an isolated cell stimulated by an applied electric field, the conundrum of generating a set of interacting cells remains less understood. Clearly, if one wishes to further exploit the potential of tissue engineering applications, a standardization of the methodology for characterization of engineered tissues is warranted. In that respect, electromechanical modeling of the transmembrane potential-dependent cell membrane capacitance can help further support and advance the field. Several experimental papers have studied the problem of membrane capacitance coupling (CaC) between cells [32, 33]. However, there is no numerical study aiming to compare the effects of proximity-induced ED and CaC between cells. The ability to control membrane CaC by modulating bioelectric properties such as ITV would be an invaluable tool for understanding how the electrical dynamics of a cell inside a tissue are affected by close boundaries. Here, we focus on an important question: how cell packing and proximity influence the effective mechanical and electrostatic properties of cell assemblies and tissues? So far, our answers to this question have been informal or incomplete. On the simulation side, lattices containing dilute concentration of cells [13] and randomly positioned cell assemblies [12, 34] addressed some of the issues with the interaction between cells. However, at what length scale electrostatic interactions among multiple cells become inconsequential and how does such length scale correlates with the mechanical strain state under electric field excitation remain unknown. While consideration of a large number of randomly distributed cells is physically more relevant for realistic conditions in a tissue, it is also important to isolate the individual effects of a set of interacting pair of cells in a first stage since it is computationally unfeasible to carry out calculations for large systems containing hundreds of cells. A major challenge is therefore to predict how a changing environment affects membrane CaC and alters cell shape. Computational Model: Here, we consider the specific assumptions and limitations: (1) the case of steady-state electric field excitation is first considered with magnitude E = 1.9 kV/cm. (2) a distinctive feature of the surface of a cell membrane in an aqueous environment is that it is negatively charged. As shown above, a constant everywhere on the cell membrane resting potential Vrest = −0.07 V superimposes to V m when an external electric field E is applied to the cell; (3) the cell is described as an elastic membrane surrounding an isotropic and homogeneous cytoplasm that is elastic; (4) our model does not capture the thin double layer of mobile cations adjacent to the extra-intracellular membrane surfaces; and (5) the membrane and cytoplasm are purely elastic materials with respective Young’s moduli set to Y m = 19 MPa and Y c = 1 kPa. We also assume that the ED of the cell does not degrade the mechanical properties of the membrane. Section 5.2 exploits the Rm C m analysis to understand the membrane charge dynamics. Within this setting, the simulation procedure for calculating C m and ITV, V m , builds on Sect. 5.2. Along those lines,

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5 Computational Approaches

numerical simulation of the difference between the ED force for a pair of cells and its counterpart for a single reference cell is performed. For all the simulations, the electric field is oriented along the z-direction as illustrated in the left panel of Fig. 5.6. This figure shows an example of a 2-cell configuration with distance r between the two cells and orientation angle denoted by θ which refers to the angle of the line joining the cell centers of mass relative to the electric field direction. The nominal values of the membrane thickness of the undeformed spherical cell is d m0 = 5 nm, cell radius (in the undeformed state) R = 5 μm, membrane conductivity σ m = 5 × 10–7 Sm−1 , membrane permittivity ε0 εm = 4.4 × 10–11 Fm−1 , cytoplasm conductivity σ c = 0.2 Sm−1 , cytoplasm permittivity ε0 εc = 7 × 10–10 Fm−1 , extracellular conductivity σ e = 0.2 Sm−1 , extracellular permittivity ε0 εe = 7 × 10–10 Fm−1 , membrane Young modulus Y m = 19 MPa, cytoplasm Young modulus Y c = 1 kPa, in coherence with the values given in Table 4.1. More specifically, two 2D square shaped configurations are designed (1-cell and and 2-cell, respectively) with respectively three and five subdomains: cytoplasm, membrane, and extracellular medium for the 1-cell configuration and cytoplasm

Fig. 5.6 (Left) A schematic diagram illustrating the geometry of the problem (not to scale). Two cells separated by a distance r oriented by an angle θ relative to the electric field direction. The orientation angle θ refers to the angle of the line joining the cell centers of mass relative to the electric field direction. As shown in this figure, by considering different values θ we can assign arbitrary anisotropies to the pair of cells configuration (from [14]). The computational box h has size = 120 μm; (right) Schematic of a typical square computational domain filled with the ECM to study the amplitude of the MST at point a' . The two cells are initially separated by distance r (not to scale). The orientation angle denoted by θ refers to the angle of the line joining the cell centers of mass relative to the electric field direction applied along the y-direction. Right and left boundaries are insulated while the top and bottom walls are prescribed with an applied potential. Dirichlet boundary conditions at the top and bottom boundaries are justified as the electric field is applied through discrete electrodes, a and a' define the points on the outer and inner layer of the cell membrane allowing us to calculate the ITV at the pole. Surface charge distributions of ± 6.2 × 10–4 Cm−2 were added to both surfaces Ain and Aout in order to account for a resting potential of −70 mV, from [14]

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197

1, cytoplasm 2, membrane 1, membrane 2 and extracellular medium for the 2cell configuration. Cytoplasms and membranes are assumed to be elastic homogenous phases and, for simplicity, the impact of the extracellular medium mechanical response is not considered. The electric field is applied using Dirichlet boundary conditions on top and bottom horizontal boundaries (ideal planar electrodes) such as V (z = 0) = 0 and V (z = h) = Eh, where E is the applied field modulus. Electric insulation boundary conditions (n·J = 0)2 are applied on the vertical faces of the system while current conservation boundary conditions are applied to all other interfaces in the system. Our results are based on the solving of Eqs. (5.17)–(5.23) [14]. We develop a two-stage computational algorithm for which the electrical part of the simulation is first solved then followed by its mechanical part. The ITV, V m , is computed by subtracting the cytoplasm–membrane interface potential (V m-in ) to the membrane– extracellular medium interface potential (V m-out ), i.e. V m = V m-out -V m-in . The absolute value of the V m difference between the 1-cell and∮ 2-cell configurations is computed |Vm2cells −Vm1cell |dl ∮ cell , where cellref dl on the contour of the reference cell and defined as ref∮ Vm1 dl cellref

cell

denotes the integration over the contour of the reference cell, and the subscripts 1cell and 2cells respectively refer to 1-cell and 2-cell configurations. The excess ED force exerted on the cytoplasm–membrane interface of the cell contour between the 1-cell ∮ ∮ and 2-cell configurations is defined as

cell ref

∮

|FED |2 cells dl− cell |FED |1cell dl ref ∮ |FED |1cell dl cell

and the absolute

ref

cell

|(|FED |2cells −|FED |1cell )|dl

value of the excess ED force reads as ref ∮ cell ref |FED | dl . The membrane thick1cell ness d m is locally computed by subtracting the position of all points on the deformed inner membrane contour to their corresponding image points on the deformed outer membrane contour, i.e. pairs of (a,a’) and (b,b’) points shown in Fig. 5.6. We define δ m = d m0 − d m, where d m is the effective membrane thickness under electric field excitation. The absolute value of the δ m difference between the 1-cell and 2-cell ∮ configurations is defined as

cell ref

∮

|δm 2cells −δm 1cell |dl

cell ref

δm 1cell dl

.

As a side remark on the influence of material parameters on ED, we don’t expect a significant effect due to Y m on θ c since the excess ED force is computed in the electrical part of our algorithm which is performed at an earlier stage of its mechanical part. Results and Discussion: Our results are discussed from two perspectives. Firstly, the main results of our analysis are shown in Fig. 5.7 as function of the parameters r and θ, the other parameters are held fixed. The separation distance-orientation angle diagram displayed in Fig. 5.7 shows the non-monotonic attenuation-amplification transition of the excess ED force at a critical value of the orientation angle θ c suggesting that anisotropy induced by the orientation angle has a significant influence on the relative sign of the ED force. Lower (resp. larger) values of θ tend to favor attenuation (resp. amplification) of the ED force for a pair of cells compared to the ED force for a single cell. 2

See Chap. 4 footnote 2.

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5 Computational Approaches

Fig. 5.7 Regions in the separation distance-orientation angle diagram for the excess ED force. The gray region indicates the region in the r − θ plane for which the absolute value of the excess ED force is 1, the cell is more polarizable than the ECM. The opposite limit, ν < ν α , occurs when cell behaves as a dielectric having low losses (low conductivity) and thus limiting the polarization of the intracellular medium. Since membrane charging occurs very rapidly compared to the oscillation time scale of the electric field [35], the electric field consequently remains confined into the ECM. On the contrary, if the cell is less polarizable than the medium, the electric field will essentially remain confined. The ECM becomes more polarizable than the cell. Furthermore, in Sect. 5.3 the analysis of the excess ED force for a pair of cells and its counterpart for a single reference cell allows us to determine a separation distance-orientation angle diagram providing evidence of a separation distance beyond which the electrostatic interactions between a pair of biological cells become inconsequential for the ED. Here, we seek to improve our theoretical understanding of the elastic membrane ED of a reference cell in close proximity of a neighboring cell under alternating electric field excitation by exploiting the anisotropic perturbation of the local electric field distribution. We also seek to unravel the role of frequency in modulating the attenuation-amplification transition of ED force which has not been fully considered in the literature. It is important to explore such possibilities and possible means to distinguish among these and other models so we can eventually learn the true nature of ED in cell assemblies.

5.4 Modelling Cell Membrane ED by Alternating Electric Fields

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Computational Model: Based on these motivations and observations, we consider a pair of cells in close proximity. The resulting setup is sketched in the right panel of Fig. 5.6. To reduce the computational complexities, we have only considered 2D geometries. However, the models presented in this study can be easily extended to 3D objects. More specifically, two 2D square shaped configurations are designed (single-cell and 2-cell, respectively) with respectively three and five subdomains. In Fig. 5.8, the single-cell model represents the reference cell. In the 2-cell model, we consider a pair of cells suspended in a square domain and in close proximity (Fig. 5.8), where L is the dimension along the x and y directions respectively. This dimension is found to be sufficiently large to have non-zero interactions between the periodic images of the lattice. We do not consider interfacial distributions of free charges although this can be a possible option to deal with charged interfaces even if in a realistically cellular environment, the free charges are distributed spatially rather than on the surface. Our results are again based on the solving of Eqs. (5.17)– (5.23), with the change in Eq. (5.18) to include the displacement current density J = σ E + 2π jvD, where D accounts respectively for the electric displacement, and ν denotes the frequency of the electric field. As we see, physics can depend very strongly on Vm , which is computed by subtracting the cytoplasm–membrane interface potential (V m-in ) to the membrane–extracellular medium interface potential (V m-out ), i.e. V m = V m-out − V m-in . Here we are only interested at the ITV at the pole V maa’ for the reference cell displayed in Fig. 5.8. Also note that we assume a resting potential Vrest set to − 0.07 V which is constant everywhere on the cell membrane. When an external electric field E is applied to the cell the ITV Vm superimposes to Vrest . The frequency dependent excess MST is defined at point a’ of Fig. 5.8 and 1cell −MST(ν)2cells , where subscripts 1cell and 2cells respectively correspond reads MST(ν) MST(ν=10Hz)1cell to the single-cell and 2-cell configurations. The cytoplasm is characterized by a complex (relative) permittivity εc (ν) = εc + σ c /(2π jε0 ν) covered by a confocal membrane characterized by a complex (relative) permittivity εm (ν) = εm + σ m / (2π jε0 ν) suspended in a continuous ECM characterized by a (relative) permittivity εe (ν) = εe + σ e /(2π jε0 ν). For illustrative purpose, the two cells are separated by a distance r/R = 0.2. The cells are subjected to an electric field for given amplitude set to 0.815 kVcm−1 . Material parameters can be found in Table 4.1 and in [14]. However, five comments are in order. Firstly, the overall membrane width is variably reported to be anywhere between 4 and 10 nm due to the numerous types of lipids and proteins. A value in the range 4–5 nm is most representative of the membrane shaved off from its outer and inner protrusions. Furthermore, the bilayer thickness of the membrane is critical in hydrophobic matching and it has been suggested that cholesterol is the principal modulator of bilayer thickness in eukaryotic cells. 5 nm is the standard value used in most numerical simulations dealing with model membranes exposed to electric fields in the archival literature [35, 36, 45, 46]. Secondly, in some cases in-plane ordering, especially in highly and anisotropically curved membrane regions, might suggest properties that have the potential to affect vesicle shape but geometrical constructions of the type presented in [46] need some refinement to be viable under electric field excitation. Thirdly, when considering a membrane as a 2D object its

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mechanical properties in the absence of anisotropies can be characterized by the Young’s modulus and the Poisson’s ratio according to continuum elasticity theory. In the archival literature, neither Young’s modulus nor Poisson’s ratio have been determined separately so far (even if the Young’s modulus and Poisson’s ratio are interrelated by formula that incorporate the bending rigidity [47]). The Poisson’s ratio in the gel phase of a coarse grained lipid membrane model has been found to be close to 0.5 [48]. The analysis of volume compressibility of lipid membranes within the framework of linear elasticity theory for homogeneous thin fluid sheets shows that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit 0.5 [49]. Fourthly, one of the basic assumptions (limitations) of our study is to consider linear, homogeneous and isotropic membranes and cytoplasm. In point of fact, our next goal will be to apply these simulation tools to deal with μs and subμs electroporation for which the cross-membrane transport and the local electrical properties are central, and linear mechanical behavior can be safely assumed due to force amplitudes and shortness of the considered timescales [50, 51]. Within this perspective, it is known that when the ITV exceeds a certain value, hydrophilic membrane pores greater than a particular size become favorable because they provide lower energy states than the intact lipid bilayer which holds otherwise more capacitive energy. Fifthly, one should keep in mind that the electrical time scales are several orders of magnitude smaller than the viscous relaxation time. Thus, based on the decoupling of the electric and hydrodynamic equations, our algorithm solves a set of equations within the scope of quasi-static mechanical equilibrium [52]. Results and Discussion: Our results are discussed from three perspectives. Firstly, we examine the impact of conductivity ratio Λ on ITV at the pole. We display two different choices of Λ, corresponding to increasing or decreasing σ e and σ c is constant. We present our results in Fig. 5.9 as a function of frequency for the single-cell case. For a single-cell configuration, ITV can be expressed as V m (ω,t,ψ) = 3 √ RE0 cos(ψ) cos(ωt − arctan(ωτ )), where E 0 , ψ and τ respectively account 2 2 2 1+ω τm

for the magnitude of the exogeneous field, the angle between the electric field direction and position on the membrane contour, and the membrane charging time defined ˜ (σc +2σe ) as τm ∼ . In Fig. 5.9, we display the location of να and ν β by arrows with = RCm2σ c σe corresponding relaxation times τα ∼ 1ms and τβ ∼ 0.1 μs, respectively. The top panel of Fig. 5.9 depicts the strong decrease in ITV for a single cell, as the frequency is increased from kHz to several MHz, in accordance with earlier results [53]. We observe a blueshift of V m at the β-dispersion frequency when the conductivity of the ECM is increased. This is due to the fact that if the intracellular medium is more conductive than the ECM, the cell is more polarizable than the medium. Complementary results ([14]) indicate a blueshift of the α-dispersion frequency in the graph of V m as the membrane conductivity is increased. This observation is consistent with a higher tangential mobility of ions on the membrane surface decreasing the associated characteristic dispersion timescale and shifting the value of ν α toward higher

5.4 Modelling Cell Membrane ED by Alternating Electric Fields

203

Fig. 5.9 a ITV at the pole for the reference single-cell as a function of frequency of the electric field. Solid (resp. dotted) lines correspond to Λ set to 15 (resp. 0.1); b As in a for the 2-cell configuration [14]. Blue (resp. red) lines correspond to θ = 0 (resp. θ = π /2), from [14]

frequency value. The bottom panel of Fig. 5.9 displays the ITV corresponding to the 2-cell case. Asymmetry is always important at low frequencies, ν < ν α , since Fig. 5.9 shows a 24% increase (resp. 27% decrease) in the magnitude of V m when the two cells are aligned (resp. perpendicularly) with the electric field direction in accordance with the fact that the membrane represents a barrier limiting the polarization of the intracellular medium at low frequencies [36]. However, evidence suggests that at higher frequencies, ν > ν β , V m changes abruptly with a noticeable blueshift of the displacement of the V m decay for a large conductivity ratio. Overall, these results show the extreme sensitivity of proximity-induced capacitive coupling arising concomitantly when the magnitude of the ED force increases as the distance between cells is decreased ([14]) and the spatial anisotropy becomes important. Secondly, because ITV is only a single parameter, it is likely that the true richness of ED can be fully appreciated by considering the MST driving cytoplasmic remodeling under electric field excitation. An exciting result bearing on this issue is the frequency-sensitive modulation of the ED force induced by the local ITV: at low frequencies, ν < ν α , Fig. 5.10 shows that there is a 47% decrease (resp. 53% increase) in the magnitude of F ED when the two cells are aligned (resp. perpendicularly) with the electric field direction while at high frequencies, ν > ν β , the ED force profile shifts to progressively higher frequencies as Λ is decreased. Upon further inspection, our results quantitatively confirm the quadratic variation of the MST as a function of V m and its dependence to variations in proximity factor

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5 Computational Approaches

Fig. 5.10 a MST at the pole for the reference single-cell as a function of frequency of the electric field. Solid (resp. dotted) lines correspond to Λ set to 15 (resp. 0.1); b As in a for the 2-cell configuration ([14]). Blue (resp. red) lines correspond to θ = 0 (resp. θ = π /2), from [14]

r/R as reported in [14]. Additionally, our work also indicates that the elastic fields emanating from Maxwell stress tensors are highly localized (Fig. 5.11). Upon application of an electric field energy does redistribute to maintain consistency boundary conditions through a perturbation of line fields for the single-cell and 2-cell configurations. Notice that Fig. 5.11 reinforces the trend shown in Figs. 5.9 and 5.10 indicating that the frequency of the applied electric field controls the modulation of electrical cues and ED force. Comparing these results for the single-cell configuration at two values of Λ and fixed frequency (Fig. 5.11a) with the evolution of the aspect ratio (defined as the ratio of semi-major axis b to semi-minor-axis a of the ellipsoid) of the cell deformation induced by the electric field as a function of frequency (left panel of Fig. 5.11b) indicates that shape anisotropy has a much weaker influence on ED of cell membrane compared to the anisotropy induced by the orientation angle itself. Notice that the decrease of the aspect ratio decrease beyond the α-dispersion frequency is consistent with the slight reduction in the intensity of the MST shown in Fig. 5.10 resulting to a decrease of the deformation. Furthermore, one sees explicitly in the left panel of Fig. 5.11b a minimum of the aspect ratio as the field frequency approaches the β-dispersion frequency for when Λ < 1. This is due to the frequency dependent sign inversion of the local net electric field imbalance across the membrane (which is the main driver behind the reduction of its local absolute value and thereby of the local intensity of the MST), occurring for a more polarizable ECM than the cytoplasm. We observe in the right panel of Fig. 5.11b how the deformation of the reference cell for a 2-cell configuration and anisotropy impact

5.4 Modelling Cell Membrane ED by Alternating Electric Fields

205

Fig. 5.11 a Effect of increasing the rotation angle θ on the local distribution of the cell displacement arising from ED forces for the single-cell and 2-cell configurations. The electric field is vertically oriented and frequency is fixed to 10 MHz. Black arrows (with normalized intensity) represent the direction of the local displacement field and the color bar shows the total displacement. b Evolution of the deformation induced by the electric field in terms of the aspect ratio b/a (defined as the ratio of semi-major axis b to semi-minor-axis a of the ellipsoid) as a function of frequency of the electric field. The solid and dotted lines correspond to Λ = 15 and Λ = 0.1 respectively, while the red and blue curves correspond to values of θ set to π /2 and 0, respectively, from [14]

electromechanical coupling compared to the single-case configuration. Interestingly, the two cases (θ = π /2 and Λ = 0.1, θ = 0 and Λ = 15) considered in Fig. 5.11b suggest that sensitive tuning of the mutual attraction between cells can be achieved at particular orientations of the cells to the field direction, with varying both Λ and frequency. It is also informative to discuss our results with those dealing with the ED of a single (quasispherical in the undeformed state) giant unilamellar vesicle in ac electric fields, see e.g. [54]. One main difference between the two studies is that the inner media (cytoplasm) is described via continuum elasticity theory rather than fluid mechanics. Despite such difference, we see similar trends of the ellipsoidal deformation in uniform alternating electric field as a function of the frequency due to the decoupling of the electric and hydrodynamic modelling. Varying the

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5 Computational Approaches

ECM conductivity can induce morphological deformations, i.e. prolate-oblate transitions as a function of Λ and frequency which closely resemble those displayed in Fig. 5.11b. Since the cytoplasm’s mechanical behavior is described by an elastic modulus which is four orders of magnitude lower than that of the membrane, and because the membrane thickness is three orders of magnitude smaller compared to the cell’s radius, increasing Y c by one order of magnitude produces a significant impact on the amplitude of deformation induced by the electric field and the resulting aspect ratio. We must however keep in mind that increasing the value of Y c induces an increase of the effective modulus of the core(cytoplasm)-shell(membrane) system (and consequently, its resistance to deformation) leading to a reduction of the aspect ratio and the amplitude of the total displacement by two orders of magnitude. Our results are also in accordance with the analytical calculations reported by Ye [55] showing that a vesicle under dc electric field excitation (2 kVcm−1 ) exhibits similar trends in prolate-oblate deformation transitions, and where decreasing membrane conductivity of a few orders of magnitude can result in oblate geometries when the ECM is more conductive than the cytoplasm. Thirdly, to investigate the implication of the anisotropy induced by the orientation angle on the MST distribution we present the evolution of the frequency dependent attenuation-amplification transition of the excess MST taken at the pole of the reference cell (point a’ in Fig. 5.8) in Fig. 5.12. Such frequency-orientation angle diagram allows us to determine the attenuationamplification transition of the MST at a given value of the proximity factor r/R. As was shown in [14], lower (resp. larger) values of θ tend to favor attenuation (resp. amplification) of the excess MST for a pair of cells compared to the MST for a single cell in low frequency (ν < ν α ) regime. Frequency has a significant impact on the transitional behavior for a large value of the conductivity ratio Λ suggesting that shape anisotropy has an even weaker influence on ED of cell membrane compared to the anisotropy induced by the orientation angle when considering intracellular conditions of greater electrical polarizability than those of the ECM. When we dial the angle θ towards a critical value from either side of the frequency-orientation angle diagram the behavior of the attenuation-amplification transition of ED force diverges. This latter property makes the ED force the most interesting parameter for applications as it allows confinement and manipulation of cells using alternating electric fields. The key point to take away from the above is that the modulation of electrical cues and MST by the frequency (from hundreds of kHz to tens of MHz) of the applied electric field provides an extremely rich toolkit for manipulating cells. Our model is able to evidence that shape anisotropy has a much weaker influence on ED of cell membrane compared to the anisotropy induced by the orientation angle itself. The behaviors of the attenuation-amplification transition of MST in a frequencyorientation angle diagram could not have been anticipated without detailed calculation. Measuring physical and in particular biomechanical properties at a single-cellscale level is a difficult and often challenging task [56]. These results provide several testable hypotheses of how membrane polarization functions in a variety of cellular shapes involving ED. At least in principle, the above-mentioned differences in MST

5.5 Effective Dielectric Properties of Cells

207

Fig. 5.12 Characterizing the anisotropy of the attenuation-amplification transition of the excess MST taken at the pole of the reference cell as a function of frequency of the electric field for values of Λ ranging from 0.1 (bottom) to 15 (top). The color bar represents the excess MST at the pole of the reference cell in percentage. The blue and red regions correspond respectively to attenuating and amplificating proximity effect induced by the presence of the neighboring cell. A cubic spline interpolation algorithm in Python programming language was used to plot the diagram. The simulation data on the y-axis correspond to values of θ = i × π /10, with i = 0,…,5 and data on the x-axis were logarithmically sampled at a rate of 8 points per decade, from [14]

distribution illustrated in Fig. 5.10 might be exploited to create mechanical-based targeting strategies for discriminating between tumor and healthy cells, since the former are about 70% softer than the latter [57].

5.5 Effective Dielectric Properties of Cells Topic: Dielectric spectroscopy is a powerful technique for studying the bioelectrical dynamics in electrically excitable cells [58–66]. Here, we investigate the relaxation behavior by simulating the dielectric response of simple models of eukaryotic cells to quasistatic exogenous electric fields. The treatment presented here provides

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5 Computational Approaches

V1

ε2

ε1

L ∂V =0 ∂n y Z

ε3

R

∂V =0 ∂n

e

x

V2 -0.5

0

0.5

Fig. 5.13 Illustration of a cross section of an infinite 3D parallel, infinitely long, identical, circular cylinder modelling a three-phase heterostructure made of a CS inclusion (phases 3: intracellular fluid (cytoplasm), and 2: low-conductivity very thin membrane) hosted in a conductive matrix (phase 1: extracellular fluid). The difference of potential imposed in the y-direction is V 1 (1 V) − V 2 (0 V), from [10]

a comprehensive comparison between several types of infinite periodic systems containing CS cylindrical inclusions. Although real systems are 3D, axisymmetric and finite, it is convenient to treat them as being 2D, periodic and infinite since the simple forms of the 2D structures allow for fast simulation and for the development of analysis techniques that would not be possible in more complicated 3D systems because they are prohibitively computer time and memory demanding. Periodic boundary conditions are used to render the system infinite (Fig. 5.13). What interests us here is how relaxation time is determined by shape and symmetry of CS structures. We systematically quantify the importance of interactions between neighboring cells by the influence of the surface fraction of inclusion on the characteristic relaxation time and relaxation strength. We also examine the implications of a transition from a normal to abnormal shape, which maintain the original surface of the CS structure, on the relaxation features. Variations in the shape and dimensions of the most numerous cells in the blood, i.e. erythrocytes, are useful in diagnosis of medical disorders [67] and eventually affect the remarkable elasticity properties and mobility of these cells [68]. Computational Model: Proceeding along the lines of Refs. [69–72] and making use of the BIE method, we have performed quasistatic calculations of the effective permittivity ε = ε' − jε'' . The six selected designs (shown in Fig. 5.14) with D∞h symmetry those we focus on consist of either: (a) an isolated core–shell inclusion (hereafter denoted as structure a, (b) same as a but with an elliptical inclusion whose long axis is oriented along the x-axis (structure b), (c) one fourth of a core–shell inclusion disposed at the opposite corners of the square unit plus an inclusion at the center (structure c), (d) same as c albeit dealing with elliptical core–shell inclusions whose long axis is oriented along the x-axis (structure d), (e) a biconcave (normal

5.5 Effective Dielectric Properties of Cells

209

Fig. 5.14 Schematics of the CS models that exhibit shape isotropy (a) and (c), and shape anisotropy (b) and (d). The e, f designs deal with the transition biconcave (e design)-abnormally shaped (f design) CS structures. The e, f designs were inspired by earlier works dealing with dramatic changes in red cell shapes, from [10]

2a R 2b

(a)

(b)

(c)

(d)

h e

r

t (

m M

H (e)

e (f)

resting shape) disk with the long axis oriented along the x-axis structure (i), and (f) a disk with rounded tips distributed over the disk surface (abnormal shape) (structure j). We used (c) and (d) designs compared to the (a) and (b) designs because the close proximity of other particles can strongly affect the electrostatic interaction between particles. In addition, the (b)–(d) designs are interesting because they are polarization sensitive. This is a consequence of the typical change of shape under applied or internal stress which results in a distorsion of the cell dimension in the direction of strain, thus rendering the CS structure anisotropic. The (e), (f) designs were inspired by earlier works dealing with dramatic changes in red cell shapes [67, 68]. Note that these simple models do not include effects such as the counterion layer at the plasma membrane surface which eventually become important under certain conditions, e.g. pump-leak mechanism, EP and for the understanding of the α-dispersion found at frequencies below 10 kHz. The square unit cell of a characteristic dimension L consists of a core (phase 3)-shell (phase2) inclusion hosted in a matrix (phase 1) is schematically shown in Fig. 5.1. We applied periodic boundary conditions at the outer boundary to ensure consistency of the solution over the computational domain. Let us introduce some notation and relevant length scales which characterize such

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5 Computational Approaches

three-phase composite structures: {[ε1' − jε1'' , L], [ε2' − jε2'' , e], [ε3' − jε3'' , R or (a,b) or (h, H, r) or (m, M, t)]}. Although the inclusion may be complicated in shape and may have several homogeneous constituents, we assume that it can be composed of a circular (resp. elliptical) dielectric particle of radius R (resp. of lengths of the short axis a and of the long axis b) coated with a lossy phase of thickness e (with reference to Figs. 5.13 and 5.14). The (e), (f) designs require three geometric parameters (Fig. 5.14e): the diameter H, the height at the center h, and the maximum height r. The (h) design is defined by m, M, and t with reference to Fig. 5.14f. The surface 2 for structures a fraction of the core–shell inclusion is denoted as Φ, e.g. ϕ = n πR L2 π ab (n = 1) and c (n = 2), and ϕ = n L2 for structures e (n = 1) and d (n = 2). The surface fraction dependence is realized by increasing the square unit size. Without loss of generality we use units where L = 1. The cell of a typical structure is 2D along the unit cell’s axis geometry, permittivity is invariant along the z axis. It is assumed here that the unit cell which has no free charges or currents. In our case, the effective (relative) permittivity along the direction corresponding to the applied field, i.e. ε = εy , can be determined by integration via 1 ε= (V2 − V1 )2

¨

(( εk (x, y)

S

∂V ∂x

)2

( +

∂V ∂y

)2 ) dxdy,

(5.25)

where k = 1, 2, 3, and S is the surface of the unit cell. In the following, we choose 1 , the complex relative permittivity values of the phases which are ε1 = 80 − j 2πFε 0 σ2 ε2 = 11.3 − j 2πFε0 , with σ 2 spanning a typical range for real biological cells (Table 0.5 4.1), and ε3 = 80 − j 2πFε . A similar aqueous phase is chosen to represent the 0 cytoplasm containing large amounts of salts, proteins, nucleic acids and various structures (nucleus, vacuoles) [73, 74] and the suspending medium which is approximated as a highly conducting salt solution with a large concentration of dissolved organic material. Ionic transport through the membrane is regulated by transmembrane channel proteins [75], and under normal circumstances, the membrane can be regarded as highly nonconducting. Similar conditions have been employed by different authors [76, 77]. The specific membrane conductivity is constant over the entire cell perimeter (or, equivalently over the surface of the infinite 3D cylindrical cell). Both the membrane thickness and the geometric factors defining the cytoplasm must be specified to define a particular model. For the simulations reported here, we impose the set of values e = 5 nm, R = 5 μm, h = 1 μm, r = 2.4 μm, H = 7.5 μm, m = 0.22 μm, t = 0.26 μm, and M = 0.2 μm. In the simulations, the frequency F of the applied electric field was in the range between 102 and 109 Hz. We restricted our present computation to ϕ < 0.25. Results and Discussion: We start our investigation by considering the frequencyε' and ε'' of structure a. In dependent variations of)/ ( ' / Fig. 5.15a we show normalized ' ' ε (defined as ε − ε∞ Δε) as a function of F Fc for a fixed ϕ (set to 0.1) and ' denote the static (dc) and the limiting highthree choices of σ2 , where εs and ε∞ ' is the relaxation strength which frequency permittivity, respectively, Δε = εs − ε∞

5.5 Effective Dielectric Properties of Cells

1.0 0.8 0.6 0.4 0.2 0.0

211

(a)

6

10 5 (b) 10 4 10 3 10 2 10 1 10 0 10-1 10 -2 10 -3 10 -4 10 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 F/Fc ) ( ' /Δε is Fig. 5.15 (Color online) a The normalized real part of the effective permittivity ε' − ε∞ / 1 , plotted as a function of F Fc for structure a. We assumed that e = 5 nm, R = 5 μm, ε1 = 80−j 2πFε 0

σ2 0.5 and ε3 = 80 − j 2πFε ε2 = 11.3 − j 2πFε . Symbols are: (white square, black) σ 2 = 10–5 Ω−1 m−1 , 0 0 –6 −1 −1 (white circle, red) σ 2 = 10 Ω m , and (×, green) σ 2 = 10–7 Ω−1 m−1 . Φ = 0.1. The solid curve is a result from the best fit from the Debye response at each value of σ 2 . b Same as in a for the normalized imaginary part of the effective permittivity ε'' /Δε as a function of F/Fc , from [10]

is related to the amount of dipoles involved in the relaxation process, and F c denotes the characteristic frequency at which the permittivity is halfway between its lowand high-frequency values. '' Figure 5.15b / shows the corresponding scaling modelling for the normalized ε '' (defined as ε )/ Δε). Two results stand out instantly. Firstly, a remarkable attribute of ( ' Δε data set is that in spite of the different σ2 values, the data points the ε' − ε∞ from our numerical simulations fall on the same sigmoidal ( ' )/S curve. Let us now ' ε Δε shows a plateau − ε comment on previous related work. In Fig. 5.15a, ∞ ( )/ ' Δε continues to increase at low frequencies. In real cell suspensions, ε' − ε∞ to very high values at low frequencies [58]. Some authors have argued that this behavior is related to other relaxation mechanisms than the β-type considered here [58]. Secondly, the most remarkable features of the ε'' spectra are its large monotonic decrease (power-law frequency behavior) for the range of σ2 used in this work. This behavior is due to the background conductivity and is not related to the dielectric relaxation of this core–shell structure. However, we noticed the appearance of a change in the slope (“hump”) close to F/F c = 1 as Φ > 0.2; the magnitude of this hump becoming more pronounced as Φ is increased. Quantification of the dielectric relaxation properties in conductor–insulator composites was presented in Refs. [58–60]. Very recently, Asami [62] attempted to quantify the effective dielectric dispersion in a spherical model of biological cells using CC relaxation terms. To parameterize the effective dielectric function we use the dipolar Debye response

212

5 Computational Approaches ' ε = ε∞ +

( ) ( ) Δε Δε σ σ ωτ Δε ' = ε∞ + + −j − j , 1 + iωτ ωε0 1 + ω2 τ 2 1 + ω2 τ 2 ωε0 (5.26)

/ where τ = 1 2π Fc is a characteristic dipole relaxation time. The effective conductivity σ and Fc are the adjustable parameters while σ 2 is kept constant. The best fits of ε' and ε'' are undistinguishable from the Debye prediction. Furthermore, the quality of the fit to the numerical data is maintained over the entire frequency range (Fig. 5.15). As the CS structure is changed from a to b we have performed a systematic surface fraction ϕ sweep up to 0.5 and it appears that similar trends are obtained (not shown). In the panels / of Fig. 5.16, we summarize the surface fraction dependence of Δε, F c , and σ σ1 , where σ 1 denotes the conductivity of the extracellular substrate, determined from the fits of our data to Debye equation, respectively. The observed monotonic decay (bottom panel of Fig. 5.16) in σ with increasing Φ is likely a result of the MWS-type interfacial polarization mechanism and is consistent with earlier observations of Asami for spheroidal cells [61]. We emphasize that, on the experimental or numerical side, it is customary to confront conductivity data with a virial expansion [16] 1.6 1.4

Δε (×104)

1.2 1.0

(d) b/a=2 (d) b/a=1/2 (d) b/a=1/3 (e) ⊥ (e) // (f)

(a) (b) b/a=3 (b) b/a=2 (b) b/a=1/2 (b) b/a=1/3 (c) (d) b/a=3

0.8 0.6

Fc (MHz)

0.4 0.2 2.6 2.4 2.2 0.8 0.6 0.4 0.2

σ /σ1

Fig. 5.16 (Color online) Top panel: Variation of the dielectric strength as a function of the surface fraction of inclusion Φ ( Fc. The top panel of Fig. 5.18 illustrates the patterns of induced potentials inside and outside the structure e with the uniform applied field oriented perpendicularly to the long axis of the cell. It is apparent that there is a notable potential drop across the membrane between the interior of the cell (shown as the dotted line) and the extracellular medium; the larger frequency the smaller potential difference across the membrane. For frequency far below F c , the membrane capacitance is charged, and the field lines no longer penetrate the membrane. If one increases the frequency of the field, there is no significant change in the potential inside the cells. For F > F c , the potential is uniform inside the structure (structures e and f have a constant potential inside the cell); the field lines can penetrate into the cell which becomes electrically invisible. Shape anisotropy is responsible for the larger values of the potential at low frequencies when the field is aligned with the long axis of the cell; Fig. 5.18, middle

Fig. 5.18 (Color online) Top panel: Local electric potential distribution as a function of dL , with distance origin placed at the center of the structure e that is situated in a uniform electric field vector oriented perpendicularly along the long axis of the core–shell inclusion, evaluated for F/Fc 1(≈ 20), (triangle, green). V is expressed in μV if all distances are in units of micrometers. The dotted vertical line corresponds to the surface of the core–shell structure; (middle panel): Same as in top panel for structure e with the electric field vector oriented along the long axis of the core–shell inclusion for F/Fc < 1 (≈ 0.04), (white square), F/Fc = 1, (white circle), and F/Fc > 1(≈ 4) (triangle); (bottom panel): Same as in top panel for structure f for F/Fc < 1 (≈ 0.11), (white square),F/Fc = 1, (white circle), and F/Fc > 1(≈ 11) (triangle). Φ = 0.2, from [10]

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5 Computational Approaches

panel. It is also interesting to notice that for structure f the potential drop at the surface occurs a at distance d which is intermediate between those corresponding to structure e for the two states of polarization previously considered; Fig. 5.18, bottom panel. We have therefore the interesting result that depending on the field polarization the cellular shape has considerable influence on the potential distribution inside the cell. Several comments are in order. This finding is consistent with the decrease of the ITV of a 3D model of biconcave erythrocyte exposed to an applied ac electric field, polarized perpendicularly to the long axis of the cell, of nominal intensity 1 Vm−1 and frequency of 1800 MHz. The ITV was also computed for cells and cell chains of various diameters and lengths with applied fields parallel to their long axes; we note that larger clusters of cells were claimed to support larger changes in ITV.

5.6 Electrostatic Forces Between Biological Cells Topic: One of the ultimate goals of biology is to be able to understand and control biochemical reaction systems. The interface between physics, materials science, and biology is currently enjoying a heyday, both as a fruitful discipline for fundamental interest and in an increasing array of compelling technological applications, e.g. cell replacement therapy. In view of potential applications in medical treatments and/ or diagnostics there has been strong activity surrounding the problem of dielectric characterization of biological materials [78–84], e.g. normal and diseased tissues, blood. Because an electromagnetic wave interacts with tissue and cell at the molecular level, it can provide molecular specificity and investigation of tissue function. From a general perspective, it is challenging to obtain unequivocal results for the intercellular EF owing to the fact that the processes involved occur over many time and length scales, and the fundamental mechanisms behind this interaction are still unresolved, even 50 years after of intensive research. However, using the CS model of biological cells has led to a pivotal development for understanding their effective dielectric response since it represents a minimal model of reduced complexity. Similarly a multitude of results have also been obtained for the mechanical response of cells thanks to the progress made in the development of methods that can resolve forces in the range of pN. While the simplicity of CS models has led to a deeper understanding of universal behavior of biological cells it poses an important question as to whether the local behavior of EF can be understood and characterized within a simple universal framework. It is a tantalizing question whether the membrane EP mechanism involves an electrically induced lateral component of EF on the polar head groups of the lipid bilayer of the membrane. Note that in such approaches the specific details of the local structure of the cell and its environment that determines large-scale behavior are ignored. In some instances, fuller theoretical treatments of different biological ingredients need be taken into account, e.g. charge diffusion at the membrane’s surface. However, solving the full force field in multicellular systems is a formidable task. With the advent of numerical methods, strong interest has been

5.6 Electrostatic Forces Between Biological Cells

217

kindled in the evaluation of the EF between two polarized objects (see Appendix …). It is well known that biological cells stressed by an external applied field undergo mechanical deformation and remodeling caused by electro-mechanical Maxwell stresses. These stresses appear at the internal and external biomembrane interfaces due to the differences in electrical properties of the membrane and the surrounding and internal fluids. The Maxwell forces exerted on the interface between two dielectrics are directed from the dielectric with a higher permittivity towards the dielectric with a lower permittivity. For a spherical cell, the polarization charge density on the internal membrane/cytoplasm interface is higher than the charge density on the external membrane/extracellular medium. The difference in the polarization charge densities is accompanied by differences in the magnitude of the electromechanical stresses on either side of the membrane. This results in the appearance of a net radial force [85]. In fields of sufficient strength, nonspherical cells will align themselves with the field and will aggregate and line up along the field lines, i.e. a pearl-chain effect. Schwan [86] showed that the threshold for such an effect is ∝ R−1.5 , where R denotes the cell radius. Field strengths needed to produce it are of the order of 10 kVm−1 for cells of 1 μm radius. There has been palpable progress in the development of techniques and tools to address the calculations of the EF between polarisable objects over the last decade. Specifically, Liu et al. [87] found that the dynamic coordination of mechanical forces and cell-cell adhesive interactions are critical to the maintenance of multicellular integrity. When implementing their numerical simulations, Angelini et al. [88] found that multicellular mechanical cooperativity guides collective cell migration. Collective motion of cell cultures is a process of great interest, as it occurs during morphogenesis, wound healing, and tumor metastasis. During these processes cell cultures move due to the traction forces induced by the individual cells on the surrounding matrix. Trepat et al. [89] measured for the first time the traction forces driving collective cell migration and found that they arise throughout the cell culture. Based on a surface charge method, Doerr and Yu [90] calculated the EF between an arbitrary number of charged dielectric spheres. They noted that the magnitude (relative to point charges in an infinite solvent) of attractive EFs decreased while the magnitude of repulsive EFs is increased (again, relative to point charges in an infinite solvent). These results raised the question of why a dense system of charged cells does not simply aggregate. Recently, Tian et al. [91] used an analytical approach to address the problem of evaluating the EF between two polarized inhomogeneous particles. Their study points towards the importance of the inhomogeneities as key factors for determining the CFs. However, due to their mathematical complexity, these approaches have been limited until now to the case of spherical particles and/or a narrow range of frequencies. Unique physical features of CF are studied as function of field frequency and gap distance between cells. The distance between cells corresponding to the CF shows both anisotropic and universal features. Aside from being of fundamental interest, these observations pave the way towards the deposition of biological cells under field excitation in an architecture that could create functional tissue. Recent

218

5 Computational Approaches

theoretical modeling [92] has suggested that the RTA transition between two polarized particles can be described by universal features. That, is when the gap distance D between the two particles is close to the crossover D* corresponding to the RTA F is of the form F ∝ |D − D∗ |α , and when D → 0 F exhibits a D−β scaling behavior. The two critical exponents α and β have been ab initio computed for a variety of particle shapes and types (metal or insulator). At the current time, no theory exists that can explicitly predict the values of α and β. It is also noting that CFs and RTA transitions defied observation so far. While a numerical demonstration cannot be a substitute for a mathematical proof, these results provide evidence that an analytical perturbation technique which allows prediction of the EF parameters (location of the crossover, scaling, asymptotics) could be effective. Computational Model: The computational domain of each system we shall study is given in Fig. 5.19 for three configurations with 6 cells. Recent developments have shown that cells can tune their own stiffness which can increase dramatically when large external forces are applied and can result in compression and rupture of the cellular membrane [93]. Here, we model the cells with an infinite stiffness (not deformable). We will assume throughout that the time dependence of the electric field excitation, assumed / to be directed along the x direction, is proportional to exp(−jωt), where f = ω 2π is the frequency of the wave ranging from 105 to 109 Hz. The unit cell length of 25 μm corresponds to ≈ 10–4 × λ at the maximum frequency of 1 GHz. In this long-wavelength limit related to the quasistatic approximation, the electric field is irrotational. For an applied harmonic electric field the instantaneous EF on a particle is obtained by integrating the MST as mentioned earlier. Fig. 5.19 Graphical representation of the 6-cell arrangements. These systems represent appropriate models, from a dielectric point of view, of biological cell arrangements where the shell represents the cell membrane and the aqueous phase represents the extracellular medium. In all cases, the dimension parameters r = 5 μm, R = 25 μm and L = 50 μm were held constant, while D was varied from 5 μm to 1 nm. The number indicates the position of each cell in the cylindrical computational domain, from [12]

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Results and Discussion: We begin by examining the relative importance and overall effect of the gap distance between cells and configuration of cells on the spectral fingerprint of EF. To aid the discussion, we will show three variations in each of which we will hold constant D and σ 1 and vary the distance between cells by considering different pairs of cells. The results are summarized in Fig. 5.20. Depending on the frequency of the applied electric field oriented along the (x, y, z)-axis we find that the EF between two cells (1–2, 3–4, and 5–6) can be classified as attractive or repulsive with maximal attraction or repulsion, e.g. when f = 20 MHz, D = 1 nm, and σ 1 = 10–3 Sm−1 . There are a few matters that must be noted about the RTA transitions as depicted in Fig. 5.20. The physics underlying the sequence of RTA transitions is derived by analogy with the available analytical data for charged spheres, i.e. non-uniform redistribution of charges at the surfaces of the two cells considered [92, 94]. Our computations which predict that there are several CFs that divide the attractive and repulsive regimes over the frequency range explored (10–1 MHz–1 GHz) are consistent with prior studies [92, 94] which suggested that the intercellular attraction EF increases with decreasing the intermembrane separation. On the same plot we show how the CFs vary with the symmetry of the cell configuration. The CFs does not reflect the rotational symmetry of the CS model of individual cell per se but rather the symmetry of the cell configuration. As is expected, negligible change in the spectral dependence of the y- and z-component of the EF is observed across the range of frequencies and σ 1 studied herein (Fig. 5.20). Additionally, we predict that it is possible to sweep continuously the EF from negative (attraction) to positive (repulsion) values by changing the field frequency. Interestingly, many CFs can be observed in this graph. We note that the oscillatory nature shown in this graph is consistent with the earlier results of Tian and co-workers [91]. At this point, it is appropriate to mention an interesting connection between the above described behavior and the crossover between attractive and repulsive interactions in dissimilar charged bodies immersed in electrolyte solutions [95]. Different scaling regimes, i.e. exponentially screened or long-range power law, of the electrostatic interactions were predicted depending on the salinity and surface charge densities. To further investigate this anisotropy, we have calculated the current flow paths in the simulation domain for the configurations of cell studied above. The results of the simulations are graphed in Fig. 5.21 for three different frequencies and 6 cells. It is also possible to visualize the MST for two frequencies (1 and 30 MHz) in Fig. 5.22. Several general features can be observed. Again for comparison we call attention to the observation of different patterns for various values of σ 1 . But the key point to note about these graphs is that when far enough apart (D = 5 μm), the two cells act like uniformly polarized objects, which repel one another when the charges are alike. But as they get closer (D = 1 nm), they become polarized, i.e. one particle elicits an image charge in the other by pushing like charge away. This is accompanied, as in the previous cases, by a short-range RTA transition. We now come back to a question posed by Doerr and Yu [90]. How is it that a dense system of cells does not aggregate? It turns out that attributes such as effective permittivity that depend upon

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Fig. 5.20 a Frequency dependence of the x component of the instantaneous EF between two CS structures modelling a pair of spherical cells (1 and 2), extracted from the FE simulations. The three phases of each cell model are the cytoplasm (50 + jσ3 /ωε0 and σ3 = 0.53 Sm−1 ) covered by a confocal membrane shell (9 + jσ2 /ωε0 and σ2 = 10–6 Sm−1 ), and the external liquid (ε1 = 80 + jσ1 /ωε0 ). The external field is applied to the system in the x direction. The data corresponds to the cell configuration displayed in Fig. 5.19 for the EF acting on cells 1 and 2, with D = 1 nm. The blue (triangle) (resp. red (O) and black (X)) lines correspond to the case, with σ 1 = 10–3 Sm−1 (resp. 0.12 and 1 Sm−1 ). The solid (resp. dashed) line corresponds to the EF acting on cell 1 (resp. cell 2), respectively. b As in a for the EF along the y-direction between cells 3 and 4. c As in a for the EF along the z-direction between cells 5 and 6, from [12]

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Fig. 5.21 Details of the streamline plot illustrating the current flow paths in the simulation domain corresponding to the 6-sphere configuration (Fig. 5.19). a f = 105 Hz, σ 1 = 10–3 Sm−1 , b f = 107 Hz, σ = 10–3 Sm−1 , c f = 109 Hz, σ 1 = 10–3 Sm−1 , d f = 105 Hz, σ 1 = 0.12 Sm−1 , e = 2 × 107 Hz, σ 1 = 0.12 Sm−1 , f f = 109 Hz, σ 1 = 0.12 Sm−1 , g f = 105 Hz, σ 1 = 1.0 Sm−1 , h f = 2 × 107 Hz, σ 1 = 1.0 Sm−1 , i f = 109 Hz, σ 1 = 1.0 Sm−1 , from [12]

the spatial average behavior of all the cells are apparently unaffected by nanoscale structure, whereas EF, CF, and RTA that involve collective behavior of only a small number of cells do show sensitivity to small length scales. These materials typically feature many cells that are crowded together in close contact, and are both jammed so that each cell is fully constrained by its neighbors, and disordered so that these constraints vary greatly among particles. The ab initio simulations presented so far strongly suggest that the RTA transition phenomenon is largely associated with an asymmetric electrostatic screening at very small separation between cells. This observation has direct application to the problem of electrostatics in biomolecular systems [96]. The surface of the cell is negatively charged for nearly all cells because of the predominance there of negatively charged groups, e.g. carboxylates, phosphates. This results in positive ions being attracted from the extracellular medium to the surface to form double layers [42]. This approach may have also potential for exploring biological cells with more complicated topology and giant unilamellar vesicles; thirdly, one other direction in which we have already begun to extend this work is the characterization of membrane mediated interactions of many nanoparticles adsorbed on cell membranes [97]. As a final remark, we would like to point out that these findings may have also important implications for both numerical calculations and experiments, in EF of multicellular systems, i.e. tissues [98, 99].

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Fig. 5.22 a The solid arrows indicate the MST for the configuration shown in Fig. 5.19 and spheres 1, 2, 3, and 4 (xy plane) at f = 1 MHz with σ 1 = 10–3 Sm−1 , and D = 5 μm. b As in a for f = 1 MHz with σ 1 = 10–3 Sm−1 , and D = 1 nm, c As in a for f = 1 MHz with σ 1 = 0.12 Sm−1 , and D = 1 nm, d As in a for f = 1 MHz with σ 1 = 1.0 Sm−1 , and D = 1 nm, e As in a for 30 MHz with σ 1 = 10–3 Sm−1 , and D = 5 μm, f As in a for f = 30 MHz with σ 1 = 10–3 Sm−1 , and D = 1 nm. g As in a for f = 30 MHz with σ 1 = 0.12 Sm−1 and D = 1 nm, h As in a for f = 30 MHz with σ 1 = 1.0 Sm−1 , and D = 1 nm, from [12]

5.7 Strain Energy in Multicellular Environments Topic: Now, we examine in detail the physics ideas which may be used to obtain a multiphysics model of the strain energy distribution in cell assemblies. EF in complex structures cannot be understood unless the structure is understood. The ability of controlling tensegrity in biological materials has generated widely debated mechanisms on the mechanical forces between cells in tissue based on energetic, time scale, and cell number considerations [11–13, 100–104]. From the vantage point of what is seen in experiments, this is an important question since it is difficult to quantify stress in cells due to their complex shapes and internal structures. Tissue and cell level architecture prevent disruptions from occurring, either by shielding cells from damaging levels of force, or, when this is not possible, by promoting safe force transmission through the cell membrane via protein-based cables and linkages. Here, as a didactic model of a complex multicellular system, we consider 7-cell groups embedded in an extracellular medium. The motivation for studying these assemblies lies in their potential to help answer current theoretical and experimental questions regarding the collective electromechanical response of deformable spherical cells in proximity to each other, to assess how the intracellular distance and cell compactness can affect the strain energy distribution, and to provide clues for where to search for

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simple tissue models which are efficient to provide theoretical predictions for classic biological research such as electroporation scenarios. Computational Model: Each cell is assumed to be heterogeneous CS spherical structure with a viscoelastic membrane. This important additional feature in our model is largely based on reports showing evidences that most living cells show a viscoelastic deformation under mechanical forces [105]. Here, the DeBruin and Krassowska self-consistent model can be solved locally on the cell surface area for which the local values of the ITV become the input variable [100]. In our simulations, the surface patterns of the electric field, surface charge density, MST, ITV, and pore density can be obtained at specific times, i.e., at the beginning, during and the end of the electrical stimulus. In order to be concrete, and for illustration purposes, we first consider an assembly of seven cells in specific configurations (Fig. 5.23) that are exposed to a positive polarity trapezoidal voltage pulse which delivers an average electric field of strength E 0 = 5 kV/cm. First, the electromagnetic equations are solved for electric potential, and then MST is calculated at the cell membrane. The MST distribution over the cell surface is an input parameter of the viscoelastic model, which solves the time evolution of cell shape. The polarization charge redistribution induces stresses and changes the initially spherical surface of the cell into a prolate-like shaped surface. Typically, EP takes place when ITV exceeds a threshold V ep above which electrically conductive pores start forming in the membrane [13]. The continuum multiphysics model for calculating the strain energy in multicellular assemblies is performed in three steps. Firstly, under the assumption that cells can be described by CS structures the spatial distribution of the electric potential in a subdomain of the cell is solved by making use of Poisson’s equation −∇ · ((σ + ε)∇V) = 0, where ∂t denotes the time derivative and σ and εε0 are the conductivity and permittivity of the subdomain, respectively. The generated fields polarize the cells leading to emergence of surface charge distributions that locally correlate with enhanced electric field regions and thus electrical forces are exerted on the cell membrane that are calculated using MST, Eq. (5.20). The surface electric field is obtained from the solution of electric potentials that are coupled at the cell membrane through the specific boundary condition given by n.J = d1m (σm + εm )(Vint − Vext ), where n is the unit vector normal to the boundary surface, J is the electric current density, σm is the membrane conductivity, d m is the membrane thickness, and ‘int’ and ‘ext’ denote the cell interior and exterior sides, respectively. A resting potential of − 50 mV is set. The second stage in the procedure is to model the cell deformation. From this prospective, the resulting stress distribution is coupled to the structural model for calculation of membrane displacement u. In the case at hand, the strain energy was calculated by solving first Eq. (5.21). The membrane is modelled with a viscoelastic material while the inner and outer regions of the cell are modelled with an elastic material. The stress leads to deformation in all media (elastic and viscoelastic). The strain energy ˝ of the cell membrane can be calcuS.γ dΩ/2) where γ is the strain lated for both deformable materials as E mem/med = and d Ω is volume or surface element. The final stage is to focus on the pore dynamics

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Fig. 5.23 Schematic of the eight canonical examples of 7-CS arbitrarily fixed in space cell configurations and subjected to an electric field pulse. In the a1–a4 configurations, all cells have the same radius set to 6 μm and the intercellular boundary-boundary distance, d b-b , with respect to the central cell is varied from 0.625 μm (a1), 1.25 μm (a2), 2.5 μm (a3) to 5 μm (a4). In the b1–b4 configurations, cells have a radius distribution leading to a broader d b-b distribution with a minimum set to 0.5 μm with respect to the nearest neighbor. The side of the computational cube domain is 50 μm. Cell-size and distance distribution information are listed in [13]

during pulsing. For simplicity, we assume a fixed pore radius (0.75 nm). We calculate the pore density in the membrane, N, based on the highly non-linear dependence on ITV, i.e. Equation (3.7) with V EP = 0.258 V, which alters(the initial cell membrane ) conductivity by adding the following term σep (t) = N (t) 2πr 2 σp dm /(πr + 2dm ) . With this result in hand, the number of pores involves surface integration˜of pore NdA. density over the total cell membrane surface area, so we obtain Npores = Likewise the total membrane surface area for the configuration is 3.16 × 10–9 m2 . Results and Discussion: It is useful for us to first focus on the trapezoidal voltage pulse excitation (Fig. 5.24a) of 100 μs duration. Unless specifically noted, the electric field is oriented along the z-axis. Other types of well-defined electric stimuli, depicted in Figs. 5.24b–d, will be considered later. As an example, Fig. 5.25 shows the evolution of ITV exceeding the electroporation threshold voltage V ep which is

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Fig. 5.24 Different electric pulse stimuli used in this study: a unipolar pulse, b bipolar pulse, c same as in b but differing in the width and depth of the second pulse, d same as b but differing in the width and depth of the second pulse. Here V = 25 V applied over the distance 50 μm leading to an average field of 5 kV/ cm during the time interval t 0 = 100 μs, from [13]

shown by regions colored in red for three fixed times during the application of the voltage pulse. In the “switch on” state, as time goes by, the red region in proportion to the total area increases from the poles and subsequently decreases because the ITV decreases. It is worth observing that these 3D plots are consistent with the anisotropic dependence of the ITV, i.e. the induced ITV varies locally with the position on the membrane with the highest established |ITV | at the poles of the cell, and minimal |ITV | established around the equator, so only the poles are electroporated. It appears that pore density persists in all areas where ITV is above threshold but as the pulse falls, ITV quickly decreases from the poles (Fig. 5.25) while pore density decreases more slowly outside the simulation temporal window. With this understanding we proceed to estimate the temporal dynamics of the maximum value of the electric field norm on the cell membrane (Fig. 5.26a), the total number of pores (Fig. 5.26b) assuming a fixed pore radius set to 0.75 nm, and the relative pore area (Fig. 5.26c). It turns out that the observed behavior of the electric field norm of the Ai configurations is qualitatively similar to those of the Bi configurations but the larger amplitude of the electric field evidenced for configurations B2 and B3 is attributed to the proximity (coupling) effect. In point of fact, the smallest d b-b value in these configurations occurs at a specific angle of the electric field orientation leading to large local field enhancement. The number of pores exhibits a transition that is abrupt as the voltage pulse is switched on (Fig. 5.26b). The plateau in the number of pores dynamics can be accounted for by the much slower rate of pore destruction than the rate of pore creation [106–110]. Pore persistence when the pulse is switched off has been described and analyzed in [110]. Histograms of the fractional pore area are not distinguishable for configurations Bi most likely because the average value of d b-b does not vary much between the Bi configurations. In contrast, the histograms shown in the left panel of Fig. 4c are somewhat more discriminant for configurations Ai because the average d b-b value varies much more among the Ai configurations. Be this as it may, pore area is suggestive to be more sensitive to symmetry rather than proximity effect. Next, we analyze the effect of symmetry and proximity effects on the strain energy distribution of the interior and exterior of the cell and viscoelastic membrane

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Fig. 5.25 The 3D plots represent the regions for configuration B2 corresponding to ITV ≥ V ep = 0.258 V (shown in red, otherwise blue) when pulse 1: a rises, b is maintained on, and c falls (see details in [13])

(Fig. 5.27a, b) for the different configurations shown in Fig. 4. Firstly, there is a general pattern in these graphs in the “switch on” state, i.e. a narrow peak during a few tens of μs followed by a broad plateau. Secondly, it is noticeable that in the “switch off” (post-pulse) state a second peak is observed with a time scale which can range from 150 to 200 μs. This observation does not arise from the dominance of the mechanical relaxation process over the electrical charge relaxation since it was also observed in our earlier study dealing with elastic deformation of cells after removal of

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Fig. 5.26 a The maximum value of the electric field norm on membranes, b the total number of pores, and c the fractional pore area occupied on the cell membrane for the different configurations studied excited with pulse 1. The dashed lines indicate the beginning and end of the pulse, from [13]

the electric stimulus [13]. The incomplete shape recovery in the “switch off” appears to be consistent with the power-law dynamics of cell deformation discussed in Ref. [105]. The “switch off” signal is likely observable in currently AFM-based, scanning probe and confocal fluorescence microscopy experiments under in vivo experiments [107]. Figure 5.27c shows the maximum value of the membrane displacement from its original position. Particular attention is now paid to explore the mechanical response to a bipolar pulse stimulus (Fig. 5.28). It is worth noting that these calculations are performed for configurations A3* and B2 which have an identical average value of d b–b of 3 μm [the star in A3* means that it is neither a A3 nor a A4 configuration since the average d b–b value lies in between the value for A3 (2.5 μm) and that for A4 (5 μm)]. Interestingly, there is a rather clear peak corresponding to the zero-crossing of voltage excitation when the electrical stimulus is oriented along the y-axis. Additionally, it is interesting to note that this “switch on” peak signal can be larger by a factor of 8 than the corresponding peak signal when the electrical stimulus is oriented along the x-axis that. In the right panel of Fig. 5.28, we show the corresponding behavior for the B2 configuration. We also point out that the “off” state associated with bipolar pulse stimuli is clearly visible for pulse 2 and 3 excitations of the assemblies containing random distributions of cells.

228 Fig. 5.27 Simulation results highlighting trends of the strain energy in 7-cell configurations versus time when the different configurations are excited by pulse 1. The dashed lines indicate the beginning and end of the pulse. a Strain energy for the interior and exterior of the cell, b strain energy for the viscoelastic membrane, and c the maximum value of the membrane displacement for the different configurations studied. Physiological conditions (T = 310 K), from [13]

Fig. 5.28 Strain energy of the membrane as a function of time when configurations A3* and B2 are excited vertically (along z-axis, blue line) and horizontally (along y-axis, red line) by a pulse 2, b pulse 3, and c pulse 4, respectively. T is defined by physiological conditions (T = 310 K). The dashed lines indicate the beginning and end of the pulse, from [13]

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The studied heterogeneous configurations present a case in which the local enhancement of the electric field, deformation of the cell, strain energy and relative area occupied by the pores are sensitive to multiple phenomena that can be difficult to deconvolve. The other remarkable aspect is that we report a “switch off” (post-pulse) phenomenon for several kinds of electrical stimuli observed in different cell configurations. Within our framework it is possible to obtain also some insight into the parameters affecting the temporal dynamics of the local enhancement of the electric field, the surface charge density, the polarization distribution, the relative deformation, the strain energy and the pore area extent within the cell membrane. Here, we are interested in cellular hardware (structure and electromechanical properties) but we have had little to say about cellular software (information processing capabilities). A comprehensive study of how they interplay to control cell form and function in tissues, i.e. how the mechanical signals are transmitted and potentially transduced into intracellular biochemical signals, is useful because it can open newer doors to be used for 3D engineered tissues.

5.8 An Approach to Electrical Modeling of Realistic Multicellular Structures Before EP Topic: In the same vein as in the previous sections we consider multicellular systems but with irregular cell shapes and nearby cells, and attempt to model the ITV before EP [12]. There are myriad difficulties in modelling biological tissues, e.g. by incorporating realistic cell shapes and different length scales ranging from several nm to hundreds of μm. Despite vast hurdles in “writing” organ-specific tissues, rapid adnvances in 3D imaging and computational… now allow to “read” these tissues with unprecedented sptial and temporal detail. Here we rely on numerical methods as the problem does not have any obvious analytical solution. There have been several attempts to address this question, particularly in the context of EP. A previous approach in this direction was performed by making use of transport lattices (see Sect. 2.4.2) and Kirchoff’s laws [111]. In [111], each cell is elongated with an irregular asymmetric shape and contains a model for a nucleus, and only nearest neighbor connections are taken into account. The authors in [111] argue that electric field amplification through current or voltage concentration changes with frequency, exhibiting significant spatial heterogeneity until the microwave range is reached, where cellular structure becomes almost “electrically invisible”. In addition, they find that membrane EP exhibits significant heterogeneity but occurs at invaginations and cell layers with tight junctions [111]. This is an interesting scenario which has yielded substantial insights into the pore-based effects of EP, but has several questionable aspects. One is that V m and the number of pores as well, will vary with the cell position in a spatially extended assembly of cells. Thus, some differences are expected with the EP of a single cell. Another is that ED is not considered. But the most problematic feature of this solution is that it lacks a mechanistic hypothesis of

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what is causing the effective membrane electrical conductivity increase, whereas we know that electric field coupling between cells also play a critical role. Some work has been done on addressing this issue, see in particular Refs. [11–14]. When studying the induced ITV, a realistic geometry will provide a better representation than a simplified model when a cell is exposed to external electric fields. However, when one wants to study the ITV for a cell which is part of a cell cluster or a tissue, the results obtained from a single isolated cell are not accurate as we show later. As described in Refs. [11, 13], a two-cell model for which the cells are close to each other but not touching, will give different results for the induced ITV under the same excitation conditions compared to the ITV when one of the cells is removed from the model. Here, we invoke a method of obtaining a realistic model for multiple cell assemblies for which cells are very close to each other. In order to model accurately the geometry of such complex cell morphologies, specific techniques must be used. A more realistic 3D model can be constructed from a sequence of cross sections taken with a light or fluorescence microscope or preferably a confocal microscope. As was shown in [112], bitmap images of separate cell planes can be easily manipulated to extract the contours (i.e. cell membrane) and connect planes together to obtain a 3D model of the cell using MATLAB® numerical computing environment. This method provides a good 3D approximation of realistic cell geometries and even though they use a small number of planes to reconstruct the geometry, the obtained results give valuable insight into differences between a realistic shape and a spherelike approximation. With today’s advances in microscopy and computer software, the amount of detail in a single cell can be greatly improved. Computational Model: A fluorescent image of epithelial cells is obtained with a microscope from a microfluidic platform [12]. Since the microscope and the staining method did not provide sufficient resolution for automatic shape detection, a manual approach is taken. The bitmap image is imported into GIMP 2.8 GIMP 2.83 and a paintbrush tool is used to trace the edges of cells and nuclei (Fig. 5.29a). Overall, 91 cells and nuclei are traced and the original bitmap image was removed from the study within GIMP. Bitmap canvas is resized to a smaller size, leaving some extra space around the traced cells, to a final size of 3000 × 2156 pixels. Cell domains are colored in blue, nuclei in yellow, and the extracellular medium in grey for easy manipulation and segmentation. The final image was imported into MATLAB® and converted into a 3D matrix, where three separating masks are used to extract separate subdomains for cytoplasm, nucleus and ECM (Fig. 5.29b). The cells, together with the nuclei, cover 59% of the total surface area of the region analyzed. A relatively large bitmap can be used in order to provide sufficient detail for each cell and nucleus, which in turn produced an extremely large number of edge elements for all domains. To remedy this issue, a minimum distance of 3 pixels between two edge nodes is used, reducing the overall number of nodes while still providing well detailed, but smoother, domain edges (Fig. 5.29c). In the FE mesh in the results presented in Fig. 5.29c, the number of 3

GNU Image Manipulation Program, freeware, www.gimp.com; [12].

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Fig. 5.29 Illustrtaing the different steps taken when transforming a microscopy bitmap image into an FE mesh. a Fluorescent microscopy bitmap image with traced cell edges and traced nuclei boundaries, which were also colored in white. b Final bitmap image of the cells and their nuclei, used to create the geometry used in an FE mesh. The numbers 1–6 denote the cells used in a separate model (see details in [12]). c Zoom of a part of the entire geometry used in the FE mesh, illustrating the dense edge (cell and nucleus membrane) meshing. The blue color represents the nuclei

edge elements ranges from 192 to 1178, with 436 being the average for the cell membrane and from 35 to 164, with 90 being the average for the nucleus membrane. The final mesh consists of 8 × 105 domain elements, 5.1 × 104 boundary elements and 1.7 × 106 degrees of freedom solved for. To extract the ITV for the cell membrane and the NE, three types of subdomains form the computational domain: the ECM, the interior of the cytoplasm, and the nucleus. Using this scheme, a potential difference across the membrane could be extracted by subtracting the potentials on both sides i.e. V ECM − V cyt for the cell membrane ITV and V cyt − V nuc for the nucleus envelope ITV. To avoid large-scale differences between the entire simulation domain and the cell membrane (~200 μm versus 5 nm) leading to meshing and convergence problems, a distributed impedance boundary condition was used to model the cell membrane and nucleus envelope. This allows to model a thin sheet of resistive material connected to a reference potential V ref without actually building and meshing the thin layer by:

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[ ] ( ( ) ) ) ∂( 1 σm Vcyt − VECM + ε0 εm Vcyt − VECM , ncel · J ECM − J cyt = dm ∂t [ ] ( ) ) ) ( ∂( 1 Vnuc − Vcyt , nnuc · J cyt − J nuc = σnm Vnuc − Vcyt + ε0 εnm dnm ∂t

(5.29) (5.30)

where n is the outward-facing normal on each boundary, J ECM , Jcyt, and J nuc are local current densitie at each node in the ECM, cytoplasmic, and nucleus subdomains. V ECM , V cyt , and V nuc are the potentials at each boundary within the ECM, cytoplasm, and nucleoplasm, respectively. The nominal values of the parameters used in the modelare those listed in Table 4.1. For generating the electric field, a terminal and a ground boundary condition were used on the top and the bottom of the extracellular domain, respectively. Two different pulses were modeled using a piecewise function: one unipolar IRE pulse with duration of 100 μs (Fig. 5.30a) and a bipolar (HFIRE) pulse consisting of a 1 μs positive voltage followed by a 1 μs zero (no) voltage, followed by a 1 μs negative voltage (Fig. 5.30b). Results and Discussion: Besides a novel approach to modeling realistic cell geometries, our interest lies in the response of the model to two different types of pulses (IRE and HFIRE) [12]. Another simulation was run with only six cells in the model in order to investigate the difference between the induced ITV when cells are tightly

Fig. 5.30 a Input terminal voltage for the unipolar (IRE) pulse; b Input terminal current for the unipolar (IRE) pulse; c Input terminal voltage for the bipolar (HFIRE) pulse; d Input terminal current for the bipolar (HFIRE) pulse, from [12]

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Fig. 5.31 The ITV variation along the membrane of a cell (Fig. 5.29b, cell 4) when it is isolated (black solid line) and surrounded by other cells (blue dashed line) during the IRE pulse (t = 20 μs), compared to the analytical calculation based on steady-state Schwan’s equation (Eq. 2.1) with constant radius (dotted line, circle marker), from [12]

packed together and when there are no cells in the immediate vicinity of the chosen cells. The chosen cells to remain in the model are marked in Fig. 5.29b. The ITV was extracted for both the cells and nuclei. Here, we focus on the six cells that were also studied separately when isolated from the other cells. Coherent with previous results for the potential and current density distributions and the fact that a dense packing of cells disrupts the uniform gradient, which can be observed when the (six) cells are isolated ([12]), the ITV values for the packed and isolated scenario are different. Typical distribution of ITV along the cell membrane is shown in Fig. 5.31. In case of an isolated cell the ITV is significantly larger compared to the packed cell situation. The mean absolute difference along the cell membrane is 1.676 V and the maximum absolute difference is 2.820 V. Figure 5.31 illustrates the ITV values 10 μs into the IRE pulse and if we take the entire pulse into account, the maximum absolute difference is 3.186 V and the mean absolute difference is 1.598 V. To see how these experimental values compare to those obtained with the ideal case, we first calculated the equivalent radius of the cell, i.e. the circle radius having same area as the cell area. Using the equivalent radius, the ITV is calculated using the steady-stae Schwan’s equation (Eq. 1.1). The results show that the ITV values using an ideal cell geometry are closer to the ITV values for the isolated cell. Finally, it is of interest to visualize the locations of the membranes where ITV is above the EP threshold (Fig. 5.32). During the IRE pulse (Fig. 5.30a), most of the porated parts of the membranes are located at the top and the bottom of the group of cells with some porated areas vertically in the middle of the group. At first glance, the situation is similar during the HFIRE pulse (Fig. 5.32b). If one looks carefully at the region left to the center of Fig. 5.29b, which correspond to the same time step, there is a region with high current density where the amount of porated regions is relatively low. Otherwise, as stated before, during the HFIRE pulse, significantly more membrane regions reach the EP threshold compared to the IRE pulse, which is partly due to the higher pulse voltage. Figure 5.32 corresponds to the time step

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at the middle of the pulses and the amount of porated regions is relatively constant throughout the pulse. Only during the rise (resp. fall) of the pulses the amount of porated regions gradually increases (resp. decreases). Thus, using high voltage and short duration, it is possible to porate more cells and also target cell nuclei while at the same time the detrimental thermal effects and unwanted muscle contractions are decreased [113]. While we consider only 2D geometries, our results can be leveraged to 3D geometries using similar protocol and confocal microscopy images. Additionally, our model can be extended to incorporate the pore formation, i.e. by considering SNK model). These results open the door to many applications dealing with the electric response of tissues and multicellular arrangements with cells of complex geometry and topology. Fig. 5.32 a Parts of cell membranes where |ITV | ≥ 1 V for the IRE pulse (t = 60 ms) is reached; b Parts of cell membranes where |ITV | ≥ 1 V for the HFIRE pulse (t = 1.5 μs) is reached (red) and |ITV | < 1 V (blue), from [12]

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5.9 An Electromechanical Boundary for the Cell Membrane ED Topic: The inherent complexity of biomaterials is well recognized; they are multiscale, multi-science systems, and bridging a wide range of temporal and spatial scales [13, 114]. As we have seen, the CS structure, i.e. a dielectric nanometric membrane surrounded by conducting cytoplasm and extracellular medium (ECM), has been quite successful, notably for evaluating the ITV. While this model is an important object of study that features various electromechanical properties of cell membrane it is likely too simplified to describe real biological cells. Simulations, until now, have largely focused on a rather narrow picture of the complexity of the membrane (dielectric, elastic continuum object). However, as FE simulations are becoming increasingly sophisticated, the most widely used methods used to analyze ED and EP issues exposes serious drawbacks in dealing with electromechanical properties at very different length scales (membrane thickness, cell size, and customized tissue scaffold) and need to be improved. This is especially true for multiscale problems that require frequent remeshing to track large structural deformations, i.e. upon application of an electric field energy does redistribute to maintain consistency boundary conditions through a perturbation of line fields. Furthermore, this feature renders the computation very costly in terms of computation time and memory requirements. In addition, there is the added complexity involved in dealing with the study of multiple interacting physical properties, i.e. electromechanical coupling [115]. Over the years, the literature suggests several ways which can be implemented numerically, and a general approach is illustrated in Fig. 5.33a. In the case of discretizations based on FE, the dominant error is due to the mesh adaptation scheme used in the simulations. In the context of continuum scale theory, one elegant approach consists to solve large heterogeneous structures without scale separation assumption by introducing relevant Dirichlet boundary conditions which can be satisfied on the capacitive elastic membrane. There have been several attempts to address this question, particularly in the context of electric characterization of cell membrane. To perform ITV, V m , calculations on realistic cell shapes, Pucihar and coworkers [116] put forward a model to replace the membrane by a boundary condition on the cytoplasm. Since it avoids meshing the membrane this approximation permits a significant decrease of the computational time, discretization cost and used memory for the calculations. Another more recent variational approach for calculating the spatial distribution of electric field without meshing the membrane is seen in [117]. In [41], the ED dynamics of lipid vesicles under direct current electric field excitation is investigated where the vesicle membrane is represented as a massless immersed boundary. In Refs. [118], the immersed boundary method is used to simulate the ED and electrohydrodynamics of a vesicle in Navier–Stokes leaky dielectric fluids under various electric field excitations. The vesicle membrane is modeled as an inextensible elastic interface which is characterized by electric capacitance and conductance. Within the leaky dielectric framework and the piecewise constant electric properties in each fluid, the electric stress can be treated as an interfacial

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Fig. 5.33 a Exact geometry of our composite model with the physical membrane [full model (FM)]. The stress (force per unit area) is directed from the dielectric with a higher permittivity (cytoplasm and ECM) towards the dielectric with a lower permittivity (membrane). Components (1), (2), and (3) represent respectively the cytoplasm, membrane, and ECM. In the cylindrical coordinate system, any point of the xOy Cartesian plane can be described via a single r coordinate as axisymmetric conditions are assumed for the rectangular computational domain, where g and h are the dimensions along the z and r directions respectively. The electric field is oriented along the z-direction, ψ denotes an angle relative to the electric field direction. The application of an electric field is performed using Dirichlet boundary conditions on top and bottom horizontal surfaces such as V (z = 0) = 0 and V (z = h) = E.h, where E denotes the applied field modulus; b As in a for the TLA-based approach: the idea is to replace the exact physical conditions by approximate numerical conditions that connect the solution on the two sides of the homogeneous thin-layer (physical membrane) boundary, therefore avoiding the need to mesh the layer and solve the exact ED problem, t and n denote respectively the unit vectors characterizing the local tangential and normal directions. The computational box is assumed to be relatively large (h = 10R, g = 5R) in each configuration, where R denotes the cell (vesicle) radius (in the undeformed state), from [14]

force so that both the membrane electric and mechanical forces can be formulated within this immersed boundary method. In any event, it would appear that, while the construction of these boundary methods is a remarkable result, the fact that only the electric constraints are dealt with somewhat limits their potential for ED investigations. For this reason, a complementary and generic approach was introduced and validated for simulating electromechanical phenomena that enables the simulation of models spanning several spatial and temporal scales [14]. The key novelty of [14] is the introduction of an explicit thin-layer approximation (TLA) to simultaneously solving the electrical potential and calculating the MST and cell displacement. Here, an important advantage is that the geometries can be arbitrarily thin while permitting arbitrarily large structural deformations. Computational Model: Our analysis focuses on two different cell configurations comprising either a single cell (2D axisymmetric geometry shown in Fig. 5.33. The displacement is modelled by applying an electric field along the z direction. The calculation of the electrical part of the simulation is being realized by making use

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Fig. 5.34 Algorithm for computing cell deformation in TLA and FM models and calculating the ED coupling in the TLA model. Both models follow the same steps. Solving of the electrical part is followed by considering the mechanical part. The arrows illustrate the input versus output causal relationships of the algorithm. The blue arrow which represents the electromechanical mechanical coupling before shape change and electric potential calculation is implemented only in the TLA model (from [14])

of Eqs. (5.17)–(5.20). All materials being assumed to be isotropic, conductivities and permittivities are represented by scalars. On the other hand, the set of electrical equations is to be supplemented by the mechanical part of the simulations, i.e. Equations (5.2)–(5.23). The details of the continuity equations for the different interface scan be found in [14]. Practically, the algorithm which is implemented is displayed in Fig. 5.34. Results and Discussion: The cell membrane was defined with a boundary condition in the TLA model or with a physical thickness in the FM model. In the latter case, the mesh generator is able to mesh the membrane domain but the number of mesh elements must be much larger than in the TLA model to achieve the same precision [14]. To check the agreement between both models we focus on the norm of the electric field, the value of the T nn component of the MST on the membrane of the reference cell, and the transmembrane imbalance of these quantities [14]. We next turn to examine the impact of the electromechanical coupling characterizing the ED force during the deformed state of the cell, for which the prolate spheroidal shape of the cell has for effect to increase the membrane polarization, and consequently the local value of V m . Here, we want to outline that the simple elasticity based analysis can be exploited to construct a complete picture of the local stress state surrounding the membrane. We show in Fig. 5.35 a comparison between selected deformed states of an initially (undeformed) spherically-shaped cell at three different times. In Fig. 5.35a, b, V m and the MST are calculated within a non-deforming mesh (no electromechanical coupling), while Fig. 5.35c is obtained with the TLA model by considering electromechanical coupling, i.e., the mesh deformation impacts the time dependent V m and, in turn, the MST.

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Fig. 5.35 Total cell displacement at three different times. The trapezoidal applied pulse [14]) has an intensity set to 1 kVcm−1 : a FM results with no electromechanical coupling, b TLA results with no electromechanical coupling, and c TLA results with electromechanical coupling. The color bar shows the total displacement and the black arrows represent the direction of the displacement (from [14]). Change b in c

In addition to the challenge of consistently generating low computational time simulations, another well-documented test problem in the literature [119, 120] can be used for the validation of the method outlined above. Besides determining the dependence of the local electric field and MST of the deformed cell, current strategies for analyzing ED focus on measuring the morphology change which is a generic feature of cell deformation. The ED theory predicts the degree of deformation b/a induced on the reference cell when applying the trapezoidal electric pulse ([14]) and electrical conductivity ratio Λ of the cytoplasm to the extracellular medium (ECM). Refs. [119, 120] worked out the implication of the more or less conductive ECM constraint for the aspect ratio. Their result is that the cell evidences an oblate deformation when Λ < 1 and prolate when is oblate when Λ > 1. This is clearly consistent with the results displayed in Fig. 5.36. As can be seen in the TLA-based calculations shown in Fig. 5.36 are in good agreement with the experimental data on giant unilamellar vesicules [119, 120], which are representative models of cells since they have similar dimension and can be individually observed and followed with video microscopy. Thus remarkably, we confirm these approximate results are relevant to describe the large cell deformations under electromechanical excitation. In fact, the TLA procedure by integrating ED in the analysis gives us a large flexibility since the physical parameters in the model we present are linked to scales we know should be associated with deformation, bending rigidity, and elastic stretching mechanisms.

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Fig. 5.36 Evolution of the aspect ratio characterizing the ellipsoidal deformation of an initially spherical vesicle as a function of time. We present our results in terms of benchmark points for vesicle deformation since we compare experimental results (black dots) [120] and numerical results (blue solid line) based on the TLA approach or the FM model (red solid line). In [120], a non electroporated vesicle in low conducting aqueous environment is subjected to an electric pulse of duration 250 μs and intensity 1 kV/cm [120]. The parameters used in the TLA model for producing this figure are summarized in Table 4.1 (from [14])

5.10 Electrical Coupling of the Cell Membrane and Nucleus Envelope Topic: Structural biology has dramatically changed our understanding of the biological self-organizing. We seek to obtain a quantitative understanding of biological systems; how they result from mechanics and energetics, and the self-organization of their constituents; how they are perturbed in disease and change over evolution. The past several decades have evidenced that alterations in the mechanical properties and morphology of cell nuclei are have been linked with numerous disease states, e.g. forces applied to the nucleus can affect the accessibility of the transcription machinery and the conformation of nucleoskeletal proteins, or cell contractility [121, 122]. Furthermore, the mechanistic properties of nuclei also facilitate healthy cell functions through transduction of mechnical stress into biochemical signals that alter gene transcription [121]. Here we introduce a dynamical approach which involves a double CS model of single biological cell since the biophysical mechanisms coupling between the cell membrane and NE are not fully understood. Before we embark on the analysis, it seems appropriate to establish the context with a brief account of the NE properties. The nucleus is the largest organelle in the cell and its position is dynamically controlled in space and time, although the functional significance of this dynamic regulation remains unclear. Nuclear movements are mediated by the cytoskeleton which transmits pushing or pulling forces onto the NE. At present, we have very few insights into the electromechanical properties of the NE (Fig. 5.37) which separates the nucleus from the cytoplasm in eukaryotic cells

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[121, 122]. The NE is constituted by a double membrane, the outer membrane of which is contiguous with the endoplasmic reticulum. The ONM and INM are compositionally distinct, as their physical properties. The ONM and INM are two concentric lipid bilayers, each ~4 nm thick, separated by the ~30–50 nm-wide perinuclear space [121, 122]. The NE comprises outer and inner leaflets as for the cell membrane, however the two leaflets are separated by an average distance which one order of magnitude larger than the cell membrane thickness. Over the last decade significant effort has been dedicated to addressing this question, particularly in the context of the nano-architecture of the NE, and notably its pore topology [123]. For example, cells can alter the rheology of their microenvironment by directly applying forces to the surrounding ECM [124]. Caille and coworkers are able to quantify the mechanical properties of the nucleus, when a single endothelial cell is subjected to compression between glass microplates [124]. Their technique allows measurement of the uniaxial force applied to the cell and the resulting deformation. The forces necessary for the compression of endothelial cells or isolated nuclei are in the range of 1–10 × 10−7 N [124]. The elastic modulus found for the cytoplasm of both round and spread cells is on the order of 500 Nm−2 . The elastic modulus of nuclei in round and spread cells is found to be around 5000 Nm−2 , on average 10 times more rigid than the cytoplasm, and the value found for the elastic modulus of isolated nuclei is around 8000 Nm−2 . The cell nucleus, which is normally substantially stiffer than the surrounding cytoplasm, may impose a major obstacle when cells encounter narrow constrictions in the interstitial space, the ECM, or small capillaries [125]. The NE possesses unique mechanical properties which enables it to resist cytoskeletal forces [125]. Thanks to a numerical model combining the CM, NE and cytoskeleton described by a random bundle of linear springs (actin filaments) connecting the CM and NE at nodal points, Ujihara and coworkers [126] show that the cell and nucleus are elongated in the stretched direction (Fig. 5.38). This computational model depicts a cell as a combination of various spring elements in the framework of the minimum energy concept. CM and NE are both modeled as shells of a spring network that express elastic resistance to changes in bending, stretching, and surface area. The interaction between the NE and the CM is expressed by a potential energy function with respect to the distance between them. Incompressibility of a cell is assured by a volume elastic energy function. Furthermore, Ujihara and coworkers [126] show that the total elastic energy is dominated by the energy stored in the actin filaments. Actin filaments that are randomly oriented before loading tend to become aligned, passively, in the stretched direction. This symmetry-breaking mechanism depends on the cytoskeleton network (filament density and orientation, and node number). Large nuclear deformations can cause localized loss of NE integrity, leading to DNA damage and eventually cell death [125]. During cell migration, the nuclear and cytoskeletal dynamics coordinate together at the leading edge and trailing end. This leads in complex changes in position and shape, which in turn affects cell polarity and shape. Therefore, modelling the nuclear deformation and the release of strain energy stored during deformation are important issues for understanding

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Fig. 5.37 (Top) Schematics of the nucleus organization (from [121]): The NE is composed of several elements, the most prominent of which are the ONM and INM. They are separated by the perinuclear space, a regular gap of 30–50 nm. The ONM is continuous with the endoplasmic reticulum membrane. The INM is in contact with the nuclear lamina (fibrous network). The NE is punctuated by around a thousand NPC, cylindrical invagination about 100 nm across, with an inner channel about 40 nm wide that allow RNA and proteins to move between the cytosol and the nucleus in either direction. The nuclear lamina is a 10- to 20-nm-thick protein meshwork associated with the INM and gives structural support to the nucleus. The nucleolus is a spherical structure found in the cell’s nucleus whose primary function is to produce and assemble the cell’s ribosomes. The nucleolus is also where ribosomal RNA genes are transcribed. A ribosome is an intercellular structure made of both RNA and protein, and it is the site of protein synthesis in the cell. (Bottom) A putative schematic view of the NE (from [121]): The cytoskeleton is connected to the nucleus through the LINC complex [e.g. SUN (Sad1 and Unc83) and KASH (Klarsicht, Anc1, Syne homology) domains]. The NPC perforates the NE providing a pathway between the nucleoplasm and the cytoplasm

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Fig. 5.38 Simulation of the deformation behavior during a tensile test and corresponding experimental study (left). The cell is modeled as an assembly of discrete elements including a CM, NE and cytoskeletal filaments (CSK). The nodes of the cytoskeletal filaments are fixed at one side of the CM, while those at the opposite side are moving on the direction of cell stretching (right). The load-deformation curve obtained from the simulation shows a significant increase in stretching load with deformation of the cell and lies within a range of experimentally obtained load-deformation curves. The length scale allows one to estimate the strain values (from [126])

how the nucleus and NE respond to stresses. In healthy cells, the nucleus is typically an order of magnitude stiffer than the cell cytoskeleton and on timescales relevant to cell migration (typically, minutes) appears to be elastic since it regains its original shape after deformation within seconds of force removal [124]. As has been noted recently in the literature [127], volume of healthy eukaryote cells is strongly correlated with their nuclear volume occupying approximately 8% of the cellular volume. To explain the relationship between cell and nuclear volumes, it has been suggested that the NE is connected to the extensile cytoplasmic actin filaments and microtubules [127], or it can be related to the nuclear lamina. Hence, it is crucial to better understand electromechanical interactions between the cell membrane and NE by enabling the entire cell to act as a mechanically coupled system [128, 129]. This could provide a physical mechanism to transmit information from one part of the cell to another. Although electromechanical processes have been contemplated, unambiguous observations with clear mechanistic interpretations have been lacking. Part of the physical challenge comes from the difficulty of spatially mapping local voltage inside the cytoplasm. One of the ways this can be achieved is to work directly with an extended Schwan’s model which represents the electrical characteristics of the cells by a distributed equivalent circuit (Fig. 2.3). There have been few attempts [130, 131] to address this question, particularly in the context of EMB. Here, we show that, under some very general and plausible assumptions about the cell constituents which are parameterized by resistors and capacitors, such model can provide a parsimonious explanation for capactive coupling as well as for ac filtering behavior of the biological cell. Motivated by the above considerations, it is worthwhile to ask whether the potential difference at the NE, V n , changes significantly during electromechanical excitation. this vein, there are many attractive questions dealing with RC network approaches of the cell. We would like in this study to take a stab at two questions. Firstly, which structural and physical parameters influence the NE

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electrical characteristics? Secondly, what is the role of the nuclear pore density in modulating the electrical behavior of the NE? In an extended study [131], we further test the accuracy of a two CS physical model by FE simulations and explore how stress transfer from the cell membrane to the cytoskeletal network to the nucleus is effective when the cytoskeleton is modelled as a set of elastic flexible filaments. Computational Model: From a phenomenological point of view, we would like first to discuss a representative RC network that incorporates the double CS structure as displayed in Fig. 5.39b in comparison to the RC combination of the single CS structure of an idealized spherical biological cell (Fig. 5.39a). It is convenient to work with COMSOL® Multiphysics which provides an analogue circuit simulator and solving Kirchoff’s laws. In the simulation, we apply an ac voltage excitation with amplitude set to 1 V and frequency ranging from 1 kHz to 1 GHz. It is worth noting that there three main limitations associated with our RC

Fig. 5.39 a Schematic diagram illustrating a resistive-capacitive combination of the single CS structure of an idealized spherical biological cell (light grey area). The CM (not at scale) is characterized by capacitance C m . The cytoplasm is described by a resistance Rc and the external medium is identified by a resistance Re . If Rm can be ignored, Kirchhoff’s circuit law reads Vm (2 + 2π jνRc Cm ) = V0 , where V 0 denotes the input voltage. b Same as in a for the two-shelled model for describing the electrical coupling between the cytoplasm (light grey area) and nucleoplasm (dark gray area). The NE (blue, not at scale) is characterized by impedance Z n . Resistances RL , RNE, Rc1 , Rc2 characterize respectively the nucleoplasm, NE, and two cytoplasm regions. In that case, it is straightforward to obtain V n by changing Rc (Fig. 1a) in 2Rc1 +(Rc2 (RN +2Zn ))/(Rc2 + RN + 2Zn ) (with 1/Zn = 1/RNE + 1/(RL + 1/(jπ νCn )) and if resistance RL can be omitted). c The NE impedance contains pure resistance RL representing the conducting lumen connected with lipid double layer capacitors of capacitance C n . However, when neglecting the voltage drop across this resistance RL , the characteristic time associated with the capacitive charging of the NE can be ( ) expressed as τn ≈

1 4

Rc1

(Cn (Rn +Rc1 ) ) 1 1 R +Rn +4 R c2

NE

[131]

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model. First, the resting potentials of the CM and NE are neglected. If needed, i.e. this is important to understand the ionic electrodiffusion in electrically stressed biological cells, voltage generators can be easily added to the RC circuit. Second, unless otherwise specified, the CM and NE are considered as passive, i.e. ion channels and pumps are ignored, and static in space and in time (equilibrium state with no fluid-dynamic stress). Constraints arise from ED, since electrostriction can wipe out deviations from isotropy of the electro-deformed cell if V m is not too large [13, 15, 16]. In this paper, we show explicitly that ED can be also dealt with by considering active elements in the RC network. Third, all materials are homogeneous, continuous and isotropic: they are characterized by scalar materials parameters, i.e. the permittivity and conductivity of materials are isotropic. In principle, V m is a complex number made up of a modulus and a phase, but in the subsequent discussion we choose to discuss its modulus only. In this section we use the physiological and anatomical data listed in Table 4.1. Results and Discussion: Several of the most important features of theelectrical behavior of the cell can be understood readily from the simple RC model displayed in Fig. 5.39b [130–132]. At the moment we are concerned with the simple case when Rm can be ignored. We observe striking differences between in the spectral behavior of V m and V n which results from unequal distribution of charges across the NE due to selective barrier permeability. Figure 5.40 shows that CM, when exposed to ac electric fields, acts as a low pass filter whereas NE acts as a wide and asymmetric bandpass filter. Although the former might be explained by the β-relaxation of the CS modelling of the single cell, it is also consistent with the cutoff frequency ∝ 1/τm estimated from Schwan’s model, which shields interior cell regions from the exogeneous electric field [13, 18, 24]. The NE bandpass filter transmits frequencies defined by a nominal center frequency (CF n ) and bandwidth (full width half maximum (FWHM) ≈ νmax ) while cutting both shorter and longer frequencies. It is also rather remarkable that V n tends to be small compared to V m . Within Schwan’s model, ( )−1 2 Kirchoff’s circuit law predicts that the spectral signal Vm = V20 1 + (2π ντm )2 / is a Lorentzian line shape function if Rm can be ignored. Furthermore, it is argued that the electrical voltage across the NE results from intranuclear fixed electrical charges, diffusion across INM and ONM, and selective transport through the NPC [125]. It helps also to examine the detailed form of V n . One derives V n by changing Rc (Fig. 1a) in2Rc1 + (Rc2 (RN + 2Zn ))/(Rc2 + RN + 2Zn ), with1/Zn = RL can be 1/RNE + 1/(RL + 1/(jπ νCn )), and if resistance ( )(omitted. Importantly, ) √ 1 the modulus of V n is given byVn = VB0 1 − √ 1 , where 1+(2πντm )2 1+(2πντn )2 ( ( )) ( ( ) ) c1 τ = A/B with A = Rn2Cn + Rc12Cn RRc2n + 1 andB = 2 RRc2c1 + RRc2n + 1 RRNE , respectively. The utility of this expression lies chiefly in identifying theoretically useful discriminators among different physical parameters affecting the CF n and FWHM. Given the high NPC density (Table 4.1), we expect that the NE does not lead to any significant electrical resistance. Given the high NPC density (Table 4.1)

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Fig. 5.40 Modulus of the transmembrane potential, V m , and NE voltage drop, V n , as a function of frequency of the ac electric field. Solid (resp. open) dots indicate the values of V m (resp. V n ) derived from solving)(numerically Kirchhoff’s laws. The dashed lines show the fits ( ) to Vn =

V0 B

1− √

1 1+(2π ντm )2

√

1 1+(2π ντn )2

. The solid lines show the fits of the data to a

Lorentzian profile. We note that works rather well (at least at the level of 99% accuracy) in the range of frequencies explored. We caution the reader that the scale units of V m and V n are different. Arrows denote the corresponding scale of V m and V n [131]

and that the nucleus areal resistance is of the order of 10–4 Ω m2 , i.e. much smaller than the cell membrane areal resistance, we expect that the NE behaves as a leaky dielectric [16]. We also observe that the nucleus areal resistance is of the order of 10–4 Ω m2 , i.e. much smaller than the cell membrane areal resistance. We want also to extend our analysis to allow for an estimation of the time scale, τ n , for charging the NE which serves as a direct probe of the capacitive coupling between the CM and NE. This point has not been considered yet in the literature. In order to evaluate τ n we apply Kirchoff’s circuit law in the time domain to the electrical circuit displayed in Fig. 5.39b. A simple formula turns out to well approximate τ n when the V n can be neglected (checked numerically [131]) and we impact of R(L on the value of ) getτn ≈

1 4

Rc1

(Cn (Rn +Rc1 ) ) 1 1 R +Rn +4 R c2

, where the capacitor C n and different resistances Rn ,

NE

Rc1 , Rc2 , and RNE are those illustrated in Fig. 5.39. This result is important because it opens the window for describing the electrical coupling between the CM and the NE. As a confirmation of the results derived previously and to determine the fine tuning of the FWHM of the NE voltage drop, we compare several metrics when several structural and environmental triggers are varied. We illustrate this in Fig. 5.41 with color shadings. To obtain these plots, we dissect the individual effects of the structural and environmental triggers, by changing one parameter at a time and keeping the rest as a constant.

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Fig. 5.41 Panels showing metrics characterizing the spectral behavior of the NE (τ m /τ n, ν β , CF n , ν max, ν min, νmax /νmin , and V nmax ) plotted as function of the: a NE conductivity σ NE , b capacitance per unit area C˜ m , nucleus volume to cell volume ratio V˜ , cytoplasm conductivity σ c , cell radius r cell , and nucleoplasm conductivity to cytoplasm conductivity ratio σ n /σ c . For this calculation we dissect the individual effects of the structural and environmental triggers, by changing one parameter at a time and keeping the rest as a constant

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Overall, different structural parameters of the cell have different degrees of influence on the spectral behavior of the NE. Increasing (resp. decreasing) σ NE , σ c et σ n /σ c has for effect to increase (resp. decrease) the effective conductivity of the intracellular medium, and correspondingly an increase (resp.) decrease of the CF n . It is also remarkable that the normalized FWHM, νmax /νmin and CFn show similar ˜ σ NE , σ n /σ c vary whereas νmax / variations when the nucleus characteristics, i.e. V, νmin and CFn show opposite variations when the cell characteristics except those dealing with the nucleus, i.e. σ c , C˜ m , rcell , vary. In the former case, it is important to note that the spectral behavior of CF n follows that of ν β, whereas ν β remains mostly unchanged in the latter case. The reason underlying the broadening of the V n spectrum as the cell radius gets larger is readily understood on the basis of the SM and is concomitant to the blueshift of the β relaxation frequency of the cell since τ m scales linearly with r cell . Another obvious fact to note is that V nmax varies significantly with larger V˜ and smaller σ NE . From the panels shown in Fig. 5.41, it ˜ which are relevant parameters can be concluded that larger values C˜ m , r cell , and V, for discriminating between healthy and pathological conditions, correspond to smaller values of CFn, larger V nmax values and narrower spectral V n profiles. Three comments are in order. Firsty, the changes to structural and environmental triggers are decided a priori, but that raises the question of how one might design changes to these parameters in order to achieve a desired V n spectrum. Physically, the patterns shown in Fig. 5.41 provide a measure of the fact that the concomitant change of the structural and environmental triggers could be confused by their respective impact, since all of them contribute to the spectral characteristics of V n . Secondly, Appendix 3 discusses in detail the effects of thermal noise in CE and NE potentials. This is an interesting subject to deal with since the stability under thermal noise is an important feature of the Schwan mode, on the one hand, and because internal thermal noise presents a primary hazard to the reliable functioning of the cell apart from the constraints it receives from the ECM, and thus can be a discriminating metric between healthy and pathological cells, on the other hand. Thirdly, there are more complicated issues of the role of nucleoplasm. The most important one is that the nucleus interior is a crowded chemical space and a fragile soft material in which the effects of biochemistry, molecular crowding and physical forces are complex and inseparable [125, 133]. In particular, liquid-like nuclear condensates, e.g. nucleoli, nuclear speckles, and Cajal bodies have been observed [125]. These components interact in multiple contexts and studying the physical interplay between biomolecular organization and nuclear mechanics is of paramount importance for understanding nuclear organization. Microrheological approaches provide a direct means of interrogating the material properties of the nucleus and subnuclear structures [134]. Little is known about how the mechanical rigidity and assembly dynamics of actin filaments impact the large scale cellular mechanics properties. It is not objectively satisfactory to lack such understanding. These remarkss highlight the challenge of dealing with a simple modelling of architectural heterogeneities due to the cytoskeletal network of actin filament bundles. Future studies should address if this prediction can be tested experimentally. These considerations emphasize the significance of experiments to look for the electromechanical properties of the nucleus.

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Generic Final Remarks Finally, I’d like briefly to discuss a few questions that definitely belong on the agenda of future EMB physics. There’s quite a lot that physicists might once have hoped to derive or explain based on first-principles of physics, for which that hope now seems far away. Here I explore some sources of these limitations and the role different sorts of explanations might play in filling the voids. Even though many of the models reported in this chapter are rudimentary since they ignore a lot of morphological details of cells and cell-cell interactions, they are instrumental in defining the field of computational modelling of EMB. A keen understanding of the advantages and limitations of these models is necessary. As was said judiciously, “Essentially, all models are wrong, but some are useful” [135]. Of course, the real story is more complicated and interesting. There are many points to debate in this analysis which is filled with subtleties that most biologists would agree we do not yet fully understand. Something that has gone undiscussed to this point, is that most of the current continuum models of the literature treat the cell as homogeneous and isotropic, i.e. there is no difference in its stiffness or other mechanical properties when the force direction is changed. In this case, a local stress would radidly decay as the distance increases. It isn’t, of course. Indeed, superficially the distribution of matter in the membrane appears to be anything but uniform [136]. In addition, the electric conductivity can be expected to vary along different axes and over time. The ionic strength dependence of the PB can be considered [137], but is often ignored. The absence of lipid domains and proteins in the membrane, roughness of the membrane asymmetry in the membrane, influence of cytoskeleton on the mechanical properties of the membrane, and a unique kinetic behavior for pores are commonly inherent assumptions and simplifications of these models. Fortunately, our nearly total ignorance concerning the random nature of pore nucleation in the membrane does not bar us from modeling its dynamical evolution of some assumed initial pore distribution. That’s because the dominant interaction on large scales is electrostatics. A priori, one might consider all kinds of assumptions about the initial pore seeds, and over the years several assumptions have been proposed [138]. Most physicists are not, and should not, feel entirely satisfied with such approximations. The parameters appearing in the standard EP models do not fully describe the fundamental electromechanical behavior of simple CS structures such as that describe in Fig. 2.1. Due to the too simple ontology of these continuum approaches we are left wondering how many supplementary parameters appear necessary to make a working description of existing observations. The answer will notably depend on details of membrane physics at very small time and length scales. Thus, there are genuine, exciting opportunities for carrying the EMB analysis of membrane parameters further. Good computational models actually inspire new experiments and this is really a virtuous circle that will provide real-world data needed for the hard-working modellers. Epilogue: Challenges Ahead Before the paradigm shift enabled by molecular biology, cell and tissue electromechanical studies were a major aspect of research in biology. Now, exploring the

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EMB properties of biological materials is re-emerging at the forefront of biological and biomedical research. In this monograph, we discussed the pillars of EMB, such as cell ED and EP by making use of crucial simplifications. We moved on to computational approaches to test the predictions from these fundamental analyses, such as tests probing models that attempt to explain CMP. As we hope this book demonstrates, there is a tremendous amount of work yet to be done in order to fully realize and exploit EMB’s scientific potential and its clinical application. The growing literature on the importance of PB leads us to advocate the EMB paradigm that combines physics, computation, biology and mechanics as building bricks. The systems studied so far have yielded a richness in their physical properties that considerably exceeded the initial expectations. Does EMB retains the scientific vitality that has characterize it for the lest decades? The most important product of our knowledge is ignorance-educated ignorance. One must be educated in order to be able to ask the right questions. A scientific field requires at two things to remain vital. First, it requires questions, interesting questions, important questions, and accessible questions. Second, it requires new experimental instruments and techniques that can be used to probe and answer these questions. As a final send-off of this book, let us identify a few directions in which physicists, biophysicists, and engineers can provide a greater understanding of the foundations of EMB. Hoefully, we expect that some of the questions that we can ask at this point of time are as exciting as they have ever been, if not more so. Regarding test of EMB, the most burning questions all involve the development of more complex and realistic models cell models. While the set of simplified CS cell models described in this book is extensive, it is also incomplete. Some omissions are by design-the major issue we have left to address is to fully consider what the heterogeneity of real membrane and cytoplasm, as highlighted in Fig. 2.1, and indicated widely in the literature [139–142], implies about the EMB mechanisms. Perhaps, one of the most important is the construction of an EMB modelling that incorporates the asssociation of the cytoskeleton with the PB, i.e. the membrane and the cytoskeleton should be considered as a composite material, e.g. cytoplasmic rheology can be treated as a biphasic material consisting of a porous elastic solid meshwork (cytoskeleton, organelles, macromolecules) bathed n an interstitial fluid (cytosol) [143]. One way to address this issue is to consider that a biological cell is a material with complex properties arising from a composite and highly dynamic, non-equilibrium intracellular biopolymer network-the cytoskeleton [143]. It is now well recognized that the asymmetrical distribution of charged lipids between the two leaflets of the cell membrane, and laterally within the leaflets, plays an important role in many cellular processes. Charged lipids in particular create an electrostatic zeta potential that not only differs on the ECM and intracellular membrane interface but can also result in distinct membrane charge patterns. Thus, a complex and integrated picture should is needed in which charged lipids, ions in solution and transient protein interactions are in a dynamic equilibrium [144]. Lipid composition determines rigidity and fluidity of membranes. Consequentially, changes in the concentration of individual lipids not only alter membrane composition, but also the force required to deform the membrane and the time that is required to enrich specific lipids at such curved

250

5 Computational Approaches

sites, i.e., membrane viscosity. Although the fundamental role of mechanics is now well appreciated and the mechanisms have begun to be understood, our knowledge has so far been limited to enable the control of cell mechanical response. Mechanical forces have been recently shown to be transmitted from the ECM to the cell nucleus, where they deform the nucleus and eventually impact the overall cell machinery [145]. Considering specific characteristics of electrostatics in extreme confinement situations such as in nanopores is also important since inside a nanochannel, ions are no longer surrounded by a homogeneous fluid, and their interaction potential may be affected by the dielectric properties of the confining medium. This phenomenon was pointed out decades ago by Parsegian in the case of ions crossing PB [146], therefore it can challenge continuum approaches of fluid transport in molecular scale confinement situations (electropores and nucleopores). There are critical gaps in our understanding of nanoscale hydrodynamics and thermodynamics in the so-called “single digit nanopores” with nominal diameters MHz as displayed in Fig. 5.44b. These observations suggest that this effect is due to the shielding effect of external layers of charges. To expand on this point, we compare the local electric field distribution for various frequencies (labelled by capital letters in Fig. 5.44), the same parameters as above, and surface charge densities as shown in Fig. 5.45. To illustrate this point, these simulations support the interpretation that the interior of the cell is screened by the outer membrane from the applied electric field below the β relaxation frequency of the cell. In addition to the features discussed, potential perturbations are localized near the outer membrane. This is consistent with the fact that, at higher frequencies (>1 MHz), the membrane can no longer screen the interior of the cell (Fig. 2.4). Up to this point, our analysis has assumed the circular cylindrical shape of cells displayed in Fig. 5.13. Below, we specify the rounded geometry adopted in our model of the biological cells. The type of structures we have in mind are Cassinian curves along which the product of the distances to two given points, called foci, is a constant. The explicit expression for these curves is [172]

Appendix 1: Computing the ITV for an Isolated Cell Model

259

Fig. 5.45 Two-dimensional plots of the local electric field (red) distribution for various frequencies (represented by positions a, b, c, and d in Fig. 4), from [171]

/ r(θ ) = ±

a2

/ cos(2(θ + π/2)) ± b4 − a2 sin2 (2(θ + π/2)).

(5.32)

The a and b’s defining the shape of the Cassinian curves are given in the caption of Fig. 5.46. All parameters are kept the same as above, except that we choose to work with a constant Φ = 0.2, i.e. volume-conserving variations in shape. The membrane thickness was set to 5 nm. It turns out that similar approximation of biological cell has been used to describe the dielectric spectra of aqueous suspensions of nonspheroidal biological cells. These Cassinian structures appear in the upper panel of Fig. 5.46 We focus on the scenario sketched in Fig. 5.46 as realized by the sequence of We begin apparent transformations (a)↔(b)↔(c)↔(d)↔(e)↔(f)↔(g)↔(h)↔(i). / by reporting some characteristics of the ITV for θ = π 2. This leads to Figs. 4a, b which compare the spectra of |Vm | and φ m . Analysis of our data show that differences between the spectra for the nine structures are visible only for frequencies below the β relaxation frequency of the cell (≈ MHz, i.e. Figure 3.1).

260

5 Computational Approaches

(a)

(b)

(c)

(f)

x

400

A

(d)

(g)

(e)

(h)

x

(i)

(a)

B

200

C

x

D

x

0 0 (b)

/4

(a) (b) (c) (d) (e) (f) (g) (h) (i)

/2 10

2

10

4

F (Hz)

10

6

10

8

Fig. 5.46 Upper panel: The (a)–(i) designs deal with the cross-sections of the Cassinian core–shell structures. (a) a = 0 μm and b = 10 μm, (b) 2a = 5 μm and b = 10 μm, (c) 2a = 10 μm and b = 10 μm, (d) 2a = 15 μm and b = 10 μm, (e) 2a = 16 μm and b = 10 μm, (f) 2a = 17 μm and b = 10 μm, (g) a = 18 μm and b = 10 μm, (h) 2a = 19 μm and b = 10 μm, (i) a = 20 μm and b = 10 μm. The shell (phase 2) is not visible in the figures. Lower panel: a The modulus of the ITV, |Um |, (shown in logarithmic scale) spectra obtained from FE simulations for the nine structures. Φ = 0.2, θ = π /2, and σ 2 = 10–6 Ω−1 m−1 , b Same as in a for the phase φ m (from [171])

Appendix 2: Two-State Model of a Random Assembly of CS Spherical Structures One important feature of stochastic 3D models of sphere packings is that beyond explaining the tissue crowding, it can be used as a two-state model in which a local membrane contains either no pore or some number of minimum size pores. One of the main purposes of this appendix is to argue that by using a parsimonious, computationally simple stochastic 3D model of sphere packing of three sizes (Fig. 4), one can obtain, by computing the changes in electric conductivity due to EP, results which are consistent with those of Ref. [173]. This model can be used to predict the fraction of electroporated cells p of a tissue by monitoring the membrane electric conductivity as a function of the exogenous electric field magnitude. This prediction is shown to agree well with experimental observations [173]. The tissue is modelled with a random assembly of CS spherical structures with three cell radii 8, 10, and 12 μm. Even if real tissues (Fig. 5.47a), however, are not composed of rigid and immobilized cells we will restrict the present analysis to undeformable sphere-shaped cells.

Appendix 2: Two-State Model of a Random Assembly of CS Spherical …

261

Fig. 5.47 a Loose connective tissue composed of adipose cells. The ECM space is a connected space and is characterized by a high tortuosity. The source of image is [20]. b A specific realization of our multicellular tissue model consisting of randomly positioned spherical CS structures with three cell sizes: 8, 10, and 12 μm. c Spatial patterns of the local increase in membrane electric conductivity illustrating the spatial distribution of hot spots in randomly positioned CS structures. The electric conductivity is computed 4 μs after the onset of the field with magnitude of 1 kVcm−1 > E th = 0.45 kVcm−1 . The cell density is set to 28 vol%. The color bar on the right gives the values of the membrane conductivity (from [173])

Since the cell positions in the computational box (Fig. 5.47b) is random, we must generate many realizations with prescribed statistical properties and subsequently evaluate averages to obtain accurate estimates of physical parameters of interest [173]. The algorithmic method consists of placing the CS spherical particles in a random manner without any overlap between them employing an internal MATLAB function, which uses a lagged Fibonacci random numbers generator. The general recipe can be used to construct various models, depending on the cell radius and number assignments [173]. An example of such geometric model is shown in Fig. 5.47b along with the corresponding local distribution in membrane electric conductivity (Fig. 5.47c). We shall use the SNK model of EP for a single cell. For a uniform external electric field, with magnitude E and rise time much lower than the membrane charging time, is applied along the x-axis. The collective (macroscopic) tissue response is determined by the local electric field experienced by the individual cells at the microscopic level. Ref. [173] showed that his tissue model provides a satisfactory description of the electrical properties. Firstly, an increase of the average electrical conductivity calculated for 15 realizations of the tissue model shown in Fig. 5.47 with time produced by E = 1 kVcm−1 > E th = 0.45 kVcm−1 should be interpreted as a plausible evidence of the EP phenomenon [174]. Secondly, the larger cells are electroporated first, and areas of cell membranes that undergo EP occur in regions where polarization is maximized. Thirdly, the fraction of electroporated cells p was calculated for 5 different realizations of the tissue model, with each model having the same cell density, and the values of p were then averaged.

262

5 Computational Approaches

100

20 vol.% 28 vol.% 33 vol.%

p (%)

80 60 40 20 0 0.2

0.4

0.6

0.8

1.0

-1

E (kVcm ) Fig. 5.48 Fraction of electroporated cells p, as a function of electric field magnitude E for cell densities of (filled black square) 20 vol%, (filled black circle) 28 vol%, and (filled black triangle) 33 vol%. The results are the averages of 5 random distributions of ternary CS tissue models for a given cell density (Fig. 4). The curves are the fits of the hyperbolic tangent law to the calculated data. The field duration is 100 μs (from [173])

Figure 5.48 compares the values of p(E) for 3 cell densities ranging from 20 vol.% to 33 vol.% for field duration of 100 μs. These results lead to the important conclusion that EP initiates at higher E when cell suspension is denser. It is important to note that our calculations are in good agreement with the experimental observations of [175], where higher cell densities required higher field magnitudes to obtain the same fraction of EP. Physically, dipolar couplings which are effective at the macroscopic scale are important when the polarization is not isotropic. We should underline that from a physical point of view, it is natural to expect a large anisotropic σ in the EP since only a fraction of the cell’s area is electroporated during the field exposure (Fig. 5.47c). As can be clearly observed in Fig. 5.48, the sigmoid / p(E) = (1 + tanh(β(E − Ec ))) 2,

(5.33)

where β is a constant and E c is a crossover field, seem to be a universal characteristic for the range of densities investigated in this study. It is worth noting that the curves shown are coherent with the modelling and experimental study of the diffusion-driven transmembrane transport of small molecules caused by electropermeabilization [176]. They are also consistent with other choices of the sigmoidal function [177]. with Although we came across this finding serendipitously, we would like to question whether this equation explains data better than a simple empirical curve. In fact, this universality can be interpreted in the framework of two-state systems commonly encountered/in several contexts in statistical physics, e.g. the archetype models are the spin-1 2 case of paramagnetism and the polarized light [178], according to which the order parameter follows a hyperbolic tangent law as a function of the exciting field. The above remarks are relevant to our work because the

Appendix 3: Thermal Noise in Cell Membrane and Nuclear Envelope …

263

electroporated and non-electroporated cell membranes subjected to the applied electric field can be viewed as the two states in the linear response regime. Taken together, these simulations qualitatively reproduce the features of the hyperbolic tangent law describing the crossover from non-electroporated to electroporated states. Reducing the cell density in the computational domain results in a decrease of E c whereas β is found to be constant and equal to 0.11 (kV/cm)−1 . In closing we observe that cells in tissue are likely to be electroporated at a lower electric field than cells in suspensions. Additionally, based on this(modelling)/study, one can estimate a 10% increase in the anisotropy ratio, Δσ = σxx − σyy σxx ,4 due to the application of electric field (duration 100 μm) above the electroporation threshold (0.4 kVcm−1 ) up to 1.2 kVcm−1 [179, 180]. These findings are consistent with the experimental data on the field-induced anisotropy dependence of the electrical conductivity due to cell membrane EP using magnetic resonance EIT [181].

Appendix 3: Thermal Noise in Cell Membrane and Nuclear Envelope Potentials Although the potential importance of noise for biological function was appreciated a long time ago,5 the development of single-cell-analysis methods in the past decade allowed the direct observation of noise in diverse organisms. It is surely one of the triumphs of evolution that Nature discovered how to make highly accurate molecular machines in such a noisy environment [182]. Noise caused by stochastic fluctuations in deterministic equivalent circuit models of biological cells is now appreciated as a central aspect of cell function behavior within the context of a thermodynamic perturbation-response analysis [182]. As was underlined by Weaver and Astumian [183], any observable response of a cell submitted to an exogeneous electric field implies that the field causes changes greater than those due to thermal random fluctuations events (JohnsonNyquist noise). This noise originates from the thermal equilibrium of the cell within the conducting ECM. For a given bandwidth, the root mean square (RMS) of the fluctuation voltage of a resistor R reads, (δ V¯ )2kT = 4kTRΔF, where ΔF is the bandwidth in hertz over which the noise is measured, and the overbar denotes thermal averaging [184]. For a 1010 Ω resistor at room temperature and a 10 kHz bandwidth, the RMS noise voltage is typically 10–2 mV. Ideal capacitors, as lossless devices, do ↔

Since the conductivity σ tensor is independent of the precise boundary conditions imposed on the ⎤ ⎡ σxx 0 0 ↔ electrical potential, we can choose those conditions such that σ = ⎣ 0 σyy 0 ⎦, in the Cartesian 0 0 σyy

4

↔

coordinate system defined by the dielectric axis. The σxx , and σyy components of σ can be easily performed by FE simulations [180]. 5 For more than a billion years, thermal energy fluctuations have driven interactions of small structures and molecules, enabling them to find the best interaction state in time and space.

264

5 Computational Approaches

not have thermal noise, but as commonly used with resistors in an RC circuit, the combination has what is called kTC noise. The noise bandwidth of an RC circuit is ΔF = 1/(4RC). When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the resistance R drops out of the equation. This is because higher R decreases the bandwidth as much as it increases the noise. The RMS noise voltage generated in such a filter is (δ V¯ )2kT = 4kT /Cm .6 Now considering the spherical cell model displayed in Fig. 2.3 for which the capacitance m is given by Cm = ∈0 ∈dmmAm , we get (δ V¯ )2kT = πεkTd 2 . Based on the values listed in Table 0 εm R 4.1, the estimate of the RMS variation in the ITV due to thermal fluctuations at 300 K is 10–1 μV. Now, for the case of a perfectly insulating spherical cell membrane, for with the ITV ( 23 ER) is given by Eq. (1.1), one obtains the estimate of the minimum / electric field to which a cell can respond by equating the ITV to (δ V¯ )2kT leading to 2 Vm−1 . Based on this back-of-the-envelope calculation, Weaver and Astumian [182] showed that a lower limit for a minimum detectable field imposed by competition between an applied electric field and thermal noise is very small compared to the resting field Vrest /dm which is of the order of 107 Vm−1 . It has furthermore been argued by Chen and Gillis that the study of the different sources of noises (thermal and 1/F) of membrane capacitance measurements allows determination of optimal frequencies for capacitance measurements [185]. One of the important questions is if this kind of calculation can be also relevant for understanding possible effects of weak electric fields in two-shelled mechanistic models of a biological cell including a NE. The construction of such models for describing the electromechanical coupling between the cell membrane and NE is a research area that has progressed significantly and garnered a great deal of interest in recent years (Sect. 5.9). The cell membrane, when exposed to ac electric fields, acts as a first-order low pass filter with a time constant τm which is in the range 0.1–1 μs, corresponding to the frequency range 1–10 MHz to be compared with a noise spectrum bandwidth ΔF of several Hz. Little has been studied on how the NE impacts the frequency spread when it acts as a first-order bandpass filter [131]. It would be also good to have a discussion relating the width of the spectrum with the aspect ratio of an electrodeformed initially spherical cell.

6

This equation can be alternatively derived by considering the capactor as a system with one degree of freedom and applying the equipartition theorem. In some circumstances it is the fluctuation / ) ( / )1/2 ( probability function which is of interest. It reads p(V ) = 2C π kT exp − 21 CV 2 kT , where the pre-exponential factor normalizes the integral of p(V ) for 0 ≤ V ≤ ∞.

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Glossary of Acronyms

1D One spatial dimension 2D Two spatial dimensions 3D Three spatial dimensions AFM Atomic force microscopy AI Artificial intelligence ATP Adenosine triphosphate CC Cole-Cole CaC Capacitive coupling CEH Canham-Evans-Helfrich CF Crossover frequency CMP Cell membrane permeabilization CS Core-shell DBK DeBruin-Krassowska DC Davidson-Cole DEP Dielectropheresis DNA Deoxyribonucleic acid DS Dielectric spectroscopy ECM External cellular medium ED Electrodeformation EF Electrostatic force EIT Electric impedance tomography EL Electrostriction EMB Electromechanobiology EP Electroporation EROT Electrorotation FDTD Finite-difference time-domain FE Finite element GEVI Genetically encoded voltage indicator GIMP GNU image manipulation program © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9

283

284

GUV Giant unilamellar vesicle HFIRE High-frequency irreversible electroporation HN Havriliak-Negami INM Inner nuclear membrane ITV Transmembrane voltage KKW Kohlrausch-Williams-Watt MC Membrane capacitance MD Molecular dynamics ML Machine learning MR Membrane resistance MST Maxwell stress tensor MWS Maxwell-Wagner-Sillars NE Nuclear envelope NPC Nuclear pore complexe ONM Outer nuclear membrane PB Phospholipid bilayer PDE Partial differential equation RNA Ribonucleic acid RTA Repulsion-to-attraction SNK Smith-Neu-Krassowska TLA Thin layer approximation

Glossary of Acronyms

Author Index

A Abdelgawad, M., 238 Abiror, I.G., 74, 76, 116, 120 Abramowitz, M., 117, 129 Agrawal, A., 39 Agrò, A.F., 75, 77 Aifantis, K.E., 33, 223, 227 Airoldi, P., 152, 253 Akimov, S.A., 75, 76, 119 Akinlaja, J., 112, 113, 114 Alber, M., 251, 252, 253 Alberts, B., 18, 19, 108, 207, 212 Aleksanyan, M., 75, 76, 81, 121, 123 Alenghat, F.J., 39 Alfonso, M., 82, 120, 201, 225 Allen, T.W., 216 Alraies, Z., 48 Aluru, N., 250 Alvarez, O., 109 Amaral, L.Q., 248 Ambrosone, L., 73, 82, 141, 142 Ammari, H., 148 Andelman, D., 219 Andersen, O.S., 216 Anderson, J.A., 181 Anderson, J.G., 217 Andrews, L.C., 158 Angelini, T.E., 12, 14, 38, 208, 209, 217, 253 Anh Pham, T., 250 An, S.S., 247 Apollonio, F., 23 Arakelyan, V.B., 74, 76, 116, 119, 120 Aranda, S., 200 Arena, C.B., 105, 124

Arsenin, V.Y., 139 Arsenovic, P.T., 249 Arshady, R., 221 Asaka, K., 257 Asami, K., 23, 73, 142, 200, 201, 203, 207, 211, 214, 256, 257 Asbury, C.L., 146 Asnacios, A., 32, 42 Astumian, R.D., 263 Atajanov, A., 250 Avallone, R., 143 Avelin, J., 148

B Baffou, G., 157 Bahrami, A.H., 221 Bailly-Maitre, B., 257 Bai, X., 195 Bajd, F., 263 Balland, M., 32, 42 Bao, G., 251 Barnett, A., 19, 114–116, 235 Bartol, T.M., 150, 151 Bastian, P., 150, 151 Basu, B., 11, 14, 22, 28, 31, 47 Bathula, K., 249 Batishchev, O.V, 75, 76, 119 Baum, B., 48, 225, 227 Baumgart, T., 35, 239, 240, 241 Bazant, M.Z., 250 Becker, F.F., 78 Beckers, F., 105 Beebe, S.J., 107, 108, 120, 131, 180 Belehradek Jr, J., 77

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9

285

286 Bellini, F., 143 Bellomo, N., 102, 131, 179 Benett, W.F.D., 180 Bénit, P., 157 Ben-Yaakov, D., 219 Benz, R., 19, 22, 23, 105 Bernardi, W., 1, 68 Beroual, A., 181, 185, 186, 188, 208, 213, 244 Berret, J.-F., 43 Bertholom, P., 80–82 Beskok, A., 109 Beta, C., 242 Beuzard, Y., 80–82 Bhargava, A., 48 Bhonsle, S.P., 124 Bianca, C., 102, 131, 179 Bia, P., 120, 195, 222 Bidaye, S., 120 Blacmore, P.F., 180 Bladergroen, M.R., 75, 77 Blakely, B.L., 39 Blanchette, C., 255 Blankschtein, D., 250 Bloom, M., 36, 39 Bobanovi´c, R., 145, 151 Böckmann, R.A., 75, 77, 83, 84, 112 Boghdady, C.-M., 47, 48 Bohinc, K., 114 Bonakdar, N., 33, 223, 227 Bord, F., 168 Bostoen, C., 43 Böttcher, C.J.F., 125–127, 132, 138, 142 Boudou, T., 14, 70, 71, 225 Box, G.E.P., 248 Boynard, M., 80–82 Brangwynne, C.P., 12, 14, 38, 208, 209 Briels, W.J., 124 Brighton, C.T., 195 Brill-Karniely, Y., 33, 36 Brodland, G.W., 48, 81, 82, 225, 227, 249 Broedersz, C.P., 47 Broers, J.L.V., 249 Broggi, G., 143, 144 Brooks, N.J., 65 Brosseau, C., 4, 14, 19, 21, 23, 24, 27, 29, 31, 42, 45, 46, 68, 75, 76, 81, 82, 103, 104, 108, 110, 115, 120–127, 131, 138, 141, 142, 145, 148, 152, 154, 158, 181, 185, 186, 188–209, 211–215, 218–239, 242–245, 250, 256–258, 260–264 Browaeys, J., 32, 42

Author Index Brown, D.A., 77 Brownell, W.E., 109, 115 Brown, T.D., 33, 36, 37 Brozena, A.H., 250 Bryant, G., 14, 27, 28, 30, 31, 45, 74, 75, 200 Buescher, E.S., 107, 108, 120, 131 Buganza Tepole, A., 251, 252, 253 Burak, Y., 219 Burke, B., 240 Butler, J.P., 33, 35, 39, 42, 43, 217, 247, 253 Byrne, H., 47

C Cadene, M., 216 Calcabrini, A., 168 Camarda, M., 143, 144 Cametti, C., 73, 82, 141, 142 Canham, P.B., 37, 38, 40, 73 Cannon, W.R., 251, 252, 253 Caratelli, D., 195, 222 Carslaw, H.S., 158 Caruso, G., 143, 144 Casciola, M., 23, 115, 117, 120, 121 Cattin, C.J., 48 Ces, O., 65 Cevc, G., 11, 14, 23, 24, 47 Chahine, N.O., 255 Chait, B., 216 Challis, L., 157 Chang, D.C., 216 Chang, Y.T., 157 Charras, G.T., 33, 194, 202 Chaudhuri, O., 207 Chauviere, A., 32, 33, 35, 40, 42, 82 Chen, C., 195 Chen, C.S., 14, 36, 39, 70, 71, 217, 225, 240, 242 Chen, D., 152 Chen, H.Y., 157, 253, 254 Chen, J., 216, 238 Chen, P., 264 Chen, P.-Y., 33 Chen, X., 81, 82 Chernomordik, L.V., 74, 76, 114, 116, 117, 120, 124 Chiapperino, M.A., 120, 195, 222 Chien, W.-Y., 238 Chizmadzhev, Y.A., 115–117, 119–121 Chrétien, D., 157 Ciccotti, G., 150 Claycomb, J. R., 2, 150–152

Author Index Coggan, J.S., 150, 151 Cohen, A.E., 35, 239, 240, 241 Cohen, D.M., 39, 217 Cole, K., 122, 140 Cole, K.S., 70, 71, 72, 207 Cole, R., 122, 140 Concha, M.L., 48 Conte, V., 48, 225, 227 Conway, D.E., 249 Corradi, V., 250 Corsetti, J.R., 195 Coster, H.G.L, 74, 112, 145 Cranston, P.G., 48, 225, 227 Crichton, B.H., 217 Croce, R.P., 105, 112, 155 Crocker, J.C., 32, 33, 36, 42, 47 Cruz, J.M., 152, 153, 156 Cumings, J., 250 Cunill-Semanat, E., 112 Cunningham, J., 157, 158 Cuschieri, A., 175 Cuvelier, D., 48

D Dao, M., 43 Davalos, R.F., 124 Davalos, R.V., 3, 105, 118, 124, 234 Davey, C.L., 210, 213 Davey, M.R., 152, 153 Davidson, D., 122, 140 Davies, A., 32, 42, 47 DeBruin, K., 114, 115, 119 DeBruin, K.A., 115, 119, 120, 123, 148, 151, 222, 223 Degiogio, V., 143 de Groot, B.L., 83, 84, 112 Dejneka, A., 253 Delbrück, M., 24, 25, 33, 36 Delemotte, L., 75, 77 Demuro, A., 19 Deng, L., 247 Deng, P., 117–119 den Otter, W.K., 124 Derjaguin, B.V., 115 Dermol-Cerne, J., 82, 108, 110, 115, 120, 123, 131 De, S., 251, 252, 253 Deserrno, M., 202 De Silva, N.S., 48 Desprat, N., 32, 42 De Vita, A., 105, 112, 155 Di Biasio, A., 73, 82, 141, 142

287 Dillon, R., 146, 200 Dimitrov, D.S., 219, 221 Dimova, R., 15, 16, 25, 28–30, 75, 76, 81, 110, 112, 121, 123, 194, 200, 205, 238, 239 Ding, Y., 195 Dobson, C.M., 181 Doerr, T.P., 217, 219 Dokukin, M., 206 Dong, S., 120 Douglas, T.A., 118 Doyle, R.L., 78, 79 Draper, N.R., 248 Drasdo, D., 47 Drew Bennett, W.F., 248 Driscoll, M.K., 48 Dubey, A.K., 11, 14, 22, 28, 31, 47 Du, E., 43 Dular, P., 235 Dunn, A.R., 33, 35, 40, 118, 222 Duperray, A., 32, 41, 42 Dura-Bernal, S., 251, 252, 253 Durr, N.J., 251, 252 Dutta-Gupta, S., 11, 14, 22, 28, 31, 47 Dutta, P., 146, 200 Dutz, S., 157 Du, X., 2 E Eastbrook, I.D., 48 Eid, J., 202 Eisenberg, R.S., 20, 22, 24, 38, 39, 73 Elani, Y., 65 Eldar, A., 263, 264 Elimelech, M., 250 El-Khoury, R., 157 Ellisman, M.H., 150, 151, 257 Elowitz, M.B., 263, 264 Engelman, D.M., 70 Engquist, B., 180 Epsztein, R., 250 Espinosa, H.D., 251 Esser, A.T., 19, 24, 31, 74–76, 123, 124 Essone Mezeme, M., 31, 45, 46, 82, 124, 135, 141, 142, 145, 158, 181, 185, 188, 192, 195, 200, 222, 230, 256–258, 260–263 Etienne, J., 32, 41, 42 Evans, E., 36, 37, 39, 40 Evans, E.A., 202 Everitt, C.T., 22, 73 Eyckmans, J., 14, 70, 71, 225 Eydelnant, I.A., 146

288 F Fabry, B., 32, 33, 39, 42, 43, 223, 227 Falvo, M.R., 247 Faucher, S., 250 Fazakas, C., 14, 222 Fear, E.C., 11, 23, 30, 45, 125, 138, 207 Feldman, Y., 71, 73, 125, 136–138, 142, 207, 211 Féréol, S., 32, 42 Fernandez, J.M., 124 Fernandez, P., 32, 34, 39, 42, 218 Filev, P.D., 23–26, 30, 31, 45, 46, 74–76, 106, 115, 119, 125, 131, 132, 125, 193 Fiolka, R., 48 Fitzgerald, P., 257 Fletcher, D.A., 3 Forgacs, G., 254 Fošnariˇc, M., 114 Foster, K.R., 11, 23, 28, 30, 71, 125, 200, 201, 207, 211 Foster, P.J., 48, 103, 150 Fouracre, R.A., 217 Fourkas, J.T., 250 Fourn, C., 208 Fox, P.M., 107, 108 Franck, C., 39 Franke, W.W., 239, 240, 241 Fredberg, J.J., 25, 32, 33, 39, 42, 43, 47, 217, 247, 253 Freeman, S.A., 116, 120 Frenkel, D, 84 Fricke, H., 70, 71, 148, 207 Friddin, M., 65 Friedl, P., 33, 47 Fritzsche, M., 33, 194, 202 Führ, G., 210 Fuller, B., 78, 79 Funatsu, T., 157 Fung, Y.C., 80 Fütterer, C., 33–36, 39

G Gabriel, B., 75 Gagnon, H., 45, 47 Galassi, V.V., 248 Galimzyanov, T.R., 75, 76, 119 Gallet, F., 32, 42 Gao, H., 157, 253, 254 Gao, Z., 157, 253, 254 Garcia-Arcos, J.M., 48 Garcia-Diego, F.J., 152, 153, 156

Author Index Garcia, P.A., 124 Garikipati, K., 251, 252, 253 Garner, A.L., 157 Gascoyne, P.R.C., 78, 143, 145, 146, 152 Gatenby, R., 157, 158 Gaus, K., 249 Gawad, S., 152, 154–156 Gedde, M., 208, 209 Geddie, W.R., 238 Gelb, I.D., 195 Genin, G.M., 39 George, E.P., 145, 152 Gerdesmeyer, L., 48 Gerhardt, M., 242 Gerthoffer, W.T., 247 Gerum, R., 33, 223, 227 Ghadiri, E., 254 Gianulis, E.C., 120 Gielis, J., 195, 222 Gilboa, E., 263 Gil, D., 33 Gillis, K.D., 264 Gimsa, J., 143–145, 200, 210 Glaser, R.W., 114, 116, 117, 124 Glogauer, M., 42, 43 Goldberg, A., 125 Goldberg, E., 82, 120, 201, 225, 248 Goldmann, W.H., 39 Gomez, M., 146 Gonzalez-Granado, J.M., 48 González, J.E., 253 Gopinath, A., 106, 118, 131, 132, 155, 249 Gorter, E., 70 Goster, H.G.L., 152 Gota, C., 157 Govind Rajan, A., 250 Gowrishankar, T. R., 19, 24, 30, 31, 42, 45, 46, 74–76, 81, 119, 123, 124, 135, 225, 229 Goyette, J., 249 Gó´zd´z, W., 201 Graber, Z.T., 35, 239, 240, 241 Granot, Y., 125 Grant, M., 35, 117, 121 Green, D.R., 257 Green, N.G., 143, 146, 152, 154–156 Greige-Gerges, H., 202 Grendel, F., 70 Grubmüller, H., 83, 84, 112 Guardo, R., 45, 47 Guilak, F., 82, 181 Guillemin, M.T., 80–82 Guillet, R., 80–82

Author Index Gundersen, G.G., 249 Gupta, A., 152 Gupta, R., 195 Gurtin, M.E., 108 Gurtovenko, A.A., 180, 183 Guz, N., 206

H Hagness, S.C., 181, 244 Ha, H.H., 157 Haider, M.A., 82, 181 Halanski, M.A., 33, 222 Hamill, O.P., 77, 78 Hanai, T., 257 Handorf, A.M., 33, 222 Hannezo, E., 217 Harada, Y., 157 Hardin, C.C., 253 Harootunian, A., 253 Harris, A.R., 33, 34, 104, 194, 202 Hartinger, A.E., 45, 47 Haudensschild, D., 255 Havriliak, S., 122, 140 Hawkins, R.J., 48 Haydon, D.A., 22, 73 Heimburg, T., 109, 157, 194 Heisenberg, C.P., 33 Helfrich, P., 37, 38 Helfrich, W., 200 Helm, V., 102, 131 Helman, S.I., 42, 45 Heminemannc, M., 48, 103, 150 Hénon, S., 32, 42 Henslee, B.E., 194 Hergt, R., 157 Herrmann, H., 146 Higgins, M.L., 78, 79 Hille, B., 150, 151, 210 Himms-Hagen, J., 156 Hintsche, M., 242 Höber, R., 70 Hobson, C.M., 247 Hochmuth, R.M., 110, 113 Hodgkin, A.L., 70, 71 Hoekstra, A.G., 16, 235 Hoffman, B.D., 32, 33, 36, 42, 47 Hoffman, E.K., 152 Holdbrook, D.A., 83, 84 Holmes, D., 152, 154–156 Holzhapfel, G.A., 44, 47 Holzzapfel, C., 202 Honig, B., 17, 18, 21, 22, 24, 25, 72

289 Hossan, M.R., 146, 200 Hosseinkhani, H., 254 Ho, S.Y., 111, 131 Houben, F., 249 Howard, J., 48, 103, 150 Howard, R., 157, 158 Hrynkiewicz, A.Z., 33 Huang, C.J., 253 Huang, H., 40, 124 Huang, J.P., 217, 219 Hubere, G., 48, 103, 150 Huber, F., 33–36, 39 Huestis, W.H., 208, 209 Hu, J.-G., 73 Humphrey, J.D., 44, 47, 78, 79 Hunt, B., 251, 252 Hunt, T., 18, 19, 108, 207, 212 Hu, Q., 109, 157, 180 Huston, M.S., 48, 225, 227 Hu, W.F., 235 Huxley, A.F., 70, 71 Hu, X., 194 Hu, Y., 124 Hyuga, H., 74

I Icard, D., 32, 42 Igliˇc, A., 114, 201 Ikada, Y., 195 Inada, N., 157 Ingber, D.E., 33, 35, 36, 39, 240, 242, 249 Ingólfsson, H.I., 250 Inoue, T., 261 Intes, X., 251, 252 Irimajiri, A., 73 Isambert, H., 19 Israelachvili, J., 221 Israelachvili, J.N., 117 Ito, K., 258 Iyer-Biswasf, S., 48, 103, 150 Jaalouk, D.E., 146 Jacinto, A., 48, 225, 227 Jacobs, H.T., 157 Jacques, S.L., 16 Jadidi, T., 202 Jaeger, J.C., 158 Jakab, K., 254 Jakobsson, E., 37, 38 Janko, K., 19, 22, 23 Janmaleki, M., 254 Janmey, P., 242 Jansen, K.A., 47

290 Jarasch, E.D., 239, 240, 241 Jastroch, M., 157 Jerome, J.W., 152, 253 Jewett, M.A.S., 238 Jiang, Y., 216 Jin, J., 181, 184, 185, 187, 191 Johnson, J.B., 263 Jones, T.B., 109, 143, 144, 145, 146, 152–156 Jonieztz, E., 255 Jonscher, A.K., 126 Jordan, C.A., 258 Joshi, R., 109 Joshi, R.P., 84, 117, 157, 180 Jossinet, J., 45 Jraij, A., 202 Jullien, L., 157

K Kaatze, U., 71, 73 Kaazempur-Mofrad, M.R., 40 Kakorin, S., 83, 84, 112 Kalaparthi, V., 206 Kalashnikov, N., 47, 48 Kaler, K.VI.S., 155 Kamm, R.D., 40, 124 Kandušer, M., 23, 25, 124 Kang, B., 157, 253, 254 Kang, H., 148 Kang, K.H., 152, 152, 155 Kaplan, D.L., 18, 115 Karcher, H., 40 Kardos, T.J., 175 Karniadakis, G., 251, 252, 253 Karplus, M., 216 Kasimova, M.A., 23, 75, 77 Käs, J., 33–36, 39 Kasza, K.E., 12, 14, 38, 208, 209 Keiper, J.S., 15, 16 Keipert, S., 157 Kell, D.B., 210, 213 Keller, D.X., 150, 151 Keller, J.B., 148 Keuer, K.D., 216 Khalid, S., 83, 84 Kimmel, E., 42, 194, 195 Kinosita Jr, K., 74 Kirsch, S.A., 75, 77, 84 Kitagawa, N., 73 Knöpfel, T., 253 Knorr, R.L., 110, 112, 123, 194 Koch, A.L., 78, 79

Author Index Koenderink, G.H., 12, 14, 38, 47, 208, 209 Koert, U., 216 Kokta, V., 45, 47 Kollmannsberger, P., 32, 33, 39, 42 Kondev, J., 10–12, 16, 17, 19, 20, 23, 24, 33–38, 242 Kotnik, T., 2, 20, 23–25, 28–31, 47, 82, 108, 110, 115, 120, 121, 123, 131, 135, 142, 145, 151, 230, 235, 242, 244, 262 Koutsouris, D., 80–82 Kozlov, M.M., 37 Krähenbühl, L., 181, 185, 186, 188, 235, 244 Kralj-Igliˇc, , V., 114, 201 Kralj, S., 201 Kranjc, M., 263 Krassowska, W., 23–26, 30, 31, 45, 46, 74–76, 106, 114–117, 119–121, 123, 125, 131, 132, 125, 148, 151, 193, 222, 223 Kraus, W., 48 Krizbai, I.A., 14, 222 Kroeger, J.H., 35, 117, 121 Krohne, G., 239, 240, 241 Krumlauf, R., 261 Kubo, M., 157 Kucsko, G., 157 Kuhl, E., 44, 47, 251, 252, 253 Kuhn, M., 33, 223, 227 Kulik, H.J., 250 Kumar, R., 11, 14, 22, 28, 31, 47 Kumar Sinha, S., 124 Kummrow, M., 200 Küppers, G., 19, 110, 118 Kurniawan, N.A., 47 Kuzmin, P., 221 Kuzmin, P.I., 75, 76, 119

L La Rosa, P.S., 263 Lafyatis, G.P., 194 Laidler, P., 33 Lai, M.C., 235 Lamacchia, C.M., 120, 195, 222 Lammerding, J., 40, 146 Lamy, M.T., 248 Landau, L., 148 Landau, L.D., 19, 35, 42, 79, 108, 125, 132, 143, 144, 152 Langer, R., 254 Langowski, J., 244

Author Index Lanzanò, L., 143, 144 Lasic, D.D., 11 Lasquellec, S., 14, 23, 24, 27, 29, 82, 103, 108, 110, 115, 120, 123, 131, 181, 185, 188, 189, 190–208, 222–228, 230, 235–239, 244 Latorre, R., 109 Lau, A.W., 32, 42, 47 Law, R.V., 65 Leckband, D., 221 Lediju Bell, M.A., 251, 252 Lee, A., 216 Lee, G.B., 19, 23, 27, 194, 244 Lee, I.S., 195 Lee, L.J., 194 Lee, R.T., 40, 124 Lee, Y.-K., 117–119 Legant, W.R., 39 Leguébe, M., 262 Lehner, S., 48 Leikin, S.L., 114, 116, 117, 124 Lekka, M., 33 Lekki, J., 33 Lele, T.P., 39 Lelievre, J.C., 80–82 Lemmon, C.A., 249 Lennon-Duménil, A.M., 48 Lenormand, G., 25, 32, 33, 39, 42, 47 Leontiadou, H., 112 Levin, M., 18, 115 Levine, S.E., 195 Levine, Z.Q., 225 Levy, A., 250 Lew, R.R., 253 Lewis, T.J., 11, 13, 14, 18–20, 194 Liang, W., 19, 23, 27, 194, 244 Li, C., 120 Li, D., 152, 152, 155 Lifshitz, E., 148 Lifshitz, E.M., 19, 35, 42, 79, 108, 125, 132, 143, 144, 152 Lim, C.T., 43 Lin, A.Y.-M., 33 Ling, H., 193 Lin, H., 19, 24–26, 30, 114, 115, 120 Lin, L., 157 Lin, R., 117 Lipowsky, R., 200, 221 Lippert, A., 33, 223, 227 Lister, J.D., 74–76 Liu, D., 175 Liu, H., 120

291 Liu, J., 12, 14, 38, 43, 157, 208, 209, 253, 254 Liu, L., 19, 23, 27, 194, 244 Liu, Z., 217 Li, W. J., 19, 23, 27, 33, 194, 222, 244 Loew, L.M., 210 Lo, P.K., 157 Lomakin, A.J., 48 Loots, G.G., 255 Lopreore, C.L., 150, 151 Lorenz, C.D., 181 Lorich, D.G., 195 Loveless, A.M., 157 Lubensky, T.C., 32, 42, 47 Luckey, M., 69 Lu, J., 255 Lukin, M.D., 157 Lunov, O., 253 Lu, Z., 238 Lv, Y., 120 Lynch, M., 48, 103, 150 Lynch, P.T., 152, 153 Lytton, W.W., 251, 252, 253

M Maccarrone, M., 75, 77 Maˇcek-Lebar, A., 75 MacGregor, S.J., 217 MacKinnon, R., 216 MacKintosh, F.C., 47 Mahadevan, L., 33, 194, 202 Mahaworasilpa, T.L., 145, 152 Maher, M.P., 253 Majumdar, A., 250 Maksym, G.N., 42, 43 Manel, N., 48 Maniotis, A.J., 36, 240, 242 Mantegazza, F., 143 Marchese, J., 82, 120, 201, 225 Mareschal, M., 105 Marguet, D., 157 Mark, A.E., 112 Markin, V.S., 124 Markwald, R.R., 254 Marques, L.B., 42, 45 Marrink, S.J., 112, 250 Marshall, G., 82, 120, 201, 225 Marsh, D., 11, 14, 23, 24, 47 Martinac, B., 77, 78 Martin, C., 141, 142, 250 Martinez, G., 141, 142 Martin, G.T., 105, 157, 158

292 Mashagh, A., 202 Maskarinec, S.A., 39 Mathias, R.T., 20, 22, 24, 38, 39, 73 Matter, H.P., 48 Matzke, A.J.M., 240, 244, 247 Matzke, A.M., 240, 244, 247 Maurer, P.C., 157 Mauri, A.G., 152, 253 Maxwell, J.C., 140 Ma, Y., 249 May, S., 114 McAdams, E.T., 45 McCaffrey, L., 47, 48 McCarty, O.J.T., 16 McCormack, K., 253 McCulloch, A.D., 82, 83, 255 McEldrew, M., 250 McNeil, P.L., 256 Meissner, R., 42, 45, 46 Mejdoubi, A., 148, 181, 185, 188, 208, 209, 211–214, 215 Menger, F.M., 15, 16 Mert Terzi, M., 202 Mesarec, L., 201 Mescia, L., 120, 195, 222 Meyers, M.-A., 33 Microfluidics, D. Li., 120 Miklavˇciˇc, D., 2, 20, 23–25, 28–31, 45–47, 75, 82, 108, 110, 115, 117, 120, 121, 123, 124, 131, 135, 141, 142, 145, 151, 158, 230, 235, 242, 244, 260–263 Milestone, W., 157 Milewicz, D.M., 47, 78, 79 Miller Jr., J.H., 2, 150–152 Miller, J.S., 39 Millet, E., 217 Mintzer, R.A., 74–76, 116, 193 Miodownik, M., 48, 225, 227 Miranda, J.M., 210 Mir, L.M., 3, 77, 124, 234, 262 Mironov, V., 254 Mittal, G.S., 111, 131 Mittelmeier, W., 48 Mi, Y., 120 Miyazaki, H., 240, 242 Moeendarbary, E., 33, 34, 104, 194, 202 Mok, S., 47, 48 Molina, M., 48 Mollinedo-Gajate, I., 253 Monticelli, L., 202 Montooth, K.L., 48, 103, 150 Mooney, D.J., 33, 35, 36, 48, 207

Author Index Moraes, C., 47, 48 Morgan, H., 143, 146, 152, 154–156 Morshed, A., 146, 200 Morshed, B.I., 109, 131 Morss, A., 194 Mosgaard, L.D., 109 Moulding, D.A., 33, 194, 202 Mouritsen, G., 36, 39 Movahed, S., 120 Mueller, R., 70 Mukherjee, P., 251 Müller, D.J., 48 Müller, K., 33–36, 39 Müller, T., 210 Muñoz-Pinedo, C., 257 Muñoz, S., 210 Murovec, T., 82, 152, 154, 181, 185, 188, 195, 218–222, 229–234, 250 Mussauer, H., 200 Mussivand, T., 109, 131 Musso, N., 143, 144

N Nader, G.P.F., 48 Nagai, T., 157 Nagle, J.F., 202 Nakamura, M., 240, 242 Nakano, M., 157 Nanavati, C., 124 Narang, J.D., 249 Narasimhan, S., 48, 225, 227 Navajas, D., 42, 43, 247 Neagu, A., 254 Neal II, R.E., 124 Needham, D., 37, 39, 110, 113 Needleman, D.J., 48, 103, 150 Negami, S., 122, 140 Negulescu, P.A., 253 Nehorai, A., 263 Nelson, C.M., 217 Neu, J.C., 74–76, 114–117, 119, 120, 121 Newman, T.J., 32, 42 Neumann, E., 74, 83, 84, 112, 116, 258 Nganguia, H., 200, 212 Nguyen, J., 80, 81 Nicholls, A., 17, 18, 21, 22, 24, 25, 72 Nicholson, G.L., 15, 73, 74 Nicolas, L., 235 Nielaba, P., 105 Noguchi, H., 85 Noh, H.J., 157 Noh, M., 157

Author Index Noll, N.A., 249 Noy, A., 250 Numann, R., 253 O O’Brien, T.J., 105 Oberhauser, A.F., 124 Ochoa, M., 251, 252 Okabe, K., 157 Olins, D.A., 146 Or, M., 42 Orlowski, S., 77 Ott, A., 32, 34, 39, 42, 218 Ou-Yang, Z.C., 73 P Pachenari, M., 254 Pakhomov, A.G., 120 Pakhomova, O.N., 120 Paoletti, C., 77 Park, H., 157 Park, S.H., 251 Parker, I., 19 Parsegian, A., 250 Pashley, M., 117 Pastushenko, V.F., 114, 116, 117, 119, 120, 124 Patel, S., 124 Pathak, N., 251 Patino, C.A., 251 Paul, R., 155 Pavlin, M., 31, 45, 46, 75, 82, 124, 135, 141, 142, 158, 260, 261, 262 Pavlov, K.V., 75, 76, 119 Pavselj, N., 75 Pedro de Souza, J., 250 Pegoraro, A.F., 242 Pennes, H.H., 157, 158 Perdikaris, P., 251, 252, 253 Perkins, G.A., 257 Perrussel, R., 235 Peterlin, P., 200 Pethig, R., 18, 31, 71, 108, 118, 125, 136, 137, 143, 144, 152–156, 207 Petralia, S., 143, 144 Petridou, N.I., 33 Petrie, R.J., 48 Petzold, L., 251, 252, 253 Phillips, K.G., 16 Phillips, R., 2, 10–12, 16, 17, 19, 20, 23, 24, 33–38, 242 Piechocka, I.K., 47

293 Piel, M., 48 Pierro, V., 105, 112, 155 Piggot, T.J., 83, 84 Pigolotti, S., 48, 103, 150 Pilwat, G., 115, 116 Pinto, I.M., 105, 112, 155 Pitaevskii, L.P., 152 Plaksin, M., 194, 195 Pliquett, U.F., 105, 157, 158 Pods, J., 150, 151 Pohl, H.A., 18, 24, 30, 31, 143, 144 Poignard, C., 124, 235, 262 Polk, C., 152 Pollack, S.R., 195 Pollard, T.D., 102, 131 Polya, G., 148 Polyakova, T., 253 Poole, K., 249 Powell, K.T., 193 Pozrikidis, C., 216 Prasanna Misra, R., 250 Preziosi, L., 32, 33, 35, 40–42, 82 Price, A., 33, 35, 40, 118, 222 Prodan, C., 73, 141, 207, 211 Prodan, E., 73, 141, 207, 211 Puc, M., 262 Pucihar, G., 20, 23–25, 28–31, 45–47, 82, 110, 112, 115, 124, 131, 135, 141, 142, 158, 230, 235, 260, 261, 262 Pulgar, E., 48 Pullarkat, P.A., 218 Puzenko, A., 207, 211 Q Qiang, Y., 43 Quan, C., 253 R Raicu, V., 73, 125, 136–138, 142 Raizer, A., 42, 45 Raj, C.D., 253 Rajendran, K., 253 Rak, M., 157 Ramachandran, I., 249 Ramaekers, F.C.S., 249 Ramos, A., 19, 42, 45, 46 Rauch, P., 33–36, 39 Ravichandran, G., 39 Rawicz, W., 37, 40 Rebersek, M., 23, 25, 82, 108, 110, 115, 120, 123, 124, 131 Rec, L.J., 107, 108

294 Reed, M., 250 Reig, G., 48 Rems, L., 2, 23, 25, 75, 77, 82, 108, 110, 115, 117, 120, 121, 123, 124, 131 Reza Rahimi Tabar, M., 202 Ricca, B.L., 3 Ricci, J.-E., 257 Riemann, F., 115, 116 Rigneault, H., 157 Riske, K.A., 15, 16, 25, 28–30, 200, 205, 238, 239, 248 Robertson, J.D., 70, 75 Robertson, J.L., 124 Rodenfelsl, J., 48, 103, 150 Rolandi, M., 146 Rols, M.P., 75, 112, 124 Romano, A., 143, 144 Ronceray, P., 48, 103, 150 Rönicke, S., 33–36, 39 Rosato, N., 75, 77 Rose, J.K., 77 Rosenheck, K., 112, 116 Rosi, A., 168 Roux, B., 216 Roux, K.J., 240 Rowat, A.C., 12, 14, 38, 146, 208, 209 Roy, S., 109, 115 Rabussay, D.P., 175 Rubinsky, B., 3, 125, 234 Rudin, D.O., 70 Ruiz, S.A., 217 Russo, G.I., 143, 144 Rustin, P., 157 Ruta, V., 216 Ryabov, Y., 207, 211

S Sabri, E., 14, 19, 21, 23, 24, 27, 29, 42, 75, 76, 81, 104, 121, 123, 181, 185, 188, 190–194, 196–207, 230, 236–239, 242, 243, 244, 245, 264 Sacco, R., 152, 253 Sachs, F., 112, 113, 114 Sachse, K., 242 Saez, P.J., 48 Saffman, P.G., 24, 25, 33, 36 Safran, S.A., 219 Salgado, J., 112 Salipante, P.F., 110, 112, 123, 194 Salou, P., 181, 185, 188, 208, 209, 211, 212, 214, 215 Sanabria, H., 2, 150–152

Author Index Sancho, M., 141, 142, 210 Sandersius, S.A., 32, 42 Sansom, M.S.P., 250 Sareni, B., 181, 185, 186, 188, 244 Saville, D.A., 143 Scaife, B.K.P., 125–127, 132, 138, 141, 142 Schaefer-Ridder, M., 74 Schatzman, M., 235 Scheer, U., 239, 240, 241 Scheffler, I.E., 257 Scheiner, A., 157, 158 Schnau, J., 33–36, 39 Schneider, W., 33, 223, 227 Schnelle, T., 210 Schoenbach, K.H., 84, 107, 108, 117, 120, 131, 180 Schönke, J., 150, 151 Schrödinger, E., 85 Schwan, H.P., 10, 11, 19, 20, 23, 27, 28, 30, 71, 125, 200, 201, 207, 211, 217 Schwartz, M.A., 47, 78, 79 Schwegler, E., 250 Scoretti, R., 124 Sebastian, J.L., 210 Seddon, J.M., 65 Seki, Y., 33 Sekine, K., 257 Seknowski, T.J., 150, 151 Selberg, J., 146 Seldes, R., 195 Sengel, J.T., 19, 75, 111, 113, 114 Sens, P., 2 Seol, Y., 235 Serra-Picamal, X., 253 Sersa, I., 263 Seyedour, S., 254 Seyyed-Allaei, H., 202 Shafiee, A., 254 Shagoshtasbi, H., 118, 119 Shakiba, N., 238 Shamoon, D., 23, 27, 82, 103, 108, 110, 115, 120, 123, 131, 181, 185, 188, 189, 195, 200, 222–228, 230, 235, 244 Shams, M., 109, 131 Shankarb, S., 48, 103, 150 Shapira, E., 194 Shayan, S.B., 254 Sheng, Y., 33, 35, 39 Shi, Z., 35, 239, 240, 241 Shil, P., 120 Shoham, S., 194, 195 Sihvola, A.H., 125, 126, 142, 148, 208, 213

Author Index Sillars, R.W., 140 Silve, A., 124, 262 Simonsen, L.O., 152 Singer, S.J., 15, 73, 74 Siwy, Z., 250 Skalak, R., 202 Sloan, S.R., 193 Sloot, P.M.A., 16, 235 Smit, B., 84 Smith K.C., 74–76 Smith, B.A., 37, 40 Smith, J.T., 251, 252 Smith, K.C., 19, 24, 31, 114, 116, 117, 119, 120, 121, 225 Sniadecki, N.J., 217 Snoeckx, L., 249 Soba, A., 82, 120, 201, 225 Sokirko, A.I., 114, 116, 117, 124 Sokolov, I., 206 Somers, K., 107, 108 Son, R.S., 225 Song, C., 253 Song, P., 157, 253, 254 Sosinsky, G.E., 150, 151 Souza, P.C.T., 250 Sowers, A.E., 258 Spector, A.A., 109, 115 Spiró, Z., 33 Spörrer, M., 33, 223, 227 Srivastava, N., 48 Stachura, Z., 33 Stange, M., 242 Stark, R.H., 107, 108 Stegun, I., 117, 129 Steinhardt, R.A., 256 Steinhauser, E., 48 Stewart, C.L., 240 Stewart, D.A., 19, 30, 46, 119, 124, 135, 229 Stoeckenius, W., 70 Stone, H.A., 35, 152, 239, 240, 241 Storm, C., 33, 47 Strano, M., 250 Stride, E., 33, 194, 202 Stuchly, M.A., 11, 23, 30, 45, 125, 138, 207 Suarez, C., 82, 120, 201, 225 Sukhorukov, V., 200 Sun, S.X., 242 Sun, T., 152, 154–156 Sun, Y., 80, 81, 238 Sundelacruz, S., 18, 115 Superfine, R., 247 Suresh, S., 43

295 Sweeney, D.C., 118, 124 Szegletes, Z., 14, 222 Szego, G., 148 Szent-Györgyi, A., 45

T Taflove, A., 181, 244 Takahashi, Y., 142 Takashima, S., 142 Takasu, M., 85 Talpasanu, I., 143, 146 Tambe, D.T., 253 Tan, J.L., 217 Taranejoo, S., 254 Tarek, M., 2, 23, 84, 115, 117, 120, 121 Tavassoly, I., 48, 103, 150 Teissié, J., 75, 112, 124, 262 Tellides, G., 47, 78, 79 Tesla, I., 75, 77 Tewari, A., 11, 14, 22, 28, 31, 47 Theriot, J., 10–12, 16, 17, 19, 20, 23, 24, 33–38, 242 Thiam, H.R., 48 Thomas, C.B., 255 Thompson, D.W., 2, 68, 69 Thompson, S.M., 42, 45 Thompson, T.E., 192, 261 Thrasher, A.J., 33, 194, 202 Thutupallir, S., 48, 103, 150 Tian, L., 251, 252 Tian, W.J., 217, 219 Tieleman D.P., 83, 84, 112, 180, 181, 248, 250 Tien, H.T., 70 Tikhonov, A.N., 139 Timoshkin, I.V., 217 Tireell, D.A., 39 Titov, D.V., 48, 103, 150 Tolpekina, T.V., 124 Toupin, R.A., 79 Traek, M., 114 Trantidou, T., 65 Trepat, X., 25, 32, 33, 39, 42, 47, 217, 247, 253 Treut, G.L., 48, 103, 150 Tsai, K.Y., 39 Tschoegl, N., 40 Tschumperlin, D.J., 247 Tsong, T.Y., 74 Tuncer, E., 139

296 U Uchiyama, S., 157 Ujihara, Y., 240, 242 Ulrich, F., 48, 225, 227 Ung, W.L., 146 Ursell, T., 2 Ušaj, M., 23, 25, 124

V Vahovska, P.M., 110, 112, 123, 194 Vajrala, V., 2, 150–152 Valic, B., 235 Valleriani, A., 242 Valon, L., 33, 194, 202 van den Engh, G., 146 Van Helvert, S., 33, 47 Varga, B., 14, 222 Váró, G., 14, 222 Vasikoski, Z., 19, 24, 31, 74–76, 123, 124 Vattulainen, I., 180, 183 Vaziri, A., 106, 118, 131, 132, 155, 249 Végh, A.G., 14, 222 Veldhuis, J., 48, 225, 227 Vendruscolo, M., 181 Venugopalan, G., 3 Verdier, C., 32, 33, 35, 40–42, 82 Vernier, P.T., 123, 225 Vernon, D., 35, 117, 121 Vidyasagar, P.B., 120 Vienken, J., 202 Vining, K.H., 33, 35, 36, 48 Virga, E.G., 201 Viswanadham, S., 180 Vlahovska, P.M., 202 Voldman, J., 143 Volynsky, P.E., 75, 76, 119 Voyer, D., 124

W Wachner, D., 143–145, 200 Wachsmuth, M., 244 Wada, S., 240, 242 Wagner, K.W., 140 Wakabayashi, H., 74 Waldeck, W., 244 Wallace, M.I., 19, 75, 111, 113, 114 Wang, J., 2, 48, 103, 150 Wang, L., 175 Wang, M.A., 116, 120 Wang, N., 33, 35, 39 Wang, S., 2

Author Index Wang, X., 78, 143, 145, 146, 152 Wang, X.B., 143, 145, 146, 152–156 Wang, Y., 19, 23, 27, 74, 194, 244, 250 Wang, Z., 175 Wasserman, M.R., 217 Weaver, J.C., 18, 19, 24–26, 30, 31, 42, 45, 46, 74–76, 81, 105, 114–121, 123, 124, 135, 157, 158, 193, 225, 229, 235, 263 Wei, G.W., 152 Wei, Y., 80, 81 Weikl, T.R., 221 Weinan, E., 180 Weitz, D.A., 12, 14, 38, 146, 208, 209, 217, 242, 253 Welf, E.S., 48 Wescott, W.C., 70 White, S.M., 192, 261 Wiggins, P., 2 Wilhelm, I., 14, 222 Wilke, N., 248 Wilson, J.H., 18, 19, 108, 207, 212 Wolfe, J., 14, 27, 28, 30, 31, 45, 74, 75, 200 Wolff, Y., 68, 69 Wu, Y., 242 X Xiao, S., 120 Xu, J., 120 Xu, J. J., 157, 253, 254 Y Yadava, N., 257 Yang, E., 208, 209 Yang, H., 157 Yang, J.M., 157 Yang, M.T., 217 Yang, R., 120 Yang, S., 250 Yang, X., 48, 103, 150 Yao, C., 2, 120 Yao, N.Y., 157 Ye, H., 206 Yi, J., 251, 252 Yih, T.C., 143, 146 Young Park, C., 253 Young, Y.N., 200, 212, 235 Yu, K.W., 217, 219 Yu, L., 33, 35, 39, 238 Yu, M., 19, 24–26, 30, 114, 115, 120 Yu, X., 14, 70, 71, 225 Yu, Y.K., 217, 219

Author Index Z Zablotskii, V., 253 Zaman, M.H., 253 Zecchi, K.A., 109 Zhang, H.Z., 257 Zhang, L., 2 Zhang, M., 120 Zhang, N., 263 Zhang, R., 157, 253, 254 Zhang, T.Y., 117 Zhang, Z., 2 Zhao, X., 120 Zhao, Y., 19, 23, 27, 120, 194, 244 Zhbanov, A., 250

297 Zhelev, D.V., 221 Zheng, Q., 152 Zheng, Y., 80, 81 Zhitnyak, I.Y., 48 Zhou, E.H., 253 Zhou, Q., 2 Zhou, Y., 33, 222 Zhu, R., 249 Ziegler, M.J., 123 Zimmerman, U., 19, 105, 110, 115, 116, 118, 200, 202 Zuk, P.J., 152 Zullino, S., 23 Zwerger, M., 146

Subject Index

A Areal resistance, 19, 245

B Bending, 15, 36–38, 47, 73, 103, 110, 202, 238, 240 Bilayer membrane, 14, 74, 109, 114, 194

C Canham-Evans-Helfrich, 37, 38, 73 Capacitance, 17, 19–23, 27, 78, 83, 109, 119, 140, 146, 192–194, 235, 243, 246, 264 Capacitive coupling, 203, 245 Cell, 1–3, 9–25, 27–48, 67–85, 101–115, 117–120, 123–125, 128, 131, 133–135, 141–143, 145, 146, 149–152, 154–157, 179–182, 186, 188–251, 253–264 Cell membrane permeabilization, 2 Constriction channel, 80 Core-shell, 11, 132, 208, 210–215, 260 Creep compliance, 40 Crossover frequency, 145, 194 Current density, 31, 114, 141, 142, 201, 223, 233 Cytoplasm, 2, 11, 13, 15–21, 23, 24, 27–32, 34, 35, 43, 47, 69, 73–75, 78, 80, 108, 119, 125, 131, 132, 142, 143, 151, 157, 190, 194–197, 200–202, 204–206, 208, 210, 217, 220, 230–232, 235, 236, 238–243, 246, 249, 257

Cytoskeleton, 10, 17, 32, 35, 36, 39, 42, 47, 74, 77, 79, 80, 102, 103, 124, 131, 180, 239–243, 248, 249

D Dashpot, 33, 34, 40–42 Dielectric spectroscopy, 70, 71, 125, 136, 207 Dielectropheresis, 143 Dispersion, 27, 70–72, 137, 138, 140, 145, 202, 204, 209, 211 Double-layer, 18, 22, 71, 117, 152, 194, 195, 221, 243, 256

E Edge energy density, 115, 116 Effective medium, 27, 67, 73, 188 Elasticity modulus, 35, 40, 103 Electrical conductivity, 29, 30, 70, 132, 200, 230, 238, 261, 263 Electric pulse, 3, 28, 31, 74–77, 105, 107, 108, 115, 117, 120, 123, 225, 238, 239 Electrodeformation, 1, 150 Electromechanobiology, 1–4, 9, 12, 16, 48, 67, 68, 71, 78–82, 85, 101, 102, 105, 106, 142, 146, 180, 181, 183, 242, 248–252, 254, 255 Electroporation, 1, 16, 114, 200, 202, 223, 224, 263 Electrorotation, 78, 143 Electrostatic force, 106, 125, 152, 216 Electrostriction, 23, 244

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Brosseau, Physical Principles of Electro-Mechano-Biology, Studies in Mechanobiology, Tissue Engineering and Biomaterials 25, https://doi.org/10.1007/978-3-031-37981-9

299

300 Electro-thermal effect, 156, 158 Equivalent circuit model, 23, 76, 124, 263 External cellular medium, 10, 11, 13, 14, 18, 22–24, 28, 32, 33, 39, 44, 47, 72, 73, 77, 78, 103, 108, 112, 114, 125, 131, 132, 143, 145, 151, 157, 196, 200–202, 206, 230–232, 235, 236, 238, 240, 247, 249, 261, 263 Extracellular medium, 10, 11, 19, 27, 117, 190, 194, 196, 197, 201, 215, 217, 218, 221, 222, 230, 235, 238

F Filament, 15, 17, 35, 79, 80, 240, 242, 243, 247 Finite-difference time-domain, 73, 181, 184 Finite element, 4, 183 Fluorescence, 19, 28, 30, 73, 75, 123, 157, 227, 230, 252, 253 Full membrane. See 000 Full width half maximum, 244

I Ion channel, 2, 3, 12, 14, 18, 20, 21, 32, 38, 70, 77, 83, 84, 131, 152, 155, 182, 244, 254

K Kirchhoff’s circuit law, 243 Kohlrausch-Williams-Watt, 122

L Laplace equation, 30, 127, 190 Lipid bilayer, 14–16, 22, 74, 84, 105, 111, 113, 115, 202, 216, 240 Lumped circuit, 27, 45

M Machine learning, 3 Maxwell stress tensor, 79, 108, 152, 156, 191, 204 Maxwell-Wagner-Sillars, 24, 28, 42, 70, 126, 127, 140, 141, 145, 154, 200, 212 Mechanical stress, 1, 14, 32, 33, 40, 79, 155, 217, 253, 254 Mechanotransduction, 3, 17, 32 Membrane, 2–4, 9–29, 31–39, 42, 43, 46, 48, 68–78, 80, 82–85, 102, 103,

Subject Index 105–121, 123–125, 131, 142, 146, 150–152, 157, 179–182, 189–204, 206, 208–210, 215–218, 220–225, 227–231, 233–237, 239–243, 245, 248–251, 253, 255–261, 263, 264 Membrane capacitance, 19, 20, 22, 23, 29, 70, 80, 108, 189, 193, 195, 198, 215, 264 Mesh, 16, 35, 184, 187, 189, 230, 231, 235–237 Molecular dynamics, 3, 12 Multiphysics, 2–4, 81, 83, 106, 146, 152, 179, 189, 222, 223, 243, 252, 253, 255 Multiscale, 2, 3, 16, 34, 47, 73, 81–85, 102, 106, 179, 180, 235, 252

N Nuclear envelope, 13, 263 Nuclear pore complexe, 15 Nucleus, 12, 13, 15, 17, 25, 31, 32, 34, 36, 48, 69, 78, 82, 102, 124, 135, 137, 143, 157, 182, 194, 210, 229–232, 239–243, 245–247, 250

P Permittivity, 19, 22, 24, 25, 27, 28, 71, 72, 116, 118, 125, 127, 128, 131–133, 136–145, 148, 149, 153–155, 186–188, 191, 194, 196, 200, 201, 208, 210, 211, 213, 214, 217, 219, 223, 236, 237, 244, 254, 256 Phospholipid bilayer, 194 Poisson-Nernst-Planck, 117, 121 Poisson ratio, 35, 202 Polarization time constant, 28 Pore, 3, 12, 13, 15, 16, 19, 24–26, 28, 31, 39, 46, 75–78, 80, 82–85, 105–108, 110–121, 123, 124, 146, 150–152, 180–182, 189, 193, 202, 223–225, 227, 229, 234, 240, 243, 248, 250, 251, 253, 260 Protein, 1–3, 10–18, 20, 21, 34–39, 42, 43, 70–75, 77, 78, 80, 102, 108, 110, 112, 124, 157, 201, 210, 222, 239, 241, 248–250, 253

R Repulsion-to-attraction, 182

Subject Index Resistance, 19, 20, 23, 27, 37, 38, 70, 103, 117–119, 193, 206, 240, 243–245, 264 Resting potential, 18–20, 195, 196, 201, 223, 244

S Scale, 1–3, 9, 12–14, 16, 20, 22, 29, 30, 32, 33, 35–37, 40–43, 45, 47, 48, 78, 79, 81–85, 101–106, 108–111, 119, 120, 123, 126, 131, 135, 144–146, 152, 155–157, 180–183, 192, 195, 196, 200, 202, 206, 209, 213, 216, 221, 222, 226, 229, 231, 235, 236, 238, 242, 243, 245, 247, 248, 250–252, 254, 255, 257, 258, 260, 262 Smith, Neu and Krassowska, 119–121, 123, 124, 146, 234, 261 Smoluchowski equation, 76, 84, 119, 121, 180 Spring, 33, 34, 38, 40–42, 119, 240 Strain energy, 43, 103, 117, 118, 182, 222, 223, 225, 228, 229, 240 Stretching, 33, 35–38, 110, 194, 238, 240, 242

301 Surface tension, 26, 75, 106, 110, 115–117, 119, 121, 123, 148

T Thermal noise, 18, 247, 263, 264 Thin-layer approximation, 236–239 Time constant of membrane charging, 17, 24, 28 Tissue, 1–3, 9, 10, 32, 33, 43–48, 67–73, 75, 77–79, 81–83, 85, 102–105, 108, 110, 124, 125, 146, 150, 180–182, 195, 216, 217, 221–223, 229, 230, 234, 235, 248, 251, 254, 255, 260–263 Torque, 132, 143, 145, 146, 153, 154, 156 Transmembrane voltage, 17, 31, 108 Transport lattice, 45, 46, 76, 124, 229 Two-state model, 260

V Vesicle, 15, 16, 74, 120, 123, 182, 201, 205, 206, 221, 235, 236, 239 Viscoelasticity, 44 Viscosity, 24, 35, 36, 40, 43, 106, 117, 250