Dissipation and Control in Microscopic Nonequilibrium Systems

Description

This thesis establishes a multifaceted extension of the deterministic control framework that has been a workhorse of nonequilibrium statistical mechanics, to stochastic, discrete, and autonomous control mechanisms. This facilitates the application of ideas from stochastic thermodynamics to the understanding of molecular machines in nanotechnology and in living things. It also gives a scale on which to evaluate the nonequilibrium energetic efficiency of molecular machines, guidelines for designing effective synthetic machines, and a perspective on the engineering principles that govern efficient microscopic energy transduction far from equilibrium. The thesis also documents the author’s design, analysis, and interpretation of the first experimental demonstration of the utility of this generally applicable method for designing energetically-efficient control in biomolecules. Protocols designed using this framework systematically reduced dissipation, when compared to naive protocols, in DNA hairpins across a wide range of experimental unfolding speeds and between sequences with wildly different physical characteristics. Supervisor's Foreword Acknowledgment Contents 1 Introduction 1.1 Molecular Machines 1.1.1 Kinesin 1.1.2 ATP Synthase 1.2 Nonequilibrium Statistical Physics 1.3 Overview of This Thesis 1.4 Contributions to This Thesis References 2 Theoretical Background 2.1 Mathematical Preliminaries 2.1.1 Random Variables, Probabilities, and Characteristic Functions 2.2 Nonequilibrium Dynamics 2.2.1 Master Equation 2.2.2 Fokker-Planck Equation 2.2.3 Langevin Equation 2.3 Stochastic Thermodynamics 2.4 Fluctuation Theorems 2.5 Entropy and Information Theory 2.6 Control in Microscopic Nonequilibrium Systems 2.6.1 Linear-Response Theory Linear-Response for Step Perturbations Linear-Response for Time-Dependent Protocols 2.6.2 Generalized Friction Tensor 2.6.3 Minimal-Work Control Protocols 2.7 Model Systems 2.7.1 Harmonic Trap 2.7.2 Periodic Potential 2.7.3 Fast-Switching Potential References Part I Experimental Tests of Nonequilibrium Theory 3 DNA Hairpins I: Equilibrium 3.1 Introduction 3.2 Experimental Setup 3.3 Equilibrium Sampling 3.3.1 Estimating the Generalized Friction Coefficient 3.3.2 Designing Protocols Model Selection and Information Criterion References 4 DNA Hairpins II: Nonequilibrium 4.1 Unfolding/Refolding Force identification 4.2 Excess Work Measurements 4.2.1 Excess Power in Designed and Naive Protocols 4.2.2 Cycle Work and Dissipation in Designed and Naive Protocols 4.3 Protocol Work Ratios 4.3.1 Excess Work Ratio for Variable Bin Widths 4.3.2 Protocol-Work Ratio for DNA Hairpins References 5 DNA Hairpins III: Conclusions 5.1 Alternative Hairpin Sequence 5.2 Mean-Variance Trade-Offs for Excess Work 5.3 Alternative Buffer Conditions 5.4 Discussion References Part II Dissipation in Nonequilibrium Systems Through the Lens of Control Theory 6 Stochastic Control 6.1 Introduction 6.2 Revisiting Linear Response 6.3 Protocol Ensembles 6.3.1 Expansion of the Excess Power 6.3.2 Lower Bound on Excess Work 6.4 Model Ensembles 6.4.1 Periodic-Potential Ensemble Exact Solution in the Zero-Barrier Limit (βE = 0) Numerical Results for Non-zero Barriers (βE ≠0) 6.4.2 Stochastically Driven Protocols 6.5 Discussion References 7 Optimal Discrete Control 7.1 Introduction 7.2 Background 7.3 Infinite-Time Work 7.4 Nonequilibrium Excess Work 7.4.1 Nonequilibrium Excess Work: Linear Response for Time-Dependent Protocols 7.5 Minimum-Work Protocols 7.5.1 Minimum-Work Protocols for a Single Control Parameter 7.6 Harmonic Trap 7.6.1 Infinite-Time Limit 7.6.2 General Solution: Finite-Time Work 7.7 Periodic Potential 7.8 Discussion References 8 On Dissipation Bounds 8.1 Introduction 8.2 Discrete Stochastic Protocols 8.3 A Cost for Control 8.4 Harmonic System 8.4.1 Timescale-Separated Limit 8.4.2 Nonequilibrium Excess Work 8.4.3 General Dissipation Bound 8.5 Discussion References Part III The Nonequilibrium Physics of Autonomous Machines 9 Free Energy Transduction 9.1 Introduction 9.2 Strongly Coupled Multi-component Systems 9.2.1 Entropy Production 9.2.2 Excess Work 9.3 Classes of Upstream Dynamics 9.3.1 External Control Parameter 9.3.2 Thermodynamically Complete System 9.4 Model System 9.4.1 Excess Power Does Not Equal Entropy Production 9.4.2 Excess Power Can Become Negative 9.4.3 Entropy Production in Thermodynamically Complete or Incomplete Systems 9.5 Discussion References 10 Hidden Excess Power 10.1 Introduction 10.2 Coarse-Grained Representations of Mechanochemical Systems 10.3 Hidden Excess Work in Molecular Machines 10.3.1 TSS Excess Work 10.3.2 Nonequilibrium Excess Work 10.4 Model Systems 10.4.1 Linear-Transport Motor 10.4.2 Rotary Motor 10.5 Discussion References 11 Conclusions and Outlook 11.1 Outlook 11.2 Final Remark References A Code and Data A.1 Master Equation A.1.1 Trajectory Simulation A.2 Langevin Equation A.2.1 Underdamped Dynamics A.2.2 Overdamped Dynamics A.3 Coupled Discrete and Continuous Dynamics B DNA Hairpins B.1 Folding Forces B.2 Alternative Excess Work Measures C Stochastic Control C.1 Generalization of Lower Dissipation Bound C.2 Disagreement Between Theoretical Predictions and Numerical Results C.3 Equivalence of Ensembles D Optimal Discrete Control D.1 Expansion of the Relative Entropy D.2 Harmonic Trap: Exact Result E On Dissipation Bounds E.1 Generalized Friction for Gamma-Distributed Dwell Times E.2 Generalized Friction for Gamma-Distributed Dwell Times: Harmonic-Trap E.3 Average Step Number for Uniform Jump Rates F Free Energy Transduction F.1 Detailed Derivation of Transduced Additional Free Energy Rate F.2 At Steady State, Excess Power Equals Heat Flow G Hidden Excess Power G.1 Expansion of the TSS Work G.2 Nonequilibrium Excess Work in Autonomous Systems G.3 Simulation Details: Linear-Transport Motor References