ENERGY-SPEED-ACCURACY TRADEOFFS IN A DRIVEN, STOCHASTIC, ROTARY MACHINE

Transcription

1 ENERGY-SPEED-ACCURACY TRADEOFFS IN A DRIVEN, STOCHASTIC, ROTARY MACHINE by Alexandra Kathleen Kasper B.Sc., McMaster University, 2015 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF PHYSICS FACULTY OF SCIENCE c Alexandra Kathleen Kasper 2017 SIMON FRASER UNIVERSITY Summer 2017 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

2 Approval Name: Degree: Title: Examining Committee: Alexandra Kathleen Kasper Master of Science (Physics) Energy Speed Accuracy Tradeoffs in a Driven, Stochastic, Rotary Machine Chair: Malcolm Kennett Associate Professor David Sivak Senior Supervisor Assistant Professor John Bechhoefer Supervisor Professor Nancy Forde Internal Examiner Associate Professor Date Defended: August 3, 2017 ii

3 Abstract Molecular machines are stochastic systems capable of converting between different forms of energy such as chemical potential energy and mechanical work. The F 1 subunit of ATP synthase couples the rotation of its central crankshaft with the synthesis or hydrolysis of ATP. This machine can reach maximal speeds of hundreds of rotations per second, and is believed to be capable of nearly 100% efficiency in near-equilibrium conditions, although a biased cycling machine is a nonequilibrium system and therefore must waste some energy in the form of dissipation. We explore the fundamental relationships among the accuracy, speed, and dissipated energy of such driven rotary molecular machines, in a simple model of F 1. Simulations using Fokker-Planck dynamics are used to explore the parameter space of driving strength, internal energetics of the system, and rotation rate. A tradeoff between accuracy and work as speed increases is found to occur over the range of biologically relevant timescales. We search for a way to improve this tradeoff by applying approximations of dissipation minimizing protocols and find a reduction in both work and accuracy, yet accuracy drops less than the work does, leading to an overall decrease in the ratio of work to accuracy. Keywords: Molecular machines; Fokker-Planck dynamics; Nonequilibrium tradeoffs iii

4 Acknowledgements I would like to thank David Sivak for allowing me to truly experience the process of research. I fell down a few rabbit holes along the way but your calm confidence that I would pull it together kept me going. You have helped me hone my physics intuition and reaffirmed my passion for aesthetics in academic figures. I would also like to thank all the members of the Sivak group for their patience during the months of Berry curvature and valuable discussions during group meetings. I would especially like to thank Steve Large and Aidan Brown for pointing me in the right direction and agreeing there is no obvious solution to my problems. The biophysics community at SFU is also deserving of my thanks - you create such a supportive atmosphere inclusive of students and faculty. I acknowledge the financial support of Simon Fraser University through the C.D. Nelson Multi-Year Fellowship and the Natural Sciences and Engineering Research Council of Canada through the Canada Graduate Scholarship. My family has provided me with unconditional support since I began my academic journey after high school. My mother sparked my curiosity during my first science experiments as a toddler and has continued to support me in my pursuit of physics, even though it isn t chemistry. My grandparents have continued to inspire me with their passion for learning and close following of the theoretical physics community - I am certain they know more about string theory than I do. My brother inspires me with his patience and persistence and reminds me of the importance of family and friends. Finally, I want to acknowledge the phenomenal support of Chapin Korosec. I truly could not have asked for a better study buddy and life partner. iv

5 Dedication To my grandparents and life-long learners, Alan and Brenda Holvey, for reminding me to never forget the physics. v

6 Contents Abstract Acknowledgements Dedication Contents List of Figures iii iv v vi ix 1 Introduction Molecular Machines Thermal Fluctuations Reynolds Number of Molecular Machines F o F 1 ATP Synthase Structure and Function Mechanochemical Coupling and Efficiency Energy-Speed-Accuracy Tradeoffs Motivations and Goals Theoretical Framework Nonequilibrium Thermodynamics Example: The Three-State Cycle Work, Heat, and Driving Driving of Cycles vi

7 CONTENTS vii Work Accumulation Example: Quadratic Trap Work-Speed-Accuracy Tradeoffs Optimal Control Friction and Minimum-Dissipation Driving Protocols Ratio of Naive and Optimal Excess Work Model System Goals of the Model System Definitions Work and Heat System Dynamics Continuity Equation Accuracy Periodic Steady State and Rotational Symmetry Methods Numerical Fokker-Planck Dynamics Flux Calculation and Accuracy Periodic Steady State Stability Energy and Time Scales Setting the Time Scale Energy Scales Friction Calculation Optimal Protocol Calculation Results Naive Driving Protocols Accuracy and Work vs. Speed Accuracy and Work Friction and Minimum-Dissipation Driving Protocols Minimum-Dissipation vs. Naive Driving Protocols

8 CONTENTS viii 5.4 Parameter Ranges: Discussion and Limitations Conclusions Significance ATP Synthase Artificial Machines Future Work Extending the Model Adaptive Time Step Friction Calculation Higher-Order Corrections to Minimum-Dissipation Protocol Experimental Bibliography 65 Appendices 71 A Quadratic Approximations 72

9 List of Figures 1.1 ATP Synthase Three State Example Energy Landscape Stability Friction Schematic Probability Distributions Naive Accuracy and Work versus Speed Naive Work and Accuracy Ratio, A= Naive Accuracy and Work Naive Flux and Power Friction, Velocities, and Protocol Average Friction Minimum-Dissipation Accuracy and Work versus Speed Minimum-Dissipation Accuracy and Work Minimum-Dissipation Flux and Power Naive and Minimum-Dissipation Ratios ix

10 Chapter 1 Introduction The second law of thermodynamics states that the universe evolves towards disorder. Yet when we look at life on both the macroscopic and microscopic scale, we observe incredible structure and seemingly directed and repeatable development. How can we address this apparent paradox of physics demanding spontaneous disorder and biology demonstrating an ability to evolve increasingly complex organisms over millions of years? In Chapter 2, we will delve further into the limitations of the laws of thermodynamics, but here we will briefly dissolve the present paradox by pointing out that the law of increasing disorder applies to closed systems evolving towards equilibrium. That is to say, on average, the universe as a whole becomes more disordered with time. However, the cell is not an isolated system: chemicals and heat are exchanged with the surroundings, adding to the disorder of the universe. Additionally, a living cell is not in equilibrium and in fact death, as famously declared by Schrödinger, is to decay into thermodynamical equilibrium [1]. Furthermore, thermodynamics deals with the average behaviour of a system, and in this case the second law states that on average the entropy of the universe increases. Yet the energetics inside the cell are sensitive to thermal fluctuations on the scale of 1 k B T =4.114 pn nm at room temperature, and deviations from average do indeed occur. An individual process may increase the organization of the cell, but at the cost of increasing disorder elsewhere and consuming energy to do so. Life is essentially a continuous battle against disorder and the soldiers are molecular machines: squishy, nanoscale proteins continuously bombarded by thermal fluctuations and fueled by chemical energy. Developing a model of a mechanochemically coupled molecular machine requires aban- 1

11 CHAPTER 1. INTRODUCTION 2 doning the intuition we have built from interacting with our macroscopic world. In the next section, we will build an intuition about the intracellular environment and the physics governing the molecules responsible for keeping us away from the unfortunate state of thermal equilibrium. More specifically, the level of noise in the form of thermal fluctuations produces large deviations from average behaviour. Furthermore, molecular machines operate in the low-reynolds regime, where inertia is negligible. Finally, understanding the energy scales of work and subsequent dissipated heat required to drive the cycle allows the quantification of efficiency. By creating a basic model that is tested against the observed behaviour of biological molecular machines, we can begin to play with the intrinsic energetics of the system and search for optimal molecular designs and driving protocols. Our model focuses on rotary machines and is inspired by F 1 ; therefore, this chapter concludes with an overview of ATP synthase. 1.1 Molecular Machines Molecular machines are found in every form of life including eukaryotes such as plants and animals but also prokaryotes: single-celled organisms that have no intracellular organelles or even a nucleus. The variability between life forms is so great that molecular machines are among the few common elements that appear to be crucial for sustaining life, such as a cellular membrane and a genetic code in the form of DNA. Molecular machines are a class of protein complexes capable of converting between two types of energy, for example chemical potential and mechanical motion. Adenosine triphosphate (ATP) is their most common source of chemical energy, releasing approximately 20 k B T of energy when a phosphate bond is hydrolysed, producing adenosine diphosphate (ADP). Molecular machines can be classified into two main groups: linear walkers and rotary systems. Linear walkers such as the myosin, kinesin, and dynein families achieve processive motion by hydrolysing ATP to step along linear tracks within the cell [2]. Kinesin and dynein walk along microtubules: long protein filaments that create a network within the cytoplasm [3]. Kinesin is responsible for a variety of crucial tasks including facilitating cellular division and transporting vesicles and organelles within the cell [4, 3, 5]. Dynein is also capable of carrying cargo and is crucial to the mobility of hair-like structures called cilia found on the surface of eukaryotic cells [6, 3]. Myosin interacts with actin, another

12 CHAPTER 1. INTRODUCTION 3 protein-based filament that together with the microtubule network forms the cytoskeleton within cells [3]. Myosin and actin are central to the mechanism of muscle contraction [3]. The disruption of the function of linear walker motor proteins has been associated with diseases such as cancer [4] and numerous other disorders [6]. While the geometry of linear walkers and rotary motors may distinguish their behaviour, the mechanism of mechanochemical coupling remains an elusive yet common problem when trying to understand how they function. Mechanochemical coupling refers to the energy transduction between chemical energy, often stored in the γ phosphoanhydride bond of ATP, and mechanical work in the form of linear or rotary motion. The two common examples of rotary machines are ATP synthase and flagella motors in bacteria. Both of these systems are associated with a cellular or organellar membrane in order to achieve rotation relative to a component tethered in the membrane. As the name suggests, ATP synthase synthesizes ATP and is therefore crucial to the functioning of a cell, since many processes ATP. For example, it is estimated that it DNA replication in E. coli costs the equivalent of ATP molecules [7]. Flagellar motors are found in bacteria and generate the torque responsible for turning the flagellar filaments and propelling the cell. Single molecule experiments can measure the time scales and forces that are relevant to molecular machines. Molecular machines hydrolyse ATP at rates of tens to hundreds of Hertz, or roughly 1 ATP per 10 ms [2, 8]. Furthermore, one type of kinesin has been shown to be capable of processive walking up to a maximal load force (stall force) of 8 pn [5]. Additionally, ATP synthase has a stall torque of roughly 30 pn nm/rad [9]. Furthermore, some of these machines are believed to be capable of nearly 100% efficiency, meaning the liberated chemical energy is nearly perfectly converted into mechanical work or the final chemical product [2, 10]. The motivation for developing a theoretical framework to describe the operation of molecular machines should be apparent from their importance to sustaining life, intrigue as an energy transducer, and inspiration for artificial nanoscale machines Thermal Fluctuations Thermal fluctuations refer to random deviations from the average system state, arising in systems with non-zero temperature and quantified by k B T, where k B is Boltzmann s con-

13 CHAPTER 1. INTRODUCTION 4 stant and T is the temperature of the surroundings. Fluctuations in the conformation or position of molecules originate from stochastic collisions with the surrounding medium. Biological systems like proteins in cells are surrounded by an aqueous environment composed of water and the other intracellular molecules. For the purposes of modelling the influence of the intracellular environment, it is often assumed to be homogeneous with the mechanical properties of water. Thermal fluctuations are incorporated into models in the form of Gaussian-distributed delta-correlated forces: at each instant in time the system is subjected to a random force drawn from a Gaussian distribution with zero mean and a standard deviation proportional to k B T. At higher temperatures, larger random forces are increasingly likely. Thermal fluctuations are the underlying cause of Brownian motion and molecular diffusion processes. At room temperature, k B T =4.114 pn nm. This is comparable to the energetic barriers between different conformational states of proteins and therefore thermal fluctuations result in significant stochastic motion Reynolds Number of Molecular Machines The Reynolds number captures the dominant forces in a particular system and is defined as the ratio of the inertial to viscous forces: Re = Lvρ η, (1.1) where L is the length scale of the object, v is the speed, and ρ and η are the density and viscosity of the surrounding medium, respectively. For the rotary machine ATP synthase, the Reynolds number of a 2 µm actin filament rotating at 100 rotations per second in a fluid with the properties of water is approximately Furthermore, the linear walker kinesin pulling cargo with radius 100 nm at 1 µm/s has a Reynolds number of The Reynolds numbers of nanoscale machines are an important consideration when modelling biological machines as well as when designing synthetic nanomachines [11]. At low Reynolds numbers (Re 1) the behaviour of the system depends solely on the forces that are acting at that moment and does not remember the past through inertia [12]. In this overdamped regime, behaviour is instead determined by the instantaneous applied force while the force history has no lasting effect and therefore continuous forcing is required

14 CHAPTER 1. INTRODUCTION 5 for forwards motion. An object that is not presently being pushed will come to rest nearly instantaneously. 1.2 F o F 1 ATP Synthase F o F 1 ATP synthase is a molecular machine that synthesizes up to 90% of the ATP produced in the cell [10, 13]. The complex consists of two components: F o and F 1. F o spans a membrane, commonly the mitochondrial membrane in eukaryotes, and harnesses energy by allowing protons through the membrane. The energy arises from the electrochemical gradient maintained across the mitochondrial membrane by other chemical processes in the cell. In non-eukaryotic cells, ATP synthase is found in the cell membrane itself, since the cell does not have organelle membranes. F o is coupled to F 1 through the rotation of a central crankshaft and drives, at three sites in F 1, the chemically unfavourable synthesis reaction of ATP from ADP and inorganic phosphate. The machine can also operate in reverse, consuming ATP to maintain a proton gradient. Furthermore, the units can uncouple, and F 1 can be observed to operate as an ATP-burning motor (ATPase) when supplied with ATP Structure and Function F o F 1 ATP synthase is found in the membranes of bacteria, mitochondria in eukaryotic cells, and chloroplasts in plant cells. As seen in Figure 1.1, F o is membrane-bound while F 1 sits beside the membrane. The F o unit has 3 types of subunits: a, b, and c. A single a and two b subunits make up the stationary portion and remain fixed in the membrane. Depending on the species, F o has between 10 and 14 copies of the c subunit, which form the c-ring [14]. The a subunit and c-ring complex are involved in the proton transfer across the membrane; however, the exact mechanism remains unknown [15, 13]. F 1 is made up of a barrel structure with three-fold symmetry (α 3 β 3 ), along with γ,δ and ε subunits. The α and β subunits alternate to form a hexamer (6-component structure) with a barrel-like arrangement. There is one catalytic site in each β subunit, near the interface with the neighbouring α subunit. This is where ATP synthesis or hydrolysis occurs. Co-axial to the barrel structure is the γ subunit, also called the crankshaft or γ-shaft. The ε subunit rotates with the γ-shaft and couples with the c-ring. The δ subunit tethers the α 3 β 3 barrel to the

15 CHAPTER 1. INTRODUCTION 6 a c n Lipid bilayer b 2 F 1 : 3 3 F o : ab 2 c n Figure 1.1: Schematic diagram (not to scale) of ATP synthase. F o is composed of ab 2 c n, where n indicates that the c-ring has a variable number of subunits between different species (typically [14]). F 1 is composed of α 3 β 3 γδε. Solid components (α 3 β 3 γ) indicate the subunits typically used in experiments studying isolated F 1. two b subunits of F o. The rotating pieces (c-ring in F o and γε in F 1 ) are together referred to as the rotor, and the stationary components (ab 2 in F o and α 3 β 3 δ in F 1 ) the stator. In most biological situations the F o F 1 complex produces ATP; however, the system is capable of operating in reverse by using ATP to generate a protonmotive force. This is common in the fermentation process in anaerobic bacteria [16]. Furthermore, the F 1 unit can detach from F o and operate as a stand-alone reversible system. When supplied with ATP, F 1 operates as a motor, sometimes called ATPase because it consumes ATP, and rotates the γ-shaft in the opposite direction to that during synthesis.

16 CHAPTER 1. INTRODUCTION 7 Due to the difficulty of working with integral membrane proteins, most experimental work on ATP synthase has focused on only the F 1 unit [8, 9, 17]. The rotation of ATP synthase was originally predicted purely from its structure and known biochemical properties, best presented by Paul Boyer in 1993 [18]. The rotation of the central crankshaft was directly observed in 1997, confirming the prediction that the F 1 unit is indeed a rotary machine [8]. However, these experiments were performed on only a portion of the F 1 unit (α 3 β 3 γ) and observed rotation due to hydrolysis. The rotation of the c-ring was directly observed in 1999, confirming that the c-ring rotates with the γ- shaft [19, 20]. These experiments were again performed in the ATP hydrolysis direction. In 2004, Itoh et al. successfully produced ATP with isolated F 1 (α 3 β 3 γ complex) by mechanically driving rotation in the synthetic direction using a rotating magnetic field [21]. This experiment confirmed both the reversibility of the machine and that ATP production does not depend on the presence of the F o unit Mechanochemical Coupling and Efficiency ATP synthase is frequently said to be nearly 100% efficient and have perfect mechanochemical coupling [9, 10, 22]. In these cases, efficiency refers to the energy transduction, or transfer, between energy stored in the electrochemical gradient across the membrane and the free energy stored in the phosphate-phosphate bond of ATP. This energy transfer is facilitated by the mechanical torque exerted on the γ-shaft. The ratio of protons let through the membrane by F o to ATP molecules produced in F 1 (H + /ATP) is one way to quantify the mechanochemical coupling. In [10], the authors argue that in the case of perfect coupling the ratio (H + /ATP) should match the ratio of c subunits to β subunits, c/β, since one proton is shuttled through the membrane for each c subunit. They observe a perfect match between these ratios and were additionally able to resolve the energy stored in the electrochemical gradient leading to the conclusion of 100% energy conversion. These experiments had rotation rates of less than 1 rotation per second. In order to obtain accurate measurements in single molecule experiments, the system is often driven slower than in vivo. ATP synthase is believed to operate at hundreds of Hz in living cells [23], and it remains unclear how its operation and efficiency might degrade at these higher speeds. Another approach to estimating energy transfer efficiency is comparing the torque gen-

17 CHAPTER 1. INTRODUCTION 8 erated on a large filament attached to the γ-shaft in isolated F 1 to the energy liberated by ATP hydrolysis. Imaging experiments have confirmed that the γ-shaft rotates in three 120 steps, corresponding to the three catalytic sites on the barrel [14, 17, 24]. Additionally, sub-steps of 80 and 40 have been resolved and mapped to intermediate states in the chemical pathway: ATP binding/adp release and P i release, respectively [25, 17]. It is well accepted that one full rotation of the γ-shaft corresponds to three identical catalytic processes. The free energy change upon hydrolysis of ATP is approximately 20 k B T under physiological conditions and therefore one hydrolysis (ATP consuming) cycle corresponds to 60 k B T (247 pn nm) of free energy liberated from ATP. The chemical free energy is used to generate a torque on the γ-shaft which pulls the attached filament through the viscous surroundings, and the energy ultimately gets dissipated as heat into the surroundings. By estimating the torque and therefore the work output of the machine, experimentalists can estimate the efficiency. The work output is given by the product of the torque and the angle swept out. Experimentalists have attached a large actin filament to the γ-shaft and measured work output of 240 pn nm per cycle [8, 22], concluding that nearly 100% of the chemical energy was transferred into mechanical rotation. This particular experiment observed rotation rates between 0.1 and 8 rotations per second. Certainly, experiments can be designed to measure the mechanochemical coupling and efficiency of ATP synthase with impressive precision; however, all experiments so far that find perfect coupling or 100% efficiency are conducted using stall forces or thermodynamically equilibrated states. ATP synthase operates at hundreds of rotations per minute in vivo and can reach maximum speeds of 350 revolutions per second at high temperatures and surplus chemical reactants [23]. Does its efficiency degrade at these higher speeds in living cells? 1.3 Energy-Speed-Accuracy Tradeoffs The optimization of a process requires a clear definition of what aspects one wishes to optimize. We might hypothesize that a cell optimizes for output of a certain reaction while being constrained by dissipation. For example, ATP synthase needs to produce a high number of ATP per second but is limited by the free energy available from the proton gradient across the mitochondrial membrane. Optimizing for speed may lead to decreases

18 CHAPTER 1. INTRODUCTION 9 in the accuracy of a process and increase the work required. In this way, one expects machine performance to face a three-way tradeoff between the speed, accuracy, and energy (in the form of dissipation). Tradeoffs of this nature have been explored in different cell biological systems. For example, recent works have demonstrated a tradeoff between speed and errors in molecular proofreading processes [26], a bound on precision set by the power and speed of physical communication channels [27], and energy-speed-accuracy tradeoffs during environmental sensing [28] and sensory adaptation [29]. 1.4 Motivations and Goals Molecular machines are impressively efficient stochastic energy transducers. These nanoscale objects operate out of equilibrium and harness thermal fluctuations to achieve directional operation. In order to define and explore the frontier of operating molecular machines, we have developed a minimalistic model of a driven rotary machine inspired by the F 1 unit of ATP synthase. We vary the driving strength, rate of rotation, and intrinsic barriers between states to explore the tradeoffs between energy, speed, and accuracy. We use energy to refer to the work required to drive rotation and accuracy to refer to the response of the system to driving. The Smoluchowski equation (the overdamped Fokker-Planck equation) is used to evolve probability distributions on time-dependent energy landscapes. The search for the frontier of machine operation leads to the question of whether there is some optimal way to drive the system, instead of a naive constant-speed protocol. Theoretical work predicting dissipation-minimizing driving protocols is applied to our model and assessed for applicability. We create a minimal model independent of the molecular details of the F 1 system, so that our model can be applied to other rotary machines (such as bacterial flagella motors) and inspire the design of synthetic machines.

19 Chapter 2 Theoretical Framework In order to explore the nonequilibrium behaviour of a model of a molecular machine, we must first build our tool kit of theory and intuition. This chapter sets the stage by introducing ideas from thermodynamics and statistical mechanics and discussing the additional considerations of nonequilibrium systems. We build definitions of work and heat applicable to our system, in order to quantify the effect of driving and consider the predictions of work accumulation in a simple nonequilibrium system. Finally, we explore the theory of a generalized friction, leading to an approximation of protocols that minimize excess work. 2.1 Nonequilibrium Thermodynamics The laws of thermodynamics were developed in the 19th century in parallel with the development of large machines like the steam engine and internal combustion engine. Questions such as energy transfer using temperature gradients and predictions of efficiency in these large systems drove the development of classical thermodynamics, which describes the bulk behaviour of large systems near equilibrium. Limiting oneself to only studying behaviour near equilibrium is a practical decision, since any isolated system, when left unperturbed, will settle into the stationary state called thermal equilibrium. In systems with fast relaxation times, one can also suppose that the system relaxes so quickly into its new equilibrium state that it spends an insignificant amount of time transitioning between equilibrium states. To discard the messy details of nonequilibrium behaviour resulting from changing external conditions, many classical thermodynamics problems start with the assumption that 10

20 CHAPTER 2. THEORETICAL FRAMEWORK 11 changes happen asymptotically slowly, meaning it takes an infinite amount of time for the process under study to occur. This is called the quasistatic limit, and in conjunction with the thermodynamic limit, which assumes the system has an infinite number of particles, we can begin to see why the study of biological processes, which occur in small systems in finite time, requires careful consideration before broadly applying thermodynamic relations. Statistical mechanics explores the connection between macroscopic properties and the underlying behaviour of individual molecules. While thermodynamics predicts average properties, statistical mechanics asks about the distribution of microscopic states, sometimes called microstates. One of the most important results from statistical mechanics is the Boltzmann distribution, which describes closed systems in thermal equilibrium. If the energy of a microstate is ε i, the probability of finding the system in that microstate is proportional to e ε i/k B T, where k B is the Boltzmann constant and T is the temperature of the system. Thermal fluctuations are the origin of deviations from average. An important property of this distribution is that every state has a non-zero probability of occurring, and the likelihood of observing a higher energy-state increases with temperature. Nonequilibrium thermodynamics is a comparatively recent area of research and is still evolving. In general, the aim is to describe systems that are not in thermal equilibrium, and most progress to date has been made with systems near equilibrium. The motivation for wanting to understand nonequilibrium systems should be apparent when considering the quasistatic and thermodynamic limits and their contradiction with reality: most systems, including all biological systems, are changing on finite time scales and are of finite size. There are three common ways in which nonequilibrium systems are modelled: during relaxation to thermal equilibrium, while being subjected to driving, and when detailed balance is broken. These situations are not mutually exclusive and indeed the response of a system subject to driving is defined by its relaxation behaviour Example: The Three-State Cycle To continue our discussion, let us become more concrete and consider a physical system of finite size, for example a particle with three microstates, with respective energies ε 1, ε 2, and ε 3. The (macroscopic) state of the system can be defined as the set of occupancy probabilities of each microstate. If we left the system connected to a thermal reservoir at

21 CHAPTER 2. THEORETICAL FRAMEWORK 12 constant temperature, it would eventually reach a steady state in which the probability of occupying each microstate is constant. The particle can still switch microstates, but the probability of being in each microstate is in agreement with the Boltzmann distribution, and the system is said to be in thermal equilibrium. In general, the system could be out of equilibrium in a few ways which we detail below. Relaxation First, it could be the case that it has not yet relaxed to thermal equilibrium and is instead in a transient state. For example, consider starting with 100% probability in microstate 1 and then subsequently exchanging, or hopping, to occupy other microstates. In this thesis we are not interested in transient behaviour, and instead limit ourselves to study systems in steady state, meaning the statistics of the system are not dependent on when we are conducting observations. The steady state is desirable because we are interested in the system once it has lost memory of the initial conditions. In general, the steady state of a system may depend on the initial conditions. One would want to consider all possible steady states of the system to build a complete picture of the average behaviour to ensure our description captures the expected observations, independent of initial conditions or time of observation. In this case, we would check that the system reaches the same steady-state distribution whether the system starts with 100% probability in microstate 1, 2, or 3. Driving Driving is the second way the system can be out of equilibrium. Driving refers to changing the energy of the microstates. For example, we could increase the energy of microstate 1 relative to some baseline energy: ε 1 ε 1 ε 0, where ε 0 is a baseline microstate energy used as a reference point. In general, we can introduce the definition of a control parameter, λ, that characterizes the driving. In this case, λi = ε i and therefore the elements of λ are the deviations of each microstate s energy from ε 0. The protocol Λ(t) contains the history of λ, and in general the current state of the system depends on the entire history of the control parameter. The system only remains out of equilibrium while λ is changing and until it relaxes into thermal equilibrium for the new value of λ. For example, consider the evolution in Figure 2.1. The system starts with ε 1 = ε 2 = ε 3 = ε 0 = 5k B T and the

22 CHAPTER 2. THEORETICAL FRAMEWORK 13 equilibrium state of this system has equal probability in each state. The (average) energy of the system is defined as: E = P i ε i, (2.1) i where P i is the occupancy probability of the ith microstate and the sum is over all possible microstates. Therefore the energy of the starting equilibrium state is E = 1 3 ε ε ε 3 = 5k B T. The energy of microstate 1 is then increased by ε 1 = 3k B T and the system energy is now E = 1 3 8k BT k BT k BT = 6k B T. Next, the system relaxes to the Boltzmann e distribution: P 1 = 8 e 2%,P e 5 +e 5 +e 8 2 = P 3 = 5 49%. The final equilibrium e 5 +e 5 +e 8 energy of the system is E = k BT k BT k BT = 5.06k B T. In the case of continuous driving, the system does not relax to equilibrium before the control parameter is again updated, leaving the system perpetually out of equilibrium. Figure 2.1: Driving a three-state system. Initially, the system is in equilibrium with three equal-energy microstates. Work is then done to change the energy of microstate 1. Next, heat is released as the system relaxes into the new equilibrium distribution. All energies ε i are in units of k B T. Breaking Detailed Balance The third way to obtain a nonequilibrium system is by breaking detailed balance. The previous discussions of relaxation and driving assumed detailed balance is obeyed at a fixed value of control parameter: there is no net direction of probability flow intrinsic to the dynamics [30]. When detailed balance is broken, there must be either cycles (periodic

23 CHAPTER 2. THEORETICAL FRAMEWORK 14 boundary conditions) or open boundary conditions. Both of these boundary conditions allow net probability flow through the system, either by coming around the cycle or by being created at one end and flowing out the other. Consider again the system with three microstates: instead of defining ε i for each microstate, we can introduce transition rates between each state, where P i k i, j is the rate of i j and k i, j is the rate constant. If the system obeyed detailed balance, there would be no net flow between any two microstates once steady state is reached: P i k i, j = P j k j,i. (2.2) This case corresponds to equilibrium because the system is identical under time reversal, also referred to as microscopic reversibility [31]. By contrast, one could also build a cyclic system that does not obey detailed balance, but rather settles into a nonequilibrium steady state (NESS) with a net probability flux through the system. This is achieved by imposing transition rates that bias one direction, inducing net movement in the system even at steady state. For example the transition rate constants going clockwise are made larger than the counter-clockwise direction for each transition such that: P i k i,i+1 > P i+1 k i+1,i. In the case of a NESS, the occupancy probabilities are still static since the probability flowing into a microstate from all other microstates equals the probability flowing out of the microstate (a condition known as balance), but in general the occupancy probabilities do not match those of the detailed balance construction [32]. The connections between NESSs and driven stochastic systems, called stochastic pumps, are further discussed in [33] but are not explored in this thesis Work, Heat, and Driving The first law of thermodynamics states that energy cannot be created or destroyed, and therefore changes in system energy must be due to energy flow between the system and the environment. The change in system energy can be split into two parts: E = W + Q, (2.3) where W is the work and Q is the heat. Work is defined as the system energy change due to changes in the microstate energies, whereas heat is associated with the change of the

24 CHAPTER 2. THEORETICAL FRAMEWORK 15 system energy due to moving between microstates [31]. The sign convention used here defines positive work and heat as energy flow into the system. In the three-state cycle example in Figure 2.1, the first step increases the energy of the top microstate and the occupancy probabilities do not change, therefore the energy change is due to work being done on the system. In the second step, the energies of the microstates remain fixed, but the probabilities relax to the equilibrium distribution, therefore this decrease in energy is associated with heat flowing out of the system. Note that the work required to change the energy of a microstate depends on how many particles are present: where N is the total number of particles in the system. W = NP i ε i, (2.4) i It is convenient to think of system driving as two distinct steps: an instantaneous change in microstate energies associated with work and subsequent relaxation associated with heat. However, continuous driving means that the control parameter does not pause after each infinitesimal change to allow the system to relax to equilibrium. Instead, the system relaxes on a continuously changing landscape, and in general the occupancy probabilities at time t do not match the Boltzmann distribution for λ(t). In some situations, we may expect the system to lag behind the driving and perhaps match the Boltzmann distribution for some earlier value of λ(t). This is referred to as an endoreversible process: the system and surroundings are in equilibrium at any instant, though not necessarily with each other [34]. Furthermore there is no steady state since the system is continuously responding to changing λ, and in general we expect the deviation from equilibrium to depend on how fast the driving occurs. 2.2 Driving of Cycles Let us restrict our discussion of driving to cycles of period τ such that the perturbation is periodic: λ(t) = λ(t + τ). Machines can be modelled as systems being driven through a cycle of states, and indeed molecular machines have also been described with this framework [5, 17, 35, 36, 9]. A system in equilibrium is reversible and therefore will have an equal probability of completing forwards and backwards cycles, achieving no net progress. Forwards can be used to refer to the desirable direction of operation, for example

25 CHAPTER 2. THEORETICAL FRAMEWORK 16 the ATP synthesis direction in F 1. In general, a successful machine can be defined as a system that achieves more forwards cycles than backwards. Biological machines operate out of equilibrium and can achieve directed motion at an energetic cost in one of two ways: experiencing periodic driving or breaking detailed balance [33] Work Accumulation In the previous section we introduced the concept of a control parameter that quantifies how the system is perturbed. According to Equation 2.3, the energy of the system can change through work done on or by the system and heat flowing between the system and a thermal reservoir. As the cycle time τ approaches infinity, the system is driven in the quasistatic limit and it is assumed that the system remains in equilibrium the entire time. Therefore, the system returns to exactly the same state after a complete cycle. The work performed during this isothermal (constant temperature), reversible process is equal to the free energy difference over one cycle, W = F = 0. In the case of finite-time driving, the process is no longer reversible and the work required to drive one cycle is expected to exceed 0, in accordance with the Clausius Inequality [37]: W F. (2.5) The excess work is defined as the work required to drive the system during the nonequilibrium protocol, above and beyond the work required to drive the system in the quasistatic case [38]. Some of this excess work will be dissipated as heat during the cycle while the rest will be stored in the system and is associated with the deviation from equilibrium. An analytic expression for nonequilibrium excess work is generally not possible except in the simplest cases and in the linear-response regime. Work fluctuation theorems describing the probability distribution of work have been developed for near-equilibrium systems [31, 39, 40, 41, 37, 42]. A simple model of non-equilibrium driving is a particle in a moving quadratic trap. This simple model can be applied to Example: Quadratic Trap An analytical expression for the excess work accumulated during non-equilibrium driving is not generally available. Such an expression is derived for a Brownian particle dragged

26 CHAPTER 2. THEORETICAL FRAMEWORK 17 by a quadratic trap in [43]. This example summarizes their derivation for excess work as a function of driving strength and speed. Their result serves as a guide for more complicated constructions of driven systems including our model. Their model considers a particle being pulled through a thermal medium of temperature 1/β (k B = 1) by a time-dependent quadratic potential of form (Equation 4 in [43]): U(x,t) = k 2 (x ut)2, (2.6) where k is the trap strength, x is the spatial coordinate, and u is the speed of the trap motion. The particle is assumed to have Langevin (over-damped, stochastic) dynamics. The corresponding equation of motion is (Equation 5 in [43]): ẋ = k (x ut) + η, (2.7) mγ where mγ is the coefficient of friction (drag) and η is delta-correlated white noise with variance 2/β mγ. Due to the noise term, individual trajectories are not identical and therefore will have different amounts of work accumulated. Furthermore, the initial conditions can be assumed to be drawn from a distribution of states, also leading to work varying between individual trajectories. The control parameter of the system is the location λ = vt of the minimum of the trap. F=0 for any trap repositioning, since the energetics of the equilibrium distribution are independent of λ = vt. Therefore all accumulated work is excess work. For an individual trajectory, the power (rate of work accumulation) is (Equation 7 in [43]): Ẇ = U(x(t),t) t = uk(x(t) ut) (2.8) = uky, where y x ut is the position of the particle relative to the minima of the trap. They then consider the dynamics of the probability distribution representing an ensemble of particles using the Fokker-Planck equation (Equation 9 in [43]): P(y,w,t) t = k yp(y,w,t) + u P(y,w,t) + uky P(y,w,t) 2 y y W P(y,w,t) βmγ y 2. (2.9)

27 CHAPTER 2. THEORETICAL FRAMEWORK 18 Fokker-Planck equations will be further discussed in Section 3.3, but here we can see a partial differential equation describing the evolution of the joint probability distribution of the position and work accumulated as a function of time. By solving this equation, the authors arrive at the transient and steady-state expressions for the expectation values of position, work, and their respective variances and covariance. The steady-state (limit in asymptotically long times) expressions are (Eq. 17 in [43]): y(t) mγu k W(t) u 2 mγt σ 2 y (t) 1 βk σ 2 W (t) 2u2 mγt β (2.10a) (2.10b) (2.10c), (2.10d) where the angled brackets indicate average values. Note that the equilibrium distribution is a Gaussian with mean y = 0 (centred within the trap) with variance 1/βk, and the nonequilibrium distribution is also a Gaussian with variance 1/βk but centred at y = mγu/k. The nonequilibrium distribution is the equilibrium distribution corresponding to a past value of the control parameter Work-Speed-Accuracy Tradeoffs In Section 2.2.1, we found that divergence from equilibrium leads to accumulated excess work, which can be interpreted as the energetic cost of driving the system out of equilibrium. Furthermore, the results for dragging a quadratic trap indicate that the cost increases with speed of driving, leading to a tradeoff between the excess power required to drive the system and the speed. While it is not expected that the quadratic trap results apply generally, the qualitative feature of requiring increased work with increased speed does agree with our intuition about nonequilibrium driving. As speed increases, the instantaneous distribution differs more dramatically from the equilibrium distribution. Let us now consider a cyclic version of the quadratic trap system in [43]: instead of moving forwards on an infinite plane, consider a trap travelling around a ring. We can first consider the quasistatic limit: assuming the probability distribution is in equilibrium for every value of λ, the centre of the distribution is λ and on average the probability travels with the trap. As the driving

28 CHAPTER 2. THEORETICAL FRAMEWORK 19 speed increases, the system has less time to respond to the change and significant probability will be left behind rather than travelling with the trap. This represents an individual realization in which the trap travelled around yet the system did not follow. It is therefore important to quantify the accuracy of driving. We define the accuracy as the probability that flows through the system over one cycle. In this case, we can consider the net probability flow per lap of the trap. The idea of a cyclic driving can be mapped to the F 1 system. If we were to consider the rotary system of F 1, accuracy could be a measure of how many ATP molecules were produced per rotation of γ. Since it is known that 3 ATP are produced per cycle during 100% efficient operation, an accuracy of 67% would mean 2 ATP are produced per rotation of γ. While current experiments have so far confirmed F 1 is capable of operating near perfect efficiency, it is reasonable to allow for the possibility of slipping or lagging at higher rates of rotation. Accuracy is a measure of the functionality of the machine: a low accuracy means the system is not accomplishing its function, in this case producing ATP. 2.3 Optimal Control We use optimal control to refer to the strategic design of a driving protocol to minimize some cost function. For example, a driving protocol can be optimized for excess work, meaning the optimal protocol is a minimum-dissipation procedure. There are various approaches to finding such work-minimizing protocols including exact solutions for a small number of simple scenarios [44], solving the Burgers equation [45], or approximating via the generalized friction [38]. Here we focus on the generalized friction approach. It has already been proposed that an understanding of minimum-dissipation protocols may contribute to an understanding of the efficiency of molecular machines, including ATP synthase [38, 46, 47]. However, a universal framework for determining the minimumdissipation driving of any nonequilibrium system is yet to be developed because we require an understanding of how nonequilibrium systems respond to external perturbations. Early work into optimal control was confined to macroscopic systems in the quasistatic limit and related the minimum dissipation to thermodynamic length, a measure of the distance between equilibrium states [48, 49]. If a system is to be driven between two equilibrium states, the process can be discretized into a finite number of steps. The quasistatic limit

29 CHAPTER 2. THEORETICAL FRAMEWORK 20 assumes the system equilibrates at each step before the next step is taken. An investigation into the optimal time allocation for each step concluded that the optimal time allocation is proportional to the largest relaxation time of the system [49]. This early connection to relaxation time scales hints at the fundamental connection between nonequilibrium dynamics and equilibrium behaviour. The motivation for optimal control designed to optimize for work is saving energy by driving the system in the most efficient matter. However, in the quasistatic limit, the system is driven asymptotically slowly, and therefore the work accumulation is equivalent to the change in free energy of the initial and final equilibrium distributions for any driving protocol. In other words, the excess work vanishes as τ approaches infinity, and there is no benefit of using the minimum-dissipation protocol. Instead, we are interested in efficiency gains from using minimum-dissipation driving protocols when the cycle time is finite. As introduced in the previous section, a finite τ means that we cannot assume our system will match the equilibrium distribution at a given control parameter. Near-equilibrium approximations are the next step beyond quasistatic assumptions and towards a general nonequilibrium framework. Near-equilibrium systems can be treated in the linear-response regime, meaning the system responds linearly to external forces. The linear-response approximation is expected to hold in systems with fast relaxation times [50]. The earlier macroscopic theories using thermodynamic length have been expanded to microscopic near-equilibrium systems through the development of a generalized friction coefficient that approximates system response to nonequilibrium driving [38, 51] Friction and Minimum-Dissipation Driving Protocols The excess power along any protocol can be explicitly defined and has a simple form in the linear-response approximation [38]. In this regime, the excess power, or rate of accumulating excess work, for a one-dimensional control parameter is given by: [ ] [ ] [ ] dλ dλ dλ 2 P ex (t) = ζ (λ(t)) = ζ (t), (2.11) dt t dt t dt t [ where ζ (t) is the generalized friction coefficient and dλ dt is the control parameter velocity ]t at time t. In general the friction depends on the value of the control parameter and therefore varies over the protocol. Note that in this linear-response approximation the excess power is

30 CHAPTER 2. THEORETICAL FRAMEWORK 21 a function only of the instantaneous values of the control parameter and control parameter velocity. In general the nonequilibrium work accumulation would depend on the entire history of the system. The total excess work accumulated over the protocol is the integral of the excess power: W ex = τ 0 dtp ex (t). (2.12) The time integration can be converted into a spatial integral considering a segment of constant velocity between two control parameter values: W ex = λ2 λ 1 dλζ(λ) dλ dt, (2.13) and therefore W ex is proportional to the control parameter velocity. This agrees with the intuition that faster driving increases the excess work per cycle. Furthermore, the dissipation is inversely proportional to the protocol duration. The generalized friction coefficient takes the form of a tensor in the case of multidimensional control parameters. In the rest of our discussion, we will limit ourselves to one-dimensional control parameters but note that the theory supports extension to multiple dimensions. The friction can be thought of as a characterization of the resistance of the system to the changing control parameter. It can be seen by considering Equation 2.11 that the excess power is greatest in regions of highest friction for constant velocity protocols. The minimum-dissipation protocol modulates velocity so as to accumulate work at a constant rate throughout the protocol. The friction is given by the time-integrated force covariance [38]: ζ (λ) β 0 dt δf(0)δf(t) λ, (2.14) where δf(t ) F(t) F λ is the deviation of the force due to the control parameter from the average force at equilibrium for control parameter value λ. The friction is calculated from the equilibrium force fluctuations, meaning the system is held at a fixed control parameter value to calculate the corresponding friction. By obtaining the friction for all values of λ in the protocol λ, we can then calculate the theoretical excess work. The appeal of this approach is that we can approximately calculate the excess work due to nonequilibrium driving from instantaneous quantities calculated from equilibrium fluctuations.

31 CHAPTER 2. THEORETICAL FRAMEWORK 22 In order to gain more intuition about the friction coefficient, we can consider an alternate decomposition: where δf 2 is the equilibrium force variance, and ζ (λ) = βτ relax δf 2 λ, (2.15) τ relax is the integral relaxation time of the system. 0 dt δf(0)δf(t) λ δf 2 λ (2.16) The form of Equation 2.15 tells us that the friction is expected to be higher in regions of slow relaxation and large force variance and smaller in regions of fast relaxation times and small force variance. The former high-friction case is associated with transitions over energy barriers: the control parameter is such that there are multiple states with significant probability. The latter case corresponds to a single minimum into which the system rapidly equilibrates. Linking the friction back to the excess power, more excess work is accumulated in regions of barrier crossing, when the system is most susceptible to changes in the control parameter. A minimum-dissipation protocol minimizes the excess work accumulated over the driving cycle. Since dissipation is proportional to the friction and inversely proportional to the protocol time, one can minimize the dissipation by spending more time in regions of high friction and less time in regions of low friction. Optimal protocols can be shown to modulate the control parameter velocity so as to accumulate excess work at a constant rate over the entire protocol [38]. Another intuitive rationale for slowing down at the barriers comes from considering the role of thermal fluctuations in the single-particle Langevin dynamics picture. Fokker- Planck dynamics correspond to the ensemble behaviour of a single particle obeying Langevin dynamics. The earlier the particle hops into the next well, the less work is required. When the control parameter is centred over a barrier, the height of the barrier between the two neighbouring states is minimized, and therefore the probability of hopping over due to a thermal kick is maximized. Spending more time where the thermal transition rate is high decreases the average transition time, thereby decreasing the work. The minimum-dissipation protocol is obtained from the friction by applying the condi-

32 CHAPTER 2. THEORETICAL FRAMEWORK 23 tion that the optimal velocity is inversely proportional to the square root of the friction: dλ MD dt ζ 1/2, (2.17) where the proportionality constant is determined by the protocol time τ and the distance between initial and final values of λ. The label MD is used to denote variables pertaining to the minimum-dissipation protocol. It has been demonstrated that these dissipationminimizing trajectories are geodesics in thermodynamic space. Other works explore the geometrical interpretations of this formalism in more detail [51, 52, 38]. A useful geometric result is that the shape is invariant with protocol time. In other words, the shape of the minimum-dissipation protocol predicted by this linear-response approximation is independent of τ: stretching or compressing the protocol to the desired total cycle time preserves the optimization Ratio of Naive and Optimal Excess Work The universal geometry of minimum-dissipation driving protocols (i.e. having τ-independent shape) leads to the result that, in the linear-response approximation, the ratio of the excess work for naive (constant velocity) protocols and the minimum-dissipation protocols depends only on the friction over the control parameter space and not on the absolute cycle time. In the linear-response approximation, the ratio of works is given by: W naive ex W MD ex = ζ, (2.18) ζ 1/22 where the bar indicates an average over the range of λ in Λ [50]. This result can be used to predict the gain from using the minimum-dissipation protocol and will be tested in this thesis.

33 Chapter 3 Model System Computer simulations and models of real systems allow researches to assess their assumptions about the mechanism of operation. As experimental methods advance, models must be reassessed to ensure a match between both macroscopic and microscopic behaviour. Models and simulations of F 1 have been developed in parallel with experiments. Many recent experimental papers conclude with a proposed model [14, 17] while theorists build their models based off experimental evidence and propose experiments to better elucidate or test the details of their models [53, 18, 20, 36, 54, 55, 56, 57]. 3.1 Goals of the Model The present model was designed as a minimalistic, generalizable model of a rotary machine subject to driving. Numerical methods were used to evolve the Fokker-Planck equation describing the time-dependent probability distribution of the state of the system. The present model aims to capture the essential behaviour of the F 1 machine while being subjected to driving in the form of an external force applied to the γ-shaft in the synthetic direction. The goal is to explore the tradeoffs of work, speed, and accuracy in a driven rotary machine, and therefore the microscopic details and specific conformational changes of the F 1 machine are not included the model. The intention is to create a model that is generalizable to any nanoscale rotary machine and not limited to the biologically accurate behaviour of F 1. Furthermore, the model should only include physically meaningful parameters such as driving strength and internal resistance to driving and have the least number of parameters 24

34 CHAPTER 3. MODEL SYSTEM 25 necessary to capture the essential behaviour. The development of the model was guided by the following ideas: N Minima The transformation of a continuous ensemble of system states into a finite number of preferred conformations is a popular way to model protein dynamics. It is known that F 1 has three identical catalytic sites and it is believed that one complete rotation then corresponds to potentially 3 chemical reactions [53]. Additionally, the γ-shaft moves in discrete steps, believed to correspond to chemical events [24, 14, 17, 25]. The present model assumes three preferred angles to map to the simplest reduction of F 1, but in general N energetic minima could be introduced. Two Coupled Systems F 1 is described as a mechanochemically coupled machine, meaning the chemical reactions of ATP hydrolysis/synthesis occurring in the αβ barrel are coupled with the mechanical rotation of the γ-shaft. Additionally, the driving by F o, magnetic tweezers or other torquegeneration methods introduces a bias on the position of the γ-shaft, essentially creating another layer of coupling. The present model follows the state of the system as a result of driving, where the state of the system is captured by a single angular position. There are two ways to interpret the system: Perfect coupling between the chemistry and γ-shaft: the system state represents the angle of the γ-shaft and the chemistry is assumed to follow along. The γ-shaft is free to rotate or not, depending on the strength and speed of driving. Perfect coupling between the γ-shaft and driving: the system state represents the chemical coordinate with one jump of 120 corresponding to a single chemical event. The angle of the γ-shaft is assumed to perfectly follow the control parameter value while the chemistry is free to proceed or not, depending on the strength and speed or driving.

35 CHAPTER 3. MODEL SYSTEM 26 No Chemical Bias The concentrations of ATP, ADP and P i in the surrounding environment affect the rate and net direction of ATP catalysis (synthesis and hydrolysis) by F 1. The present model assumes that the chemical concentrations are fixed at equilibrium values such that no net rotation occurs in the zero-driving case. Exact, Autonomous Driving The model assumes that the driving protocol is precisely defined prior to the protocol: there is no uncertainty in the value of the control parameter at a particular time. Furthermore, autonomous refers to the protocol not being a feedback process that depends on the system state: the value of the control parameter is set independent of the state of the system and is completely defined prior to the start of the protocol. Over-damped Dynamics Due to the low Reynolds number of F 1 calculated in Section 1.1.2, it is valid to assume the rotating γ-shaft has no inertia and the system obeys overdamped dynamics. In fact, removing the large actin filament would further decrease the Reynolds number. 3.2 System Definitions The time-dependent energy landscape, herein referred to as the potential, captures both the intrinsic properties of F 1 and the effect of the external driving by summing two components to generate the total potential. The intrinsic potential, shown in Figure 3.1a, represents the internal mechanics of F 1 and therefore in the minimal representation has three minima at 0, 120, and 240 degrees, representing the three catalytic states. The barriers between each state have a maximum height A. The exact form of our potential is U intrinsic (θ) = A 2 ( [ ( 1 + sin 3 θ + π )]). (3.1) 2

36 CHAPTER 3. MODEL SYSTEM 27 The time-dependent driving potential, U driving (θ,t) = k 2 ( 1 + sin [θ π ]) 2 λ(t), (3.2) has a single minimum located at the position of λ(t). The driving strength is captured in the parameter k which is the value of the driving potential exactly half a rotation away from λ(t). Both A and k are in units of k B T. The driving potential approximates driving by a magnetic field: a magnetic dipole feels a force sinusoidal in the angle to the magnetic field. The minimum of the driving potential is set by the driving protocol Λ. In the case of naive driving, λ(t) has a constant angular velocity of 2π/τ. The minimum-dissipation protocol is calculated from the friction, as further described in Section 4.3. (a) (a) (b) (b) A Energy A k k 2 /3 4 /3 π 2 π 4 π 5 π π 2 π π 2 π (t) Position of Figure 3.1: The two components of the potential defining the system. (a) The intrinsic potential has three minima and energy barriers of height A. (b) The time-dependent driving potential has a single minimum at the value of λ(t) and a height k Work and Heat The energetics of the system are set by the potential and the energy of the system is therefore an average over the probability distribution P(θ,t): E(t) = 2π 0 dθ P(θ,t)U(θ,t). (3.3)

37 CHAPTER 3. MODEL SYSTEM 28 The work is defined as the change of energy when the potential is updated, and the heat is defined as the change in energy when the system relaxes on the static landscape. Because of the discrete nature of simulations, these two processes are distinguishable substeps within one update of time. These definitions match those introduced in Section System Dynamics The dynamics of a stochastic process can be described by Langevin dynamics with the general form: dv = U (x,t) m 2γ dt γvdt + dw(t), (3.4) βm where U (x,t) is the spatial derivative of the total potential, v is the velocity of the particle, m is the mass, γ is the viscous drag or friction coefficient with units of inverse time, β 1/k B T, k B is Boltzmann s constant, T is the temperature of the surroundings, and W(t) is the standard Wiener process [30]. To better envision the impact of the Wiener process, note that the discrete approximation used for numerical integration is dw(t) dtn (0,1) where N (0,1) is the standard normal distribution with 0 mean and unit variance. An example of a Wiener process is simple diffusion. dw(t) and its pre-factor are sometimes written simply as η(t) and called the Langevin force or noise term. η(t) is a Gaussian random variable with zero mean and delta function correlations: η(t)η(t ) = 2γ mβ δ(t t ), (3.5) where δ(t t ) is a Dirac delta function. The Langevin equation provides a means to obtain a single trajectory of the stochastic process - since random fluctuations are introduced through the noise term, each individual simulated trajectory will be unique. One would need to generate an ensemble of trajectories in order to obtain statistics on the macroscopic properties of the system. The Fokker-Planck equation is a partial differential equation used to describe the evolution of a probability distribution for a system subject to Langevin dynamics. The system is defined such that it has the same deterministic forces, drag, and diffusion that are defined in Equation 3.4. In the overdamped limit, the system is assumed to have no inertia and the

38 CHAPTER 3. MODEL SYSTEM 29 left side of Equation 3.4 is therefore set to zero [58]. Such a system is sufficiently described by a reduced form of the Fokker-Planck equation called the Smoluchowski equation: P(x,t) t 2 = x [A(x)P(x,t)] + 1 [B(x)P(x,t)], (3.6) 2 x2 where P(x,t) is the probability distribution of position as a function of time, A(x) is the drift term and B(x) is the diffusion term. The exact forms of A(x) and B(x) can be derived by considering the dynamics over a time t such that t is small enough that only small changes have occurred, yet long enough that the process remains Markovian. In this limit, A(x) = X t B(x) = ( X)2 t (3.7a), (3.7b) and therefore the average and variance of the change in time t is sufficient to define the long-time dynamics. The drift term is called such because A(x) is actually the drift velocity of the system, resulting from the applied driving potential. The drift velocity v drift is obtained by equating the driving force (defined as the negative spatial derivative of the applied potential) with the frictional force, resulting in a net zero force on the system: 0 = F net (3.8a) = U (x,t) mγv drift (3.8b) v drift = U (x,t) mγ = D U (x,t) k B T (3.8c) (3.8d) = A(x,t), (3.8e) where U (x,t) is the spatial derivative of the total potential. Note that a time-dependent driving potential results in a time-dependent definition of A(x,t). Secondly, B(x) can be derived by considering Equation 3.6 in the absence of driving (A(x,t) = 0) and noting that this is in fact the diffusion equation: P(x,t) t 2 = x 2 [B(x)P(x,t)] = D P(x,t) x 2, (3.9)

39 CHAPTER 3. MODEL SYSTEM 30 where D is the diffusion coefficient and is assumed to be constant in space. Therefore, B(x,t) is a constant and can be defined using the Einstein relation: B = 2D = 2 k BT mγ. (3.10) Finally we arrive at the Smoluchowski equation defining the system dynamics in terms of physical parameters: P(x,t) t Continuity Equation = D [U (x,t)p(x,t)] + D 2 P(x,t) k B T x x 2. (3.11) The Fokker-Planck equation evolves a probability distribution, necessitating that probability be conserved and leading to the continuity equation for probability density: P(x,t) t = J(x,t) x, (3.12) where J(x,t) is the probability flux. Intuitively, if we imagine a boundary at position x is experiencing net flux of probability away, the probability must be getting depleted. From this definition of flux, we obtain an equation for the probability flux: 3.4 Accuracy J(x,t) = A(x)P(x,t) 1 2 x B(x)P(x,t) = D U (x,t) k B T P(x,t) D (3.13) x P(x,t). The accuracy of the system is a measure of how well the system responds to the driving. This is quantified by the net probability flux through the system over one cycle. An accuracy of 100% means the system followed the driving perfectly, while less than perfect accuracy can be interpreted as the probability that the system follows the driving cycle. For example, an accuracy of 70% can be taken to mean that on average the system will successfully complete 70% of the cycle, for each driving cycle. More precisely, the accuracy η is defined as the integrated flux over one cycle averaged over all space: η = 2π τ 0 0 J(θ,t)dt dθ. (3.14)

40 CHAPTER 3. MODEL SYSTEM Periodic Steady State and Rotational Symmetry Since the potential is changing with time, the system is not assumed to ever be in equilibrium. As a result, the state of the system in general depends on the entire history of both the system state and the driving potential. In particular, the state of the system depends on the initial conditions. In all simulations, the system starts in equilibrium with λ(t = 0) = 0 and driving drags the system into a nonequilibrium state. There is thus a transient phase that can last many cycles during which subsequent cycles are not identical. However, we are not interested in this transient behaviour and therefore evolve the system until each cycle is identical, referred to as the periodic steady state (PSS). Squared differences between probability distributions can be used to assess whether the following two conditions are met. While the probability is changing over the course of one cycle, the probability at a particular time in the cycle is identical cycle to cycle, hence the term periodic. Specifically, the first condition for whether the PSS has been reached is P(θ,t) = P(θ,t + τ), (3.15) for all t. Furthermore, the system is known to be three-fold symmetric, and therefore no one angular state should be preferred over the course of an entire cycle. The PSS should therefore be independent of the initial conditions. In other words, the second condition is that the PSS at time t + τ/3 should exactly match a 120 rotation of the PSS at time t: P(θ,t + τ/3) = P(θ 2π/3,t). (3.16) In general, the PSSs do not converge on computationally accessible time scales. In the case of large barrier heights, the transition rate is so low that the probability remains concentrated in the initial well. This is resolved by excluding cases in which less than 5% of the probability flows around per cycle. This is rationalized by the argument that a machine with barrier heights A so high as to inhibit cycling is not of interest. Convergence to the PSS can also be assessed by tracking the heat and work flows. As introduced in Section 2.2.1, a portion of the excess work can accumulate in the system, and thus another test for the PSS is that all excess work is dissipated as heat and there is no net accumulation of energy in the system.

41 Chapter 4 Methods The Fokker-Planck equation and the specific form of the Smoluchowski equation in Equation 3.11 are partial differential equations describing the evolution of a probability distribution on a continuous, time-dependent, energy landscape. Due to the time dependence of the energetics, neither an exact solution nor an analytic approximation of the system dynamics is tractable. Numerical methods using a computer must therefore be used to evolve the system. Finite difference methods and spectral methods are the two most common approaches to numerically solving partial differential equations. Spectral methods are ideal for high spatial resolution in multiple dimensions; however, we chose the simpler approach of finite difference for our one-dimensional system and concluded that the computational cost of sufficient resolution did not warrant switching to more advanced methods [59]. 4.1 Numerical Fokker-Planck Dynamics We chose to use the finite differences method with explicit time integration to evolve the probability distribution. In order to be implemented numerically, the partial derivatives in the Smoluchowski equation (Equation 3.11) must be replaced by discrete approximations. Explicit time integration calculates the future state of the system using current values, while implicit methods involve solving an equation for the current and future state. We chose an explicit method for its simpler implementation. The approximations used in these simula- 32

42 CHAPTER 4. METHODS 33 tions are given by [59]: x [U (x,t)p(x,t)] U (x + x,t)p(x + x,t) U (x x,t)p(x x,t) 2 x (4.1a) 2 P(x + x,t) + P(x x,t) 2P(x,t) P(x,t) x2 ( x) 2, (4.1b) where U (x,t) is the spatial derivative of U(x,t) (the exact derivative from continuous calculus, not the numerical approximation). The model discretizes the spatial coordinate in order to define the probability distribution; however, the true continuous form of U(x) is used because the applied potential is continuous. The complete calculation of the update to the probability distribution is then given by: P(x,t + t) =P(x,t) + D t 2k B T x [U (x + x,t)p(x + x,t) U (x x,t)p(x t,t)] + D t [P(x + x,t) + P(x x,t) 2P(x,t)]. ( x) 2 Periodic boundary conditions are used to match up the system at x = 0 = 2π. (4.2) Flux Calculation and Accuracy The continuous form of the flux in Equation 3.13 must also be approximated with a discrete difference method. In this case, the flux through a region of width x over time t is the desired quantity. The desired flux is then given by: ( J(x,t) = D U (x,t)p(x,t) k B T P(x + x,t) P(x x,t) 2 x ) t x. (4.3) The accuracy, as defined in Equation 3.14, is obtained by taking the average of the cumulative fluxes at each discrete position in the system. By construction, this scheme exactly preserves probability normalization Periodic Steady State Convergence to the PSS is assessed once per cycle using both criteria introduced in Section 3.5. The squared difference (residual sum of squares) RSS N between the distributions

43 CHAPTER 4. METHODS 34 at λ = 0 is calculated for the current cycle and the previous cycle. If RSS N is less than 10 8, RSS 240 is calculated. RSS 240 is the squared difference between the current distribution for λ = 0 and the distribution for λ = 240 rotated forwards by 120. If RSS 240 is less than 10 5, the next cycle is recorded as the PSS. This means the cumulative flux and work are initialized at the beginning of the PSS cycle and their final values are used as the accuracy and work per cycle Stability The approximation of a continuous differential equation with discrete updates requires that the spatial and temporal bin sizes are selected such that the propagation is correctly resolved on the lattice leading to a stable solution. The Fokker-Planck equation takes the form of a wave equation: u t = v u x + 1 ( x) 2 2 u, (4.4) 2 t and therefore the Courant-Friedrichs-Lewy stability criterion is used to determine stability [59]. The intuition of this stability criterion comes from considering the differencing scheme used to update the system at each time point. The probability P(x,t) depends on three points in the past: P(x x,t t),p(x,t t), and P(x + x,t t). In the wave view of the equation, information in the form of the state of the wave travels at speed v. In other words, the state at position x can only be affected by past events that are less than v t away. This creates a cone of influence spreading back from the point P(x,t) encompassing all points affecting its new value. In the case of the Fokker-Planck equation, x and t must be chosen such that the entire cone of influence is captured within the discrete points being used in the differencing scheme, i.e. P(x x,t t)and P(x + x,t t). The Courant-Friedrichs-Lewy stability criterion requires: v t x < 1, (4.5) and comparison with Equation 3.11 yields v = DU (x,t)/k B T. Therefore in this specific system the maximal time step is set by: k B T t < x DU, x {0,2π},t {0,τ}. (4.6) (x,t)

44 CHAPTER 4. METHODS 35 The maximal value of U (x,t) is 3 2 A + 1 2k and therefore the maximal stable time step is proportional to x and k B T, and inversely proportional to D, A, and k. All simulations were checked against the stability criteria: t < 1 k B T x 2D 3 2 A + 2 k. (4.7) To maintain consistency, the same time step was used in all simulations, but the stability criteria was checked for each combination of parameters to ensure this choice was sufficiently small to be used in all cases. Figure 4.1: The cone of influence, shown as the shaded triangular region, must not go beyond the 3 points at t t affecting the point p(x,t). 4.2 Energy and Time Scales In order to relate the results of the simulations to the behaviour of biological systems, the energy and time scales must be compared with those of real systems. The motor F 1 can reach maximal rates of 350 revolutions per second [23]. Depending on the specific tether and driving method used, experimentalists are able to conduct single molecule driving at rates of 0-15 Hz [9, 21]. Corresponding torques of 0-70 pn nm/rad are estimated to be