Statistical Physics for Biological Matter - Graduate Textbook

Statistical Physics for Biological Matter Wokyung Sung Graduate Texts in Physics Graduate Texts in Physics Series editors Kurt H Becker Polytechnic School of Engineering Brooklyn USA JeanMarc Di Meglio Université Paris Diderot Paris France Sadri Hassani Illinois State University Normal USA Bill Munro NTT Basic Research Laboratories Atsugi Japan Richard Needs University of Cambridge Cambridge UK William T Rhodes Florida Atlantic University Boca Raton USA Susan Scott Australian National University Acton Australia H Eugene Stanley Boston University Boston USA Martin Stutzmann TU München Garching Germany Andreas Wipf FriedrichSchillerUniversität Jena Jena Germany Graduate Texts in Physics Graduate Texts in Physics publishes core learningteaching material for graduate and advancedlevel undergraduate courses on topics of current and emerging fields within physics both pure and applied These textbooks serve students at the MS or PhDlevel and their instructors as comprehensive sources of principles definitions derivations experiments and applications as relevant for their mastery and teaching respectively International in scope and relevance the textbooks correspond to course syllabi sufficiently to serve as required reading Their didactic style comprehensive ness and coverage of fundamental material also make them suitable as introductions or references for scientists entering or requiring timely knowledge of a research field More information about this series at httpwwwspringercomseries8431 Wokyung Sung Statistical Physics for Biological Matter 123 Wokyung Sung Department of Physics Pohang University of Science and Technology Pohang Korea Republic of ISSN 18684513 ISSN 18684521 electronic Graduate Texts in Physics ISBN 9789402415834 ISBN 9789402415841 eBook httpsdoiorg1010079789402415841 Library of Congress Control Number 2018942003 Springer Nature BV 2018 This work is subject to copyright All rights are reserved 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 express 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 Cover Image DNA chromosomes and genes Courtesy National Human Genome Research Institute This Springer imprint is published by the registered company Springer Nature BV The registered company address is Van Godewijckstraat 30 3311 GX Dordrecht The Netherlands To my lifelong companion Jung Preface This book aims to cover a broad range of topics in extended statistical physics including statistical mechanics equilibrium and nonequilibrium soft condensed matter and fluid physics for applications to biological phenomena at both cellular and macromolecular levels It is expected to be a graduatelevel textbook but can also be addressed to the interested seniorlevel undergraduates The book is written also for those interested in research on biological systems or soft matter based on physics particularly on statistical physics One of the most important directions in science nowadays is physical approach to biology The tremendous challenges that come widely from emerging fields such as biotechnology biomaterials and biomedicine demand quantitative physical explanations A basic understanding of biological systems and phenomena also provides a new paradigm by which current physics can advance In this book we are mostly interested in biological systems at a mesoscopic or cellular level which ranges from nanometers to micrometers in length Such biological systems com prise cells and the constituent biopolymers membranes and other subcellular structures This biosoft condensed matter is subject to thermal fluctuations and nonequilibrium noises and owing to its structural flexibility and connectivity manifests a variety of emergent cooperative behaviors the explanation of which calls for novel developments and applications of statistical physics Students and researchers alike have difficulties in applying to biological prob lems the knowledge and methods they learned from presently available textbooks on statistical physics One possible reason for this is that in biology the systems consist of complex soft matter which is usually not included in traditional physics curricula Typical statistical physics courses cover ideal gases classical and quantum and interacting units of simple structures In contrast even simple bio logical fluids are solutions of macromolecules the structures of which are very complex The goal of this book is to fill this wide gap by providing appropriate content as well as by explaining the theoretical method that typifies good modeling namely the method of coarsegrained descriptions that extract the most salient features emerging at mesoscopic scales This book is of course in no way com prehensive in covering all the varied and important subjects of statistical physics vii applicable to biology I went to great effort to incorporate what I consider to be the essential topics which of course may reflect my own personal interests and lim itations The major topics covered in this book include thermodynamics equilib rium statistical mechanics soft matter physics of polymers and membranes nonequilibrium statistical physics covering stochastic processes transport phe nomena hydrodynamics etc More than 100 problems are given alongside the text rather than at the end of the chapters because they are a part of the text and the logical flow these problems some of which are quite challenging to solve will help readers develop a deeper understanding of the content A number of good textbooks have recently been written under the titles of physical biology biological physics and biophysics A number of these books give excellent guides to biological phenomena illustrated in the quantitative language of physics In some of these books biological systems and phenomena are first described and then analyzed quantitatively using thermodynamics and statistical physics Following biospecific topics physicsoriented readers might struggle to build systematically and coherently on the basics their own understanding of nonspecific concepts and theoretical methods which they may be able to apply to a broader class of biological problems In this book another approach is taken that is nonspecific basic methods and theories with detailed derivations and then biological examples and applications are given The book is based on lectures I gave to graduate students at POSTECH in a course under the title of Biological Statistical Physics It is my hope that by attempting to fill this aforementioned gap I can at the very least help students and researchers appreciate and learn the immense potential of statistical physics for biology particularly for biological systems at mesoscopic scales Pohang Korea Republic of Wokyung Sung viii Preface Acknowledgements I owe a great debt of thanks to a number of my teachers and colleagues that I have been influenced by and associated with throughout my scientific career Profs Yun Suk Koh Koo Chul Lee David Finkelstein George Stell John Dahler Harold Friedman Norman March Philip Pincus Man Won Kim Alexander Neiman Dmitri Kuznetsov Tapio Ala Nissila Michel Kosterlitz Kimoon Kim Byung Il Min Jongbong Lee Nam Ki Lee and Jaeyoung Sung I also wish to extend my thanks to a number of previous graduate students of mine Pyeong Jun Park Yong Woon Kim Kwonmoo Lee JaeHyung Jeon Won Kyu Kim Jaeoh Shin and in particular Ochul Lee who helped me with formatting the manuscript and drawing the figures in the book I would like to express my deep gratitude to Springers Editorial Director Dr Liesbeth Mol and Prof Eugene Stanley who suggested and encouraged me to attempt this daunting task It is with pleasure to acknowledge the support of Institute of Basic Science for SelfAssembly and Complexity ix About the Author Wokyung Sung is Professor Emeritus at Pohang University of Science and Technology POSTECH where he taught and researched in the fields of statistical physics and biological physics for about 30 years He obtained his Bachelor of Science at Seoul National University and PhD at the State University of New York at Stony Brook He has been working mostly on a variety of biological matter and processes at the mesoscopic level using statistical physics of soft matter and stochastic phenomena In particular he pioneered the theory of polymer translocation through membranes engendering a whole new field in biological and polymer physics He is a member of the Journal of Biological Physics editorial board and was an editor in chief in the period 20072009 For his seminal contri butions to science in particular to statisticalbiological physics Prof Sung was awarded a Medal of Science and Technology bestowed by the Korean Government in 2010 He also served as a director of Center for Theoretical Physics at POSTECH and the Distinguished Research Fellow at Center for Self Assembly and Complexity Institute of Basic Science in Pohang Professor Sung was a visiting scientist and professor at Oxford University the Jülich Research Center University of Pennsylvania and Brown University xi Contents 1 Introduction Biological Systems and Physical Approaches 1 11 Bring Physics to Life Bring Life to Physics 1 12 The Players of Living Selforganizing Structures 2 13 Basic Physical Features Fluctuations and Soft Matter Nature 4 14 About the Book 5 Further Reading and References 6 2 Basic Concepts of Relevant Thermodynamics and Thermodynamic Variables 7 21 The First Law and Thermodynamic Variables 8 211 Internal Energy Heat and Work The First Law of Thermodynamics 8 212 Thermodynamic Potentials Generalized Forces and Displacements 9 213 Equations of State 14 214 Response Functions 15 22 The Second Law and Thermodynamic Variational Principles 16 221 Approach to Equilibrium Between Two Systems 17 222 Variational Principles for Thermodynamic Potentials 18 Examples Biopolymer Folding 20 Nucleation and Growth A Liquid Drop in a SuperCooled Gas 21 Further Reading and References 23 xiii 3 Basic Methods of Equilibrium Statistical Mechanics 25 31 Boltzmanns Entropy and Probability Microcanonical Ensemble Theory for Thermodynamics 26 311 Microstates and Entropy 26 312 Microcanonical Ensemble Enumeration of Microstates and Thermodynamics 28 Example TwoState Model 28 Colloid Translocation 32 32 Canonical Ensemble Theory 34 321 Canonical Ensemble and the Boltzmann Distribution 34 322 The Energy Fluctuations 37 323 Example TwoState Model 39 33 The Gibbs Canonical Ensemble 41 FreelyJointed Chain FJC for a Polymer Under a Tension 42 34 Grand Canonical Ensemble Theory 44 341 Grand Canonical Distribution and Thermodynamics 45 342 Ligand Binding on Proteins with Interaction 47 Further Reading and References 49 4 Statistical Mechanics of Fluids and Solutions 51 41 PhaseSpace Description of Fluids 51 411 N Particle Distribution Function and Partition Function 51 412 The MaxwellBoltzmann Distribution 53 42 Fluids of Noninteracting Particles 57 421 Thermodynamic Variables of Nonuniform Ideal Gases 57 422 A gas of Polyatomic Moleculesthe Internal Degrees of Freedom 60 43 Fluids of Interacting Particles 61 431 The Virial ExpansionLow Density Approximation 61 432 The Van der Waals Equation of State 63 433 The Effects of Spatial Correlations Pair Distribution Function 65 44 Extension to Solutions CoarseGrained Descriptions 69 441 SolventAveraged Solute Particles 69 442 Lattice model 72 Further Reading and References 73 xiv Contents 5 CoarseGrained Description Mesoscopic States Effective Hamiltonian and Free Energy Functions 75 51 Mesoscopic Degrees of Freedom Effective Hamiltonian and Free Energy 75 52 Phenomenological Methods of CoarseGraining 77 6 Water and BiologicallyRelevant Interactions 81 61 Thermodynamic Properties of Water 81 62 The Interactions in Water 84 621 Hydrogen Bonding and HydrophilicHydrophobic Interaction 84 622 The Coulomb Interaction 85 623 IonDipole Interaction 88 624 DipoleDipole Interaction Keesom Force 90 625 Induced Dipoles and Van der Waals Attraction 92 63 Screened Coulomb Interactions and Electrical Double Layers 94 631 The PoissonBoltzmann Equation 95 632 The DebyeHückel Theory 96 633 Charged Surface Counterions and Electrical Double Layer EDL 98 Further Readings and References 102 7 Law of Chemical Forces Transitions Reactions and Selfassemblies 103 71 Law of Mass Action LMA 104 711 Derivation 104 712 Conformational Transitions of Biopolymers 107 713 Some Chemical Reactions 108 Dissociation of Diatomic Molecules 108 Ionization of Water 108 ATP Hydrolysis 109 714 Protein Bindings on Substrates 110 72 Selfassembly 111 721 Linear Aggregates 113 722 TwoDimensional Disk Formation 115 723 Hollow Sphere Formation 116 Further Readings and References 119 8 The Lattice and Ising Models 121 81 Adsorption and Aggregation of Molecules 122 811 The Canonical Ensemble Method 122 812 The Grand Canonical Ensemble Method 124 Contents xv 813 Effects of the Interactions 125 814 Transition Between Dispersed and Condensed Phases 127 82 Binary Mixtures 129 821 Mixing and Phase Separation 129 822 Interfaces and Interfacial Surface Tensions 132 83 1D Ising Model and Applications 133 831 Exact Solution of 1D Ising Model 133 832 DNA Melting and Bubbles 136 833 Zipper Model for DNA Melting and Helixto Coil Transitions 139 Further Reading and References 142 9 Responses Fluctuations Correlations and Scatterings 143 91 Linear Responses and Fluctuations FluctuationResponse Theorem 143 92 Scatterings Fluctuations and Structures of Matter 149 921 Scattering and Structure Factor 150 922 Structure Factor and Density FluctuationCorrelation 151 923 Structure Factor and Pair Correlation Function 152 924 Fractal Structures 156 925 Structure Factor of a Flexible Polymer Chain 157 Further Reading and References 159 10 Mesoscopic Models of Polymers Flexible Chains 161 101 Random Walk Model for a Flexible Chain 162 1011 Central Limit Theorem CLTExtended 164 1012 The Entropic Chain 166 Example A Chain Anchored on Surface 168 The Free Energy of Polymer Translocation 170 102 A Flexible Chain Under External Fields and Confinements 171 1021 Polymer Greens Function and Edwards Equation 172 1022 The Formulation of PathIntegral and Effective Hamiltonian of a Chain 173 1023 The Chain Free Energy and Segmental Distribution 176 1024 The Effect of Confinemening a Flexible Chain 178 1025 Polymer BindingUnbinding Adsorption Desorption Transitions 182 103 Effects of Segmental Interactions 185 1031 Polymer Exclusion and Condensation 185 1032 DNA Condensation in Solution in the Presence of Other Molecules 188 xvi Contents 104 Scaling Theory 191 Example The First Nuclear Bomb Explosion 191 Sizes and Speeds of Living Objects 192 PolymerAn Entropic Animal 193 Further Reading and References 194 11 Mesoscopic Models of Polymers Semiflexible Chains and Polyelectrolytes 195 111 Wormlike Chain Model 195 112 Fluctuations in Nearly Straight Semiflexible Chains and the ForceExtension Relation 200 1121 Nearly Straight Semiflexible Chains 200 1122 The ForceExtension Relation 201 1123 The Intrinsic Height Undulations Correlations and Length Fluctuations of Short Chain Fragments 205 1124 The Equilibrium Shapes of Stiff Chains Under a Force 208 113 Polyelectrolytes 209 1131 Manning Condensation 210 1132 The Charge Effect on Chain Persistence Length 212 1133 The Effect of ChargeDensity Fluctuations on Stiffness 215 Further Reading and References 216 12 Membranes and Elastic Surfaces 219 121 Membrane Selfassembly and Phase Transition 220 1211 Selfassembly to Vesicles 220 1212 Phase and Shape Transitions 222 122 Mesoscopic Model for Elastic Energies and Shapes 223 1221 Elastic Deformation Energy 223 1222 Shapes of Vesicles 226 123 Effects of Thermal Undulations 228 1231 The Effective Hamiltonian of Planar Elastic Surface and Membranes 228 1232 Surface Undulation Fluctuation and Correlation 230 1233 Helfrich Interaction and Unbinding Transitions 238 Further Reading and References 239 13 Brownian Motions 241 131 Brownian MotionDiffusion Equation Theory 242 1311 Diffusion Smoluchowski Equation and Einstein Relations 242 Contents xvii 132 Diffusive Transport in Cells 247 1321 Cell Capture 247 1322 Ionic Diffusion Through Membrane 252 1323 A Trapped Brownian Particle 255 133 Brownian MotionLangevin Equation Theory 257 1331 The Velocity Langevin Equation 257 1332 The Velocity and Position Distribution Functions 260 1333 A Brownian Motion Subject to a Harmonic Force 262 1334 The Overdamped Langevin Equation 266 Further Readings and References 267 14 Stochastic Processes Markov Chains and Master Equations 269 141 Markov Processes 269 1411 Probability Distribution Functions PDF 269 1412 Stationarity Time Correlation and the WienerKhinchin Theorem 270 1413 Markov Processes and the ChapmanKolmogorov Equation 274 142 Master Equations 277 1421 Derivation 277 1422 Example Dichotomic Processes 278 1423 Detailed Balance 280 1424 OneStep Master Equations 282 Random Walk 282 Poisson Process 283 Linear OneStep Master Equation 285 Reactions 286 Further Reading and References 289 15 Theory of Markov Processes and the FokkerPlanck Equations 291 151 FokkerPlanck Equation FPE 291 1511 Derivation 291 1512 The FPE for Brownian Motion 293 152 The Langevin and FokkerPlanck Equations from Phenomenology and Effective Hamiltonian 295 1521 FPE from OneStep Master Equation 297 153 Solutions of FokkerPlanck Equations Transition Probabilities and Correlation Functions 299 1531 Operators Associated with FPE 299 1532 Eigenfunction Method 300 1533 The Transition Probability 304 1534 TimeCorrelation Function 305 1535 The Boundary Conditions 306 xviii Contents 1536 The Symmetric Double Well Model 307 Further Reading and References 310 16 The MeanFirst Passage Times and Barrier Crossing Rates 313 161 First Passage Time and Applications 313 1611 The Distribution and Mean of Passage Time 314 1612 Example Polymer Translocation 318 162 The Kramers Escape Problem 320 1621 Rate Theory FluxOver Population Method 321 1622 The Kramers Problem for Polymer 322 Further Reading and References 325 17 Dynamic Linear Responses and Time Correlation Functions 327 171 TimeDependent Linear Response Theory 328 1711 Macroscopic Consideration 328 1712 Statistical Mechanics of Dynamic Response Function 331 1713 FluctuationDissipation Theorem 334 172 Applications of the FluctuationDissipation Theorem 336 1721 Dielectric Response 336 1722 Electrical Conduction 338 1723 FDT Under Spatially Continuous External Fields 340 1724 Density Fluctuations and Dynamic Structure Factor 342 Further Reading and References 346 18 NoiseInduced Resonances Stochastic Resonance Resonant Activation and Stochastic Ratchets 347 181 Stochastic Resonance 348 1811 Theory 348 1812 Biological Examples 352 Ion Channel 352 Biopolymers Under Tension 354 182 Resonant Activation RA and Stochastic Ratchet 355 1821 Model 356 Example Rigid Polymer Translocation Under a Fluctuating Environment 358 Stretched RNA Hairpin 360 183 Stochastic Ratchets 360 Further Reading and References 361 19 Transport Phenomena and Fluid Dynamics 363 191 Hydrodynamic Transport Equations 364 1911 Mass Transport and the Diffusion Equation 365 1912 Momentum Transport and the NavierStokes Equation 366 Contents xix 1913 Energy Transport and the Heat Conduction 370 1914 Boltzmann Equation Explains Transport Equations and TimeIrreversibility 372 192 Dynamics of Viscous Flow 373 1921 A Simple Shear and Planar Flow 373 1922 The Poiseuille Flow 375 Blood Flow Through a Vessel The FahraeusLindqvist Effect 377 1923 The Low Reynolds Number Approximation and the Stokes Flow 379 1924 Generalized Boundary Conditions 382 1925 Electroosmosis 384 1926 Electrophoresis of Charged Particles 386 1927 Hydrodynamic Interaction 388 Further Reading and References 390 20 Dynamics of Polymers and Membranes in Fluids 391 201 Dynamics of Flexible Polymers 392 2011 The Rouse Model 393 2012 The Zimm Model 398 Segmental Dynamics 402 202 Dynamics of a Semiflexible Chain 404 2021 Transverse Dynamics 405 2022 Chain Longitudinal Dynamics and Response to a Small Oscillatory Tension 410 203 Dynamics of Membrane Undulation 414 204 A Unified View 418 Further Reading and References 421 21 Epilogue 423 Surmounting the Insurmountable 423 Additional Topics 426 Index 427 xx Contents Symbols A A Surface area a Unit length ionic radius the unit area attraction strength B Magnetic field B2 Second virial coefficient b Bond energy unit length in a polyelectrolyte C Heat capacity C ci Concentration threedimensional concentration D Diffusion constant intersurface distance D Noise strength Df Fractal dimension DT Thermal diffusion constant E Internal energy E Eext External electric field er Local energy density F f Helmholtz free energy Helmholtz free energy density fi Generalized force F f Force FQ Effective Hamiltonian or the free energy function associated with Q fRt Random force G gG Gibbs free energy Gibbs free energy per particle variable Gibbs free energy Gr r0N Polymer Greens function gr gr Pair distribution function radial distribution function H Enthalpy H Hamiltonian h h Planck constant Undulation height J Jn Flux number flux vector K Kinetic energy Ke Entropic spring constant Ks Stretch modulus xxi KT Isothermal compressibility k Wave vector kB Boltzmann constant ke kq Spring constant in the beadspring model L Langevins function evolution operator L þ Adjoint evolution operator LFP FokkerPlanck operator l Segmental length step length dipole length lB Bjerrum length lp Persistence length M Magnetic moment M m Mass M Microstate N N Number of particles n n Number density fluctuating number density n0 Concentration at standard state n1 Concentration at the bulk O rð Þ Oseen tensor P p Pressure P Probability distribution function PDF p Momentum dipole moment Q Heat charge mesoscopic degrees of freedom QN Configuration partition function Chap 4 q Charge coordination number wave number stochastic variable q Wave vector R Radius RG Radius of gyration Re Reynolds number r Position vector distance vector rn The position of nth bead S s Entropy entropy density S q ð Þ Structure factor S x ð Þ Power spectrum S q x ð Þ Dynamic structure factor s Arc length T Absolute temperature TX Periodicity t Time U ri f g u ri ð Þ External potential energy U q ð Þ Drift u Unit tangent vector fluid velocity V v Volume Velocity W Work applied to the system w E ð Þ Density of states xxii Symbols Symbols Xi Xi X Generalized displacement macroscopic and microscopic Z Z Canonical partition function z Valence of ions singleparticle partition function fugacity α Polarizability β 1kBT Γt Systems phase space point γ Surface tension E System energy ε Electric permeability internal energy density ε0 Electrical permeability in vacuum εw Electrical permeability of water ε Binding energy ζ ζs Friction coefficient surface friction coefficient η Shear viscosity Θ Strength of thermal noise θ Coverage of protein Chaps 2 3 polar angle Chap 3 κ κ Bending rigidity curvature modulus κG Gaussian modulus κG Curvature modulus for sphere κT Heat conductivity λ Thermal wavelength wave length linear charge density λD Debye screening length μ Chemical potential ξ Correlation length ρr ρer Mass density charge densities σ Pressure tensor or stress tensor surface force density τ Mean first passage time MFPT correlation or relaxation time τR τZ Rouse time Zimm time τK Kramers time τp Momentum relaxation time Φri φrij Interaction potential energy φ Azimuthal angle potential energy φs Surface potential χ Magnetic susceptibility χP Static electric susceptibility χt The dynamic response function Ω Grand potential Chaps 2 3 solid angle Chap 3 ω Frequency xxiii Chapter 1 Introduction Biological Systems and Physical Approaches Open the door open the door the Flower Thunder and Storm be the only way the Flower open the door Seo Jung Ju In January 1999 at the dawn of the new millennium Time Magazine devoted the majority of its coverage to a special issue entitled The Future of Medicine The cover story began as follows Ring farewell to the century of physics the one in which we split the atom and turned silicon into computing power Its time to ring in the century of biotechnology Despite the tremendous importance of life science and biotechnology nowadays as the above statements proclaim at this stage their knowledge appears to be largely phenomenological and thus undeniably calls for fundamental and quantitative understandings of the complex phenomena It will be timely to ring in the century of a new physical science to meet this challenge 11 Bring Physics to Life Bring Life to Physics Biological Physics or Biophysics is a new genre of physics which has attempted to describe and understand biology Despite a few important achievements such as unravelling DNAs doublehelical structure by James Watson and Francis Crick using Xray diffraction biological physics as the fundamental and quantitative Fig 11 Physics and biology Between them lies a mountain called biological physics or physical biology On the axis toward you is chemistry Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158411 1 science of biological phenomena has had rather a slow growth and is yet in its infancy There are dramatic differences between two sciences physics and biology in study methods and objects Physics by tradition pursues unity and universality in underpinning principles and quantitative descriptions for rather simple systems Biology in contrast used to deal with variety and specificity and seek qualitative descriptions for very complex systems Physics and biology represent two opposite extremes of sciences so presence of a seeminglyinsurmountable barrier between them is not a surprise Fig 11 From the view point of physics biological systems have enormously complex hierarchies of structures that range from the microscopic molecular worlds to macroscopic living organisms In this book major emphasis is focused on the mesoscopic or cellular level which covers nanometer to micrometer lengths in which cells and their constituent biopolymers membranes and other subcellular structures are the main components of interest Fig 12 Cells consist of nanometer and micrometer sized subcellular structures which appear to be enormously complex yet exhibit certain orders for biological functions the phenomenon what we call biological selforganization The flexible structures incessantly undergo thermal motion and in close cooperation with each other and the environment play the symphony of life 12 The Players of Living Selforganizing Structures Biopolymers are the most essential functional elements which can be appropriately called the threads of life Among them DNA is the most important biopolymer which stores hereditary information The monomers of DNA called nucleotides form two complementary chains in double helices encoding genetic information Cilia Mitochondrion Lysosome Rough endoplasmic reticulum Golgi apparatus Nucleus Cell membrane Microtubules Ion channel Fig 12 A biological cell is the elementary factory of life with selforganizing micronano scale internal structures Several key organelles are drawn 2 1 Introduction Biological Systems and Physical Approaches At first glance DNA appears to be quite complex as it winds to form chromosomes but it reveals a fascinating hierarchy of ordered structures It is remarkable that although a cells DNA may be as long as a few meters it can miraculously be packed into a nucleus that is only a few micrometers in size Fig 13 Proteins are also important biopolymers Proteins are chains of monomers called amino acids interconnected via a variety of interactions in water The interactions cause proteins to fold into the native structures that have the lowest energies among a vast variety of configurations Mother Nature accomplishes with ease the protein folding into the native structures in which they perform biological functions Understanding this mystery remains yet an important challenge in biological physics Another dramatic example of selforganization occurs at a biological mem brane which we may call the interface of life Fig 14 A lipid molecule lipid which is the basic constituents of the membrane is composed of a hydrophilic head and hydrophobic tails The lipids spontaneously selfassemble into a bilayer forming a barrier to permeation of ions and macromolecules thus providing the most basic function of a biological membrane For certain functions of life like Chromosome Chromatin loops Nucleosomes DNA Double helix Fig 13 DNA folded and packed within a nucleus in a multiscale hierarchy from doublestranded duplex to chromosome Lipid molecule Ion channel cytosol Fig 14 A cross section of a cell membrane with associated ion channels and proteins 12 The Players of Living Selforganizing Structures 3 neural transmissions and sensory activities certain specific ions must pass through the membrane For this reason Nature dictates some certain proteins to fold into the membrane and form a nanomachine called an ion channel to regulate passage of ions The information of the channel structures is given gradually but compre hensive physical understanding of how they work is yet to be achieved 13 Basic Physical Features Fluctuations and Soft Matter Nature The preceding overview has implied that the biological components selforganize themselves to function To perform the biological selforganization they often cross over the energy barriers that seem to be insurmountable in the view point of simple physics To this end there are two physical characteristics that feature in the mesoscopic biological systems introduced above The first one is their aqueous environments and thermal fluctuations therein The water has many outstanding properties among all liquids Its heat capacity is almost higher than any other common substance meaning that it functions as a heat reservoir with negligible temperature change The most outstanding property of water is its di electric constant around 80 that is much higher than those of other liquids Because of this water can reduce electrostatic energy of the interaction to the level of thermal energy These unique properties of water originate microscopically from hydrogen bonding between water molecules This bonding is also a relatively weak interaction even though the bonding can be broken due to thermal fluctuations it causes longrange correlation between water molecules As a result the liquid water manifests a quasicritical state where it responds collectively and sensitively to external stimuli Another physical characteristics is the structural connectivity and flexibility the systems may have the features that are not seen in traditional physics Although interactions between monomers eg the covalent bonding between two adjacent nucleotides in a DNA strand can be as large as or larger than several electron volts eV the chain as a whole displays collective motions and excitations of energy as low as in the order of thermal energy kBT 0025 eV Such a low energy is commensurate with weak biological interactions eg hydrophobic hydrophilic the Van der Waals and the screened Coulomb interactions between two segments mediated by water Thus thermal agitations can easily change conformations shapes of the biological components and at the long times when the equilibrium is reached minimize their free energies at the temperature of the surrounding examples include conformational transitions such as DNAprotein folding lipid selfassembly and membrane fusion The conformation emerges as a new primary variable and conformation transition becomes the central problem for biological physics The biological systems in mesoscale characterized by the soft interconnectivity and weak interactions may appropriately be called the 4 1 Introduction Biological Systems and Physical Approaches biosoft condensed matter To this matter a thermal fluctuation with energy of the magnitude kBT may come as a thunderstorm it adds to the disorder in ordinary matter but may assist biological matter to surmount the barriers for selforganization The biological systems in vivo function out of equilibrium driven by external influences Due to the macromolecular nature and the viscous backgrounds the dynamics of biological components at mesoscales is usually dissipative slow yet stochastic The biological dynamics can be modelled as generalized Brownian motion not only with the internal constituents fluctuating while interacting with each other but also with external forces that can fluctuate often far from equilib rium It was found that thermal fluctuations or internal noises do not simply add to the disorder of the system but counterintuitively contribute to the coherence and resonance to external noises In short the basic physical features behind bio logical selforganization are thought to be thermal fluctuations and nonequilibrium stochasticity combined with soft matter flexibility and weak interactions 14 About the Book This book addresses the basic statistical physics for biological systems and phe nomena at the mesoscopic level ranging from nanometer to cellular scales Because of thermal fluctuations and stochasticity probabilistic description is inevitable The statistical physics description for such biological systems requires a systematic way of characterizing the complex features effectively in terms of relevant degrees of freedom what we call coarse graining The book first deals with equilibrium state of matter starting with thermody namics and its foundational science statistical mechanics To illustrate its practical utility we apply statistical ensemble methods to relatively simple but archetypal systems in particular twostate biological systems We then present the application of statistical mechanics to both simple and complex fluids the playgrounds for biological complexes We introduce the method of coarsegrained description for the emerging degrees of freedom and the associated effective Hamiltonians We then devote several chapters to the general physical aspects of water weak inter actions between the objects therein and to reactions transitions and selfassembly The lattice and Ising models are presented to deal with a number of twostate problems such as molecular binding on substrates and biopolymer transitions We then describe how the responses to a stimulus and a scattering on matter are related with the internal fluctuations and their spatial correlations In two chapters on poly mers we adapt statistical physics to mesoscopic descriptions of flexible and semi flexible polymers their conformationalentropic properties exclusioncollapse confinementstretching and electrostatic properties etc The next chapter is devoted to mesoscopic description of membranes in terms of the shapes and curvatures 13 Basic Physical Features Fluctuations and Soft Matter Nature 5 The other part of the book is devoted to nonequilibrium phenomena Dynamics of biological systems is essentially the nonequilibrium process often with their soft matter nature displayed The basic methods include a stochastic approach in which the mesoscopic degrees of freedom undergo the generalized Brownian motions We start with the EinsteinSmoluchowskiLangevin theories of Brownian motion which are extended within the framework of Markov process theory the master equation and the FokkerPlanck equation are discussed and applied to biological problems The thermallyinduced crossing over free energy or activation barriers is discussed using the rate theory and mean first passage time theory The response of a dynamic variable to timedependent forces or fields is introduced along with underlying time correlation function theories FluctuationDissipation Theorem A thermal fluctuation when optimally tuned will be shown to induce coherence and resonance to a small external driving Also an emphasis is placed on the fluid backgrounds and its own hydrodynamics and transport phenomena The dynamics of biological soft matter such as simple polymers and membranes interacting hydrodynamically in a viscous fluid often anomalous due to the structural con nectivity is then described Further Reading and References J Knight Physics meets biology Bridging the culture gap Nature 419 244246 2002 H Frauenfelder PG Wolynes RH Austin Biological Physics Rev Mod Phys 71 S419S430 1999 R Phillips SR Quake The biological frontier of physics Phys Today 59 5 2006 Biological Physics Books Examples R Phillips J Kondev J Therio Physical Biology of the Cell Garland ScienceTaylor and Francis Group 2008 P Nelson Biological Physics WH Freeman 2007 K Sneppen G Zocchi Physics in Molecular Biology Cambridge University Press 2006 D Ball Mechanics of the Cell Cambridge University Press 2002 M Daune Molecular Biophysics Oxford University Press 1999 MB Jackson Molecular and Cellular Biophysics Cambridge University Press 2006 W Bialek Biophysics Searching for Principles Princeton University Press 2012 TA Waigh The Physics of Living Processes A Mesoscopic Approach Wiley 2014 H Schiessel Biophysics for Beginners A Journey through the Cell Nucleus Pan Stanford Publishing 2014 D Andelman Soft Condensed Matter Physics in Molecular and Cell Biology Ed by WCK Poon Taylor and Francis 2006 JA Tuszynsky M Kurzynski Introduction to Molecular Biophysics CRS Press 2003 6 1 Introduction Biological Systems and Physical Approaches Chapter 2 Basic Concepts of Relevant Thermodynamics and Thermodynamic Variables A macroscopic or a mesoscopic system contains many microscopic constituents such as atoms and molecules with a huge number of degrees of freedom to describe their motion Thermodynamics1 seeks to describe properties of matter in terms of only a few variables arguably being the allaround basic area of sciences and engineering including biology Thermodynamics and thermodynamic vari ables characterize states of matter and their transitions phenomenologically without recourse to microscopic constituents In this chapter we summarize what we believe to be the essentials that will serve as references throughout the book The link between this phenomenological description and microscopic mechanics is provided by statistical mechanics beginning next chapter When a macroscopic system is brought to equilibrium where its bulk properties become timeindependent they can completely be described by a few variables descrip tiveof the statecalledthestatevariablesFor examplethemacroscopicpropertiesofan ideal gas or of an ideal solution at equilibrium can be described by the pressure or the osmotic pressure p volume V and absolute temperature T eg for a mole of them the equation of state is pV ¼ RT where the R is the universal gas constant The ther modynamic state variables are either extensive or intensive Extensive variables are proportional to the size of the system under consideration intensive variables are independent of the system size for example the gas volume V and internal energy E are extensive whereas the pressure p and the temperature T are intensive Here we briefly summarize the universal relations beginning with the first law of thermodynamics By a universal relation we mean the relation independent of the systems microscopic details We introduce the basic thermodynamic potentials 1Contrary to what the nomenclature implies thermodynamics mostly deals with the equilibrium state of matter at macroscale so often is also coined as thermostatics The second law of ther modynamics however is concerned with nonequilibrium processes approaching equilibrium the rigorous treatment of which is treated in the area called nonequilibrium thermodynamics S R de Groot and P Mazur Nonequilibrium Thermodynamics 1984 Courier Corp In chemistry or biochemstry communities biological thermodynamics include the chemical kinetics and reac tions eg Biological Thermodynamics D T Haynes 2008 Cambridge University Press Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158412 7 from which we can find the various thermodynamic variables From the second law of thermodynamics we discuss nature of the processes leading to equilibrium which are governed by variational principles for the thermodynamic potentials relevant to ambient thermodynamic conditions 21 The First Law and Thermodynamic Variables 211 Internal Energy Heat and Work The First Law of Thermodynamics Here we consider the changes of thermodynamic state variables controlled by quasistatic processes which are ideally slow so as to retain the equilibrium state Quasistatic processes are reversible ie can be undone First consider the net energy of the system called the internal energy E which is conserved in a system that does not exchange matter or energy with the environment called an isolated system Because E is given uniquely by other state variables Yi the independent variables E EY1 Y2 the state variable E is also a state function with its infinitesimal change dE being an exact differential dE sum i partial E partial Yi dYi 21 The first law of thermodynamics is simply the statement of energy conservation involving various forms of energies It says dE dQ dW 22 where dQ and dW are respectively the infinitesimal heat and the infinitesimal work applied to the system by certain external agents Equation 22 says that its internal energy increases if it is heated and decreases if the work is done by it Unlike the internal energy both of the heat and work cannot be solely described by the present state variables but depend on the processes through which they are changed As such their infinitesimal changes denoted by d signify inexact differentials which depend on the paths or histories of the processes taken For example consider a quaistatic cyclic process of a gas undergoing an expansion process 1 2 and compression 2 1 returning to its initial state 1 under a pressure Fig 21 The cyclic change of the work defined by dW p dV is not vanishing but given by the shaded area In contrast the cyclic change of the internal energy a state variable with its differential being exact denoted by dE E1 E1 is zero In a similar manner the cyclic change of the heat is not vanishing Fig 21 The relation between pressure p and volume V for a cyclic process consisting of a reversible expansion 1 2 and a reversible contraction 2 1 on a gas In this cyclic process the system does work by the amount given by the shaded area dQ 0 23 However according to Rudolf Clausius for any cyclic change controlled to be reversible dQ T 0 24 where T is a state variable called the temperature From the equation an exact differential of a state function S called entropy is defined as dS dQ T 25 Entropy which is the central concept in thermodynamics and in various aspects of biological processes will be discussed later repeatedly P21 Show that for an ideal gas or solution of one mole for which E 3RT 2 and pV RT are known 25 for a reversible process of changing the volume and temperature is dS 3R 2T dT R V dV which is indeed an exact differential The entropy change from V1 T1 to V2 T2 is S 3R 2 lnT2 T1 R lnV2 V1 Although derived for a reversible process because S is a state variable this relation is independent of the thermodynamic paths taken between the initial and final states so that S it is applicable to any processes including irreversible one that connects the same initial and final states 1 and 2 212 Thermodynamic Potentials Generalized Forces and Displacements Now consider the work in detail it can be generated by various agents such as external forces and fields acting on the system dW μ dN sum i fi dXi 26 The first term on the right is the chemical work involving the chemical potential μ necessary to increase the number of particles N of the system by unity For a mixture of m component particles it can be generalized to sum k1 to m μk dNk where k denotes the species In the second term fi is a generalized force or a field and Xi is a thermodynamic conjugate to it called a displacement Table 21 The first three generalized forces and displacements in the table are mechanical while the last two examples are electromagnetic fi are intensive state variables whereas Xi are extensive state variables For illustration consider a onecomponent system m 1 with one generalized force fi and the associated displacement Xi The most familiar case is a particle system such a gas or a colloidal solution confined within a volume by a pressure for which fi p Xi V For a stretched chain the tension f and the length of extension X are such a forcedisplacement pair Table 21 Using the relations 25 and 26 the first law of thermodynamics 21 can be written in terms of state variables S N and Xi dE dQ dW T dS μ dN fi dXi 27 Representing S as the primary variable 27 can be rewritten as dS 1T dE μT dN fiT dXi 28 which expresses S as a state function of independent state variables E N and Xi S SE N Xi Equation 28 being an exact differential the following relations are obtained 1T partial S partial EXi N 29 Table 21 Examples of generalized forces and the conjugate displacements Systems Generalized forces intensive variables fi Xi Generalized displacements extensive variables Fluid Pressure p V Volume String Tension f X Length of extension Surface Surface tension γ A Surface area Magnet Magnetic field B M Magnetization along the field Dielectrics Electric field E P Polarization along the field μ T S NEXi fi T S XiEN where the subscripts in the partial differentiations indicate the variables that are held fixed Equations 29211 mean that once S is obtained as a function of independent variables E N and Xi it can generate their thermodynamic conjugates T μ and fi by taking the firstorder partial derivatives with respect to the independent variables Functions obtained by taking firstorder partial derivatives over thermodynamic potentials will be called the firstorder functions Equations 29211 show how the basic intensive variables are related to the entropy Equation 29 is a fundamental thermodynamic relation that defines the temperature the ratio of an increase of the entropy with respect to the energy increase is a positive quantity 1 T Equation 210 tells us that the chemical potential μ is a measure of the change of entropy when a particle is added to the system without an external work and change of internal energy Equation 211 defines the generalized force fi that acts in the direction to decrease the entropy with E N fixed In a gas or a solution the force is the pressure p compressing the system to keep it from increasing its entropy For a polymer string it is the tension force f to extend it Fig 22 A thermodynamic potential is a state variable that describes the systems net energy from which all other variables can be derived One example is the internal energy we have considered another one is the Helmholtz free energy defined by F E TS If we consider this as the primary thermodynamic potential 27 is transformed to dF dE TS SdT fi dXi μ dN which indicates that F is the state function that depends on the state variables T Xi and N ie F FT Xi N It can generate thermodynamic relations for the firstorder variables S F TXi N Fig 22 Two kinds of forces pressure p force per unit area on the gas to keep its volume as V and extensional tension f on a polymer to keep its extension as X The forces act in the directions in which to decrease the entropy fi F XiT N μ F NT Xi The internal energy is then obtained from the Helmholtz free energy E F TS F TF TXi N T2 F T T For systems controlled by a displacement Xi eg for a fluid confined within a volume or a string kept at a constant extension the Helmholtz free energy is the thermodynamic potential of choice S in this representation depends on T as well as on Xi and N in contrast to 28 Since ²F xj xk ²F xk xj one can also obtain the Maxwell relations for the second order variables S Xi fi T fi N μ Xi S N μ T P22 Consider the enthalpy defined by H E pV as a primary thermodynamic potential and obtain the thermodynamic relations for the first and second order variables P23 Consider that a strip of rubber is extended quasistatically to a length X Show how the force of extension or the tension is expressed in terms of the free energy Find the Maxwell relations Another useful representation is the one in which the Gibbs free energy G F fi Xi is the primary thermodynamic potential From 212 its differential is given as dG dF fi Xi SdT Xi dfi μ dN The Gibbs free energy is the thermodynamic potential that depends on three independent variables T fi and N ie G GT fi N For a onecomponent system because N is the only extensive variable among the three the extensivity of G requires that GT fi N NgT fi where gT fi is the Gibbs free energy per particle In this representation the firstorder thermodynamic variables are derived as S G TN fi Xi G fiT N μ G NT f gT fi The chemical potential is the Gibbs free energy per particle for a onecomponent system which is independent of the number of particles number thus dμT fi S N dT Xi N dfi For systems controlled by the generalized force fi the Gibbs free energy is a convenient thermodynamic potential Because experiments on fluids are usually performed under constant pressures the Gibbs free energy is often chosen as the primary thermodynamic potential Lastly let us consider the grand potential as the primary thermodynamic potential which for a onecomponent system is defined by Ω F μN Its differential dΩ SdT fi dXi N dμ is obtained by using 212 so that Ω has the independent variables T Xi and μ Consequently S Ω TX μ fi Ω XiT μ N ΩμTX Noting that G μN ie Ω F G fi Xi fi can also be obtained directly from Ω as fi ΩXi P24 The relation Ω fi Xi can be generalized to the case where there are multitude of conjugate pairs Consider a liquid droplet in a gas In this case the grand potential is given by Ω pg Vg pl Vl γA where pg Vg and pl Vl are the pressures and volumes of the gas and liquid phases respectively γ is surface tension in the interfacial area A 213 Equations of State One of the most important tasks of equilibrium statistical mechanics is to obtain the thermodynamic potentials explicitly for specific systems as functions of their own independent variables From this procedure the firstorder variables are obtained and related to yield the equations of state The most wellknown example is the equation of state that relates the pressure p with the volume V of a onemole ideal gas or an ideal solution pV RT An approximate equation of state for nonideal fluids that includes the interparticle interactions is the Van der Waals equation of state p aV2V b RT where a and b are the constants that parametrize interparticle attraction and repulsion respectively The equation of state that describes ideal paramagnets is Curies law MB CT where B is a magnetic field along a direction M is the magnetization induced along the direction and C is the Curie constant which is materialspecific Due to the mutual interactions between the magnetic moments within it a paramagnet undergoes a phase transition at a temperature called the critical temperature Tc to a ferromagnet for which an approximate equation of state is MB CT Tc Another example which is of biological importance is the equation for the force f necessary to extend a DNA fragment by an amount X f AT141 XL2 14 XL where L is a contour length and A is a constant P25 Calculate the Helmholtz free energy of the Vander Waals gas What is the chemical potential What is the isothermal compressibility P26 Using 236 a Find the Helmholtz free energy F of the DNA as a function of X At what value of X is the free energy minimum b By how much does the entropy change when the DNA is quasistatically extended from X 0 to X L2 at a fixed temperature T c If you increase the temperature slightly by ΔT with the extension force held fixed as f how would the extension X change 214 Response Functions The properties of a material can be learned by studying how it responds to small external influences The response of the system to a variation of temperature is given by a response function called heat capacity C dQdT T ST Using 27 and 216 the heat capacity of a material with fixed N measured at fixed volume is given by CV T STV ETV T 2FT2V which means that the constantvolume heat capacity CV can be obtained from either S or E The fact that the CV is the secondorder derivative of the thermodynamic potential F implies that CV yields higherlevel information than can be afforded by the firstorder variables As we will reveal CV is directly related to the intrinsic energy fluctuations of the systems and identifies thermallyexcited microscopic degrees of freedom that underlie Other response functions of interest that we will study are isothermal compressibility KT 1VVpT and magnetic susceptibility χT MBT which are secondorder thermodynamic functions related to the systems volume and magnetization fluctuations respectively Chap 9 22 The Second Law and Thermodynamic Variational Principles The state variable entropy S first introduced by Clausius in 1850 is defined by 25 in terms of the heat reversibly exchanged at an absolute temperature T However strictly speaking most spontaneous processes that occur in nature are not reversible but pass through nonequilibrium states For example consider a gas that undergoes free expansion Experience tells us that the infinitesimal change of heat in the spontaneous irreversible processes is less than that given by 25 δQ T δS where δ denotes the differential indicating an irreversible change Therefore for an isolated system that does not exchange heat with the outside δQ 0 δS 0 This formulates a form of the second law of thermodynamics for an isolated system a spontaneous process occurs in such a way that the entropy increases to its maximum δS 0 which is just the equilibrium state The entropy is identified as a measure of the systems disorder as will be shown in next chapter This fundamental law sets the directions for natural phenomena to take the time arrow allowing us to distinguish the future from the past This variational form of the second law for the entropy can be extended to the variational principles for other thermodynamic potentials to have approaching equilibrium Table 22 as we shall see It is mistakenly perceived that living organisms defy the second law because they can organize themselves to increase the order ie they live on negative entropy called negentropy Whereas the entropy maximum is referred to an isolated system at equilibrium the living being is an open system which can exchange both energy and matter with its environment For example the entropy of a biopolymer undergoing folding decreases while that of the surrounding water increases in such a way that the entropy of the whole if isolated increases as will be shown below Furthermore the living organisms in vivo usually function far from equilibrium The equilibrium thermodynamics is nevertheless applied to biological systems in vitro which are either at or near the equilibrium state 221 Approach to Equilibrium Between Two Systems We first use the 2nd law of thermodynamics to study the approach to equilibrium between two systems at contact and the conditions of the equilibrium Consider an isolated system composed of two subsystems A and B partitioned by a movable wall which allows the exchange of matter as well as energy Fig 23 Suppose that each of the subsystems is at equilibrium on their own but not with respect to each other and evolve irreversibly towards the total equilibrium through the exchanges During an infinitesimal process the net entropy change of the isolated system is given by δS δSA δSB SAEA SBEA δEA SAVA SBVA δVA SANA SBNA δNA 243 where δEA δVA δNA are respectively the changes of the internal energy volume and particle number of subsystem A Because the net energy net volume and net particle number are all fixed in the isolated system these changes are equal to δEB δVB δNB respectively Then noting SBEA SBEB SBVA SBVB SBNA SBNB along with the relations 29211 and following the second law the net entropy should increase until the maximum δS 1TA 1TB δEA pATA pBTB δVA μATA μBTB δNA 0 244 Suppose for a moment that there is only an energy exchange while both of each volume and particle number are fixed δVA δNA 0 Then the inequality in 244 means that TA TB leads to δEA 0 that is the energy flows from A to B ie form a hotter to a colder place The entropy maximum δS 0 is reached when TA TB 245 The equality between the temperatures is the condition for thermal equilibrium between the two subsystems in contact which is named as the zeroth law of thermodynamics With this thermal equilibrium established we let the partition be movable and pA pB with no exchanges of the particles Then 244 leads to δVA 0 meaning that by the pressure difference the system A expands until the pressures are equalized pA pB 246 By considering an exchange of particles one can also show that the particles flow from the system of higher chemical potential to that of lower chemical potential until they reach the chemical equilibrium where μA μB 247 Because δEA δVA δNA are independent of each other each term in parentheses in 244 vanishes at the equilibrium so the above three equations called the condition of thermal mechanical and chemical equilibrium respectively are simultaneously satisfied at the equilibrium 222 Variational Principles for Thermodynamic Potentials Now suppose that a subsystem A considered above is much smaller than B so that the latter forms a heat bath kept at temperature T throughout Fig 24 Considering the subsystem A as our primary system a polymer for example to study we drop the subscript A The infinitesimal change of total entropy δST of the isolated system A B is given by δST δS δSB δS δQT δS δE δWT δF δWT 248 Here δQ is the differential heat given to system A by the bath at the fixed temperature T by the first law δQ δE δW Using the second law δST 0 248 tells us that δF δW ie δF is the minimum of the reversible work done on the system by the bath If the systems displacement and number of particles are kept as fixed then δW fiδXi μδN is zero and δF 0 249 This is a famous variational principle stating it again if the system at a fixed T has fixed Xi and N but is left unconstrained its Helmholtz free energy decreases spontaneously to its minimum as the system approaches equilibrium For example a biopolymer which keep its extension X as fixed and thus undergoes no work conforms itself in a way to minimize its Helmholtz free energy Often the systems are under a fixed generalized force fi eg in a gas at atmospheric pressure or a polymer chain subject to a fixed tension In this case δF δW δF fiXi 0 leading to δG 0 250 ie the Gibbs free energy of the system with T kept at fixed fi but otherwise unconstrained decreases until it approaches the minimum namely the equilibrium The biopolymer subject to constant tension conforms itself to minimize the Gibbs free energy A spherical vesicle blown by a pressure can have an optimal radius to minimize it See 1221 Finally consider an open system in which the number of particles can vary but the displacement and chemical potential μ not to mention the temperature are fixed In this case δST 0 with 248 leads to δF δW δF μN δΩ 0 251 it is the grand potential that is to be minimized There are many situations where the numbers of systems constituent units vary eg phase transitions reactions and selfassemblies Table 22 Constrained variables and associated thermodynamic principles Systems Thermodynamic variational principle Isolated system with fixed N E Xi Entropy S maximum Closed system with fixed N T Xi Helmholtz free energy F minimum Closed system with fixed N T fi Gibbs free energy G minimum Open systems with fixed μ T Xi Grand potential Ω minimum Listed in Table 22 are the summary of the variational principles for the thermodynamic potentials to be optimized and their independent state variables conditioned to be fixed These variational principles can be applied to any systems kept at a fixed temperature the presence of the enclosing adiabatic wall in Fig 24 is immaterial because the wall can be placed at an infinite distance away from the systems in question As will be shown throughout this book the variational principles will be of great importance in determining the equilibrium configurations of flexible structures at a fixed temperature as typified by biomolecule and membrane conformations at body temperature Strictly speaking these variational potentials should be distinguished from the equilibrium thermodynamic potentials F G dealt in Sect 21 which are just extrema of the variational ones This is will be done whenever necessary hereafter by using different scripts eg F for F G for G Examples Biopolymer Folding A biopolymer subject to thermal agitation in an aqueous solution undergoes foldingunfolding transitions For this case the combined system of the polymer and the liquid bath can be regarded as an isolated system According to the second law δST δS δSB 0 Let us consider the transition from an unfolded state to a folded state at a fixed temperature Folding means an increase of the order which as will be shown next chapter signifies δS 0 hence δSB 0 The entropy of the liquid bath increases because during the folding process the water molecules unbind from the polymer and will enjoy a larger space to wander around that is a larger entropy Following the thermodynamic variational principle the free energy change of the polymer in contact with the heat bath then should satisfy δF δE TδS 0 this equation leads to δE TδS and following δS 0 as shown above δE 0 which implies that E decreases due to the folding of the polymer In biological systems conformation transitions such as this folding transition are numerous at body temperature Fig 25 Polymer unfoldingfolding transition that occurs above and below the critical temperature Tc T Tc T Tc Fig 26 A liquid drop L in a supercooled gas G at a fixed temperature Because of the interfacial tension γ the liquid pressure pl should be higher than the gas pressure pg Spontaneous processes at a fixed T occur whenever the free energy of the system decreases δF δE TδS Tc TδS 0 252 where Tc δEδSδF0 is the critical temperature Therefore if T Tc the transition to the ordered phase δS 0 occurs whereas if T Tc the transition to the disordered phase δS 0 occurs These are examples of a multitude of biopolymer conformational transitions many more of which will be studied later Nucleation and Growth A Liquid Drop in a SuperCooled Gas Nucleation is localized formation of a thermodynamic phase in a distinct phase There are numerous examples in nature they include ice formation supercooling within body fluids selforganizing and growth process of molecular clusters and protein aggregates Here we include a simple case of nucleation and growth of a liquid drop in a supercooled gas A gas supercooled below its vaporization temperature is in a metastable state giving way to a more stable equilibrium phase that is a liquid In the process of condensation phase transition of the whole system into a liquid a droplet of liquid spontaneously nucleates and grows in the supercooled gas Because the gas and liquid are free to exchange the molecules and energy both of chemical potential and temperature are equal in each phase that is uniform throughout the entire system The pressure in each phase however cannot be same if the effects of interface are included Because the chemical potential as well as the temperature and total volume are given as fixed we choose as the primary thermodynamic potential the grand potential Ω pgVg plVl γA 253 where pg Vg and pl Vl are the pressures and volumes of the gas and liquid phases respectively γ is surface tension in the interfacial area A To minimize the surface contribution γA the liquid drop should reduce its surface area to the least possible value and thus become spherical The grand potential change associated with formation of a spherical drop with the varying radius r is Fig 27 The grand potential ΔΩr of forming a spherical droplet of radius r in a supercooled gas ΔΩr 4πr³3 Δp 4πr²γ 254 where we noted that total volume Vg Vl remains constant With the fact that the liquid pressure is higher Δp pl pg 0 the profile of ΔΩr is depicted by Fig 27 The mechanical equilibrium between the surface tension and volume pressure is reached when Ωr 0 namely r rc where rc 2γΔp 255 This is called the YoungLaplace equation But the above is an unstable equilibrium condition at the critical radius rc the grand potential is at the maximum Fig 27 to reduce ΔΩ the droplet will either shrink and vanish leading to a metastable gas phase r 0 or will grow to infinity transforming the entire system into the liquid phase For the nucleus to grow beyond rc the energy barrier of the amount ΔΩc 16πγ³3Δp³ 256 must be overcome Ubiquitous thermal fluctuations however enable the nucleus to cross over the barrier and the metastable supercooled gas to transform to a liquid This model of nucleation and growth can be applied to a host of the first phase transitions eg condensation of vapor into liquid including cloud formation phase separations and crystallizations P27 As another example consider the pore growth in a membrane For a circular pore of radius r to form in a planar membrane it costs a rim energy 2πrλ while losing the surface energy πr²γ Discuss how the pore growth and stability depend on the line and surface tensions λ and γ Further Reading and References Many textbooks on thermodynamics have been written To name a few AB Pippard Elements of Classical Thermodynamics Cambridge University Press 1957 HB Callen Thermodynamics and an Introduction to Thermostatistics 2nd edn Paper back Wiley 1985 EA Guggenheim Thermodynamics An Advanced Treatment For Chemists And Physicists 8th edn North Holland 1986 W Greiner L Neise H Stokër Thermodynamics and Statistical Mechanics Springer 1995 D Kondepudi I Prigogine Modern Thermodynamics From Heat Engine to Dissipative Structures Wiley 1985 DT Haynie Biological Thermodynamics Cambridge University Press 2001 GG Hammes Thermodynamics and Kinetics for Biological Sciences Wiley 2000 Many textbooks on statistical physics include chapters on thermodynamics Further Reading and References 23 Chapter 3 Basic Methods of Equilibrium Statistical Mechanics In principle the macroscopic including thermodynamic properties of matter ultimately derive from the underlying microscopic structures Because the exact mechanics for a huge number of constituent particles is out of question one is forced to seek statistical methods The fundamental idea of statistical mechanics starts from the notion that an observed macroscopic property is the outcome of averaging over many underlying microscopic states For a microcanonical ensemble of an isolated system at equilibrium we show how the entropy is obtained from information on the microstates or from the probabilities offinding the microstates Once the entropy is given the first order thermodynamic variables are obtained by taking derivatives of it with respect to their conjugate thermodynamic variables as shown in Chap 2 We then consider the microstates in canonical and grand ensembles of the system which can exchange energy and matter with the surrounding kept at a constant temperature From the probability of each microstate and the primary thermodynamic potentials for the ensembles all the macroscopic properties are calculated Statistical mechanics also allows us to obtain the information on the fluctuations of observed properties about the averages which providesdeeperunderstanding ofthe structuresof matter The standard ensemble theories of equilibrium statistical mechanics will be outlined in this chapter In applying such methods to biological systems we face a shift of its old paradigm of relating the macroscopic properties to the microscopic structures Unlike ideal and simple interacting systems covered in typical statistical mechanics text books biological systems are too complex to be explained directly in terms of the small molecules or other atoministic structures Nevertheless the structures and properties can be observed on nanoscales thanks to various singlemolecule experimental methods which are now available Certain nanoscale subunits or even larger units rather than small molecules can emerge as the basic constituents Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158413 25 and properties Throughout this chapter we demonstrate the applicability of statistical mechanics for numerous mesoscopic biological models involving these subunits 31 Boltzmanns Entropy and Probability Microcanonical Ensemble Theory for Thermodynamics 311 Microstates and Entropy A macrostate of a macroscopic system at equilibrium is described by a few thermodynamic state variables We consider here an isolated system with specified macrovariables namely its internal energy E its number of particles N and generalized displacement Xi such as its volume see Table 21 for the definitions The number N is usually very large for the system consisting of one mole of gas the number of molecules N is the Avogadro number NA 6022 x 1023 and is often taken to be infinity thermodynamic limit in macroscopic systems Many different microstates underlie a given macrostate The set of microstates under a macrostate specified by these variables E N Xi constitutes the microcanonical ensemble For illustration consider a onemole classical gas that is isolated with its net energy E and enclosing volume V Microscopic states of the classical gas are specified by the positions and momenta of all N particles There are huge virtually infinite number of ways microstates that the particles can assume their positions and momenta without changing the values of E N V of the macrostate Each of these huge number of microstates constitutes a member of the microcanonical ensemble Suppose that the number of microstates also called the multiplicity belonging to this ensemble is WE N Xi Then the central postulate of statistical mechanics is that each microstate M within this ensemble is equally probable PM 1 WE N Xi 31 This equalapriori probability is the leastbiased estimate under the constraints of fixed total energy This very plausible postulate is associated with another fundamental equation that relates the macroscopic properties with the microscopic information the socalled Boltzmann formula for entropy SE N Xi kB ln WE N Xi 32 where kB 138 x 1023 J K 138 x 1018 erg K is the Boltzmann constant Equation 32 is the famous equation inscribed on Boltzmanns gravestone in Vienna Fig 31 and is regarded as the cornerstone of statistical mechanics It proclaims that the entropy is a measure of disorder S 0 at the most ordered Fig 31 The gravestone of Ludwig Boltzmann in Vienna where the famous formula S kB ln W is inscribed state where only one microstate is accessible W 1 the irreversible approach to an entropy maximum is due to emergence of most numerous microstates ie most disordered state which is attained at equilibrium Furthermore it tells us that once W is given in terms of the independent variables E N and Xi all the thermodynamic variables can be generated by SE N Xi using 29211 For an alternative useful expression for the entropy imagine that M virtually infinite replicas exist of the system in question Suppose that the number of replicas that are in a microstate state i is ni Then the number of ways to arrange n1 systems to be in state 1 and n2 systems to be in state 2 etc is WM M n1 n2 33 Consider that the values of ni are so large that the Stirling approximation ln n ni ln ni ni is valid Then by noting that Σi ni M ln WM M ln M M Σini ln ni ni Σi ni ln ni M 34 and the entropy of the system is given by the total entropy of the M replicas divided by M S 1 M kB ln WM kB Σi Pi ln Pi kB ΣM PM ln PM 35 where Pi ni M is the probability of finding ni replicas out of M This entropy is expressed in terms of Pi which can be interpreted as the probability PM for microstate M of a single replica system It is in the form of the information entropy SI K Σi Pi ln Pi introduced by Shannon where here with K kB In the microcanonical ensemble of the system in which P ℳ f g ¼ 1W the entropy is indeed given by S ¼ kB ln W Within the information theory the probability and thermodynamic entropy at equilibrium are the outcomes of maximization of the information entropy Shannon 1948 Jaynes 1957 P31 Show that the probability distribution that maximizes the entropy 35 under a constraint P ℳ P ℳ f g ¼ 1 is the microcanonical probability P ℳ f g ¼ 1W 31 Use the method of Lagranges multiplier 312 Microcanonical Ensemble Enumeration of Microstates and Thermodynamics The designation of microstates depends on the level of the description chosen Let us consider a system composed N interacting molecules In the most micro scopic level of the description where the system is described quantum mechani cally involving molecules and their subunits such as atoms and electrons the microstates are the quantum states labeled by a simultaneously measurable set of quantum numbers of the system which are virtually infinite At the classical level of description the microscopic states are specified by the Nparticle phase space ie the momenta and coordinates of all the molecules as well as their internal degrees of freedom For both of these cases enumeration of total number W of microstates in a microcanonical ensemble would be a formidable task Example TwoState Model In many interesting situations however the description of the system need not be expressed in terms of the underlying quantum states or phase space Consider a system that has N distinguishable subunits each of which can be in one of two states A simple example is a linear array of N sites each of which is either in the state 1 or 0 Fig 32a Such twostate situations occur often in mesoscopic systems that lie between microscopic and macroscopic domains The two state model not only allows the analytical calculation although seemingly quite simple it can be applied to many different interesting problems of biological significance Of par ticular interest are biological systems that consist of nanoscale subunits for example Fig 32b the specific sites in a biopolymer where proteins can bind via selective and noncovalent interactions and c the basepairs in doublestranded DNA that can close or open Now let us consider as our microcanonical system an array of N such subunits eg a biopolymer with N binding sites or a Nbase DNA each of which has two states with different energies For simplicity we neglect the interaction between subunits Due to thermal agitations the subunits undergo incessant transitions from 28 3 Basic Methods of Equilibrium Statistical Mechanics Fig 32 Two state problems a Linear lattice with each site that is either in the state 1 or 0 Two biological examples of two state subunits b Sites in a biopolymer double stranded DNA for this case bound by a protein or not The binding sites are marked dark c DNA with base pairs in closed slashed and open looped states an energy state to the other What is the entropy of the array and what is the probability at which each state occurs in a subunit The M here are chosen to be the mesoscopic states represented by a set ni n1 n2 nN where ni is the occupation number of the ith subunit ni is either 0 or 1 depending on whether the subunit is unbound or bound with the energy ε0 or ε1 respectively The net energy is E Σi1 to N 1 ni ε0 ni ε1 N0 ε0 N1 ε1 36 where N0 and N1 N N0 are the number of subunits belong to the energy states ε0 and ε1 respectively Because E is determined once N0 and N1 are given WE N of the total microstates in a microcanonical ensemble that is subject to the net energy E and total number N is the number of ways to divide N sites into two groups N0 unbound sites and N1 bound sites W N N0 N1 37 Following Boltzmann the entropy on this level of description is expressed as S kB ln W kB N ln N N N0 ln N0 N0 N1 ln N1 N1 NkB N0 N ln N0 N N1 N ln N1 N 38 where the Stirlings formula N N ln N N is used assuming that N0 N1 and N are large numbers Note that in microcanical ensemble theory the primary thermodynamic potential S should be expressed as a function of the given independent variables N and E expressing N0 and N1 in terms of N and E yields N0 Nε1 EΔε and N1 E Nε0Δε where Δε ε1 ε0 SE N kB Nε1 EΔε ln Nε1 ENΔε E Nε0Δε ln E Nε0NΔε 39 Using 29 the temperature is expressed as 1T SENxi kBΔε ln Nε1 EE Nε0 310 from which we can express the internal energy E in terms of temperature T E Nε0 eβ ε0 ε1 eβ ε1eβ ε0 eβ ε1 311 where β 1kB T The probability that a subunit will be in the state n 0 is P0 WN1 Eε0WN E N1N01 N1 N0 N1N N0N 312 where the equala priori probability 1WN E 31 of finding any one of the subunit with ε0 is multiplied by WN1 Eε0 which is the number of ways that the remaining energy can be distributed among the other N1 subunits The result 312 is very obvious In a similar way one can find P1 N1N 313 Substituting the expression for E 311 into N0 Nε1 EΔε and N1 E Nε0Δε yields Pn NnN eβ εnn01 eβ εn n01 314 This is the singlesubunit Boltzmann distribution It signifies that the higher energy state is less probable unless excited by very high thermal energy kB T β1 Each probability can be rewritten explicitly as P0 11 eβ Δε 315 P1 eβ Δε1 eβ Δε 11 eβ Δε 1 P0 316 The relative probability of finding state 1 relative to state 0 is eβ Δε If we put the unbound and bound state energies of the subunit to be 0 and ε respectively the probability of the bound state is given by P1 11 eβ ε Sβ ε 317 Sβ ε called the sigmoid function Fig 33 is typical of the transition probability in twolevel systems When ε0 P0 P1 12 ie the open and closed states are equally probable When ε kB T P1 1 ie a site or base pair tends to be mostly bound If there were attraction between subunits P1 rises more sharply at a given temperature than the sigmoid This cooperative binding will be studied in detail in Chap 8 in the context of DNA basepair opening or denaturation In terms of single subunit probability 314 the energy 311 is expressed by E N Σn01 εn Pn 318 meaning that the internal energy is given by the thermal average The entropy 38 is expressed as S N kB Σn01 Pn ln Pn 319 Fig 33 The sigmoid function Sβ ε For low temperature β ε 1 the function rises sharply at ε0 and becomes unity for large ε For a biopolymer with N binding sites bound by Np N proteins the entropy is S N kB θ ln θ 1θ ln1θ 320 where θ NpN is coverage of the proteins This is the wellknown entropy of mixing two components When only one state exists θ1 or θ0 then the entropy of mixing is 0 When the two states are equally probable ie θ12 the entropy is at the maximum P32 Show that the Helmholtz free energy is given by F N kB T lneβ ε0 eβ ε1 P33 Find the chemical potential of the system Solution Because the primary thermodynamic potential is SE N X the chemical potential is given by μ T SNEX as a function of E and N If we obtain it by taking a derivative on 320 with respect to N it would be wrong because the entropy is not explicitly expressed as a function of the independent variables E and N The twostate model can be applied to a host of biological transitions between two states such as coiled and helix states BDNA right handed and ZDNA lefthanded states in addition to the examples mentioned above The model can be applied even to the higher levels biological phenomena such as the ion channel gating transitions from an open to a closed state ligand binding on receptors and much more Colloid Translocation As another example of the two state transitions consider translocation of colloidal particles from one place to the other Consider identical colloidal particles Fig 34 initially confined within the chamber on the left pass through a narrow pore in the partitioning membrane toward the right chamber Suppose that the internal energy does not change during this translocation process The number of microstates with N1 particles translocated to the right is given by WN N1 N1 N N N1 N1 321 Fig 34 Colloidal particles translocating from a chamber to another through a pore beween them The probability with which N1 particles exist in the right chamber is given by PN1 WN N1 N1 N10N WN N1 N1 WN N1 N1 2N where we use N10N WN N1 N1 N10N N N N1N1 2N PN1 is the binomial distribution for N1 shown by Fig 35a The average is N1 N10N N1PN1 N2 and the variance is N1 N12 N10N N1 N12 PN1 N4 Fig 35 a The probability distribution PN1 of number of particles that translocate to the right side b The entropy associated with translocation SN1 For large N PN1 or WN N1 N1 shows a sharp peak at N1 N 2 Fig 35a because root mean squared rms deviation or standard deviation of N1 ΔN1 N1 N1212 N12 2 is much smaller than N This means that in real situations of large N this sharplypeaked state with N1 N 2 dominates over all other possibilities as is observed at equilibrium Thermodynamically this is the equilibrium state where the entropy S kB ln W kBN ln N N N1 ln N N1 N1 ln N1 has the maximum kB N ln 2 This means that the second law of thermodynamics forbid all the particles initially placed on the left to translocate toward the right even in infinitely long time We have demonstrated that the basic postulates of equala priori probability and Boltzmann entropy lead to a clear and satisfactory construction of a statistical mechanical method for finding statistical and thermodynamic properties The results derived above the thermodynamics and probabilities are obtained for the microcanonical ensemble of isolated systems in which the total energy and total number are regarded as fixed Despite these constraints these microcanonical ensemble theory results are equal to those for the natural situations where these variables fluctuate provided that the standard deviations or root mean squares of the fluctuations are much smaller than their averages As we will show next thermodynamic variables can be calculated more easily by considering ensembles in which the constraints on fixed variables E Xi and N are relaxed 32 Canonical Ensemble Theory Due to the constraints of fixed total energy E and total number of particles or subunits N the number of available microstates in a microcanonical ensemble is difficult to calculate for the most of nontrivial systems In what is called a canonical ensemble the constraint is relaxed by considering that the system in question is put into a heat reservoir or bath of size much larger than the system size at a fixed temperature so that the macrostate is characterized by its temperature T instead of its energy E and by N and X in addition The systems energy by exchange with the reservoir can take any of the accessible energy values Fig 36 The canonical ensemble is the collection of many microstates of a macrosystem characterized by its temperature T N and Xi To retain the temperature as fixed the system is put into a contact with a heat bath of the same temperature energy ET as depicted in Figs 24 and 36 Then the number of all the accessible microstates in the total system is WTET M WEM WBET EM where M signifies the summation over all accessible microstates of the system each having the energy EM WEM is its number of microstates WBET EM is the number of the microstates of the heat bath given that the system has the energy EM In the each one of the microstates counted in 326 is equally probable a priori by the postulate 31 so that the probability that the system will be in a specific state M WEM 1 is PM WBET EM M WBET EM To go further we note that WBET EM exp 1 kB SBET EM and the systems energy EM is much smaller than the total energy ET or the reservoir energy ET EM Consequently the exponent above is expanded as exp 1 kB SBET EM exp 1 kB SBET EM ET SBET exp 1 kB SBET EM T where the relation 29 SBEE 1 T is used From 327329 we find an important relation PM eβEM ΣM eβEM where still β 1kB T This relation means that the probability of finding a system at a temperature T to be in a microstate M depends solely on the systems energy EM and decays exponentially with it following the socalled Boltzmann factor eβEM This canonical distribution is valid to the system in equilibrium at a fixed temperature T independently of its size It should be noted that the system need not be large enough to assure its statistical independence from the thermal bath as wrongly claimed in some textbooks This fundamental relation can be derived in various ways One way is by maximizing the information entropy under constraints as given by the following problem P34 By maximizing the information entropy 35 S kB ΣM PM ln PM subject to constraints ΣM PM 1 and ΣM EM PM E find that PM is given by the canonical distribution 330 Use the method of Lagranges multiplier EM being a fluctuating energy that depends on the microstates or degrees of freedom M is identified as the Hamiltonian HM Thus we express the probability in a more conventional form PM eβHM ZT N Xi The normalization factor ZT N X ΣM eβHM is called the canonical partition function or partition sum Including the multiplicity WEM of states that have energy EM the partition function is also given as ZT N X ΣEM WEM eβEM Thus the probability for the systems to have the energy EM is proportional to WEM eβEM not to the Boltzmann factor eβEM which refers to the probability for the system to be at a microstate M Given the probability various thermodynamic variables of the system can be obtained First the internal energy is the average energy of the system given by E H ΣM HM eβHM Z ΣM eβHM Zβ ln Z β Using the relation E T2 FTT 216 we can identify the Helmholtz free energy FT N X kB T ln Z In this way by using the thermodynamic relations involving the derivatives with respect to F 213215 the partition function can generate all the thermodynamic variables P35 Consider a simple model where DNA unbinding of the double helix is like unzipping of a zipper a base pair bp can open if all bps to its left are already open as shown in the figure below The DNA has N bps each of which can be in one of two states an open state with the energy 0 and closed state with the energy ε a Find the partition function b Find the average number of open bps when ε 04 kB T 322 The Energy Fluctuations The energy distribution of macroscopic systems in canonical ensemble is a sharp Gaussian around the average energy To show this consider that values of the microstate energy E are continuously distributed with density of states wE over a range dE so that the partition function 333 can be written as Z dE wE eβE which implies that probability distribution of the energy within the range dE is PE wE eβE Z eβE T SE Z eβFE Z where FE E T SE E kB T ln wE is the free energy given as a function of an energy E Because eβT SE increases and eβE decreases with E we expect that PE is peaked at E where FE is minimum Around the minimum FE can be expanded FE E T SE 12 T 2 SE E2 E E2 FE 1 2 T CV E E2 In the above we used 2 SE E2 E 1T 1 T2 CV along with SE E 1T and T E 1 E T 1 CV 238 Finally we obtain PE eβFE exp 1 2 kB T2 CV E E2 The probability distribution for the energy E which is allowed to exchange with the bath at temperature T is Gaussian with a mean E E E and a rms deviation ΔE E E212 TkB CV from the mean The energy distribution PE is peaked at the mean E which minimizes he free energy FE to FE F Because E and CV are extensive quantities that increase with system size N the relative peak width ΔE E scales as N12 Therefore on a macroscopic scale PE is very sharp and looks like a delta function about the mean PE δE E Fig 37 For this reason when measuring the energy E of a macroscopic system we observe negligible fluctuations about the mean which as the outstandingly probable outcome Because energy fluctuation is practically absent in this case the canonical ensemble yields the same thermodynamics that the microcanonical one does Fig 37 The distribution of the energy E in a macroscopic system is sharply peaked around the average energy E E Even a macroscopic system experiences the energy fluctuation ΔE although very small compared with E An important lesson here however is that even the macroscopic variables fluctuate although imperceptibly The fluctuations are consequence of the intrinsic universal thermal motion of microscopic constituents inherent in systems at a nonvanishing temperature The relative effect of the fluctuations increases as the system size decreases as dramatically visualized in Brownian motion The canonical ensemble results could differ significantly from the microcanonical results as the system size gets small Therefore when considering mesoscopic systems of small system sizes an appropriate type of ensembles must be chosen carefully to meet the actual situation Water has a distinctively high heat capacity so that its temperatures remain nearly constant For biological systems bathed in an aqueous solvent the canonical ensemble including the Gibbs and grand canonical ones shown next are a most natural choice to take 323 Example TwoState Model As a simple example we revisit the twostate model of independent N subunits that was studied earlier in a microcanonical way The Hamiltonian is derived from 36 Hni 1niε0niε1 341 i1 N where ni the occupation number of the ith subunit can be either 0 or 1 The probability of the microstate that is the joint probability that all subunits are in the state n1 n2 nN simultaneously is given by Pni expβHni Z Z1 expβ 1niε0 niε1 342 i1 N where Z expβHni expβ 1niε0 niε1 343 ni ni0 1 i1 N expβ1 niε0 niε1 N i1 ni0 1 eβε0 eβε1N is the partition function In deriving it the two summations in the second expression above was exchangeable The binomial expansion of 343 expresses the partition function as Z N eβε0N0 ε1N1 344 N10 N0N1 where N0 N1 are the numbers of empty and occupied subunits respectively NN0N1 represents the number of microstates for the state that has net energy ε0N0ε1N1 37 The 342 implies the obvious statistical independence of subunits Pni Pn1 Pn2 PnN 345 where Pni eβ1niε0 niε1 eβεn 346 n0 is the probability for the subunit to be in the state ni this is identical to 314 The calculation of thermodynamic variables is straightforward The Helmholtz free energy is F kBT ln Z NkBT lneβε0 eβε1 347 which is obtained in a more straightforward way compared with the microcanonical theory From the free energy we obtain the entropy ST N FT NkB lneβε0 eβε1 Nε0eβε0 ε1eβε1 Teβε0 eβε1 348 F T E T and the internal energy E Nε0eβε0 ε1eβε1 eβε0 eβε1 349 which can be also directly derived from 334 All of thermodynamic quantities derived coincide with those of the microcanonical ensemble which is no surprise because we considered the thermodynamic limit of large numbers using the Stirlings formula in microcanonical calculations P36 Referring to the problem of colloid translocation if each particle loses energy by E when passing through the pore to the right at what configuration is the probability maximum Find the probability that N1 particles are on the right while N2 N N1 particles are on the left and the associated entropy 32 Canonical Ensemble Theory 41 As the name implies canonical ensemble theory provides the most standard method by which the microstate probabilties and the thermal properties are evaluated In later chapters it will be used to study diverse systems ranging from small molecular fluids to polymers and membranes and to study a multitude of phenomena such as transitions cooperative phenomena and selfassembly Although versatile the canonical ensemble condition of fixed X and N can make analytical calculations difficult in some situations In the following we consider other ensembles where one of the two variables is free to fluctuate 33 The Gibbs Canonical Ensemble Now a system in contact with a thermal bath is subject to a generalized force fi which is kept at constant so that the systems Hamiltonian is modified to HgM HM fiXiM 350 Here the generalized displacement XiM the conjugate to the force fi is a thermally fluctuating variable The system is specified by the macroscopic variables T fi N and the underlying microstates constitute the so called Gibbs canonical ensemble The microstate M occurs with the canonical probability PM eβHgM Zg T fi N eβHM βfiXiM Zg T fi N 351 where Zg T fi N eβHM βfiXiM 352 M is the Gibbs partition function Examples are a magnet subject to a constant magnetic field and a polymer chain subject to a constant force which is discussed below The average displacement in this ensemble is given by Xi XiM M XiM eβHM βfiXiM M eβHM βfiXiM Zg βfi Zg kBT fi ln Zg T fi N 353 In view of the thermodynamic identity Xi fi G 223 the Gibbs free energy is identified as GT fi N kB T ln ZgT fi N 354 from which all the thermodynamic variables are generated as explained in Chap 2 FreelyJointed Chain FJC for a Polymer Under a Tension A simple model for a flexible polymer is the freelyjoined chain FJC consisting of N segments each with length l which can rotate by an arbitrary angle independently of each other Fig 38 How much is the chain stretched on average by an applied tension Due to the thermal agitation of the heat bath in the absence of the applied tension the freely jointed chain segments are randomly oriented and thus the corresponding chain Hamiltonian H does not depend on the segment orientation ie is trivial In the presence of an applied tension f acting on an end rN with the other end held fixed at the origin r0 the Hamiltonian is given by Hg M f rN f Xi M f sumn1N l un f sumn1N l cos thetan 355 The microstates of the FJC here is M u1 u2 uN where un is the unit tangent vector of the nth segment oriented with polar angle thetan along the axis of the applied tension The partition function is ZgT f N int dOmega1 int dOmegaN ebeta f sumn l cos thetan int dOmegan ebeta f l cos thetan N 4 pi sinhbeta f l beta f l N 356 Here Omegan is the solid angle of the nth segment with respect to the direction of the force int dOmegan int11 d cos thetan int02 pi d phin where phin is the azimuthal angle Using 353 the average value X of the endtoend distance of the chain along the axis X sumn l cos thetan is given by Fig 38 A freelyjointed chain extended to a distance X under a tension f XN l cothbeta f l 1beta f l Lbeta f l 357 where Lx is the Langevins function Now we ask ourselves the inverse question what is the tension f necessary to keep the endtoend distance as X Because X is given as fixed and f is a derived quantity this problem in principle should be tackled by the canonical ensemb e theory However it is quite complicated to impose the constraint of fixed extension X in the analytical calculation Because the forceextension relation for a long chain is independent of the ensemble taken the 357 provides the solution with interpretation f as the derived average tension which is written as the inverse of the Langevins function f kB T l L1 XN l 358 and is depicted by Fig 39 Let us first consider the case of small force beta f l 1 or f kB Tl Because Lbeta f l beta f l 3 357 leads to XN l f l 3 kB T 359 which one can alternatively put as f 3 kB T N l2 X where f is the force necessary to fix the chain extension as X This is the wellknown Gaussian chain behavior 1020 where the force is linear in the extension the domain within the broken ellipse in Fig 39 Its temperature dependence implies that it is an entropic force the restoring force f is directed towards the origin X0 where the entropy is the maximum Next we consider the opposite extreme where f kB Tl Because cotbeta f l 1 XN l 1 1beta f l in 357 from which one obtains the entropic force to keep an extension X f kB Tl 1 XN l1 360 Fig 39 Tension f necessary to keep the extension as X in a freelyjointed chain The tension is entirely the entropic force An infinite force is required to extend the chain to its full length N l at which the chain entropy is zero P37 What are the Gibbs and Helmholtz free energies for the chain extended with the tension f and the distance X for the case f kB Tl Solution Because f partial Fpartial X we integrate the 360 over X to find the Helmholtz free energy FX T N N kB T ln 1 XNl where the irrelvant constant is omitted On the other hand the Gibbs free energy is Gf T N FX T N fX N f l kB T lnfl kB T where F and X are expressed as functions of f Alternatively G is directly obtained from the partition function expression Gf T N N kB T ln ebeta f l ebeta f l beta f l N kB T ln ebeta f lbeta f l P38 A biopolymer is composed of N monomers each of which can assume two conformational states of energy epsilon1 and epsilon2 and coressponding segmental extension lengths l1 and l2 respectively Calculate the partition function When a tension f is applied to the both ends what would be the extension X 34 Grand Canonical Ensemble Theory When a system is in contact with a thermal bath its number of particles can fluctuate naturally as its energy does Because the system is at equilibrium with the bath the temperature and chemical potential of system are the same as those of the bath The microstates of the system compatible with this macrostate of given temperature T chemical potential μ and displacement X constitute the grand canonical ensemble Fig 310 Fig 310 The grand canonical ensemble of a system is characterized by its temperature T chemical potential μ and displacement Xi To retain the temperature and chemical potential as fixed the system is put into a contact with a heat bath of the same temperature and chemical potential 341 Grand Canonical Distribution and Thermodynamics The distribution of an underlying microstate M of the system with the energy HM and particle number N is derived using logic similar to that for the canonical ensemble PM eβHMμN ZGTμXi 361 where ZGTμXi ΣM eβHMμNM ΣN0 ΣMN eβHMμN ΣN0 eβμN ZN 362 is the grand canonical partition function Here ΣMN is the summation over the microstates of the system with N given of which the canonical partition function is ZN The average number of particles in the system is given as N N ΣM NM eβHMμNM ΣM eβHMμNM ZGZG βμ 363 The grand canonical ensemble theory is useful for systems in which the number of particles varies ie for open systems The fluctuation in the number of particles in the system about the mean N N is ΔN2 N2 N2 2 ZG ZG βμ2 ZG ZG βμ2 2 ln ZG βμ2 N β μ 364 where 363 is used Because N β μ is an extensive quantity the rms deviation ΔN ΔN212 scales as N12 Consider that N is very large Then one can show the distribution over the number of the particles is very sharp Gaussian around N N which dominates the partition sum ZGT μ Xi ΣN0 eβμN ZN C eβμN ZN 365 where C is a constant independent of N This domination allows the grand potential to be given by ΩT μ Xi kB T ln ZGT μ Xi μN F 366 Starting from this thermodynamic potential the average number entropy and entropic force are generated as given earlier 228231 S ΩT N Ωμ fi ΩXi Ω Xi The fluctuation of particle number given by N μ 364 can be related to mechanical susceptibility of the system eg isothermal compressibility of the system To see this we note that N dμ Xi dfi 225 for an isothermal change so N μ NTXi Xi fi NTXi 367 Consider the right hand side of the above equation for the fluid systems where Xi V and fi p In view of p pT n N V V p NTV V2 N p VTN 1 n kT 239 Therefore 364 leads to the relative fluctuation for the number ΔN N n kB T KT12 N12 368 which evidently tells us that the isothermal compressibility KT is always positive and further that the relative fluctuation is negligible for a system with large N But for mesoscopic systems the relative fluctuation can be quite sizable The relation 368 can be applied to for example a membrane in equilibrium with its lipids in a solution If the stretching modulus Ks 1213 corresponding to the inverse of the mechanical susceptibility is quite small then the number N of lipids in a membrane with its average N being not very large can show large relative fluctuations Fig 311 The configurations of ligand binding on two sites of a protein that contribute to the grand canonical partition function expressed in 369 342 Ligand Binding on Proteins with Interaction As an example to show the utility of the grand canonical ensemble theory we consider systems of molecules or ligands such as O2 that can bind on two identical but distinguishable sites in a protein eg myoglobin hemoglobin Fig 311 How does the average number of bound ligands depend on their ambient concentrations Compared with a similar problem of twostate molecular binding treated in Sect 31 there is an important difference earlier the system of interest was a biopolymer with fixed N binding sites whereas the system in question here is the bound ligands whose number N can vary In this case the grand partition function is expressed as ZG ΣN0 eβμN ZN Z000 z Z110 z Z101 z2 Z211 369 where Zmnmn is the canonical partition function with m and n ligands bound on two sites and z eβμ is the fugacity of a ligand If the energy in the bound state is ε 0 and the interaction energy is φ Z000 1 Z110 Z101 eβε and Z211 eβ2εφ so ZG is given as ZG 1 2 z eβε z2 eβ2εφ 370 Using 363 the coverage per site is θ 12 N 12 z z ln ZG z eβε z eβ2εφ 1 2 z eβε z2 eβ2εφ 371 If φ 0 so that two sites are independent of each other the coverage is θ z eβε 1 z eβε 1 eβεμ 1 372 To find μ consider that at equilibrium the chemical potential of the bound particles is the same as that of the unbound particles in the bath Because the unbound particles form an ideal gas or solution with density n their chemical potential is given by μ μ0T kB T lnn n0T 373 as will be shown in next chapter μ0T is the chemical potential of the gas at the standard density n0T Equating the chemical potentials we obtain Fig 312 Ligand binding isotherm The coverage θ increases with the ambient density n at a given temperature The attraction φ 0 between the bound particles enhances the coverage θ over that of the Langmuir isotherm solid curve The repulsion φ 0 lowers the coverage θ 11 n₀n eβε µ₀ nn nT 374 where nT n₀T eβε µ₀T 375 is purely a temperaturedependent reference density The Langmuir isotherm solid curve in Fig 312 shows how the coverage increases as the background density or concentration n increases at a temperature nT is the crossover concentration at which the coverage is 12 If the bound particles interact 371 can be written as θ nn nT n φ 376 For an attractive interaction such that eβφ 1 θ is higher and thus n is less than that for the Langmuir isotherm Fig 312 because of the attraction binding is enhanced On the other hand when the interaction is repulsive φ 0 the binding is reduced These interesting effects due to the interaction are called the cooperativity P39 Find the rms fluctuation in coverage How are they affected by the interaction between the binding ligands Further Reading and References The original references for the Shannon Entropy and the Information theory are C Shannon A mathematical theory of communication Bell Syst Tech J 27 379423 1948 ET Jaynes Information theory and statistical mechanics Phys Rev Series II 1064 620630 1957 For more details on basic concepts on ensemble theories see standard textbooks on statistical mechanics in graduate level as exemplified below M Kardar Statistical Physics of Particles Cambridge University Press 2007 W Greiner L Neise H Stocker Thermodynamics and Statistical Mechanics Springer 1995 RK Pathria Statistical Mechanics 2nd edn ButterworthHeinemann 1996 M Plischke B Bergersen Equilibrium Statistical Physics 2nd edn Prentice Hall 1994 K Huang Statistical Mechanics 2nd edn Wiley 1987 DA McQuarrie Statistical Mechanics Universal Science Books 2000 L Reichl A Modern Course in Statistical Physics 2nd edn WileyInterscience 1998 M Toda R Kubo N Saito Statistical Physics I Equilibrium Statistical Mechanics Springer 1983 GF Mazenko Equilibrium Statistical Mechanics Wiley 2001 G Morandi E Ercolessi F Napoli Statistical Mechanics An Intermediate Course 2nd edn World Scientific 2001 Further Reading and References 49 Chapter 4 Statistical Mechanics of Fluids and Solutions Biological components function often in watery environments Biological fluids are either water solvent or various aqueous solutions and suspensions of ions and macromolecules with which virtually all chapters of this book are concerned In this chapter we start with a review of how the canonical ensemble method of statistical mechanics can be used to derive some basic properties of simple classical fluids that consist of small molecules We derive the wellknown thermodynamic properties of noninteracting gases either in the absence or in the presence of external forces For dilute and nondilute fluids we study how the interparticle interactions give rise to the spatial correlations in the fluids which affects the thermodynamic behaviors These results which are essential for a simple fluid for its own can be extended to aqueous solutions of colloids and macromolecules eg the results of dilute simple gas can be directly applied to dilute solutions We outline coarsegrained descriptions in which the solutions are treated as the fluids of solutes undergoing the solventaveraged effective interactions As a particularly simple but useful variation we shall introduce the lattice model 41 PhaseSpace Description of Fluids 411 N Particle Distribution Function and Partition Function Consider a simple fluid consisting of N identical classical particles of mass m each with no internal degrees of freedom The fluid is confined in a rectangular volume V with sides Lx Ly Lz and kept at a temperature T For a classical but microscopic description the microstate M of the system is specified by a point in 6N dimensional phase space Γ p₁ r₁ pᵢ rᵢ pN rN pᵢ rᵢ where pᵢ rᵢ are the Springer Nature BV 2018 51 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158414 threedimensional momentum and position vectors of the ith particles The particles are in motion with the Hamiltonian Hpᵢ rᵢ Kpᵢ Urᵢ Φrᵢ 41 Here Kpᵢ ⁿᵢ₁ pᵢ²2m is net kinetic energy of the system Urᵢ ⁿᵢ₁ urᵢ is the net external potential energy where urᵢ is one body potential energy of particle i Φrᵢ ᵢⱼ φrᵢ rⱼ the net interaction potential energy which is the sum of NN 12 pairwise interaction potential energies between particles positioned at rᵢ and rⱼ φrᵢ rⱼ φrᵢⱼ The canonical microstate distribution 331 for this system is the N particle phasespace distribution function Ppᵢ rᵢ 1N1h³ᴺeβHpᵢrᵢZₙ 42 This is the joint probability distribution with which the N particles have their all positions and momenta at p₁ r₁ pᵢ rᵢ pN rN simultaneously The partition function Zₙ is given as the 6Ndimensional integral Zₙ 1N1h³ᴺ dΓ eβHΓ 1N1h³ᴺ dp₁ dr₁ dpN drN eβHpᵢrᵢ 43 Here the Plancks constant h is introduced to enumerate the microstates in phase space The phase space volume for a particle in three dimension is h³ due to the underlying quantum mechanical uncertainty principle that forbids a simultaneous determination of the position and momentum of a particle the 3D Nparticle phase space volume should be divided by h³ᴺ This kind of consideration to appropriately count the number of states depends on the level of the description that defines the states and is not essential for thermodynamic changes as we will see below More importantly the division factor N is inserted to avoid overcounting states of the N identical particles which are indistinguishable with respect to mutual exchanges A close look at the integral whose hyperdimensionality may seem overwhelming allows the factorization Zₙ Zₙ⁰Qₙ 44 where Zₙ⁰ 1NVᴺh³ᴺ dp₁ dpN eβKpᵢ 45 is the partition function of the particles with no mutual interactions and no external fields and QN 1VN dr1 drN eβUri Φri 46 is the configuration partition function that includes the effects of the potential energies Z0N is readily calculated by noting the factorization Z0N 1N h3N VN dpi eβ pi22mN 1N h3N VN 2 m π β3N2 47 where dp eβ p2 2m dpx eβ px2 2m dpy eβ py2 2m dpz eβ pz2 2m 2 m π β32 48 and dp eβ p22m 2 m π β12 The ideal gas partition function Z0N is then written as Z0NT N V 1N V λT3N 49 where λT h2 2 π m kB T12 410 is called the thermal wavelength 412 The MaxwellBoltzmann Distribution From this canonical distribution and partition functions given above the statistical and macroscopic properties of the classical fluids at a temperature can be found in a great variety Let us start with the famous MaxwellBoltzmann distribution for the particle velocity The mean number of particles with the momentum between p1 and p1 d p1 and at the position between r1 and r1 d r1 is given by fp1 r1 d p1 d r1 where fp1 r1 N d p2 d r2 d pN d rN Ppi ri N N h3N ZN d p2 d pN eβ i1N pi2 2m d r2 d rN eβ Uri Φri 411 Here we used 42 and inserted N into the numerator as the number of ways to assign a particle with the subscript 1 Integrating over the momenta yields fp1 r1 Pp1 nr1 412 where Pp1 2 m π β32 eβ p12 2m 413 and nr1 N VN QN d r2 d rN eβ Uri Φri 414 Integrating 412 over r1 yields fp1 N Pp1 415 Therefore fp1 d p1 is the number of molecules that have a momentum between p1 and p1 d p1 and Pp is a particles momentum probability distribution or probability density from which the wellknown MaxwellBoltzmann MB distribution of velocities can be found Φv m3 Pp 2 π m β32 eβ m v2 2 2 π kB T m32 em v2 2 kB T 416 The prefactors ensure the normalizations dp Pp 1 and dv Φv 1 The MB distribution is a Gaussian distribution in velocity Fig 41 and applies universally to thermalized particles at equilibrium Because the phase space distribution 42 is factorized into a momentumdependent part and a positiondependent part the MB distribution is independent of the intermolecular interaction strength and so may also be valid to structured molecules in a liquid phase where their centerofmass translational degrees of freedom are decoupled with the internal degrees of freedom Fig 41 The MaxwellBoltzmann distribution function for xcomponent velocity The most probable velocity is zero Each component of the velocity is statistically independent of every other component Φv Φxvx Φyvy Φzvz 417 where Φαvα 2 π kB T m12 em vα2 2 kB T 418 In the MB distribution the average velocity component is zero vα d vα vα Φαvα 0 419 so is v Also vα2 d vα vα2 Φαvα kB T m 420 so that the average kinetic energy of a particle is 12 m v2 12 m vx2 vy2 vz2 32 kB T 421 It means that each of the three translational degrees of freedom has energy of kB T 2 which is a special case of the equipartition theorem stating more generally that the energy in thermal equilibrium is shared equally among all degrees of freedom that appear quadratically in the total energy Although the average velocity of a particle is zero the average speed is not We note that the probability that the speed has the value between v and vdv is Φv 4 π v2 dv Dv dv which defines the MB speed distribution function Fig 42 Fig 42 The Maxwell Boltzmann speed distribution function curve The most probable speed at temperature Ti is not zero but vp 2kB Ti m12 Dv 4π m 2πkB T32 v2 emv2 2kB T 422 The average speed is then calculated to be v from 0 to vDv dv 8kB T πm12 423 The most probable speed where the probability is the maximum given by the condition dDvdv 0 is vp 2kB T m12 424 The peak of the speed distribution increases as the square root of temperature and the right skew means there an appreciable fraction of molecules have speed is much higher than vp The water molecules that belong to the highspeed tail of the distribution can escape the surface of water because of this removal of highenergy molecules the average speed of the remaining molecules ie their energy temperature decreases Thus evaporation of water alone is cooling process which can be balanced by heat transfer from the environment to retain the water temperature The evaporation process makes rain possible P41 What is the probability that a nitrogen gas molecule on surface of the earth can escape the gravisphere Assume that the temperature throughout is 300 K P42 Suppose that water molecules escape a planar surface of a liquid water if its energy exceeds the average 3kB T 2 Calculate the cooling rate of the liquid Now going back to the 414 nr is recognized as the number density or concentration of the molecules at position r In the absence of all potential energies external and interactional it can be shown to be uniform nr N V n This also holds true for a fluid of particles that are mutually interacting with an isotropic potential φr φr but in the absence of the external potential where the fluid is translationally invariant and homogeneous Below we consider the alternative case in which interaction is absent but external potentials exist to make the fluid nonuniform 42 Fluids of Noninteracting Particles 421 Thermodynamic Variables of Nonuniform Ideal Gases When Φri 0 the configuration partition function 46 reduces to QN q1N 425 where q1 1V dr eβur 426 so 414 becomes nr neβur 427 The nonuniform fluid density follows the Boltzmann distribution For a gas under uniform gravity directed downward along the z axis uz mgz we get nz neβmgz nezz 428 which is none other than the barometric formula It means that thermal agitation allows the gas to overcome gravitational sedimentation It is because the characteristic altitude z kB T mg of the density decay increases with T and decreases with m At T 300 K z of O2 m 32 g mol 532 1026 kg molecule is 795 km and the z of H2 m 2 g mol 332 1027 kg molecule is 127 km this inverse relationship between z and m means that at high altitude light gases are more abundant than heavy gases This prediction is not strictly valid because T and g vary with altitude Also we note that the barometric formula can be applied to sedimentation of colloidal particles suspended in a solvent provided that the mass is modified in such a way to incorporate the buoyancy and hydration For thermodynamic properties the partition function 44 is calculated easily using 49 and 425 ZN 1 N V λ3N q1N 429 The Helmholtz free energy is obtained as FT V N kB T ln ZN kB T N lnV q1 N λ3 1 430 where Stirlings formula is used In the absence of the external potential FT V N kB T N lnV N λT3 1 431 If volume V is taken to be microscopically large enough to contain many molecules but macroscopically very small so that it can be regarded as a point located at r we note that q1 eβur Then the local free energy density in the presence of the potential is given by fr F V kB T nr lnnr λ3 1 nr ur 432 where nr is number density of the nonuniform fluid It is straightforward to obtain the first order thermodynamic variables from the free energy First the pressure of the gas confined in a box of the volume V is given by p F V TN N kB T 1 V V ln q1 433 In the absence of an external force it is reduced to the wellknown ideal gas equation of state p N kB T V n kB T 434 If the external potential is present the pressure ie the force per unit area on the enclosing wall depends on its normal direction and is therefore not isotropic P43 For the gas under a uniform gravity along the zaxis the pressure acting on the wall normal to zaxis is given by pz F Lx Ly LzTN Show that unless mgLz kB T this differs from px and py both of which are N kB T V Considering an infinitesimal volume that encloses the point r we find the local pressure is positiondependent but isotropic pr nr kB T 435 The entropy is given by SN V T F T N kB lnV q1 N λ3 52 N u T 436 where u drur eβur dreβur In the absence of the external potential 436 reduces to SNVT kB N lnV Nλ3 52 kB N lnV N 32 lnT constant The local entropy in the presence of the external potential is sr SV kB nr lnnr λ3 52 In addition the internal energy is obtained as E F TS 32 N kB T Nu The internal energy E is the sum of the average translational kinetic energy 3N kB T 2 and the average potential energy Nu and can be obtained alternatively from the relation E ln ZN β Considering the enclosing volume around the point r to be small we obtain the obvious result for local energy density energy per unit volume er 32 nr kB T nr ur The overall chemical potential is obtained as μ F N kB T lnn λ3 q1 whereas the local chemical potential μr is μr ur kB T k lnnr λ3 The second term is the contribution from the entropy The condition of equilibrium within the fluid μr constant yields the earlierobtained result nr neβur where n is the density at which u 0 The heat capacity at fixed volume is CV E T 32 N kB N T u which indicates that each particle has three translational degrees of freedom that are thermally excited 422 A gas of Polyatomic Moleculesthe Internal Degrees of Freedom A polyatomic molecule consists of two or more nuclei and many electrons In addition to the translational degrees of freedom of the center of the mass the molecule has the internal degrees of freedom arising from rotational vibrational molecular motions and electronic other subatomic motions At room temperature T 300 K two rotational degrees of freedom in diatomic molecule can fully be excited and therefore contribute kB to heat capacity per molecule The partition function of an ideal gas of polyatomic molecules including the internal degrees of the freedom may be written as ZN 1N V q1 λ3 ziTN where ziT is the partition function from the internal degrees of the freedom per molecule In the absence of an external potential the chemical potential is μ F N kB T lnn λ3 ziT kB T lnn λ3 fiT where fi kB T lnziT is the free energy from the internal degrees of freedom in a single molecule In general chemical potential can be written as μ μ₀T kB T lnn n₀T Here the subscript 0 denotes a standard or reference state at which the density and chemical potential are n₀T and μ₀T respectively At the standard state the 2nd term concentrationdependent entropy in 447 vanishes so μ₀T is the intrinsic free energy of a single polyatomic molecule that includes such an extreme as a long chain polymer For solutes the standard density n₀T is usually taken to be 1 mol concentration M which is an Avogadro number Na per 1 L liter P44 Consider a classical ideal gas of N diatomic molecules interacting via harmonic potential φri rj kri rj2 2 Calculate the Helmholtz free energy entropy and heat capacity What is the mean square molecule diameter ri rj212 43 Fluids of Interacting Particles Now we focus on the particles that have no internal structures but have mutual interaction Φri i j φri rj where the interaction potential is isotropic φri rj φri rj φrij Considering the Hamiltonian Hpi ri Kpi Φri the partition function is given by ZN 1 N h3N dp1 dr1 dpN drN eβ i1N pi2 2m i j φrirj ZN0 QN 1 N V λ3N QN where QN 1 VN dr1 drN eβ i j φrirj is the configurational partition function of N interacting particles P45 A lot of biological problems is modelled to be onedimensional for an example protein or ion in motion along DNA As a useful model Möbius et al 2013 consider Tonks gas which is a collection of N particles in the interval 0 x L mutually interacting pairwise through a hard core repulsion φx for x σ and φx 0 for x σ Calculate the configuration partition function QN and the onedimensional pressure acting at an end 431 The Virial ExpansionLow Density Approximation We first consider dilute fluids where the interparticle interactions can be regarded as a perturbation We start by rewriting QN as QN VN dr1 drN i j 1 fij where fij eβ φrij 1 is a function that is appreciable only when rij is within the range of potential which we regard as short For dilute gases the value of fij is small and serves as a perturbation in terms of which we perform expansion ij 1fij 1ij fij ij kl fij fkl 451 We consider the case of a dilute gas in which the first two terms in 451 are included Then QN 1VN dr1drN 1 ij fij 11VN dr1 drN ij fij 452 1N2 2V dr21 f12 where we note the number of interacting pairs is NN12 N22 and dr1 dr2 dr3 drN dr1 dr21 dr3 drN VN1 dr21 This leads to the total partition function and free energy ZN ZN0 1N2 2V dr21 f12 453 FF0 kB T ln1N2 2V dr21 f12 F0 kB T N2 2V dr21 eβϕr12 1 454 F0 kB T N2 V B2 where the superscript 0 denotes the ideal gas part and B2 is the second virial coefficient B2 12 dr21 eβϕr12 1 2π dr r2 eβϕr 1 455 The pressure is obtained by differentiating the free energy with respect to volume pp0 B2 N2 V2 kB T 456 This is the second order approximation of the density or virial expansion for the pressure pkB TnB2 n2 B3 n3 457 where B3 is the third virial coefficient that includes threebody pairwise interactions involving fij fik fjk Likewise the free energy is expanded as below FF0 kB T N2 V B2 kB T2 N3 V2 B3 458 432 The Van der Waals Equation of State We now make an approximation that is useful for nondilute fluids and derive the vander Waals equation by statistical mechanical methods The intermolecular pair potential ϕr can in many cases be separated into two parts a harsh shortrange hardcore repulsion for rσ and a smooth relatively longrange attraction for r σ where σ is the hardcore size or the diameter of molecules A typical example is the LennardJones potential Fig 43 ϕLJ r4εrσ12 rσ6 459 Then the second virial coefficient 455 is expressed as the sum of two integrals each representing the hardrepulsion and softattraction part B2 2π 0σ dr r2 1eβϕr σ dr r2 1eβϕr 460 Fig 43 The LennardJones potential ϕLJ r4εσr12 σr6 In the first integral the exponent eβϕr is negligible for rσ where the potential sharply rises to infinity so that the integral is evaluated as 2πσ33 b For r σ ϕr is a weak attraction effectively so that eβϕr 1βϕr yielding the second integral as akB T where a2π σ r2 ϕr dr 461 Then the second virial coefficient is given as B2 bakB T b1ΘT 462 where the ΘakB b is the parameter called the Boyle temperature If T Θ then B2 0 the repulsion dominates the attraction overall contributing positively to the pressure and free energy If T Θ then B2 0 and the gas behaves ideally For T Θ and B2 0 the attraction dominates the repulsion contributing negatively to them Then we rewrite 456 as pkB T n1bnan2 kB T 463 Although we derived 463 for a dilute gas we seek a way to extend the equation to denser fluids This we do by replacing 1bn by 1bn1 which yields the same pressure at low densities but an infinite pressure as bn 2πnσ33 approaches to 1 characteristic of incompressible liquids The resulting equation is the Van der Waals equation of state pan2 n kB T1bn 464 Although by no means exact this equation is valid for dense gas and even liquids and is useful for explaining the gasliquid phase transition A morejustified way of deriving it without invoking the low density approximation at the outset is the mean field theory MFT In MFT the interactions of all the other particles on a particle is approximated by a onebody external potential called a mean field thus reducing a manybody problem to a onebody problem That is a particle is thought to feel a mean uniform field given by the excluded volume b and the attraction of the strength 2aV per pair which is the volume average of the attractive potential The hardcore repulsion and softweak attraction are the key features that well characterize the liquid state and gas state respectively The partition function 429 then is expressed in the form ZN frac1N left fracV Nblambda3 rightN expbetaN222aV ZN0 leftfracV NbVrightN expleftfracbeta N2 aVright 465 This equation yields all thermodynamic variables including the Van der Waals pressure equation The free energy internal energy entropy and chemical potential are obtained as F F0 NkB T ln1 nb fracN2 aV 466 E E0 fracN2 aV 467 S S0 NkB ln1 nb 468 mu mu0 kB T leftln1 nb fracnb1 nb right 2na 469 respectively Here the quantities superscripted by 0 are those of an ideal gas 433 The Effects of Spatial Correlations Pair Distribution Function Now we consider a nondilute fluid that has arbitrary density From 448 and 449 the internal energy of the system is obtained E fracpartialpartial beta ln ZN frac32 N kB T langle Phi rangle 470 where the average interaction energy is langle Phi rangle frac1VN QN sumi j int ldots int dr1 ldots drN phiri rj ebeta sumi j phiri rj frac1VN sumi j iint dr dr phir r frac1QN int ldots int dr1 ldots drN deltari r deltarj r ebeta sumi j phiri rj fracNN12 V2 int dr dr phir r gr r 471 Here we note langle deltari r deltarj r rangle frac1QN int ldots int dr1 ldots drN deltari r deltarj r ebeta sumi j phiri rj and define gr r fracV2NN1 sumi j langle deltari r deltarj r rangle frac1nN sumi j langle deltarij r r rangle 472 gr r is called the pair distribution function and is applicable to any one of NN12 pairs This is the probability of finding a particle at a position r given another particle placed at r relative to that for an ideal gas it provides a measure of the spatial correlation between a pair of particles In the absence of an external potential this function as well as the potential is isotropic phir phir gr gr so we derive the energy equation fraclangle Phi rangleN fracN12 V2 int dr r dr phir r gr r 2 pi n int0infty dr r2 phir gr 473 where r is the radial distance between the pair Fig 44 The average number of particles at a distance between r and r dr from a particle put at an origin r 0 is Fig 44 The radial distribution function gr is given in such a way that the average number of particles within a shell dr of the radius r form the central particle is 4 pi r2 gr n dr dNr 4 pi r2 gr n dr so gr for this isotropic case is appropriately called the radial distribution function Next we consider the pressure In the absence of an external potential the pressure on the wall of the container is independent of its shape so we will assume it is a cube of size L The pressure is given by p kB T fracpartialpartial V ln ZN kB T fracpartialpartial V ln VN QN 474 To extract Vdependence VN QN is rewritten as VN QN L3N int ldots int dr1 ldots drN ebeta sumi j phirij L 475 in terms of the dimensionless length eg ri riL rij rijL where rij ri rj We take the derivative with respect to volume V L3 p kB T fracpartial3 L2 partial L ln L3N QN 476 which by noting fracpartialpartial L ln L3N QN frac3NL beta sumi j leftlangle fracd phirijd rij fracrijL rightrangle 477 is finally expressed as p n kB T frac2 pi3 n2 int0infty dr r3 fracd phirdr gr 478 which indicates the pair distribution gr or the radial distribution gr plays the central role in determining thermodynamic properties of simple fluids Furthermore gr 472 provides the most essential knowledge on the configurations of the interacting particles When the separation r becomes much larger than the potential range gr approaches the ideal gas limit gr rightarrow infty 1 which indicates that particles are not spatially correlated In contrast as a result of the hard core repulsion gr rightarrow 0 0 In the low density limit of an interacting fluid one can envision only a two particle interaction for gr so that gr ebeta phir Theoretical studies of dense fluids and liquids have centered around analytical and computational investigations of the pair distribution function and on developing a variety of approximation schemes For the LenardJones potential at a liquid density gr shows damped oscillations around 1 Fig 45 with peaks at integer multiples of σ and troughs at halfinteger multiples of σ this feature is called the shortrange order At a distance r σ gr is zero because the two particles cannot overlap due to harsh repulsion At r σ a distance of close contact gr tends to peak this means that two particles caged at contact is in the most probable and stable state because surrounding particles of high density fluid constantly hitting and thereby the two particles do not have chance to be separated In contrast gr is at a minimum at r 15 σ when two particles tend to be most unstable to background agitations and least likely to stay in contact The probability increases again when r 2 σ where two particles tends to be stable because they are separated by just distance for another particle to fit between them The oscillation in probability persists with decaying amplitude gr can be interpreted as the probability of finding another particle at a distance r from one so we may write gr eφeffrkBT 479 where φeffr is the effective interaction potential energy between two particles φeffr is the reversible work needed to bring the two particles from the infinite distance to r In dilute gas it is just φr the bare interaction between the two because the presence of a third molecule is negligible φeffr is called the potential of mean force which at liquid density oscillates between negative and positive values due to the influences of surrounding molecules as explained above The pair distribution function is directly related via Fourier transform to the structure factor of the system This is a central topic to study for the structure of matter in condensed phase and can be determined experimentally using Xray diffraction or neutron scattering In the Chap 9 we will study this in detail 44 Extension to Solutions CoarseGrained Descriptions 69 44 Extension to Solutions CoarseGrained Descriptions 441 SolventAveraged Solute Particles We have been considering a simple fluid of onecomponent particles moving in a vacuum However in biology we consider solute particles such as ions and macromolecules immersed in water which itself is a complex liquid that undergoes anisotropic molecular interactions We remind ourselves that at equilibrium the momentum degrees of freedom of all the particles and molecules are usually separated and become irrelevant Yet the statistical mechanics involves complex situations in which the configurations of all particles in mixtures ie solutions solute as well as solvent must be considered including all interactions A simple approach to bypass this formidable task is to highlight the solute particles while treating the solvent as the continuous background whose degrees of freedom are averaged Fig 46 To describe this formally we write the total interaction energy as the sum ΦVrV ΦUrU ΦVUrV rU Here ΦV ΦU are the interaction energies among the solvent particles and solute particles respectively and ΦVU is the interaction energy between the solvent and solute particles with rV rU representing the solvent and solute particle positions The configuration partition function is given by Q drVdrU expβΦVrV ΦUrU ΦVUrV rU 480 where drV drv¹ drv² drU dru¹ dru² Then we can write Q drU expβΦUrU drVexpβΦVrV ΦVUrV rU drU expβΦeffrU 481 70 4 Statistical Mechanics of Fluids and Solutions where ΦeffrU ΦUrU ΔΦUrU 482 with the solvent averaged part of the potential ΔΦUrU kBT ln drV expβΦVrV ΦVUrV rU 483 In this formulation the total partition function is integrated over the solvent degrees of freedom with the remaining solute particles left to interact with one another with the effective interaction ΦeffrU 482 which is different from the bare interaction ΦUrU by ΔΦUrU 483 This solvent averaged effective potential also called the potential of the mean force is temperaturedependent This coarsegrained description is typical in colloid science As a simple example the effective interaction between two ions of charges q1 and q2 at a distance r12 in water is given by the Coulomb interaction φr12 q1q24πεwr12 which is about 180 of the Coulomb interaction in vacuum because the dielectric constant εw of water a temperaturedependent quantity is about 80 times that of the vacuum For N identical solute particles the starting point for the statistical description is the partition function ZU 1 Nv0N dru¹ dru² expβΦeffrU UrU 484 where UrU is an effective external potential energy of the solute The elementary volume v0 is introduced to count the states it is the volume allocated per particle so the entire volume V includes V v0 states per particle In the absence of the potentials the partition function is ZU⁰ 1 Nv0N dru¹ dru² 1N Vv0ⁿ 485 which gives the number of ways to place N identical noninteracting particles in the volume V The 485 differs from 49 in that λT³ is replaced by v0 Because of this replacement the partition function yields the thermodynamic quantities of an ideal solution that with v0 put to be independent of temperature exclude contributions from the translational momentum degree of freedom as shown below by E 0 in particular The Helmholtz free energy of the ideal solution FT V N NkBT lnNv0V 1 486 number of monomers such that there is no correlation between the beads this process gives rise to a flexible chain with a new coarsegrained continuous curve frfðsÞg The relevant level of the description is often guided by measurement An example is the endtoend distance of the polymer to describe its conformation Q ¼ R Fig 51c While the primary degree of freedom Q dictated by the measurement and observation to make can be easily identified it is often formidable to derive FðQÞ from 54 in general In many cases FðQÞ can be adopted directly from a macroscopic phenomenological energy or the probability of Q Because the endend distance R of a long flexible chain is distributed in Gaussian the associated free energy FðRÞ will be harmonic as shown in Chap 10 The method of coarsegraining in terms of Q can be regarded as an art of cartoondrawing which captures the salient and emergent behaviors But it is constrained to yield quantitative agreement with experimental measurements CoarseGraining CoarseGraining CoarseGraining a b c Fig 51 Schematic diagrams of coarse graining for the particles in a solution a coarsegraining into lattice model b a polymer coarsegrained into a semiflexible string c a flexible polymer coarsegrained into the linearly connected beads and to an entropic spring extended by a distance R 78 5 CoarseGrained Description Mesoscopic States Effective Despite their complex natures many biological phenomena can be described effectively in terms of phenomenologically observed states that emerge beyond the complexity of the underlying microstates In many cases of the mesocopic level biological systems we consider throughout this book we will use this method Twostates we introduced as biological microstates in Chap 3 exemplify such mesoscopic states The definition of mesoscopic depends on the perspective If the perspective is macroscopic then these meso states are relatively microscopic Throughout this book thus either one of the notations and Q for the state and correspondingly one of and FðQÞ for the Hamiltonian will be adopted depending on the perspective 52 Phenomenological Methods of CoarseGraining 79 Chapter 6 Water and BiologicallyRelevant Interactions Water is abundant and ubiquitous in our body and on earth Despite its critical importance in life and compared with the spectacular development of modern physics fundamental understanding of its physics is surprisingly poor In principle statistical mechanics is expected to explain its physical properties in a quantitative detail but is quite difficult to implement due to the relative complexity of water molecules and the nonisotropic interactions among them The statistical mechanics study for water is rare and limited Dill et al 2005 Stanley et al 2002 Instead of the statistical mechanics we give a semiquantitative sketch of basic thermal properties of water and the hydrogen bonding that underlies the unique charac teristics of water We also introduce the biologically relevant interactions between objects in water They are hydrophilic and hydrophobic interactions the electrostatic inter action among charges and dipoles and Van der Waals interactions In many cases the electrostatic interactions turn out to be weak with the strength comparable to the thermal energy kBT and much less upon thermalization due to the screening effects of waters high dielectric constant and the ion concentration These weak inter actions facilitate conformational changes of biological soft matter such as polymers and membranes at body temperature 61 Thermodynamic Properties of Water Liquid water has many properties that are distinct from other liquids One of waters most wellknown anomalies is that it expands when cooled contrary to ordinary liquids At atmospheric pressure when ice melts to form liquid water at 0 C the density increases discontinuously and then the liquid density continues to increase until it reaches a maximum at 4 C Fig 61a This behavior leads to a wellknown consequence that a lake freezes topdown from the surface on which Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158416 81 the ice floats whereas the bottom of the lake remains at 4 C Children skate on the icy surface while fishes swim over the watery bottom The phase diagram Fig 61b shows how the ice vapor and water liquid phases exist as functions of temperature T and pressure p The curved solid lines indicate coexistence of the different phases at equilibrium They meet at the triple point about 001 C and 0008 atmospheric pressure atm where the three phases coexist The coexistence line of liquid and vapor terminates at the critical point ðT ¼ 378 K p ¼ 218 atmÞ Near this point the interfaces of coexisting liquid and vapor become unstable and fluctuate widely showing a variety of divergent response behaviors called the critical phenomena The critical phenomena that occur in diverse matter have been one of central problems in modern statistical physics but are beyond the scope of this book In Fig 61b each of phasecoexistence solid lines is given by the ClausiusClapeyron equation dp dT ¼ Ds Dv ð61Þ 8 6 4 2 0 2 4 6 8 10 10000 09999 09998 09997 09180 Density gmL Temperature C 09170 a b 1 triplepoint critical point water vapor ice water liquid 378 100 001 0 218 P atm Temperature C 0008 water liquid ice Fig 61 The phasediagrams of water a The density of water increases discontinu ously as it undergoes the phase transition from solid ice phase to liquid phase In the liquid phase the density is maximum at 4 C b Pressure in atmospheric pressure units atm versus temperature in Celsius The solid lines repre sent the coexistence between two different phases of water The dashed curve is the phase coexistence between ordinary liquids and their vapors 82 6 Water and BiologicallyRelevant Interactions 62 The Interactions in Water 621 Hydrogen Bonding and HydrophilicHydrophobic Interaction The remarkable properties of water discussed above derive from its unique molecular structure and to hydrogen bonding HB among water molecules In a water molecule an oxygen atom is covalentbonded with two hydrogen atoms by sharing electrons But the oxygen atom has much greater affinity for electrons than the hydrogen atoms making the molecule polar with a high dipole moment Fig 62a HB is the electrostatic attraction between hydrogen containing polar molecules in which electropositive hydrogen in one molecule is attracted to an electronegative atom such as oxygen in another molecule nearby Fig 62b The HB in water has strength of a few kJmole which is much weaker than covalent or ionic bonds but much stronger than the generic nonHB bonds between small molecules This is the reason why the heat of vaporization boiling point and surface tension are relatively high in water Furthermore in water HB forms a network with large orientation fluctuations of the molecules that can be correlated over a long range The large fluctuations and longrange correlation hint at waters high response functions susceptibilities such as high dielectric constant and high heat capacity somewhat likened to the phenomena near the critical point HBs occur in both inorganic molecules and biopolymers like DNA and proteins b a Fig 62 a The dipole moment of a water molecule b Hydrogen bonding dashed line between water molecules Fig 63 The hydrophilic inter action The negatively charged polar heads of lipid molecules in a micelle attracts water mole cules 84 6 Water and BiologicallyRelevant Interactions The attractive interaction between water and other polar or charged objects is called hydrophilic interaction For example charged parts of an object are attracted to the oppositely charged parts of the water dipoles Fig 63 This is an important reason why water is such a good solvent Hydrophobic interaction in contrast is an indirect interaction between non polar objects in water The association of water molecules on nonpolar objects is entropically unfavorable because of restriction of the water molecule orientation on the interface When two nonpolar objects come in contact there is a strong gain of entropy due to reduction of the entropically unfavorable intervening region from which the water molecules are released this process eventually induces aggregation of the nonpolar objects Fig 64 The phase separation of fat in water is a good example of this particular interaction The hydrophobic interactions in part enable the folding of proteins because it allows the protein to decrease the surface area in contact with water It also induces phospholipids to selfassemble into bilayer membranes 622 The Coulomb Interaction The water medium affects fundamentally the interaction between two ions Phenomenologically the interaction between two ions of charges q1 and q2 sepa rated by a distance r12 is just the Coulomb interaction u12 ¼ q1q2 4pewr12 ð64Þ where ew is the electric permeability of water As mentioned in Sect 44 this effective interaction is formally obtained by integrating averaging over all the Fig 64 Hydrophobic interaction Two nonpolar objects upon approaching to contact liberate water molecules between them into the bulk where they have more entropy and hydrogen bonding Nature favors this and drives the contact namely hydrophobic attraction 62 The Interactions in Water 85 Chapter 8 The Lattice and Ising Models As introduced in Chap 4 the lattice model is a highly coarsegrained model of statistical mechanics for particle systems with builtin excludedvolume interac tion The model can address the structural and thermodynamic properties on length scales much larger than molecular size To incorporate the configurational degrees of freedom of manyparticle systems the system is decomposed into identical cells over which the particles are distributed With the shortrange interaction between the adjacent particles included this seemingly simple model can be usefully extended to a variety of problems such as gasto liquid transitions molecular binding on substrates and mixing and phase separation of binary mixtures For the particles that are mutually interacting in two and three dimensions we will intro duce the mean field approximations The lattice model is isomorphic to the Ising model that describes magnetism and paramagnettoferromagnetic transitions We study the exact solution for the Ising model in one dimension which is applied to a host of biopolymer properties and the twostate transitions Fig 81 Lattice model The substrate or volume is decom posed into many cells each of which either occupies a particle or not Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg10100797894024158418 121 Further Reading and References KA Dill S Bromberg Molecular Driving Forces 2nd edn Garland Science 2011 M Plischke B Bergersen Equilibrium Statistical Physics 3rd edn 2006 AW Adamson AP Gast Physical Chemistry of Surfaces 6th edn Wiley 1997 AA Hyman CA Weber F Jülicher Liquidliquid phase separation in biology Annu Rev Cell Dev Biol 30 3958 2014 J Palmeri M Manghi N Destainville Thermal denaturation of fluctuating DNA driven by bending entropy Phys Rev Lett 99 088103 2007 J Palmeri M Manghi N Destainville Thermal denaturation of fluctuating finite DNA chains the role of bending rigidity in bubble nucleation Phys Rev E 77 011913 2008 O Lee W Sung Enhanced bubble formation in looped short doublestranded DNA Phys Rev E 85 021902 2012 142 8 The Lattice and Ising Models Chapter 11 Mesoscopic Models of Polymers Semiflexible Chains and Polyelectrolytes Most biopolymers are semiflexible they can bend and undulate Mechanically they are characterized by finite values of their persistence lengths lp the scales below which the chains can be regarded as straight Fig 111 For example the persis tence length of doublestranded DNA is about 50 nm while that of actin filament is about 20 lm For the length scale much longer than the persistence length the chain appears to be flexible to which the models presented earlier can be applied This chapter covers basic mesoscopic conformations their fluctuations and elastic behaviors of semiflexible chains and polyelectrolytes that are either free or subject to external forces and constraints 111 Wormlike Chain Model We start with construction of the effective Hamiltonian for a free semiflexible chain As mentioned earlier the effective Hamiltonian can be taken from the macroscopic phenomenological energy which for a semiflexible chain is the energy required to bend an elastic string with a locally varying curvature Fig 111 Mesoscopic conformations of polymer chains with different persistence lengths lp L is the contour length Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584111 195 F Oosawa Polyelectrolytes Marcel Dekker New York 1971 T Odijk Polyelectrolytes near the rod limit J Polym Sci 15 477 1977 J Skolnick M Fixman Electrostatic persistence length of a wormlike polyelectrolyte Macromolecules 10 944 1977 JL Barrat JF Joanny Advances in Chemical Physics Polymeric Systems vol 94 Wiley 2007 R Podgornik VA Parsegian Chargefluctuation forces between rodlike polyelectrolytes pairwise summability reexamined Phys Rev Lett 80 1560 1998 VA Bloomfield DNA condensation by multivalent cations Biopolymers 44 3 269 1997 BY Ha AJ Liu Counterionmediated attraction between two likecharged rods Phys Rev Lett 79 1289 1997 NV Hud KH Downing Cryoelectron microscopy of k phage DNA condensates in vitreous ice the fine structure of DNA toroids Proc Natl Acad Sci USA 98 14925 2001 HG Garcia P Grayson L Han M Inamdar J Kondev PC Nelson R Phillips J Widom PA Wiggins Biological consequences of tightly bent DNA the other life of a macromolec ular celebrity Biopolymers 85 2 2006 WK Kim W Sung Charge density coordination and dynamics in a rodlike polyelectrolyte Phys Rev E 78 021904 2008 WK Kim W Sung Charge density and bending rigidity of a rodlike polyelectrolyte effects of multivalent counterions Phys Rev E 83 051926 2011 G Ariel D Andelman Persistence length of a strongly charged rodlike polyelectrolyte in the presence of salt Phys Rev E 67 011805 2003 A Caspi et al Semiflexible polymer network a view from inside Phys Rev Lett 80 1106 1998 T Baba et al Forcefluctuation relation of a single DNA molecule Macromolecules 45 2857 2012 Further Reading and References 217 Chapter 12 Membranes and Elastic Surfaces An essential component of a cell is a biological membrane or biomembrane it forms and modulates an interface of the cell and cells various internal com partments called organelles acting as a selectively permeable barrier between them Biomembranes consist mostly of phospholipid lipid bilayers and the associated proteins The bilayer is about 5 nm thick being selfassembled from lipid molecules each with a hydrophilic head and hydrophobic tails The lipids in a fluid membrane can move laterally within the bilayer organizing themselves to adopt the phase or the shape at equilibrium corresponding to free energy minimum There are two kinds of membrane proteins that perform a variety of cellular functions integral proteins such as ion channel all or part of which span the bilayer and peripheral proteins which lie outside the core of the bilayer see Fig 121 Fig 121 A cell membrane and its constituents such as phospholipid molecules and membranebound proteins including ion channels A phospholipid molecule is composed of a hydrophilic head and hydrophobic tails Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584112 219 In this chapter we study the thermomechanical aspects of the membrane with a particular focus on its mesoscopic fluctuations and conformations at equilibrium and shape transitions Although they are in reality very complex and heterogeneous in this introduction we will consider the proteinfree homogeneous membranes or membrane fragments that are amenable to statistical physics analysis 121 Membrane Selfassembly and Phase Transition The membrane is composed of many species of lipids proteins and cholesterols depending upon its functions The lipid which is the major component has a polar head group connected with hydrophobic chains When dispersed in an aqueous solution depending on their concentrations the lipid molecules assemble to form monolayers called the micelles and bilayers in the forms of vesicles and planar membranes Figure 122 depicts the various forms of the aggregates 1211 Selfassembly to Vesicles Of particular interest are the bilayer membranes The lipid chains line up side by side with their tails clustered together within the bilayer due to their hydrophobic interactions and with their heads interfacing with water due to hydrophilic attractions Such amphiphilic interactions among lipid heads and tails are much weaker than the directattraction or covalent bond that drives formation of two dimensional structures studied in Chap 7 Despite this difference and complex molecular architectures of the lipids the general statistical thermodynamic theory put forward in Chap 7 can nevertheless be applied to basic understanding of vesicle selfassembly As we learned in Sect 72 which we briefly recapitulate below the game rule of the selfassembly is to minimize the free energy culmi nating in establishment of the chemical potential balance ln ¼ l1 between a lipid bound in aggregates of n lipids nmers and a lipid unbound in solution Closed bilayer membranes vesicles tend to form more easily than planar membranes when the bending energy cost of forming a closed membrane can be monolayer micelle a bilayer vesicle b c planar membrane bilayer Fig 122 Lipids selfassembled to a micelle with singletailed lipids a a vesicle b and a planar membrane c 220 12 Membranes and Elastic Surfaces Chapter 13 Brownian Motions In previous chapters we were mostly concerned with the equilibrium state of matter Although the equilibrium statistical physics is relevant to studying the biological structures and conformations at body temperature the living processes operate out of equilibrium For nonequilibrium phenomena historically there are two pillars of statistical physics One is the kinetic theory of Boltzmann and Maxwell a groundbreaking work in nonequilibrium statistical mechanics that described the transport properties of gases on the basis of molecular motions The other one is the Brownian motion theory developed by Einstein Smoluchowski and Langevin and others which initiated stochastic descriptions of fluctuations in matter If for mulated on the basis of microscopic dynamics these two approaches converge Here we start with the stochastic approach to matter because of our primary interest in mesoscopic level surpassing atominstic details In this chapter and later ones we discuss the Brownian motion and extend the idea to describe the stochastic dynamics of biological systems and even other complex systems for which the microscopic Hamiltonian cannot be defined In 1827 botanist Robert Brown looking through a microscope found that particles in pollen grains were undergoing random and incessant motion in water He attributed this to the very nature of living embodied in the old philosophy of vitalism However this motion called the Brownian motion was subsequently observed in the grains of inorganic substances This was a great curiosity at the time but was suspected to be an outcome of the basic constituents of matter that are susceptible to thermal agitation In 1905 Albert Einstein published a paper that explained how the Brownian particles stochastically move surrounded by water molecules This explanation of Brownian motion served as a confirmation that molecules and atoms actually exist and was further verified experimentally by Jean Perrin in 1908 Fig 131 In this chapter the Brownian motion or diffusion equation theories of Einstein and Smoluchowski are given with a number of biological applications More Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584113 241 general theory via the Langevin equation is then given to describe the stochastic behaviors of the Brownian motion that differ in characteristic time scales 131 Brownian MotionDiffusion Equation Theory 1311 Diffusion Smoluchowski Equation and Einstein Relations Understanding that the Brownian motion is an incessant continuation of random jumps Einstein derived the equation for the probability density Pðr tÞ of a Brownian particle to be found at a position r and time t Pðr tÞ t ¼ Dr2Pðr tÞ ð131Þ Here the D is the diffusivity or the diffusion constant given by D ¼ hl2i 6s ð132Þ s is the jump time which is chosen to be macroscopically small but microscopically large enough that the motions after the time are mutually independent In the time interval s the particle is displaced by a distance l that is statistically distributed with Fig 131 The Brownian motion is depicted in the cover page of Atoms Ox Bow Press 1923 authored by Jean Perrin whose experiments on Brownian motion laid a foundation on atomicity of matter 242 13 Brownian Motions M Denny Air and Water The Biology and Physics of Lifes Media Princeton University Press 1993 DA Doyle J MoraisCabral RA Pfuetzner A Kuo JM Gulbis SL Cohen BT Chait R MacKinnon The structure of the potassium channel molecular basis of K conduction and selectivity Science 280 5360 1998 K Lee W Sung A stochastic model of conductance transitions in voltagegated ion channels J Biol Phys 28 279287 2002 JB Johnson Thermal agitation of electricity in conductors Phys Rev 32 97 1928 H Nyquist Thermal agitation of electric charge in conductors Phys Rev 32 110 1928 268 13 Brownian Motions H Risken The FokkerPlanck Equation Methods of Solution and Applications 2nd edn Springer Berlin Heidelberg New York 1989 W Ebeling IM Sokolov Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems World Scientific Publishing Co Pte Ltd 1992 DT Gillespie Markov Processes An Introduction for Physical Scientists Academic Press San Diego 1992 R Kubo M Toda N Hashitsume Statistical Physics II Nonequilibrium Statistical Mechanics 2nd edn Springer Berlin Heidelberg 1991 R Zwanzig Nonequilibrium Statistical Machanics Oxford University Press Oxford 2001 Further Reading and References 311 Chapter 16 The MeanFirst Passage Times and Barrier Crossing Rates 161 First Passage Time and Applications The first passage time FPT is the duration that a stochastic variable takes to approach a given threshold for the first time for example the duration for a random walker shown in Fig 161a to reach the cliff for the first time The first passage time problem is important in an enormous variety of situations to name a few transport reaction and targeting processes In particular it is of paramount importance in chemistry and biology where the rates of chemical reactions or conformational transitions are basic a b Fig 161 a Random walk in the region X between a reflecting ðqR Þ and a absorbing ðqAÞ boundary b a noiseinduced escape of a dynamical state from the region X q0 is the initial state Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584116 313 separating a long DNA fragment from short fragments using the channel shown as Fig 164 The long DNA will escape the well of battling confinement preferring to pass through a narrow constriction and advance to uncontested entropy space A short chain cannot Further Reading and References LE Reichl A Modern Course in Statistical Physics WileyVCH Verlag GmbH 2016 NG Van Kampen Stochastic Processes in Physics Chemistry Elsevier BV H Risken The FokkerPlanck Equation vol 18 Springer Series in Synergetics pp 6395 S Redner A Guide to FirstPassage Processes Cambridge University Press 2001 P Hänggi P Talkner M Borkovec Reactionrate theory fifty years after Kramers Rev Mod Phys 62 251 1990 W Sung PJ Park Polymer translocation through a pore in a membrane Phys Rev Lett 77 4 783 1996 I Goychuk P Hänggi Ion channel gating a firstpassage time analysis of the Kramers type PNAS USA 996 35523556 2002 T Chou MR DOrsogna First Passage Problems in Biology World Scientific 2014 HX Zhou Rate theories for biologistsResearchGate Q Rev Biophys 432 219293 2010 P Reimann GJ Schmid P Hänggi Universal equivalence of mean firstpassage time and Kramers rate Phys Rev E 601 R1R4 1999 PG De Gennes Coilstretch transition of dilute flexible polymers under ultrahigh velocity gradients J Chem Phys 60 5030 1974 J Han SW Turner HG Craighead Entropic trapping and escape of long DNA molecules at submicron size constriction Phys Rev Lett 83 1688 1999 Fig 164 The flexible poly mers are confined within a well Only the long chains can cross the narrow con striction toward to the open space of high entropy 162 The Kramers Escape Problem 325 Chapter 17 Dynamic Linear Responses and Time Correlation Functions Although seemingly stationary matter in equilibrium spontaneously fluctuates due to microscopic degrees of freedom thermally excited therein Even the macroscopic properties for example the length of a rod or the polarization of a dielectric fluctuate although imperceptibly on a finer time scales the time series of these properties looks stochastic with the variances reflecting the intrinsic response of the matter to a small external influence as we studied in Chap 9 Although apparently random the timeseries signals at different times are correlated at a close look In this chapter we will find that the time correlation is directly related to the response of the system to a timedependent perturbation namely the fluctuationdissipation theorem In particular how the time correlation decays is same as how the nonequilibrium state relaxes after removal of the perturbation From the knowledge of the time correlations a variety of the associated dynamic response functions and transport coefficients can be obtained a b Fig 171 a An RNA hairpin under a stretching force provided by an optical tweezer b the time series of endtoend distance of the RNA hairpin for various stretching forces Republished from Stephenson et al 2014 PCCP permission conveyed through Copyright Clearance Center Inc Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584117 327 Further Reading and References L Onsager Reciprocal relations in irreversible processes I Phys Rev 37 405 1931 L Onsager Reciprocal relations in irreversible processes II Phys Rev 38 2265 1931 R Kubo M Toda N Hashitsume Statistical Physics II Nonequilibrium Statistical Mechanics Series in SolidState Sciences Springer Berlin 1998 P Martin AJ Hudspeth F Jülicher Comparison of a hair bundles spontaneous oscillations with its response to mechanical stimulation reveals the underlying active process Proc Natl Acad Sci 9825 2001 W Stephenson et al Combining temperature and force to study folding of an RNA hairpin Phys Chem Chem Phys R Soc Chem 16 906 2014 346 17 Dynamic Linear Responses and Time Correlation Functions Chapter 18 NoiseInduced Resonances Stochastic Resonance Resonant Activation and Stochastic Ratchets Our world is replete with noises In common sense a noise is a nuisance that blocks coherence you feel annoyed with ambient sound noises when listening to music In this chapter we will study a counterintuitive phenomenon called stochastic resonance SR where a periodically modulated perturbation or signal too weak to be detected can be enhanced by adding the random noise to a nonlinear system Fig 181 The noise with an optimal strength can be instrumental rather than harmful in driving synchrony and resonance There exists another noiseinduced phenomenon the resonant activation RA where the rate of the noiseinduced transition is maximized by a modulation of an external signal at an optimal rate Biological systems in cellular level live on a variety of noises the ambient temperature in particular Due to their flexibility manifested on mesoscopic scale some biological complexes may utilize the ambient noises for their biological transitions and functions As we have seen in Chap 16 thermal fluctuations in such softcondensed matter facilitate the barrier crossing seemingly difficult to surmount typically assisted by conformational transitions Added to this phenomenon the SR and RA can provide essential physical mechanisms for inducing coherence and order in noisy and dissipative environments out of equilibrium Fig 181 A cartoon describing the phenomenon of stochastic resonance A weak signal can be enhanced in a dissipative media by an ambient noise at the optimal strength Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584118 347 voltagegated ion channel undergoes conformational transitions between a closed state and open state depending on the membrane potential Fig 185a The dynamics of ion channel transitions and the associated transmembrane ion transport is an enormously complicated problem requiring multiscale descriptions In a most coarsegrained description of voltagegated channels the single relevant degree of freedom q tð Þ can be chosen as the position of the gating charge repre sentive of positively charged helices within the channel which is believed to be the major component of voltage sensor An increase of membrane potential makes this gating charge move toward the extracellular side triggering a conformational transition to the open conductive state In this coarsegrained picture the gating charge can be considered to be a Brownian particle hopping between two confor mational states In the presence of a noisy macromolecular and fluid environment the centerofmass position q of the gating charge is subject to its complex free energy landscape with the free energy parameters such as activation barrier sensitively depending on temperature For a guinea pig ileal muscle channel for which data on the parameters as well as the rates are available a double well free energy model for the two state transitions was constructed Parc et al 2009 An important feature here is that the transition rates are not Arrheniuslike because of the temperaturedependent activation barrier With a weak oscillating voltage added to a constant potential across the membrane a simulation of the gating charge dis placement showed its power spectrum Sf x ð Þ indeed manifested the SR peak at the driving frequency x ¼ X The peak height Sf X ð Þ is maximum at an optimal noise strength which is found to be just the body temperature TSR ¼ 320 K of the guinea pig Fig 185b The ion channel owing to the flexible structures opens and closes in a maximum coherence with the oscillating membrane potential at the body temperature This suggests that the body temperature is not accidental but possibly an outcome of natures selection to make it a good noise essential for living Fig 185 a A schematic picture of a voltagegated ion channel With a membrane potential applied the gating charge positively charged helix shifts to the extracellular side inducing the channel to open b The peak in the power spectrum Sf X ð Þ for the gating charge flow emerges around 320 K in a guinea pig ileal muscle channel Adapted from Parc et al 2009 181 Stochastic Resonance 353 Biopolymers Under Tension RNA molecules are biopolymers that carry and relocate hereditary information of vital importance The RNA folds into unique three dimensional conformation called tertiary structure by sequential binding of an essential secondary structure named as RNA hairpin The singlemolecule experiments showed how the RNA hairpins subject to a stretching force provided by optical tweezers undergo conformational changes from folded to unfolded states Fig 186 A Brownian dynamic simulation of the foldingunfolding trajectories of a model 22nucleotide P5GA RNA hairpin under a constant force f indicates that the free energy as a function of the extension z is bistable Fig 186b The mean folding and unfolding times are the Kramers times Fig 186 a P5GA RNA hairpin under a stretching force exerted by an optical tweezer b A Brownian dynamic simulation on a model 22nucleotide P5GA RNA hairpin shows that depending on the force f the free energy of the extension is bistable with two conformational states a folded state at the extension zF and a unfolded state at zU and the transition state barrier top at zTS c The time trajectories of the extension under a time dependent tension f þ df cos Xt with f ¼ 17 pN df ¼ 14 pN from a Brownian dynamic simulation The transition from the unfolded state to the folded state synchronizes to the periodic driving of the resonant period 102 ms while it is incoherent to the oscillations with smaller and higher periods Adapted from Kim et al 2012 354 18 NoiseInduced Resonances Stochastic Resonance WK Kim W Sung How a single stretched polymer responds coherently to a minute oscillation in fluctuating environments an entropic stochastic resonance J Chem Phys 137 074903 2012 PJ Park W Sung A Stochastic model of polymer translocation dynamics through biomem branes Int J Bifurcat Chaos 8 927 1998 CR Doering JC Gadoua Resonant activation over a fluctuating barrier Phys Rev Lett 69 2318 1992 RD Astumian P Hänggi Brownian motors Phys Today 5511 33 2002 RD Astumian M Bier Fluctuation driven ratchets molecular motors Phys Rev Lett 72 1766 1994 AB Kolomeisky Motor Proteins and Molecular Motors CRC Press 2015 362 18 NoiseInduced Resonances Stochastic Resonance Chapter 19 Transport Phenomena and Fluid Dynamics Most systems in nature are dynamic that is change in time In nonequilibrium processes there are flows transports of mass momentum and energy from one place to the other If a system is near equilibrium the transports occur in such ways that the distributions of the mass momentum and energy which are nonuniform and time dependent are relaxed to the equilibrium where there are no flows One example is the diffusion of particles from a crowded region to a less one The equilibrium state represents a stationary state The other stationary state is the nonequilibrium steady state where there are constant flows driven by external means For example a rod whose ends are maintained at two different temperatures is in a steady state with a constant heat flow from a high temperature end to a lower one The temperature gradient in the rod is the driving force for the heat flow Biological complexes are bathed in aqueous environments Over the scales much longer than the mean free length between collisions of the solvent molecules the solvents can be treated as continuous fluids The complex fluids such as solutions of biopolymers and cells probed over a certain long length scale can also be treated as continua For these cases the hydrodynamic description of transport in terms of densities of fluids conserved quantitiesthe mass momentum and energy is very useful The governing dynamics for these hydrodynamic variables which is also called hydrodynamics or fluid mechanics is widely applicable to the problems not only in basic sciences but also in engineering disciplines For biological organisms in particular fluid motion is something with which they must contend a factor to which their design reflects adaptation Vogel 1984 In this chapter we study basic principles and apply them to some important fluid flows which allow analytical treatments Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584119 363 Chapter 20 Dynamics of Polymers and Membranes in Fluids The dynamics of biological softcondensed matter biosoft matter such as biopolymers membranes and cells has at mesoscale several features which are not seen in ordinary matter consisting of particles One is the soft matter structural connectivity although the strengths of its atoministic interactions are in the order of eV or higher interconnected as a whole it can undergo collective motions with the energies in the order of or less than thermal energy Despite the shortrange interconnectivity among near neighbors the biosoft matter at room temper ature can be correlated over long distances as we have studied in Chaps 1012 Also we studied in Chaps 16 and 18 that it can move cooperatively in thermally fluctuating backgrounds and susceptibly in response to external fields The biological complexes live usually in viscous aqueous environments the background fluids impart dissipation but mediate hydrodynamic interactions HI between segments in the complexes In contrast to the structural connectivity HI is long ranged adding the unique cooperativity to dynamical behaviors In this chapter we study the interplay of the structural connectivity and hydro dynamic interaction in soft matter dynamics The basic method for the dynamics is a stochastic approach in which each internal constituent mesoscopic subunit undergoes Brownian motions while interacting with one another and with the fluid environment As standard models that allow analytical understandings we consider flexible chains then semiflexible nearlystraight polymers and planar membranes The nonspecific physical features that are obtained from the relatively simple systems can give valuable insights into the dynamics of more complex biological soft matter under flows and constraints Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584120 391 Chapter 21 Epilogue For one Chrysanthemum to bloom the Thunder so must have cried again within the dark cloud Seo JungJu Surmounting the Insurmountable A cell is a playground for various extraordinary events what we may call biological selforganizationsThe basiccomponents saybiopolymersmembranes ionchannels and even their aqueous environments have very complex structures and yet show unusual cooperative behaviors which go beyond the scope of traditional physics It is grossly hopeless to solve microscopic equations of motion for the enormous number of atoms and molecules that constitute the biological matter and even to treat them collectively using the traditional statistical mechanics The standard analytical methods of statistical mechanics have been implemented mostly for simple systems such as ideal gases and magnets and simple interacting units How can we cope with the biological Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg101007978940241584121 423 transition and activates crossing the free energy barrier that may exist The bio logical dynamics at mesoscale due to the overdamping solution background is dissipative and slow However we saw a most striking phenomenon that the noise when tuned at an optimal strength can induce the maximal coherence and resonance of the transitions and barrier crossing dynamics of the system to an external timedependent signal albeit very week The dissipative fluid background in cell is not merely a passive medium but an active structure that signals a nonequilibrium noise therein to resonate with the systems transition dynamics This phenomenon of stochastic resonance and related resonant activation can be nonspecific physical paradigms pointing to the critical role of the thermal fluctua tions and external noises these oftenneglected degrees of freedom which may not be seen phenomenologically play such magic Part II After all two eminent features boost these kinds of unique interplay between the systems and the backgrounds One is the background waters many outstandingly high susceptibilities in particular the high dielectric constant that facilitates various transitions of the system by reducing the electrostatic interaction energies therein to the level of thermal energy or below If the systems are the soft matter such as polymers and membranes another key feature is their structural connectivity and flexibility which gives rise to cooperativity and low energy excitations In parallel with the weak interactions mentioned above under the fluctuating aqueous envi ronments the biological soft matter can undergo whatever transitions and surmount the seemingly unsurmountable barriers at body temperature The thermal noises may come as random thunderstorms to the soft matter but at optimal conditions may lend a helping hand with accomplishing the biological selforganizations As a way to bypass the virtually impossible task of deriving mesoscopic descriptions for a biological complex from underlying microscopics the wellknown classical phenomenology can fortuitously be used with an input of the fluctuations For example the effective Hamiltonian of a DNA fragment is the classical elastic energy of bending with the curvature promoted to be fluctuating degrees of freedom Chap 11 Another example is the Langevin equation which is obtained by adding noises to macroscopic equations of motion Chaps 13 and 15 If we allow their charge densities to fluctuate and correlate two objects with equal net charges can attract rather than repel to minimize the free energy Coulomb interaction This explains how the charge fluctuations induce DNA collapse Chap 11 and membrane adhesion Classical phenomenology such as elasticity mechanics electricity and hydrodynamics can thereby be revived to adapt to some biological phenomena by endowing the variables with stochasticity It is akin to how the quantum fluctuation phenomena can be realized by replacing classical variables by operators Surmounting the Insurmountable 425 Additional Topics Throughout this book we have studied the selected themes of statistical mechanics softmatter physics and related areas that I believe to form a coherent basis for applications to a variety of biological phenomena mostly on mesoscales It is rather a kind of extended statistical physics book than of biological physics or biophysics book As such there can be many important biophysics topics that were not addressed particularly on molecular scales and system levels Also the theoretical methods and biological examples that are covered may be relatively simple To cope with the higher complexity the basic physical premises need to be further revised and expanded for example the concepts of spatial homogeneity and tem poral stationarity may not be valid for crowded cell environments I hope this project will nevertheless give an example of the first step toward the challenging and timeconsuming endeavor to build up paradigms of a new fusion science by surmounting barriers between biological and physical sciences Within biological statistical physics there are a number of topics that I initially intended to cover nonMarkovian and anomalous dynamics molecular motors and the applications of stochastic thermodynamics and fluctuation theorem in their infancy To date the present version is the best I could try with limited time To incorporate these topics with coherence and harmony in the future edition remains a challenge 426 21 Epilogue Index A Absorbing BC 306 315 A chain anchored on surface 168 Actin filaments 209 Activation barrier 308 353 Active mechanism 335 Active structure 425 Adenosine diphosphate ADP 109 Adenosine triphosphate ATP 109 Adjoint 315 Adsorbates 122 Adsorbent 122 Adsorption 122 Adsorptiondesorption transition 183 Adsorption isotherm 122 Aerial organism 249 Amino acids 3 Amphiphilic interactions 220 Angleaveraged interaction 89 91 Anions 95 Anomalous behaviors 404 Anomalous dynamics 417 Arc length 196 Arrhenius law 322 Arrheniuslike 353 Athermal 355 Athermal noises 360 ATP hydrolysis 109 Average magnetization 135 B Backward FP operator 300 Barometric formula 57 Base pair 28 136 Base pairing energy 139 Bayes rule 270 Beadspring model 174 Beadspring the Gaussian chain model 393 Bending energy 117 224 Bending modulus 196 Bending rigidity 224 Bernoullis law 368 Bilayer 3 219 Bilayer membrane 85 Binding energy 113 Binomial distribution 33 283 285 Binomial expansion 39 Biological complexity 423 Biological Physics or Biophysics 1 Biological selforganization 2 4 423 Biomolecular motors 361 Biopolymer folding 20 Biopolymers 2 Bio soft condensed matter 5 Birth and death 282 285 Bjerrum length 86 Blood 377 Body temperature 20 137 353 Boltzmann constant 26 Boltzmann distribution 30 34 57 Boltzmann entropy 34 Boltzmann equation 372 Boltzmann factor 36 95 Boltzmann formula for entropy 26 Bond energy 126 129 Born energy 86 Boundary condition BC 202 306 BraggWilliams approximation 126 Brownian motion 5 6 241 249 Brownian particles 366 Springer Nature BV 2018 W Sung Statistical Physics for Biological Matter Graduate Texts in Physics httpsdoiorg1010079789402415841 427 Brown Robert 241 Bubble 137 199 289 Bulk viscosity 369 C Canonical ensemble 34 122 Canonical partition function 36 Cations 95 Cell capture 247 Cell division 222 Cellrich fluid region 377 Central Limit Theorem CLT 163 244 258 Chain conformations 171 Chain connectivity 139 180 319 Chain flexibility 161 184 ChapmanKolmogorov Equation CKE 275 Charge correlation function 149 Charge density 95 Chargedensity fluctuation 215 Charge neutrality 95 99 Chemical energy 361 Chemical equilibrium 18 Chemical force balance 104 Chemical potential 10 11 13 44 59 71 Chemical work 10 Chemoreception 250 Cholesterols 220 Classical phenomenology 425 ClausiusClapeyron equation 82 CM diffusion 400 Coarsegrained description 70 76 353 392 Coarsegrained model 121 Coarse graining 5 230 Coexistence line 82 Coherence 5 Cohesion energy 221 Cohesive energy 117 Coilstretch transition 324 Coiltoglobule transitions 188 Collective diffusivity 345 Collective excitations 149 Collective motions 391 Colloid 32 Colored noise 279 Compressibility 153 Compressibilityrelation 153 Concentration 56 Condensation 128 Condensed phase 127 128 Conditional probabilty distribution 270 Conditions of the equilibrium 17 Configurational partition function 53 61 122 Conformational adaptability 324 Conformational states 278 Conformational transitions 136 162 Conformations 4 220 Conformation transition 4 Conjugate 41 Conserved quantities 363 Constant 400 Continuity equation 364 Contour length 196 Convection 366 Convective drift current 245 Convective momentum flux 367 Convective time derivative 365 Cooperative effect 127 139 141 Cooperative phenomena 133 Correlation function 135 206 215 406 Correlation function of local density fluctuations 147 Correlation length 136 148 Correlation time 272 Couette flow 374 Coulomb interaction 70 85 Counterion 98 Coupled damped oscillators 395 Covariance 270 Coverage 32 47 122 125 Critical aggregation concentration 113 Critical concentration 117 221 Critical condition 127 Critical phenomena 82 Critical temperature 21 128 Cucurbiturils 116 Curvature 196 Curvature energy 226 Curvature modulus 117 118 221 Curvature tensor 225 Cyclic process 8 Cytoskeletal filaments 111 113 D Debye 96 Debye function 159 DebyeHückel equation 96 DebyeHückel limiting law 98 DebyeHückel theory 98 Debye length 96 100 Debye model 336 Decays 285 de Gennes P G 193 Degrees of freedom 36 55 Denaturation 137 Density fluctuation 149 151 Density of states 37 281 Density spatiotemporal correlation function 343 428 Index Designation of microstates 28 Detailed balance 280 Dichotomic noise 279 Dichotomic process 278 314 356 359 Dielectric constant 4 70 83 167 Dielectric continuum 88 Diffusion constant 242 Diffusion current 248 Diffusion equation 243 283 291 Dilute fluids 61 Dipoledipole interaction 90 Dipole moment 87 Directed motion 360 Disk formation 115 Disorder 26 Disperse phase 127 128 Displacement 10 Dissociation 108 Divergence theorem 364 DNA 2 DNA condensation 188 DNA melting 136 Double helices 2 DoubleStrand DS state 137 Doublestranded DNA 28 Double well potential 348 Drag 381 Driving force 245 348 Dynamic response function 328 350 Dynamic structure factor 343 410 E Edwards equation 173 Effective Hamiltonian 5 76 166 201 230 232 424 Effective Hamiltonian of the chain at the segmental level 174 Effective Hamiltonian of the membrane 223 Effective potential of the stochastic process 296 Effective temperature 335 Egg fertilization 222 Eigenfunction 173 301 Eigenfunction expansion method 299 Eigenstate 308 Eigenvalue 173 225 301 308 309 Eigenvector 225 Einstein Albert 241 Einstein relation 245 Electric Double Layer EDL 99 101 385 Electric permeability 83 95 Electric susceptibility 89 145 337 Electrokinetic effects 384 Electrolyte 95 Electroosmosis 384 Electrophoresis 386 Electrophoretic velocity 387 Electrostatic analogy 250 Electrostatic bending energy 213 Electrostatic persistence length 214 Emergent behavior 167 319 Emerging degrees of freedom 5 Emission of quanta 282 Endocytosis 227 Endtoend distance 42 Energy density 59 370 Energy density flux 371 Energy dissipation 330 337 343 371 Enthalpy 12 Entropic chain 166 Entropic force 43 Entropic spring 174 Entropic spring constant 167 Entropic SR 414 Entropy 9 11 20 26 Entropy density 71 Entropy of mixing 32 Equala priori probability 26 30 34 Equations of state 14 Equilibrium 7 Equilibrium constant 106 Equilibrium sate 304 Equipartition of the energy 202 265 Equipartition theorem 55 197 232 Ergodicity 272 Euler buckling instability 209 Exact differential 8 10 Excluded volume 186 Excluded volume effect 123 185 Excluded volume interaction 121 Exocytosis 227 Extension 201 Extensive variables 7 External force density 367 External signal 347 F FahraeusLindqvist effect 377 Faxens law 382 Fermis golden rule 280 Ferromagnetic transition 135 Ficks law 243 First law of thermodynamics 8 10 First Passage Time FPT 313 Fission 222 Flexibility 4 Index 429 Flexible polymer chains 392 Flipping 361 Flory exponent 187 Flows 380 Fluctuating barrier 357 Fluctuating degrees of freedom 425 Fluctuation 46 220 FluctuationDissipation Theorem FDT 260 334 411 Fluctuationinduced attraction 93 Fluctuationinduced interaction 238 Fluctuation theorem 426 Fluid mechanics 363 Fluid membrane 223 Flux 245 Flux over population method 321 FokkerPlanck 424 FokkerPlanck dynamics 295 315 Fokker Planck Equation FPE 291 356 FokkerPlanck operator 299 Folding and unfolding times 360 Folding of proteins 85 Foldingunfolding transitions 20 Forceextension 204 Forceextension relation 201 Form factor 151 154 Fourier mode 343 Fourier transform 151 152 202 230 264 405 FP operator 300 FPT distribution 314 Fractal 156 Fractal dimension 156 Free diffusion 408 Free energy 4 230 Free energy change of the reaction 104 Free energy density 58 71 Free energy function 76 166 318 Free energy landscape 131 Free energy of the chain 169 Free energy of the confinement 178 Free energy of translocation 319 FreelyJoined Chain FJC 42 198 Freelyjointed chan model 167 Frequencydependent conductivity 338 Frequencydependent diffusivity 345 Frequencydependent elecric permeability 337 Frequency dependent electric susceptibility 337 Frequencydependent response function 350 Frictional force 258 Friction coefficient 244 Fugacity 47 126 Fundamental solution 168 Fusion 222 227 G Gastoliquid phase transition 128 148 Gating charge 353 GaussBonnet theorem 225 Gaussian 38 45 258 Gaussian chain 43 164 Gaussian curvature energy 225 Gaussian distribution 165 Gaussian level approximation 202 Generalized boundary condition 382 Generalized diffusion equation 345 Generalized force 10 Generalized Langevin equation 276 Generalized spring constant 419 Generating function 283 284 288 Genus number 225 Gibbs ensemble 226 Gibbs free energy 12 19 41 226 Gibbs partition function 41 Glassy systems 335 Globular conformation 322 Globule 188 Good noise 353 Good solvent 186 GouyChapman length 100 Grand canonical ensemble 44 124 153 Grand canonical partition function 45 Grand partition function 47 Grand potential 13 19 21 46 125 GreenKubo relation for selfdiffusion 340 Ground state dominance approximation 182 Growth process 21 H HagenPoiseuilles law 376 Hair bundle cells 335 Hamaker constant 94 239 Hamiltonian 36 39 134 144 Harmonic order 229 Heat 8 Heat capacity 15 59 83 Heat conductivity 371 Heat of vaporization 83 Heat reservoir or bath 34 Height undulations 205 Helfrich interaction 238 Helixcoil transition 140 Helmholtz free energy 11 12 19 37 40 57 75 130 HelmholtzSmoluchowski relation 385 Hermitian operator 301 430 Index Hollow sphere 116 Homogeneous phase 131 Hydrodynamic description 363 Hydrodynamic equations 243 Hydrodynamic friction 381 Hydrodynamic Interaction HI 388 391 401 416 Hydrodynamic radius 401 Hydrodynamics 363 Hydrodynamic variables 364 Hydrogen bonding 4 84 Hydrophilic attractions 220 Hydrophilic head 3 219 Hydrophilic interaction 85 Hydrophobic chain 220 Hydrophobic interaction 85 220 Hydrophobic tail 3 219 I Ice age 349 Ideal chain 159 162 164 Ideal gas 58 Ideal gas partition function 53 Ideal solution 70 Identical particles 52 Image method solution 257 Image solution method 169 Incoherent 349 Incompressible flow 365 366 370 Incompressible mixture 129 Induced polarizability 89 Inelastic scattering 150 343 Inexact differentials 8 Inflection point 127 141 Information theory 28 Initiation energy 139 Inorganic phosphate Pi 109 Inphase response 340 Intensive variables 7 Interacting particles 61 Interface energy 223 Interfaces 132 Interfacial area 21 Internal degrees of freedom 60 Internal energy 8 12 59 In vitro 17 In vivo 5 Ion channel 4 252 278 352 Iondipole interaction 88 Ionic transport 252 Irreversible processes 16 Ising model 126 133 Isolated system 8 16 26 Isothermal compressibility 16 46 145 J Johnson noise 264 JohnsonNyquist theorem 265 Joint probability 39 356 K Keesom force 90 Kinesin motors 110 Kramers escape problem 320 KramersMoyal expansion 292 Kramers problem for polymer 322 Kramers rate 322 Kramers time 320 348 Kubo formula 338 Kuhn length 162 198 L Low Reynolds number 380 Langevins function 43 88 Langevin equation 257 392 Langmuir isotherm 48 111 Laplace transform 287 Lattice model 72 77 121 134 Law of Mass Action LMA 106 LenardJones potential 68 Length fluctuations 205 Length of extension 10 LennardJones potential 63 Level of the description 28 Lever rule 132 Light scattering 149 Linear aggregates 113 Linearized PoissonBoltzmann equation 96 Linear response theory 207 235 350 410 Line charge density 210 Line tension 115 133 Lipid 3 219 Liquid droplet 14 Local entropy 59 Local osmotic pressure 71 Local pressure 58 Local radius of curvature 196 London dispersion force 93 Longitudinal dynamics 410 Longitudinal fluctuation 207 208 Longrange spatial correlation 416 Lorentzian 279 Lotkas law 297 M Macroscopic properties 25 Macroscopic system 7 26 Macrostate 26 Magnetic susceptibility 16 145 Index 431 Magnetization 10 15 145 MarkoSiggia model 145 Manning condensation 210 Markov chain 274 Markov process 6 269 274 348 Master equation 277 MaxwellBoltzmann MB distribution 54 261 MaxwellBoltzmann MB speed distribution 55 Maxwell construction 128 Maxwell relations 12 Mean curvature modulus 224 Mean field 64 95 Mean Field Approximation MFA 126 130 226 Mean field theory 95 Mean first passage 313 357 Mean Squared Displacement MSD 402 Mean squared EED 198 Mean square fluctuation 143 Mechanical equilibrium 22 Melting 137 Melting point 141 Membrane 2 3 219 Memory friction 276 Mesoscopic length scales 161 Mesoscopic states 29 Metabolites 247 Metastable state 21 Micelles 220 Microcanonical ensemble 26 28 Micropipette 227 Microscopic boundary layer 382 Microscopic degrees of freedom 75 Microscopic displacement 144 Microscopic fluctuations 424 Microscopic local number density 147 Microstates 26 423 Microtubules M 110 Mixing entropy 122 130 Molar 105 Molar concentration 97 221 Molecular motors 361 426 Momentum density 366 Momentum density flux tensor 367 Monomer concentration 177 193 Multiplicity 26 Multivalency 212 N Nanoscale subunits 25 28 Native structures 3 Navier boundary condition 377 383 NavierStokes equation 368 370 Nearest neighbor 134 Negentropy 17 Nernst potential 254 Neutron scattering 68 149 Newtons law of viscous flow 369 Noiseassisted phenomenon 361 Noise strength 259 317 351 NonArrhenius 322 Nonequilibrium noise 279 355 Nonideal fluids 14 NonMarkovian process 276 Nonpolar molecules 92 Nonuniform fluid 57 58 Nucleation 21 Nucleotides 2 Number density 56 O Occupation number 29 39 OdijkSkolnickFixman OSF 214 One step processes 282 Onsagers regression theorem 333 Open systems 45 Optical tweezer 214 328 Orientation correlation function 237 OrnsteinUhlenbeck process 267 275 298 396 Oseen tensor 389 Osmotic pressure 100 Overdamped Langevin equation 266 293 P Pair distribution function 66 152 Paretos law 297 Partition function 39 53 57 61 134 189 Pathintegral 173 Periodic boundary condition 202 Perrin Jean 242 Persistence length 162 195 197 212 237 pH 109 Phase boundary 83 Phasecoexistence 82 Phase delay 329 Phase diagram 82 Phase separation 131 Phase space 28 51 294 Phasespace distribution function 52 Phase transition 15 133 Phospholipid 219 Physiological conditions 138 Planck formula 281 432 Index Poiseuille flow 375 PoissonBoltzmann PB equation 95 Poisson distribution 284 288 Poisson process 283 Polar head 220 Polarizability 92 Polarization 10 145 Polar molecule 90 Polyelectrolytes PE 209 Polyethylene 161 Polymer bindingunbinding transition 182 Polymer chain 19 Polymer globule 184 Polymer Greens function 168 171 Polymer translocation 318 Polypeptides 209 Poor solvent 187 Population 282 Pore growth 22 Power amplification 351 Powerlaw 297 Power law decay 157 Power spectrum 279 335 Preaveraging approximation 399 Pressure 10 67 367 Primary degrees of freedom 77 Primary thermodynamic potential 11 Primary transverse relaxation time 409 Principal curvatures 224 Probability Distribution Function PDF 185 269 Protein folding 3 Protein motors 361 Proteins 3 220 Q Quantum coherence 333 Quantum states 28 Quasistatic processes 8 R Radial distribution function 67 157 Radius of gyration RG 158 164 200 Random force 258 Random walk 162 282 Rare events 285 Ratchet 360 Reactions 5 285 286 Reactions coordinate 320 Real polymer chain 159 Receptors 247 Red Blood Cells RBC 377 Reflecting BC 306 315 Relaxation time 271 416 Relevant degrees of freedom 5 76 Resonant activation 347 356 Response function 15 84 143 147 Rest potential 254 Reversible 8 Reynolds number 379 RNA hairpin 328 354 Root Mean Squared RMS deviation 34 38 45 Rotational friction coefficient 382 401 Rotational relaxation time 397 Rouse model 393 Rouse modes 395 Rouse time 396 Rudolf Clausius 9 S Salt 95 Scaleinvariance 157 Scaling law 159 192 Scaling theory 191 Scatterings 149 Schrödinger equation 173 301 Schrödingerlike equation 302 303 Screening length 98 Second law of thermodynamics 34 372 Second virial coefficient 62 186 Segmental distribution 176 Selfassembly 5 111 Selfavoiding walk 185 Selfconsistent field 171 Selfsimilar structure 156 Semiflexible polymers 404 Shape fluctuations 228 Shape transitions 220 222 Shear flow 368 Shear viscosity 369 Shortrange order 68 Sigmoid function 31 SignaltoNoiseRatio SNR 265 352 Signal transduction 222 Simple fluid 51 SingleStrand SS state 137 Slippage 383 Slow decay dynamics 407 Small Angle Neutron Scattering SANS 157 188 Small Angle Xray Scattering SAXS 157 188 Smoluchowski equation 244 245 266 291 293 Solid angle 88 Solute 69 Solvation energy 86 Index 433 Solvent 69 Solvent averaged effective potential 70 Spatial correlation 66 Spatial homogeneity 152 Spherical vesicle 19 Spherical virus 154 Spontaneous process 16 21 Spontanoeus curvature 224 Stacking interaction 138 Standard density 60 Standard free energy 106 Standard internal energy 106 Standard state 60 State function 8 State variables 7 Static structure factor 343 Stationarity 271 Stationary process 271 Stationary solution 278 288 Stationary state 271 280 363 Statistical ensemble 5 Statistical mechanics 5 7 25 Stern layer 102 Stiff polymer 113 Stirling approximation 27 Stochastic differential equation 258 297 Stochastic process 269 Stochastic Resonance SR 347 349 413 425 Stochastic thermodynamics 426 Stochastic variable 269 Stoichiometric coefficient 104 StokesEinstein relation 246 Stokes flow 380 Strength of the noise 258 Stress tensor 369 Stretch modulus 145 224 Structural connectivity 4 391 Structure factor 68 151 152 154 199 Subcellular structures 2 Subdiffusion 417 Substrate 122 Supercooled gas 21 Supramolecular aggregates 111 Surface growth 282 Surface potential 97 Surface pressure 123 Surface tension 10 21 83 123 133 223 Susceptibility functions 143 Symmetric double well 307 T Taylor GI 190 Telegraphic process 278 Temperature 9 11 Tension 10 Tether 227 The fluctuationresponse theorem 144 The Hill equation 111 The mesoscopic level 5 Thermal diffusion constant 371 Thermal energy 4 83 Thermal equilibrium 18 Thermal fluctuation 4 162 239 425 Thermal neutron 149 Thermal noise 258 Thermal undulation 204 234 238 Thermal wavelength 53 Thermodynamic conjugate 10 Thermodynamic limit 40 Thermodynamic potential 11 Thermodynamics 5 7 Thermodynamic variables 8 The second law of thermodynamics 16 Theta H temperature 186 Theta solvent 186 The Van der Waals equation of state 14 Third virial coefficient 63 Time arrow 17 372 Time correlation function 273 299 310 392 396 Time irreversibility 372 Tonks gas 61 Topological invariant 225 Transfer matrix 134 Transition probability 274 277 304 Transition rates 278 Translational degrees of freedom 60 Translational invariance 152 231 Translocation 32 170 Transports 363 Transverse fluctuation 205 Trivalent cations 215 Turbulence 379 Two dimensional polymers 111 Twofluid model 377 Twostate model 28 39 Twostate transitions 140 307 U Unbinding 141 Unbinding transitions 238 Undulation 230 Undulation correlation 231 Undulation time correlation 414 416 V Valence 4 Van der Waals equation of state 64 434 Index Van der Waals attraction 93 VanHove correlation function 345 Van t Hoff equation 107 Variance 33 Variational principle 17 19 Velocity distribution 260 Velocity moments 372 Velocity relaxation time 245 Vesicle 220 227 Viscous fluid 373 Voltage sensor 353 Volumetric flow rate 376 378 W Water 4 39 81 Weak signals 352 White noise 258 WienerKhinchin theorem 273 335 Wiener process 266 Work 8 WormLike Chain WLC 77 196 404 X Xray diffraction 68 Y YoungLaplace equation 22 227 Z Zigzag motion 252 Zimm model 398 Zipfs law 297 Zipper model 139 Index 435