c 2006 by prasanth sankar. all rights...
TRANSCRIPT
c© 2006 by Prasanth Sankar. All rights reserved.
PHENOMENOLOGICAL MODELS OF MOTOR PROTEINS
BY
PRASANTH SANKAR
M. S., University of Illinois at Urbana-Champaign, 2000
DISSERTATION
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2006
Urbana, Illinois
Abstract
Motor proteins produce directional movement and mechanical output in cells by converting
chemical energy into mechanical work. Innovative applications of physics experimental tech-
niques to the study of motor proteins have generated a wealth of data. By careful analysis
and modeling of this data, we try to gain a quantitative understanding of this biological
system. We have proposed a way to incorporate the biochemical data on motor proteins
into the modeling and thus minimize the number of fitting parameters and increase the rel-
evance of the models. Mechanical understanding of motor proteins are obtained by using
an experimental technique of observing the motion of a large cargo elastically attached to
it. On modeling the effect of the elastic motor-cargo link on the motion of the cargo, we
see that the conclusions on motor mechanism derived from such techniques are dependent
on the size of the cargo as well as the stiffness of the link. By proposing a new modeling
scheme, we devise a unified description of such experiments for various motor proteins. This
description is based on the picture that the motor thermal fluctuations are rectified in a pre-
ferred direction by means of barriers whose heights are modulated by the chemical reactions
taking place at the motor.
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Acknowledgments
I would like to thank my advisor Yoshitsugu Oono for his constant encouragement, guidance,
and support. I would also like to thank Satwik Rajaram for carefully reading the manuscript
and suggesting improvements, and Bojan Tunguz for being a nice office mate all these years.
This work was supported in part by teaching assistantship from University of Illinois, and
summer research assistance from University of Illinois Campus Research Board.
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Table of Contents
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motor Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 ATP: The Energy Source . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Types of Motor Proteins . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Experimental Studies of Motor Proteins . . . . . . . . . . . . . . . . . . . . 61.3.1 Biochemical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Structural Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Single Molecule Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Our Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Road Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 2 Phenomenology of Motor Protein Kinesin . . . . . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Theoretical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 General Considerations on Phenomenological Modeling . . . . . . . . 18
2.2 Empirical Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Proposed Kinetic Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Kinetic Steps and States . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Modeling Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Explanation of the Mechanochemical Data . . . . . . . . . . . . . . . . . . . 332.4.1 Force-velocity relation . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Velocity-ATP Concentration Relation . . . . . . . . . . . . . . . . . . 362.4.3 Force-Randomness Relation . . . . . . . . . . . . . . . . . . . . . . . 372.4.4 Randomness-ATP Concentration Relation . . . . . . . . . . . . . . . 392.4.5 Force-Run Length Relation . . . . . . . . . . . . . . . . . . . . . . . 402.4.6 Run Length-ATP Concentration Relation . . . . . . . . . . . . . . . . 412.4.7 Velocity and Run Length Variation with ADP . . . . . . . . . . . . . 42
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.1 Uniqueness of Kinetic Scheme . . . . . . . . . . . . . . . . . . . . . . 442.5.2 Motor Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.3 Is There a Power Stroke in Kinesin Force Production? . . . . . . . . . 452.5.4 How Unique is the model? . . . . . . . . . . . . . . . . . . . . . . . . 47
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Chapter 3 Effects of the Elastic Motor-Cargo Link on Motor Transport 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Motor-Cargo System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Variation of Motor Velocity with Link Stiffness . . . . . . . . . . . . . . . . . 533.4 Stokes Efficiency of Motor-Cargo System . . . . . . . . . . . . . . . . . . . . 593.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 4 Interpretation of Single Molecule Experiments of Motor Proteins 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Motor-Cargo System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 An Effective Cargo Only Representation of a Coupled Motor-CargoSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Mathematical Formulation of Effective Cargo Only Representation . . 684.3 Explanation of Mechanochemical Data . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 F1-ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.2 Kinesin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.3 Myosin V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 5 Affinity Switch Model for Rotary Motor F1-ATPase . . . . . . 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Algorithmic Description of the Model . . . . . . . . . . . . . . . . . . . . . . 845.4 Comparison With Analytically Solvable Model . . . . . . . . . . . . . . . . . 865.5 Comparison of Model Predictions with Empirical Results . . . . . . . . . . . 875.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 6 Summary and Open Questions . . . . . . . . . . . . . . . . . . 93
Appendix A Solution of the Kinetic Model . . . . . . . . . . . . . . . . . . 96
Appendix B Stochastic Energetics . . . . . . . . . . . . . . . . . . . . . . . 102
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Author’s Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
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Chapter 1
Introduction
1.1 Motivation
Ever since the application of X-ray crystallography to the determination of the structure of
DNA, incorporation of physical techniques into biological research have produced a steady
stream of advances in the field of cellular and molecular biology. This laid the foundation for
one of the most exciting insights into the machinery of life, the structure-function relation
which postulated that the function of nucleic acids and proteins are coded in their three
dimensional structure. Attempts to determine the static structure of proteins (the protein
folding problem) and understanding the structural contribution to function dominated the
attention of physicists initially contributing to biological understanding. At the same time,
physical techniques and theoretical tools which were developed to solve complex problems in
a quantitative way were also incorporated into the available tools of biological research. This
produced a new realm of quantitative understanding which was a break from the qualitative
descriptions which characterized biological research.
Development of these new experimental techniques and their application to the study
of function of proteins have generated a wealth of quantitative data. With this, a new
consensus has emerged that to have a complete understanding of a biological phenomenon,
theoretical and experimental collaboration between biologists and physicists is needed. Ac-
cumulated experimental data allow proposals of qualitative hypotheses. These proposals are
substantiated by using quantitative models based on the understanding of underlying physics
and quantitative agreement of such models with available data. Such models are useful in
1
extracting further information from experimental data, making predictions regarding the
unknowns and thus directing further experimental studies.
While proposing such models, we must pay due attention to certain peculiarities of
biological systems, nature of required understanding, and the way physical approaches should
be modified while approaching a biological problem. Understanding of a biological system is
complete only if its function is explained in terms of underlying physics and the interaction
of the system with its environment. Even the simplest constituents of biological systems
such as individual protein molecules are made up of many atoms. This makes application of
direct reductionist methods of physics which seeks to explain the a phenomena in terms of
known interactions of underlying atomic constituents less successful. It is difficult to bridge
the timescale of atomic motions occurring in picoseconds or less to the biologically relevant
dynamic processes of the system as a whole which occur in microseconds or slower. Another
traditional physics approach of explaining relevant phenomena in terms of coarse-grained
statistical descriptions also needs modification when applied to biological systems. Since
the function is coded in relevant substructures of the system, coarse-graining should retain
these structural domains which may be unique to individual systems. In addition, biological
systems are in nonequilibrium. Though the statistical descriptions of equilibrium systems
are well developed, the nonequilibrium extensions are not.
Motor proteins such as myosins and kinesins which are used for directional transport
in cells constitute one such biological system in which development and application of new
experimental techniques have generated a wealth of information at various time and length
scales [1]. At a fundamental physical level, these are energy transducers which convert
available chemical energy in cells to mechanical energy which is used for directional motion.
The mesoscopic size of the system forces them to do this energy conversion while being
influenced by interactions with the thermal environment. The physically relevant quantities
are the speed at which these motors move, the mechanical force generated, the role of
structural elements in the energy conversion, the efficiency of chemical to mechanical energy
2
conversion, etc. An important question is how the system uses or minimizes the effects of
thermal fluctuations in various processes of molecular machines. It is also interesting to
know whether the function of different motors can be explained by a unified mechanism at
a coarse-grained level. In this thesis, by incorporating relevant experimental information on
the chemical processes, biochemical and structural information and the dynamics of relevant
functional subdomains, we propose models which attempt to explain the working of motor
proteins in a quantitative way. While proposing these models, attempts are made to minimize
the number of fitting parameters by utilizing available biochemical and structural data on
motors.
1.2 Motor Proteins
Proteins are a class of macromolecules. Individually or in association with other macro-
molecules they perform a variety of biological functions needed to sustain life. Many proteins
act as enzymes catalyzing chemical reactions. Others have structural and mechanical roles
in the cellular machinery. Proteins are made of amino acids. Depending on the sequence of
amino acids constituting them they can have unique three dimensional structures in cellular
environment.
Of the many required cellular functions, some involve directional movement and transport
of macromolecules and supramolecular complexes. Without specialized motors, efficient
transport of large complexes over eucaryotic cells is impossible. Motor proteins are such
molecular motors which produce directional movement in cells and mechanical output by
converting chemical energy into mechanical work [1].
1.2.1 ATP: The Energy Source
Motor proteins produce work by converting the free energy liberated by the hydrolysis of
adenosine triphosphate (ATP) to mechanical energy. ATP is an organic compound com-
3
posed of adenine, the sugar ribose, and three phosphate groups. The breaking of the bond
connecting the last phosphate ion to the rest of the molecule by the following hydrolysis
reaction creates an adenosine diphosphate (ADP) and inorganic phosphate Pi.
ATP + H2O ↔ ADP + Pi, (1.1)
The breaking of the phosphate bond has a high negative free energy of reaction. The
total free energy released depends on the standard free energy, ∆G0, and the concentrations
of ATP, ADP and Pi. It is given as
∆G = ∆G0 + kBT log[ADP][Pi]
[ATP], (1.2)
where ∆G0 = −54 × 10−21J/mol is the standard free energy change at pH 7, and [ATP],
[ADP] and [Pi] are the concentrations of ATP, ADP and Pi, respectively, in molarity, kB is
the Boltzmann constant and T is the absolute temperature.
Since the typical length scales involved in motor proteins are nanometers (nm), and the
forces generated are of the order of piconewtons (pN), it is convenient to use these units
instead of SI units. In these units, the standard free energy change ∆G0 = −54 pNnm,
kBT = 4.1 pNnm at room temperature. Under physiological conditions, [ATP]∼ 10−3 M,
[ADP]∼ 10−4 M, and [Pi]∼ 10−3 M, ∆G∼ −90 pNnm. ∆G is negative under all practical
conditions, implying that the reaction always proceed in the direction of hydrolysis, and
that spontaneous net synthesis of ATP never happens in solution. However in the absence
of any enzymes the reaction rate to the right for the above reaction is very low and it is this
stability that makes the phosphate bond such an ideal high-energy source.
4
1.2.2 Types of Motor Proteins
Diversity of cellular functions are realized through the evolution of many different classes of
motor proteins. Latest estimate suggests there are 100 different motor proteins in eucaryotic
cells performing various tasks. Different motor proteins can be classified according to their
specific functions [2] .
Kinesins are a family of motor proteins involved in cargo transport along microtubule
tracks [3]. Microtubules are formed by the polymerization of αβ tubulin heterodimers. They
have a hollow cylinder shape with diameter of 24 nm with tubulin dimers oriented parallel
to the cylinder axis. In addition, they have a polarity which allows fixing the direction of
motor motion with respect to the track as moving towards the plus end or the minus end of
microtubule.
Of the many members of kinesin family, some of them such as kinesin-1 form dimers and
move processively (motor takes many steps before detaching from the track) toward the plus
end of microtubule. Ncd is another kinesin which dimerizes and moves toward the minus end
of microtubule nonprocessively. There are monomers such as Unc104, KIF1A which are also
processive motors. In addition to transporting cargo such as vesicles, they are also involved
in cell division and microtubule assembly. Defective functioning of kinesins are implicated
in neurological disorders, cancer and various other diseases.
Myosins are a family of motor proteins involved in muscle contraction as well as cargo
transport along actin filaments [4]. An actin filament is a left-handed helix of actin monomers
with a periodicity of 36 nm. Similar to microtubule they also have a polarity. The thick
filaments in a muscle are composed of myosin II, and by sliding with the thin filaments of
actin, they produce the muscle contraction force. Myosin V dimerizes and moves processively
toward the plus end of actin whereas myosin VI dimers move towards minus end. Both are
involved in cellular transport.
Dyneins are another class of motor proteins [5]. One type of dynein powers cilia and
5
flagella (structures with a core made of microtubules which oscillate at high frequency by
virtue of force generated by dynein molecules arrayed along microtubule). Another form of
dynein, cytoplasmic dynein, is involved in the transport of organelles along microtubules.
Polymerases such as DNA and RNA polymerases, move along the strands of DNA in
order to replicate them and to transcribe them into RNA, respectively [6].
In addition to the linear motors above there is a class of rotary motors known as ATPases
which are involved in transport of ions across membranes. A prominent member of rotary
motors is F1-ATPase [7] which is involved in the synthesis of ATP from ADP and Pi using the
proton flow across the membrane as the energy source. Similarly, flagellar motors involved in
the directed motion of bacteria also use ion gradient across membrane as the energy source.
Another class of motors are involved in packaging of viral DNA into protein capsids [8].
In this thesis we will be considering kinesin, myosin V, and F1-ATPase.
1.3 Experimental Studies of Motor Proteins
Experimental characterization of motor proteins can be categorized into the general cate-
gories of biochemical, structural, mechanical and optical studies.
1.3.1 Biochemical Studies
Biochemical studies assume that, while converting chemical energy of ATP to mechanical
work, motors cycle through a series of long lived states with different positions along the
track. After completion of a cycle, the motor advances by one step. These states depend on
the state of the nucleotide (ATP or its hydrolysis intermediates) bound to it and whether the
motor is attached to the track. The state transitions are represented by chemical transitions
of the form,
X0
k1⇀↽
k−1
X1 ⇀↽ · · ·Xi−1
ki⇀↽
k−i
Xi ⇀↽ · · ·Xn, (1.3)
6
where Xi is an intermediate state of the motor and ki are the rates at which one state
transforms to another. In addition to identifying these long lived states, biochemical studies
also seek to determine the rate constants of state transition. A summary of application of
biochemical methods to the study of motor proteins is given in [9].
1.3.2 Structural Studies
Successful crystallization of protein molecules allows the determination of three dimensional
structures using X-ray crystallography. Structural characterization of nucleotide and track
binding regions and the regions connecting these two have contributed greatly to the under-
standing of motor function. The goal of such studies is to identify the possible structural
transformations that the motor undergoes while hydrolyzing ATP, and generating force and
movement. For this, it is sometimes possible to crystallize the intermediates with different
nucleotide bound states and determine their structures [10]. Difficulties with crystallization
mean there is not enough structural data to give a complete picture of motor conformational
changes. Sometimes low resolution cryo-EM images [11] are used to fill the gaps in structural
knowledge.
Kinesin: Figure 1.1 shows the crystal structure of kinesin dimer with both heads bound
to ADP [12]. The structure comprises two motor heads connected through a coiled-coil
domain. The head is attached to this coiled coil through a structurally important region
called the neck linker (β9 and β10 in the figure). Structures of kinesin bound to microtubule
are not yet determined and the docking of known structure to low resolution cryo-EM images
are used to infer the structure of microtubule bound states and possible structural changes
associated with microtubule binding.
Myosin: Structural studies of myosin are more advanced than kinesin since more hydrol-
ysis intermediate structures available. Crystal structures of myosin II with no nucleotide or
bound to ADP or ADP-vanadate have been reported. Structures of myosin bound to actin
have not yet been reported. Here also docking of known structures to low resolution cryo-
7
Figure 1.1: Crystal structure of kinesin dimer [12].
EM images are used to infer the possible changes associated with actin binding. Fig. 1.2
shows the known crystal structures of myosin II and the structural changes associated with
nucleotide changes [13]. One of the most important insights gained from crystal structures
of myosin is that there is a lever like region (grey helix in Fig. 1.2) which rotates in response
to the changes in nucleotide states.
F1-ATPase: Another motor whose crystal structure determination has provided further
insights into its force generation is the F1-ATPase motor. As shown in Fig. 1.3, the motor
structure consists of α3β3γδǫ subunits [14]. The central γ subunit can rotate inside a cylinder
made of three α and three β subunits arranged alternately. ATP binding sites are located
primarily on a β subunit. Crystal structures show that, depending on the state of ATP
present at the pocket, β subunit can be in either a closed state or open state.
In addition to knowing the structures of individual motors, comparative studies of struc-
8
Figure 1.2: Known crystal structures of myosin II and a proposed schematic representationof possible structural changes during the ATP hydrolysis [13].
tures of different motors are also useful. It is seen that the core of myosin and kinesin
structures have similar structure. This structural similarity suggests nucleotide binding, hy-
drolysis, and release of hydrolysis products may trigger similar motions in motor proteins.
Thus it is possible that both motors have similar mechanisms. Such structural similarities
are encouraging. It is possible that understanding the function of one motor will help in
identifying the functional mechanism of other structurally similar motors.
1.3.3 Single Molecule Studies
The advent of in vitro motility assays [15] allowed the study of motion produced under
biochemically defined conditions using one or a few proteins. Such single molecule studies of
motor proteins can be classified into two general categories. Mechanical studies monitors the
motion of the motor against an external force applied to it. Optical studies with fluorescence
9
Figure 1.3: Schematic representation of F0F1-ATPase structure. Crystal structure is knownonly for F1 part (the top part of the figure) [14].
tags use fluorescent probes attached to the motor to determine distances involved in the
motion of the motor as a whole or structural changes within the motor.
Mechanical studies: Motor proteins typically produce piconewton forces. To observe
the motion of motors against a load, such piconewton forces can be applied to it using an
optical trap (a focused laser beam) or by attaching the motor to a very fine glass needle. In
a typical optical trap experiment shown in Fig. 1.4, the motor is attached to a large bead
which is trapped in the laser beam. The motion of the bead is tracked at high resolution
(to have accurate detection of bead position, beads which are many times motor size are
used in experiments) [16]. To move the bead away from the trap center, motor has to exert
a force on it to overcome the force exerted on the bead by the trap. Optical trap studies
allow the determination of the variation of motor velocity with external load acting on it,
the maximum work that the motor can produce, and the step size of motor displacements.
10
Another variation of this set up uses a large cargo attached to the motor and the motion of
this cargo is observed. In this case, the motor work is measured as the work done against
the viscous drag produced by the cargo.
Application of optical trapping experiments on kinesins have shown that they move on
microtubule by taking 8 nm steps, and generates a maximum force of 8 pN [16]. For myosin
V, the step size is determined as 36 nm and the maximum force produced is 2 pN [17].
Figure 1.4: Optical trap setup used to study the load dependence of kinesin motion [16]
Optical studies with fluorescence tags: Fluorescence resonance energy transfer (FRET)
studies [18] are useful in characterizing the structural changes involved in motor operation.
Here a donor and an absorber fluorophore are attached at two different locations on the pro-
teins. If there is a structural change which modifies the distance between these fluorophores,
the changes in emission spectra of the donor can be used to determine the distance change
involved. Recent advances have allowed tracking of a single fluorophore attached to the
11
motor with very high resolution. For example, fluorescence imaging at nanometer accuracy
(FIONA) [19] is able to track the location of a fluorophore to nanometer resolution. Appli-
cation of FRET studies to kinesin shows that the neck linker undergoes a conformational
change. Application of FIONA to kinesin and myosin V has shown that the dimers move in
a hand-over-hand fashion in which the two dimers alternate their relative position.
1.4 Theoretical approaches
The goal of any theoretical approach is to understand the mechanism by which motors
convert chemical energy from ATP binding, hydrolysis and product release into mechanical
work. The pioneering work was done by A. F. Huxley [20] to explain muscle contraction. A
cross-bridge model involving relative sliding of actin and myosin filaments was proposed as
causing muscle contraction. In the model, myosin molecules in the filament moves back and
forth about an equilibrium position as a result of thermal fluctuations. On reaching near a
binding site on actin, the strained myosin binds to it. The relaxation of the strain causes
the sliding of the actin filament. On completion of actin sliding, myosin detaches from it
and the cycle repeats.
The determination of crystal structures and the identification of possible structural
changes occurring during ATP hydrolysis has given rise to newer approaches to modeling
motor operation. At the microscopic level, molecular dynamics simulations seek to explain
motor operation in terms of the atomic structures and forces involved. Atomic resolution
structures are used for detailed calculations of molecular energetics and dynamics. The
ultimate aim is to understand how the conformational changes that engender the motion
are produced by ATP and its hydrolysis. The presently available computing power is not
yet sufficient to bridge the time scale of atomic motions to functionally relevant timescales.
However, there is a simulation method which seeks to accelerate microscopic dynamics by
the application of external forces [21]. This approach is ideally suited to the study of chem-
12
ical reaction paths. It has clarified the details of the energetics of ATP hydrolysis at the
nucleotide pockets of known motor crystal structures [22]. Still, it is difficult to study the
actual conformational changes during the motion cycle of motors.
The time scale limitations of full molecular dynamics simulations have given rise to
intermediate mesoscopic models which use some sort of abstraction of structural details as
the starting point. In a class of models known as power stroke models, the protein structure
is abstracted as a machine made up of a spring-like or elastic element to produce force, a
lever to amplify the force, and a latch to regulate nucleotide binding or release. Fig. 1.5
shows such a representation of the operation of kinesin motors [23].
Figure 1.5: Kinesin motor as a molecular machine [23].
These power stroke models postulate the storage of energy of ATP hydrolysis in the
spring. The relaxation of this stored energy through the motion of the lever is postulated as
the mechanism driving the motor forward. β closing models in F1-ATPase [24], neck linker
docking models of kinesin [25], and lever arm models of myosin [10] are examples of this
modeling approach.
Another class of models identify a few long-lived states in the mechanochemical cycle
13
of the motor and assume stochastic transitions with mean kinetic rates between them [26].
These models assume the whole motor as a particle that can go through a certain number
of states with different positions along the track. After completion of the cycle the motor
advances by one step. All the stochastic transitions between states are assumed to be
reversible. In presence of an external load acting on the motor, the kinetic rates are modified
as dependent exponentially on the load. The load dependence and the rate constants are
determined by curve fitting to the experimentally determined mechanochemical data on the
motor such as its load dependent velocity.
1.5 Our Point of View
As summarized above, the presently available models capable of explaining the mechanochem-
ical data, the power stroke models, imagine motors as miniature versions of macroscopic
engines working by means of levers, springs, etc. In this picture, ATP binding, hydrolysis,
or product release induces conformational changes in the protein that under load create
strain. The strain drives movement of any load attached to the motor, and this movement
which is referred as working stroke of the motor, relieves strain. For such a model, the
free energy difference between the pre- and post-work stroke states must be comparable to
motor work output. Though this is possible, there are no clear experimental confirmations
of this assertion. For the putative work stroke of kinesin, the conformational change of neck
linker, experiments show that the associated free energy change of nearly 8 pNnm [27] is
much less than the motor output which is greater than 60 pNnm. For myosin, the work
stroke is associated with lever arm rotation, but as of now there are no clear experimental
observations which show that there is a significant free energy change associated with this.
Similar is the case for the assigned work stroke of F1-ATPase, the closing of the β structure.
On the other hand, it is well recognized that physics of small particles in solution are
strongly affected by viscous drag and thermal noise which dominate the inertial forces which
14
determine system behavior of macroscopic systems. In ratchet systems (for a review, see
[28]), the thermal fluctuations are rectified in a preferred direction by maintaining the system
at nonequilibrium provided there is an asymmetry built in the system. For track binding
motors, it is assumed that the interaction between motor and track produces an asym-
metric potential. Application of such simple ratchets to the quantitative explanation of
mechanochemical data of motor proteins have been less successful till now. Velocity and en-
ergy conversion efficiency obtainable from simple ratchet models are much lower than that
of motor proteins. Nor are they able to explain the crucial role of the structural changes in
motor force generation.
In this thesis we will show that models without any explicit motor driving can explain
the mechanochemical data on motor proteins quantitatively with a suitable chemically tight-
bound ratchet model. The ingredients needed in the models are, (i) incorporation of bio-
chemical experimental data on motors, (ii) role of an elastic link that connects the motor to
load, (iii) reinterpretation of the role of conformational changes as modulating the attach-
ment and detachment of motor to the track in a preferred direction.
1.6 Road Map
In Chapter 2, a phenomenological scheme to incorporate the experimental data on motors
into modeling is proposed. The resulting model is used to explain the mechanochemical data
on kinesin motor. It is found that mechanochemical data alone cannot be used to determine
the unique motor force production mechanism. Indirect experimental observation of using a
large cargo elastically attached to the motor is identified as the possible explanation for the
inadequacy of mechanochemical data to identify the motor mechanism. In Chapter 3 the role
of the elasticity of the link connecting motor to cargo is studied using simple motor models.
It is shown that motor velocity and energetics are dependent on the link flexibility. Chapter
4 proposes a unified scheme incorporating motor biochemical data and motor-cargo link.
15
The resulting scheme is applied to explaining the mechanochemical data on motor proteins
kinesin, myosin V, and F1-ATPase. In Chapter 5 incorporating the effects of motor-cargo
link and its flexibility, a simple model (a null model) is proposed for the F1-ATPase motor
which can explain the available mechanochemical data. Chapter 6 summarizes the results
of this thesis.
16
Chapter 2
Phenomenology of Motor ProteinKinesin
2.1 Introduction
As introduced in the previous chapter, active transport of molecules and molecular com-
plexes are needed inside cells because of their sheer size. Motor proteins serve as the engines
for this intracellular transport. Kinesins are one such family of motor proteins that move
uni-directionally along the microtubule while hydrolyzing ATP. An interdisciplinary effort
involving different experimental fields and theoretical approaches are needed to have an un-
derstanding of the most interesting aspect of molecular motors: its mechanism of converting
chemical energy gained from the fuel molecules such as ATP into mechanical energy.
Experimental characterization of these motors can be divided into two general categories.
One set of experiments mainly seek qualitative characterization of the motors. Structural
studies involving crystallization, identifying structural changes during motor motion, bio-
chemical characterization of intermediate motor states and their correlation with ATP hy-
drolysis processes and qualitative descriptions of the way motor moves during its operation
fall in this category. The other set of experiments (mechanochemical experiments) measures
the motor output, its velocity, force production, and the efficiency of energy conversion.
2.1.1 Theoretical Approaches
A detailed understanding will involve describing the motor operation at a microscopic level in
terms of atomic motions. Molecular dynamics (MD) simulations involving motor structure,
17
interaction between motor and ATP and its hydrolysis intermediate, chemical details of ATP
hydrolysis process, and motor track interaction will give such a detailed description. Such an
approach has not yet been very successful since the time scale of molecular motor motions
is still much longer than the time scale routinely available by the current MD.
At the other end of description are the abstract models. Here, structural and biochemical
details of motor operation are neglected and motors are modeled as point particles. In
ratchet models [28], rectification of thermal fluctuations of motors by asymmetric potentials
produced by motor-track interaction is identified as responsible for motor motion and force
generation. In another class of models [26, 29, 30, 31, 32], a set of abstract chemical states
localized along different track positions are introduced. The motor motion is identified
with stochastic transitions between these states. An example is the modeling by Fisher
and Kolomeisky [26], which tries to explain the mechanochemical data on kinesin. This
presents an abstract kinetic scheme with four intermediate states with exponentially load
dependent rate constants. Although the scheme is able to fit the data with the aid of more
than fifteen fitting parameters, it is difficult to translate such a scheme into biochemically
relevant statements concerning the internal mechanism of the motor operation. Besides, it
is hard to incorporate structural or other biological information into the scheme.
2.1.2 General Considerations on Phenomenological Modeling
As described above, theoretical contributions to the understanding of motor proteins, other
than detailed molecular dynamics simulations, try to propose abstract models. Ability to
fit the output from the quantitative experiments are used as a measure of success of these
models. There are two chief obstacles in making a phenomenological description. One is that
the mechanochemical data on motors alone do not adequately constrain the space of possible
models as explicitly noted by Duke and Leibler [33]. The other difficulty is the number of
parameters: if there were many fitting parameters (say, more than 10), even if we could fit
the kinetic scheme to empirical results, it would be difficult to check whether the obtained
18
set of parameters is the best choice for a given kinetic scheme (let alone the verification of
the scheme itself). Therefore, we wish to minimize the number of adjustable parameters as
much as possible before trying to explain the empirical mechanochemical data quantitatively.
To this end, based on the intermediate states and substeps of motor stepping inferred from
structural and other experimental data, a possible kinetic scheme can be constructed first
(without referring to the availability of rate constants). Then the biochemical experimental
results can be used to determine the phenomenological rate constants appearing in the
scheme. The descriptive power of such a scheme can be tested by trying to explain the
mechanochemical data quantitatively. In essence, if successful, this will produce a kinetic
scheme which may be regarded as a phenomenological summary (or a minimal model, so to
speak) of biochemical and mechanochemical empirical results.
Therefore, it is desirable to construct good mesoscopic models of molecular motors that
can facilitate understanding of single molecule experiments (e.g., [34, 35, 36]) at the time
scale of microseconds or longer. We feel the first step in this direction is the incorporation of
qualitative information on motors into modeling. The general approach can be summarized
as follows. Identify the relevant qualitative information on motors. Propose a model which
incorporates the identified information, and the additional assumptions regarding the motor
force production, and check the agreement of model results with quantitative measurements
on motors. The methodology of phenomenological modeling of kinesin is developed in this
chapter. While formulating the phenomenological model it is assumed that all the empir-
ical data are really reliable. As we will see later, some structure-related empirical results
are not very reliable. This makes phenomenological models less effective in giving a clear
understanding of motor mechanism.
19
2.2 Empirical Facts
As already discussed in the introduction, our strategy is to make a kinetic description max-
imally utilizing currently available biochemical and structural information, and then to try
to reproduce the mechanochemical experimental results quantitatively with minimal fitting
parameters. Therefore, we first outline the experimental results relevant to the kinesin dimer
procession along the microtubule (Mt).
(i) Observation of kinesin motion using optical traps while it is transporting a cargo shows
that a kinesin dimer takes rapid 8 nm forward steps [35] spaced with (often long) waiting
periods. The periodicity of microtubule track is also 8 nm. There can be occasional backward
steps whose probability increases with increasing external load on the cargo [36, 37]. For
low loads each 8 nm step takes place with consumption of a single ATP molecule [38, 39].
The kinesin dimer processivity (i.e., the capability to take many consecutive steps on the
track before detaching from it) is due to the coordination of two monomer heads [40, 41].
Direct observation of stepping of each head by fluorescent tags shows that kinesin moves by
a hand-over-hand mechanism in which the two heads switches their relative position on the
track [42]. There are strong evidences indicating that the two consecutive steps are left-right
asymmetric (or without left-right symmetry) [43].
(i) is the basic observation and our kinetic scheme is based on: the coordination of two
heads.
(ii) The 8 nm step can be resolved into fast and slow substeps, each corresponding to a
(cargo) displacement of 4 nm. This was first demonstrated by Higuchi et al. [44] and has
been fully confirmed by Nishiyama et al. [36]:
a. The duration of the faster substeps is about 50 µs and is insensitive to the force (for 3-8
pN).
b. The second substep duration is variable (120 ∼ 300 µs for higher loads).
It is known that each head has four major states depending on the state of nucleotide
20
ATP or its hydrolysis intermediates that are attached to the kinesin head: E: without any
nucleotide, D: with an ADP, T: with an ATP, and DP: with a hydrolyzed ATP (before
releasing the phosphate ion Pi). We can approximately describe the state of a double-
headed kinesin as (A, B), where state A is the state of the trailing head (closer to the minus
end of Mt) and state B that of the advanced head. Also let us denote by the underline the
binding to Mt. Thus, e.g., (DP, D) implies that the trailing head is with a hydrolyzed ATP
and is attached to Mt, and the advanced head is with an ADP but not attached to Mt. We
use X for a not specified state (this does not mean that any state is allowed).
(iii) The reactivity of kinesin head with nucleotides may be summarized as follows:
a. if a kinesin head is free from Mt, releasing ATP or accepting ATP is not easy. This
exchange becomes much easier when the head is attached to Mt [40].
b. ATP → ADP + Pi (without releasing the phosphate ion) becomes difficult when the head
is attached to Mt; only when kinesin is a dimer is the release of phosphate ion possible [40].
c. Exchange of ADP is not strongly affected by the presence or absence of Mt. Under the
presence of ADP, (E, E) is not observed [45].
d. The T and E states have nearly the same binding strength to Mt [46, 45]. D is significantly
weaker than E or T [45]. DP is intermediate between T and D [46].
(iv) Transition from (X, E) to (X, T) triggers the force production.
a. ATP binds to the motor with a second order rate constant 2±0.8µM−1s−1 [47, 48] and
this is followed by the 4 nm (cargo) displacement, which is completed within 50 µs [36].
b. It is strongly suggested [41] that X here is actually DP. (As yet there is no direct
experimental demonstration, but many researchers assume the existence of such a state as
(DP, T) [25].
(v) Detachment of one of the heads occurs in (DP, X).
a. Hancock and Howard [41] have inferred that for a kinesin monomer the detachment occurs
in the DP state.
b. Hackney [40] suggests that phosphate ion dissociation precedes ADP dissociation. The
21
measured phosphate ion release rate from the monomeric kinesin is very small compared
with the case of dimers [41, 48]. Therefore, the phosphate release process must be assisted
by the other head.
c. Study of the kinetics of a mutant defective in ATP hydrolysis [49] suggests that the
hydrolysis of ATP in the rear head is required for the strong attachment of the front head
to Mt.
d. The step (forward or backward) is tightly coupled to the consumption of one ATP and
the waiting state for both seems the same. The stalling state is interpreted to be where the
probabilities of forward and backward steps are equal [36].
e. The measurement by Nishiyama et al. [36] and Carter and Cross [37] of the ratio of
forward to rear steps as a function of load shows that the number of backward steps increase
as a function of load.
(vi) a. The ATP binding leads to two sequential isomerizations, the second of which reorients
the neck linker relative to the Mt surface [50].
b. The neck linker docks, pointing in the forward direction, to the kinesin catalytic core in the
T or DP state, but is not docked in the E or D state; this transition has an enthalpy change
of ∆H ∼ −200 pNnm [51]. It is also known that the ATP hydrolysis is not a prerequisite
of neck linker docking [51]. This fact combined with a suggests that neck linker docking
precedes ATP hydrolysis [50].
c. For a related protein KIF1A the neck linker docks in the T state but not in the D state
[52].
d. Under the condition that both heads are attached to Mt, if the neck linker of the trailing
head is docked, it is sterically impossible for the front head to have a docked neck linker
[25, 53].
e. Tomishige and Vale [54, 55] have shown that when both heads are bound to Mt, the two
neck linkers (each with length 4 nm) must be in opposite-directing conformations, pointing
backward in the front head and forward in the rear. When the length of the neck linker is
22
decreased by two residues, processivity is lost. Inserting a 4 nm polyproline segment between
the neck linker and the coiled coil allows the motor to move processively with 16 nm steps
[54].
All these imply that (DP, T) assumed above must have transient substates (denoted with
1 and 2) with respect to the neck linker conformation.
(vii) Transition from (X, D) to (X, E) is the transition from the singly to doubly bound
conformation; there is a consensus about this process [41, 47, 56].
a. The Kawaguchi-Ishiwata experiments [57] have shown that with a head in the T state
and the other in the D state the motor is singly attached to Mt.
2.3 Proposed Kinetic Scheme
2.3.1 Kinetic Steps and States
Based on the summary in the preceding section, we propose the following kinetic scheme for
the kinesin dimer processivity (Fig. 2.1):
(DP, E)k1[ATP]
⇀↽k−1
(DP, T)k2⇀↽
k−2[P](D, T)
k3⇀↽
k−3(T/DP, D)1k4⇀↽
k−4(T/DP, D)2
k5⇀↽
k−5[ADP](DP, E).
Here, [ATP] denotes the ATP concentration, [ADP] the ADP concentration, and [P] the
phosphate ion concentration. Detailed justification will follow the outline of our scenario.
Our scenario may be outlined as follows. Supporting empirical facts summarized in the
previous section are specifically mentioned with key steps.
(1) After ATP goes into the front empty head attached to Mt, state (DP, T) is formed (cf.
(iva)). However, the neck linker does not dock to the motor core of the front head with ATP
immediately (cf. (vid)).
23
Figure 2.1: Kinetic scheme for kinesin dimer processivity. The nucleotide state of each headis denoted by, E: nucleotide free; T: with ATP; DP: hydrolyzed ATP, D: with ADP. Dockedand undocked portions of neck linker are denoted by thick and thin lines respectively.
(2) Next, release of a phosphate ion from the rear head (cf. (va, b)) and its detachment from
microtubule occur in a concerted way, and (D, T) is reached. The cooperativity needed for
accelerated phosphate release in the dimer (vb) comes from the attempts of the neck linker
of the front head in T state to dock.
(3) This allows the completion of docking of the neck linker of the front head to the core (cf.
(vid)) and the rear head moves a distance of 8 nm forward and is poised to move forward
further (perhaps slightly ahead of the previous front head that is still attached to Mt; the
cryo-electron microscopy studies of Hoenger et al. [58] suggests such a state). This is our
interpretation of the power stroke proposed by Vale et al. (cf. (vi)), and is correspondent to
the fast substep (iia). The still Mt-bound head (the previous front head) is in the T or DP
state with the docked neck linker (cf. (vib)) (likely to be in the DP state according to (iv)).
24
This is the state (T/DP, D)1.
(4) From this state the unattached head in the D state (the previous rear head) diffuses
forward a distance of another 8 nm to state (T/DP, D)2, which is just before reattaching
of the new front head to the new position on Mt. Due to the steric constraint, the neck (or
load) diffusion is required for the diffusing head to reach the new position on Mt. The still
attached now-rear head is in the T or DP state (with the docked neck linker; very likely to
be in DP [41, 49]). This corresponds to the second substep (iib). Strictly speaking, this is
our interpretation, but is the simplest interpretation of the facts summarized in (ii).
(5) The release of ADP and reattachment to Mt occurs to produce (DP, E), which corre-
sponds to the same state as the starting state but having advanced 8 nm. This is the waiting
state for ATP.
(6) The backward steps at higher loads is taken into account by assuming that once ATP goes
in (step (1)) the attempts of the front head linker to dock produce an internal tension and
either of the heads can detach from microtubule (vd). There is a higher probability for the
rear head to detach (ve). A possible kinetic scheme for backward steps is given in Discussion.
Once the rate constants are known, with the aid of the standard procedure summarized
in [59] the velocity, randomness, and their [ATP] and [ADP] as well as force dependences
can be determined.
The detailed explanation and justification of the states and processes follow.
(DP, E)
Supporting facts for this state to be the waiting state for a step have already been summarized
in (iv) and (va) in the preceding section.
The transition from this state to the next state (DP, T) involves the binding of ATP to
the front head with a second order rate constant k1.
(DP, T)
In this state the neck linker of the rear head is being docked but that of the front head in the
25
T state has not yet been docked (cf. (vid)). The binding of ATP is assumed to be contingent
to a displacement of some part(s) of the front head. This displacement may correspond to
the one observed in KIF1A due to Kikkawa et al. [52]. Another observation that may have
relevance to this displacement is that the kinesin head has an intrinsic bias towards forward
displacement independent of the neck linker [25]. If the monomer velocity v measured by
Inoue et al. [60] is interpreted as due to a biasing displacement of δ associated with ATP
binding followed by diffusion, then we have v = kδ, where k is the ATP turnover rate. Their
data are approximately compatible with δ =1 nm.
Observations by Higuchi et al. [44] have demonstrated that there is a waiting step after
ATP binds the front head and before the motor starts to produce force. An interpretation
is that the initial binding of ATP is a kind of collision complex, and it requires some dis-
placements of the parts of the head to come into the (DP, T) state which is the state that
allows the subsequent phosphate ion release from the rear head and its detachment from
microtubule (the change to (D, T) ).
This state is likely to be identified with the starting point of the neck linker docking
(see Discussion). From this state (D, T) is formed with a rate constant k2, and a reverse
transition to the initial state (DP, E) that releases the bound ATP [41, 47, 56] with a rate
constant k−1. Another process that can happen especially under high load is the detachment
of the front head in T state and kinesin taking a backward step (vd, e).
(D, T)
This state is obtained by the release of a phosphate ion from the rear DP state (cf. (vb)),
and detachment of the rear head from microtubule into the D state assisted by the front
head (cf. (va, b)).
The driving force for this change is interpreted to come from the initial stage of neck
linker docking to the front head core.
Further neck linker docking happening at the front head causes a forward transition to
(T/DP, D)1 state with a rate constant k3. Reattachment to microtubule and phosphate ion
26
rebinding produces (DP, T) state.
(T/DP, D)1
This state is obtained through the neck linker docking to the front head. This process
corresponds to the first substep mentioned in (iia) [25].
The hydrolysis of ATP may not have taken place as summarized in (v). The neck linker
of the T/DP state is docked and that of the D state is undocked (cf. (vib)). Gilbert et al.
and Ma and Taylor as well as Rice et al. [47, 56, 51] have shown that ATP hydrolysis is not
a prerequisite for moving one single step.
It is observed that the free energy change associated with neck linker docking is small
(∼ 1-2 kBT) [35].
Note that the enthalpy change for docking is very negative (△H ∼ −200 pNnm (vb)).
There is a forward transition from this state to (T/DP, D)2 by the forward movement of
the detached head (in the D state, the former rear head) in the D state to the next binding
site. Also there is a reverse transition to (D, T) (by the undocking of the neck linker).
(T/DP, D)2
The new front head which is now in the D state is near the next binding site but not yet
attached to it tightly. The rear head is in the T/DP state (likely to be in DP) with the neck
linker docked (cf. (vib)). It is possible that the front head in this state is attached weakly
to the next binding site.
The experimental evidence for such a state comes from all the chemical studies [47, 56]).
Ma and Taylor [56] propose that the D state on Mt can exist in two states, one weakly bound
and the other strongly bound. The observations by Sosa et al. [61] show that the D state is
weakly attached to Mt and is very flexible. Other observations [46, 57] suggest that the D
state is weakly attached to Mt.
There is a forward transition from this state to (DP, E) by the detachment of ADP from
the front head. Also there is a reverse transition to (T/DP, D)1 by the detachment of the
weakly bound front head.
27
Returning to (DP, E)
This state is recovered by the attachment (or weak to strong attachment) of the front D head
of (T/DP, D)2 to Mt and subsequent (or almost concerted) release of ADP. This release is
fast (100-300 s−1, which is sensitive to kinesin species) and completes the transition from
a weakly bound to strongly bound head [46, 57]. In this state the neck linker of the rear
head is docked and points forward, while the neck linker of the front head that is not docked
points rearward (cf. (vie)).
This is the waiting state for the next 8 nm movement, but can get back to state (T/DP,
D)2 (very likely to be (DP, D)2, that is, the nucleotide is hydrolyzed) by the attachment of
ADP to the nucleotide-free head E. Ma and Taylor [56] propose that this happens with a
second order rate constant ∼ 1.5-4 µM−1s−1. The observations of Yajima et al. [62] showing
that the velocity and processivity of the motor decreases with increased concentration of
ADP also points to the presence of this reversibility.
2.3.2 Modeling Details
The choice of the rate constants (see Table 2.1 for a summary) and the general mathematical
expression of the proposed kinetic scheme are summarized here with supporting arguments.
(DP, E) to (DP, T)
In the presence of an external force F (> 0 implies the force against procession), the second
order rate for ATP binding is assumed to be k1(F ) = k1[ATP] exp(−Fδ/kBT ), where kB
is the Boltzmann constant, T the absolute temperature (chosen to be a room temperature
300K), and [ATP] the ATP concentration. Here, δ describes the initial displacement associ-
ated with T binding.
For the reverse transition from state (DP, T) to state (DP, E) the rate constant is k−1.
A possible unified picture of the ATP binding and the nature of this substep are given in
Discussion.
There is a general consensus on the value of the second order rate constant for ATP bind-
28
ing, k1 [47, 56] in the absence of external force: k1 = 2±0.8 µM−1s−1 according to Gilbert
et al. [47]. The reverse transition rate in the absence of external force is: k−1 = 71 ± 9 s−1
according to Gilbert et al. [47]. We have chosen k1 = 2 µM−1s−1, k−1 = 71 s−1 and δ = 1.2
nm. As already discussed in the entry of (DP,T), δ ≃ 1 nm is consistent with the data by
Inoue et al. [60].
(DP, T) to (D, T)
For low loads this transition corresponds to the phosphate ion release and detachment of the
rear head from microtubule. We assume both these processes happen in a concerted way
with a net rate constant k2. Ma and Taylor [56] show that the maximum ATPase rate is
100s−1, which is k2k5/(k2 + k5) if k5 is the ADP release rate. The measured value of k5 is in
the range 100-150 s−1. We adopt k5 = 150 s−1, giving k2 = 300 s−1.
At higher loads as mentioned before there is a finite probability for the front head to
detach and a backward step to take place. The ratio of the probability of forward (Pf ) to
backward (Pr) steps may be written as
Pf
Pr
= e(GT−GDP−Fδd)/kBT (2.1)
to be consistent with detailed balance condition with exponential load dependence, where
GT is the free energy change due to the detachment of the head in T state, GDP that in DP
state and δd accounts for the force dependence of head detachment rate from the track. We
adopt GT − GDP ∼ 8kBT that is obtained by the thermodynamic consideration on phos-
phate ion release (details are given in Discussion). The assumption behind this is that the
rear head detachment which happens together with phosphate ion release is assisted at least
in part by the free energy change due to phosphate ion release. The displacement distance
δd ∼4 nm is chosen, because Pf/Pb = 1 when F is the stalling force (∼ 8 pN). Such a value
is compatible with the experimental observation of the force dependence of unbinding in the
29
T and E state by Uemura et al. [45] who get δd in the range of 3-4 nm.
The reverse transition rate which is proportional to the phosphate ion concentration [P]
is written as k−2[P]. There are no measurements of k−2 available. As explained below in
4.G, a study of phosphate ion dependence of the velocity can be used to estimate k−2 to be
∼ 0.1 µM−1s−1.
(D, T) to (T/DP, D)1
This transition corresponds to the fast substep of Nishiyama et al. [36]. It is shown that this
takes place within 50 µs (as observed by the displacement of a 0.2 µm diameter bead attached
to the neck coiled-coil) and is independent of the external force (for their observation range
3-8 pN). We interpret that the actual length of this step is 3 nm with the experimental
limitations of step resolution taking the 1 nm of ATP binding as part of a 4 nm step. Our
molecular picture of state (T/DP, D)1 is that once the rear head is detached, the neck-linker
docking proceeds immediately and this docking moves the neck coiled coil forward by 3 nm
and this places the previously trailing head slightly ahead of the previously advanced head
(see Fig. 1).
A possible explanation for the insensitivity of this step to the external force is that the
docking of the neck linker produces forces much larger than even the stalling force of the
motor which is in the range of 6-8 pN. The docking process is probably due to zipping of
the neck linker to the motor core. The very high △H ∼ −200 pNnm associated with this
conformational change could be a source of a very large force. Note that in this picture
the time taken will be inversely proportional to the friction constant, implying its direct
proportionality to radius of the bead. (The radius of the head ∼ 3 nm is much less than
the radius of the beads used in mechanochemical experiments which is of the order of 0.1
µm, so that the displacement of the head can be neglected as a rate determining process in
comparison to that of the bead.)
The rate constant k3 is taken as the inverse of the above mean reaction time. For the
30
explanation of the mechanochemical data of Visscher et al. [16] who use beads of diameter
0.5 µm, the rate constant k3 = (1/50 µs) × 0.2/0.5 = 8, 000s−1 (i.e., scaled according to the
diameter). As mentioned earlier the free energy decrease associated with neck linker docking
is −△G ∼1-2 kBT . Using this and the fact that ratio of the forward and the backward rate
constants satisfies k3/k−3 = exp[−(△G−F×d)/kBT ], with d = 3 nm being the displacement
against the load F , we can estimate the reverse rate constant k−3 for getting back to the
(D, T) state. We choose −△G ∼2 kBT (The results are not very sensitive to this choice as
long as it is in the range 1-2 kBT ).
(T/DP, D)1 to (T/DP, D)2
It is assumed that this step corresponds to the free diffusion of the previous rear head over
a distance of 8 nm, if no external load exists. Reflecting and absorbing boundary conditions
are chosen at the two ends to model the diffusion process. In presence of an external load
applied to the bead attached to the coiled-coil as in mechanochemical experiments, the rate
constant is taken as the inverse of the mean first passage time for this bead to move against
the load by a distance of 4 nm. Here, we assume that the link connecting the motor and
cargo is rigid. For the front head to bind to the forward binding site on Mt, the bead has
to diffuse forward by 4 nm. Once the bead reaches this position, the forward head would be
in a favorable position to attach to the binding site on Mt. With the displacement d =4 nm
we get the mean first passage time as [32],
d2
Db
[
eFd/kBT − (1 + Fd/kBT )]
(Fd/kBT )2, (2.2)
where Db is the diffusion constant of the bead to which results are sensitive at high loads, but
not very at smaller loads. For the explanation of the mechanochemical results of Visscher
et al. [16] we use Db = 8.79 × 10−13m2/s, estimated from the bead diameter of 0.5 µm, and
the viscosity of solution ∼0.01poise. k4(F ) is taken to be the reciprocal of the above mean
passage time.
31
The front head, which is attached to microtubule after the completion of the diffusion
process discussed above, can detach from Mt, making the reverse transition to (T/DP, D)1.
This process is assumed to have a rate constant, k−4 which is close to the detachment rate
from microtubule. From the results of Hancock and Howard [41], the detachment rate for
the head with ADP from microtubule is 1.01 s−1 ± 0.28 and we take k−4 to be 1.01 s−1.
(T/DP, D)2 to (DP, E)
For this transition, Gilbert et al. [47], who use Drosophila kinesin, give the rate k5 = 300
±100 s−1. Ma and Taylor, who use human kinesin, give the rate 100-150 s−1. Gilbert et al.
attribute this difference to the different types of kinesin used. To reproduce approximately
the measured velocity vs. force data of Visscher et al., who used squid kinesin, we have to
choose k5 ≃ 150 s−1 (See also the section on (DP, T) to (D, T) transition above). The speed
of kinesin procession depends strongly on this parameter, because ADP release is the factor
determining the maximum speed. This process is sensitive to the molecular structure as can
be exemplified by the fast fungal kinesin required for hypha elongation of Neurospora crassa
with a large nucleotide binding pocket [63]. Thus, k5 is a species-dependent parameter.
Therefore, we must treat this as an adjustable parameter. Notice, however, that the selected
value is within the known range and is a very natural one. For the reverse transition the
rate constant k−5 = 1.5-4.5 µM−1 s−1 according to Ma and Taylor [56]. We choose k−5 = 2
µM−1s−1. The model results are not sensitive to this choice of k−5.
Detachment from Mt
Hancock and Howard [41] measured the detachment rates of two-headed kinesin from mi-
crotubule at zero load in different nucleotide states. The measured rates are, 1.01±0.28 s−1,
1.67±0.50 s−1, 0.0009±0.0002 s−1 and 0.0010±0.0004 s−1 for states D, DP (from state D with
excess phosphate ions), E and T (with ATP and AMP-PNP), respectively. Experiments by
Coppin et al. [34] have shown that the detachment rate of kinesin from microtubule increases
with external load. In our scheme, the intermediate states involve states analogous to E (for
(DP, E)), T (for (DP, T) and (D,T)) and DP (for (T/DP, D)1 and (T/DP, D)2) states. We
32
choose the detachment rates for each of the above states at zero load as, 0.001 s−1, 0.001 s−1,
and 1.7 s−1, respectively. The run length is sensitive to the last kinetic parameter, so just
as in the case of k5, strictly speaking, we treat this as an adjustable parameter. However,
notice that the chosen value 1.7 s−1 is very close to the average value cited above.
To account for the force dependence of the detachment, each of these rates are multi-
plied by an exponential term e0.25F (F in pN) corresponding to an interaction range of 1 nm
between kinesin head and microtubule. The above rate constants and the exponential term
produces good agreement with the experimental data on processivity.
Note on the low-load data of Yajima et al.
Yajima et al. [62] uses kinesins fused to gelosin to determine the low load variation of
kinesin velocity and runlength with ATP, ADP and phosphate ion concentration. For the
explanation of this data the previous rate constants and mechanisms are modified slightly,
owing to the fact that they are using a different kinesin (rat kinesin) and there are no beads
attached to kinesin. The necessary modifications are: changing the ATP attachment rate
k1 to 4.5 µM−1 s−1, and detachment rate from Mt in the DP state to 3.2 s−1. The diffusion
step is taken to be the escape time for the head to travel 8 nm and the fast substep rate is
taken to be > 20,000 s−1. Note that at low loads neither the velocity nor the run length is
sensitive to these rates. The ADP reattachment rate is taken as 4.5 µM−1 s−1. The other
rate constants have the same value used in the previous case.
2.4 Explanation of the Mechanochemical Data
We will show below that the kinetic scheme constructed in the above through distilling the
known biochemical and structural information explains the mechanochemical experimental
results of kinesin dimer processivity quantitatively.
It should be emphasized that except for
(i) the choice of the species-dependent ADP release rate k5 = 150 s−1, and the DP state
33
Process Rate constant Reference
ATP binding (k1) 2 µM−1s−1 [47]ATP unbinding (k−1) 71 s−1 [47]Phosphate ion release (k2) 300 s−1∗ [56]“Neck linker docking” (k3) 20,000 s−1# [36]ADP unbinding (k5) 150 s−1 [56]ADP binding (k−5) 2 µM−1s−1 [56]Detachment from microtubulein state D (k−4) 1.01 s−1 [41]Detachment from Mt in state DP 1.7 s−1 [41]Detachment from Mt in state T 0.001 s−1 [41]Detachment from Mt in state E 0.001 s−1 [41]
* This is not a directly measured rate but one inferred from the measured maximum ATP turnover rate and
the measured ADP detachment rate. See the (DP, T) to (D, T) section below for details.
# This is when a bead of 0.2 µm diameter is attached to the neck coiled-coil.
Table 2.1: The adopted rate constants. Except for k4 and the detachment rate from Mtin the DP state, they are chosen from the literature (representative or average values arechosen). The rate constants k2, k5 and the detachment rate from Mt in the DP state areadjusted but are still within the error bars of the reported data; we say we have adjustedthese values simply because they are not the mean (center) values given in the literature.
detaching rate from Mt (1.7 s−1),
and
(ii) the following four qualitative information:
(a) The ATP binding rate is force-dependent (e−0.3F consistent with the displacement δ = 1.2
nm as discussed before, and F in pN),
(b) The neck linker docking is insensitive to load,
(c) The ratio of forward to reverse steps is of the form Pf/Pr = e8−F (F in pN) that has
also been discussed already,
(d) The detachment rate of kinesin from Mt increases with external load (assumed to be
e0.25F , F in pN),
all the parameters and the structure of the kinetic scheme are determined without fitting
to any of the available mechanochemical experimental results. Even the chosen values to
the adjustable parameters in (i) are within the empirically obtained ranges. The agreement
34
of our prediction and the available mechanochemical results is comparable to that of the
descriptions as Fisher and Kolomeisky [26] with sheer fitting parameters.
2.4.1 Force-velocity relation
0
200
400
600
800
1000
0 2 4 6 8
Ve
loci
ty (
nm
/s)
Force (pN)
[ATP]=2,000µM
[ATP]=5µM
Figure 2.2: The force-velocity relationship for ATP concentrations of 2,000 µM and 5 µM.The results due to our kinetic scheme and experimental data of Visscher et al., (Visscher etal., 1999) are given. If we define the stalling force as the force that reduces velocity below 5nm/s, it is of the order of 6 pN for [ATP] = 5 µM and 8 pN for [ATP] = 2,000 µM. Positiveforce increases the velocity only for intermediate and low levels of ATP concentrations. Theindependence of velocity for assisting force under high [ATP] is a prediction of our kineticscheme.
The variation of velocity with force at two different [ATP] is shown in Fig. 2.2. For
comparison, the experimental data of Visscher et al. [16] are also given. See the appendix for
an expression of velocity in terms of force dependent rate constants and ATP concentration.
Clearly, the velocity obeys the Michaelis-Menten type formula as a function of [ATP].
35
The velocity predicted by our kinetic scheme agrees with the experimental data of Viss-
cher et al. [16]; the velocity saturates for low loads and drops to low values at higher loads.
There is another published velocity-force relation by the Vale group [34]. Qualitatively, their
results exhibit this saturation effect and agree with our functional shape for higher [ATP]. At
high ATP and with intermediate to large forces, the diffusion step is the only rate limiting
and force dependent step, and this range of the curve is supposed to be explained by this
diffusion process.
At saturating ATP levels, the effect of forward (i.e., assisting) forces is negligible, whereas
at lower [ATP] the velocity increases with force. This is a feature not yet verified experi-
mentally, although there are certain indications in this regard in the experimental data of
Coppin et al. [34].
From the plots it is obvious that the stalling force (defined as the force for which velocity
falls below a certain threshold) of the motor increases with increasing [ATP]. If we choose
the threshold as 5 nm/s, the stalling force varies from ∼6 pN for [ATP] = 5 µM to ∼8 pN
for 2,000 µM. This is in agreement with the data presented by Visscher et al.
2.4.2 Velocity-ATP Concentration Relation
Figure 2.3 exhibits the variation of velocity with [ATP] for four different load levels. For
loads of 1.05 pN, 3.59 pN and 5.63 pN the predictions due to our kinetic scheme agree with
the experimental data of Visscher et al. [16]. For zero load case the experimental data of
Yajima et al. [62] is used for comparison and the agreement is good. The maximum velocity
is found to decrease as the load increases. The velocity at very low [ATP] is also found
to decrease with increasing force. This is indicative of the fact that the Michaelis-Menten
constant increases with force in agreement with the results of Visscher et al. [16].
A very low ATP concentration data can be found in Hua et al. [38]: for 5 nM of ATP
the rate is 0.09 ± .01 nm/s, and for 400 nM of ATP the rate is 5.3 ± 0.9 nm/s. Our model
gives 0.07 nm/s and 5.1 nm/s for respective concentrations.
36
1
10
100
1000
1 10 100 1000 10000
Ve
loci
ty (
nm
/s)
[ATP] (µM)
F=5.63 pNF=3.59 pNF=1.05 pN
F=0 pN
Figure 2.3: Velocity-[ATP] relation. Experimental data of Visscher et al., [16] are given forloads of 1.05 pN, 3.59 pN and 5.63 pN. Data for 0 pN are taken from Yajima et al. [62]. Theplot is of the Michaelis-Menten form with the Michaelis-Menten constant increasing withload.
2.4.3 Force-Randomness Relation
Randomness is defined by [16]
r = limt→∞
〈x2(t)〉 − 〈x(t)〉2d〈x(t)〉 =
2D
vd, (2.3)
where x(t) is the load position at time t, 〈 〉 is the sample average, D is the effective diffusion
constant of the motor, v its velocity, and d the step length. It is suggested that randomness
is a measure of the number of rate limiting steps affecting the motion of the motor. Fig.
2.4 shows the variation of randomness with force at a saturating (2,000 µM) [ATP]. It was
found that the there is a significant disagreement with the data of Visscher et al. at large
forces unless backward steps are taken into account.
37
0
0.5
1
1.5
0 2 4 6 8
Ra
nd
om
ne
ss
Force (pN)
[ATP]=2,000µM
Figure 2.4: Randomness vs force for [ATP] = 2,000 µM. Experimental data of Visscheret al. [16] for 2,000 µM is also given. For 2,000 µM and for low forces the value of r isapproximately 0.5 and the value approaches unity for high forces.
Our kinetic scheme has (i.e., the available biochemical information suggests) two rate
determining step candidates for low loads and saturating [ATP], the ADP release step and
phosphate ion release step. At saturating ATP levels, the ATP binding rate cannot be rate
limiting (4,000 s−1 for [ATP] = 2,000 µM and with low forces) and is much greater than
the ADP release rate (100-150 s−1) and phosphate ion release rate (>300 s−1). The fact
that these two rates are nearly of the same magnitude indicates constant randomness value
near 0.5 at low loads. At larger forces the diffusion step also becomes rate limiting and the
randomness approaches 1. The presence of backward steps at higher loads accounts for the
randomness values greater than 1.
38
0
0.5
1
1.5
2
1 10 100 1000 10000
Ra
nd
om
ne
ss
[ATP] (µM)
F=5.69 pNF=3.59 pNF=1.05 pN
Figure 2.5: Randomness with [ATP] for loads of 1.05 pN, 3.6 pN and 5.69 pN and theexperimental data taken from Visscher et al. [16]. See text for explanation.
2.4.4 Randomness-ATP Concentration Relation
Figure 2.5 exhibits the variation of randomness with ATP concentration. For intermediate
and high ATP concentrations the data agree with the experimental results of Visscher et al.
[16]. For low forces, at low ATP concentrations, irrespective of the load, the ATP binding
process is the sole rate limiting process and gives a randomness value should be close to
1. As the ATP concentration is increased, the ATP binding, and ADP and phosphate ion
detachment rates becomes rate limiting and the randomness approaches 0.33, and at higher
ATP concentration, ADP and phosphate ion detachment rates becomes rate limiting and
randomness approaches 0.5. The higher value of randomness at larger force is due to the
presence of substantial number of back steps.
39
2.4.5 Force-Run Length Relation
0
500
1000
1500
2000
0 2 4 6 8
Ru
n le
ng
th (
nm
)
Force (pN)
[ATP]=2,000µM[ATP]=5µM
Figure 2.6: Variation of run length with force for [ATP] = 2,000 µM and 5 µM and experi-mental data taken from Schnitzer et al. [35]. The result of our kinetic scheme for these two[ATP] values are almost indistinguishable.
Figure 2.6 exhibits the variation of run length with load for two different [ATP]. For
comparison, the experimental data of Schnitzer et al. [35] are also given. The predictions
due to our kinetic scheme agree with the experimental data in the intermediate and large
force ranges. There is some discrepancy at very low loads, but is likely due to the large
uncertainty in the experimental results (See the next paragraph).
The predicted run length does not strongly depend on the ATP concentration. At low
[ATP] the (DP, E) state dominates the mechanochemical cycle whose detachment rate is
very small. This smallness compensates for the low velocity associated with low [ATP]. The
above conclusion is in agreement with the recent experiments of Yajima et al. [62] which
show that for low forces the run length is independent of [ATP], and disagrees with [35].
40
2.4.6 Run Length-ATP Concentration Relation
0
200
400
600
800
1000
1200
1400
1 10 100 1000 10000
Ru
n le
ng
th (
nm
)
[ATP] (µM)
F=5.6 pNF=3.6 pNF=1.1 pN
F=0 pN
Figure 2.7: Run length vs [ATP] for loads of 0 pN, 1.1 pN, 3.6 pN and 5.6 pN and experi-mental data taken from Yajima et al. [62] for 0 pN and Schnitzer et al. [35] for loads of 1.1pN, 3.6 pN and 5.6 pN. The run length is found to decrease not significantly for low loadseven for very small levels of ATP concentration.
Figure 2.7 exhibits the variation of run length with [ATP] for four different loads. For
comparison, experimental data of Schnitzer et al. [35] for loads of 1.1 pN, 3.6 pN and 5.6
pN and Yajima et al. [62] for 0 load are also given(the fact that 1.1 pN data of Schnitzer et
al. has greater randomness value than the 0 pN data of Yajima et al. might be due to the
different kinesin used). It is found that the run length saturates with higher levels of [ATP]
and the saturated value decreases with increasing force. Although the general shape of this
curve is in agreement with Schnitzer et al. [35], there is one notable difference. Our kinetic
scheme shows that at low loads even at very low [ATP] the processivity does not decrease
significantly. This is in agreement with more recent results of Yajima et al. [62]. In our
41
kinetic scheme this happens because at low [ATP] the (DP, E) state is dominant from which
detachment rate is small. Since the velocity also is low in this case, the net processivity
(velocity/detachment rate) is not significantly different from the higher ATP level cases.
2.4.7 Velocity and Run Length Variation with ADP
0
200
400
600
800
1000
0 500 1000 1500 20000
250
500
Ve
loci
ty (
nm
/s)
Ru
n le
ng
th (
nm
)[ADP](µM)
Run length
Velocity
Figure 2.8: Velocity and run length vs [ADP] in the presence of 2,000 µM ATP with 0 pNforce, and experimental data taken from Yajima et al. [62]. Both the velocity and the runlength are found to decrease with added ADP.
Figure 2.8 exhibits the variation of velocity and run length of the motor with increasing
[ADP] in presence of saturating ATP levels and low force. The prediction of our kinetic
scheme is in agreement with Yajima et al. [62] as seen in the figure. Both the velocity and
the processivity are found to decrease with the increasing ADP level. This result can be
interpreted in the following way. The binding of ADP to kinesin in state (DP, E) slows down
the net ADP detachment rate from state (T/DP, D)2. As noted earlier the velocity is very
42
sensitive to this rate. For processivity a similar mechanism is also in operation. The binding
of ADP increases steady state probability of state (T/DP, D)2 from which detachment rate
is high compared to the low rate detaching from state (DP, E), resulting in the decrease in
processivity.
Yajima et al. also studied the variation of velocity and run length for two different
phosphate ion concentrations. If the phosphate ion concentration is increased, the velocity
is found to decrease slightly with no change in run length. Fitting the velocity data by
keeping all the rate constants fixed except k−2 gives the phosphate ion rebinding rate k−2 ∼
0.1 µM−1s−1. This rebinding slows down the net forward transition from (DP, T) state to
(D, T) state and explains the velocity decrease with increasing phosphate ion concentration.
The fact that the run length is not affected is explained by the fact that the rebinding of
phosphate ion does not alter the microtubule binding strength. (It takes kinesin from one
strongly bound state (D, T) to another strongly bound state (DP, T)).
2.5 Discussion
It has turned out that we can use the biochemical data for the ATP hydrolysis process and the
overall biochemical scheme is consistent with the one given by biochemical experiments [47].
Our scheme also incorporates structural and other physical information of motor operation
and is capable of explaining mechanochemical data, while at the same time being compatible
with biochemical data.
We have tried to distill a kinetic scheme from the available biochemical and structural
information, and have given a detailed account of the scenario we believe most likely. Our
basic kinetic scheme follows the general scheme for motors summarized by Bustamante et
al. [64]. We have assumed (as the most natural scheme compatible with almost all the
available biochemical and structural information) strict coordination of the two heads and
tight coupling of motor motion to ATP. The obtained scheme explains the mechanochemical
43
data of kinesin quantitatively. That is, the available biochemical information augmented
with certain details on neck linker docking has turned out to be sufficient to explain the
mechanochemical data quantitatively. The scheme also demonstrates the compatibility of
the experimental information about the parts of the motor, say, the neck linker described in
Rice et al. [51], with overall functioning of the motor.
2.5.1 Uniqueness of Kinetic Scheme
As mentioned before, the waiting step for forward and backward steps are the same, that
is the ATP binding. On ATP binding and the associated displacement of the front head, if
the rear head is detached, a forward step takes place. On the other hand if the front head
in ATP state is detached, a backward step is produced. An approximate scheme for such a
step is
( , DP, E) ⇀↽ ( , DP, T) ⇀↽ ( , DP, T) ⇀↽ (DP, DP/D, )
⇀↽ (DP, D/E, ) ⇀↽ (DP, E, ),
where the empty sites have been explicitly denoted.
Since there are 8 states T, T, DP, DP, D, D, E, E for each head, a possibility is that
we must consider the transition table among 64 states at least for a single kinesin dimer as
mechanically (or blindly) done in [65]. However, the sequence T → DP → D → E never
changes (reversing this reaction sequence is generally very hard), and with or without Mt
only a couple of transitions occur without difficulty. They are T → DP, DP → D, E → D
without Mt (however, note that T is hard to realize and so is DP). With Mt T → DP (vc),
D ↔ E (iiic), and E → T (iva) can occur. Furthermore, ease of detachment (iiid) tells us
that essentially only D → D is easy. Therefore, no parallel kinetic path other than the one
we proposed can contribute significantly.
44
2.5.2 Motor Energetics
The energetics of the kinesin dimer may be inferred as follows:
If we assume cellular conditions with [ATP] = 4,000 µM, [ADP] = 20 µM and the phosphate
ion concentration 2,000 µM, the free energy supplied by the ATP hydrolysis is about 25 kBT
(= 100 pNnm). Using the chemical kinetic constants for ATP hydrolysis process [47] and
the microtubule binding equilibrium constants [46], free energy changes involved in each of
the kinetic steps can be estimated. The energy transduction can be explained as follows.
(1) Major sources of free energy available are from binding of ATP (∼ 5 kBT ), phosphate
ion release (∼16 kBT ) and strong binding to microtubule (∼8 kBT ). (2) The ATP binding
energy is transfered to the load by means of partial neck linker docking. (3) Nearly half of
phosphate ion release energy is used to detach the rear head from microtubule and the other
half is dissipated. (4) Diffusing front head is captured by microtubule and the interaction is
stabilized by its strong binding. In this energetic picture the maximum available energy for
work is nearly half of ATP hydrolysis energy and this can explain the efficiency measured
in stall experiments which is near 50%. Since the detachment of the rear head is coupled
to phosphate release with an available free energy of ∼ 8 kBT , this justifies the assumption
that GT − GDP ∼ 8kBT .
2.5.3 Is There a Power Stroke in Kinesin Force Production?
For a motor displacement to be characterized as a power stroke, two conditions have to be
satisfied. There should be a motor conformational change associated with this displacement,
and the free energy change associated with this conformational change should be at least as
large as the maximum work produced by this displacement. Such a large free energy change
implies that the load dependence of the time taken to complete this step will be independent
or weakly dependent on load for low loads. Vale et al. [25] has proposed the neck linker
docking as producing the power stroke. Rice et al. [51] has shown that the free energy
45
change associated with neck linker docking is small (∼ −10 pNnm). But Nishiyama et al.
[36] has shown that the time taken to complete the first 4 nm step is independent of load
for small loads. These seemingly contradictory information can be reconciled as follows.
We have suggested that the initial ATP binding to state (DP, E) is a kind of collision
complex. The power stroke is, as suggested by Vale and coworkers, driven by the neck
linker docking. We know the enthalpy change for this docking is very negative (∆H ∼ −200
pNnm), but the Gibbs free energy change is small (∼ −10 pNnm). This implies that
the entropy change is also very negative, almost compensating the potential energy gain
due to docking. The most natural interpretation of this entropy decrease is by a precise
conformation required for the neck linker docking; relatively unconstrained conformation of
the linker must assume a ‘dockable’ arrangement.
The microscopic picture we have behind our scheme is as follows. The state (DP, T) is
actually this prerequisite state for docking (or for the zipping of the neck linker) or the initial
stage of zipping (nucleated stage). For zipping, some sort of nucleation is needed. This is
probably the initial zipping of a small portion of the neck linker close to the motor core.
This initial zipping produces sufficient conformation change to induce the phosphate ion
release and subsequent detachment of the rear head. Thus, the state (D, T) is formed. The
neck linker continues to dock to the motor and this docking moves the motor forward. This
is the subsequent fast substep to (T/DP, D)1. Thus, the force dependent barrier between
(DP, E) and (DP, T) is interpreted as the largely entropic barrier for the preparation of the
dockable state. The relaxation from the top of this barrier in the forward direction into the
deep energy valley (large negative ∆H = deep energy valley; notice that the valley looks
deep from the top of the barrier, but its absolute depth is small as given by ∆G) can explain
the force independence of step in the forward direction, despite the total free energy change
associated with this process being small.
Note that the above interpretation precludes the existence of a power stroke in kinesin
operation. For the fast forward displacement of 4 nm associated with neck linker docking,
46
the reverse step is also fast unlike in the power stroke picture. On completion of 4 nm
substep if motor takes a fast diffusive step forward, it is prevented from going backward due
to the head attachment to track. As shown below, such a picture can explain the energetics
of motor force production. The role of neck linker docking is interpreted as producing a bias
in the forward direction, not as a power stroke.
2.5.4 How Unique is the model?
At the beginning of the paper we quoted Duke and Leibler [33] asserting that the mechanochem-
ical data alone do not adequately select models. Thus, we have extensively used available
biochemical data. The scheme proposed by biochemical observations is unique. Modeling
of mechanical motion as due to 4 + 4 nm steps with the first one as a load independent
step produced by neck linker docking, the scheme that we obtained is unique and can ex-
plain all the available mechanochemical data. This uniqueness is contradictory to Duke and
Leibler [33] assertion. This is due to the fact that Nishiyama et al. [36] could probe the
mechanical stepping with higher resolution and give more information than that obtained
by mechanochemical measurements of motor velocity. Still, the fact that mechanochemical
experiments observe the motion of a large cargo attached to the motor elastically, not the
motor motion itself, makes such substep measurements unreliable. For example, there is a
recent observation [37] claiming that there are no substeps in the 8 nm step.
With slight modifications of kinetic constants it is found that the following schemes could
also explain mechanochemical data.
(1) We can assume that both the substeps in Nishiyama et al. [36] correspond to the rectified
diffusion of bead. In the first step, the bead moves against the load by a distance of 4 nm
and then the neck linker docks, preventing the backward motion of the bead. In the second
substep the bead moves over another 4 nm and the rebinding of the front head to microtubule
prevents its going back (as the model adopted in this paper).
(2) We may also assume that there is no substep associated with neck linker docking. The
47
bead moves against the load by 8 nm and is prevented from going back by the binding of
the front head to microtubule.
The inability of mechanochemical measurements to distinguish between the different pos-
sible schemes described above can be attributed to the nature of experimental observation.
In the single molecule mechanochemical experiments such as optical trap experiments, a large
cargo elastically attached to the motor is observed to infer the motor mechanism. This may
mean the unique mechanism of motor stepping and force production is not resolved, thus
allowing more than one scheme to fit the data. A first step in trying to get more information
out of single molecule experiments will be to understand the effects of the size differences
of the motor and cargo and the elasticity of the motor-cargo link on the mechanochemical
observations.
In the next chapter we study the effects of motor-cargo link on the motor velocity and
efficiency. This leads to the conclusion that since mechanochemical experiments cannot
observe any detailed motion of motor itself, a simple mechanism involving motor diffusion
combined with modulation of motor-track interaction depending on the nucleotide state of
the motor and possibly one or more structural changes of the motor is enough to explain the
mechanochemical data. If one wishes to understand the motor mechanism, one must devise
experiments to reject this null model.
48
Chapter 3
Effects of the Elastic Motor-CargoLink on Motor Transport
3.1 Introduction
As discussed at the end of Chapter 2, the attempts to develop motor phenomenology require
a careful study of the effects of motor-cargo link. Attempts to explain the mechanochemical
data on kinesin in the previous chapter showed that more than one motor schemes are
compatible with these data, virtually without fitting parameters, if we remove marginally
informative experimental results. We have seen that this marginality is closely connected to
the nature of the present mechanochemical experiments: they can observe only large probes
(cargoes) attached to the motor. If we take this limitation into account, it is hard to select
the mechanism uniquely. This may render attempts to use these data to understand the
motor mechanism less useful unless we have a clearer understanding of the limitations of
these experiments. In this chapter using simple motor models we will try to understand the
effects of the elasticity of the link on mechanochemical measurements such as motor velocity
and energetics.
In addition, such a study has relevance is understanding the cellular function of motors.
The primary function of motor proteins such as kinesins and myosin V is the intracellular
transport carrying cargo that are often much larger than themselves along particular tracks
[1]. The motor and cargo are connected by a flexible link (for example, the measured stiffness
of the link for kinesin is ∼0.1 pN/nm [66]).
The study of the effect of the motor-cargo link has direct relevance to interpreting
mechanochemical experimental data often used to study the motility and force generation
49
of motor proteins [16]. In optical trap experiments, as shown in Fig. 1.4 in Chapter 1, the
motion of an optically trapped bead elastically attached to the motor is analyzed to obtain
information about the motor motion. The motor can be made to move against a force ap-
plied to the bead and the kinetic features of the bead motion are interpreted in terms of
motor properties (e.g., [26]). In such studies, the effect of the link stiffness and the bead
diffusion constants are absorbed into effective kinetic rates for motor transition between
different states and are obtained by fitting to the experimental data. The spatiotemporal
resolution limits of the experimental setup mean to resolve the motion of the motor accu-
rately, a large bead has to be used in optical trap experiments. Another example is the
experimental studies of F1-ATPase motor [67] in which motion of a long fluorescent actin
filament elastically connected to the motor is observed to infer motor mechanism. In these
experiments, it is hoped that the observation of cargo motion can be used to infer the details
of motor mechanism such as the nature of force production, and efficiency. An example is
the dissipative experiments such as that of F1-ATPase, where the motor output to the cargo
is dissipated. It is found that to have a high motor efficiency (the Stokes efficiency of [24]
defined as the ratio of the cargo dissipation to motor input energy) the force experienced
by the cargo filament should be a constant. Thus, it is suggested that the motor itself pro-
duces a constant force output. As we show below, the velocity of the motor and thus the
Stokes efficiency are dependent on the link stiffness, so such a conclusion about the motor
mechanism itself is likely to be misleading (at best).
A previous study in this direction [68, 69] considered the variation of motor velocity with
the stiffness of the link when the motor is modeled as a ratchet. Detailed analytic results
for motor velocity were given for the asymptotic regions of large and small link stiffness and
cargo diffusion constant. Here we will consider the effect of the link on the efficiency of
the motor-cargo system and the limitations caused by such a setup in inferring the motor
mechanism.
50
3.2 Motor-Cargo System
Since we are interested in having a physical understanding of the effect of motor-cargo link
on motor operation, we will consider simpler physical representations of motor mechanism
rather than detailed descriptions as done in the previous chapter. There can be two types of
simple motor representations depending on the coupling between energy consumption and
mechanical motion. In tight coupling models, consumption of one unit of energy (hydrolysis
of one ATP molecule in motor protein case) produces one mechanical step. The other simple
representation of motor mechanisms is in terms of loosely coupled thermal ratchets. Using
such a loosely coupled ratchet models is at variance with the reality of actual motors, but as
an illustrative system capturing physics of the directional motion of a particle in a thermal
environment, the model is useful to the understanding of real motors. There are many
possible physical realizations of the ratchet [28]. We adopt the simplest scheme proposed
in [70, 71]. The motor is modeled as a strongly damped particle moving in a (spatially
asymmetric) ratchet potential that fluctuates between two different states 1 and 2 with
potential Φ1 and Φ2, respectively. Fig. 3.1 shows the motor-cargo system moving in such a
potential. The motor-cargo link is modeled by a harmonic spring.
C MKs
γ1
a
u
γ2
Φ1
Φ2
Figure 3.1: The motor-cargo system modeled as a fluctuating ratchet with potentials Φ1 andΦ2. u is the potential maximum height, a < ℓ/2 denotes the potential asymmetry. γ1 andγ2 are the transition rates of the potential between the two states. Motor (M) and Cargo(C) are connected by a harmonic link of stiffness Ks.
51
The system can be described by a set of Fokker-Planck equations of the form:
∂P1
∂s= −∂J1,m
∂X− ∂J1,c
∂Y− Γ1(X)P1 + Γ2(X)P2, (3.1)
∂P2
∂s= −∂J2,m
∂X− ∂J2,c
∂Y− Γ2(X)P2 + Γ1(X)P1, (3.2)
where X is the motor coordinate, Y the cargo coordinate, Pi(X,Y, s) the probability density
for state i ∈ {1, 2} at time s, Γi(X) the transition rate or fluctuation rate of the potentials
when the motor is at X. Here, the motor and the cargo fluxes in state i, Ji,m(X,Y, s) and
Ji,c(X,Y, s), respectively, are given by
Ji,m = −Dm
(
∂Pi
∂X+
1
kBT
∂Vi
∂XPi
)
, (3.3)
Ji,c = −D′
c
(
∂Pi
∂Y+
1
kBT
∂Vi
∂YPi
)
. (3.4)
In these formulas, kB is the Boltzmann constant, T is the temperature, Dm is the motor
diffusion constant, D′
c is the cargo diffusion constant, and the potential Vi is given by
Vi(X,Y ) = Φi(X) +1
2Ks(X − Y )2 − FY, (3.5)
where Ks is the stiffness of the link, and F is the external force acting on the cargo.
At steady state,
−∂J1,m
∂X− ∂J1,c
∂Y− Γ1(X)P1 + Γ2(X)P2 = 0, (3.6)
−∂J2,m
∂X− ∂J2,c
∂Y− Γ2(X)P2 + Γ1(X)P1 = 0. (3.7)
In a steady state, the boundary conditions to be satisfied are continuous periodic boundary
conditions in the X-direction for probability densities and fluxes: Pi(X+ℓ, Y +ℓ) = Pi(X,Y ),
and Ji,r(X+ℓ, Y +ℓ) = Ji,r(X,Y ) for r = m or c. Here, the spatial periodicity of the potential
52
is ℓ. In the Y -direction, the probability densities are assumed to vanish at infinity. To have
a numerical solution, a simple finite difference scheme was found to be sufficiently accurate
to obtain the steady state.
From now on, we will work with the following dimensionless quantities: x = X/ℓ, y =
Y/ℓ, v = V/kBT , φi = Φi/kBT , Dc = D′
c/Dm, γi = ΓiDm/ℓ2, t = sℓ2/Dm, ks = Ksℓ2/kBT
and f = Fℓ/kBT .
3.3 Variation of Motor Velocity with Link Stiffness
Before studying the efficiency of the motor we will study the general behavior of the motor-
cargo system. Fig. 3.2 exhibits the variation of the motor velocity with the link stiffness
obtained by numerically solving the Fokker-Planck equations (3.6) and (3.7). We see that
the variation is non-monotonic and for moderate values of the potential transition rates
(here, γ1 = γ2 = γ as a representative case) there is an optimal stiffness of the link which
maximizes the motor velocity.
The numerical results can be understood intuitively with the aid of the case with u ≫ 1
that can be analytically studied. Assume that the motor is in state 2 at t = 0. The
probability distribution in state 2 is localized at the potential minima because u ≫ 1. Once
the motor makes a transition from state 2 to 1, the motor starts diffusing freely. Let p(x, y, t)
be the joint motor-cargo probability in state 1 at time t with the initial motor position at
x = 0. The probability of the motor being captured around x = 1 when the potential
switches back to 2 is given by
Pf =∫
∞
adx
∫
∞
−∞
dy∫
∞
0dt p(x, y, t)R(t), (3.8)
53
0
0.075
0.15
0.225
0.75 1.25 1.75 2.25
Vel
ocity
log10(ks)
γ=1.6γ=160
Figure 3.2: Variation of motor velocity of the motor with the stiffness of the link for threedifferent transition rates. The potential parameters are, u = 20, a = 0.3, Dc = 0.1, andexternal load f = 0.
and the probability of being captured around x = −1 is
Pb =∫
−(1−a)
−∞
dx∫
∞
−∞
dy∫
∞
0dt p(x, y, t)R(t), (3.9)
where R(t) is the waiting time distribution for the state to return from 1 to 2 and may be
modeled by a Poisson process: R(t) = γe−γt.
The time evolution of p(x, y, t) is given by the coupled diffusion equation,
∂p
∂t=
∂
∂x
(
∂p
∂x+
∂v0
∂xp
)
+ Dc∂
∂y
(
∂p
∂y+
∂v0
∂yp
)
, (3.10)
with v0(x, y) = (1/2)ks(x−y)2. To estimate the velocity, we need a time dependent solution
to the coupled diffusion equation above. The coordinate transformation z = (Dcx+y)/(Dc+
54
1) and r = x − y allows the separation of these equations as,
∂Pz(z, t)
∂t= De
∂2Pz(z, t)
∂x2, (3.11)
and
∂Pr(r, t)
∂t= Dt
∂
∂r
(
∂Pr(r, t)
∂r+
∂v0(r)
∂rPr(r, t)
)
. (3.12)
where De = Dc/(1 + Dc) and Dt = 1 + Dc. Using the time dependent solutions for the dif-
fusion equation, and the Ornstein-Uhlenbeck process, we get the solution for the probability
distribution as
p(x, y, t) =∫
∞
−∞
dx0
∫
∞
−∞
dy0 p(x, y, t|x0, y0, t0) p(x0, y0, t0), (3.13)
where
p(x, y, t|x0, y0, t0) = C1exp
[
−(
(Dct(x − x0) + Dmt(y − y0))2
α+
((y − x) − δ(y0 − x0))2
β
)]
,
(3.14)
with S(t, t0) = 1 − exp (−4(t − t0)/τ), τ = 2/ksDt, α = 4De(t − t0), β = 2S(t, t0)/ks,
δ = exp(−2(t − t0)/τ), Dct = Dc/Dt, Dmt = 1/Dt and C1 is a normalization constant. The
initial distribution p(x0, y0, t0) for sufficiently high barrier in state 1 can be approximated as
p(x0, y0, t0) = C2δ(x0 − xi) exp(
−ks(y0 − x0)2/2
)
, (3.15)
where C2 is a normalization constant and δ(x0 − xi) is the Dirac delta function.
For sufficiently large γ, R(t) can be approximated as a Dirac delta function, ∼ δ(t−1/γ).
Thus, the velocity of the motor can be approximated as v ∼ (γ/2)(Pf − Pb). Using the
analytically obtained p(x, y, t), we obtain
v ∼ γ
4[Erfc(
√γ1a) − Erfc(
√γ1(l − a))] , (3.16)
55
where Erfc is the complementary error function and γ1 = 1/[α + (1/Dt)2(β + (δ − 1)2ǫ)]
with α = 4De/γ, β = 2(1 − exp (−4/γτ)), τ = 2/ksDt, δ = exp(−2/γτ), ǫ = 2/ks, De =
Dc/(1 + Dc) and Dt = 1 + Dc. Figure 3.3 illustrates the velocity dependence on the link
stiffness given by (3.16). It is qualitatively similar to the behavior in Fig. 3.2; due to a
stiffer potential the main features are exaggerated in Fig. 3.3.
0.12
0.14
0.16
0.18
0.2
-1 0 1 2 3 4 0
0.2
0.4
0.6
0.8
1
Vel
ocity
(γ=
2)
Vel
ocity
(γ=
10)
log10(ks)
γ=2γ=10
Figure 3.3: Variation of the velocity of the motor determined in the analytically solvablelimit with the link stiffness ks for transition rates γ = 2 and 10. The cargo diffusion constantDc = 0.1. The potential asymmetry a = 0.3.
The various regimes in Fig. 3.3 can be understood as follows.
∗ Small transition rate and small stiffness: Here, the waiting time for motor in state 1 is
long (∼ 1/γ). The motor initially localized at x = 0 has enough time to spread. Because
wide spreading is facilitated by the small stiffness of the link, when the state changes from
1 to 2 there is nearly the same probability of motor being captured at the forward (x = 1)
and backward sites (x = −1). Thus, the motor velocity is small.
∗ Small transition rate and intermediate stiffness: As the link stiffness is increased, the width
of the motor probability distribution in state 1 decreases. This and the potential asymmetry
of state 2 cause the motor to be captured with much larger probability by the forward site
56
than the backward site when transition to state 2 is made. Consequently the motor velocity
increases.
∗ Small transition rate and large stiffness: The motor distribution in state 1 remains sharply
peaked even after the long diffusion time. This decreases both the probabilities of the motor
being captured by the forward and backward sites when transition to state 2 occurs. Thus,
the motor velocity decreases.
∗ Large transition rate and small stiffness: Although the waiting time for motor in state 1
is not long, still the motor in state 1 can spread enough thanks to the weak link to feel the
asymmetry of the potential in state 2 when transition to state 2 occurs. Thus the motor can
efficiently proceed.
∗ Large transition rate and intermediate and large stiffness: the short diffusion time and
larger stiffness combine to produce a sharply peaked motor probability in state 1 and de-
creases the forward capture probability when state changes to 2. Motor remains at the same
position for most of the state changes. This causes the velocity to decrease.
For the ratchet model, the cargo effect on motor motion can be interpreted as modifying
the effective forward or backward transition probabilities of the motor. For small stiffness,
the relative probabilities of forward and backward motor steps are not much affected by the
link to the cargo, so the motor velocity is not significantly affected by the cargo irrespective
of its size. Increasing the link stiffness reduces the motor fluctuations and localizes the
motor position especially if the cargo is large. This can be beneficial in ratchet models for
which energy input is dependent on the motor position, because efficient supply of energy
to a localized motor may realize. This can, however, be disadvantageous if the motor needs
spatial exploration by thermal fluctuation to obtain energy supply.
As seen in Chapter 2 this happens for tightly coupled models of motors for which energy
supply is localized along certain track positions. Efficient diffusion of the motor allows it to
reach the supply point quickly, but the stiff link with a large cargo hinders this motion. We
consider three such scenarios shown in Fig. 3.4: in Case (1) energy is supplied continuously
57
along the motor coordinate. Cases (2) and (3) require diffusive search by the motor for the
completion of a step. Fig. 3.5 shows the variation of velocity with the link stiffness obtained
by solving the associated Fokker-Planck equations numerically.
In Case (1), since there is continuous dragging of the motor to the forward direction,
motor velocity is independent of the link stiffness and is given by v = △µ/(1+Dc). In Cases
(2) and (3), however, the increasing stiffness of the link decreases the motor velocity. This
decrease is understandable if the localizing of the motor due to the increasing stiffness of the
link is taken into account. For low stiffness, the motor starting at the left is able to diffuse
easily and reach the other end. Increased link stiffness makes this process difficult and slows
down the motor. A similar scenario must be operational in the cellular conditions in which
motor carries cargo which is usually much larger than themselves. The motor-cargo link
must be sufficiently flexible for the transport to be sufficiently fast. Similarly, the velocity
measured in optical trap experiments will also be dependent on the size of the bead used
and the trap stiffness.
C
12
3∆µ
¼l½l l
M
Figure 3.4: △µ is the free energy which is released to the motor during one step of its motion.In (1) the energy is gradually released along the motor coordinate. (2) and (3) involve freediffusing regions within the step.
58
0.2
0.45
0.7
0.95
0.75 1.25 1.75 2.25 2.75
Ve
loci
ty
log10(ks)
a=1a=0.5
a=0.25
Figure 3.5: Variation of the velocity of the motor with the stiffness of the link for threedifferent ways of free energy release within a step. The potential parameters are, △µ = 9,Dc = 0.1, and external load f = 0.0.
3.4 Stokes Efficiency of Motor-Cargo System
Molecular motors often drag cargoes that are much larger than themselves. In the absence
of a measurable external force applied to the motor it is difficult to give a thermodynamic
definition of motor work output and thus to define its efficiency. This is also the case
in the common experimental setup used in the study of motor in which a large element
(cargo) is attached elastically to the motor and its observed motion is used to infer the
motor mechanism. An example is the study of the F1-ATPase motors in which a large
actin filament attached to the motor is observed [67]. In these experiments the measurable
quantities are the rate at which energy is supplied to the motor and the velocity of the
attached element. Using these two observables, Oster and coworkers [24] proposed a new
definition of motor efficiency called the Stokes efficiency defined as the ratio of the rate of
mechanical dissipation of the energy at the cargo 〈Qc〉 and the motor energy input rate
〈Rm〉 (Here, all the quantities are dimensionless. Energy rates are in units of kBTD′
c/ℓ2 and
59
velocity is in units of D′
c/ℓ).
ηStokes =〈Qc〉〈Rm〉
, (3.17)
where
〈Qc〉 = ζc〈v〉2 (3.18)
with ζc = 1/Dc being the cargo friction constant and 〈v〉 its average velocity.
Oster et al. has shown that the Stokes efficiency can give information about the nature of
motor energy output. For example, a high Stokes efficiency implies a constant motor energy
output without any spatial irregularities in the potential. However, the conclusion is true
only when one neglects the link between the motor and the cargo.
To calculate the Stokes efficiency we need the energy supply rate. Each change of po-
tential from φ1 to φ2 supplies energy φ2 − φ1 ≡ v2 − v1. Therefore, the mean rate of energy
input to the motor is
〈Rm〉 =∫ 1
0dx∫
∞
−∞
dy(v2(x, y) − v1(x, y)) · (γ1P1(x, y) − γ2P2(x, y)). (3.19)
Figure 3.6 shows that the Stokes efficiency for the thermal ratchet described in Section
3.2 is dependent on the stiffness of the link and there is an optimal stiffness which maximizes
the Stokes efficiency for γ = 1.6. The explanation for this peak and variation is the same
as that for the velocity in the preceding section. The effective motor transition rates to the
forward or backward site is dependent on the potential transition rate and the link stiffness.
A much more serious effect of the link stiffness can be seen in the case of the tightly
coupled model discussed at the end of Section 3.3. In this case the Stokes efficiency is defined
as v2/Dcr△µ, where r is the rate of motor energy input and △µ the free energy input per
step. For unit step length, r = v and hence the Stokes efficiency is given as v/Dc△µ. Figure
3.7 shows the variation of the Stokes efficiency with the link stiffness corresponding to the
three cases in Fig. 3.4. We see that the softer link increases the Stokes efficiency for a given
60
0
0.002
0.004
0.006
0.008
0.01
0.75 1.25 1.75 2.25
Sto
kes
Eff
icie
ncy
log10(ks)
γ=1.6γ=160
Figure 3.6: Variation of the Stokes efficiency of the motor for the ratchet model with thestiffness of the link for two different transition rates. The potential parameters are, u = 20,a = 0.3, Dc = 0.1, and external load f = 0.
energy release pattern within a step. This dependence on the stiffness of the link imposes a
serious limitation on the quantitative theoretical argument based on the Stokes efficiency on
the motor mechanism. For example, suppose the measured Stokes efficiency is around 0.45.
The theoretical argument in [24] disregarding the link effect would conclude that a = 1/2
(under the assumption that the potential has only one sloped portion). Actually, however,
any value of a between 1/2 and 1/4 is possible dependent on the stiffness. If we use realistic
(i.e., rather soft) link, a ≃ 1/4 is likely.
If we use stochastic energetics formalism [72], a more detailed energetics study can be
performed. For completeness sake, some results are summarized in Appendix B.
3.5 Discussion
We have demonstrated that motor velocity, and efficiency depend on the motor-cargo link
and cargo diffusion constant significantly. As shown explicitly, quantities such as the Stokes
61
0
0.2
0.4
0.6
0.8
1
0.75 1.25 1.75 2.25 2.75
Sto
kes
Eff
icie
ncy
log10(ks)
a=1a=0.5
a=0.25
Figure 3.7: Variation of the Stokes efficiency of the tight coupled motor with the stiffness ofthe link for three different ways of free energy release within a step. The potential parametersare, △µ=9, Dc =.1, and external load f = 0.0
efficiency depend on the motor-cargo link, so the inference one draws from Stokes efficiency
alone about the nature of motor force production may be wrong if the effect of cargo size and
link stiffness are not modeled explicitly. This indicates that discussing the motor mechanism
based on the mechanochemical experiments requires caution. Incidentally, we found that for
a simple loose coupling ratchet model of the motor there is an optimal stiffness of the link
which maximizes the velocity and the efficiency of the motor. For a tightly coupled model of
the motor, both the velocity and efficiency increase as the stiffness of the link is decreased.
As suggested by the above observations, the nature of information one gets out of optical
trap experiments on motor proteins are dependent on the experimental setup. This combined
with the observation of Chapter 2 that these experiments are not good at resolving the
mechanism of motor force production suggests that there might be a unified interpretation
of these experiments which can be obtained without modeling motor mechanism. In the
next chapter, we attempt to develop such a unified interpretation of these single molecule
experiments.
62
Chapter 4
Interpretation of Single MoleculeExperiments of Motor Proteins
4.1 Introduction
Single molecule experiments are currently one of the most powerful methods for the physical
understanding of motor proteins [1, 16, 73]. In one class of experiments motion of a cargo
(an optically trapped bead much larger than the motor) elastically attached to a motor is
analyzed to obtain information about the kinetic and energetic aspects of the motor motion
[16, 74]. A variation of this setup observes an attached cargo (a fluorescent element) in a
viscous medium [75, 76]. Because the observable is the mechanical motion of the cargo, as
shown in the previous chapter, modeling of the motor-cargo system is required to infer the
motor mechanism and energetics from the observed results. Such models may be formulated
by assuming particular motor mechanisms. If the model results are in agreement with
experimentally measured quantities such as motor velocity, efficiency, etc. [26, 29, 31, 24],
these assumed mechanisms are taken as representing the actual motor mechanism.
As shown in Chapter 3, motor properties such as velocity and efficiency are dependent
on the size of the cargo and the stiffness of the motor-cargo link. Thus, the results of models
which neglect the effect of motor-cargo link (for e. g. [32, 26]) should be treated with caution
even if they show agreement with experimental measurements such as motor velocity. In
this chapter we will show that if we take into account the time scale distinction between the
motor and the cargo fluctuations, then:
(1) we can devise a unified description of single molecule experiments of various molecular
motors;
63
MM M
Cargo Cargo Cargo
Figure 4.1: A large cargo is connected to motor M through a long elastic link. The dashedvertical line shows the starting position of the cargo.
(2) no information about the details of the driving force generated by the motor is needed
for quantitative understanding of single molecule experiments so far obtained.
4.2 Motor-Cargo System
In the system used in the single molecule studies as shown in Fig. 4.1, a large cargo is
connected elastically to the motor which moves along a track with evenly spaced binding
sites. The motor moves along a linear track with evenly spaced binding sites. A typical
motor step involves a chemical process (e.g., ATP binding) at one binding site, unbinding
of the motor from the site, its diffusive motion biased in the forward direction by a driving
potential or by some other mechanism, and rebinding at the next binding site which is
stabilized by another chemical process (e.g., the release of products of ATP hydrolysis).
Forward motion of the motor stores energy in the elastic link connecting it to the cargo and
it is the tensile force in the link that drives the cargo forward.
A single molecule mechanochemical experiment typically monitors the cargo stepping
with nanometer resolution when a load is applied to it using an optical trap or by the
viscosity of the surrounding medium. In Chapter 3, we modeled the elastic link as a har-
64
monic spring studied the cargo response when the motor potential is modeled as a fluctu-
ating ratchet. Such a description cannot be used for a quantitative understanding of single
molecule experiments, because it assumes that chemical reaction time scales are much faster
than the mechanical motion time scales.
In the typical single molecule experimental setup the dimensions of the cargo is ∼1 µm
and that of the motor ∼1 nm. Thus, the Einstein relation tells us that the ratio of the
diffusion time scales (which are proportional to the dimension) of the motor and the cargo
is about 103. This implies that the fluctuation time scales which are orders of magnitude
different.
While motor jumps between one track binding position to next, it is working against the
strain in the link connecting it to the cargo and storing energy in the link by stretching it.
The time taken for this motor diffusive motion against the strain in the link can be estimated
as [1] τ = τ0 exp(E)/Eα, where E is the required energy storage (in units of kBT ), τ0 is the
diffusive time scale of 0.1 µs, and α = 1.5 (if energy is stored in a harmonic spring) or 2
(if stored in a linear potential). The energy stored in the link can be estimated using the
maximum force generated by the motor (also called the stalling force). For kinesin, using
the known stalling force which is in the range of 7 pN [16], we estimate τ < 1 ms, which is
well within the order of slowest chemical step completion time and is compatible with the
observed cargo velocities (this implies motor diffusion time scales does not affect the cargo
velocity). For the F1-ATPase, the known stalling force gives E ≃ 80 pNnm [73]. Here, using
the experimental information [78] that single step of 120 degree consists of substeps of 80
and 40 degrees, the diffusive time scale again is less than 1 ms.
65
4.2.1 An Effective Cargo Only Representation of a Coupled
Motor-Cargo System
As shown in Chapter 2, to ensure tight coupling (consumption of one ATP molecule to pro-
duce one step) a particular reaction in ATP hydrolysis cycle occurs only when the motor
position is within a small range around a particular locations on the track (modulo peri-
odicity) [1]. When the motor reaches a reaction position, the cargo is still lingering in the
position corresponding to the previous reaction position of the motor, because due to its
much smaller size the motor moves much faster than the cargo. This stretches the elastic
link connecting the motor and cargo. While the motor is waiting for the next chemical
reaction to occur, the strain in the link pulls the cargo forward. Before the motor moves to
the next reaction position, cargo reaches its equilibrium position corresponding to the motor
position for the just completed reaction. Therefore, the occurrence of various chemical reac-
tions associated with nucleotide states/motor-track binding states and the cargo positions
are roughly one-to-one correspondent. We may use these cargo locations as the ‘checking
positions’ for reactions. (These checking positions depend on external forces, etc., but we
need only relative distances between successive checking positions which are expected to
be independent of external forces, etc.). This suggests that the kinetic constants obtained
from the biochemical observations of the motor alone can be used in the explanation of the
experiments. Between such ‘checking positions’ the cargo is pulled by the motor. The time
scales of the motor and the cargo are very different and the slowest time scale is introduced
by the chemical reaction time scale (essentially motor waiting time along fixed track posi-
tions). This allows a simplified representation of the coupled motor-cargo motion as that
of the cargo alone in an effective potential exerted on it by the strain in the motor-cargo
link. For reasonably elastic links this potential can be approximated as a linear potential
U = fy made by the cargo-motor link (y > 0 is the usual moving direction), where f (< 0)
is the average tensile force in the link, irrespective of the details of the potential acting on
66
the motor [68].
As is seen below the cargo only model with a constant force may be justified with a
singular perturbative reduction of the motor-cargo full model augmented by a further sim-
plifying approximation that can be checked numerically. Let the cargo position be y and the
motor position x. Since the time scale of x is much faster, the coupled stochastic differential
equations has the following form with a small parameter ǫ > 0:
dx = [−A(x − y) + F (x)]dt +√
2AdB,
dy = −ǫA(y − x)dt +√
2ǫAdB′, (4.1)
where A > 0 corresponds to the stiffness of the link connecting motor and cargo, F (x) is the
potential experienced by the motor during its stepping due to structural changes introduced
by ATP hydrolysis processes or due to motor track interactions, B and B′ are independent
Wiener processes and units are chosen such that kBT = 1. If as in our case F allows a stable
fixed point corresponding to the next motor position, the equation for y reads, to the lowest
nontrivial order,
dy = −ǫA(y − 〈x〉y)dt +√
2ǫAdB′, (4.2)
where 〈x〉y is the average position of x when the cargo position is y. Since F exerted on the
trapped motor is strong, 〈x〉y is almost y-independent and is the next reaction position. Up
to this point the approximation is a systematic expansion with respect to ǫ (we could employ
the standard averaging method). In our approach, to simplify further the term −ǫA(y−〈x〉)
is replaced with an appropriate average −f (where f (< 0) is the average tensile force in
the link). Needless to say, with this approximation we lose stepwise sample motions of the
cargo, but still overall average results can be recovered. The magnitude of required f is later
shown to be realistic. One benefit of this constant force assumption is that the model can
67
be solved analytically. In Chapter 5, this constant force assumption is further verified by
the simulation of combined motor-cargo system.
Thus, the observed motor-cargo motion can be modeled as the cargo diffusion under
constant force (due to the link tensile force + external force on it, which need not be a
potential force) between appropriate ‘checking positions’ that correspond to the chemical
reactions. The above consideration allows us to disregard the details of the potential (in
other words, the mechanochemical experimental results are expected not to reflect motor
mechanisms) experienced by the motor during its motion between binding sites on the track.
The time scale separation between the slower chemical reaction times (usually of the order
of tens of milliseconds or longer) and motor diffusion times (even for nondriven diffusive
motion) makes this consideration reasonable.
4.2.2 Mathematical Formulation of Effective Cargo Only
Representation
A unified framework to explain the observable dynamics is illustrated for the following
mechanochemical scheme with two substeps which can describe the mechanochemistry of
F1-ATPase:
(a)k′1[ATP ]
⇀↽k−1
(b) Substep1⇀↽
(c)k2⇀↽
k′
−2[P ](d)
k3⇀↽
k′
−3[ADP ](e) Substep2
⇀↽(f).
Here the forward and backward kinetic rate constants are k′
1, · · · , k′
−3 and [ATP], [P] and
[ADP] are the concentrations of ATP, phosphate ion and ADP, respectively. Once ATP
binds to the motor in state (a) located at x1 = 0, state (b) is formed and motor detaches
from the track. Motor makes a substep of length x2 and forms state (c) with hydrolyzed ATP
attached to it. When the motor displacement is complete and the cargo reaches the next
‘checking position,’ phosphate ion P is released and state (d) is formed. With the release of
68
ADP state (e) is formed and motor is again detached from the track. On completion of the
next substep the starting state of the motor with one step (of length xl) advanced state (f)
is formed.
To treat the diffusion and chemical reaction processes in a unified fashion, state proba-
bilities are introduced. Let Px (for x = a, · · ·, e) be the probability of the cargo with the
motor being in state x which we assume is localized within a length scale ∆l along the track
with uniform probability density Px/∆l. In state (b) or (e) the cargo can move: ρb(y) and
ρe(y) are the probability densities of the cargo at angle y for states (b) and (e), respectively.
Time evolution of probabilities can be described by the following equations:
dPa
dt= −k1Pa + k−1Pb + J ′
a,
dPb
dt= −k−1Pb + k1Pa − Jb,
∂ρb(y)
∂t= −∂Jb(y)
∂y,
dPc
dt= −k2Pc + k−2Pd + J ′
c,
dPd
dt= − (k3 + k−2) Pd + k2Pc + k−3Pe,
dPe
dt= −k−3Pe + k3Pd − Je,
∂ρe(y)
∂t= −∂Je(y)
∂y, (4.3)
where k1 = k′
1[ATP], k−2 = k′
−2[P] and k−3 = k′
−3[ADP]. Jx and J ′
x are, respectively, the
outgoing and incoming fluxes at state x. The cargo diffusive motion is described by the
Fokker-Planck equations with the flux Jx(y) in the above equation given as
Jx(y) = −Dc
(
∂ρx(y)
∂y+
1
kBTfρx(y)
)
, (4.4)
for x = b or e. Here, f is the effective driving force experienced by the cargo. If there is a
load fL acting on the cargo, f in the above expression must be replaced by f − fL. In this
69
model the boundary conditions to solve partial differential equations are Dirichlet conditions
for concentration matching. However, in the steady state, we are interested in the system
with a periodic boundary condition, so continuity of probabilities and fluxes are required.
In a steady state, all the time derivatives vanish. Integrating dJb/dy = 0 from 0 to
y gives Jb(y) = J , where J is the steady probability flux. With the boundary conditions
ρb(0) = Pb/∆l, ρb(x2) = Pc/∆l, and the flux continuity requirement Jb(x) = J = Jb = J ′
c
we obtain
Jb(y) = −Dc e−βfy d
dy
(
ρb(y) eβfy)
= k1Pa − k−1Pb. (4.5)
Integrating this from y = 0 to x2 gives
(Pc/∆l) eβfx2 − (Pb/∆l) = − 1
Dc
∫ x2
0
eβfydy × (k1Pa − k−1Pb). (4.6)
Analogous calculation for Je(y) yields
(Pa/∆l) eβfxl − (Pe/∆l) eβfx2 = − 1
Dc
∫ xl
x2
eβfydy × (k3Pd − k−3Pe). (4.7)
With the normalization condition
Pa + Pb +∫ x2
0ρb(y)dy + Pc + Pd + Pe +
∫ xl
x2
ρedy = 1 (4.8)
we have a closed set of equations for steady state probabilities and the steady flux J .
Though cumbersome, analytical solution of the above equations can be obtained very
easily with the aid of, e.g., Mathematica. The exact expression for the motor velocity v
(= xlJ) is given as
v = C
(
1 − eβfxlk−1k−2k−3
k1k2k3
)
, (4.9)
where the coefficient C is a function of Dc, effective motor driving force f (< 0) and the
reaction rates. It has an explicit analytic formula which is very long and hence is not shown
here. Note that if there is an external load fL acting on the cargo, the effective driving force
f is replaced by f − fL and this way external load enters into velocity expression.
70
Since we know that Pi release rate is much faster than ADP release rate and both
processes occur at the same motor location, we combine these two processes into a single
process with forward rate k∗
2 and reverse rate k∗
−2 and get the following expression for motor
velocity v as
v = C
(
1 − eβfxlk−1k
∗
−2
k1k∗
2
)
, (4.10)
where the coefficient C has the following explicit form,
C−1 =1
k1k2β2Dceβfx2∆l[β2 Dc f 2 (eβ f x2 (k1 + k∗
2) + e2 β f x2 (k1 + k−1) + eβ f xl (k∗
2 + k∗
−2)
+ eβ f (x2+xl) (k−1 + k∗
−2)) ∆l + (e2 β f x2 (k1 − k−1) (k∗
2 − k∗
−2) + eβ f xl (k1 − k−1)
× (k∗
2 − k∗
−2) + eβ f x2 (−2 k1 k∗
2 + k∗
2 k−1 + k1 k∗
−2) + eβ f (x2+xl) (k∗
2 k−1 + k1 k∗
−2
− 2 k−1 k∗
−2)) ∆l + β f (Dc (−(eβ f x2 (k1 + k∗
2)) + e2 β f x2 (k1 − k−1) + eβ f xl (k∗
2 − k∗
−2)
+ eβ f (x2 + xl) (k−1 + k∗
−2)) + ∆l (e2 β f x2 (k1 + k−1) (k∗
2 − k∗
−2) ∆l + eβ f xl (k1 − k−1)
× (k∗
2 + k∗
−2) ∆l − eβ f x2 (2 k1 k∗
2 ∆l + k∗
2 k−1 ∆l + k1 k∗
−2 ∆l + k1 k∗
2 xl) + eβ f (x2 + xl)
× (k∗
2 k−1 ∆l + k1 k∗
−2 ∆l + 2 k−1 k∗
−2 ∆l + k−1 k∗
−2 xl)))]. (4.11)
4.3 Explanation of Mechanochemical Data
The general scheme illustrated above can explain quantitatively the load dependence of
velocity of F1-ATPase, kinesin, and myosin V obtained by optical trap as well as viscous
load experiments with the aid of the published reaction rates and a single fitting parameter:
f , the effective driving force exerted by the motor on the cargo which is a constant for a
given motor.
71
Table 4.1: The adopted rate constants for F1-ATPase (from Pake et al. [79])
Process Rate constant
ATP binding 2.08 µM−1s−1
ATP unbinding 270 s−1
Phosphate ion release 2030×105 s−1
Phosphate ion binding 0.81 µM−1s−1
ADP unbinding 490 s−1
ADP binding 8.9µM−1s−1
4.3.1 F1-ATPase
F1-ATPase is a part of F0F1-ATP synthase. It consists of a ring-like structure α3β3 and
a central shaft γ that can be rotated using the free energy of ATP hydrolysis and thus
produce work with high efficiency [76]. The motor rotation (which is identified with the
γ rotation) is tightly coupled to the consumption of ATP. The energetics of this motor is
studied by attaching a large fluorescent actin filament of varying length to the rotating shaft
and observing its rotation [73]. It is also known that each motor step of 120 degrees consists
of substeps of 80 and 40 degrees [78]. The experimentally measured chemical rate constants
are given in Table 4.1. The load on the motor is that of viscous drag acting on the filament.
The load variation of motor velocity can be explained by above scheme with substeps with
available kinetic constants from biochemical experiments. Fig. 4.2 shows the variation of
velocity of the motor with the increasing length of the attached actin filament. Even though
[73] presents data for different ATP concentrations, most of the data were obtained by using
ATP regenerating system which did not monitor the concentrations of ADP and Pi. There
was only one data set which monitored the concentrations of ADP and Pi and this data
set was quantitatively explained by our model. The other data could also be explained by
assuming particular realistic ADP and P concentrations. The diffusion constant Dc was
estimated as [76] Dc = kBT/γ with γ = (4π/3)ηL3/log(L/2r)) − 0.447, where L is the
length of the actin filament, η the viscosity whose value is taken as that of viscosity of water
72
0.01
0.1
1
10
100
0 1 2 3 4
Ro
tatio
n r
ate
(r
p s
)
Actin length (µm)
Figure 4.2: Variation of actin rotation rate with increasing length. [ATP] = 20 µM , [ADP]= 1 µM and [P] = 100 µM . Solid line is the results from the analytic model with an effectivedriving torque of −40 pNnm for both substeps and ∆l = 5 degrees.
(10−3Ns/m2), and r = 5 nm the radius of the actin filament. It is found that results are
not very sensitive to the exact value of ∆l as long as it is less than 10 degrees.
4.3.2 Kinesin
Kinesin hydrolyzes one ATP while moving along the microtubule track by one step (xl = 8
nm) [16]. There are two approaches to study the load dependence of the velocity of kinesin.
[16] uses an optical trap setup which can apply a constant load to a bead of diameter 0.5
µm attached to the motor. Dependence of motor velocity on the external load for different
concentrations of ATP was studied. It is also observed that at very high loads the motor
can make reverse steps [37]. The probability of these reverse steps are very low and for
loads away from stalling they can be neglected. In this case, the model with the following
mechanochemical scheme is able to explain the load variation with external load.
(a)k′1[ATP ]
⇀↽k−1
(b) Mechanical motion⇀↽
(c)k2⇀↽
k′
−2[P ](d)
k3⇀↽
k′
−3[ADP ](e)
73
Table 4.2: The adopted rate constants for kinesin (from Cross [80])
Process Rate constant
ATP binding 2 µM−1s−1
ATP unbinding 80s−1
Phosphate ion release 300s−1
Phosphate ion binding 0.25 µM−1s−1
ADP unbinding 300s−1
ADP binding 4.5 µM−1s−1
The experimentally measured rate constants given in Table 4.2 were used to explain the
velocity data.
Figure 4.3 shows the variation of velocity of kinesin with load for two different ATP
concentrations. The same model also explains the variation of the velocity of the motor
with changing ATP concentration [16] as well as the case of motor response to a viscous
load as in [75]. Also note that as shown in [81] the velocity of the motor does not change
significantly when a forward load is applied to the motor. In this case, even if the forward
load accelerates the mechanical motion, the net motor stepping rate will be limited by the
chemical kinetic rates which are independent of load.
The data in [16] were obtained by using ATP regenerating system which did not monitor
the concentrations of ADP and P concentrations. This makes comparisons with experimental
data difficult especially for low ATP concentration. For [ATP] = 5 µM , to fit the data it had
to be assumed that the concentration of ADP present also increases with external load. For
[ATP] = 2 mM, we chose [P] = 100 µM and [ADP] = 10 µM to get the stall force in agreement
with experimental data. For the [ATP] = 5 µM, with the above kinetic constants and ADP
and P concentrations, the load dependent velocity decrease was much less pronounced than
that of experimental data. Velocity stays nearly constant for a range of loads (up to 4 pN)
and then suddenly decreases. To get the curve above which fits the data we assumed that
74
0
200
400
600
800
1000
-6 -4 -2 0 2 4 6 8
Velo
city
(nm
/s)
Load (pN)
[ATP]=2mM[ATP]=5µM
Figure 4.3: Variation of velocity of kinesin with load for [ATP] = 2mM and [ATP] = 5 µM.Experimental data of [16] is also shown. Driving force f = −6 pN, and ∆l = 1 nm.
the unmonitored ADP concentration also increases with load fL as, [ADP] × exp(2fLβ).
Here the diffusion constant Dc of the cargo is obtained as Dc = kBT/6πηa, where η is taken
as the viscosity of water (10−3Ns/m2) and a = 0.25 µm, the radius of the bead used in [16].
4.3.3 Myosin V
Myosin V walks processively along actin with step lengths of 36 nm [42]. [82] measured the
variation of velocity of this motor with an external load using an optical trap setup with bead
diameter of 0.356 µm. Using the experimentally determined rate constants given in Table
4.3, the chemical scheme without substeps used to explain kinesin data can quantitatively
explain the velocity variation with load for myosin V as well. Figure 4.4 shows the variation
of velocity of the motor with load.
75
Table 4.3: The adopted rate constants for myosin V
Process Rate constant Reference
ATP binding 0.9 µM−1s−1 [83]ATP unbinding 1.2s−1 [84]Phosphate ion release 220s−1 [85]Phosphate ion binding 0.01 µM−1s−1 [84]ADP unbinding 12s−1 [83]ADP binding 12.6 µM−1s−1 [86]
0
100
200
300
400
500
-5 -4 -3 -2 -1 0 1 2 3
Ve
loci
ty (
nm
/s)
Load (pN)
Figure 4.4: Variation of velocity with load for myosin V at [ATP] = 2mM. Experimentaldata from [82] is also shown. It was assumed that ADP and P were also present with aconcentration of 5 µM. ∆l = 1 nm and the driving force f=−1.75 pN.
76
4.4 Discussion
A simple scheme to interpret the single molecule experimental data on motor proteins derived
from optical trap and viscous load experiments is proposed. A significant feature of the
scheme is that it treats diffusion and chemical reactions in a unified fashion. With a fitting
parameter (the constant driving force exerted by motor on cargo, which is a given constant
for a given motor and linkage independent of the experimental condition) the available motor
experiments for F1ATPase, kinesin, and myosin V are reproduced quantitatively. Also, it
is straightforward to modify the scheme to include the possibility of backward steps (the
fact that the probability of backward steps is very low at loads below stalling and in viscous
experiments implies that incorporation of backward steps in our scheme is not necessary to
explain the available experimental data).
One requirement for the application of the scheme is the availability of experimentally
determined biochemical rate constants. In the absence of known kinetic constants, it is
still possible to assume reasonable rate constants (or treat them as fitting parameters). An
example is the application of the scheme to the recently published results on viral DNA
packaging motor φ29 [87]. With reasonable rate constants the observed velocity variation
with load can be explained.
In the modeling we have assumed that the effective driving force exerted by the motor on
the cargo is constant. This is an approximation that is strictly valid at very low stiffness of the
link connecting them. As shown in Chapter 5, numerical simulation of the complete motor-
cargo system with a realistic link demonstrates that, the linear potential approximation is
quantitatively reliable over a wide range of link stiffness.
The most important conclusion is that models which do not assume anything about the
details of motor force production mechanisms can explain the single molecule mechanochem-
ical experimental data. The orders of magnitude differences in motor and cargo fluctuation
time scales and the presence of a link connecting motor and cargo imply that the single
77
molecule experiments cannot probe the details of motor force production. The fact that
models which assume a particular motor mechanism (such as the existence of a power stroke)
can explain single molecule mechanochemical experimental results does not guarantee the
existence of power stroke in the actual motor operation. Additional supporting information
from experiments which probe the motor motion, or conformational changes directly are
necessary.
In the next chapter, we will test the conclusions of this chapter by modeling the combined
motor-cargo system (i.e., without reducing it to a cargo only model) of the rotary motor
F1-ATPase.
78
Chapter 5
Affinity Switch Model for RotaryMotor F1-ATPase
5.1 Introduction
As described in Chapter 1, F1-ATPase is an ATP factory in cells [7]. To make this Chapter
self contained we will restate some of the experimental observations on this motor and general
modeling approaches to motor proteins. The motor consists of the threefold symmetric outer
ring α3β3 which surrounds the shaft γ that lacks three fold symmetry (as shown in Fig. 1.3
in Chapter 1) [88]. If the γ shaft is rotated, the resultant mechanical work drives the
ATP synthesis by combining ADP and inorganic phosphate Pi. Conversely, utilizing ATP
molecules, F1-ATPase can rotate as a molecular motor, using the free energy available from
ATP hydrolysis [88]. Its successful crystallization [89] and advancements in single molecule
experimental techniques [73, 78] greatly facilitated physical characterization of this motor.
Observation of a large cargo attached to the γ subunit has shown that coordinated ATP
binding and hydrolysis processes at three binding pockets at the α-β interface rotate the γ
rotor. One revolution is made up of three discrete steps of 120 degrees, each of which is
tightly coupled to the consumption of one ATP molecule and is made up of substeps of 80
and 40 degrees [78]. It is also known that the β structural subunit changes its conformation
[88] depending on the state of nucleotide (ATP or its hydrolysis intermediates ADPPi, the
hydrolysed ATP, ADP and inorganic Pi) at the pocket.
Development of ratchet models [20, 28, 90, 91] have provided physical insights into the
operation of mesoscopic motors working in a thermal environment. For a system maintained
at nonequilibrium, thermal fluctuations can be rectified to produce directional motion by
79
using asymmetric potentials. Application of abstract ratchet models to specific biological
motors has been less successful in producing quantitative agreement with mechanochemical
data. Models which attempt to produce quantitative agreement have relied on identifying
a conformational change within the motor which converts the free energy associated with
ATP hydrolysis into mechanical work needed for cargo motion. Such power stroke models for
F1 motor [24, 92, 93, 94] identify the open to closed conformational change of β, following
ATP binding and hydrolysis, as pushing the γ forward. The underlying assumptions are
that there are no strong binding interactions between β and γ and that the asymmetric γ
structure produces steric interactions which convert the closing motion of β into forward γ
motion.
Though such models are appealing for their simplicity and correspondence with macro-
scopic motors with levers and crank-shafts, the underlying assumptions are indirect interpre-
tations of known experimental data. There are localized specific amino acid residues along
both β and γ [95, 96], replacement of which by any non-specific residues causes the the mo-
tor to lose its activity completely or partially. These specific residues have opposite charges,
allowing for strong binding of γ onto β. Even in the absence of γ and ATP, β can close
spontaneously [21, 97]. There are no clear experimental data indicating large free energy
change associated with closing of β. The above two observations can be used to question the
assumption that the conformational change of β alone produces the necessary free energy
change required to drive γ forward. In addition, as shown in Chapters 3 and 4, the fact
that a model can explain the mechanochemical data does not mean the force production
assumption of the model is necessarily correct.
The system is of mesoscopic size operating in a thermal environment. Therefore, even
without explicit driving, if the motions in wrong directions are appropriately checked by
conformational changes in β, there is a possibility that the desired motion of γ in the allowed
time scales is accomplished by thermal fluctuations alone. The conformational changes of β
provide affinity switches between γ and β as well as barriers checking wrong motions and
80
thus acting as the rectifiers of thermal motion of γ. In this chapter, we demonstrate that this
scenario can quantitatively explain single molecule experimental results without any fitting
parameter.
5.2 Model
The structural data on F1 can be abstracted as the motor having three nucleotide pockets
interacting with γ that lacks three-fold symmetry. A γ rotation step consists of two substeps
of 80 degrees and of 40 degrees [78]. Between these substeps, the γ is in an immobile waiting
state and the steps are triggered by changes in the occupancy of the ATP binding pocket.
Biochemical data shows that the first substep follows ATP binding and the second follows
Pi release and possibly ADP release [98]. Below, T, E, DP, D imply, respectively, as in the
previous chapters, the occupancy of the pocket with ATP, nothing (empty), hydrolyzed ATP
before releasing Pi, and ADP. Structural data shows that the conformation of β depends on
the state of nucleotide in the pocket. It is in a closed conformation for the T state, half open
conformation in the DP state and open in the E state [89].
We use the following assumptions:
(1) the immobile waiting state of γ is produced by the strong β-γ binding interaction con-
trolled by the state of nucleotide in the pocket; (2) the orientation of γ and the chemical
reactions associated with ATP hydrolysis are tightly coupled (a particular reaction in the
nucleotide pocket is realizable only when it makes a right relative angle with γ); (3) the
conformational changes of β associated with nucleotide changes acts as an affinity switch
between β and γ or as barriers which blocks the ‘wrong’ motions of γ. According to the
experimental observations [102, 99] γ in a waiting state cannot be pulled in the backward
direction even with applying large forces on it. A natural interpretation of these facts is
that there are barriers preventing wrong directional motions. In addition, such barriers are
logically required to ensure tight coupled chemical reactions associated with ATP hydrolysis
81
to occur in a correct sequence; (4) in between such binding sites or barriers γ undergoes
Brownian rotation due to thermal noise.
Based on these assumptions we propose the following scheme (see Fig. 5.1) to explain the
mechanism of this motor. The waiting state is (a). On binding of ATP, γ is released from
T DP
E
DP
T
DP D
T
DPT
T
DP DP
T
E
T
DP
ATP binding
80 deg rotation
Phosphate release ADP release
40 deg rotation
(a) (b) (c) (d) (e) (f )
(a)
(b)
(c)
(d)
E
Figure 5.1: Model for the rotation of F1-ATPase. The three nucleotide binding pockets arelabeled according to the state of occupying nucleotide. As the motor cycles through states(a) · · · (f), γ shaft (the triangle at the center) rotates by 120 degrees in substeps of 80 and40 degrees. The lower part of the figure shows the potential experienced by the γ elementin the different motor states.
the ring (b) and undergoes thermal fluctuations. There is a barrier preventing the backward
motion of γ beyond the starting point. When thermal fluctuation of γ takes it to 80 degrees
(c), γ is trapped in another potential well produced by the strong γ-β interaction. The
previously bound ATP is hydrolyzed at this location forming DP bound state. Pi release
at the DP bound β gives the D bound β (d). On ADP release from this β (e), γ is again
released from the well and starts undergoing Brownian motion. There is a potential barrier
82
that prevents the Brownian motion to go back from the 80 degree position. If a 40 degree
rotation of γ is completed (f), it is in the potential well of the waiting state and 1/3 rotation
is complete.
The lower part of Fig. 5.1 shows the energetic picture of the model. γ starts in a
state strongly bound to β (the leftmost well of the potential curve (a)). On binding of
ATP, conformational changes at the β-γ interface modify the β-γ interaction potential curve
(b). γ now diffuses forward because of the barrier preventing the backward motion. Upon
moving forward by 80 degrees, it binds strongly to β (c). On Pi and ADP release, further
conformational changes at β-γ modifies the potential curve to (d), allowing the free diffusion
of γ forward by another 40 degrees. γ cannot go back due to the barrier on the left. After
completion of the 40 degree diffusion, γ again falls into the right well by forming strong
binding interactions with β. In the model the entire free energy of ATP hydrolysis is used
to alter the β-γ interaction.
The above model uses a tri-site reaction mechanism in which all three nucleotide binding
pockets are occupied while the motor takes a 120 degree step. Such a reaction scheme is not
completely inferable from the presently available experimental data. There are proposed bi-
site schemes which are compatible with the same experimental information [88]. Note that
our results are not affected by the exact nature of the scheme as long as any alternate scheme
also assumes that ATP binding produces 80 degree substep and product release produces
the remaining 40 degree substep.
In cargo observation experiments [73, 78], it is seen that the motor can drive cargoes
that are many times motor size. In this case, the cargo diffusion times without any driving
from motor is known to be much longer than the time scales at which the motor operates.
Using the fact that γ to which cargo is attached is elastic and is capable of storing energy
[100, 101], our model can be made compatible with this observation of motor driving cargoes
many times its size.
83
5.3 Algorithmic Description of the Model
The model can be summarized for one step as follows:
(a)k1[ATP]
⇀↽k−1
(b) Substep1⇀↽
(c)k2⇀↽
k−2[P](d)
k3⇀↽
k−3[ADP](e) Substep2
⇀↽(f).
Here, k1, · · · , k3 are the kinetic constants associated with ATP binding and release of hy-
drolysis products, [ATP], [P], and [ADP] are the concentrations of ATP, Pi, and ADP,
respectively. In the following, β = 1/kBT , where kB is the Boltzmann constant and T the
temperature, ∆l ∼ 5 degrees, ks the stiffness of the link connecting motor and cargo (in-
cluding the stiffness of γ) which is modeled as a harmonic spring, Dm the motor diffusion
constant, Dc the cargo diffusion constant. x is the γ angle coordinate, y the cargo angle
coordinate, and t time. Here, the state of the motor is denoted by (x±), where x denotes
the nucleotide states (a-e in Fig. 5.1), and ± denotes the state of γ: + implies that γ is
movable diffusively and − implies that γ is fixed to the ring (not all the combinations make
sense; e.g., (a+) or (b−) does not exist). The elementary evolution of time step dt is taken
as 10−7s. In the following algorithm, the cargo position y is updated at every time step as
y(t + dt) = y(t) − βDcks(y − x)dt +√
2DcdtNy, (5.1)
where Ny is a Gaussian random number with zero mean and unit variation.
(A1) Start from state (a−): x = 0, y = 0 at time t = 0.
(A2) Let n be a uniform random number in [0, 1]. When the motor position is x < ∆l:
if n < k1dt, go to state (b+); if n > k1dt but n < k1dt + k−3dt, go to e− (binding of ADP
instead of ATP at x = 0) ; otherwise, the state remains unchanged.
(A3) For (b+) update the motor positions according to
x(t + dt) = max{0, x(t) − βDmks(x − y)dt +√
2DmdtNx(mod 120 deg)}, (5.2)
84
which includes the barrier constraints. Nx is a Gaussian random number with zero mean
and unit variation.
(A4) When the motor coordinate is x > x2 −∆l (x2 = 80 degrees), the motor completes the
first substep (its position is updated to x2) and the state changes to (c−).
(A5) In state (c−) if n < k2dt, go to state (d−) (Pi unbinds from motor). Otherwise, the
state is not changed. x is not changed (x = x2).
(A6) In state (d−) if n < k−2[Pi]dt, go to (c−) (rebinding of Pi). x is not changed; if
n > k−2[Pi]dt but n < k3dt + k−2[Pi]dt (ADP release) go to state (e+).
(A7) For (e+) update the motor positions according to
x(t + dt) = max{x2, x(t) − βDmks(x − y)dt +√
2DmdtNx(mod 120 deg)}, (5.3)
which includes the barrier constraints. (A8) In state (e+) if the motor position is x < x2+∆l,
and if n < k−3[ADP]dt, go to state (d−) (ADP rebinding). x = x2; if n ≥ k−3[ADP]dt stay
in (e+) and update x according to (A7).
(A9) In state (e+) if x > xl − ∆l (xl = 120 degrees), then x = xl and the state becomes
(f−) = (a−); the 120 degree step is completed; if x < xl − ∆l, stay in (e+) and update x
according to (A7).
It is possible to have external load fL acting on the cargo. In this situation, in y position
update expressions of the above algorithm, the force experienced by the cargo ks(y − x) is
replaced with ks(y − x) − fL.
Table 5.1 gives the values of kinetic rate constants used in the simulation. The remaining
constants are chosen as follows.
∗ The stiffness of the link ks is 40 pNnm. The choice is consistent with the equilibrium
fluctuations given in [102].
∗ The γ rotational diffusion constant Dm is chosen to be ∼4000 rad2/s. The model results
85
Table 5.1: The adopted rate constants taken from Panke and Rumberg [103]
Process Rate constant
ATP binding 2.08 µM−1s−1
ATP unbinding 270 s−1
Phosphate ion release 2030×105 s−1
Phosphate ion binding 0.81 µM−1s−1
ADP unbinding 490 s−1
ADP binding 8.9 µM−1s−1
are not sensitive to the actual value chosen. Any value greater than 1000 rad2/s gives the
saturating motor hydrolysis rate in the absence of any cargo (In terms of the viscosity η of
water, the radius r = 0.9 nm of γ and its length L = 6 nm, the rotational diffusion constant
Dm of γ is given as kBT/4πηr2L ≃ 108 rad2/s, so the adopted value of 4000 rad2/s is nearly
105 times as large as this number. Effectively, we are assuming a viscosity of 105 times that
of water to account for the possible weak interactions between γ and the α-β ring. This
is an outrageously big friction and the actual friction may be smaller. However, as noted
above, a faster diffusion constant does not alter the result quantitatively. Therefore, the
choice here is rather technical: to have a reasonably large time step for simulation smaller
diffusion constants are convenient).
∗ For the experiment involving the observation of an actin filament attached to γ as a cargo,
its diffusion constant is estimated with the aid of Dc = kBT/[(4π/3)ηL3/log(L/2r)− 0.447],
where L is the length of the actin filament, η is the viscosity of water, and r = 5 nm is the
radius of the actin filament.
5.4 Comparison With Analytically Solvable Model
In Chapter 4, we developed an analytically solvable limit of the elastically coupled motor-
cargo system, when the size of the cargo is many times that of the motor so that there is
a clear time scale separation between motor and cargo fluctuations. It was shown that the
86
coupled description of motor and cargo can be reduced to that of the cargo alone as being
driven by an effective motor driving torque. Using the simulation results in this chapter, it
is shown that the simplifying approximation of constant driving force in the analytic model
is quantitatively reliable, since the analytical results agree with simulation results as well as
with experimental results.
5.5 Comparison of Model Predictions with Empirical
Results
Figure 5.2 exhibits typical displacement records of cargo for varying ATP concentrations
obtained by the simulation of the model of ATP hydrolysis (Section 3). Most of the time
the motor is waiting for ATP binding or product release. Once ATP binds to the motor,
it makes quick diffusive forward steps during which there is very little cargo displacement.
Once motor reaches the next binding site and is waiting for ATP binding, the cargo is
dragged forward.
Figure 5.3 exhibits the variation of the cargo rotational rate with ATP concentration.
In this case, in order to explain the rotational rate at low ATP concentrations a different
ATP binding rate had to be used; the values used were k1 =27 µ M−1s−1 for [ATP] = 0.02
mM and 22 µ M−1s−1 for [ATP] < 10 µM. These values were taken from the analysis of
waiting time data given in [73]. With k1 = 2.08µ M−1s−1 at low ATP concentrations waiting
times occasionally exceeded 10 s. It is possible that in the actual single molecule experiment
such data were neglected, because they were indistinguishable from such long waiting times
caused by the formation of contacts between the actin rod and the surface or other such
irregularities. This probably is the reason for an effective higher binding constant.
Figure 5.4 exhibits the actin rotation rate as a function of the length of the actin rod. In
this case concentrations of ATP, ADP and Pi were monitored [73]. The simulation data and
the results from the analytical model (using the effective driving torque f of −40 pNnm for
87
-200
100
400
700
1000
1300
0 5 10 15 20 25 30
Rota
tional A
ngle
(D
egre
es)
Time (s)
[ATP]=0.02 µM[ATP]=0.06µM[ATP]=0.2 µM[ATP]=0.6 µM
Figure 5.2: Displacement records for different ATP concentrations. [ADP] = 0 µM and [Pi]= 100 µM. The length of actin filament is 1 µm.
both 80 and 40 degree substeps. This value of f is compatible with the stiffness of the link
as well as the experimental observations) agree with the experimental data. Since our model
reproduces the rate-load relation, the efficiency of the motor is very high for our model.
Figure 5.5 exhibits the variation of actin rotation rate with increasing actin length at an
ATP concentration of 20 µM.
Figure 5.6 shows the variation of the actin rotation rate with increasing externally im-
posed torque for two different ATP concentrations. The analytical model predictions for a
driving torque of −35 pNnm is also shown. As of now there are no experimental data for
external torque. We expect the recent advances in designing magnetic traps makes such an
experiment feasible in the near future.
88
0.01
0.1
1
10
0.01 0.1 1 10 100
Rota
tional R
ate
(r.
p.s
)
[ATP] (µM
exptl. data
Figure 5.3: Variation of the cargo rotational rate with ATP concentration. The length ofthe actin filament is 1 µm. [ADP] = 0 µM and [Pi] = 100 µM. Experimental data from [73]is also shown.
5.6 Discussion
We have introduced a simple model to explain the F1 motor with the following ingredients:
(1) the modulation of biding affinity between γ and β elements by the state of nucleotide
during the ATP hydrolysis process, (2) the mechanical motion of γ caused by thermal fluc-
tuation without any driving, and (3) experimentally inferred barriers checking motions in
wrong directions. It is shown that by incorporating the known biochemical data on the
motor, the model can quantitatively explain the available mechanochemical data without
adjustable parameters.
In most single molecule experiments that allow imposing external forces on the motor,
a large cargo (probe) is attached to the motor shaft, so the actual motor motion is never
directly observed. As seen in Chapters 3 and 4, in such situations the velocity of the motor
is dependent on the cargo size and stiffness of the link connecting them, irrespective of
the nature of the potential exerted by the motor. The indirectness of motor observation
89
0.25
1
4
1 2 3
Rota
tional r
ate
(r.
p.s
)
Actin Length (µm)
[ATP]=2mM[ATP]=2mM (expt. data)
Figure 5.4: Variation of actin rotation rate with increasing actin length. [ATP] = 2 mM,[ADP] = 10 µM and [Pi] = 100 µM. Solid line shows the result from the analytic model withan effective driving torque of −40 pNnm for both substeps and experimental data from [73]is also shown.
may be the reason why many different models [24, 92, 93, 94] are compatible with available
quantitative experimental results.
It is possible to put a fluorescent probe directly on the shaft, but for the currently
available probes the space-time resolution of the obtained results is not good enough for
inferring any mechanism of the motor.
In the model, we assumed that the motion of γ is entirely due to thermal fluctuations.
It is possible to modify the minimal model to incorporate the driven motion of γ due to
some potential slope mimicking, e.g., the so-called power stroke. We found that no such
modification is needed to explain the mechanochemical data. So it is difficult to support the
current models with special driving mechanisms if the only criterion of their goodness is an
agreement with single molecule experiments.
If ATPase uses essentially simple diffusion to rotate, then substeps produces faster ro-
tation. Thus we can expect that ATPase without substeps should be considerably slower
90
0.125
0.25
0.5
1
2
4
1 2 3
Rota
tional r
ate
(r.
p.s
)
Actin Length (µm)
[ATP]=0.02mM
Figure 5.5: Variation of actin rotation rate with increasing length. [ATP] = 20 µM, [ADP]= 1 µM and [Pi] = 100 µM. Solid lines are the results from the analytic model with aneffective driving torque of −40 pNnm for both substeps.
than F1-ATPase. V1 motor [104] behaves as such a motor which, similar to F1, uses ATP
hydrolysis to drive 120 degree rotation of a shaft. It is known that these motor steps do not
involve substeps [104]. On applying our model without any substeps to this motor, we find
that the motor velocity can be explained with reasonable rate constants of the order of that
of F1 (there are no known rate constant measurements for this motor).
Possible ways of proceeding from the current status that cannot reject the simple model
are:
(1) Experimentally demonstrate that there are no strong interactions between γ and β.
(2) Experimentally demonstrate that barrier heights cannot be modulated by nucleotide
changes alone.
(3) Attach small probes directly to γ, find its fluctuations in the absence of any cargo.
(4) Identify some mutations which make a motor that can rotate as fast as or faster than F1
but with no substeps.
(5) Using a highly sensitive magnetic trap, determine the load dependence of substep com-
91
1e-04
0.001
0.01
0.1
1
10
0 10 20 30 40
Rota
tional r
ate
(r.
p.s
)
Load (pN-nm)
[ATP]=2mM[ATP]=0.02mM
Figure 5.6: The variation of motor rotational rate with increasing external torque. [ADP]= 1 µM and [Pi] = 100 µM and length of actin filament L = 1 µm. The solid lines are theanalytical model predictions for a driving torque of −35 pNnm for both substeps.
pletion times. This can say something about possible barriers.
(6) Measure the position dependence of the stalling forces.
Since (1) and (2) check the main ingredients of the minimal model, needless to say, they
directly test the simple model without any driving. However, as discussed already, structural
and other experimental data suggest that these demonstrations are not possible. (3) may
be an ideal experiment, but sufficient space-time resolution of such small probes is currently
impossible. (4) could strongly support active driving of γ by some potential formed by the
ring. (5) and (6) crucially depend on the progress of magnetic trap technology; at present,
we feel these experiments are not feasible, but they are the most direct single molecule
experiment conceivable.
92
Chapter 6
Summary and Open Questions
In this thesis, through careful analysis and modeling of available experimental data we have
tried to develop a phenomenological understanding of motor proteins. We have demonstrated
the following:
(1) In Chapter 2 we have developed a phenomenology of motor protein kinesin. We have
distilled a kinetic scheme from the available biochemical and structural information. This
has allowed us to develop a phenomenology compatible with the available data, ignoring the
flexibility of motor-cargo link. For the kinesin dimer, preferential detachment ( attachment)
of one head from (to) the track dependent on the conformation of the linker (neck linker
precisely) connecting the two heads, and the state of nucleotide at both heads is identified
as the biasing mechanism which introduces directionality. At the same time, we found that
biochemical data and mechanochemical data still cannot single out a unique motor force
generating scheme.
(2) In Chapter 3 by modeling the motor as simple tight and loosely coupled ratchets, we
have showed that the measured velocity of a large cargo attached to it elastically depends
on the stiffness of the link and the cargo diffusion constant. We have also showed that
the efficiency measures which are proposed as tools to identify the details of motor force
production also depends on the link stiffness. This suggests that inference of the details of
motor force production from cargo observations alone should be treated with caution. It is
possible to have qualitatively wrong inferences.
(3) In Chapter 4 we have proposed a scheme to interpret the single molecule mechanochem-
ical experimental data on motor proteins. The scheme is capable of treating motor diffu-
93
sion and chemical reactions in a unified fashion. The model can quantitatively explain the
available mechanochemical data for F1ATPase, kinesin, and myosin V with just one fitting
parameter. A significant finding is that models without any detail of motor force production
mechanisms suffice to reproduce mechanochemical experimental results, if the cargo used in
these experiments are much larger than motor and the motor-cargo linkage is sufficiently
flexible.
(4) In Chapter 5 a simple model has been given to explain the motor F1-ATPase with
the following ingredients: (1) the modulation of binding between the rotary element γ and
the stator element β by the state of nucleotide during the ATP hydrolysis process, (2) the
mechanical motion of γ caused by thermal fluctuation without any driving, and (3) ex-
perimentally inferred barriers checking motions in wrong directions. The resultant model
incorporating the biochemical data has turned out to be able to describe the mechanochem-
ical data quantitatively without any adjustable parameters.
We have considered the mesoscopic models with the least ad hoc features compatible
with available experimental data. The model may be regarded as the null model, because
any experiment that can give a nontrivial statement about the motor mechanism must be
able to produce a result that cannot be explained by the model. We hope in the near future
experiments with better spatio-temporal resolutions will be able to reject the null model.
Then, minimal revisions of the null model to incorporate the new experimental results will
stimulate the next experimental progress. In addition to the explanation of mechanisms
of individual motors, such studies may tell us something about universal features of motor
proteins in general.
Another problem which can be approached at the mesoscopic level is understanding the
biasing mechanisms in motors which introduces directionality. In our studies, careful analysis
of available experimental data suggested that for kinesin dimer a possible biasing mechanism
is preferential detachment from track or attachment to track of a motor head depending on
the nucleotide state and the conformation of the neck linker. For F1-ATPase, it is preferential
94
accessibility to binding sites on the stator (β) by the rotor (γ), depending on the state of
nucleotide on the stator. A common chemical description for both of these mechanisms is
the modulation of barrier height by the changes in nucleotide state of the motor or some
other conformational change. Physical descriptions of such biasing mechanisms in terms of
coarse-grained modeling of motor structures will be useful in the future.
Another unresolved question in the motors that we have considered is whether the opera-
tion is due to power stroke or due to rectification of thermal fluctuations. As we have showed,
currently available mechanochemical data is not sufficient to resolve the motor mechanism.
A clear resolution of this is possible, if we can estimate the free energy changes associated
with the conformational changes implicated in motor force production. Advances in com-
puting power and molecular dynamics simulation methods will be useful in resolving this
issue.
95
Appendix A
Solution of the Kinetic Model
In this Appendix we present the expressions for the velocity v, the randomness r, and the
run length L obtained by the general method summarized by Elston [59]. d denotes the step
length (= 8 nm). The 5-state model of Fig. 2.1 can be represented by the following chemical
kinetic scheme, where Mi represents a particular motor state.
M1
k1⇀↽
k−1
M2
k2⇀↽
k−2
M3
k3⇀↽
k−3
M4
k4⇀↽
k−4
M5
k5⇀↽
k−5
M1 (A.1)
The time evolution equations of such a scheme is given by
dFj
dt= LFj + L+Fj−1 + L−Fj+1, (A.2)
where index j denotes the spatial step number and Fk for k = j, j ± 1 is defined as
Fk =
(M1)k
(M2)k
(M3)k
(M4)k
(M5)k
,
(A.3)
96
L =
−(k1 + k−5) k−1 0 0 0
k1 −(k−1 + k2) k−2 0 0
0 k2 −(k−2 + k3) k−3 0
0 0 k3 −(k−3 + k4) k−5
0 0 0 k4 −(k−4 + k5)
,
(A.4)
L+ =
0 0 0 0 k5
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
(A.5)
L− =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
k−5 0 0 0
. (A.6)
Using the generating function defined as
P (z, t) =j=∞∑
j=−∞
zjF (j, t), (A.7)
the time evolution equations can be written as
dP
dt= LP + zL+P +
1
zL−P = A(z)P. (A.8)
97
where
A(z) =
−(k1 + k−5) k−1 0 0 zk5
k1 −(k−1 + k2) k−2 0 0
0 k2 −(k−2 + k3) k−3 0
0 0 k3 −(k−3 + k4) k−5
1zk−5 0 0 k4 −(k−4 + k5)
. (A.9)
As shown by Elston, the velocity and effective diffusion constant can be obtained as,
v = dλ′
0(1), (A.10)
Deff =d2
2(λ′′
0(1) + λ′
0(1)). (A.11)
In the above expression, λ0 is the largest eigenvalue of matrix A(z), and the primes denotes
derivatives with respect to z evaluated at z = 1. An easy way to calculate these derivatives
is as follows. Write down the characteristic equation of A(z). Using the fact that the largest
eigenvalue is zero, we can write,
λ0(1 + α) = αλ′
0(1) +α2
2λ′′
0(1) + ...., (A.12)
Also, substitute z = 1 + α in the characteristic equation. Equating the coefficients of α to
be zero, the derivatives of λ can be easily determined.
For the five state models that we use, using the above technique, the expressions for
velocity and randomness can be obtained.
Let us define
98
p1 ≡ k3k4(f)k5[ADP] + k3k4(f)k−1 + k3k5k−1 + k4(f)k5k−1 + k4(f)k5k−2[P]
+k4(f)k−1k−2[P] + k5k−1k−2[P] + k5k−1k−3(f) + k5k−2[P]k−3(f)
+k−1k−2[P]k−3(f) + k3k−1k−4 + k−1k−2[P]k−4 + k−1k−3(f)k−4
+k−2[P]k−3(f)k−4 + k1[ATP](k4(f)k5 + k4k−2[P ] + k5k−2[P]
+k5k−3(f) + k−2[P]k−3(f) + (k−2[P] + k−3(f)k−4 + k3(k4(f) + k5 + k−4)
+k2(k3 + k4(f) + k5 + k−3(f) + k−4)) + (k−1k−2[P] + k4(k−1 + k−2[P])
+k−1k−3(f) + k−2[P]k−3(f) + (k−1 + k−2[P] + k−3(f))k−4
+k3(k4(f) + k−1 + k−4))k−5[ADP] + k2(k−3(k5 + k−4) + (k−3(f) + k−4)k−5[ADP]
+k4(f)(k5 + k−5[ADP]) + k3(k4(f) + k5 + k−4 + k−5[ADP])), (A.13)
p2 ≡ k−1(k3k4(f)k5 + k4(f)k5k−2[P] + k−2[P]k−3(f)(k5 + k4)) + k1[ATP](k3k4(f)k5
+k5k−2[P](k4(f) + k−3(f)) + k−2[P]k−3(f)k−4 + k2(k5(k4(f) + k−3(f))
+k−3(f)k−4 + k3(k4(f) + k5 + k−4))) + (k−1k−2[P](k4(f) + k−3(f))
+(k−2[P]k−3(f) + k−1(k−2[P] + k−3(f)))k−4 + k3k−1(k4 + k−4))k−5[ADP]
+k2(k3k4(f)k5 + k−3(f)k−4k−5[ADP] + k3(k4(f) + k−4)k−5[ADP]), (A.14)
p3 ≡ k1[ATP]k2k3k4(f)k5, (A.15)
p4 ≡ k−1k−2[P]k−3(f)k−4k−5[ADP]. (A.16)
In terms of these quantities
v0 = d(p3 − p4)/p2, (A.17)
r0 = 1 +2
(p3 − p4)
(
p4 −p1(p3 − p4)
2
p22
)
. (A.18)
99
If the probability of back steps is taken into account the formulae are modified as,
v = v0.(Pf − Pr), (A.19)
r =1
1 − 2Pr
+ (r0 − 1)(1 − 2.Pr), (A.20)
where Pr and Pf are the probability of forward and backward steps, respectively. The steady
probabilities PX for the intermediate state X read:
P(DP, E) = (k2k3k4(f)k5 + k−1(k3k4(f)k5 + k5k−2[P](k4(f) + k−3)
+k−2k−3(f)k−4))/p2; (A.21)
P(DP, T) = (k1[ATP](k3k4(f)k5 + k5k−2[P](k4(f) + k−3(f)) + k−2[P]k−3(f)k−4)
+k−2[P]k−3(f)k−4k−5[ADP])/p2; (A.22)
P(D,T) = (k1[ATP]k2(k4(f)k5 + k−3(f)(k5 + k−4))
+(k2 + k−1)k−3(f)k4k−5[ADP])/p2; (A.23)
P(T/DP, D)1 = (k1[ATP]k2k3(k5 + k−4)
+(k2k3 + k−1(k3 + k−2))k−4k−5[ADP])/p2; (A.24)
P(T/DP, D)2 = (k1[ATP]k2k3k4(f) + (k2k3k4(f) + k4(f)k−1(k3 + k−2)
+k−1k−2k−3(f))k−5[ADP])/p2. (A.25)
If the detachment rate from state X is denoted by kX , the net detachment rate during a
mechanochemical cycle is given by
kdet = k(DP, E)P(DP, E) + k(DP, T)P(DP, T) + k(D, T)P(D, T)
+k(T/DP, D)1P(T/DP, D)1 + k(T/DP, D)2P(T/DP, D)2, (A.26)
100
and the run length is given by
L = v/kdet. (A.27)
101
Appendix B
Stochastic Energetics
Stochastic energetics due to Sekimoto [72] aims at developing an energetic picture of Langevin
dynamics. It can be used to determine the energetic quantities of motor-cargo system defined
as follows. In a particular state (say, 1) the motor is described by the Langevin equation
−∂v1/∂x + (−γmdx/dt + ξ(t)) = 0, (B.1)
where ξ(t) is the Gaussian noise with zero mean and 〈ξ(t)ξ(t′)〉 = (2/γm)δ(t− t′). The work
done by the heat bath on the motor is [72] given by
D = −∫ tf
ti(−γm
dx
dt+ ξ(t)) ◦ dx(t) =
∫ tf
ti−∂V1(x, y)/∂x ◦ dx(t), (B.2)
where ◦ denotes Ito’s circle product. Taking the average of D with the probability of state
obtained by solving the Fokker-Planck equation and adding the contributions from states 1
and 2, we get the motor heat exchange rate with the environment as,
〈Dm〉 =∫ 1
0dx∫
∞
−∞
dy
(
−∂V1(x, y)
∂xJ1m(x, y) − ∂V2(x, y)
∂xJ2m(x, y)
)
. (B.3)
Similarly, the heat exchange rate of the cargo with the environment is
〈Dc〉 =∫ 1
0dx∫
∞
−∞
dy
(
−∂V1(x, y)
∂yJ1c(x, y) − ∂V2(x, y)
∂yJ2c(x, y)
)
. (B.4)
102
The cargo output rate is the same as the work done by the cargo against the load,
〈W 〉 = −∫ 1
0dx∫
∞
−∞
dyF (J1c(x, y) + J2c(x, y)) . (B.5)
The efficiency of the system is defined as the ratio of the cargo output rate and the motor
input rate,
ηeff =〈W 〉〈Rm〉
. (B.6)
At steady state it can be shown that
〈Rm〉 = 〈Dm〉 + 〈Dc〉 + 〈W 〉, (B.7)
Using the numerical solution of Fokker-Planck equations and the above definitions, we get
0
2
4
6
8
0.75 1.25 1.75 2.25
Inp
ut/
Dis
sip
atio
n R
ate
s
log10(ks)
<Rm><D.
m><D.
c>
Figure B.1: Variation of the energetic parameters-the motor energy input rate (〈Rm〉), themotor dissipation rate (〈Dm〉) and the cargo dissipation rate (〈Dc〉) – with the stiffness ofthe link. The potential parameters (for notations see Fig. 3.1) are, u = 20, a = 0.3, Dc =0.1, γ = 1.6 and external load f = 0.
the energetics. Fig. B.1 shows the variation of energetic parameters with the link stiffness
103
in the absence of any external force. The behavior of energetic quantities can be understood
intuitively if we consider the limit u ≫ 1 as in the velocity explanation. Transition from
state 2 to 1 hardly supplies any energy, because motor spends most of its time close to the
potential minima in state 2. Hence, the net energy input rate can be approximated as
〈Rm〉 = (γ/2)∫
∞
−∞
dx∫
∞
−∞
dyφ2(x)P (x, y, 1/γ), (B.8)
where P (x, y, 1/γ) is the probability distribution of being in state 1. Since∫
∞
−∞dyP (x, y, 1/γ)
0
0.001
0.002
0.003
0.004
0.75 1.25 1.75 2.25
Eff
icie
ncy
log10(ks)
γ=1.6γ=160
Figure B.2: Variation of the efficiency of the motor with the stiffness of the link for twodifferent transition rates. The potential parameters (for notations see Fig. 3.1)are, u = 20,a = 0.3, Dc = 0.1, and external load f = -0.4.
is a Gaussian distribution peaked near x = 0, the greater the spatial spread of this Gaus-
sian, the greater the input energy rate, since this allows motor to make transitions to high
energy regions of state 2 when the motor state changes from 1 to 2. This is the case for
very low stiffness of the link. As the stiffness of the link is increased, the width of the motor
probability distribution narrows and the motor input rate decreases, since more and more of
the motor transitions will be to the low energy regions of state 2. The same reasoning also
104
explains the decrease of motor dissipation with increased stiffness. For very low stiffness,
only a small fraction of motor input is transmitted to the cargo before being dissipated at
the motor. As the stiffness is increased, more and more of the motor input is transmitted to
the cargo and dissipated there.
Figure B.2 shows the efficiency of the motor (B.6) as a function of the link stiffness. For
moderate transition rate (γ), there is an optimal stiffness of the link which maximizes the
efficiency. The velocity variation for this transition rate also shows a peak at this stiffness
and the same explanation for velocity peak can explain the efficiency peak. At this stiffness,
due to increased velocity, motor is able to transmit more power to the cargo.
105
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Author’s Biography
Prasanth Sankar was born on Nov. 1, 1976 in Kerala, India. He received MSc. degree in
Physics in 1998 from the Indian Institute of Technology, Kharagpur. He received his M.
S. in Physics, from the University of Illinois at Urbana-Champaign in 2000. While at the
University of Illinois he was supported by teaching and research assistantships.
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