The Edge of Thermodynamics: Driven Steady States in Physics and Biology by Robert Alvin Marsland III Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 c Massachusetts Institute of Technology 2017. All rights reserved. ○ Author ................................................................ Department of Physics May 24, 2017 Certified by............................................................ Jeremy L. England Cabot Career Development Associate Professor of Physics Thesis Supervisor Accepted by ........................................................... Nergis Mavalvala Curtis and Kathleen Marble Professor of Astrophysics Associate Department Head of Physics 2 The Edge of Thermodynamics: Driven Steady States in Physics and Biology by Robert Alvin Marsland III Submitted to the Department of Physics on May 24, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract From its inception, statistical mechanics has aspired to become the link between bi- ology and physics. But classical statistical mechanics dealt primarily with systems in thermal equilibrium, where detailed balance forbids the directed motion characteris- tic of living things. Formal variational principles have recently been discovered for nonequilibriumsystemsthatcharacterizetheirsteady-statepropertiesintermsofgen- eralized thermodynamic quantities. Concrete computations using these principles can usually only be carried out in certain limiting regimes, including the near-equilibrium regime of linear response theory. But the general results provide a solid starting point for defining these regimes, demarcating the extent to which system’s behavior can be understood in thermodynamic terms. I use these new results to determine the range of validity of a variational proce- dure for predicting the properties of near-equilibrium steady states, illustrating my conclusions with a simulation of a sheared Brownian colloid. The variational principle provides a good prediction of the average shear stress at arbitrarily high shear rates, correctly capturing the phenomenon of shear thinning. I then present the findings of an experimental collaboration, involving a specific example of a nonequilibrium struc- tureusedbylivingcellsintheprocessofendocytosis. Ifirstdescribethemathematical model I developed to infer concentrations of signaling molecules that control the state of this structure from existing microscopy data. Then I show how I performed the inference, with special attention to the quantification of uncertainty, accounting for the possibility of “sloppy modes” in the high-dimensional parameter space. In the final chapter I identify a trade-off between the strength of this kind of structure and its speed of recovery from perturbations, and show how nonequilibrium driving forces can accelerate the dynamics without sacrificing mechanical integrity. Thesis Supervisor: Jeremy L. England Title: Cabot Career Development Associate Professor of Physics 3 4 Acknowledgments This interdisciplinary work would not have been possible without the generous col- laboration of a large number of people with a wide range of expertise. First of all, I would like to thank Prof. Jeremy England for his mentorship over these past five years, and for nurturing such a vibrant intellectual environment in our lab. It has been wonderful to see the group grow and mature almost from its very beginnings, and I owe a great deal of my own personal growth during this time to its rich atmo- sphere of genuine friendship and serious scholarship. I am also grateful for the hard work of the administrative staff – especially to Catherine Modica, who has been so helpful and encouraging from the beginning of my degree to the end. I need to express my deep appreciation for the hospitality of Prof. Tomas Kirch- hausenandhisgroup, whohavepatientlybroughtmeuptospeedonsomeofthemost exciting topics in cell biology, and transformed me from a physicist curious about bi- ology into a biophysicist. Special thanks go to Kangmin He, who welcomed me into his core postdoctoral project, and performed all the experiments that form the basis of my fourth chapter. I also had many valuable conversations with Ilja Kusters, Ben- jamin Capraro, Gokul Upadhyayula, and Joe Sarkis about scientific issues related to this project. I want to thank Catherine McDonald as well, for always being ready to help with any logistical issue. I am grateful to Tal Kachman for introducing me to Python and the iPython Notebook. The numerical work and data analysis in Chapters 3 - 5 would have been much more challenging and time-consuming without these helpful tools. For the sim- ulations of Chapter 3, I want to acknowledge the contribution of Benjamin Harpt, an undergraduate research assistant who helped me finish that project and start explor- ing some possible extensions. The model of Chapter 5 matured through discussions with another undergraduate researcher, Arsen Vasilyan, and with postodoc Sumantra Sarkar. Finally, I want to thank my friends who have generously given me feedback on this manuscript and related papers. Jordan Horowitz, Todd Gingrich, Gili Bisker and 5 Zachary Slepian have each dedicated time to reading and commenting on my drafts, and have dramatically improved the quality of the final product. 6 Contents 1 Introduction 11 2 The Edge of Linear Response 15 2.1 Introduction: A Driven Ideal Gas . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Overdamped Model . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Periodic Steady State . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 A Variational Principle . . . . . . . . . . . . . . . . . . . . . . 19 2.1.4 Range of Validity . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Microscopic Reversibility . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Microcanonical Derivation . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Environment Entropy . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4 Path Ensemble Averages . . . . . . . . . . . . . . . . . . . . . 30 2.3 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Markov Jump Process . . . . . . . . . . . . . . . . . . . . . . 34 2.3.2 Langevin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4 Coarse-Grained Steady-State Distribution from Forward Statistics . . 37 2.4.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.2 Cumulant Expansion . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Extended Linear Response . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 Phenomenological Equations and Fluctuation Trajectories . . 45 2.5.2 Work Statistics for Linear Dynamics . . . . . . . . . . . . . . 46 7 2.5.3 Linearity and Nonlinearity . . . . . . . . . . . . . . . . . . . . 48 2.5.4 Nonlinear Correction . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.5 Degree of Nonequilibrium . . . . . . . . . . . . . . . . . . . . 50 2.6 Application to Other Models . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.1 Transport of Energy and Particles . . . . . . . . . . . . . . . . 52 2.6.2 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Shear Thinning in Brownian Colloids 59 3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.1 Flow-Induced Steady State . . . . . . . . . . . . . . . . . . . . 60 3.1.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.3 Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Probabilities and Work Statistics . . . . . . . . . . . . . . . . . . . . 64 3.2.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2 Physical Intuition . . . . . . . . . . . . . . . . . . . . . . . . . 68 4 Regulating Disassembly in Clathrin-Mediated Endocytosis 71 4.1 Biological Background . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.1 Auxilin and Hsp70 . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Phosphoinositides and Fission Sensing . . . . . . . . . . . . . 75 4.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 Auxilin-based Sensors . . . . . . . . . . . . . . . . . . . . . . 79 4.2.2 Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Data Interpretation and Averaging . . . . . . . . . . . . . . . . . . . 84 4.3.1 Deterministic Modeling of Stochastic Processes . . . . . . . . 84 4.3.2 Biological Heterogeneity . . . . . . . . . . . . . . . . . . . . . 86 4.4 Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.1 Sensor Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.2 Phosphoinositide Kinetics . . . . . . . . . . . . . . . . . . . . 90 4.4.3 Clathrin Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4.4 Summary of Modeling Assumptions . . . . . . . . . . . . . . . 94 8 4.5 Data Fitting and Sensitivity Analysis . . . . . . . . . . . . . . . . . . 94 4.5.1 Bayesian Statistics . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.2 Estimating Cost Function . . . . . . . . . . . . . . . . . . . . 97 4.5.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . 98 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.6.1 PIP Concentrations . . . . . . . . . . . . . . . . . . . . . . . . 102 4.6.2 Enzyme Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.6.3 Sensor Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Accelerating Kinetics through Nonequilibrium Driving 109 5.1 The Trade-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Active Dissociation Model . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2.1 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2.2 Mean-Field Interactions . . . . . . . . . . . . . . . . . . . . . 114 5.2.3 Coarse-Grained Rates . . . . . . . . . . . . . . . . . . . . . . 116 5.3 Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1 Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.3 Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.4 Cost of Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6 Conclusions 131 A Microscopic Reversibility in the Langevin Equation 143 A.1 Langevin Equation for Brownian Motion . . . . . . . . . . . . . . . . 144 A.2 Colloid Simulation with Externally Imposed Flows . . . . . . . . . . . 146 B Dual Processes in Multiple Dimensions 149 9 C Perturbative Calculations of Work Statistics for Nonlinear Macro- scopic Dynamics 155 C.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 155 C.2 Driving by Imposed Flow . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.3 Driving by Thermal/Chemical/Mechanical forces . . . . . . . . . . . 160 D Physical Justification of Mean Wall Stress 163 D.1 Stress in Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . 164 D.2 Mapping to Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . 166 D.3 Obtaining the Induced Charge on the Conducting Plate . . . . . . . . 167 D.3.1 Contribution of Variations about the Mean . . . . . . . . . . . 169 D.3.2 Contribution of the Mean Charge . . . . . . . . . . . . . . . . 170 D.4 Mapping Back to Hydrodynamics . . . . . . . . . . . . . . . . . . . . 171 10