ENERGY-SPEED-ACCURACY TRADEOFFS IN A DRIVEN, STOCHASTIC …summit.sfu.ca/system/files/iritems1/17509/etd10321_AKasper.pdf · Molecular machines are stochastic systems capable of converting

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 forFair Dealing. Therefore, limited reproduction of this work for the purposes ofprivate study, research, criticism, review, and news reporting is likely to be

in accordance with the law, particularly if cited appropriately.

ii

Approval

Name: Alexandra Kathleen Kasper Degree: Master of Science (Physics) Title: Energy Speed Accuracy Tradeoffs in a Driven,

Stochastic, Rotary Machine Examining Committee: 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

Abstract

Molecular machines are stochastic systems capable of converting between different forms

of energy such as chemical potential energy and mechanical work. The F1 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 F1. 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 rele-

vant 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

Acknowledgements

I would like to thank David Sivak for allowing me to truly experience the process of re-

search. 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 reaf-

firmed 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. Nel-

son 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 jour-

ney 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

Dedication

To my grandparents and life-long learners, Alan and Brenda Holvey, for reminding me to

never forget the physics.

v

Contents

Abstract iii

Acknowledgements iv

Dedication v

Contents vi

List of Figures ix

1 Introduction 11.1 Molecular Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Thermal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Reynolds Number of Molecular Machines . . . . . . . . . . . . . . 4

1.2 FoF1 ATP Synthase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Structure and Function . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Mechanochemical Coupling and Efficiency . . . . . . . . . . . . . 7

1.3 Energy-Speed-Accuracy Tradeoffs . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Motivations and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Theoretical Framework 102.1 Nonequilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Example: The Three-State Cycle . . . . . . . . . . . . . . . . . . . 11

2.1.2 Work, Heat, and Driving . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Driving of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

vi

CONTENTS vii

2.2.1 Work Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Example: Quadratic Trap . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Work-Speed-Accuracy Tradeoffs . . . . . . . . . . . . . . . . . . . 18

2.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Friction and Minimum-Dissipation Driving Protocols . . . . . . . . 20

2.3.2 Ratio of Naive and Optimal Excess Work . . . . . . . . . . . . . . 23

3 Model System 243.1 Goals of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 System Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Work and Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Periodic Steady State and Rotational Symmetry . . . . . . . . . . . . . . . 31

4 Methods 324.1 Numerical Fokker-Planck Dynamics . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Flux Calculation and Accuracy . . . . . . . . . . . . . . . . . . . . 33

4.1.2 Periodic Steady State . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Energy and Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Setting the Time Scale . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.2 Energy Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Friction Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Optimal Protocol Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Results 415.1 Naive Driving Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Accuracy and Work vs. Speed . . . . . . . . . . . . . . . . . . . . 43

5.1.2 Accuracy and Work . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Friction and Minimum-Dissipation Driving Protocols . . . . . . . . . . . . 50

5.3 Minimum-Dissipation vs. Naive Driving Protocols . . . . . . . . . . . . . 57

CONTENTS viii

5.4 Parameter Ranges: Discussion and Limitations . . . . . . . . . . . . . . . 59

6 Conclusions 606.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.1 ATP Synthase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1.2 Artificial Machines . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2.1 Extending the Model . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.2.2 Adaptive Time Step Friction Calculation . . . . . . . . . . . . . . 63

6.2.3 Higher-Order Corrections to Minimum-Dissipation Protocol . . . . 63

6.2.4 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Bibliography 65

Appendices 71

A Quadratic Approximations 72

List of Figures

1.1 ATP Synthase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Three State Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Energy Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Friction Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Naive Accuracy and Work versus Speed . . . . . . . . . . . . . . . . . . . 44

5.3 Naive Work and Accuracy Ratio, A=0 . . . . . . . . . . . . . . . . . . . . 45

5.4 Naive Accuracy and Work . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.5 Naive Flux and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Friction, Velocities, and Protocol . . . . . . . . . . . . . . . . . . . . . . . 51

5.7 Average Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.8 Minimum-Dissipation Accuracy and Work versus Speed . . . . . . . . . . 54

5.9 Minimum-Dissipation Accuracy and Work . . . . . . . . . . . . . . . . . . 55

5.10 Minimum-Dissipation Flux and Power . . . . . . . . . . . . . . . . . . . . 56

5.11 Naive and Minimum-Dissipation Ratios . . . . . . . . . . . . . . . . . . . 58

ix

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 kBT =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

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 gov-

erning 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 pro-

duces large deviations from average behaviour. Furthermore, molecular machines operate

in the low-Reynolds regime, where inertia is negligible. Finally, understanding the en-

ergy 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 ener-

getics of the system and search for optimal molecular designs and driving protocols. Our

model focuses on rotary machines and is inspired by F1; 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 kBT 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 proces-

sive 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


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 be-

haviour, 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 phosphoanhy-

dride 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 3.5×108 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.

1.1.1 Thermal Fluctuations

Thermal fluctuations refer to random deviations from the average system state, arising in

systems with non-zero temperature and quantified by kBT , where kB is Boltzmann’s con-


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 com-

posed 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 kBT . 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, kBT =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.

1.1.2 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 2 ×10−5. Furthermore, the linear walker ki-

nesin pulling cargo with radius 100 nm at 1 µm/s has a Reynolds number of 1 ×10−7.

The Reynolds numbers of nanoscale machines are an important consideration when mod-

elling 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


for forwards motion. An object that is not presently being pushed will come to rest nearly

instantaneously.

1.2 FoF1 ATP Synthase

FoF1 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: Fo and F1. Fo 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. Fo is coupled to F1 through the rotation of a central

“crankshaft” and drives, at three sites in F1, 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 F1

can be observed to operate as an ATP-burning motor (ATPase) when supplied with ATP.

1.2.1 Structure and Function

FoF1 ATP synthase is found in the membranes of bacteria, mitochondria in eukaryotic

cells, and chloroplasts in plant cells. As seen in Figure 1.1, Fo is membrane-bound while

F1 sits beside the membrane. The Fo 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, Fo 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]. F1 is made up

of a barrel structure with three-fold symmetry (α3β3), 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 α3β3 barrel to the


✏

�

a

b2

cn Lipid bilayer

↵ ↵�

�F1 : ↵3�3��✏

Fo : ab2cn

Figure 1.1: Schematic diagram (not to scale) of ATP synthase. Fo is composed of ab2cn,

where n indicates that the c-ring has a variable number of subunits between different species

(typically 10-14 [14]). F1 is composed of α3β3γδε . Solid components (α3β3γ) indicate the

subunits typically used in experiments studying isolated F1.

two b subunits of Fo.

The rotating pieces (c-ring in Fo and γε in F1) are together referred to as the rotor,

and the stationary components (ab2 in Fo and α3β3δ in F1) the stator. In most biological

situations the FoF1 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 F1 unit can detach from Fo and operate as a stand-alone reversible

system. When supplied with ATP, F1 operates as a motor, sometimes called ATPase because

it consumes ATP, and rotates the γ-shaft in the opposite direction to that during synthesis.


Due to the difficulty of working with integral membrane proteins, most experimental work

on ATP synthase has focused on only the F1 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

F1 unit is indeed a rotary machine [8]. However, these experiments were performed on

only a portion of the F1 unit (α3β3γ) 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 F1 (α3β3γ complex) by me-

chanically 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 Fo unit.

1.2.2 Mechanochemical Coupling and Efficiency

ATP synthase is frequently said to be nearly 100% efficient and have perfect mechanochem-

ical 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 fa-

cilitated by the mechanical torque exerted on the γ-shaft. The ratio of protons let through

the membrane by Fo to ATP molecules produced in F1 (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 ro-

tation 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-


erated on a large filament attached to the γ-shaft in isolated F1 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]. Addition-

ally, sub-steps of 80◦ and 40◦ have been resolved and mapped to intermediate states in the

chemical pathway: ATP binding/ADP release and Pi 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 kBT under

physiological conditions and therefore one hydrolysis (ATP consuming) cycle corresponds

to 60 kBT (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 vis-

cous 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 thermodynami-

cally 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


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 F1 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 applica-

bility. We create a minimal model independent of the molecular details of the F1 system, so

that our model can be applied to other rotary machines (such as bacterial flagella motors)

and inspire the design of synthetic machines.

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 intro-

ducing ideas from thermodynamics and statistical mechanics and discussing the additional

considerations of nonequilibrium systems. We build definitions of work and heat applica-

ble 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 devel-

opment 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 relax-

ation 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 exter-

nal conditions, many classical thermodynamics problems start with the assumption that

10

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, some-

times 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 εi, the probability of finding the system in that microstate is pro-

portional to e−εi/kBT , where kB is the Boltzmann constant and T is the temperature of the

system. Thermal fluctuations are the origin of deviations from average. An important prop-

erty 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 bal-

ance is broken. These situations are not mutually exclusive and indeed the response of a

system subject to driving is defined by its relaxation behaviour.

2.1.1 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 ε1,

ε2, and ε3. 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


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: ∆ε1 ≡ ε1− ε0, where ε0 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 = ∆εi and therefore the elements of~λ are

the deviations of each microstate’s energy from ε0. 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 ε1 = ε2 = ε3 = ε0 = 5kBT and the


equilibrium state of this system has equal probability in each state. The (average) energy

of the system is defined as:

E = ∑i

Piεi , (2.1)

where Pi 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 = 13ε1+

13ε2+

13ε3 =

5kBT . The energy of microstate 1 is then increased by ∆ε1 = 3kBT and the system energy

is now E = 138kBT + 1

35kBT + 135kBT = 6kBT . Next, the system relaxes to the Boltzmann

distribution: P1 =e−8

e−5+e−5+e−8 ≈ 2%,P2 = P3 =e−5

e−5+e−5+e−8 ≈ 49%. The final equilibrium

energy of the system is E = 21008kBT + 49

1005kBT + 491005kBT = 5.06kBT . 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 εi

are in units of kBT .

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


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 εi for each mi-

crostate, we can introduce transition rates between each state, where Piki, j is the rate of

i→ j and ki, 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:

Piki, j = Pjk 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: Piki,i+1 > Pi+1ki+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.

2.1.2 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


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:

W = ∑i

NPi∆εi , (2.4)

where N is the total number of particles in the system.

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 sur-

roundings 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 chang-

ing 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 frame-

work [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


the ATP synthesis direction in F1. 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].

2.2.1 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 re-

quired to drive one cycle is expected to exceed 0, in accordance with the Clausius Inequal-

ity [37]:

W ≥ ∆F. (2.5)

The excess work is defined as the work required to drive the system during the nonequilib-

rium 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 equilib-

rium. 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

2.2.2 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


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 (kB = 1) by a time-dependent

quadratic potential of form (Equation 4 in [43]):

U(x, t) =k2(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]):

x =− kmγ

(x−ut)+η , (2.7)

where mγ is the coefficient of friction (drag) and η is delta-correlated white noise with vari-

ance 2/βmγ . 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 equi-

librium 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]):

W =∂U(x(t), t)

∂ t=−uk(x(t)−ut)

=−uky ,

(2.8)

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

=k2

∂yP(y,w, t)∂y

+u∂P(y,w, t)

∂y+uky

∂P(y,w, t)∂W

+1

βmγ

∂ 2P(y,w, t)∂y2 . (2.9)


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)〉 → −mγuk

(2.10a)

〈W (t)〉 → u2mγt (2.10b)

σ2y (t)→

1βk

(2.10c)

σ2W (t)→ 2u2mγt

β, (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 nonequi-

librium 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.

2.2.3 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 equilib-

rium. 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 gen-

erally, the qualitative feature of requiring increased work with increased speed does agree

with our intuition about nonequilibrium driving. As speed increases, the instantaneous dis-

tribution 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


speed increases, the system has less time to respond to the change and significant proba-

bility 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 F1 system. If we were to consider

the rotary system of F1, 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 F1 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 ap-

proaches 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 minimum-

dissipation 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 dis-

tance between equilibrium states [48, 49]. If a system is to be driven between two equilib-

rium states, the process can be discretized into a finite number of steps. The quasistatic limit


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 re-

laxation 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 ex-

panded to microscopic near-equilibrium systems through the development of a generalized

friction coefficient that approximates system response to nonequilibrium driving [38, 51].

2.3.1 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 accumulat-

ing excess work, for a one-dimensional control parameter is given by:

Pex(t) =[

dλ

dt

]

tζ (λ (t))

[dλ

dt

]

t= ζ (t)

[dλ

dt

]2

t, (2.11)

where ζ (t) is the generalized friction coefficient and[

dλ

dt

]t

is the control parameter velocity

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


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:

Wex =∫

τ

0dtPex(t). (2.12)

The time integration can be converted into a spatial integral considering a segment of con-

stant velocity between two control parameter values:

Wex =∫

λ2

λ1

dλζ (λ )dλ

dt, (2.13)

and therefore Wex is proportional to the control parameter velocity. This agrees with the in-

tuition 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 multi-

dimensional 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 sys-

tem 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

∫∞

0dt〈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 pa-

rameter 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.


In order to gain more intuition about the friction coefficient, we can consider an alter-

nate decomposition:

ζ (λ ) = βτrelax〈δF2〉λ , (2.15)

where 〈δF2〉 is the equilibrium force variance, and

τrelax ≡∫

∞

0dt〈δF(0)δF(t)〉λ〈δF2〉λ

(2.16)

is the integral relaxation time of the system.

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 accumu-

lated 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 driv-

ing 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 modu-

late 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-


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 pertain-

ing to the minimum-dissipation protocol. It has been demonstrated that these dissipation-

minimizing trajectories are geodesics in thermodynamic space. Other works explore the

geometrical interpretations of this formalism in more detail [51, 52, 38]. A useful geomet-

ric 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 indepen-

dent of τ: stretching or compressing the protocol to the desired total cycle time preserves

the optimization.

2.3.2 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 de-

pends 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 naiveex

W MDex

=ζ

ζ 1/22 , (2.18)

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.

Chapter 3

Model System

Computer simulations and models of real systems allow researches to assess their assump-

tions 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 F1 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 de-

scribing the time-dependent probability distribution of the state of the system. The present

model aims to capture the essential behaviour of the F1 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 ma-

chine, and therefore the microscopic details and specific conformational changes of the F1

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

F1. 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

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 pre-

ferred conformations is a popular way to model protein dynamics. It is known that F1 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 F1, but in general N energetic

minima could be introduced.

Two Coupled Systems

F1 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 Fo, magnetic tweezers or other torque-

generation 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.


No Chemical Bias

The concentrations of ATP, ADP and Pi in the surrounding environment affect the rate and

net direction of ATP catalysis (synthesis and hydrolysis) by F1. 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 F1 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 F1 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 F1 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

Uintrinsic(θ) =A2

(1+ sin

[3(

θ +π

2

)]). (3.1)


The time-dependent driving potential,

Udriving(θ , t) =k2

(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 kBT .

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) (b)

π 2 π

A k

(a) (b)

�(t)π3

2 π3

π 4 π3

5 π3

2 π

0 2⇡ 0 ⇡ 2⇡2⇡/3 4⇡/3

Position of �

Ener

gy

A k

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.

3.2.1 Work and Heat

The energetics of the system are set by the potential and the energy of the system is there-

fore an average over the probability distribution P(θ , t):

E(t) =∫ 2π

0dθ P(θ , t)U(θ , t). (3.3)


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 sub-

steps within one update of time. These definitions match those introduced in Section 2.1.2.

3.3 System Dynamics

The dynamics of a stochastic process can be described by Langevin dynamics with the

general form:

dv =−U ′(x, t)

mdt− γvdt +

√2γ

βmdW (t) , (3.4)

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/kBT , kB 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

mβδ (t− t ′) , (3.5)

where δ (t− t ′) is a Dirac delta function. The Langevin equation provides a means to ob-

tain 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 evo-

lution 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


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

=− ∂

∂x[A(x)P(x, t)]+

12

∂ 2

∂x2 [B(x)P(x, t)] , (3.6)

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(3.7a)

B(x) =〈(∆X)2〉

∆t, (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 vdrift 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 = Fnet (3.8a)

=−U ′(x, t)−mγvdrift (3.8b)

vdrift =−U ′(x, t)

mγ(3.8c)

=−DU ′(x, t)

kBT(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

=12

∂ 2

∂x2 [B(x)P(x, t)] = D∂ 2P(x, t)

∂x2 , (3.9)


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 = 2kBTmγ

. (3.10)

Finally we arrive at the Smoluchowski equation defining the system dynamics in terms

of physical parameters:

∂P(x, t)∂ t

=D

kBT∂ [U ′(x, t)P(x, t)]

∂x+D

∂ 2P(x, t)∂x2 . (3.11)

3.3.1 Continuity Equation

The Fokker-Planck equation evolves a probability distribution, necessitating that probabil-

ity 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:

J(x, t) = A(x)P(x, t)− 12

∂

∂xB(x)P(x, t)

=−DU ′(x, t)

kBTP(x, t)−D

∂

∂xP(x, t) .

(3.13)

3.4 Accuracy

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 ac-

curacy 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 suc-

cessfully 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

0

∫τ

0J(θ , t)dt dθ . (3.14)


3.5 Periodic Steady State and Rotational Symmetry

Since the potential is changing with time, the system is not assumed to ever be in equilib-

rium. 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 concen-

trated 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.

Chapter 4

Methods

The Fokker-Planck equation and the specific form of the Smoluchowski equation in Equa-

tion 3.11 are partial differential equations describing the evolution of a probability distribu-

tion 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 sys-

tem. 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

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

∂x2 P(x, t)≈ P(x+∆x, t)+P(x−∆x, t)−2P(x, t)(∆x)2 , (4.1b)

where U ′(x, t) is the spatial derivative of U(x, t) (the exact derivative from continuous cal-

culus, not the numerical approximation).

The model discretizes the spatial coordinate in order to define the probability distribu-

tion; 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

2kBT ∆x[U ′(x+∆x, t)P(x+∆x, t)−U ′(x−∆x, t)P(x−∆t, t)]

+D∆t

(∆x)2 [P(x+∆x, t)+P(x−∆x, t)−2P(x, t)] .

(4.2)

Periodic boundary conditions are used to match up the system at x = 0 = 2π .

4.1.1 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)

kBT− 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 cumu-

lative fluxes at each discrete position in the system. By construction, this scheme exactly

preserves probability normalization.

4.1.2 Periodic Steady State

Convergence to the PSS is assessed once per cycle using both criteria introduced in Sec-

tion 3.5. The squared difference (residual sum of squares) RSSN between the distributions


at λ = 0◦ is calculated for the current cycle and the previous cycle. If RSSN is less than

10−8, RSS240 is calculated. RSS240 is the squared difference between the current distribu-

tion for λ = 0 and the distribution for λ = 240◦ rotated forwards by 120◦. If RSS240 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.

4.1.3 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

+12(∆x)2

∆t∇

2u , (4.4)

and therefore the Courant-Friedrichs-Lewy stability criterion is used to determine stabil-

ity [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) encom-

passing 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)/kBT . Therefore in this specific

system the maximal time step is set by:

∆t < ∆xkBT

DU ′(x, t),∀x ∈ {0,2π}, t ∈ {0,τ}. (4.6)


The maximal value of U ′(x, t) is 32A+ 1

2k and therefore the maximal stable time step is

proportional to ∆x and kBT , and inversely proportional to D, A, and k. All simulations

were checked against the stability criteria:

∆t <1

2DkBT ∆x32A+ k

2

. (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 suffi-

ciently 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 F1 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


acting on the γ-shaft [9]. Finally, the energy scale of the barriers between the angular states

of F1 can be estimated by observing the angular distribution within the three wells.

4.2.1 Setting the Time Scale

The time elapsed in the simulation must be mapped to meaningful time units in order to

compare with observed behaviour of F1. The diffusion coefficient is used to map between

simulation time and real time. The Einstein relation, D = kBT/mγ , allows for a direct

conversion by equating the experimental diffusion coefficient, Dexp, with the diffusion co-

efficient defined by the parameters kBT , and mγ 1.

In an experiment using a magnetic bead tethered to the γ-shaft, the experimental fric-

tional drag coefficient γexp is

mγexp = 8πηr3 +6πηy2r , (4.8)

where η is the viscosity of the surrounding liquid (ηH2O = 10−9pN s nm−2), r is the radius

of the bead, and y is the distance between the center of the bead and the rotational axis [36]2.

Using reasonable values for r and y (y = r = 0.287µm), and assuming room temperature

experiments (kBT =4.114 pN nm), Dexp can be estimated as:

Dexp =kBTmγ≈ 30 rad2/s. (4.9)

Furthermore, the simulation diffusion coefficient is

Dsim =1

1000tsim. (4.10)

The simulation time tsim is found by equating the two diffusion coefficients:

tsim =1s

1000 ·30≈ 30µs . (4.11)

1Other texts use ζ = mγ to denote the friction coefficient. Here we explicitly write out the separatecomponents to avoid confusion with the generalized friction coefficient.

2The equation in the publication is missing the factor of r in the second term. Communication with Xuconfirmed the correction presented here.


4.2.2 Energy Scales

Our model scales energy in units of kBT , with A and k setting the molecular and driving

energetics, respectively. As mentioned above, experimentalists are capable of generating

driving forces up to 70 pN nm/rad, leading to 240 pN nm (58 kBT ) of work per cycle [9].

Furthermore, the angular distribution of the system can be used to estimate the height,

A, of the intrinsic molecular barriers. Toyabe et al. report a standard deviation of σ =

20◦ = 0.35 rad around each well, and confirmed an agreement with the Einstein relation

kwidth = kBT/σ2, leading to a value of kwidth ≈ 8kBT . A quadratic trap with a spring

constant of kwidth maps to a value of A = 29kwidth in the present model (see Appendix A).

Therefore a value of A≈ 2kBT maps the present model to the F1 system.

4.3 Friction Calculation

The friction for a particular control parameter value, λ , as defined in Equation 2.14 is an

infinite-time integral of the force covariance. A schematic of this integral is depicted in

in Figure 4.2. At large time separations, the covariance of force decays to 0 and therefore

the infinite-time integral can be approximated by truncating the integration at a sufficiently

long time, t∗. The equilibrium force fluctuations must be simulated to obtain the average

force covariance.

The friction calculation is implemented by initializing the distribution with 100% of the

probability at one position, θ0, in the system and then building the average force covariance

integral by weighting the individual integrals by the probability of starting at θ0:

〈δF(0)δF(t)〉λ = ∑θ0

〈δF(0)δF(t)〉θ0P(θ0), (4.12)

where the subscript θ0 indicates the force covariance for the starting position θ0. Since we

are seeking the equilibrium force fluctuations, the driving potential is fixed at the value of λ

and therefore the system evolves on a static landscape, relaxing towards equilibrium. The

force refers to the force of the driving potential and not the total force; therefore, the force

used in this calculation is the negative spatial derivative of the driving potential defined in

Equation 3.2. The force is the sum of the force at a particular θ weighted by the probability


t

h�F 2i

00 t⇤

⇣(�) ⌘ �

Z 1

0

dth�F (0)�F (t)i�h�F (0)�F (t)i�

Figure 4.2: Schematic diagram of the equilibrium force covariance decay with time. The

y-intercept is the force variance and t∗ is the cut-off time. As t∗→ ∞, the area under the

curve approaches the true value of ζ (λ )/β .

distribution:

F(t) = ∑θ

−U ′driving(θ , t)P(θ , t). (4.13)

The force covariance for a particular value of θ0 is calculated as:

〈δF(0)δF(t)〉θ0 = (F(0)−Feq)(F(t)−Feq), (4.14)

where the equilibrium force Feq is zero. The force covariance averaged over all starting

positions is then given by the sum of the individual covariances weighted by the probability

of starting at position θ0:

〈δF(0)δF(t)〉λ = ∑θ0

〈δF(0)δF(t)〉θ0P(θ0)eq, (4.15)

where P(θ0)eq is the equilibrium probability of being at position θ0, defined by the Boltz-

mann distribution. The calculation is truncated when

〈δF(0)δF(t∗)〉λ〈δF(0)δF(0)〉λ

< 0.0001 , (4.16)


in other words, when the force covariance has decayed to less than 0.01% of the force vari-

ance. Once this condition is satisfied, the friction is calculated by numerically integrating

under the 〈δF(0)δF(t)〉λ curve using the left rectangular method 3. This crude approxi-

mation for the integral is valid because the time increment between saved force covariance

values, ∆t∗, is sufficiently small such that t∗� ∆t∗. Thus the friction is approximated by:

ζ (λ )≈t=t∗

∑0〈δF(0)δF(t)〉λ ∆t∗. (4.17)

4.4 Optimal Protocol Calculation

The friction, ζ , is calculated at particular control parameter values, where {λ}ζ is the set

of all λ for which ζ is calculated. The optimal velocities at these particular values of

control parameter are known to be proportional to ζ−1/2 under linear response [38]. It

is convenient to define the raw velocities as vraw(λ ) = ζ−1/2(λ ). A single proportional-

ity constant, α , is required to obtain the true velocities and is obtained by applying the

constraint that the protocol takes time τ to complete a cycle from 0 to 360 degrees. By

combining these constraints, the optimal control parameter protocol can be determined by

assuming constant acceleration between the points of known velocity. To our knowledge,

the following method for calculating the minimum-dissipation protocol from a discrete set

of known friction values is an original development.

The raw times refer to the unscaled time corresponding to each value of control param-

eter in {λ}ζ and are built iteratively from

ti+1,raw =2(λi+1−λi)

vi,raw + vi+1,raw+ ti,raw , (4.18)

where t0 ≡ 0 and λi is the ith element in {λ}ζ . The raw times can then be converted into

the true times for the minimum-dissipation protocol by scaling according to

ti,MD =τ

tfinal,rawti,raw , (4.19)

3The centre rectangular (midpoint) method should have been used. The left rectangular method leads to aconstant of 1/2〈δF(0)δF(0)〉λ added to each integral, increasing the calculated value of the friction.


where tfinal,raw is the raw time corresponding to the final control parameter value. The true

velocities for the optimal protocol can then be calculated from

vi,MD =tfinal, raw

τvi,raw . (4.20)

Finally, the optimal control parameter value for any time in the protocol can be cal-

culated by applying the kinematic equation for constant acceleration between two known

points:

λ (t) = λi + vi(t− ti)+(vi+1− vi)(t− ti)2

2(ti+1− ti), (4.21)

where i is chosen such that ti < t < ti+1 and the true velocities and times for the minimum-

dissipation protocol are used, although the subscripts MD have been dropped.

Chapter 5

Results

This chapter first considers naive (constant velocity) driving protocols, then explores the

friction calculations and (approximately) minimum-dissipation protocols, and finally com-

pares naive and minimum-dissipation protocols. The naive protocols also allow us to gain

an intuition about the system behaviour before turning to the minimum-dissipation driving

protocols. It should be emphasized here and throughout this section that the minimum-

dissipation protocols calculated with these methods assume the system exists in the linear-

response regime. In general, the true protocol that minimizes work accumulation may not

be well approximated by the existing theory applied here.

We explored the system response as a function of the intrinsic barrier height, A, driving

strength, k, and driving cycle time, τ , by considering the tradeoffs between excess work,

speed, and accuracy. In both naive and dissipation-minimizing driving protocols, there is

a tradeoff between accuracy and work as speed increases. Results are compared to the

prediction for work accumulation if driven by a quadratic trap on a flat landscape that was

introduced in Section 2.2.1. We also compare ratios of naive and minimum-dissipation

excess works and do find a decrease in the ratio of work and accuracy, and therefore there

is an advantage to using the minimum-dissipation protocol. The ratio of excess works is

also compared to the linear-response prediction introduced in Section 2.3.2.

All results shown are cycle averages in systems that have reached the periodic steady

state.

41

CHAPTER 5. RESULTS 42

5.1 Naive Driving Protocols

The naive driving speed is set by the cycle time τ: v = 2π/τ . The system is initialized

as a uniform distribution with λ = 0 and λ then proceeds as λ (t) = 2πtτ. Because of the

periodic boundary conditions of the system, λ returns to 0 at the completion of each cycle.

This choice has no influence on the final results due to the periodic form of the potentials

in Equations 3.1 and 3.2. The system is evolved until it satisfies the periodic steady state

conditions described in Section 3.5. Figure 5.1 shows visualizations of the PSS distribution

for three different values of k. Driving is in the counter-clockwise direction.

(a) (b) (c)

Figure 5.1: Polar plot of the PSS distributions over a third of a cycle. Dashed lines show the

instantaneous value of λ and the correspondingly coloured solid lines are the distribution

as visualized on a polar plot. The radial axis is the probability. Both plots have rotation

rate 14 Hz and barrier height A = 4 kBT . The driving strength k is (a) 32 kBT , (b) 16 kBT ,

or (c) 4 kBT .


5.1.1 Accuracy and Work vs. Speed

Figure 5.2 shows the accuracy η , and excess work as a function of rotation rate, as well as

their ratio. The accuracy is defined as the net probability flow per cycle and characterizes

the average cycle completion per driving cycle, as in Equation 3.14. The flux is the accuracy

divided by the cycle time (η/τ). The excess work reported here is the cumulative excess

work over a single PSS cycle with protocol Λ, and since ∆F = 0 over a cycle, the work

equals the excess work.

Accuracy decreases monotonically with rotation rate for all explored values of barrier

height A and driving strength k. As the potential is changed more rapidly, the system does

not respond as strongly before the control parameter returns to its initial position.

The accuracy does not capture the influence of rotation rate on the production rate of

ATP. The flux is defined as η/τ and is therefore the average rate of probability flow, which

maps to the production rate of ATP.Tthe largest possible flux is limited by the rotation rate,

with the maximum flux (in the case of 100% accuracy) of 1/τ . Figure 5.2 shows that actual

flux peaks at intermediate rotation rate for some combinations of A and k. These peaks in

flux coincide with the crossover in accuracy.

Work peaks at intermediate rotation rates, for some combinations of A and k. Specifi-

cally, this feature only empirically occurs when k > A. The increase in work at slow rates

is expected: increasing speed increases the work because the system is being held further

from equilibrium. The exact scaling of work with rotation rate is further explored in Sec-

tion 5.1.2. In all cases with initially increasing work explored here, the work eventually

peaks and subsequently decreases as rotation rate increases further still.

The peaks in work (third row of Figure 5.2) coincide with the peaks in flux and with

the crossover in accuracy. The rotation rate was not altered with a resolution fine enough

to precisely match the work peak with the midpoint of the accuracy. The peaks in flux

and work leads to a tradeoff between speed and accuracy: for driving rates on either side

of the peak with the same amount of input work, the machine can achieve the same flux

while driven faster with lower accuracy or slower with higher accuracy. A plausible goal of

molecular machines is to complete a certain number of successful cycles in a given amount

of time. These results suggest that there are two rotation rates that lead to the same flux

and power: a slower driving rate with higher accuracy or a faster rate with lower accuracy.


0.0

0.2

0.4

0.6

0.8

1.0

η

A=0 A=2 A=4 A=8

k (kBT)

1 2 4 8 16 32

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

η/τ

(Hz)

10-2

10-1

100

101

102

⟨ W⟩Λ

(kBT)

101 102100

101

102

103

⟨ W⟩Λ/η

(kBT)

101 102 101 102 101 102

Rotation Rate (Hz)

Figure 5.2: Accuracy, flux, excess work, and work/accuracy ratio as a function of rotation

rate for naive protocols. Intrinsic molecular barrier heights, A, vary between sub-plots

while driving strength, k, varies within each sub-plot. Top: accuracy decreases as cycle

rate increases. Second row: flux varies non-monotonically as rotation rate increases. Third

row: excess work per cycle varies non-monotonically as rotation rate increases. Bottom:

the ratio of excess work and accuracy increases as both rotation rate and A increase.


Experimentalists acknowledge that they cannot distinguish between these cases from bulk

measurements of ATP synthesis rates [23].

The tradeoff between accuracy and work can be further explored by considering their

ratio, as in the lower row of Figure 5.2. Figure 5.3 shows the barrier-less (A = 0) work-

accuracy ratio from Figure 5.2, but on a linear scale. Here the ratio 〈W 〉Λ/η increases

linearly with rotation rate, with identical slopes of 1.18 kBT s for all values of k. In other

words, without molecular barriers, there is a constant work required per unit probability

pushed around the system, independent of driving strength k. This finding that barrier-less

work is independent of k agrees with the results in [43].

0 20 40 60 80 100 120 140

Rotation Rate (Hz)

0

20

40

60

80

100

120

140

160

⟨ W⟩Λ/η

(kBT)

k (kBT)

1

2

4

8

16

32

Figure 5.3: Ratio of excess work and accuracy for A = 0. The linear scaling with rotation

rate is independent of k and has a slope of 1.18 kBT s.


5.1.2 Accuracy and Work

The ensemble properties can be considered in the space of accuracy (probability flow per

cycle) and work (Figure 5.4), and their corresponding rates: flux and power (Figure 5.5).

An ideal motor should maximize accuracy and minimize work while operating at high

speeds. Aiming for the upper left region of Figure 5.4 with maximal flow per cycle and

minimal excess work does not account for the disadvantage of requiring slow cycles. Al-

ternatively, an ideal motor reaches maximal flux of 1/τ with minimal power, occupying the

upper left region of Figure 5.5.

The theory lines in Figures 5.4 and 5.5, are independently derived from the quadratic

trap driving theory in [43] and introduced in Section 2.2.2. The value of k in the present

model maps, in the limit of large k, to a quadratic trap with spring constant 2k, see Ap-

pendix A. In the limit of large k, most of the probability is within the region for which

the quadratic approximation is valid, i.e. the trap is strong enough that there is insignifi-

cant probability anywhere but near the control parameter value. The theory predicts that a

quadratic trap dragged across an otherwise flat landscape accumulates work v2mγt where v

is the speed of the trap (2π/τ in this case), mγ is the viscous friction, and t is the elapsed

time (τ). Mapping the quadratic case to the A = 0 system, the work accumulated over one

cycle is predicted to be independent of k:

Wex ≈ mγ(2π)2

τ=

kBTD

(2π)2

τ, (5.1)

with corresponding power

〈Pex〉Λ ≈ mγ

(2π

τ

)2

=kBTD

(2π

τ

)2

. (5.2)

The lines in Figures 5.4 and 5.5 were obtained by multiplying the prediction for work

and power, respectively, by the accuracy on the vertical axis. The lines therefore correspond

to the linear decrease that would occur if the work (or power) decreased proportionally to

the total flow (or flux). For example, this assumes that if 50% of the probability is driven

around in the cycle, it costs half as much work as that predicted by the quadratic trap case.

Figure 5.5 plots rates rather than cycle averages and quantifies how much is accomplished

per unit time and how much ongoing operation costs. For rates, the quadratic trap theory

lines end at different maximum flux values, corresponding to 1/τ . This contrasts with the


theory lines in the per cycle case. In both cases, decreasing the rotation rate shifts the theory

line towards the left. However, for rates, the largest flux possible is fixed by the rotation

rate 1/τ , thus emphasizing that slower is not necessarily better if there is a requirement to

maintain a certain level of production, i.e. complete cycles.

The insets in Figures 5.4 and 5.5 indicate that a linear decrease indeed occurs for the

A = 0 case, since all points for a given rate fall on the corresponding line. Furthermore, the

points for A > 0 all fall below their corresponding lines, indicating an increase in work due

to raising the intrinsic barriers that is not due to pushing proportionally more probability

around. In other words, the cost per probability is higher as A increases. This result is

more clearly seen in the lower row of Figure 5.2, where the ratio of 〈W 〉Λ/η increases

as A increases. The result that no points cross their corresponding line derived from the

quadratic trap driving indicates this is the frontier of possible operation: no combination of

A and k brings the machine closer to the region of maximal accuracy and minimal work.


10-1

10-2

10-3

10-4

10-5

100

10-2 10-1 100 101 102

10-1

10-2

10-3

10-4

10-5

100

10-2 10-1 100 101 102

Figure 5.4: Accuracy and excess work per cycle for naive protocols. Points shown are for

A = 1,2,4,8 and k = 1,2,4,8,16,32. Colour denotes rotation rate. No points cross their

corresponding line derived from the quadratic trap driving, creating a frontier. Decreasing

rotation rate pushes the frontier towards the upper left. Inset: only the A = 0 points are

plotted to show the close match with the corresponding lines.


10-2

10-1

100 101 102 103 104

100

101

102

10-3100 101 102 103 104

10-2

10-1

100

101

102

10-3

Figure 5.5: Probability flux and power averaged over one naive cycle. Points shown are

for A = 1,2,4,8 and k = 1,2,4,8,16,32. Colour denotes rotation rate. Decreasing rotation

rate pushes the frontier towards the optimal region in the upper left. Inset: only the A = 0

points are plotted to show the close match with the corresponding lines.


5.2 Friction and Minimum-Dissipation Driving Protocols

The generalized friction quantifies the resistance of the system to changes in control pa-

rameter. The exact form of the friction is a function of equilibrium fluctuations, whereas

the purportedly minimum-dissipation protocol is only guaranteed to minimize dissipation

when linear response holds. Friction and minimal-dissipation velocities in Figure 5.6 are

calculated for discrete points at 10 degree increments and are plotted with straight connect-

ing lines to guide the eye. Following the notation used in Section 4.4, {λ}ζ is the set of

positions {0,10,20...360}. The minimum-dissipation protocol (Figure 5.6, lower row) is

obtained by extrapolating between known points using the method outlined in Section 4.4,

using a time increment of τ/1000.

Friction reaches its peak values when the control parameter λ is at the intrinsic barriers

(60◦, 180◦, and 300◦), while friction is minimized in the wells (0◦, 120◦, and 240◦). This

behaviour is observed for all combinations of A and k and agrees with the intuition that

friction is maximal in regions of slow relaxation and large force variance. When λ aligns

with a well, the system relaxes quickly into this metastable state. However, when λ aligns

with the barriers, there are two equal-energy wells, and therefore the force correlation time

is maximized. Recall that the force correlation in the definition of friction refers to driving

forces only (not the total force experienced at a particular position), and therefore the two

wells on either side of λ experience sign-reversed forces in the case of λ centred on a

barrier: one side experiences only positive deviations from the average while the other

experiences only negative values.

The amplitude (difference between maximal and minimal values) of the friction is not

monotonic with k. Additionally, the average value of the friction over all angles is not

monotonic, as seen in Figure 5.7. Yet the observed behaviour does match with intuition for

the zero driving and strong driving limits. In the weak driving limit (k� A) the forces due

to driving are negligible and therefore the equilibrium fluctuations do not vary significantly

as λ is moved, and thus the friction has a smaller amplitude. For extremely strong driving

(k� A), the molecular landscape varies negligibly with λ compared to the scale of the

driving potential, so equilibrium fluctuations again do not vary significantly depending on

whether λ is aligned with a well or a barrier, leading to a decrease in amplitude again, but

this time because the energy landscape is dominated by the effect of driving. Furthermore,


0 π/3 2π/3 π 4π/3 5π/3 2π102

103

104

ζ(k

BT·s)

1

0 π/3 2π/3 π 4π/3 5π/3 2π

λ (rad)

A=2

k (kBT)

1 2 4 8 16 32

0 π/3 2π/3 π 4π/3 5π/3 2π

4

0 1/6 1/3102

103

dλ

MD

dt

(rad/s

)

0 1/6 1/3

λ (rad)0 1/6 1/3

0 1/6 1/30

1/6

1/3

λM

D/2π

0 1/6 1/3

t/τ0 1/6 1/3

Figure 5.6: Friction (top) and minimum-dissipation velocities (middle) as a function of

control parameter, and the minimum-dissipation protocols (bottom). Intrinsic molecular

barrier heights, A, vary between sub-plots while driving strength, k, varies within each sub-

plot. Note: the relaxation time for k = 1,A = 4 is exceedingly long—precluding accurate

estimation of the friction—so this data is omitted.


100 101

k (kBT)

102

103

104

⟨ ζ⟩ λ(k

BT·s)

A (kBT)

1

2

4

Figure 5.7: Spatial average of the friction as a function of driving strength k, for varying

barrier heights. The average friction increases with A and peaks at intermediate values of

k. Dashed line is the expected friction for a quadratic trap on a flat landscape.

the average value of the friction can be predicted in the strong driving limit on a flat land-

scape (A = 0). In this case, the system is approximated well by the quadratic trap driving,

which has a constant friction of mγ = kBT/D. By considering the large k behaviour in

Figure 5.7, the system does indeed converge to the value of mγ used in the simulations,

1000 kBT s, denoted by the dashed line.

The linear-response approximation of the optimal protocol yields a velocity propor-

tional to the inverse square root of the friction (Equation 2.17); hence, the maximal ve-

locities correspond to the minimum values of friction, and vice versa (Figure 5.6, middle

row). The minimum-dissipation velocities and corresponding protocol are shown only for


a third of a cycle, covering 2π/3, since the system is periodic and the other two periods

are identical. The control parameter position is interpolated by assuming constant acceler-

ation between points of known velocity. Finally, because of the τ-independent shape of the

protocol, it has been reported here as a function of nondimensionalized time: t/τ .

Figure 5.8 shows for minimum-dissipation driving cycles the accuracy, flux, excess

work, and work/accuracy ratio. These show the same qualitative features as for naive pro-

tocols ( Figure 5.2). Figures 5.9 and 5.10 report the minimum-dissipation results in the

spaces of accuracy-work and flux-power, respectively. All points are still constrained by

the apparent frontier quantified from the quadratic trap driving approximation. The next

section compares the naive and minimum-dissipation protocols directly.


0.0

0.2

0.4

0.6

0.8

1.0

η

A=0 A=2 A=4

k (kBT)

2 4 8 16 32

10-7

10-6

10-5

10-4

10-3

10-2

η/τ

(Hz)

10-2

10-1

100

101

102

⟨ W⟩Λ

(kBT)

101 102100

101

102

103

⟨ W⟩Λ/η

(kBT)

101 102 101 102

Rotation Rate (Hz)

Figure 5.8: Accuracy, flux, excess work, and work/accuracy ratio, as a function of driving

rate for the minimum-dissipation protocols predicted by linear-response theory. Intrinsic

molecular barrier heights, A, vary between sub-plots while driving strength, k, varies within

each sub-plot. Top: the accuracy decreases as cycle rate increases in all cases. Second row:

the flux changes non-monotonically as driving rate increases. Third row: excess work per

cycle changes non-monotonically as driving rate increases. Bottom: the ratio of excess

work and accuracy increases as both rotation rate and A increase.


10-1 100 101 102⟨W⟩

Λ (kBT)

10-4

10-3

10-2

10-1

100

η

Rate (Hz)

133

66

33

16

8

4

2

Figure 5.9: Accuracy and excess work per cycle for minimum-dissipation protocols. Points

shown are for A = 1,2,4 and k = 1,2,4,8,16,32. Colour denotes rotation rate. No points

cross their corresponding line derived from the quadratic trap driving, creating a frontier.

Decreasing rotation rate pushes the frontier towards the upper left.


100 101 102 103 104⟨Pex

⟩Λ (kBT/s)

10-2

10-1

100

101

102

Flux (

1/s)

Rate (Hz)

133

66

33

16

8

4

2

Figure 5.10: Probability flux and power averaged over one minimum-dissipation driving

cycle. Points shown are for A = 1,2,4 and k = 1,2,4,8,16,32. Colour denotes rotation

rate. Decreasing rotation rate pushes the frontier towards the optimal region in the upper

left.


5.3 Minimum-Dissipation vs. Naive Driving Protocols

Figure 5.11 reports the ratios between naive and minimum-dissipation protocols of the

accuracy η , excess work W , and the work-accuracy ratio W/η . Interestingly, the accuracy

is higher in the naive driving case, although significant accuracy differences only appear

in the A = 4 case. Further investigation into this behaviour is required in order to gain an

understanding of its origin. The ratio of the excess works is compared to the ratio predicted

in Equation 2.18: ζ

ζ 1/22 . The prediction is shown as the dashed lines in the middle row.

Agreement with this prediction is found in the A = 1,2 cases for low driving rates.

The ratio ζ

ζ 1/22 characterizes the spatial variability of the friction. When the spatial

variability in friction is most extreme, the minimum-dissipation protocol predicted by the

linear-response theory deviates most from naive. When the MD protocol differs more from

the naive protocol, there is a larger gain when the MD schedule is used. It is for this reason

that the largest ratio between naive and MD is when the friction is most extreme. Figure 5.6

shows that intermediate values of k lead to the most extreme spatial variability in friction.

The ratio of naive to minimum-dissipation W/η (bottom in Figure 5.11) seems to tend

towards 1 at large rotation rates for all values of A. The work differences are quite sig-

nificant: in the A = 4 case, the naive protocol requires over 40% more work than the

minimum-dissipation protocol. The barrier heights in F1 correspond well to the A = 2

case of our model. At slow driving rates, the naive protocol does nearly 20% more work

than the minimum-dissipation protocol. Finally, the bottom row explores the ratio of the

scaled work: the work per probability pushed. This quantity is arguably more meaningful

in this context, since a decrease in work during the minimum-dissipation protocol could

arise from a decrease in accuracy.

The trend of maximum differences between the two protocols at low rotation rates is

still apparent: the minimum-dissipation protocol does indeed dissipate less energy per unit

probability than the naive driving.


⌘naive

⌘MD

W naive

WMD

(W/⌘)naive

(W/⌘)MD

Figure 5.11: Ratios of accuracy, excess work, and W/η for the naive and minimum-

dissipation protocols. Accuracy (top) is higher in the naive driving case. At low rotation

rates, the naive work is larger than that for the minimum-dissipation protocol (middle). At

high rotation rates, the ratio tends to 1. However, at intermediate rotation rates there is

no clear common trend. The dashed lines show ζ

ζ 1/22 . The scaled ratio (bottom) follows

similar trends as the work ratio.


5.4 Parameter Ranges: Discussion and Limitations

The range of parameter space explored was set by a combination of computational limita-

tions, a desire to focus on regions of interesting behaviour, and biological relevance. With

the motivation of studying efficient machines, parameter combinations in which less than

5% of the probability responds to driving were deemed low priority. For this reason, anal-

ysis focused on values of A less than k. Furthermore, rotation rates faster than 133 Hz

posed computational challenges, especially during minimum-dissipation protocols, since

it pushed the limits of simulating continuous driving. Since the accuracy dropped signifi-

cantly by these top speeds and since the biological system is believed to operate at ∼ 100

Hz, this was deemed an acceptable upper bound. Regarding slowest driving rates, the com-

putational resources required to push to slower rotation rates was not set as a priority over

other analysis. Further work could and should explore the asymptotic behaviour at slow

driving. However, since this work chose to focus on biologically relevant time scales, that

study is left to a later date.

The range of A was set by computational limitations and the fact that accuracy was in-

significant in all but the strongest and slowest driving cases. In the case of strong barriers,

the relaxation time of the system is large enough to make the friction calculation compu-

tationally intractable using the strict convergence criteria applied at lower barrier heights.

One could adjust the calculation to use small time increments near the initial relaxation and

then switch to larger time steps when the dynamics become slow; however, this was not

performed.

Chapter 6

Conclusions

This research set out to explore the fundamental limitations of molecular machines by

developing a minimalistic model. Our model is general enough to capture the essential

physics of a driven stochastic rotary machine, yet is detailed enough to have a physical

interpretation that maps to biological molecular machines. Fokker-Planck dynamics were

used to numerically study the evolution of nonequilibrium probability distributions of a

driven system. The accuracy, speed, and excess work were quantified for periodic steady

state cycles. Existing theoretical work was applied to search for minimum-dissipation driv-

ing protocols in an attempt to further improve the tradeoffs.

Our results find a significant tradeoff between accuracy and work over the range of

speeds corresponding to experimental and biological rotation rates of F1. Specifically,

for the same input work, one can go faster with lower accuracy or slower with higher

accuracy. Furthermore, the frontier of machine operation in the space of accuracy and work

appears to be well predicted by the work accumulation function for a quadratic potential

on a flat landscape in the linear-response regime. Intrinsic barriers lead to an increase in

work required to drive the same amount of probability through the system. However, the

intrinsic barriers represent the discrete chemical events synthesising or hydrolysing ATP.

In order for this model to well approximate mechanochemically coupled F1, finite barrier

heights are necessary.

60

CHAPTER 6. CONCLUSIONS 61

6.1 Significance

6.1.1 ATP Synthase

The value of A = 2 in our model best approximates the barriers in F1. This class of systems

saw a significant change in behaviour as both rotation rate and driving strength are varied.

At rotation rates comparable to those used in single molecule experiments and strong driv-

ing (k > A), accuracy was at least 50%. However, at over 100 Hz, accuracy dropped to less

than 20% even for the strongest driving applied here (k = 32). Our model therefore pre-

dicts a significant decrease in accuracy when the system operates at the rates we believe are

achieved in active cells. In the A= 2 case, our model predicts that the minimum-dissipation

protocol we approximated decreases the energetic cost of driving probability through the

system, with the naive protocol costing nearly 20% more in the k = 16 case.

Comparing our naive and minimum-dissipation driving protocols to the actual mo-

tion of the γ-shaft suggests that ATP synthase’s operation is more like the minimum-

dissipation protocol. When F1 is run as an ATP-burning motor in diffusion-limited con-

ditions, the γ shaft hops between discrete angular positions, rather than proceeding at a

constant speed [14]. Furthermore, crystal structures used to connect γ’s motion with the

chemical reactions indicate that the crankshaft remains fixed at well-defined angles during

the “catalytic dwell” and while waiting for diffusion of reactants into active sites [17]. It is

plausible that γ pauses at these precise angles to hold the catalytic sites in conformations

that correspond to a midpoint between their own metastable states. Indeed, their crystal

structures resolve “half-closed” and “half-loose” states (referring to the state of β and the

αβ interface, respectively), and the authors argue that these intermediate structures are

crucial to the tripartite coupling between catalytic sites that they propose for the conforma-

tional cycle of F1.

Thinking instead about operation in the ATP synthesis direction when F1 is driven by

FO, it is also likely that non-naive protocols are a more faithful picture of γ’s behaviour.

The mechanism through which the c-ring rotates γ is not well understood, but is believed

to occur as n individual steps, where n is the species-specific number of c subunits [15].

We are not proposing that ATP synthase rotates in discrete steps because of evolutionary

forcing towards non-constant driving: every process tied to chemical reactions will have to

proceed as a series of discrete events when considered on a time scale in which individual


reactions are resolved and appear as instantaneous events. However, it could be possible

that the conformations of the waiting dwells have been optimized in agreement with the

predictions of minimum-dissipation protocols. That would mean pausing in FO conforma-

tions that create comparable probabilities for the “before” and “after” states of F1, allowing

the system to optimally harness thermal fluctuations to drive the cycle forwards.

6.1.2 Artificial Machines

Our work explored the tradeoffs of accuracy, excess work, and speed while driving a cyclic

stochastic machine. While we were inspired by the goal of driving ATP production, our

findings can apply to enzymes that catalyze other reactions or other stochastic processes

that can be quantified within a cyclic framework.

Our results agree with the intuition of dissipation-minimizing protocol design: drive

fast in regions of fast relaxation and slow in regions of high potential barriers. While we

imagined that our perturbation to the intrinsic energy landscape of the system was caused by

a magnetic field, the results are generalizable to any form of driving capable of influencing

the energetics of the system in a cyclic manner that is not time reversible: the change in

control parameter is distinguishable under time reversal.

6.2 Future Work

6.2.1 Extending the Model

Our model was deliberately designed to be generalizable and minimalistic. However, this

model could be extended to better capture the details of a specific system, such as F1. For

example, our current model couples two systems and assumes perfect coupling either be-

tween the magnet and γ or between γ and the chemistry. One could add another layer of

coupling to represent a third level of the system to loosen the assumption of perfect cou-

pling. Our current system has two levels and therefore adding a third is a natural next

step. These three levels would represent the driving, γ crankshaft, and the chemical co-

ordinate. Furthermore, our model considers one-dimensional state variables: driving and

γ are each specified by a single parameter, θ0 and angular position, respectively. If the


chemical coordinate were to be added, it might be best described as a multidimensional

state; for example, each of the three catalytic sites could be in one of several possible states

representing ATP bound, open, etc.

More compelling still would be to consider feedback to better map the model onto an

FO-driven system. If γ was instead being turned by FO, it is reasonable to believe that the

driving could sense the current state of F1 and feel resistance to progression if there is a

lag. For example, there could be a greater work cost if the driving proceeded too far ahead

of the probability, but conversely could potentially increase the accuracy.

6.2.2 Adaptive Time Step Friction Calculation

The friction calculation we used became computationally intractable for barrier heights

larger than 4, due to the long relaxation times. An algorithm using adaptive time steps

could improve the speed of the calculation without significant loss of detail. The algorithm

would use a short time step in the Fokker-Planck dynamics during the rapid relaxation after

initialization. Once the force correlation has dropped below a certain threshold and dynam-

ics are not changing as rapidly, a larger time step can be used. The initial fast relaxation

is because the system is initalized from a delta function, and therefore a sufficiently small

time step is required to avoid negative probability left in any bin.

6.2.3 Higher-Order Corrections to Minimum-Dissipation Protocol

The approximation of the minimum-dissipation protocol we used assumed linear-response.

However, linear-response is not necessarily expected to hold in our system. At low rota-

tion rates, the predictions for the ratio of naive and minimum-dissipation work agree with

the theory, suggesting that the system is in linear response at these rates (Figure 5.11).

However, at higher rotation rates, there is a deviation from the predictions.

Returning to the original derivation of the minimum-dissipation protocol, it is not ex-

pected that the protocol remains shape-invariant with driving rate. Rather, the rate of rota-

tion will influence the dissipation-minimizing protocol. By observing the PSS distributions

in the case of fast driving, the system is observed to lag behind the control parameter by a

fixed amount, as large as 30◦.


It is possible that the next leading correction to the protocol involves a velocity-dependent

shift of the control parameter trajectory: the shape of the trajectory remains the same but

the locations of maximal velocity are not aligned with the minima of the potential. Such

a shift would result in the peaks in control parameter velocity occurring after the centre

of the barrier: the control parameter rushes over the peak while significant probability is

still relaxing in the previous well, and then the control parameter slows down afterwards to

allow the system to catch up 1.

6.2.4 Experimental

As experimental techniques advance, the ability to resolve faster dynamics is possible.

Observing the rotation of γ while being driven by FO would further our understanding of

the angular states that γ occupies in vivo. The 3 (or 6) dwells that have been observed for

isolated F1 are not necessarily representative of the system when also tethered to the c-ring,

since the symmetry of the ring has a different periodicity (10-15 discrete rotation events per

complete 360 degree rotation). When both subsystems operate together as in the complete

ATP synthase, γ will feel both the torque of the c-ring and the wells of F1, perhaps leading

to a novel steady-state angular distribution.

It would also be interesting to revisit the angular distributions from past experiments.

Our results indicate that accuracy significantly decreases for weaker driving and faster ro-

tation rates. It would be interesting to revisit the weakly driven data from experiments and

distinguish between futile cycles (γ did not go around) versus occasional backwards cycles,

since both could sum to the same number of net forwards cycles, but would have different

implications about the response to driving. Results shown in [9] hint at the former case, but

a complete analysis could reveal more insight.

Finally, experimental verification of our finding that the minimum-dissipation protocol

significantly reduces the excess work could be conducted using single-molecule techniques

that already exist. By combining existing methods of measuring excess work, observing

rotation rate, and controlling the orientation of magnetic fields, it would not be beyond the

capabilities of an exceptional experimental group to conduct these experiments.

1Based on personal communication with Tomohiro Shitara and David. A. Sivak

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Appendices

71

Appendix A

Quadratic Approximations

The sinusoidal driving potential in Equation 3.2 can be approximated as a quadratic poten-

tial though Taylor expansion around θ = 0:

U(θ , t) =k2

(1+ sin

[θ − π

2−θ0

])

=12

k(1− cos(θ −θ0))

≈ k sin2(

θ0

2

)− 1

2θk sin(θ0)+

14

kθ2 cos(θ0)+O(x3) .

(A.1)

Noting that the first two terms are 0 and cos(θ0) = 1 when θ0 = 0, and matching the

quadratic term with the form of a quadratic potential yields

14

kθ2 =

12

k′θ 2 , (A.2)

where k′ is the spring constant of the quadratic approximation. Therefore k′ = k/2: our

model’s driving potential corresponds to a quadratic potential with a trap stiffness of k/2.

The sinusoidal intrinsic potential in Equation 3.1 can also be approximated by a quadratic

potential through Taylor expansion around θ = 0:

Uintrinsic(θ) =A2

(1+ sin

[3(

θ +π

2

)])

≈ 9Aθ 2

4+O(x4) ,

(A.3)

and again matching the quadratic term with the form of a quadratic potential yields

94

Aθ2 =

12

A′θ 2 , (A.4)

72

APPENDIX A. QUADRATIC APPROXIMATIONS 73

where A′ is the spring constant of the quadratic approximation. Therefore A′ = 2A/9:

each well in our model’s intrinsic potential corresponds to a quadratic potential with a trap

stiffness of 2A/9.

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