Continuous and Discrete Stochastic Models of the F

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Continuous and Discrete Stochastic Models of the F 1 -ATPase Molecular Motor

Eric Herman Gerritsma

A thesis presented in partial fulfilment of the requirements for the degree of Doctor in Philosophy in Science

Physics of Complex Systems and Statistical Mechanics

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The first principle is that you must not fool yourself and you are the easiest person to fool.

Richard Feynman

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Acknowledgments

This doctoral research was supervised by Professor Pierre Gaspard.

I thank Professor Pierre Gaspard for giving me this opportunity and for showing me what rigorous scientific research is about.

I am profoundly thankful to Dr. John W. Turner for inspiring lectures on Classical and Quantum Mechanics, Solid state Physics and Stochastic processes, and to Professor René Lefever for introducing me to Thermodynamics and to Reaction-Diffusion processes and also for inspiring historical anecdotes on the Interdis- ciplinary Center for Nonlinear Phenomena and Complex Systems of the ULB, where I have studied and made this research.

I thank my colleagues for their support during all these years.

Especially, I thank Pierre de Buyl for bringing a great Italian cof- fee machine to the group’s lounge and helpful comments on my L^ATEX and FORTRAN codes. Also, I thank Nathan Goldman for inspiring discussions. Both have had a positive influence on me.

The necessary financial support was provided by a grant from the Fond pour la formation à la Recherche dans l’Industrie et l’Agriculture(FRIA), by aPrix Fondation David et Alice van Buuren and also by Professor Pierre Gaspard, who gave me an additional six-month support.

Also, during my PhD, I had a one-year research contract in Plasma Chemistry under the direction of Professor François Reniers in the CHANIgroup at the Chemistry department.

I thank Professor François Reniers for giving me this helpful opportunity and for encouragements to pursue my PhD, as well as my colleagues of the CHANI group for their support and friendship.

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Finally, I express my deepest gratitude to my family and friends for their support and, in particular, to my mother Monique Farnetti- Gerritsma and my partner in life Sandrine Cammarata for their infallible love and encouragements.

Eric Herman Gerritsma, Genval, June 2010.

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Introduction

Molecular motors are nanoscale biological machines made of protein molecules¹ [2, 3]. Ubiquitous in the natural world, these protein assemblies can indeed be called machines or motors since they have the ability to convert energy from an external source to perform mechanical work. The mechanical motion of these molecular motors occurs by successive conformational changes, which are spatial rearrangements of their 3-dimensional structure [4–6].

Depending on the type of motor, these conformational changes will result in a stepping behaviour, a rotary motion or in a gate mechanism.

The diversity of tasks molecular motors accomplish is immense and very diverse, and yet, as is customary in the chemistry of the living, they are also extremely specific. Each motor uses a well defined source of energy to perform a well defined task.

Some molecular motors useadenosine triphosphate(ATP) as their unique source of energy, like the F1-ATPase or the myosin motor which is responsible for muscle contraction [3]. Other motors, known as ion channels, are powered by ion gradients to open and close the gates of biological cells.

Despite their diversity and specificity, molecular motors have a very important common characteristic, which is due to their nanoscale size : Their activity exhibits small random orstochasticfluc- tuations. These stochastic fluctuations are due to the inescapable influence of thermal Brownian motion [6–8].

In this doctoral thesis, we have focused our attention on the F1- ATPase molecular motor. It is our purpose to investigate both the chemical and the mechanical properties of this molecular motor.

1Recently, synthetic molecular motors have been developed [1].

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16 CHAPTER 1. INTRODUCTION First of all, let us answer the question :

Why are the mechanics and the chemistry of molecular motors stochastic processes?

1. The chemistry of molecular motors is a stochastic process because the arrival time of a substrate molecule is a random event in time.

2. The mechanics of molecular motors is a stochastic process because the thermal fluctuations of the media.

In consequence, the mechanical and chemical or chemomechan- ical properties are not deterministic anymore, but probabilistic, because of Brownian motion.

1.0.1 Discovery of Brownian motion

Brownian motion is the historical name given to the random agitation of small, micrometer sized, objects or molecules immersed in a fluid, in honor of Robert Brown.

Robert Brown was an English botanist, who sailed the world to study plant life in the early nineteenth century. Although there were several precursors that found also (independently) this phenomenon [9], he was attributed the discovery.

In 1828, Brown was looking at the fertilization process in a species of flower he discovered during an expedition. Looking at a suspen- sion of pollen in water through a microscope, he observed small particles in “rapid oscillatory motion”. The examination of the pollen of other species gave identical results.

At that time, many other scientists believed that this motion was due to a “vital force” contained by the particles. Testing this in- ference on glass and minerals² lead him to conclude that the idea of vitality was quite out of the question in the observed phenomena [9].

Scientists worked on, but it took about thirty-five years before any major contribution to the understanding of the phenomenon. The first essential contributions came, in 1863, with the work of fa- mous American mathematician Norbert Wiener who developed a

2Brown tested a fragment of the Egyptian Sphinx, considering it was deprived of any vital force, thanks to its very old age [9].

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17 theory of Brownian motion, and in 1905, thanks to Albert Einstein who published among other renowned articles, a paper were he predicts theoretically the phenomenon of Brownian motion and formulated a correct quantitative theory [10]. Paul Langevin, derived in 1908 an equation of motion for the Brownian particle [11]

still used today.

1.0.2 Early concepts of Brownian motors

It is considered a great challenge for the burgeoning field of nano- technology to design and construct motors at the microscopic scale that can use input energy to drive directed motion against the random molecular agitation [6]. A question arises naturally : How could Brownian motors extract work from random fluctua- tions?

At the macroscopic scale, we can construct a device that can extract useful work from unbiased random fluctuations, provided that the fluctuations are strong enough and that this device has a preferential direction of motion, i.e. has an asymmetric structure.

Windmills, water mills and self-winding wristwatches are examples of devices that extract work from random fluctuations [10].

The underlying mechanism of these devices is known as theratchet effect.

However, at the microscopic scale, we cannot obtain directed transport from a spatially asymmetric isothermal system, as was shown by Smoluchowski [12, 13].

The Brownian ratchet

The first major contribution to the understanding of the ratchet effect at the microscale was brought in 1912 by Smoluchowski’s Gedankenexperiment[12].

Smoluchowski imagined a microscale ratchet in a heat bath³, which was later discussed by Feynman [13]. An illustration of such a ratchet is given in Fig. 1.1. One could easily assume that the average rotation of the ratchet over a certain amount of time is not zero. This assumption is wrong. Indeed, at the microscale, the

3A heat bath is a system that remains always at constant temperature even if put in contact with other objects.

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18 CHAPTER 1. INTRODUCTION

Figure 1.1: Microscale ratchet and pawl in a gaseous heat bath.

The particles of the gas collide with the propeller at one end of the axle which causes the asymmetric sawtooth wheel at the other end to turn, lifting up the pawl and spring. The spring attached to the pawl pushes against the wheel to prevent backward rotation.

random collisions due to the Brownian motion of the particles af- fect both the propeller of the ratchet and the pawl. Smoluchowski showed that in spite of the built in asymmetry, no preferential direction of motion is possible.

The detailed analysis given by Feynman in his seminal lectures on physics [13] shows that the effect of the collisions is twofold :

1. The propeller moves randomly

2. The spring lifts up occasionally, letting the wheel turn freely.

The result of the analysis shows that the average rotation of the wheel is zero.

Feynman resolved the problem of zero average rotation by adding a second heat bath at a different temperature to the ratchet system. The temperature gradient between the two different heat baths is the necessary bias that enables the ratchet to extract work from the fluctuations it undergoes [13].

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1.1. THE STATE OF THE ART 19 Isothermal ratchets

Although the analysis of Feynman is correct, Brownian motors are not in contact with several heat baths. Indeed, Brownian motors are in an isothermal environment, in contrast to thermal motors.

The necessary bias to perform mechanical work comes from input energy, which is provided by a chemical reaction or an external force [6, 14–17]. An important and complex question arises from this consideration :

How is the mechanical motion coupled to the chemical reaction?

Molecular motors are asymmetric structures and they convert chemical energy into mechanical force and movement through conformational changes.

It is our goal in this doctoral thesis to explain how this can be achieved.

1.1 The state of the art

The contemporary study of molecular motors is constituted by two complementary but distinct approaches. The experimental approach, which is based on measures made in a laboratory under specific and well defined conditions, and the theoretical approach.

In this doctoral thesis, we have used a theoretical approach based on stochastic models.

A theorists has a twofold mission.

Firstly, his mission is to understand qualitatively the general features of molecular motors and develop mathematical and compu- tational models to explain their behaviour. Donald Knuth⁴ has an interesting opinion on this subject :

“Science is what we understand well enough to explain to a computer. Art is everything else we do.”[18]

Secondly, his mission is to apply the laws and theories of equilibrium and nonequilibrium thermodynamics and statistical mechanics on the experimental observations in order to retrieve quan- titatively information on the chemical and physical properties of these molecular motors.

4Donald Knuth is a Professor Emeritus of theArt of Computer Programmingat Standford University.

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20 CHAPTER 1. INTRODUCTION The theoretical approach can also bring light to the understanding of molecular motors by giving a comprehensive and solid framework to the experimental observations and sometimes predictions can be made.

1.1.1 Stochastic models

In stochastic models, the molecular motor is considered as a kind Brownian particle moving in a potential field. This approach is in- spired from the analysis of the fluctuations observed experimen- tally [19–28] and directly related to the field of Stochastic Physics and Statistical Mechanics.

The understanding of the statistical properties of these molecular motors is crucial to the development of potential application in nanotechnologies and is a major intellectual challenge [6, 7, 29, 30]. The analysis of their fluctuating properties, is also a major challenge, and is performed with the aid of fluctuation theo- rems [31–33].

1.1.2 Molecular Dynamics studies

Molecular Dynamics (MD) simulations have also helped consider- ably the understanding of the chemomechanical coupling of molecular motors [34, 35]. MD simulations consists of the numerical, step-by-step, solution of the classical equations of motion :

!ƒ_=md²r!_ dt²

which relates the force!ƒ_ on element⁵(=1,2, . . . , N) to its mass m_ and acceleration d²!r_/dt², where r!_ is the space vector of element . Furthermore, the forces !ƒ are usually derived from the potential energy U(r!^N) of the system, where !r^N represents the N space vectors (!r^N=r!₁,r!₂, . . . ,r!_N), in the following way :

!ƒ_ =− ∂

∂!rU

In MD studies, the potential energy functionU is of critical impor- tance. The precise evaluation of this functionU is often very diffi- cult and approximations are necessary in some cases. However,

5Usually the elements are atoms or particles.

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1.2. EXPERIMENTAL STUDIES 21 MD studies have given important contributions to the understanding of the conformational changes of the F₁-ATPase molecular motor [34, 35]

Time resolution of the dynamical simulation is also a critical factor, which depends to a great extend on computer “power”. Nanosec- ond time-resolution has recently been achieved in the molecular dynamics simulations of the F₁-ATPase molecular motor [34].

1.2 Experimental studies

Experimental studies have contributed significantly to the understanding and development of the field of molecular motors. The experiments performed are either on single-enzymes or on multi- enzyme systems.

Since the early 1990’s, many single-enzyme experiments have been performed thanks to important technological and scientific progress. In these experiments one single enzyme can be manip- ulated and its properties recorded which enables the study of the individual behaviour of such molecular motors, in contrast of average measures resulting from multi-enzyme measures. The mea- sure of average values smooths out the individual fluctuations, which are often very important at the nanoscale. These fluctuations are due to the Brownian motion of the enzymes and must be taken into account when studying their individual properties. To- day, a lot of different techniques exists to perform various types of single-enzymes experiments. We here briefly give an overview of some techniques encountered during this research.

1.2.1 High-speed imaging and laser dark-field micro- scopy

High-speed imaging and laser dark-field microscopy [36] have been used to observe direct rotation of a 40 nanometer bead attached to the γ-subunit of the F₁-ATPase [23]. In these experiments, the rotation was recorded with a charge-coupled-device (CCD) camera at speeds up to 8000 frames per second.

The principle consists of introducing a laser beam into a dark- field condenser in order to illuminate the specimen obliquely⁶.

6In this technique, only the object of interest is illuminated which produces the appearance of a dark background and a bright, highly contrasted, object on

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22 CHAPTER 1. INTRODUCTION The light scattered from the bead is then collected by the objec- tive [36].

High-speed imaging has also been associated to fluorescence [22].

1.2.2 Fluorescence

Fluorescence is also commonly used to visualize the activity of enzymes. Thanks to the highly sensitive optical detectors, experimentalists are able to count single photons emitted by adequately labeled proteins or nucleotides [26].

By fluorescent labelling an actin filament, and then by chemically attaching the actin filament to theγshaft of the F1-ATPase, experimentalists were able to observe the rotation of the actin filament during catalytic activity [22]. The images of the rotary motion of the actin filament where taken by a CCD camera coupled to an epifluorescence microscope.

1.2.3 Magnetic manipulation of the F₁-ATPase

Manipulation of a single F₁-ATPase has been performed by using magnetic tweezers [25,26,37]. In these experiments, experimentalists exert a force on a magnetic bead attached to the F₁-ATPase using a magnetic field gradient. The magnetic field gradient was obtained by using two electromagnet pairs electrically connected in a series.

The experiments using magnetic tweezers demonstrated the an- gle dependent kinetics of the catalytic states of the molecular motor by applying an external torque [25, 26, 37].

Furthermore, experimentalists were able to force the rotation of the F₁-ATPase in the opposite direction, i.e. in the direction of ATP synthesis [26]. These experiments demonstrated that the molecular motor works in a tight-coupled regime of its chemical catalysis and its rotary motion.

1.2.4 Crystallography

The crystal structures obtained of the F1-ATPase provide information on the possible conformations of the molecular motor by ac- tually trapping it in its current state [38, 39]. Several important

it.

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1.3. SYNOPSIS 23 features are consistent with the binding change mechanism of catalysis [40], in which binding of a substrate induces conformational changes. Furthermore, the obtained structures give insight on the dynamical possibilities of rotation of theγ-shaft inside the α₃β₃ barrel [38, 39, 41].

1.3 Synopsis

The thesis is organized as follows.

The aim of chapter1 is to briefly describe the state of the art and the techniques used to study molecular motors and especially the F₁-ATPase rotary molecular motor. The F₁-ATPase molecular motor itself will be described in chapter2.

In chapter 2, we discuss in detail the biochemical aspects of AT- Pase molecular motors. We focus on the ATP molecule, which is the main energy currency of living organisms [2, 3]. We will discuss briefly the mitochondrion, which is the organelle were most ATP’s are produced in living organisms.

A brief overview of the possibilities of tracing molecular evolution through ATPase motors is given. Although molecular evolution is a very different field, this subsection will hopefully give some insight to the reader on the common features, mechanisms and structures of ATPase molecular motors nature has developed.

Furthermore, we will discuss the biochemical structure of proteins.

The kinesin motor is discussed to illustrate linear ATPase motors, which are very common too. We then give a detailed overview of the F₁-ATPase motor which is the molecular motor we have studied during this doctoral thesis.

In chapter 3, we describe the necessary mathematical tools and physical concepts used throughout this work and the related literature.

In chapter 4, we discuss the general thermodynamics of molecular motor and apply them to the F1-ATPase molecular motor. Im- portant properties regarding the nonequilibrium are derived like the entropy production and the efficiency of the motor.

In chapter 5, we derive and discuss in detail the continuous 6- state model of the F₁-ATPase, which was published in the Journal of Theoretical Biology [42].

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24 CHAPTER 1. INTRODUCTION

In chapter 6, we present the two-state discrete model of the F₁- ATPase which was recently submitted for publication.

Conclusions are drawn in chapter7 and perspectives are given.

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Chapter 2

Biochemical aspects of ATPase motors

2.1 The ATP currency

The adenosine triphosphate molecule or ATP was discovered in 1929 by Karl Lohmann [43]. It is a ubiquitous and multifunctional molecule that plays key roles in the biochemistry of the living :

1. ATP is the cell main source of energy and can be seen as the fuel molecule for activated biological phenomena. Some canonical examples are :

• Synthesis of DNA and RNA [2, 3]

• Metabolic bio-synthetic reactions, i.e. gluconeogenesis, protein synthesis, . . .

• Motility, i.e. muscle contraction and motion, cellular transport and reorganisation, . . . [3, 27]

2. ATP is a co-factor¹in certain enzymatic reactions, where it ac- tivates the enzyme by giving it its terminal phosphate group². ATP is thus hydrolysed to ADP for a while but regains an inorganic phosphate group in a subsequent reaction restoring the overall balance [2]. In such enzymatic reactions, ATP is used as a kind of molecular key which triggers the function- ing of the enzymatic activity.

1Non-substrate molecule or ion that is necessary for the activation of an enzyme [2].

2This mechanism is called phosphorylation [2].

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26 CHAPTER 2. BIOCHEMICAL ASPECTS OF ATPASE MOTORS 2.1.1 Basic biochemistry of ATPase enzymes

The biochemistry of enzymes is complex, involving often many molecules at a time and well defined conditions. In fact, enzymatic reactions carry out only if all the precise working conditions of temperature, osmotic pressure and pH are exactly satisfied and if no inhibiting or denaturing molecule or ion is present. If these conditions are not satisfied, the enzyme will be progressively de- natured³or its catalytic action will be blocked [2].

In addition, besides meeting all these conditions, enzymatic reactions are chemically specific since they do not catalyse the wrong reactant(s) and produce no side product(s) [2]. In practice this means that an enzymatic reaction carries out only and only if the right molecules are in appropriate contact [2].

In the case of ATPase molecular motors like the F₁-ATPase, we consider the ATP molecule and the motor like an irreversible key-lock system. ATP is like a key that unlatches the motor and by doing so, it is hydrolyzed into adenosine diphosphate (ADP) and inorganic phosphate (P_) releasing an amount of energy ΔG_ATP which is communicated from the catalytic site to the rest of the motor.

The energy is transformed into an elastic strain which progressively releases and forces a conformational change of the motor, setting it into motion [5, 19, 23, 40].

The kinetics of hydrolysis of ATP into its products ADP and P_ is schematized by the following chemical equation :

ATP⁴⁻+H₂O−→ADP³⁻+HPO²⁻₄ +H⁺ (2.1) where all species are written in their ionic form. The inorganic phosphate HPO²⁻₄ is often represented P_ for simplicity and because it is almost always protonated in aqueous solutions according to :

PO³⁻₄ +H⁺ !HPO²⁻₄ (2.2) At chemical equilibrium [44], Eq. (2.1) of ATP hydrolysis is exactly balanced by its opposite chemical reaction, i.e. ATP synthesis. The complete reaction of hydrolysis/synthesis is thus depicted usually by :

ATP!ADP+Pi (2.3)

3The conformation of the enzyme will be perturbed and it will lose its catalytic active state.

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2.1. THE ATP CURRENCY 27 The energy changes of the reaction Eq. (2.3) will be discussed in detail in section4.3.1

Figure 2.1: Schematic representation of ATP. From left to right : 3 phosphate groups, a ribose (pentagonal sugar with 5 carbon atoms) and an adenine.

The structure of ATP is schematically depicted on Figure 2.1.

We can identify on Figure 2.1 the three different parts of the ATP molecule, i.e. an adenine composed of a hexagonal and a pentagonal aromatic ring with nitrogen atoms, a ribose and 3 phosphate groups. It is the addition or removal of the terminal phosphate group that inter-convert ATP to ADP according to Eq. (2.1).

An order of magnitude

In order to sustain in time these mechanisms, ATP has to be con- tinuously recycled. It has been estimated that, on any given day humans turn over their body weight equivalent in ATP [45].

2.1.2 The mitochondrion

The mitochondrion is the organelle that produces most of the cell’s supply of ATP and is often called the "cellular power plant".

A mitochondrion has a size ranging from 0.5-10μm in diameter.

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28 CHAPTER 2. BIOCHEMICAL ASPECTS OF ATPASE MOTORS

Figure 2.2: Schematic representation of a mitochondrion. In vivo, a proton gradient goes through the membrane embedded F_o part of ATP synthase, from the intermembrane space to the matrix.

The proton gradient will drive the rotation of theγ-shaft of the F1

part of ATP synthase.

An interesting aspect of mitochondria is that, unlike other organelles, they possess their own DNA. In fact, it was observed that mitochondria have many features in common with prokaryotes, like their reproduction cycle by cell division⁴. These observations and the subsequent hypothesis started in the late 19^th century. Botanists suggested an exogenous origin for mitochondria (and chloroplasts) which would have originally derived from aerobic bacteria (prokaryotes) that were engulfed by much larger anaerobic bacteria where they evolved and adapted in a mutual benefiting relationship (symbiosis) [46]. This powerful hypothesis and its subsequent theory is today confirmed by numerous experimental studies and is called the Endosymbiotic Theory.

4We only briefly discuss the origin of mitochondria but other organelles like chloroplasts (in photosynthetic organisms) have a similar origin.

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2.1. THE ATP CURRENCY 29 2.1.3 Molecular evolution of proton pumping ATPase

molecular motors

Proton pumping ATPases, like F-ATPases and V-ATPases, are found in all groups of present day organisms [47]. In consequence, scientists hope to gather information on the last common ancestor to all organism and better understand the origin of life on Earth.

The F-ATPases of eubacteria⁵, mitochondria and chloroplasts also function as ATP synthases, i.e. they catalyze the final step that transforms the energy available from oxidation/reduction reactions (e.g. in photosynthesis) into ATP, the usual energy currency of the cell [3] as schematized by Eq. (2.3).

These universal enzymes can be traced down and their DNA sequence precisely determined, which points out to a potential use- fulness of ATPase subunits as universal molecular marker.

It is the primary structure of these ATPases and ATP synthases that was found to be much more conserved between different groups of bacteria than other parts of their protein machinery [47]. The F-type ATPases and the V-type ATPase⁶ were found to be homologous to each other. Both are multi-subunit enzymes thatin vitro can be dissociated into two parts : a water soluble F₁or V₁portion that contains the ATP binding sites, and the F_o and V_oportion that is embedded in the membrane. The F1 and V1 portions are both composed of three copies of a catalytic and a regulatory subunits and in addition, a single copy of three minor subunits that form the connection between the catalytic complex and the membrane embedded one [2]. Their catalytic mechanism is close.

The last common ancestor was thus a highly developed organism that used DNA for the storage of genetic information. The use of DNA implies among other important organelles the presence of ribosomes and a sophisticated protein machinery not much different to that of today’s organisms. In addition, it had membranes that separated this organism from the environment and they were already energized by proton gradients [47].

The last common ancestor thus appears to have been a sophisticated organism already advanced in the evolution of life with

5The term eubacteria is used to denote the phylogenetic group containing all bacteria besides archaebacteria.

6Vaccuolar ATPase

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30 CHAPTER 2. BIOCHEMICAL ASPECTS OF ATPASE MOTORS several hundreds to thousands of different genes [47]. The evolution of these genes is to some degree conserved in the molecular record which is today of much interest for modern analysis.

2.2 Protein structure and chemistry

2.2.1 Composition and primary structure of proteins Proteins are biological molecules composed of amino acids (aa) [2], hence called biopolymers. These biopolymers are synthesised in the cell’s cytoplasm during the translation of RNA [2, 3]. All the living organisms use the same set of 20 amino acids⁷. The diversity of proteins arises from the composition and from the number of amino acids the considered protein contains⁸. The composition and the ordering of the aa chain is called the primary structure.

Each aa is chemically bounded to its two neighbours by a peptide bond [2]. Figure2.3illustrates the common backbone of all amino

Figure 2.3: Schematic representation of an amino acid. Letters stand for the corresponding chemical element. R stands for the lateral chain and lines stand for covalent bonds. The central Car- bon is called the α Carbon and is chiral since it is bound to four different elements.

acids. The difference between all amino acids remains in the nature of the lateral chain, represented byRin Fig. 2.3.

The biochemical activity is enabled by the subsequent foldings of the primary chain.

7There are some exceptions.

8A small protein is for example TRP-cage (20 aa) and a huge one is Ttn (34500 aa).

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2.2. PROTEIN STRUCTURE AND CHEMISTRY 31 2.2.2 3-dimensional shape and biochemical activity The primary chain of aa’s will naturally fold (under physiological conditions) several times and eventually attain its native conformation which is biochemically active [2]. The hierarchy of successive foldings the polypeptide goes through is composed of two types of structures⁹.

The secondary structure

The secondary structure is composed of the α helices and the β sheets.

A α helix is a right-handed helical structure composed of the primary chain of aa’s. The helix has on average 3.6 aa per 360^◦ turn and a 0.15 nm height increment along the vertical axis [2]

as illustrated Figure 2.4. The α helix is a dipole with a positively charged N terminus and a negatively charged C terminus. Figure 2.4 illustrates nicely that each carbonyl oxygen of the backbone of the helix makes a hydrogen bond with an amide nitrogen also from the backbone four residues away.

A β sheet is an assembly of β strands, which are composed of a continuous stretch of 5-10 aa [2]. Theβ strands can be parallel or anti-parallel to each other, giving rise to different types of structures. Hydrogen bonds establish between adjacent C=Oand NH groups. Figure 2.5 illustrates α helices and β sheets. Some secondary structures are known to furthermore organize themselves inmotifscalled super-secondary structures [2].

The tertiary and quaternary structures

Globular hydrophylic polypeptides of secondary structure can fold further, often driven by the burial of hydrophobic parts of some aa’s. This new conformation is known as the tertiary structure.

Once folded, the tertiary structure is also stabilized by hydrogen bonds, ionic interactions and disulfure bonds between aa’s from different secondary elements. The tertiary structure is a biologi- cally active conformation.

Many globular proteins are composed by several polypeptides,

9The Nobeprize in chemistry was awarded to Linus Pauling in 1954 for his research on the peptidic bond and the structure of proteins.

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32 CHAPTER 2. BIOCHEMICAL ASPECTS OF ATPASE MOTORS which gather into an assembly known as the quaternary structure.

The polypeptides are in this context also called protein subunits¹⁰.

2.3 Kinesins

Kinesins are motor proteins that, unlike F- or V-ATPAses, transport cargo across cells by moving along microtubule filaments [48].

There are at least 14 different types of kinesin motors in the kinesin protein superfamily [49].

The kinesin protein consist of two identical polypeptide subunits that dimerize to form a rod-shaped, coiled-coil stalk, with a cargo- binding tail at one end and twin globular domains, usually called heads, at the other [50]. Each head of kinesin is a catalytically active ATPase that attaches to microtubule with a nucleotide dependent affinity. The kinesin motor advances uni-directionally by 8 nanometer steps over the microtubule lattice [51], hydrolyzing one ATP molecule per step [52, 53]. Once attached to a microtubule, kinesin can generate hundreds of steps [20, 54] before spontaneously releasing it. It was shown by S.M. Block et al. [20]

that kinesin motors incubated in excess with silicon beads could move smoothly along a microtubule for several micrometers, but that an average of 0.7 to 3 kinesin attached to one silicon bead could only move 1.4μm. This finding clearly indicates that kinesin motors close to each other interact in some way.

The chemomechanical activity of kinesin is considered highly pro- cessive and indicates that one head is attached to the microtubule at all times [48]. The motion of kinesin thus requires at least two microtubule sites where it can firmly bind (one for the fixed head and one for the moving one). These considerations will help us to understand the mechanism by which kinesin moves along a microtubule.

2.3.1 Kinesin’s moonwalk mechanism

There are a priori two ways by which kinesin can move along a microtubule keeping one head attached¹¹ at all times. The most elegant way is walking by alternating the roles of the heads, one

10The F1-ATPase for example, is a quaternary structure, itsα and βsubunits are tertiary structures, themselves composed of secondary structures.

11In order not to fall off.

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2.4. THE F1-ATPASE IN DETAIL 33 moving and the other remaining fixed to the microtubule, like most human beings. This mechanism is described by the hand- over-hand models. The other, more lame, supposes that the kinesin’s heads keep their role, inducing a stepping behaviour close to that of a leech. This second mechanism is called the inchworm mechanism (from the motion of the larvae of the Geometer moth).

Recent experiments show that kinesin moves by an asymmetric hand-over-hand mechanism [48, 50].

2.4 The F

1

-ATPase in detail

2.4.1 The birth of F₁-ATPase

Research on the F1-ATPase molecular motor began in the early 1950’s, with the work of Efraim Racker. At that time nobody sus- pected the existence of molecular motors. At that time, Efraim Racker showed that glycolysis was dependent on the continuous regeneration of ADP and P_ by ATPase. With his group, they real- ized that this ATPase was the coupling factor that restored oxydative phosphorylation. Eventually, they provided the evidence that the coupling and ATPase activity are both catalyzed by F1.

Later, with Yasuo Kagawa, Racker identified a mitochondrial membrane factor that anchored F1 to the membrane, sensitive to the toxic antibiotic oligomycin [55]. They named this insoluble F₁- binding factor F_o, with the subscript "o" for oligomycin.

In 1974, joined by Walter Stoeckenius, Racker incorporated bacteriorhodopsin, a protein that functions as a light-driven proton pump, and FoF1-ATPase into liposomes [56]. They showed that the protons pumped out by the bacteriorhodopsin flowed back through the F_oF₁-ATPase and generated ATP from ADP and inorganic phosphate, proving Peter Mitchell’s hypothesis [57]¹². 2.4.2 Structure of F₁-ATPase

The F1 domain of FoF1-ATPase is approximately a sphere of 10 nanometer in diameter and contains the three catalytic binding sites for the substrates ADP and inorganic phosphate (see Fig.2.8).

12Peter Mitchell is a biochemist that introduced the theory of oxydative phosphorylation, by which ATP is produced from the motion of an ion gradient going through a specialized transmembrane protein (Fo) of the mitochondrion [57].

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F₁ is a water soluble complex of five different proteins with the stoichiometry 3α : 3β: 1γ: 1δ: 1ε. The minimum assembly of F₁ is(αβ)3γ. In bovine mitochondria they contain 510, 482, 272, 146 and 50 amino acids, respectively [38,58], giving a relative molecular mass of 371 000. The heterogeneity of composition of F₁, in- dicated by subunit stoichiometry, is responsible for F1’s structural asymmetry. This structural asymmetry is strengthened by the asymmetric positioning of theγ−subunit relative to theα₃β₃ sub- assembly, inducing an overall chiral environment. Asymmetry is evident in the crystal structure [38], as shown on Figure 2.7.

The sequence of the α− and β−subunits are quite homologous (20% identical) but, the catalytic binding sites are located in the β−subunits at the interface with the α−subunits, whereas the function of nucleotides in theα−subunits, which do not exchange during catalysis, is obscure. The α−and β−subunits, which have a similar fold, are arranged alternatively like the segments of an orange around a centralα−helical domain of theγ−subunit [38].

2.4.3 Rotary catalysis

In the 1960s through the 1970s, Paul D. Boyer’s¹³ developed the binding change mechanism which postulates that ATP synthesis is coupled to the a conformational change generated by the rotation of the γ-shaft of F₁ [40]. According to this mechanism, the structure of the three catalytic sites are always different, each going through a cycle of "open", "loose" and "tight" states. The crystal structure of F₁ obtained in 1994 by John E. Walkeret alconfirmed the possibility of a rotary mechanism of catalysis of ATP [38]. Re- cently, crystal structures of F1 with all three catalytic sites in different states have been obtained [58].

In 1997, Kinositaet alconfirmed directly the hypothesis of rotation by attaching a fluorescent actin filament to theγ-shaft as a visual marker [22]. In the presence of ATP, the filament of micrometer size rotated over 100 revolutions per second and a rotary torque of 40 pNnm was reached.

Visualization of the circular trajectories showed that the rotary speed is not constant and that small¹⁴, stochastic fluctuations oc-

13Paul D. Boyer was awarded the Nobel prize in Chemistry in 1997 for his contributions.

14In comparison to the angular steps.

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2.4. THE F1-ATPASE IN DETAIL 35 cur [22–24]. The stochasticity of the motion is the consequence of the nanometric size of the F₁-ATPase motor making it sensitive to the thermal and molecular fluctuations of the atomic structure of the protein and surrounding medium [59, 60]. Besides the fluctuations, a stepping behaviour is observed in the trajectories of the γ-shaft [22, 23, 61].

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Figure 2.4: Core structure of an alpha helix.

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2.4. THE F1-ATPASE IN DETAIL 37

Figure 2.5: Part of Bacillus thuringiensis (Bt) delta-endotoxin CytB.

Dark pink indicates areas of alpha-helix and gold indicates areas of beta-sheet (PDB: 1CBY).

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Figure 2.6: Schematic representation of a kinesin carrying a load along a microtubule.

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Figure 2.7: Cartoon representation of the Bovine Mitochondrial F₁- ATPase (PDB:1BMF). The outer ring is composed of 3 α and 3 β subunits alternated, and in the center theγ-shaft.

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Figure 2.8: Schematic representation of the F_oF₁ATPase.

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ADP

ADP ATP

ATP ADP

ADP

ADP ADP

ADP ATP ATP ADP

ADP ADP

ADP ADP ADP ATP

ADP

ADP ADP

ATP ATP

ADP + P

ATP

ATP ATP

ADP + P

ADP + P ADP + P

ADP + P

! = 1

! = 4

! = 2

! = 3

! = 6

! = 5

Figure 2.9: The motor cycle and its possible states : The inner ring contains three states with empty sites and three possible shaft angles. The second and third rings represent the motor cycles based on the bi- and tri-site mechanisms. The intermediate state which is here denoted ADP is either an excited state ATP^∗ of ATP, ADP·Pi, or ADP alone in the corresponding catalytic site [23]. P denotes inorganic phosphate.

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10 nm

40 nm bead

F ₁ _{ADP + P}

i

ATP

Figure 2.10: Schematic representation of the F₁ATPase motor with a bead attached to the centralγ-shaft.

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Chapter 3

Mathematical framework

In this chapter we will try to give a comprehensive overview of the mathematical framework used throughout this study and the related literature.

3.1 Basic Knowledge from Probability The- ory

Our starting point is the description of astochastic or random vari- ableand the chance orprobabilityto effectively observe its value.

3.1.1 The Stochastic Variable and its Probability A stochastic variableX is defined by a set of N possible values :

X1, X2, X3, . . . , XN (3.1) and a probability distribution over this set of values [59]. One can think of the values Eq. (3.1) as possible outcomes of an experi- ment. TheX (=1,2, . . . , N) are notper sea part of a process, i.e.

they can be independent from one another. The set of possible values Eq. (3.1) can be either discrete or continuous in a given interval depending on the nature of the considered problem.

The probability distribution associated to the set Eq. (3.1) is a non-negative function [59] :

P()≥0 (3.2)

and normalized according to :

!

P()d=1 (3.3)

43

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44 CHAPTER 3. MATHEMATICAL FRAMEWORK where the integral extends over the whole set of possible values X_ (=1,2, . . . , N) of Eq. (3.1). The probability that X has a value betweenand+dis given by :

P()d

.

3.1.2 The Conditional Probability

Let X be the random variable with N components of Eq. (3.1). Its probability density PN(1, 2, . . . , N)is also called the joint prob- ability distributionof the N variables Eq. (3.1) [59]. We can also consider a subset ofS < N variablesX₁, X₂, . . . , X_S.

Furthermore, we can attribute fixed values to the subsetXS+1, . . . , XN

and consider the joint probability of the remaining variablesX₁, X₂, . . . , X_S. This is called theconditional probabilityof X₁, . . . , X_S, conditional

on the subsetXS+1, . . . , XNhaving the prescribed valuesS+1, . . . , N

[59].

The conditional probability is denoted by :

P_S|N−S(₁, . . . , _S|_S+1, . . . , _N) (3.4)

3.2 Markov Processes

Markov processes¹ constitute a large subclass of stochastic processes that obey the Markov property [59]. The Markov property states that for any set ofn successivetimes

t1< t2< . . . < tn

one has :

P_1|n−1(_n, t_n|₁, t₁;. . .;_n−1, t_n−1) =P_1|1(_n, t_n|_n−1, t_n−1) (3.5) That is, the conditional probability densityP_1|n−1at timetn, given the value_n−1at timet_n−1(see Eq. (3.4)), is uniquely determined and is not affected by any knowledge of the values at earlier times. In other words, the Markov property states that the probability distribution of the system at the next step only depends on the current state of the system and is independent on the previous ones [59, 60].

1Called after Russian mathematician Andrey Andreyevich Markov (1856 - 1922) [62].

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3.3. THE CHAPMAN-KOLMOGOROV EQUATION 45

3.3 The Chapman-Kolmogorov Equation

The Chapman-Kolmogorov equation is an identity that follows from the condition of normalization Eq. (3.3) [59, 60]. For a trivial Markov process t1 < t2 < t3, it states that given an initial time and state (₁, t₁)and a final one (₃, t₃), the process has passed through an intermediary state (₂, t₂). Summing over all possible conditional probabilities p(2, t2 | 1, t1) and rearranging the equation one gets [59] :

p(₃, t₃|₁, t₁) =

!

d₂p(₃, t₃|₂, t₂)p(₂, t₂|₁, t₁) (3.6) All the transition probabilities must necessarily obey Eq. (3.6) [59].

3.4 The Master Equation

The master equation is an equivalent form of the Chapman-Kolmogorov equation for Markov processes [59]. The master equation is a differential equation that describes the evolution of a Markov process. Since chemical reactions are discrete phenomena, we will here focus on the discrete master equation [59, 60] which rules such processes.

The discrete master equation rules the time evolution of the probability P_n(t) to find the studied system at state n at time t considering transitions to and from neighboring sites n^% (&= n). We represent such elementary transitions by :

n!n^% (3.7)

The probabilitiesPn(t)of the discrete process satisfy a normalization condition [59, 60, 63] :

"

n

P_n(t) =1 (3.8)

which is analogous to Eq. (3.3) in the continuous case.

The discrete master equation ruling this process is given by [59, 60] :

dP_n(t) dt = ^"

n^%&=n

{W(n|n^%)Pn^%(t)−W(n^%|n)Pn(t)} (3.9)

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46 CHAPTER 3. MATHEMATICAL FRAMEWORK where W(n |n^%)is the transition probability per unit timeortran- sition ratefrom state n^% to staten, and W(n^%|n)is the transition rate of the opposite reaction, i.e. from state n to state n^%. The transition rates can’t be negative and obey necessarily the condition [59, 60] :

W(n^%|n)≥0 (3.10)

for any staten andn^%.

The master equation (3.9) is a gain-loss equation for the probability Pn(t) of state n at time t. The first term on the right-hand side in equation (3.9) is the gain due to transitions from staten^%to staten, the second term is the loss due to transitions from state n ton^% [59].

The master equation conserves the total probability : d

dt

"

n

P_n(t) =0 (3.11)

for all values oft.

In the seventies, Schnakenberg noticed that the product of the ratios²of the forward and backward transition rates over a cycle³ is independent of the states of the cycle. It can furthermore be expressed in terms of the non-equilibrium constraints imposed on the system from the environment [64] :

W(n|n^%)

W(n^%|n) =e^β(Xⁿ^−Xⁿ^%⁾ (3.12) whereXn denotes the corresponding thermodynamical potential⁴. The master equation admits in general a time-independent stationary solutiondP^st_n/dt=0. If all the thermodynamical constraints vanish, the stationary solutionP^st_n represents the equilibrium state P^eq_n , which obeys the condition of detailed balance :

W(n|n^%) W(n^%|n) = P^eq_n

P^eq_n%

(3.13)

2Schnakenberg considered several elementary reversible reaction like the one depicted in Eq. (3.7) [64].

3The term cycleis considered in the usual sense of the word. For example, the transition from staten to staten^% and back again to state nconstitutes a cycle.

4Under laboratory conditions temperature and pressure are constant so that the thermodynamical potential is a free enthalpy.

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3.5. LANGEVIN’S APPROACH TO BROWNIAN MOTION 47

3.5 Langevin’s Approach to Brownian Mo- tion

In 1908 Paul Langevin proposed an equation to account for the motion of the Brownian particle [11]. Langevin’s starting point was the equipartition theorem of kinetic energy which states that the Brownian particle has, in the  direction, an average kinetic energy equal to RT/2N, where R = 8.314472 J mol⁻¹ K⁻¹ is the gas constant,T the temperature [65].

According to classical mechanics [66], the velocity (in the  direction) of this particle is given by :

(t) = d

dt (3.14)

Considering Newton’s equation :

F = m (3.15)

= md

dt (3.16)

we can relate F, which is the total force acting on the particle, m its mass and  its acceleration, to the velocity expressed in Eq.

(3.14). We know from Newton that the forces ƒ acting on the particle can be summed up yielding the resulting total force

F=^"



ƒ_

The Brownian particle is subject to gravity which attracts it vertically to the ground but it is slowed down by the viscous drag force Fddue to its motion in the surrounding viscous media. The viscous drag force pushes the Brownian particle oppositely to the gravity force.

The viscous drag force is also calledStokes’ law, and is given by : Fd(t) =6πηr(t) (3.17) where η is the viscosity coefficient, r is the radius of the particle and  its velocity. The Brownian particle moves in an external potential field V(). The motion of the Brownian particle in this potential fieldV() induces a force−∇V opposite to the direction of motion.

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48 CHAPTER 3. MATHEMATICAL FRAMEWORK

Figure 3.1: Schematic illustration of a Brownian particle of mass m moving vertically in a viscous media. The Brownian particle experiences the force due to gravityF_gand the viscous drag force Fd.

The novel idea Langevin had was to consider the random collisions of the Brownian particle as the effect of a fluctuating time dependent forceξ(t).

Langevin’s equation for a Brownian particle in an external field is given by :

md(t)

dt =−∇V()−6πηr+ξ(t) (3.18) The fluctuating time dependent forceξ(t)is also callednoise[67].

We consider the case ofwhitenoise⁵which has the following properties :

〈ξ(t)〉 = 0 (3.19)

〈ξ(t)ξ(t^%)〉 = 2k_BTζδ(t−t^%) (3.20)

5The termwhite noiseis used in analogy with white light, in which the spec- trum of frequencies is equally distributed over the whole visible band interval and all the frequencies (colours) add up and give what we call white light.

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3.6. FOKKER-PLANCK EQUATION 49 whereζ=6πηr is the friction coefficient.

3.5.1 Over-damped Langevin Equation

Because of the viscosity of the media, no acceleration takes place, i.e.

|d

dt| )1

For this reason, the inertia term m^d_dt can be neglected in Eq.

(3.18). Taking this simplification into account, one writes : 0=−∇V()−6πηrd(t)

dt +ξ(t) (3.21) which is rearranged to its common form :

ζd

dt =−∇V() +ξ(t) (3.22)

3.6 Fokker-Planck Equation

The Fokker-Planck equation is the Chapman-Kolmogorov equation for strictly continuous processes. Furthermore, the Fokker-Planck equation is a continuous partial differential equation that describes the time evolution of the probability distributionP(, t) of the po- sition at timet [59] :

∂P(, t)

∂t =−∂J(, t)

∂ (3.23)

whereJ(, t)is the probability current given by : J(, t)≡ζ⁻¹[−∂V

∂ +ƒ]P(, t)−D ∂

∂P(, t) (3.24) given in terms of the deterministic force−∂_V+ƒ which biases the random motion and the diffusion coefficient :

D= k_BT

ζ (3.25)

The first term on the right-hand side of Eq. (3.24) is also called thedrift termand the second is called diffusion term[59].

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50 CHAPTER 3. MATHEMATICAL FRAMEWORK The probability distribution and the current are submitted to peri- odic boundary conditions :

P(0, t) =P(L, t) (3.26) J(0, t) =J(L, t) (3.27) whereL is the period of the potentialV().

If we multiply the over-damped Langevin equation Eq. (3.22) by 1/ζand add an external force ƒ we obtain :

d(t) dt = 1

ζ[−∂V()

∂ +ƒ+ξ(t)] (3.28) which we can link more easily to the probability current Eq. (3.24).

The average velocity is found by integrating the probability current Eq. (3.24) in the interval[0, L]:

! _L

0

d J(, t) =ζ⁻¹〈−∂V(, t)

∂ +ƒ〉 (3.29)

where the term proportional to the temperatureTvanishes thanks to the periodicity of P(, t) imposed by the boundary conditions Eq. (3.27). The right-hand side of Eq. (3.29) is the average of the over-damped Langevin equation Eq. (3.22). The average velocity of the molecular motor

〈(t)〉=〈d(t)/dt〉 is thus given by :

〈(t)〉=

! _L

0

dJ(, t) (3.30)

3.6.1 Chemomechanical Coupled Fokker-Planck Equa- tions

To model the chemomechanics of molecular motors, terms de- scribing the random jumps between the discrete chemical states n must be included to Eq. (3.23) :

∂tpn(, t)+∂Jn(, t) = ^"

n^%&=n

[pn^%(, t)Wn^%→n()−pn(, t)W_n→n^%()]

(3.31)

Continuous and Discrete Stochastic Models of the F