Stochastic Modeling of Motor Proteins

Stochastic modeling of motor proteins

Martin Lindén

Doctoral Thesis KTH School of Engineering Sciences Stockholm, Sweden 2008 TRITA-FYS 2008:9 KTH School of Engineering Sciences ISSN 0280-316X AlbaNova Universitetscentrum ISRN KTH/FYS/--08:9--SE SE-106 91 Stockholm ISBN 978-91-7178-897-9 Sweden

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till oﬀentlig granskning för avläggande av teknologie doktorsexamen i teoretisk fysik freda- gen den 28 mars 2008 klockan 10.00 i sal FA32, AlbaNova Universitetscentrum, Kungl Tekniska högskolan, Roslagstullsbacken 21, Stockholm.

Abstract Motor proteins are microscopic biological machines that convert chemical energy into mechanical motion and work. They power a diverse range of biological processes, for example the swimming and crawling motion of bacteria, intracellular transport, and muscle contraction. Understanding the physical basis of these processes is interesting in its own right, but also has an interesting potential for applications in medicine and nanotechnology. The ongoing rapid developments in single molecule experimental techniques make it possible to probe these systems on the single molecule level, with increasing temporal and spatial resolution. The work presented in this thesis is concerned with physical modeling of motor proteins on the molecular scale, and with theoretical challenges in the interpretation of single molecule experiments. First, we have investigated how a small groups of elastically coupled motors collaborate, or fail to do so, when producing strong forces. Using a simple model inspired by the motor protein PilT, we find that the motors counteract each other if the density becomes higher than a certain threshold, which depends on the asymmetry of the system. Second, we have contributed to the interpretation of experiments in which the stepwise motion of a motor protein is followed in real time. Such data is naturally interpreted in terms of first passage processes. Our main conclusions are (1) Con- trary to some earlier suggestions, the stepping events do not correspond to the cycle completion events associated with the work of Hill and co-workers. We have given a correct formulation. (2) Simple kinetic models predict a generic mechanism that gives rise to correlations in step directions and waiting times. Analysis of stepping data from a chimaeric flagellar motor was consistent with this prediction. (3) In the special case of a reversible motor, the chemical driving force can be extracted from statistical analysis of stepping trajectories.

Preface

This thesis presents the results of my work as a graduate student at the Department of Theoretical Physics at the Royal Institute of Technology during the years 2003 – 2008, and as a visiting student at the Institute for Physical Science and Technology, University of Maryland, College Park, during the fall of 2005. The thesis is divided into two parts. The first part gives a general introduction to the research field, and a background to the scientific work of this thesis. The second part consists of scientific papers I have coauthored. A specific introduction to the papers is given in chapter 6.

Comment on my contributions to the papers Paper I The basic modeling approach in this paper was ﬁrst formulated in a diploma project in our group 188. I further developed the model, wrote the simulation code, and performed all simulations and data analysis. I produced the ﬁgures for the paper, and took part in writing the manuscript.

Paper II I uncovered the discrepancy between the ratio of forward and backward steps in kinesin data 47,143 and an earlier theoretical description in terms of cycle completions 108, and found the correct ﬁrst passage problem. I also checked the published derivations, and took an active part in discussions during the writing of the manuscript.

Paper III I did all calculations, analyzed the stepping data, produced all ﬁgures, and took part in writing the manuscript.

Paper IV I did all calculations, produced the ﬁgures, and wrote the manuscript.

v Contents

Abstract ...... iii

Preface v

Contents vi

I Background 1

1 Introduction 3

2 Three examples of motor proteins 5 2.1 Kinesin I and intracellular transport ...... 5 2.2 The bacterial ﬂagellar motor ...... 10 2.3 Type IV pilus retraction ...... 13 2.4 Molecular fuel ...... 16 2.5 Optical trapping in motor protein experiments ...... 18

3 Modeling motor proteins 21 3.1 The ﬂashing ratchet motor ...... 22 3.2 The free energy landscape ...... 23 3.3 Modeling transition rates ...... 27 3.4 Circulation currents in steady state ...... 31

4 First passage processes 35 4.1 A discrete ﬁrst passage problem ...... 35 4.2 Steps and cycle completions ...... 38

5 Stochastic simulations 43 5.1 Simulation of chemical kinetics ...... 43 5.2 Continuous and mixed models ...... 45

6 Introduction to the papers 49

vi vii

Acknowledgments 53

Bibliography 55

II Publications 73

Part I

Background

Chapter 1

Introduction

The ability to move your hand and open this thesis is the work of an amazing molecular machinery. If we look close enough at the muscles in our fingers, for example using electron microscopy, we would see an intricate mechanism with several levels of organization. On the molecular level, muscle contraction is caused by the collective action of vast numbers of motor proteins, nanoscale machines that pull on protein filaments in our muscle fibers 92. Motor proteins (although not the specific kind in our muscle), and theoretical models to describe how they work, are the topics of this thesis. Microbiology offers many exotic examples of motion powered by motor proteins. Some bacteria swim using long screw-like propellers 29, or by driving kinks down their own phone-cord like bodies 172. Other bacteria crawl on surfaces using grappling hooks 134. The self-assembly of certain viruses involves a phase where the viral DNA molecule is packed into the small virus capsule 78, using a motor protein that can achieve packing pressures up to sixty atmospheres 177. Our own cells also rely on motor proteins for many different tasks, for example to change shape and move around, to run an intricate cellular transport system 193, and to organize the genetic material and pinch off daughter cells during cell division. The translation and transcription of genes also involve special enzymes move along the DNA and RNA molecules 3. These are just a few examples. This thesis is part of an ongoing effort to understand, on the molecular level, how these machines work. Molecular motors in solution (water, or cellular fluid) work under very different conditions than those of our everyday macroscopic world. Inertial effects are virtually absent. Instead, viscous friction and random thermal fluctuations are the dominating forces at these length scales. Small objects in solution move in a highly erratic fashion, under the influence of constant collisions with surrounding molecules coming in from all directions. Using our everyday experience, it is rather surprising that anything useful can be done in a purposeful and efficient manner under such circumstances. But millions of years of evolution have accomplished just that. Many biological machines are robust, efficient, and precise.

3 4 Chapter 1. Introduction

The challenge to understand how this is possible, in general as well as in many particular cases, is not only an interesting problem in its own right. Using biological systems, or design principles from biological systems, to invent artificial nanoscale machines is an enterprise with great technological potential. The motor proteins of many infectious bacteria are also potential targets for new antibacterial drugs 23,119. If a key part of a motor protein that is crucial for the infection process is used as a drug target, mutations to avoid the drug is likely to also destroy the motor function, rendering the bacterium harmless. The field of molecular motors, and the study of complex biomolecules in general, has been driven in the last decades by rapid developments in experimental techniques. Single molecule techniques such as advanced microscopy, various fluo- rescence methods, and optical and magnetic tweezers, make it possible to detect, trace, and manipulate single motor proteins in real time, with a steady increase in accuracy. In parallel, the methods of molecular biology make it possible to manipulate the construction elements of the motor proteins by changing their amino acid sequence, and also to discover many new variants of motors through the genome project. One major advantage with single molecule experiments on biomolecules is that they can probe the fluctuations of the system under study. Fluctuations are usually averaged out in bulk experiments, where the behavior of a macroscopic number of molecules, e.g., 1012 (the number of molecules in a 1 ml test tube at 1 nanomolar concentration) is studied. Measurements at the single molecule level helps us to appreciate the role of fluctuations in the dynamics of these ’soft’ systems. The fluctuations also contain information that might not be easily available from average values alone. Two important goals for research on motor proteins are to understand the detailed mechanisms of specific systems, and identify general principles for their operation. These are difficult problems, that require efforts from biology, chemistry, and physics. The contribution from this thesis is to develop and analyze theoretical models, that can be used to describe and interpret experimental data. Modeling of motor proteins can be done on different levels. In this thesis, we consider coarse- grained stochastic models, with many traits that can be traced back to early work on muscle contraction 87,94. The rapid progress in single molecule experimental techniques motivates investigations of new aspects of these models, for example regarding the fluctuating dynamics of single motors. Using simple models to study these problems can also give valuable opportunities to develop a deeper understanding of non-equilibrium statistical physics. It seems likely that the developments in single molecule techniques will continue. As the examples later in this thesis will indicate, the understanding of many interesting systems is highly incomplete. The endeavor to understand motor proteins, and complex biomolecules in general, can therefore be expected to continue to pose challenging problems for further research. Chapter 2

Three examples of motor proteins

This is a thesis about protein motors. What are they? What do they look like, where can you ﬁnd them, and what do they do? In this chapter, we will introduce three motors that are relevant to the research presented in this thesis. All three examples are motors that work alone, or in small groups, and produce long stretches of motion before pausing or detaching from their molecular track. With an enzymatic term, motors that repeat their working cycles many times in a row are called processive. Biology oﬀers many examples of both processive and non-processive motor proteins. One notable example of the latter is the myosin in skeletal muscle. Here, the individual motor proteins spend most of their time detached from their actin track, but occasionally bind to give a single pull before dissociating again 92. By working together in large groups, there are always some motors bound to the track to keep it from sliding away. By spending a large fraction of time dissociated from the track, a single motor does not hinder the motion produced by the other motors. This makes it possible to attain higher speed than that corresponding to the turnover rate and step size of a single motor. Having acquainted us with some examples of motors, we turn to the molecular fuel that powers their operation. Finally, we will consider an important single molecule experimental technique, optical trapping.

2.1 Kinesin I and intracellular transport

One important and well studied class of motors are responsible for intracellular transport along the cytoskeletal ﬁlaments. The cytoskeleton of eukaryote cells is a network of protein ﬁlaments that extends throughout the cytoplasm. It helps the cell to maintain its shape and carry out coordinated movements, to interact mechanically with the environment, and to organize the cellular components 3. It also acts as a network to organize intracellular transport, which is our main interest here.

5 6 Chapter 2. Three examples of motor proteins

There are three main classes of cytoskeletal filaments, microtubules, actin filaments, and intermediate filaments. There are also three main families of motors working on the cytoskeleton filaments. Kinesin and dynein motors operate along microtubules, and myosins along actin filaments. The catalytic core (ATP binding pocket) of myosins and kinesins are very similar, which suggests a common evolu- tionary origin 3, different from that of dynein. All three motor families have many members. The characteristic length scales, the persistence lengths, for thermal shape fluctuations in actin and microtubules are much longer than the individual steps of single motors. From the perspective of single cytoskeletal motors, they can be con- sidered stiff structures. However, the cytoskeleton is a highly dynamics structure, and both actin filaments and microtubules can exert forces by active polymerization at their ends 3,92. Both actin filaments and micotubules are polar. In addition to being filament specific, the different motor proteins are adapted to walk in a certain direction along its track. Intracellular transport on the cytoskeleton is usually carried out by small groups of motors that work together to move their cargo, with different motors walking in different directions 16,24,80,113,114,120,121. The rest of this section will be devoted to one of the most well studied processive protein motors, kinesin I, or conventional kinesin, and its track, the microtubule filament.

Microtubules are hollow cylinders Microtubules are long cylinders, with an outer diameter of about 24 nm. Micro- tubules are very stiff and almost straight on the cellular scale, with persistence length on the order of several mm 39. The building blocks of microtubules are very stable tubulin dimers, consisting of one α and one β subunit. During formation of microtubules, the tubulin dimers first bind head to tail to form short polymers, called protofilaments. The protofilaments then bind to each other to form a hollow tube. Each tubulin subunit is about 4.1 nm long, so the periodicity of the protofilaments is 8.1-8.2 nm 65,122,130,198. As illustrated in Fig. 2.1, the tubulin subunits in a microtubule form a regular lattice, where the protofilaments run parallel to the main axis. This comes about in the following way: When the microtubule forms, there is a small offset between adjacent protofilament strands. After 13 protofilaments, the accumulated offset is equal to the length of three monomers 7,50,63. The resulting lattice can also be described as a three start helix. Because the monomers alternate, this still allows for two different configurations, the A and B lattices illustrated in Fig. 2.1. In the A lattice, the protofilaments are organized so that the lateral interactions are mainly between α and β subunits. The B lattice contains mainly α − α and β − β interactions, except for the seam between protofilament one and 13. It turns out that the B lattice is the more favorable configuration 103,171,178, and no microtubules with A lattices has been confirmed 92. However, the microtubule structure shows 2.1. Kinesin I and intracellular transport 7

protofilaments + end (a) (b)

4 nm

α β 24 nm − end seam

Figure 2.1. Idealized structures of a microtubule filament with 13 protofilament strands of alternating α (light) and β (dark) subunits. Fig (a) and (b) shows the A and B surface lattices respectively. The B lattice is energetically favorable, with lateral α − β interactions only along the seam. There are no confirmed instances of the B-lattice. Each subunit is about 4 nm long, giving a periodicity of 8 nm along the protofilaments.

a lot of variation. Filaments with between 8 and 19 protofilaments, 2 to 4 starts, and more than one seam has been observed 50,103,155. The structure can also vary along individual filaments due to lattice defects 51. Because the tubulin dimers are asymmetric and binds head to tail, the protofilaments are polar. This polarity carries over to the whole microtubule, since the protofilaments bind parallel to each other. Microtubules grow and shrink at their ends, with different speeds, with the β tubulin monomers are exposed at the fast growing plus end. The other end is called the minus end. Cells organize and control their microtubules by providing nucleation sites on certain organelles. The plus end of the microtubules grows outwards from the nucleation site, and the polarity therefore defines local inwards and outwards direction. 8 Chapter 2. Three examples of motor proteins

stalk

heads

tail neck−linkers

Figure 2.2. Diﬀerent parts of a kinesin I heavy chain dimer.

Kinesin walks along microtubules Kinesin I, or conventional kinesin, is a two-headed motor protein that walks towards the plus end of microtubules 3. It was first discovered in the squid giant axon 194. One kinesin I molecule is composed of two identical heavy chains, and, in some species 193, with two additional associated light chains. The heavy chains wind together to form a dimer, consisting of a coiled-coil stalk with a hinge in the middle, two globular heads at one end, and a fan-like cargo binding domain at the other end (the tail). The total length is about 80 nm, and the heads are about 10 nm in diameter 91. Two short flexible polypeptide chains, called neck linkers, connect each head to the stalk (Fig. 2.2). Kinesin I is the first discovered member of a large protein family, which are related by high sequence similarity in the motor domain. Humans have about 40 distinct kinesin proteins 3. Each head is an active motor domain that attaches to the tubulin dimers on the microtubule surface. The heads can also bind and hydrolyze ATP, and the binding affinity to tubulin depends on the chemical state of the head. Once bound, the heads undergo a coordinated cycles of chemical and conformational transitions which enables the kinesin molecule to walk along the microtubule. A single kinesin molecule can take several hundred steps before it eventually falls off 92,181. The heads use binding sites on the tubulin dimers as stepping stones, so the step size of about 8.2 nm 65,181 comes from the periodicity of the microtubule protofilaments. The walk follows the protofilament strand. This gives straight paths on microtubules with 13 protofilaments, and twisted paths on microtubules with twisted surface lattices due to a different number of strands 155. It is not clear if the the two heads bind to the same protofilament, or to two adjacent ones. In any case, the binding sites seem to be periodically distributed, i.e., both heads have the same step length within the present experimental resolution 65. The exact manner in which stepping occurs is not resolved in all details, but the main features are clear. The heads function very much like our own feet when we walk 11: they alternate between 16 nm steps and waiting while the other head moves 100,208. This is called ’hand-over-hand’ motion, in contrast to the ’inchworm’ 2.1. Kinesin I and intracellular transport 9

ATP ADP

D DP 0 T D DP 0 0

8 nm

Figure 2.3. The main events in the mechanochemical cycle of kinesin I, after Howard 92. The chemical state of the heads are denoted 0 (empty), T (ATP bound), D (ADP bound), and DP (ADP and Pi bound).

model, where one head is always leading 11. Furthermore, the heads pass each other on alternating sides, so that the stalk does not get twisted during the walk. Since the kinesin dimer consists of two identical heavy chains, but lacks left-right symmetry, the right-handed and left-handed steps are slightly different. One of the heads requires slightly longer time on average to complete its step, which leads to a slight limp in the walk 12,85. Forward stepping consists of a repeating cycle of chemical reactions and conformational changes, with one ATP hydrolyzed per cycle 53,93,168. Starting from a state where the trailing head (1) is empty and bound to the microtubule, and the forward head (2) is free and with ADP bound, the main steps in one cycle are as follows 92 (Fig 2.3): Binding of ATP to head 1 allows binding of head 2 to the next tubulin binding site. Next, ATP is hydrolyzed in head 1, while ADP is released from head 2. With ADP and phosphate bound, head 1 releases from the microtubule and is brought forward. Finally, phosphate release from head 1 brings the molecule back to the starting state, with the role of the heads interchanged. In order to make it possible for the motor to take many consecutive steps without falling off the microtubule, the two heads must be tightly coordinated, so that one head does not detach before the other is securely attached. This is believed to be accomplished through one or several check points, or gates, in the reaction cycle. One example of such a gate is the release of ADP from the heads, which is very slow in solution, but greatly accelerated when the head binds to microtubules 92. Another gating mechanism is illustrated in the first reaction of Fig. 2.3, in which ADP release and attachment to the microtubule by the leading head does not take place until the trailing head binds an ATP. Somehow, the reaction rates of one head depends on the state of the other. This could be accomplished for example through internal strain, or local conformational changes in the microtubules. The gating mechanism remains one of the unresolved details in the stepping mechanism of kinesin 37,195. Under low load in vitro , single kinesin molecules steps almost exclusively forward, and reach speeds of around 1 µm/s at high ATP concentrations 47,143,168. Figure 2.4 shows a common setup to study stepping, and a sketch of a stepping 10 Chapter 2. Three examples of motor proteins

(a) optical trap (b) position bead kinesin

MT molecule ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ − ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ + cover slip time

Figure 2.4. (a) A common experimental setup to follow the motion of kinesin. A kinesin molecule is attached to a ∼ 1 µm bead, and walks along a microtubule (MT) attached to microscope cover slip. An optical trap (see Sec.2.5) is used to manipulate the bead, and exert a force on the motor. (b) Illustration of a bead position trace, projected along the microtubule (synthetic data). The kinesin molecule steps towards the plus direction, with visible forward steps. The main noise source is the Brownian motion of the bead. record. The tail of a kinesin molecule is attached to a micron-sized bead of for example polystyrene, which can be tracked with high accuracy. The kinesin heads can then bind to microtubules that are immobilized on a cover slip, walk along the microtubule if ATP is present, and drag the bead along. Using an optical trap (see Sec. 2.4), it is also possible to exert forces on the bead. If an opposing load is applied, the speed goes down, and backsteps and detachments increase in frequency. The force at which the mean velocity drops to zero, called the stall force 92, is about 8 pN, and seems to be almost independent of ATP concentration 47,143. Kinesin can be made to walk backwards by applying superstall forces. The backward velocity also increases with increasing ATP concentration under these conditions, indicating that backsteps also require ATP binding 47.

2.2 The bacterial ﬂagellar motor

Many bacteria, such as Escherichia coli , swim by rotating their flagella, thin helical filaments that extend from all around the bacterial cell 3,29–31. Research into bacterial motility is almost as old as the light microscope, and the first bacteria using flagella to swim was observed in the early seventeenth century 28. Wild type E. coli have several flagella, four on average 29. The rotation is powered by a rotary motor situated in the cell membrane, at the base of each flagellum. When rotating in the counter-clockwise (CCW) direction, as seen looking from from the flagellar tip towards the base, the flagella form bundles that work like screw-like propellers. This is accommodated by the flagellar hook, a soft structure 2.2. The bacterial flagellar motor 11

flagellum

hook CCW rotation

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Figure 2.5. Sketch of the ﬂagellar motor, adapted from Berg 29, Brown et al. 43, and 131 Manson . The stator units consist of MotB2MotA4 complexes (PomB2PomA4 for sodium driven motors), and contain channels for the ion ﬂux that powers the rotation. The MotB subunits anchor the stators to the peptidoglycan layer. The MotA subunits interact with FliG proteins attached to the cytoplasmic side of the MS ring, to generate torque.

just outside the motor, which acts as a universal joint. In this mode, a bacterium can reach swimming velocities of a few tens of µm/s 29. The swimming efficiency for this construction is very low, only about 2% of the rotational power from the motors is converted to propulsive force 48. Every few seconds, the rotation of one or several motors changes direction for some fraction of a second, and that flagellum leaves the bundle to rotate on its own, pointing in a random direction. This causes the bacterium to ’tumble’: it spins randomly until all flagella resumes counter-clockwise rotation, and then swims off in a new direction 29,57. The frequency of switching between clockwise and counter-clockwise rotation is controlled by an intricate signaling system, that senses temporal changes in the concentrations of various substances. By decreasing the tumbling frequently when the concentrations of for example nutrients increase, the bacterium can bias its random movement to search for food and good conditions 3,31.

The flagellar rotary motor is a complex structure, about 45 nm in size. It consists of about 20 different kinds of proteins. The functions and positions of many parts are quite well known, and some of them are indicated in Fig. 2.5. However, many details about how the different parts work together to produce 12 Chapter 2. Three examples of motor proteins rotation are still unknown. The power source is an inward ion flux through ion channels in stator units that span the inner cell membrane 3,29. Some species, like E. coli , use protons 102, but there are also motors powered by sodium ions 95. The motors are sufficiently similar that it is possible to exchange parts between different species, to make a proton motor run on sodium and vice versa 10. At low loads, the rotation frequency can reach about 300 Hz 29,209. Wild type motors have many torque generating units. This can be seen in resurrection experiments, where defective motors are resurrected by addition of functional stator units. These kick in one by one, resulting in stepwise velocity increments 29. So far, up to 11 such steps have been seen 157, indicating that there is room for at least 11 functional torque generating units in a motor. Torque is generated by the stator units, which consists of MotB/MotA complexes (PomA/PomB for sodium driven motors). The composition is believed to be 2B+4A, with two ion channels per stator 29,41,164. The MotB proteins anchor the stator to the peptidoglycan layer, while MotA interact with FliG subunits in the MS ring, probably working like a stepping motor. The periodicity of the FliG ring varies between 24- and 26-fold between different motors 43,131,183. Measurements of the rotational flexibility of the hook gives a rotational spring 36,38 constant of kθ ≈ 400 pN nm/rad . Together with the high rotational frequency, and the many motor subunits, this makes it very difficult to detect steps in wild type motors. However, single steps has been observed by Sowa et al. 179, using a genetically modified sodium driven motor, with only one functional motor unit. The step length was roughly d = 2π/26, consistent with the periodicity of FliG units along the MS ring. Compared to the stiffness of the hook, this step length is quite small. For the stepping experiments, performed at 23 ◦C, the standard deviation for angular fluctuations of the hook is approximately 26 rk T σ = B d ≈ 0.42d. (2.1) 2π kθ Although the overall swimming efficiency is low, the motors themselves seem to be very efficient. Several observations are consistent with a tightly coupled mechanism, operating near equilibrium at low rotation speed near the stall torque 29,205. The strongest indication is that experiments with high viscous loads give a rotation speed proportional to the ionic motive force 29,72.(The ionic motive force is the energy per ion passing through the motor, see Sec. 2.4). The torque-speed curve, which is the rotational analog of a force-velocity relation v(F ), declines very steeply near the stall torque 29. This means that the velocity at high viscous load is proportional to the stall torque, which is therefore proportional to the ionic motive force. The steep decline in the torque-speed curve continues smoothly to backwards rotation 34. Backward rotation driven by outward proton flux has also been observed 132. The exact number of protons needed per turn, or per elementary stator step, is not yet resolved. Simple estimates indicates that one proton per step for a single stator is not enough, but that two might be 29,126. Early estimates of the proton 2.3. Type IV pilus retraction 13 consumption gave around 1000 protons per turn 29,95. At 10 stators and 26 steps per stator and turn, this gives ∼ 4 protons per elementary step. Randomness analysis of resurrection experiments are consistent with about 50 rate limiting transition per stator per turn 29, which would be consistent with 2 protons per elementary step. However, this kind of analysis can only give lower bounds on the number of transitions per turn 29,110. If the motor indeed does operate reversibly, the results of paper III of this thesis might be used to shed light on this issue, provided that one can detect individual steps reliably under conditions where backsteps occur with appreciable probability.

2.3 Type IV pilus retraction

Some bacteria can move in other ways than swimming. Twitching motility is a crawling motion over a surface, and is driven by repeated extension, tip attachment, and retraction of bacterial surface filaments that work in the manner of grappling hooks (see Fig. 2.6a). The filaments are called type IV pili, and protrudes from one or both of the cell poles 134. Twitching motility was discovered by Lautrop 117, who coined the term because of the somewhat jerky fashion of this movement when viewed in suspension. The association with surface filaments was made later 84. Type IV pili are crucial for the initial adherence of certain Gram-negative bacteria to their host cells, DNA uptake, cell signaling, and microcolony formation 55,98,134,145. The classification of bacteria into Gram-negative and Gram-positive refers to the outcome of the Gram staining test, and reflects different structural properties of the cell walls. Gram negative bacteria have two cell membranes with a thin peptidoglycan layer in the periplasmic space between the membranes. Gram- positive bacteria have only one cell membrane inside the thick cell wall of peptidoglycan 3. Bacteria capable of type IV pilus retraction include several human pathogens, for example Neisseria gonorrhoeae , Pseudomonas aeruginosa , and Neisseria meningi- tidis 134. The type IV pilus system is therefore a candidate as a target for designing new antibacterial drugs 23,119. Twitching motility is also one of the motility modes for the soil bacterium Myxococcus Xanthus , which has been studied extensively due to its interesting pattern formation abilities 99,163,176. Different gene nomenclature are used in the different systems. We will use the terminology for N. gonorrhoeae , where single molecule experiments on retraction exist 128,129,137. Craig and Li 55 has compiled a partial translation table between different well studied organisms. The pili are slender filaments, up to several micrometers long 128, and with an outer diameter of only about 6 nm 54,70,147. The persistence length is about 5 µm 175. They consist of pilin subunits, PilE. Each pilin subunit has a globular head domain that builds up the outer filament surface, and a strongly hydrophobic tail hidden inside the filament. According to a recent high resolution structure 54, the N. gonorrhoeae pili are right-handed one-start helices, with ∼ 3.6 subunits per turn, and a pitch of 3.7 nm. This means that each pilin subunit contributes just 14 Chapter 2. Three examples of motor proteins

(a) bacterium

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¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ cover slip retraction

Figure 2.6. (a) Twitching motility is driven by repeated extension, attachment and retraction of type IV pilus filaments. Only one pilus is shown, although piliated bacteria can have many. (b) The pilin subunits are stored in the inner membrane after depolymerization, and can be recycled. (c) Illustration of the surface lattice of the type IV pilus filaments of N. gonorrhoeae , after Craig et al. 54. The hydrophobic tail domains of the pilin subunits (not shown) are buried inside the filament, shielded by the pilin head groups from the surrounding water. The darker subunits indicate one strand in the three start helix, while the solid line illustrates the four start helix structure. Each pilin subunit contributes about 1 nm to the total pilus length. (d) Experimental setup in retraction experiments on single retraction motors 128,129,137. The bacterium was attached to a stationary bead, and a smaller bead was held in a nearby stationary optical trap. Eventually, the bead was captured by a pilus filament and pulled towards the bacterium. 2.3. Type IV pilus retraction 15 above 1 nm to the filament length. As shown in Fig. 2.6c, the structure can also be seen as a three- or a four-start helix. Compared to earlier results 147, the general architecture is the same, but the surface lattice is considerably different. The earlier prediction of a helix with an integer number (five) of subunits per turn gives in a surface lattice with straight protofilaments, similar to microtubules. Extension and retraction of the filaments are driven by polymerization and depolymerization at the base of the pilus, and is powered by ATP hydrolysis 55,145. When not part of filaments, the pilin subunits are stored in a reservoir in the inner cell membrane, and can be recycled when new filaments are formed 98,175 (Fig. 2.6b). Extension and retraction are mediated by distinct, but similar, ATPases, called pilF (extension) and PilT (retraction). Both are supposedly localized near the inner cell membrane 145, but details about their position are not known. The retraction process of single filaments on live bacteria has been studied in a series of single molecule experiments 128,129,137, using the setup sketched in Fig. 2.6d. The maximum velocity is about 1200 nm/s, corresponding to a polymerization rate of about 1200 pilin subunits per second. The velocity is approximately load independent for loads up to about 40 pN. At higher loads, the velocity decreases, and falls off to zero at an average stall force of 110 pN, although forces up to around 150 pN have been detected. By inhibiting ATP synthesis in the cells, the effect of reduced ATP concentration by an unspecified amount could be studied. It turns out that the velocity at low load decreases, while the stall force and velocities at high loads are unaffected 128. This is consistent with the retraction being powered by ATP hydrolysis, and also indicates that the rate limiting transition at high load is not ATP binding. By varying the expression of both pili and PilT, it could be established that these characteristics correspond to one single filament, and that each filament is retracted by one PilT molecule. Given the velocity and effective monomer size, single steps could not be resolved, and so it is not known if the monomers are always removed one by one, or in larger groups. Compared to the other motors described in this chapter, the working mechanism of the PilT motor is known in much less detail, which leaves much room for speculations about its function. Crystal structures of PilT reveals a hexameric ring of identical subunits with one ATP binding motif each 71,165, as expected for members of the AAA family of motor proteins 191. The hexamer appears donut-shaped, about 11 nm in diameter, 5.5 nm high, and with an inner diameter varying between 2 and 4 nm 71. A recent study detected a highly asymmetric conformation of the PilT hexamer, which indicates that the PilT ring can undergo large conformational changes 165. It has also been suggested that the PilT molecule might encircle the base of the pilus, and work via a power-stroke mechanism similar to that of the F1-ATPase, which is another hexameric motor protein with similar catalytic sites 98,145. The work required to retract a distance of 1 nm, corresponding to a single pilin subunit, is about 150 pN nm at the highest detected loads. If the ATP hydrolysis 39,92 free energy is estimated to 20 − 25kBT , or 80 − 100 pN nm , this indicates that more than one ATP per pilin subunit is needed, at least at high loads 128. 16 Chapter 2. Three examples of motor proteins

Figure 2.7. The structure of ATP (from Wikipedia 202).

2.4 Molecular fuel

We now go on to consider the free energy content of the molecular fuel that powers the motors in the previous sections.

Free energy of ATP hydrolysis Hydrolysis of adenosine triphosphate, or ATP, is the most common energy source in cellular reactions 3. For example, it is the energy source for the cytoskeleton and pilus retraction motors described in the previously in this chapter. The free energy is harnessed by removing the outer phosphate group in a hydrolysis reaction, which can be written

[ADP]eq[Pi ]eq 5 ATP + H2O )* ADP + Pi ,Keq = ∼ 10 M. (2.2) [ATP]eq

As seen from the equilibrium constant Keq, the reaction strongly favors ADP and Pi . However, the spontaneous equilibration is very slow, which makes ATP an ideal energy carrier 92. Both ATP, phosphate and ADP exist with diﬀerent charges and ligands in the cell, and the above concentrations involve the equilibrium concentrations of all the diﬀerent species 5. Because of this, the equilibrium constant does not only depend on temperature, but also on other factors like pH, ionic strength, and concentration of free magnesium ions 4. The free energy of a hydrolysis reaction is a measure of how far the concentrations of both ATP and the hydrolysis products are from their equilibrium values, which is usually expressed in terms of the standard free energy ∆G0 = −kBT ln Keq, as

[ATP] [ATP]/[ATP]eq ∆G = ∆G0 − kBT ln = −kBT ln . [ADP][Pi ] [ADP][Pi ]/[ADP]eq[Pi ]eq (2.3) 2.4. Molecular fuel 17

At pH=7, ionic strength 0.25 M, and [Mg2+] = 0.1 mM, roughly corresponding to the conditions of eukaryotic cytoplasm 3, the standard free energy at 25 ◦C is 4

−21 − ∆G0 = 58 · 10 J ≈ 14kBT. (2.4)

In order to use ATP hydrolysis to drive other processes, living organisms maintain ATP, ADP and Pi levels far from equilibrium. Typical concentrations are [ATP], [Pi ] ∼ 1 mM, and [ADP] ∼ 10 µM, which gives a hydrolysis free energy of 39,92 about 20-25kBT . Single molecule experiments on stepping motors in vitro with a controlled hydrolysis free energy are diﬃcult and, so far, rare. One experiment on the F1 rotary 140 motor , scanned the interval 7.3kBT < −∆G < 15kBT . Many in vitro experiments on motors instead use vanishingly small concentrations of Pi and often also of ADP, thus pushing the hydrolysis free energy towards minus inﬁnity.

Free energy stored in a membrane potential Another way for cells to store energy is to build up concentration differences and electrical fields across cell membranes. The bacterial flagellar motor is powered by a ion flux across the inner bacterial membrane. The free energy difference that drives this flux is called the ionic motive force. It has two components 3,29. One is entropic, arising from concentration differences between the in- and outside of the cell. The other component comes from the electrical potential difference across the cell membrane. Both components are maintained by various metabolic processes, and are tightly regulated by the cell. Ionic motive forces are used for example to drive active transport through the membrane, and the electrical potential difference is important for the propagation of nerve signals in axons 3. The ionic motive force is conventionally defined as inside potential minus outside potential, so that a negative ionic motive force favors inward flux, and in units of an electrical potential difference 3,29. For an ion with charge +1, e.g., H+, the ionic motive force (in this case proton motive force, or pmf) is given by k T pmf = ∆ψ − 2.303 B ∆pH. (2.5) e Here, ∆ψ is the electrical membrane potential difference, which is usually nega- + 3 [H ]out tive . The term containing a pH difference, ∆pH = log10 + is the entropic [H ]in contribution, and ln 10 ≈ 2.303 comes from using base 10 logarithms in the definition of pH. As it stands, the expression for the entropic contribution is strictly speaking an approximation for dilute solutions. For non-dilute solutions, one cor- rects for this by using effective concentrations, called activities, instead of actual concentrations 133. The same applies to the above expressions for ATP. At room temperature, 2.303kBT/e ≈ 59 mV. Typical values for E. coli is to maintain an internal pH of 7.6-7.8, and a proton motive force between -140 and -170 mV, depending on growth conditions 29. The tight physiological regulation of 18 Chapter 2. Three examples of motor proteins the proton motive force is one reason to use a chimaeric sodium powered motor in some experiments 179, in which case pNa replaces pH when computing the sodium motive force (smf) using Eq. (2.5).

2.5 Optical trapping in motor protein experiments

The optical trap, also called laser tweezers, is a technique to exert forces on small particles, such as the beads depicted in Figures 2.4 and 2.6. In this section, we will outline the basic principles for a simple optical trap. Technical details, more advanced setups and design considerations, as well as discussions of other single molecule techniques, can be found in recent reviews 60,77,135,142,159,162, and references therein. The development of laser trapping started with the work of Ashkin in the early seventies 13,17. This eventually led to the development of the single-beam gradient force trap in the mid eighties 15. Trapping of living bacteria and virus particles was also demonstrated 14. The technique has since found wide applications in diverse areas of physics, biology, and biophysics, not least in the study of protein motors. Rapid developments are still in progress. An optical trap consists of a tightly focused laser beam, usually obtained by sending the beam through a microscope objective with a high numerical aperture. The high electromagnetic field gradients near the focus can polarize a dielectric bead, and exert a force on it which is proportional to the intensity gradient, and directed towards the intensity maximum in focus if the refractive index of the bead is greater than that of the surrounding medium. The beam also exerts a scattering force on the bead, placing the equilibrium position of the bead slightly down-beam of the focus spot. In biological applications, wavelengths around 1 µm are often used, since most biological materials are nearly transparent in that region. Beads in the µm range are also common. By chemically attaching the bead to the molecule of interest, one creates a handle which can be used to manipulate for example a motor molecule, and also follow its motion with high accuracy. If the molecule is tethered to some other point as well, e.g., the microscope cover slip, a force can be exerted by moving the trap or the microscope stage. Larger objects, such as single cells, virus particles, or organelles within large cells, can be manipulated directly. For small deviations from the trap center, ∼ 150 nm, the trap forms a harmonic potential to very good approximation. The spring constants are typically in the range 0.01 - 1 pN/nm, depending on parameters such as laser power, beam geome- try, material and size of the bead etc. The stiffness might be different in the parallel and perpendicular directions relative to the beam. Many experiments on motors use a stationary trap, in which the opposing load on the motor protein increases as it pulls the bead away from the trap center. There are also several methods to apply a (nearly) constant force, ’force clamp’, on a moving object by using a feed-back loop that moves the laser focus or the 2.5. Optical trapping in motor protein experiments 19 microscope stage. The position of the bead can be measured by collecting light scattered by the bead in various ways, sometimes using a separate detection laser. Detection accuracy on the nanometer scale or better, and sampling frequencies on the order of 100 kHz, are possible. Often, the limiting factor is Brownian motion of the bead. By equipartition, the thermal fluctuations of the bead position has standard p deviation kBT/κ, where κ is the total (linear) stiffness. Most biomolecules have nonlinear elasticity, and tend to stiffen with increased tension, which improves the measurement resolution under an applied load. Increased stiffness also improves the accuracy by decreasing the relaxation time of the Brownian motion, given by τrel = γ0/κ, where γ0 is the viscous drag. The above estimates come from the simplest description of a diffusion bead in a harmonic potential, described by the overdamped Ornstein-Uhlenbeck process 73,197, given by the Langevin equation

∂x(t) γ = −κx + p2k T γ ζ(t). (2.6) 0 ∂t B 0 Here, x(t) is the trajectory of the bead, given in one dimension for simplicity, and ζ(t) is Gaussian white noise, with hζ(t)ζ(t0)i = δ(t − t0). For a spherical particle at 116,180 low Reynolds number, the drag can be estimated by Stokes’ law , γ0 = 6πaη, where a is the radius of the sphere, and η is the viscosity of the solution. For water at room temperature, η ≈ 9 · 10−4 Pa s = 0.9 · 10−9 pNs/nm2. For a free bead with 1 µm diameter in a trap of stiffness 0.1 pN/nm, the relaxation time is about 85 µs. ∂2x Dropping the inertial term m ∂t2 is motivated by comparing the characteristic time for decay of kinetic energy from such a term. It is given by m/γ0, where m is the mass of the bead. For the above example bead, this time is about 0.1 µs, which is very much shorter than τrel. Recently, Brownian motion of small beads has been studied in detail, partially driven by the need to develop exact calibration methods for optical traps 32,33,127,169,185. For an exact description of the behavior at short time scales, hydrodynamic effects give important corrections to Eq. (2.6). The inertia of the surrounding fluid creates a memory effect, which can be described by a generalized Langevin equation 210. The vicinity of a surface such as the microscope cover slip can also complicate matters. Hydrodynamic effects tend to increase the viscous drag near a surface 116, and other interactions between the bead and surface can also influence the bead dynamics 62,169.

Chapter 3

Modeling motor proteins

Models of proteins can be constructed on several levels of detail, from atomistic molecular dynamics to a few discrete states in a simple chemical kinetics model. Most protein motors are large molecules, consisting of tens of thousands of atoms, whose large scale dynamics often take place on the millisecond time scale. Atom- istic molecular dynamics simulations of such large systems, in water solution, do not (yet) reach these time scales. For example, the Gromacs 196 benchmark simulation 79 of the lysozyme protein 201 in water solution proceeds at a speed of a few µs/month, using ∼20 processors in parallel. With about 160 amino acids, lysozyme is about half the size of a single kinesin motor domain 112, which is a very small motor protein. Computational approaches to get around this problem include con- strained 45 or coarse-grained 8,9 simulations. In this thesis, we will consider an approach with simple stochastic models, that represent a further level of coarse-graining. Such models are useful to describe many experimental observations of single motors, and to understand how they interact with each other and with other elements in their natural environments. The simplicity of these models also makes them useful to identify and understand general features of motor mechanisms, that might be difficult to discern in detailed atomistic descriptions. In this chapter, we will see what these models might be, how they come about, and review some of their properties. To start with, we consider a prototypical model of a Brownian motor, the flashing ratchet. We then consider the connection between an atomistic description and coarse-grained models at different levels of complexity, and see how this connection is reflected in some general properties of the coarse grained models. Finally, we describe a useful way, due to Hill 88,89, to characterize the steady state properties of discrete state models. This description also provides a background for the first passage problems to describe steps and cycle completion events in Ch. 4.

21 22 Chapter 3. Modeling motor proteins

(a) Voff = −Fx V(x) Von = V s − Fx

d(a−1) 0 ad d d(1+a) (b) [on] [off] [on] [off] . . .

Figure 3.1. (a) Illustration of the flashing ratchet mechanism. The sawtooth potential Vs has periodicity d and asymmetry factor a < 1/2. As the system enters the [off]-state, the particle is located near a potential well at x = 0 (dashed). In the [off]-state, it undergoes biased Brownian motion. If the transition back to the [on]- state happens after an appropriate time, the probability for the particle to be caught in the next forward potential well (shaded) is much higher than the probability to be caught in the next backward well. (b) Energy to turn the potential on and off could come from a progressing chemical reaction.

3.1 The ﬂashing ratchet motor

A prototype model for how to achieve directed motion on the molecular scale is the flashing ratchet mechanism 2,18,21,148,158. The mechanism is based on a particle diffusing in a one-dimensional potential Vs, as sketched in Fig. 3.1a. The potential is asymmetric and periodic, and we also allow for an applied force F . If the potential height is not too small compared to the thermal energy kBT , the particle will mostly diffuse around near the minima, although occasional jumps between neighboring minima are possible. According to the second law of thermodynamics, there can be no net motion if F = 0 in this case. However, if the potential switches between on and off, for example driven by the chemical reactions sketched in Fig. 3.1b, the particle can move, and even per- form work against an applied load. When the potential is turned off, the particle starts to diffuse with an average velocity F/γ0, where γ0 is the drag coefficient. It p also undergoes Brownian motion, with the dispersion growing like 2kBT t/γ0. If the potential is turned on again after an appropriate time, as sketched in Fig. 3.1, the particle will most probably be trapped in the same potential well. There is also a finite probability of being trapped one period forward (shaded in Fig. 3.1a), but almost no probability to be trapped one step backward. This mechanism to produce forward motion has been used to construct a separation device for DNA molecules 22, although it does not reach the efficiencies of many biological motors 21,39,148. The flashing ratchet mechanism illustrates several important ingredients for a 3.2. The free energy landscape 23 motor on the molecular scale. The energy input from the fluctuating potential could mimic the driving force from a chemical reaction out of equilibrium. The directionality is supplied by the asymmetric potential, which could mimic a periodic and polar substrate. Finally, the motion is dominated by thermal noise, which is used in a constructive way. However, all these ingredients are described in a minimalistic way. There are many ways to extend the description and add more details to it, if we have detailed information about a particular system and want to construct a quantitative model. This is discussed in the following sections.

3.2 The free energy landscape

In this section, we consider a general framework for construction of coarse-grained stochastic models, and describe how they are connected to a microscopic description.

Elimination of fast variables In an effective model, we aim to describe the dynamics in terms of a few important degrees of freedom 44,87,94,97,158. The simplest description is in terms of a chemical kinetics, i.e., random Markovian transitions between discrete states, corresponding to different chemical or conformational configurations of the motor. In the following, we will sketch the path from a detailed atomistic description to an effective coarse- grained model, and also look at a useful and common modeling level of intermediate complexity. Some steps on this path are difficult to carry out in practice. However, they still serve as an important motivation for the general modeling framework that comes out in the end: simple Markov processes and chemical kinetics. We start with an atomistic description, and denote the microscopic degrees of 44 freedom, which we call bath variables , by x1, x2,..., xN . They constitute a complete microscopic description of the system, including positions and momenta of all atoms in the protein and the surrounding solution, chemical bonds and bond angles etc. The effective degrees of freedom, or system variables 44, are denoted y1, y2, . . . yn, with n N. They describe the large scale conformations of the system, or smaller conformational details that are important for chemical reactions, such as stretching of certain chemical bonds. Collective motions that are combina- tions of several simultaneous motions are also possible, similar to normal modes. The system variables are related to the microscopic description by relations of the form yk = fk(x1, x2, . . . , xN ), which map each microscopic configuration into a point ~y = (y1, y2, . . . , yn) in the phase space of the system variables. We now assume a separation of time scales between bath and system variables, so that the bath variables relax to local thermodynamic equilibrium very quickly compared to the characteristic time scales of the system variables. They can therefore be treated as being always in the equilibrium distribution associated with the instantaneous configuration of the effective degrees of freedom 44,87,158. This results 24 Chapter 3. Modeling motor proteins in an effective model where the system variables move stochastically on a free energy landscape, and the bath variables make their presence known as friction terms and fluctuating forces 44,101,158. In principle, the free energy landscape could be calculated as a potential of mean force 136, by integrating out all the fast variables. Thus, the effective free energy V (y1, y2, . . . yn) is formally related to the microscopic Hamiltonian H(x1, x2, . . . , xN ) by an integral over the surface of constant y1, y2,..., Z −V (~y)/kBT N Y −H(x1,x2,...,xN )/kBT e = d x δ yk − fk(x1, x2, . . . , xN ) e . (3.1) k In practice, this program is difficult to carry out. Instead, coarse-grained models are often constructed by hand, with the aid of experimental information and general principles. There are some general requirements of the system variables. To account for the conversion of chemical energy into mechanical work, there must be at least two 44,101 variables : one ’chemical’ coordinate, say y1, to keep track of the consumption of fuel molecules, and one ’mechanical’ coordinate, say y2, to keep track of how far the motor has advanced. For motor proteins that function in a repetitive fashion, like the ones introduced in Ch. 2, the energy landscape is periodic in these variables. In the chemical direction, there might be an additional slope given by the chemical potential difference ∆µ between fuel reactants (e.g., ATP) and products (e.g., ADP and phosphate), such that

V (y1 + 1,...) = V (y1,...) + ∆µ. (3.2) The chemical potential diﬀerence, ∆µ per period, acts as a thermodynamic force which tends to drive the chemical coordinate in the downhill direction (increasing y1, if ∆µ < 0). Along the mechanical coordinate, we have

V (y1, y2 + d, . . .) = V (y1, y2,...), (3.3) where d is the underlying repeat distance, for example given by the periodicity of the molecular track. The other system variables describe the internal state of the motor, and move in bounded regions. The simplest choice of stochastic dynamics 73,197 for yk is a set of overdamped Langevin equations ,

∂yk ∂V p γk = − + Fk(t) + 2kBT γkζk(t), (3.4) ∂t ∂yk 0 0 where ζk(t) are Gaussian white noise processes, with hζk(t)ζj(t )i = δkjδ(t−t ), with 210 the prefactor given by the ﬂuctuation-dissipation theorem . The Fk(t)’s are the components of an external force, for example given by a load opposing the motion of the motor. These equations are equivalent to a diﬀusion for the joint probability density in the phase space of the system variables, described by a Fokker-Planck equation 73,161,197 2 ∂ X ∂ 1 ∂V (~y) ∂ kBT ρ(~y, t) = − Fk(t) + 2 ρ(~y, t). (3.5) ∂t ∂yk γk ∂yk ∂ yk γk k 3.2. The free energy landscape 25

y1 (a) (b)

1 1 [1]0 [1]1

0 [3]0

0 0 [2]0 [2]1

0 fuel consumption 0 [1]0 [1]1 chemical force

applied force −1 [3]0 y2 forward motion

Figure 3.2. A free energy landscape with two reaction coordinates. Local minima (light) correspond to metastable states. Applied chemical and mechanical forces adds and overall linear gradient to the periodic landscape. Each unit cell of the landscape contains three different states, [1], [2], and [3]. (a) The stochastic trajectory of a motor taking a full step forward while consuming one unit of fuel. (b) Division of the free energy landscape into discrete states. The boundaries between different states are indicated by dashed lines, and most probable transition paths between different states by full lines.

Converting chemical energy to mechanical work

The effective theory described so far, offers a way to understand the working mechanism of a motor in terms of structural properties of the free energy landscape. The arrangement of ridges and local minima makes it possible to couple downhill flow along the chemical coordinate to an uphill flow in the mechanical one. This is illustrated in Fig. 3.2, which shows a hypothetical free energy landscape for two reaction coordinates, with high energy being darker. As indicated by the shading, a chemical gradient driving the system upwards is added, together with an applied force opposing rightward (=forward) motion. c c The landscape has three different local minima (light), denoted [1]m, [2]m, and c [3]m. They repeat periodically in both directions, and the indices c, m indicate the period in the chemical and mechanical directions, respectively. A motor evolv- ing stochastically would spend most time in these minima, which correspond to metastable states, and make fast transitions between different states. The transitions tend to go along paths where the barriers are lowest, indicated by full lines in Fig. 3.2(b). The irregular line in Fig. 3.2(a) shows an example trajectory, the evolution of 26 Chapter 3. Modeling motor proteins which corresponds to the motor advancing one period forward, while consuming one fuel molecule. Note that two different [3] → [1] transitions are possible, associated c c+1 c c+1 with rightward ([3]m → [1]m+1) or leftward ([3]m → [1]m ) motion. Although the applied force makes the leftward transition energetically more favorable, the higher barrier associated with that transition makes the rightward transition more probable, for kinetic reasons.

Discrete states As the above description already anticipates, further elimination of continuous system variables in favor of discrete states is sometimes possible. To do this, we divide the phase space into regions (dashed lines in Fig. 3.2), and define the probability pi(t) to be in state i as the integrated probability over the corresponding region, Z n pi(t) = ρ(~y, t) d y. (3.6) ci ~y∈[i]mi As with with the elimination of bath variables, an assumed separation of time scales is needed here, namely that transitions between different states happens seldom, compared to the time scale for relaxation to internal equilibrium in the different states. In this case, we retrieve chemical kinetics, i.e., random Markovian transitions between discrete states. This is described by a discrete master equation 73,197 for pj(t), X ∂tpi(t) = wijpj(t) − wjipi(t) , (3.7) j6=i where wij is the rate for the the transition j → i. The process described by Eq. (3.7) consists of a sequence of random jumps, separated by random waiting times. The waiting time in each state is exponentially distributed with mean hτii,

−1 X ρ(τi) = hτii exp − τi/ hτii , hτii = wji. (3.8) j

Given the assumed separation of time scales, it also makes sense to deﬁne a free energy for each state, given by Z ~ V (~y) − ~y · F n Gi(F~ ) = −kBT ln exp − d y, (3.9) ci kBT ~y∈[i]mi where the external force F~ = (F1,F2,...) was included, but is taken to be orthog- onal to the chemical coordinates (F1 = 0). In the following, we will understand a force F without vector notation to mean the component F2 along the direction of motion. Mixed models are also possible. In this case, each discrete state is associated with a set of potentials, and the governing equations are coupled Fokker-Planck 3.3. Modeling transition rates 27 equations

2 X ∂ 1 ∂Vi(~y) ∂ kBT ∂tρi(~y, t) = − Fk(t) + 2 ρi(~y, t) ∂yk γk ∂yk ∂ yk γk k X + wij(~y, F~ )ρj(~y, t) − wji(~y, F~ )ρi(~y, t) , (3.10) j6=i where ρi(~y, t) is the joint probability density for the remaining system variables ~y and discrete state i, and the possible space dependence of the transition rates is written out explicitly. Typically, the discrete transitions describe chemical events like binding or hydrolysis reactions, while the continuous description is retained for mechanical motions.

Tight and loose coupling The degree of correlation between the motion along the chemical and mechanical coordinate is obviously an important characteristic of the motor. The example trajectory in Fig. 3.2(a) illustrates a productive turnover, where one fuel unit is consumed to produce one unit of forward motion. If this is always the case, the motor is said to exhibit tight coupling, and the trajectory for repetitive forward motion will follow a straight line in the free energy landscape. c c+1 If, one the other hand, the last transition had been [3]m → [1]m , the result would have been a futile turnover, where the fuel consumption did not produce any net motion. If this can happen with appreciable probability, the motor is said to exhibit loose coupling. The coupling eﬃciency is not a ﬁxed quantity, but can depend on for example applied load. It is also important to note that tight coupling for forward steps does not imply that fuel is produced during occasional backward steps. For example, the experiments on Myosin V indicate that forward stepping is tightly coupled to ATP hydrolysis 192, at least at low load, while forced backward stepping seems to be independent of ATP 75.

3.3 Modeling transition rates

If the free energy landscape is known, transition rates can be calculated using Kramers 111 or Eyring 64 rate theories, as reviewed by e.g. Hänggi et al. 83. The main difficulty in applying these results is not so much that the calculations in themselves are difficult, but that it is difficult to obtain detailed microscopic information about the free energy landscape. It is often less difficult to get experimental information about rate constants directly, at least in some limited range of conditions. In this section, we will therefore present a common phenomenological ansatz for how rate constants depend on applied load: exponentially. We will also give 28 Chapter 3. Modeling motor proteins

V(y 2 )

dijF Gij

Gj y Gi 2 ξ ξ ξ j ij i

Figure 3.3. Sketch of Kramers’ escape problem 111 with and without an applied load. The reaction coordinate is the distance along a minimal energy path, projected onto the direction of motion, y2, and ξi, ξj , and ξij denote the projected mean positions of the states and the transition state respectively, giving dij = ξij − ξj . fundamental constraints on rate constants, that can reduce the number of free parameters in a model if the diﬀerent forces are known.

Exponential force dependence One starting point is to consider a one dimensional escape problem, sketched in Fig. 3.3, which was studied as a model for chemical reactions in a seminal paper by Kramers 111. The one dimensional reaction coordinate could be the distance along one of the minimum energy paths in Fig. 3.2, projected on the mechanical coordinate y2. The reaction can happen when the particle diﬀuses up to the barrier, which is high enough for local equilibrium to be established in the well before the escape, and then either escapes or bounces back 92. This gives an escape rate which can be put in the Arrhenius form, Gj − Gij wij = Aij exp , (3.11) kBT where Gj is the free energy of state j, Gij is the height of the barrier separating states i and j, and Aij = Aji is a frequency prefactor. Applying an external load parallel to the reaction coordinate x tilts the energy proﬁle and changes the potential barrier by approximately F ∆x, where ∆x is the distance between the potential minimum in state j and the transition state at the maximum point of the barrier. Ignoring the force dependence of the frequency prefactor Aij, one therefore expects the leading force dependence of a reaction rate 3.3. Modeling transition rates 29

~ ~ wij to be exp dij · F~ , with a characteristic length dij that can be interpreted as the (vector) distance between state j and the intermediate transition state. Furthermore, if we consider the ratio of the forward and backward rates, the position of the intermediate state cancels, and we get ! w (F~ ) F~ · (ξ~ − ξ~ ) ij ∝ exp i j , (3.12) wji(F~ ) kBT where we have adopted vector notation to denote the three-dimensional mean positions of states i and j 66,104. So, within this approximation, we can interpret the exponential force dependence of the ratio as reﬂecting the diﬀerence in mean positions between the two states, although the individual mean state positions ξj might depend on force as well. Phenomenological rate models of motor proteins often assume an exponential force dependence on transition rates 66,67,107,205. However, numerical experiments with test potentials 101, shows that an exponential behavior can only be expected to hold in a limited range of forces, so the correspondence between the force dependence and distance to a transition state is not a strict one.

Detailed balance In order to do useful work, molecular motors cannot be in equilibrium, and hence do not satisfy detailed balance. Still, there are fundamental constraints on the rate constants, which are motivated by the requirement that a well defined equilibrium state must exist. Since the phase space is periodic both in the mechanical and chemical coordinates, one can make a thought experiment in which the motor is somehow confined to a finite region of the free energy landscape by infinitely high and steep walls 68. The confined motor will then relax to equilibrium within this eq region, with ρj (~y) ∝ exp(−Vj(~y)/kBT ). In equilibrium, the net flux along each transition vanishes. This leads to a constraint on the ratio of forward and reverse transition rates, eq eq wij(~y) Vj(~y) − Vi(~y) wij(~y)ρj (~y) = wji(~y)ρi (~y) ⇒ = exp , (3.13) wji(~y) kBT which must be satisfied also by the model in the infinite state space. (We have assumed that the applied force does not act on any of the eliminated system variables.) This is the familiar Boltzmann ratio, which for the discrete case takes the form w G − G ij = exp j i , (3.14) wji kBT where Gk is the free energy of state k. (This is the reason why the prefactor of ~ Eq. (3.11) is symmetric). We recover Eq. (3.12) by setting Gk(F~ ) = Gk(0) + F~ · ξk, ~ ~ and assuming that the difference in mean positions ξj − ξj is independent of force. 30 Chapter 3. Modeling motor proteins

Non-local balance conditions It is sometimes convenient to eliminate the free energies of single states from a model, and concentrate only on transition rates. The above constraints can be translated to non-local conditions on transition rates, which depend only on the chemical driving force ∆µ, and the spatial periodicity d. This is done by multiplying the several forward-backward ratios together. Let i0 )* i1 )* i2 )* ... )* iK be a sequence of allowed transitions, that starts and ends at equivalent states, but possibly in diﬀerent periods, i.e.

[i ]cK = [i ]c0+∆c , ∆c, ∆m = 0, ±1, ±2 .... (3.15) K mL 0 m0+∆m Multiplying the Boltzmann ratios for all transitions, we get K K ! Y wikik−1 X Gik−1 − Gik Gi − Gi = exp = exp 0 K . (3.16) wi i kBT kBT k=1 k−1 k k=1 Next, we use the condition (3.15), and the periodicity of the free energy landscape to compute the free energy diﬀerence Gi0 −GiK , including the inﬂuence of an external force. Z ~ V (~y) − ~y · F n GiK = −kBT ln exp − d y (3.17) cK kBT ~y∈[iK ]mK Z ~ V (~y + ∆ceˆ1 + d∆meˆ2) − (~y + d∆meˆ2) · F n = −kBT ln exp − d y c0 kBT ~y∈[i0]m0 (3.18)

= Gi0 + ∆µ∆c − F2d∆m, (3.19) where we have again used y1 as the chemical, and y2 as the mechanical reaction coordinate. Substituting back in Eq. (3.16), we get K ~ Y wikik−1 (F ) F2d∆m − ∆µ∆c = exp = e−∆G({ik})/kBT . (3.20) ~ kBT k=1 wik−1ik (F ) Equation (3.20) constitute non-local constraints along transition paths that start and end in equivalent states. All references to the underlying free energy landscape have canceled, but the relations express the fact that the free energy landscape is periodic. The work F2d∆m performed on the motor by the external load, plus the free energy −∆µ∆c released by the fuel, equals the dissipated free energy −∆G({ik}) associated with completing the reaction sequence i0 → i1 → ... → iK in steady state conditions. This quantity corresponds to the thermodynamic force of Hill 88, or the dicycle entropy of Lipowsky and Liepelt 125. A corresponding argument can be made for mixed models, and results in an expression analogous to Eq. (3.20), which is valid for each (ﬁxed) value of ~y. 3.4. Circulation currents in steady state 31

Chemical force as mass action The above relations treat the mechanical force F~ and the chemical force ∆µ on an equal footing. However, while changing the mechanical force can in principle inﬂuence all transition rates, changes in the the chemical forces act more locally. We can see this in the simple case of an ATP driven motor, by writing out the expression for a transition sequence with ∆c = 1. Using Eq. (2.3), we get

K w (F~ ) [ATP] Y ikik−1 ∆mF2d/kBT = Keq e . (3.21) ~ [ADP][Pi ] k=1 wik−1ik (F ) This expression describes a sequence where one ATP is bound, and one ADP and one Pi are released. The ratio of concentrations come from first order kinetics, with the rate for binding one molecule of species X is proportional to [X]. ADP and Pi binding in the denominator comes from the reverse sequence, which describes ATP synthesis. The chemical force from ATP hydrolysis work via mass action on the binding reactions. Those are the only points where the internal dynamics of the motor that can sense the bulk concentrations 20. The two components of the proton motive force have different characters. The entropic part, e.g., ∆pH for the proton motive force, works via mass action, while the electrical potential gradient exerts forces on charges. The difference can be detected if the two components are controlled independently 126.

3.4 Circulation currents in steady state

In discrete state models, the general modeling framework described in the preceding sections is a random walk on an infinite lattice, which is periodic in (at least) two directions. In this section, we will consider a way to understand the steady state behavior of such systems, which has been developed by Terrell Hill 88,89. This method is easiest to describe for systems with a finite number of states, which we can get by restricting our description to one unit cell in the free energy landscape, and applying periodic boundary conditions. The resulting finite state Markov process has a normalizable steady state, which do not satisfy detailed balance for an motor protein operating out of equilibrium. Instead, the steady state can be decomposed in terms of probability currents, or cycle fluxes, around all closed loops, cycles, in the state space. These fluxes also describe the frequency with which the system completes the different cycles. The frequency of completing a cycle in the forward or backward direction is related by a Boltzmann factor involving the free energy dissipation. This decomposition gives an elegant way to analyze the performance of the motor (see e.g., Liepelt and Lipowsky 123 for a recent application to kinesin). As we will see in the next chapter, it is however not easy to detect cycle completions in single molecule experiments on motor proteins. 32 Chapter 3. Modeling motor proteins

(a) [3] (b) [3] [3] [3]

(0) [2] [2] II [2] III (0) I (0) (+) (+) (+) [1] [1] [1] [1]

Figure 3.4. (a) The example model from the free energy landscape in Fig. 3.2 reduced to a kinetic diagram with three states. Each line corresponds to a reversible transition. (b) The three cycles of the diagram in (a), with positive directions indicated by arrows.

Reduction to finite state space The finite state description means that we distinguish between different conformational and chemical states, but not between equivalent states at different positions, or between different net number of consumed fuel molecules. Starting from the periodic description of Fig. 3.2 and Eq. (3.6), this can be formally accomplished by projecting the periodic coordinates (indices m, c) onto a S c single unit cell, [i] = m,c[i]m. The master equation for these projected states [i] c is obtained by the corresponding summation of the master equation for the [i]m states 124. If we attempt this operation on the example model in Fig. 3.2, we end up with the kinetic diagram of Fig. 3.4(a), a three state model, with two different c * c+1 channels between states 1 and 3, coming from the transitions [3]m ) [1]m+1 (+), c * c+1 and [3]m ) [1]m (0) in the original model.

Cycles and fluxes In the steady state of the reduced model, the rate at which the transition i → j occurs is wjipi, and the net transition flux along each transition is ∆Jij = wijpj − wjipi = −∆Jji. From the master equation for the steady state, we get equations for the transition fluxes into each state, X X ∂tpi(t) = 0 = wijpj(t) − wjipi(t) = ∆Jij. (3.22) j6=i j6=i The transition fluxes are in general not independent, but can be expressed in terms of cycle fluxes, probability currents flowing around the closed loops in the kinetic diagram. The cycles fluxes, JC , are functions of the rate constants, and take the general form 88,89 Σ J = Π − Π C = J − J ,J = Π Σ /Σ. (3.23) C C+ C− Σ C+ C− C± C± C

Here, ΣC and Σ are complicated polynomials in the transition rates, C is the cycle index, and ΠC± is the product of rate constants around the cycle in the positive 3.4. Circulation currents in steady state 33

1 [1]1 [1]0 1 y1 JII+ JIII+

−1 0 J 0 [1]−1 [1] I+ [1] JI− 0 1

JIII− JII− −1 −1 y [1]−1 [1]0 2

Figure 3.5. The diﬀerent directed cycle ﬂuxes translated back to the free energy landscape picture.

or negative direction. The directed cycle fluxes JC± are interpreted as forward and backward components of the net cycle flux. To complete the decomposition, two more steps are needed. First, the transition fluxes can be expressed in terms of the cycle fluxes 88,89, X ∆Jji = ji,C JC+ − JC− , (3.24) C∈ij where ji,C = ±1 takes care of the sign, depending on in which direction the cycle C passes through the transition ij. Second, one must show that the directed cycle fluxes JC± really correspond to actual fluxes, i.e., describe the average frequencies of forward and backward cycle completions. This has been confirmed by computer simulations 90 and later proved mathematically 86,89,106,150,153. The direction of each cycle flux depends on the dissipated free energy associated with the cycle 88,89,

JC+ ΠC+ = = e−∆G(C+)/kBT , (3.25) JC− ΠC− in agreement with Eq. (3.20). Hence, we can understand the steady state as a competition between the different cycle fluxes in the kinetic diagram. The relative magnitudes of the forward and backward component of the cycle flux are given by the dissipated free energy associated with that cycle. However, the relative magnitudes of fluxes around different loops depend in a non-trivial way on the individual rate constants. Going back to our example, the kinetic diagram in Fig. 3.4(a) has three cycles, illustrated in Fig. 3.4(b). One positive turn around each cycle corresponds to the 34 Chapter 3. Modeling motor proteins following periodic paths in the original model:

0 0 0 I+ : [1]0 → [3]−1 → [1]1, −∆GI = F2d 0 0 0 1 II+ : [1]0 → [2]0 → [3]0 → [1]0, −∆GII = −∆µ, (3.26) 0 0 0 1 III+ : [1]0 → [2]0 → [3]0 → [1]1, −∆GIII = F2d − ∆µ, i.e., cycle I+ describes independent mechanical motion perpendicular to the chemical direction, II+ describes futile hydrolysis, and III+ describes a productive turnover, coupling one consumed round of fuel to one forward step. Translating back to the free energy landscape picture, we get six different fluxes in different directions that describe the steady state behavior of the random walk in the free energy landscape, as illustrated in Fig. 3.5. Chapter 4

First passage processes

A first passage problem answers questions regarding when some event occurs for the first time. Much of the research presented in this thesis concerns experiments on stepping processive motor proteins, and asks questions such as ’when does the next step occur, and what direction is it?’ The probabilistic answer to these questions is given by a first passage problem. Step-wise motion has been observed in several processive motor proteins, for example kinesin 47,82,143,182, myosin V 52,75,160,189, cytoplasmic dynein 184 or RNA polymerase 1. Stepwise rotation in ATP synthase 61,144,173,174,190,206,207 and the flagellar motor 179. The purpose of this chapter is twofold. First , we will introduce notation and discuss two solution methods for a discrete state first passage problem. For simplicity, we will use a one dimensional random walk as an example, and refer to standard references for extensions and more details 149,156,197. Second, we will discuss how the cycle completion events in Sec. 3.4, and the waiting times between steps in a stepping motor, can be translated to first passage problems, again using the one-dimensional example. We will see that steps and cycle completions are described by different first passage problems, and that approximating one with the other might give misleading results. This observation is one of the central results of this thesis.

4.1 A discrete ﬁrst passage problem

We consider a random walk on discrete states, labeled by integers. Transitions are allowed between nearest neighbors only, as sketched in Fig. 4.1. We follow 187 the notation of Tsygankov et al. , and deﬁne um = wm+1,m, wm = wm−1,m for forward and backward transitions respectively. This leads to the master equation

d p (t) = u p (t) + w p (t) − (u + w )p (t). (4.1) dt m m−1 m−1 m+1 m+1 m m m

35 36 Chapter 4. First passage processes

wL wL+1 wL+2 wR−1 wR

L L+1 . . . R−1 R

uL uL+1 uR−2 uR−1 uR

Figure 4.1. A ﬁrst passage problem: escape from the interval [L + 1,R − 1]. Absorbing states L and R are shaded.

The system starts at time t = 0 in state i, with L < i < R. Assuming all transition rates are positive, the system will eventually undergo either the transition L+1 → L or R − 1 → R. We define FL,i(t) as the probability that state L is visited before R, and that the first visit occurs at time t or earlier. The associated density ∂ is fL,i(t) = ∂t FL,i(t), and the splitting probability πL,i is defined as the total probability that L is visited first. Similarly, we define FR,i, fR,i, and πR,i for the case R is visited first. We have the relations πL,i + πR,i = 1, and Z ∞ πK,i = lim FK,i(t) = fK,i(t)dt, K = L, R. (4.2) t→∞ 0

The conditional ﬁrst passage times are denoted τK,i, with normalized probability distributions fK,i(t)/πK,i. How can we compute fK,i(t), and πK,i?

Absorbing boundary method An intuitive way to solve the above problem is to turn the target states 0 and N into absorbing states, which the random walker can never leave. To do that, we deﬁne qj(t) as the probability that the system is in state j at time t, and that neither 0 nor N has been visited yet. These probabilities satisfy a reduced master equation on the interval [1,N −1], with the transition rates of Eq. (4.1), but with all transition rates originating outside the interval removed. Written in matrix form, this equation is d ~q(t) = M~q(t), ~q = (q (t), q (t), . . . , q (t))T , (4.3) dt L+1 L+2 R−1 where the elements of the matrix M are given by

Mkj = δk−1,juk−1 − δk,j(uk + wk) + δk+1,jwk+1, L < k, j < R. (4.4)

The initial conditions are qj(0) = δj,i, and the distribution functions are given by the probability ﬂux into the absorbing states

R−1 d X Z t f (t) = w q (t) = w q (t) ⇒ f (t) = w q (s)ds. dt L,i Lm m L+1 L+1 L,i L+1 L+1 m=L+1 0 (4.5) 4.1. A discrete ﬁrst passage problem 37

With a similar argument, we get

Z t fR,i(t) = uR−1 qR−1(s)ds. (4.6) 0

The formal solution of Q~ (t) is given by eMt~q(0), which can be written in terms of (k) ~(k) right and left eigenvectors ~a and b of M, with corresponding eigenvalues λk, as R−L−1 X eMt~q(0) = etλk~a(k)~b(k) · ~q(0). (4.7) k=1 The eigenvalues have negative real part, and the eigenvectors are normalized such ~(k) (m) that b · ~a = δm,k. (In the unlikely event that M does have a full set of linearly independent eigenvectors, the solution has a diﬀerent form, that involves polynomials of t 167.)

Moment generating functions If one is interested in, say, the splitting probabilities and mean ﬁrst passage times rather than the full distributions, it might be advantageous to use Laplace trans- forms instead of solving for ~q(t) in the time domain. This gives the moment generating functions, Z ∞ ˜ −st 1 2 2 fK,i(s) = e fK,i(t)dt = πK,i 1−s hτK,ii+ s τK,i +...,K = 0,N, (4.8) 0− 2 where the prefactor comes from the normalization, Eq. (4.2), and the moments come from a Taylor expansion of e−ts. Laplace transformation 154 of Eqs. (4.3-4.6) leads to w u sI − M~q˜(s) = ~q(0), f˜ (s) = L+1 q (s), f˜ (s) = R−1 q (s). (4.9) L,i s L+1 R,i s R−1 By expanding in s, and identifying terms in Eq. (4.8), one can compute moments analytically without having to compute the eigenvalues of M. Alternatively, one can write ~q˜(s) as a power series in s, and solve for successively higher order in s to obtain the splitting probabilities and lower moments of τK,i. The initial state i enters in the initial condition, qj(0) = δi,j.

The adjoint equations The above treatment gives the solutions for both absorbing boundaries, but only for one speciﬁc initial condition. A diﬀerent approach enables treatment of all initial states at once, for only one of the absorbing boundaries at a time. This can be derived using the observation that, given a start in state i at t = 0, the probability to be absorbed at time t + ∆t or earlier is FK,i(t + ∆t). This can either 38 Chapter 4. First passage processes be accomplished by staying in state i during a time ∆t, and then completing in a time t or faster, or by making a transition to state j 6= i, and eventually being absorbed at K from there (again, K = L, R). If ∆t is small, we get

X 2 FK,i(t + ∆t) = 1 − ∆t wji + O(∆t ) FK,i(t) j6=i,L,R X 2 + ∆t wji + O(∆t ) FK,j(t). (4.10) j6=i,L,R

After rearranging the terms and taking the limit ∆t → 0, we get the equation

d X F (t) = w F (t) − F (t) (4.11) dt K,i ji K,j K,i j6=i,L,R

= wiFK,i−1(t) + uiFK,i+1(t) − (ui + wi)FK,i(t), (4.12) for L < i < R. To get a well-posed problem, we need initial and boundary conditions. The natural boundary conditions are FL,R(t) = FR,L = 0, and FL,L(t) = FR,R(t) = 1: If starting in state R, state L cannot be reached first, while the first passage time to absorption in K is zero if starting in state K. For the initial conditions, we note that it takes at least one transition to reach the absorbing state, 2 so that FK,i(t) = wKit + O(t ) for small t, i.e., FK,i(0) = 0 for i 6= L, R. By differentiating Eq. (4.11), we see that it is also an equation for fK,i(t). In principle, the initial and boundary conditions also follow by differentiation. For short times, we have fK,i(t) = wKi + O(t) ⇒ fK,i(0) = wKi, which can be used together with the boundary conditions fK,L(t) = fK,R(t) = 0. In the Laplace domain, this becomes T ~˜ ~ (sI − M )fK (s) = fK (0), (4.13) where MT denotes the matrix transpose. Alternatively, one could use the boundary conditions fL,L(t) = fR,R(t) = δ(t), fR,L(t) = fL,R(t) = 0, in which case fK,i(0) = 0 should be used 197.

4.2 Steps and cycle completions

We now consider the ﬁrst passage problems associated with describing cycle completions and waiting times in stepping experiments. In order to make a point about steps and dwell times, rather than about any particular motor protein, we will use the discrete version of the three state example model in Fig. 3.2, and add two further simpliﬁcations. First, we ignore the c * c+1 transition [3]m ) [1]m . Second, we assume that all three states in each unit cell c have the same mean position hy2i ≈ md for the states [i]m, so that one cannot distinguish between them based on position measurements. However, the major c * c+1 step transitions [3]m ) [1]m+1, are assumed to be detectable. With this, we get 4.2. Steps and cycle completions 39

y2 w1 w2 2 2d [0]2 w0 w1 w2 1 1 1 d [0]1 [1]1 [2]1 w u 0 0 0 0 2 0 [0]0 [1]0 [2]0 u0 u1 u −1 2 −d [2]−1 u0 u1

Figure 4.2. N=3 sequential model as a periodic random walk. Note that y2 is on the vertical axis, with d being the spatial periodicity of the free energy landscape. All states in each unit cell are assumed to have indistinguishable positions, such that only the black arrows correspond to detectable changes in position. a N = 3 state sequential model 108,151,152,187, a one-dimensional random walk with periodic transition rates, sketched in Fig. 4.2. This model has only one cycle, corresponding to cycle III in Fig. 3.4b. The stepping behavior of this model is illustrated in Fig. 4.3.

Cycle completions, by definition, occur when the system returns to a previous state in the reduced diagram, after completing a loop in either direction 90. If we c take state [1]m as starting state, a forward (backward) cycle is therefore completed c+1 c−1 108,152 when the system reaches [1]m+1 ([1]m−1) for the first time . In the notation c−1 c+1 c of the previous section, we have L = [1]m−1, R = [1]m+1, and i = [1]m. For this (c) (c) setup, we denote the first passage times τ± , with probability densities ρ± (t), for exit to the left (−) and right (+), corresponding to the backward and forward cycle completion times. The corresponding splitting probabilities, describing the fraction (c) (c) c of forward and backward cycles, are π+ = 1 − π− . Note that since states [1]m c and [2]m have the same mean position, one must use some other means than the c−1 c−1 position of the motor to detect the transition [2]m−1 → [1]m−1, where the backward cycle ends. Using the methods of the previous section, one can derive the following symmetry relations

π(c) u u u ∆G ρ(c)(t) = ρ(c)(t), + = 1 2 3 = exp − III . (4.14) + − (c) w w w k T π− 1 2 3 B

Dwell times, the waiting times between steps, correspond in this simple model c c+1 to the times between occurrences of the transition [3]m → [1]m+1, giving a forward c c−1 151,187 (+) step, or [1]m → [3]m−1, giving a backward (−) step . The corresponding c−1 c+1 ﬁrst passage problem is given by, e.g., L = [3]m−1, R = [1]m+1, and the initial condition depends on the direction of the previous step. After a forward step, m m i = [1]c , and after a backward step, i = [3]c . Hence, this problem gives rise to four 40 Chapter 4. First passage processes

τ+ τ τ + τ++ y2(t) − −− −

Figure 4.3. Four conditional dwell times to describe the stepping trajectory of the example 3-state model. The black line corresponds the idealized running average of the position y2(t). Small differences in positions between states are hidden by experimental noise (gray). conditional splitting probabilities, first passage times and associated probability densities, which we denote π±±, τ±±, and ρ±±(t), as illustrated in Fig. 4.3. (We use the convention that π+− denotes the conditional probability of a backward step following a forward step.) A similar first passage problems has also been suggested to describe the transit times of small particles through channel proteins in biological membranes, under conditions when only one particle can occupy the channel at once 6,6,26. There are also symmetry relations, which in this case reads

π++ u1u2u3 ∆GIII ρ++(t) = ρ−−(t), = = exp − . (4.15) π−− w1w2w3 kBT The similarity of Eqs. (4.14) and (4.15) reflects an underlying mathematical similarity, a one-to-one mapping between the different forward and reverse trajectories 6,27,124,199. The relation between splitting probabilities can also be shown 19,152 to be a consequence of non-equilibrium fluctuation theorems 40,56,74,96,115,118,170. The results generalize to cycles of arbitrary length, and are robust against detach- ment events 108,124 and presence of non-dissipative subcycles 124,199. As discussed in Ch. 5, an overdamped one-dimensional diffusion can be approx- imated by a one-dimensional random walk in continuous time. Thus, Eq. (4.15) corresponds to a previously known symmetry relation for the mean last-touch, first- touch first passage times of a particle undergoing over-damped diffusion between two points in an arbitrary potential 35,58. More interesting, it can also be proved without the assumption of overdamped motion 27. Beside these mathematical similarities, there are several qualitative differences between using cycle completions instead of the above dwell time problem to understand and describe stepping experiments. The most striking prediction is the possibility that steps and dwell times are correlated 124,187, while cycle completions are statistically uncorrelated. This implies a simple way to extract more quantitative information from extensive stepping experiments like several kinesin experiments 47,143,182, by measuring conditional properties instead of forward and backward dwell times, and fractions of forward and backward steps. There have also been suggestions to use Eq. (4.14) to extract the dissipated free energy from stepping trajectories 179, or time-series of the number of product 4.2. Steps and cycle completions 41 molecules in an enzyme reaction 152. Since the fraction of forward and backward steps depend on single rate constants in a non-trivial manner, this could lead to systematic errors, which could be avoided if Eq. (4.15) is used instead 124,187.

Chapter 5

Stochastic simulations

Most of the coarse grained models that have been proposed to describe motor proteins are Markov processes. In this chapter, we will review some stochastic methods to simulate such models. This is a valuable complement to various analytical and numerical techniques, and can often handle complex situations where exact analytical or numerical solutions are unwieldy. Stochastic simulations were used to investigate such a complex model in Paper I, and also to check the results in Paper II. The most straight-forward way to use stochastic simulations is to generate sample realizations of the process of interest, and analyze them in much the same way as one would with repeated experimental observations. With this in mind, we start by considering Markov models with discrete states, and then move on to discuss continuous and mixed models.

5.1 Simulation of chemical kinetics

A simple start is to consider how to generate sample trajectories of a discrete Markov process in continuous time with constant transition rates. We recall Eq. (3.7) for the probability pi(t) to be in state i at time t, X ∂tpi(t) = wijpj(t) − wjipi(t) . (3.7) j6=i By deﬁnition, the probability for a transition j → i to occur in a short time interval 2 [t, t+∆t] is wji∆t+O(∆t ), which means that the probability for nothing to happen during that interval is

2 X P (no transition) = a0 + O(∆t ), a0 = 1 − ∆t wji (5.1) j6=i

This can be used to generate a sequence of states, in, at discrete times tn = n∆t. For a given initial state i0, one chooses a (small) time step ∆t, and iterates forward.

43 44 Chapter 5. Stochastic simulations

With probability a0, in+1 = in is chosen. If it turns out that in+1 6= in, in+1 is drawn from the distribution w P (i = j|i 6= i ) = jin . (5.2) n+1 n+1 n P w k6=in ki In order for this to be a good approximation, the probability for two or more transitions in the interval [t, t, ∆t] must be very small, which means that the probability for one transition to occur is also small. Thus, the price for an accurate simulation is that only a small portion of the computer time will be spent on simulating the actual events. In addition, we only get to know the jump times with a precision of ∆t. A more eﬃcient algorithm is the Gillespie method 76, which generates sample sequences of a discrete Markov process by sampling from the known distributions of waiting times and jump probabilities. An exponential waiting time with expectation P value hτii = 1/ j6=i wji can be generated as τi = − hτii ln u, where u ∈ U(0, 1) (a uniform random number between 0 and 1). Hence, if the system arrives in state i at time t0, the next transition happens at

t1 = t0 − hτii ln u. (5.3) P The next state j is chosen according as in Eq. (5.2), P (j|i) = wji/ k wki, and the process is repeated. In addition to being simple to implement, his method is exact, in the sense that we use the known exact solution to generate the samples, and get a complete speciﬁcation of the sample trajectory.

Time dependent transition rates The Gillespie method can be generalized to the case when the rate constants are time dependent. We assume that the system starts out in state i at time t0, and that the next transition is to state j and occurs at time t1. We want to generate the transition time t1, and then the destination state for a given transition time, j|t1, i. Following Gillespie’s original derivation 76, we deﬁne the reaction probability density ρj,t1|i,t0 (k, t)∆t as the probability that the transition i → k occurs in the inﬁnitesimal interval [t, t + ∆t], given that the system stayed in state i up to time t. This quantity is given by

ρj,t1|i,t0 (k, t) = wki(t)P0(t), (5.4) where P0(t) is the probability that no transition has occurred up to time t. To calculate P0(t), we can consider a ﬁrst escape time problem from state i, starting at time t0. Using the absorbing boundary method, with all states except i as absorbing boundaries, we get 197

X Z t X ∂tP0(t) = −P0(t) wki(t) ⇒ P0(t) = exp − wki(s)ds , (t ≥ t0), (5.5) k6=i t0 k6=i 5.2. Continuous and mixed models 45

and P0(t) = 1 for t < t0. From this, we see that the transition time t1 has distribution function F1(t) = P (t1 ≤ t) = 1 − P0(t), and probability density X ρt1|i,t0 (t) = −H(t − t0)∂tP0(t) = H(t − t0) wki(t)P0(t), (5.6) k6=i where H(t) is the Heaviside step function. Hence, to sample the transition time, we draw a random number u ∈ U(0, 1), and invert F1(t), i.e., solve for t1 from

Z t1 X u = F1(t1) ⇒ − ln(1 − u) = wki(s)ds, (t1 ≥ t0). (5.7) t0 k6=i

Finally, we must choose a target state. We use Bayes’ theorem 81,203, to get the desired conditional probability

ρj,t1|i,t0 (j, t1) wji(t1) P (j|t1, i) = = P . (5.8) ρt1|i,t0 (t1) k6=i wki(t1) 5.2 Continuous and mixed models

For a diffusion processes, there is no manageable way to completely specify a sample trajectory, and some kind of discretization is necessary even when an exact solution is known. More commonly, an exact solution is not known, in which case there exist various strategies for numerical approximations. We will briefly discuss two of them, based on discretization of time or space respectively, and state some results for a one-dimensional diffusion process. Following common notation in the literature on stochastic differential equations 73,105,146, we start with the Ito stochastic differential equation dyt = A(yt)dt + B(yt)dWt, (5.9) where A(y) and B(y) are drift and diffusion coefficients respectively. Here, we have generalized the Langevin equation in Ch. 3, Eq. (3.4), to allow the possibility of multiplicative noise through a space-dependent diffusion coefficient B(y). We also interpreted the noise as an Ito diffusion process. This corresponds to the Fokker- Planck equation 1 ∂ ρ(y, t) = −∂ A(y)ρ(y, t) + ∂2B2(y)ρ(y, t). (5.10) t y 2 y

If yt describes the diﬀusion of a particle, A(y) and B(y) can be related to a potential V (y) and a local, possibly space dependent, diﬀusion constant D(y). The relations are ∂V 1 A(y) = − ,D(y) = B(y)2. (5.11) ∂y 2 Depending on the physical origin of the noise, it is sometimes a more natural modeling choice to interpret the white noise using the Stratonovich stochastic integral 197. 46 Chapter 5. Stochastic simulations

In the case of additive noise, the Ito and Stratonovich integrals are equivalent. If B depends on y, as in Eq. (5.9), they are related by a transformation. For example, the Stratonovich stochastic diﬀerential equation corresponding to Eq. (5.9) can be obtained by adding an additional drift term,

1 ∂B(y ) dy = A(y ) − B(y ) t dt + B(y ) ◦ dW , (5.12) t t 2 t ∂y t t where, following Kloeden and Platen 105, the ring (◦) denotes a Stratonovich interpretation.

Time discretization Using the Ito stochastic differential equation (5.9) as a starting point, one can discretize time and derive numerical methods similar to the ones used for ordinary differential equations 73,105. Consider a time interval [0,T ], introduce a time step ∆t = T/M, discrete time points tn = n∆t, and let yn be an approximation of ytn , for n = 0, 1,...,M. The development and analysis of numerical methods for stochastic differential equations is a large and complex field in itself. We will only state some basic results and refer to, e.g., Kloeden and Platen 105, for details and further references. The simplest method is the explicit Euler method,

yn+1 = yn + A(yn)∆t + B(yn)∆Wn, (5.13) where ∆W = W − W are the increments of the standard Wiener process W . n tn+1 tn √ t The increments are√ independent N(0, ∆t) variables (normally distributed with standard deviation ∆t). Adding the lowest order correction term, we obtain the Milstein scheme 139,

1 ∂B(yn) 2 yn+1 = yn + A(yn)∆t + B(yn)∆Wn + B(yn) (∆Wn) − ∆ . (5.14) 2 ∂yn One peculiarity of numerical methods for stochastic differential equations, compared to their deterministic counterparts, is the existence of different convergence criteria 73,105. Strong convergence is a statement about how well the simulated trajectory yn follows the actual diffusion path yt. Weak convergence is a statement about how well the distribution of yM approximates that of yT . The strong order of convergence for a numerical scheme is often lower than both the order of weak convergence, and the convergence order of the corresponding scheme for deterministic equations 105. On the other hand, in many applications a close path-wise approximation is not required, in which case weak convergence is enough. Provided that the drift and diffusion coefficient are sufficiently smooth and differentiable, the Euler and Milstein schemes have strong order of convergence 0.5 and 1.0 respectively, while both have weak order of convergence 1.0 73,105. Also note that the Milstein scheme reduces to the Euler in case of additive noise, i.e. 5.2. Continuous and mixed models 47

V(x)

xn−2 xn−1 xn xn+1 xn+2 un−1 un un+1

wn−1 wn wn+1

Figure 5.1. Approximating a diﬀusion with a discrete jump process, with forward rates un and backward rates wn (unrealistically coarse discretization).

B(y) = const. Just as for ordinary differential equations, stability of the algorithm is also a concern, and stiff problems might be more efficiently solved with an implicit algorithm 105. For mixed models, one can combine time discretization of the continuous vari- able(s) with the brute force method of Eqs. (5.1, 5.2) using the same time step 42. It is also possible to use the Gillespie algorithm with time dependent rate constants for mixed models, with the time dependence in Eqs. (5.7,5.8) given by the stochastic evolution of the continuous variables 49,166. For stability and accuracy, it is important to choose a small enough time step, although a too small step would waste computer time and lead to accumulation of round-off errors. In practice, one can often estimate the relevant time scales of the problem, and choose the time step accordingly. A basic control is to make calculations with different time steps, to make sure that the chosen step is small enough for the solutions to be independent of the time step.

Space discretization A different approach is to discretize space in the Fokker-Planck equation (5.10). This could be the first step towards solving numerically for the full distribution, but can also be interpreted as approximating the continuous diffusion with a random walk on a discrete lattice 73, as illustrated in Fig. 5.1. The following scheme, due to Wang et al. 200, is suitable for potentials with discontinuities or sharp edges, for example the potential in Fig. 3.1. We consider Eq. (5.10), with constant diffusion constant D = B2/2, and potential V (y). The spatial coordinate is discretized into 48 Chapter 5. Stochastic simulations

cells with diameter ∆y, and we let pn(t) be the probability to be in the cell with midpoint yn = n∆y,

Z yn+∆y/2 pn(t) ≈ ρ(y, t)dy ≈ ρ(yn, t)∆y. (5.15) yn−∆y/2

Transitions are allowed between nearest neighbors cells. The transition rates un and wn out of cell n, in the forward and backward direction respectively, are given by

2 2 ∆Vn/(∆y) ∆Vn−1/(∆y) un = , wn = , ∆Vn = V (yn+1) − V (yn). (5.16) e∆Vn/D − 1 e∆Vn−1/D − 1 There are several reasons why this might be attractive for modeling mixed models. First, it translates the diffusive motion into the same type of dynamics as the chemical kinetics. Thus, a mixed model can be simulated without further approximations using the Gillespie algorithm. For the simple one-dimensional case, one can also exploit exact results for random walks to compute mean first passage and residence times 108,149,187, or steady state velocity and effective diffusion constant 59. An explicit matrix formula for the latter has also been derived 200. This scheme is second order accurate. As with the time discretization methods, it is important to make control calculations to make sure that the time step is small enough. A potential drawback is that an accurate description of the diffusion process requires a fine space discretization. As can be seen in Eq. (5.16), and for other schemes as well 73,200,204, this makes the average dwell time on each site very short. In that case, a simulation using the Gillespie algorithm will spend a lot of time generating fine details of the diffusion. The matrices used to compute steady state quantities may also become very large, if there are several continuous degrees of freedom or very many discrete states. If one is interested mainly in the steady state behavior, a higher order method based on local equilibrium distributions has been developed to make do with a coarser grid 204. Chapter 6

Introduction to the papers

In this chapter, we give a more speciﬁc introduction to the scientiﬁc papers in this thesis, and discuss the results and possible further developments.

Paper I The motility of certain Gram-negative bacteria is powered by repeated extension, tip attachment, and retraction of surface filaments called type IV pili. These filaments are essential for infectivity. Experiments on single retraction motors 128,129,137 show that the retraction motor, PilT, can generate forces up to 150 pN, making it one of the strongest known motor proteins. The molecular details of the retraction mechanism are largely unknown. However, some features have been identified, such as the hexameric structure of the PilT protein, and the structure of the filament. We study force generation in a simple model of interacting motor subunits, which mimic certain features of the PilT system. The model describes coupled Brownian motors which pull on a helical filament, and interact through elastic deformations of an elastic backbone. The interaction between the filament and the motor subunits are described by time-dependent, asymmetric potentials (flashing ratchets). We compute force-velocity relations in a subset of parameter space, and explore how the stall force depends on parameters such as number of motor subunits, backbone stiffness, and binding strength. We identify two qualitatively different regimes of operation, depending on the relation between the degree of asymmetry and number of motor subunits. In the transition between these two regimes, the stall force depends nonlinearly on the number of motor subunits. The model describes several qualitative features seen in experiments, such as roughly constant velocity at low loads, and insensitivity of the stall force to changes in ATP concentration. Since the publication of paper I, new results for the structure of the pilus filaments 54, see Fig. 2.6c, has appeared, which predicts a different surface lattice than previous results 70,147. The structure of the PilT ring has also been solved, and reveals some information about possible conformational changes in the hexameric

49 50 Chapter 6. Introduction to the papers ring 165. Despite this progress, construction of more realistic models of the pilus retraction machinery is difficult, due to the lack of structural information. Retrac- tion experiments under constant load, where an effective diffusion constant could be extracted, would also give valuable input and help to constrain future modeling efforts.

Paper II Increased resolution in single molecule detection techniques has made it possible to detect single steps taken by many processive motor proteins. One example is conventional kinesin, where recent experiments 47,143 have investigated forward and backward stepping under a wide range of conditions. Steps can also be detected in many other systems, e.g., myosin V 52,75,160,189, cytoplasmic dynein 184, ATP synthase 61,144,173,174,190,206,207 and the flagellar motor 179. In order to interpret data from such stepping experiments, we study processive stepping in a sequential motor model that was previously used to model single molecule experiments on kinesin and myosin V 66–69,104,107–109,186. We develop a first passage analysis, and compute general expressions for quantities like mean dwell times, fractions of forward and backward steps etc. Our approach is based on the observation that steps correspond to single transitions in discrete state models, and the results differ in several important ways from theoretical descriptions that identified steps with completed mechanochemical cycles 107,108,152. We find that the step direction and dwell time before a step depends on the direction of the previous step. Hence, compared to the cycle completion approach, our analysis predicts that more independent quantities that can be extracted from stepping trajectories with backsteps. The mean dwell times τ+ and τ−, prior to forward and backward steps respectively, are in general unequal. Instead, the conditional mean dwell times τ++ and τ−− between successive forward and successive backward steps are equal. Our analysis can be extended to models with several substeps in each cycle. We investigate an N = 2 state model numerically, and demonstrate a wide range of behavior.

Paper III In this paper, we generalize and extend the results of paper II in several ways. We consider dwell time distributions instead of average values, and a wider class of models that include multiple pathways and detachments. In analogy with recent results for cycle completions 152,199, we ﬁnd a that the symmetry relation τ++ = τ−− for conditional dwell times is true in distribution. We also ﬁnd a relation between conditional stepping probabilities that might be used to extract the free energy dissipation per cycle from stepping trajectories. The class of models for which this applies describe tightly coupled, reversible motors. This limits the number of systems where the free energy dissipation can be extracted. On the other hand, our results constitute a possible way to detect loose coupling. 51

To demonstrate our method, we analyze data from an experiment on the flagellar motor 179. Within the uncertainties in the data, the results are consistent with our predictions for dwell time symmetry and step-step correlations. The most general lesson from papers I-II is the prediction about correlation between steps in a trajectory with backward steps. This opens the possibility to extract additional quantitative information from stepping trajectories by considering conditional quantities, i.e., dividing the events according to the direction of both the previous and following step, instead of considering only forward and backward steps. Regarding the flagellar motor, further analysis is needed to draw any firm conclusions about the function of the motor subunits. Such an analysis presents both experimental and numerical challenges. In order to convert the free energy dissipation per step to number of ions per step, the sodium motive force must be measured independently. A method to measure the sodium motive force has recently been presented 126. Moreover, longer data series under stable conditions are needed, containing 10-100 −− events. Currently, stable rotation as slow as ∼ 10 Hz, with a sodium motive force of about 2kBT , has been achieved for a single stator under low load 126. The flexibility (see Eq. (2.1)) and drag of the bead and the flagellar stump makes accurate step detection very difficult 179. Step detection methods are con- stantly being developed 46,138,141, and there is no doubt room for further numerical and algorithmic improvements, and adaption to this special problem. After publication of our results for dwell time symmetry in molecular motors, it came to our attention that similar results had recently been found independently in another context, translocation of ions through ion channels 6,27.

Paper IV From basic linear algebra considerations like Eq. (4.7), one generally expects the number of exponential terms needed to describe a fist passage time distribution to be equal to the number of discrete states that the system escapes from. In practice, fast time constants might be difficult to resolve experimentally, in which case the number of terms needed to fit an empirical distribution is an estimated lower bound on the number of states. In this communication, we demonstrate that the distribution of cycle completion times, which has attracted recent theoretical interest 25,152,199, constitutes an ex- ception to the above-mentioned behavior. Due to the underlying periodicity of the first passage problem, half of the terms cancel. In light of the results of papers II- III, we also discuss the possibilities to detect cycle completion times in experiments on single enzymes, and find that a resent suggestion 152 needs to be modified.

Acknowledgments

First of all, I would like to thank my supervisor, Mats Wallin, for stimulating collaboration, and for constant encouragement and confidence in my ability. Likewise, I am grateful to Ann-Beth Jonsson, for fruitful collaboration and discussions on pilus retraction and bacterial motility. I am thankful to all members in the Statistical Physics, Condensed Matter theory, and Theoretical Biological physics groups at KTH, for the many discussions, and for teaching me to appreciate good coffee. I would especially like to thank Patrik Henelius, for sharing many insights and hands-on advice to improve my teaching, and my long-time room-mate Marios Nikolaou, to whom I am indebted for many things, not least much help with computers. Many thanks also to Michael E. Fisher and Denis Tsygankov for stimulating discussions and collaboration, and for introducing me to the problem of modeling stepping experiments. I would also like to thank Caricia Claremont, Subir Das, Madhav Ranganathan, Dave Thirumalai and his group, many other people at IPST, and William Ratcliff, who all helped make my visit to Maryland pleasant and fruitful. My deep gratitude also goes to the Royal Institute of Technology, for funding my PhD position, giving me the possibility to work full time to satisfy my curiosity. Finally, I am grateful to Ed Fontes, for encouraging me to start on a PhD, and to my family and friends.

Martin Lindén, Stockholm, Match 4, 2008

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Part II

Publications

Paper I Martin Lindén, Tomi Tuohimaa, Ann-Beth Jonsson, and Mats Wallin Force generation in small ensembles of Brownian motors

Physical Review E, 75:021908, 2006

Paper II Denis Tsygankov, Martin Lindén, and Michael E. Fisher Back-stepping, hidden substeps, and conditional dwell times in molecular motors Physical Review E, 75:021909, 2007

Paper III Martin Lindén and Mats Wallin Dwell time symmetry in random walks and molecular motors

Biophysical Journal, 92:3804–3816, 2007

Paper IV Martin Lindén Decay times in turnover statistics of single enzymes

submitted for publication