Proceedings of SMASIS 2008
2008 ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems
October 28-30, 2008, Ellicott City, Maryland, USA
SMASIS08-585
A STOCHASTIC MECHANO-CHEMICAL MODEL FOR
COOPERATIVE MOTOR PROTEIN DYNAMICS
Bogdan I. Epureanu∗
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan 48109-2125
Email: epureanu@umich.edu
Arjun Krishnan
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan 48109-2125
Email: arjunkc@umich.edu
ABSTRACT
The Peskin-Oster model for single molecule experiments on
kinesin is extended to analyze the motion of two motors cooperatively pulling a bead in a kinesin motility assay. The various
assumptions inherent in the model are tested for the case of two
motors, and justified rigorously. We formulate a Markov process similar to the single motor case, and simulate the master
equations on a restricted set {−N, · · · , N} for each motor. The
model is simulated for different initial configurations, and motors
with identical parameters are found to synchronize in expectation, both spatially and in phase. It is found that synchronization
is faster at higher loads and for higher adenosine triphosphate
(ATP) concentrations. From a physical perspective, it is shown
that the motors move to a configuration where the load is shared
equally between them.
along track-like structures called microtubules (MTs). The MTs
are hollow cylindrical structures formed by bundles of protofilaments aligned in parallel. The proto-filaments themselves are
long single-strand polymers of tubulin dimers. MTs have a specific polarity, and a set of equally spaced chemical binding sites
where motor protein attach.
A kinesin motor protein consists of two globular domains
called heads. Each of the heads is a fully functional enzyme capable of hydrolyzing adenosine triphosphate (ATP). The heads
are both attached to the end of a long coiled-coil tether. The
other end of the tether is attached to the cargo. The heads have
radii of about 10 nm, and the tether is about 60 nm long when unstretched. Kinesin consumes ATP and uses the resulting chemical energy to walk along the microtubule by moving its heads
alternately from one binding site to another. The resulting steps
are 8 nm long.
Kinesins have been studied since the 90s, and experimental
techniques have now become sufficiently precise to obtain useful data about their behavior. Their dynamics are rich and complex. First, it is surprising that a device so small can produce
directed motion in the thermal environment of the cell. Second,
the details of its chemical and mechanical cycle are sparse, and
many mechanisms governing their function are unknown. For
example, it is not known precisely how the two ATPases synchronize their enzymatic cycles to produce directed mechanical motion [1]. Third, molecular motors are seen to organize MTs into
functional remarkably regular structures during mitosis [2]. It is
also hypothesized that kinesins work cooperatively in organized
teams to move loads inside the cell. Understanding the physics
INTRODUCTION
The cell contains many mechanisms to transport nutrients, proteins and organelles, and to perform various organizational activities during essential cellular processes like mitosis.
Smaller molecules like glucose can diffuse to their target locations through the cytosol, but larger organelles like mitochondria
need to be carried from one location to other. Some cells carry
appendages such as flagella and cilia, which are moved like flexible oars for propulsion. These mechanical and transport activities are performed mainly through the use of molecular motors
(motor proteins) like dynein, kinesin and myosin, which move
∗ Address
all correspondence to this author.
1
c 2008 by ASME
Copyright form:
of such synchronized behavior in a fundamentally stochastic system is a challenging problem.
Modeling kinesin is complicated by its discrete and continuous (hybrid) dynamics and the inherent randomness due to thermal noise [3]. There are many models that treat the stepping
and kinetics as a Markov chain [4] or as a Brownian particle diffusing over energy wells [5], [6]. The first physically consistent
model that combined mechanical, kinetic and stochastic aspects
of kinesin’s cycle was the Peskin-Oster model [7]. Other notable
models that try to use detailed mechanics are Aztberger and Peskin [8], Hendricks, Epureanu and Meyhofer [9], and Derenyi
and Vicsek [10].
Current models for cooperative multiple motor behavior are
abstract, and formulated in terms of thermal ratchet models [11].
Our analysis is based on Peskin-Oster model, which is more
suited to our goal of obtaining greater physical insight into the
dynamics.
Db
∂2 p
∂2 p
Dh ∂ ∂V +
D
+
p +
h
∂x2b
∂x2h kB T ∂xh ∂xh
Db ∂ ∂V ∂p
p −
= 0, (1)
kB T ∂xb ∂xb
∂t
where p is the joint density of the bead and diffusing head
locations xb and xh , Db and Dh are the diffusion coefficients,
kB T is thermal energy, and V is the potential energy of the
system. The subscripts b and h stand for bead and head respectively.
4. As soon as ATP hydrolysis completes, the free head regains
its affinity for the MT and quickly finds a free binding site to
attach to. The potential in which the head diffuses is biased
in the forward direction thus creating a “power stroke”, i.e.,
the minimum of the biasing potential is located close to the
forward bind site.
5. Once this binding takes place, ATP hydrolysis is complete
and the by-products of the hydrolysis are ejected from the
nucleotide binding pocket. This brings the system back to
its original state S · S with both heads free of nucleotide. One
of the heads has travelled a distance of 16 nm and the bead
has moved 8 nm, the spacing on the MT.
The Peskin-Oster Model
The Peskin-Oster (PO) model and other variants based on
it [4, 7, 8, 12] have been fairly successful in predicting various
aspects of kinesin’s mechanism. The PO model is one dimensional, tracking only coordinates along the MT. The chemistry
is assumed to have only two steps, ATP binding and subsequent
hydrolysis. The bead is attached to the elastic tether, which is
assumed to obey Hooke’s law. This is an assumption that is
well-justified by the analysis of Atzberger and Peskin [8], who
obtained tether energies from experimental data. The other end
of the tether is attached to a “hinge-point” to which both motors are attached. Crystallographic studies show that the heads
are attached to a common point by their neck-linkers [13]. The
neck-linkers may be modelled as a elastic or rigid elements.
Let us define two binding states for each head as S (strongly
bound to the MT) and W (weakly bound to the MT). The
bead, hinge, bound head and free head locations are denoted by
xb , xh , xbnd and x. The hinge point is defined to be between
the two heads at all times. The various steps in the mechanochemical cycle of this model can be summarized as follows:
There are two distinct states in which we must define the
system potential energy: (1) when both heads are bound to the
MT, and (2) when one head is free and diffuses along with the
bead. The coupled diffusion of the bead and the head is difficult
to solve analytically. A key simplification made here is to separate time-scales by assuming that the head’s diffusion is much
faster than the bead’s. This is because the approximate radius of
the head is much smaller than that of the bead - a ratio of 1/1000
- implying that the diffusion coefficient is much larger. That is,
at any instant, the head is at relative equilibrium with the bead.
Then, we can write an effective potential for the bead, and
solve for its probability density. Furthermore, if the tether stiffness is large, we can consider the time-scale for the bead to reach
its equilibrium density to be smaller than the chemical time scale
(see Section A.1). This means that relative to the chemical processes, the bead can also be considered to be at its equilibrium
position at every instant.
1. The motor starts in the state S · S. In this state, the motor
is rigid and the bead undergoes a Brownian motion in some
potential well established by various elastic components of
the system. Then, xbnd1 = 0, xbnd2 = 8, xh = 4.
2. Next, ATP binds to one of the two heads, and the system
undergoes a transition S · S → S ·W , i.e., one of the heads is
now weakly bound and is free to diffuse in a potential biased
towards the next forward binding site. This biasing potential
is another modeling input. One simple choice is to assume a
quadratic potential. Suppose ATP binds to the forward head.
Then, xbnd = 8 nm, xh = 8 + x0, x = 8 + 2x0.
3. While the ATP hydrolysis is taking place, the free head and
the bead diffuse together, governed by an equation of the
Then defining a single state variable k that is allowed to take
integer (k ∈ I) and half-integer values (k ∈ H), we can designate
whether the motor has both heads bound or just one, respectively.
Constructing a Markov process over the states k, we can derive
expressions for the rate of change of the mean and variance of
the state (and hence the bead position).
2
c 2008 by ASME
Copyright PO model for a single motor, plus a few more as follows:
1. Each head equilibriates instantaneously with the bead when
diffusing. This is still valid because of the ratio of diffusion
coefficients of the head and the bead, Dh /Db → ∞.
2. The time scale of diffusion of the bead is still negligible. The
tethers are in parallel, making the effective potential nearly
twice as stiff, which makes the time-scale smaller than that
for a single motor (see Section A.1).
3. A head, when diffusing, almost instantaneously binds to a
binding site once hydrolysis completes. That is, the probability that the hydrolysis processes of both motors complete
simultaneously in the small time window required for binding is negligible.
Figure 1.
Schematic of likely typical experimental setup
1 Modeling Two Motors
Here, we extend basic PO model to characterize the motion
of two kinesin motors pulling the bead and make predictions regarding their cooperative behavior. The objective is to compare
this with single-motor experiments, explore the cooperative dynamics, and to serve as a more severe test of the model.
The PO model requires very little modification. All we have
to do is to keep track of an extra coordinate, that of the additional
motor as shown schematically in Fig. 1. The state of the system
is represented by ( j, k), where j and k, as before, can both be
either integers or half-integers. If j or k is a half-integer, one
head of the corresponding motor is unbound from the MT and
is diffusing, and hence has another coordinate associated with it,
the positions x j and xk respectively. The bead position is denoted
by xb .
There are three possible combinations of states - depending
on whether each motor has a head diffusing or not - and three
associated potentials. The potential V is a function of the head
locations (if applicable) x j , xk , bead position xb and the separation between motors k − j. The definitions of xbnd and xh j (k) are
as before, and are uniquely determined by x j(k) . One obtains
2
V (x j , xk , xb ) = f xb + K2th xh j − xbnd +
2
+ K2th xhk − xb + W (x j , xk ),
It might turn out that when the motors are both perfectly
synchronized, the hydrolysis processes might be coupled together possibly through load dependence. Including the possibility that the hydrolysis processes complete together is not
very hard. Calculating the probabilities of binding forward
and backward for each reduces to a first-passage problem in
two dimensions. In our particular case, the problem is simple
since the potentials that each head experiences are decoupled
as V (x j , xk , xb ) = V1 (x j , xb ) + V2(xk , xb ), clearly, from Eq. 2. In
general the equations are elliptic (but linear) and yield to simple
finite-differencing schemes.
2 Simulation
As for a single motor, we construct a Markov process, with
the states and transitions as indicated in Fig. 2. The only complication here is that the state-space is infinite, and the transition
probabilities of the Markov process depend on state. We have
to calculate each probability numerically, and there are limited
analytical options.
To circumvent the problem of infinite number of states, we
resort to a truncation. We choose a sufficiently large number of
binding sites {−N, · · · , N}, and make the boundaries absorbing,
by stating that once either motor reaches ±N, they are absorbed
and do not undergo any further transitions. We then simulate the
governing differential equations for the probabilities of being in
each site, given different initial configurations.
Let C j,k be the probability of being in state ( j, k). Let
Pj,k represent the probability of binding forward for a motor whether it is for the motor at j or k depends on whether they are
integers or half-integers. When the motors are both diffusing j and k are both half-integers - and hydrolysis completes, either
motor binds instantaneously and we must calculate two probabilities of binding forward for each motor P1 j,k , P2 j,k associated
with state { j, k}. The other chemical parameters are α, βb , β f
and γ, the same as in the PO model. We need to write four different types of differential equations for the four different kinds of
(2)
where f is the force on the bead, W (x) is the biasing potential and
Kth is the tether stiffness. The biasing potential W (x) is modelled
as in [8], and depends on the state { j, k}, as:

0
j ∈ I, k ∈ I



 Kbias xh − (xbnd + x0)2 j ∈ I, k ∈ H
j
k
2
2
W (x j , xk ) = Kbias
. (3)

x
−
(x
hj
bnd j + x0 ) +

2

2
 Kbias
+ 2 xhk − (xbnd j + x0 ) j ∈ H, k ∈ H
We continue to make the assumptions made in the original
3
c 2008 by ASME
Copyright " #
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Figure 2.
Possible State Transitions, where
j, k
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represents the state of a motor. Here, both
j
and
k
and assumed to be integers. Only a part of the
possible states is shown, since there are an infinite number of possible states.
states. Then, for −N + 1 < j, k < N − 1 one obtains,
dC j,k
dt
for a single motor. If there is only one diffusing head (suppose it
is k) determine the conditional (given xb ) head density ρ(xk |xb )
and probability of binding forward pt (k, xb ). The total probability Pj,k ( f ) follows immediately by using the equilibrium bead
density ρbead (xb ).
If both motors are diffusing, the conditional densities and
probabilities continue to be independent, ρ(x j |xb ), ρ(xk |xb ),
pt ( j, xb ) and pt (k, xb ) because the potential for the two motors
decouples as mentioned in Section 1. Then, the total probabilities of binding forward can be written in terms of the the effective
potential Ve f f (xb ) and bead density ρbead (xb ) as,


 αP1 j− 21 ,kC j−1/2,k +



αP2 j,k− 1 C j,k− 1 +


2
2



+α(1 − P1 j+ 1 ,k )C j+ 1 ,k +


2
2



+α(1 − P2 j,k+ 1 )C j,k+ 1 +


2
2



−2(βb + β f )C j,k
j ∈ I, k ∈ I





. (4)
= αPj− 21 ,kC j− 12 ,k +


+α(1
−
P
)C
+
1
1

j+ 2 ,k
j+ 2 ,k




βbC j,k− 1 +


2


 β f C j,k+ 1 + (βb + β f + α)C j,k j ∈ I, k ∈ H


2







βbC j− 1 ,k + β f C j+ 1 ,k +


2
2

 βbC
j,k− 1 + β f C j,k+ 1 − 2αC j,k j ∈ H, k ∈ H
2
(k+1/2)L
−V1 (xk , xb )
dxk ×
kB T
−(k−3/2)L
(5)
Z ( j+1/2)L
−V2(x j , xb ) ×
exp
dx j ,
kB T
−( j−3/2)L
Z
Ve f f (xb ) = −kB T log
P1 j,k ( f ) =
Z ∞
pt (x j , xb )ρbead (xb )dxb ,
(6)
P2 j,k ( f ) =
Z ∞
pt (xk , xb )ρbead (xb )dxb .
(7)
2
In Eq. 4, note that for each differential equation, there are
source terms from the neighboring states from which the system can enter that particular state and one decay term for the
rate at which the system leaves that particular state. So, for the
boundary equations - if | j|, |k| > N − 1 in Eq. 4 - we drop the
appropriate decay term and source terms if j, k = ±N.
The next step is to calculate the total probabilities for each
relevant state. The procedure is essentially that described in [7]
exp
−∞
−∞
There are a few homogeneities in the problem we can exploit to ease computational effort: any state ( j, k) is identical to
(0, k − j) (or (1/2, k − j) if j is a half-integer) but for an origin shift of jL. Since the motors are identical, parameters like
4
c 2008 by ASME
Copyright the probability of binding forward and the potential energy are
independent of the identity of the motor. When the motors are
both diffusing, if state ( j, k) has probabilities P1 j,k , P2 j,k , state
( j1 , k1 ) with j1 = k, k1 = j, has probabilities P2 j,k , P1 j,k because
of the obvious homogeneity. These observations significantly reduce the number of calculations.
Velocity @nmsD
600
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á
ó
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á
500
ç
á
400
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á
á
ç
300
100
á
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ç 2 Motors Init = 0
á 2 Motors Init = 4
ó 1 Motor
200
á
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0
3 Results
The two motors were simulated on a grid of binding sites
with {−50, · · · , 50} for two ATP concentrations - the same as the
single-molecule data - for a range of forces (−4.5, 9.5) pN. The
ODEs in Eq. 4 were simulated for a time such that edge-effects,
i.e., the effect of the absorbing binding sites at −50 and at 50 on
the motor dynamics was negligible. Results were obtained for
different initial configurations.
The multiple motor data is plotted against the single-motor
predictions for the two different ATP concentrations in Fig. 3.
Notice that at strongly assisting loads, the mean velocity of the
bead is lower than the single-motor, owing to the fact that at least
two ATP hydrolyses need to occur to move 8 nm. Because of the
randomness, the hydrolysis can never be perfectly synchronous.
However, because the load is shared between the two motors, the
stall force is extended to nearly twice the stall-force of a single
motor. We explore different aspects of the motor and bead dynamics in the following sections.
-4
0
-2
2
4
6
8
Force @ pND
(a) Force vs. Velocity, ATP = 1600 µM
Velocity @nmsD
40
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á
ó
ç
á
ç
á
30
ó
ç
á
ç
á
20
á
ç
ó
ç 2 Motors Init = 0
á 2 Motors Init = 4
ó 1 Motor
10
á
ç
á
ç
á
ç
ó
ó
0
-4
-2
0
2
4
6
8
Force @ pND
(b) Force vs. Velocity, ATP = 4.2 µM
Figure 3.
Two motors pulling a bead: force velocity data for different
ATP concentrations compared with single motor predictions using the
PO model. The multiple motor data is obtained for different initial motor configurations, and since the motors synchronize, the velocity predic-
3.1
Motor Dynamics
A general observation from the detailed numerical study performed is that the motor synchronize for all different initial spatial separations for high and low ATP concentrations. The reason
lies in the dependence on the states ( j, k) of the probability of
binding forward. As mentioned before, the probability depends
only on the relative separation j − k. Also, note that there are
two probabilities when j and k are both half-integers. The probabilities are plotted in Fig. 4(a) for a low and high force - clearly
these are dependent solely on mechanical parameters - with j = 0
in Fig. 4(a) and j = 1/2 in Fig. 4(b). It is seen that the lagging
motor has a much larger tendency to bind forward in both cases
(when the leading motor is bound firmly to the MT, and when it
has bound ATP and one of its heads is diffusing).
A more physical picture is shown in Fig. 5. Here, the mean
load “felt” by each motor - i.e., the extension of its tether - is
plotted as a function of time. Figure 5 shows that while the total
load, as it must be, is constant and equal to the external force,
the load on each individual motor varies until it is shared equally
on the two motors. Hence, it is unsurprising that the expected
positions of the motor locations synchronize.
A plot of the individual motor positions (Fig. 6) shows a
dynamic picture of the motor synchronization process. We obtained plots for low and high ATP concentrations and loads to
tions are nearly identical.
j = 0, k = n.
Init = n
represents the initial configuration
indicate that this synchronization happens for a large parameter
range. Note that at low ATP, it takes a longer time for the load
to be equally shared and for the motors to synchronize - this is
a consequence of the lower rate of stepping. At higher loads,
synchronization happens faster than at lower loads. The reason
this happens is indicated by the probabilities of binding forward
in Fig. 4(a), which shows that the probability of binding forward
for the leading motor falls to zero quicker at higher loads than
at lower loads. Another observation is that the leading motor
“waits” for the lagging motor to “catch up”. Again, this fact is
easily reconciled with the data from Fig. 4.
3.2 Bead Variance and Randomness
The bead position and its standard deviation is easily obtained from the relative motor positions. To calculate the randomness parameter, let xb (t) be the bead position so that,
r = lim
t→∞
5
Var[xb (t)]
Var[xb (t)]
t
1
= lim
.
E[xb (t)]L t→∞
t
E[xb (t)] L
(8)
c 2008 by ASME
Copyright 40
10
motor1 (0) = 4
motor2 (0) = 0
30
25
20
15
10
6
4
2
0
−2
5
0
motor1 (0) = −4
motor2 (0) = 0
8
Motors Location site #
Motors Location site #
35
0
0.1
0.2
0.3
0.4
−4
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time [sec]
Time [sec]
(a) Motor position (site index) vs. time for both motors, different
initial separation; f = 0.5 pN, ATP = 1600 µM
(b) Motor position (site index) vs. time for both motors, different initial separation; f = 7.5 pN, ATP = 1600 µM
16
4
motor1 (0) = −3
motor2 (0) = 0
14
motor1 (0) = −3
motor2 (0) = 0
3
Motors Location site #
Motors Location site #
12
10
8
6
4
2
2
1
0
−1
0
−2
−2
−4
0
0.5
1
1.5
2
2.5
3
−3
3.5
Time [sec]
0
0.5
1
1.5
2
2.5
3
3.5
Time [sec]
(c) Motor position (site index) vs. time for both motors, different
initial separation; f = 0.5 pN, ATP = 4.2 µM
(d) Motor position (site index) vs. time for both motors, different initial separation; f = 7.5 pN, ATP = 4.2 µM
Figure 6. Motor position vs. time for different initial separations, loads and ATP. Note that the motors synchronize under all conditions.
properties of the randomness parameter for Poisson processes)
that now both motors have to complete a step for the bead to
move 8 nm, lowering the randomness parameter.
The most direct way to calculate the randomness parameter
is to solve Eq. 4 for a long time interval, find the variance and expected value, and then take their ratio. It can be seen in Fig. 7(a)
that the randomness parameter has not quite reached its steadystate value. However, it is reasonable to believe that it will indeed
reach a limiting value if computed for a larger time interval. We
know from renewal theory [14] ( when applied to a single kinesin
motor), that the ratios E[xb (t)]/t and Var[xb (t)]/t approach constant limits as t → ∞. Since, essentially, this is also a renewal
process - but with a much more complicated reward function a similar limit is to be expected. Hence, a better estimate of the
randomness parameter can be obtained by computing the slopes
of the mean and variance of xb (t) for a large time t. This is shown
in Fig. 7(b).
The randomness parameter computed by the two methods is
shown in Fig. 3.2 and is compared with the single motor predictions. Note that the randomness parameter is lower for all forces.
A not-very-convincing way to physically justify this is to argue
that for each renewal - one motor completes a step - the bead
steps only by L/2. Next, one might argue (based on the general
4 Conclusions
We extended the Peskin-Oster model to two motors pulling
a bead to analyze cooperative behavior between multiple motors.
The model was simulated on a truncated grid for a range of loads,
ATP concentrations and other parameters.
The graphs of force vs. mean velocity of the bead show
that the velocity is lower than a single motor pulling the bead at
assisting loads. This is because the both motors have to finish
a hydrolysis before the bead steps 8 nm and the motors cannot
be perfectly in sync in a random environment. Stall forces were
seen to be extended to nearly twice the single motor range.
The motors were observed to synchronize in expected value
for the whole range of parameters simulated. The synchronization was faster at higher ATP concentrations because of the increased rate of stepping. Synchronization was also observed to
6
c 2008 by ASME
Copyright æææææææææææææææææææ
òòòòòòòòòòòòòòòòò
æ
æ
ò
8
æ f = 0.5 pN
f2,m 1, m1 (0) = −4
æ
0.8
ò f = 7.5 pN
ò
0.6
f2,m 2, m2 (0) = 0
6
f2 = 7.5pN
æ
4
ò
0.4
Load [pN ]
Probability of Binding Forward
1.0
æ
0.2
ò
æ
ò
0.0
-20
-10
2
0
ò ò òæ
ò ææææææææææææææ
òòòòòòòòòòòòòò
0
10
20
f1,m 1, m1 (0) = 4
−2
f1,m 2, m2 (0) = 0
Site Index
−4
f1 = 0.5 pN
0
0.1
0.2
1.0 ç
æ ç
æ ç
æ ç
æ ç
æ ç
æ ç
æ ç
æ ç
æ ç
æ
æ ç
æ ç
æ ç
æ ç
æ ç
æ ç
ó ç ç ç ç
ó
0.8
æ
ç
ó
æ
ò
ó
æ
0.6
æ
f
f
f
f
= 0.5
= 0.5
= 7.5
= 7.5
0.3
0.4
0.5
0.6
Time [sec]
(a) Load on each motor vs. time; motor1 (t = 0) = −4,4, motor2 (t = 0) = 0,
ATP = 1600 µM
pN, lagging
pN, leading
pN, lagging
pN, leading
8
ó
f2,m 1, m1 (0) = 0
0.4
6
æ
ò
f2,m 2, m2 (0) = −3
ó
0.2
f2 = 7.5pN
4
ò
ó
Load [pN ]
Probability of Binding Forward
(a) Probability of binding forward vs. site index k; j = 0 is fixed
ò ò ò ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò ó
ò
0.0
0
5
10
15
Site Index
2
0
(b) The two probabilities of binding forwards when both diffusing vs. site
index k; j = 1/2 is fixed
−2
Figure 4.
−4
Probabilities of motors binding to the forward site, and their
dependence on the relative states - spatial separation and phase - of the
two motors
f2,m 1, m1 (0) = 0
f2,m 2, m2 (0) = −3
f2 = 7.5pN
0
0.5
1
1.5
2
2.5
3
3.5
4
Time [sec]
(b) Load on each motor vs. time; motor1 (t = 0) = −3, motor2 (t = 0) = 0,
ATP = 4.2 µM
Figure 5.
be faster at higher loads. This was explained in terms of relative
magnitudes of the probabilities of binding forward.
Load sharing between the two motors for high and low ATP
and load. The dashed lines show the sum of the individual loads. They
are constants, as expected.
The leading motor was observed to “wait” for the lagging
motor to “catch up”. Again, the probabilities provide a good
explanation for this. The reason lagging motor waits can also
be understood from the graphs showing the loads experienced
by each motor individually. The leading motor was shown to
experience a much larger load than the lagging motor.
A Appendix
A.1 Diffusion
One result that is constantly used for overdamped Brownian
motion is related to the Ornstein-Uhlenbeck process [15]. It was
originally developed to show how the velocity of a fluid particle
relaxes to the Maxwell-Boltzmann distribution. In terms of the
velocity U, we can write Newton’s Law as a stochastic differential equation in Ito form as:
The randomness parameter was also calculated, and as expected, was lower than the single motor prediction. This is related to the fact that both motors must take a step for the bead to
traverse 8 nm. Arguing based on simple results for the Poisson
process, we can conclude that including this “extra” substep in
the multiple motor case decreases the randomness parameter.
√
dU(t) = −βU(t)dt + σZ(t) dt,
7
(9)
c 2008 by ASME
Copyright f = 0.5 pN
f = 7.5 pN
ç 1 Motor
á 2 Motors method 1
ó 2 Motors method 2
1.0
Randomness
0.9
Randomness Var[xb ]/(E[xb ] L)
ó
1.2
1
0.8
0.7
0.8
ó
á
0.6
0.4
á
ó
á
ó
á
ó
á
ó
á
ó
ó
á
ó
á
ó
á
ó
á
ç
ç
ç
ç
ç
ç
ç
ç
ç
ó
á
ó
á
á
0.2
0.6
ç
0.0
-4
0.5
0.4
0.35
-2
0
2
4
6
8
Force @ pND
0.4
0.45
0.5
0.55
Figure 8. Force vs. Randomness for two motors computed by two different methods and one motor from the PO model
0.6
Time [sec]
(a) Randomness Parameter vs. t
with of stiffness Ksp ), it follows that the particle relaxes to its
equilibrium position with a time constant given by γ/Ksp , and a
variance of kB T /Ksp .
1000
f = 0.5 pN
f = 7.5 pN
linear fits
Bead Variance Var[x b ] [nm2 ]
900
800
700
A.2 Effective Potential
The head is restricted artificially between the sites in the PO
model for convenience. Hence, the effective potential that the
bead diffuses in is no longer quadratic. To make the argument
that the diffusion time-scale is still smaller than the chemical
time-scales, we must show that the effective potential for the
bead is still nearly quadratic and its stiffness is of an order of
magnitude such that the time scale estimate (from Section A.1)
is still much smaller than the chemical time-scales.
If there are two motors, then there are two tethers in parallel,
and the bead diffuses in a potential twice as stiff, making its diffusion faster than in the single motor case. Hence, it is enough if
we show the following for the single motor case. It is easy to see
that the only case which we have to be concerned about is when
the head and the bead are diffusing. Then, we can find derivatives of the effective potential at the equilibrium point x̄b , and
with a single application of the mean value theorem, we arrive at
the following bounds:
600
500
400
300
200
100
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [sec]
(b) Bead Variance Var(xb ) vs. t and linear fits
Figure 7.
Evolution of the bead variance and randomness parameter
with time for ATP = 1600 µM
where β = γ/m with γ being the Stokes’ Law friction coefficient.
Then, we may solve the Fokker-Planck equation [16] to find that
U(t) is normally distributed with mean and variance given by:
E[U(t)] = u0 e−βt ,
Var[U(t)] =
σ2 (1 − e−2βt )
2β
d2V (xb ) (Kth L)2
>
K
−
,
th
kB T
dx2b x̄b
d3V (xb ) (Kth L)3
.
<
3
(kB T )2
dxb x̄b
(10)
.
(11)
The constant σ can be related to the thermal energy kB T
by comparing it with the Maxwell-Boltzmann distribution, and
all the quantities of interest can be most conveniently written in
terms of a diffusion coefficient, D = kB T /γ. For an overdamped
particle in an quadratic potential (corresponding to a linear spring
(12)
(13)
There are more terms in higher derivatives, but the key is
that they are of the form (Kth )m (Kth L)n /(kB T )n−1 , where m, n
are positive integers. Thus, all we require is that (Kth L)2 /(kB T )
is small. We use Kth = 0.25 pN/nm following [7] - analysis
8
c 2008 by ASME
Copyright of the experimental data in [8] showed that it’s actually about
0.15 pN/nm - the thermal energy is around 4.14 pN · nm and
L = 8 nm, which gives us a value of about 0.9 for the ratio in
question and a time-scale of about 10 µsec. Although it is rather
large, it a worst case estimate in any case. The effective potential
for a range of forces is shown in Fig. 9, along with quadratic fits
to them. It is seen that the quadratic fits are very good, and the
stiffness varies by at most 0.03 pN/nm at the highest load which
is much less than the worst-case bound established.
40
20
à
à
à
à
à
Energy @ pN×nmD
[11]
à
à
à
à
à
-5
pN, Ksp = 0.24
0 pN, Ksp = 0.22
ò 5 pN, Ksp = 0.22
æ
à
à
à
à
à
à
à
à
0
à
à
à
à
òò
òò
à
à
àà
àà
àà
àà
òò
òò
òò
ààà
à
à
[12]
æ
æ
òò
òò
òò
òò
æ
æ
òò
òò
-40
[10]
à
à
à
à
à
à
-20
[9]
æ
æ
òò
òò
òò
òò òò òò òò òò
æ
æ
òò
òò
æ
æ
æ
æ
-60
[13]
æ
æ
æ
æ
æ æ
æ
æ æ
-40
-20
0
20
40
xb
Figure 9.
[14]
Effective potentials on the bead for different loads and
quadratic fits to them
[15]
[16]
namics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo
Tether”. Bulletin of Mathematical Biology, 68 (1), pp. 131.
Hendricks, A., Epureanu, B.I., and Meyhofer, E., 2006.
“Synchronization of Motor Proteins Coupled Through a
Shared Load”. Proceedings of IMECE, 257, IMECE200615752, p. 489.
Derenyi, I., and Vicsek, T., 1996. “The Kinesin Walk: A
Dynamic Model with Elastically Coupled Heads”. Proceedings of the National Academy of Sciences, 93 (13) ,
pp. 6775.
Badoual, M., Jülicher, F., and Prost, J., 2002. “BiDirectional Cooperative Motion of Molecular Motors”.
Proceedings of the National Academy of Sciences, 99 (10),
p. 6696.
Mogilner, A., Fisher, A., and Baskin, R., 2001. “Structural
Changes in the Neck Linker of Kinesin Explain the Load
Dependence of the Motor’s Mechanical Cycle”. Journal of
Theoretical Biology, 211 (2), pp. 143.
Rice, S., Lin, A., Safer, D., Hart, C., Naber, N., Carragher,
B., Cain, S., Pechatnikova, E., Wilson-Kubalek, E., Whittaker, M., et al., 1999. “A structural change in the kinesin
motor protein that drives motility”. Nature, 402, pp. 778.
Karlin, S., and Taylor, H., 1975. A First Course in Stochastic Processes. Academic Press.
Uhlenbeck, G., and Ornstein, L., 1930. “On the Theory of
the Brownian Motion”. Physical Review, 36 (5), pp. 823.
Cox, D., and Miller, H., 1977. The Theory of Stochastic
Processes. Chapman & Hall/CRC.
REFERENCES
[1] Block, S., 2007. “Kinesin Motor Mechanics: Binding,
Stepping, Tracking, Gating, and Limping”. Biophysical
Journal, 92 (9), pp. 2986.
[2] Nedelec, F., Surrey, T., Maggs, A., and Leibler, S., 1997.
“Self-Organization of Microtubules and Motors”. Nature,
389 (6648) , pp. 305.
[3] Astumian, R., and Haenggi, P., 2002. “Brownian Motors”.
Physics Today, 55 (11), pp. 33.
[4] Fisher, M., and Kolomeisky, A., 2001.
“Simple
Mechanochemistry Describes the Dynamics of Kinesin
Molecules”. Proceedings of the National Academy of Sciences, p. 7748.
[5] Bier, M., 1997. “Brownian Ratchets in Physics and Biology”. Contemporary Physics, 38 (6), pp. 371.
[6] Reimann, P., and Hänggi, P., 2002. “Introduction to the
physics of Brownian motors”. Applied Physics A: Materials Science & Processing, 75 (2), pp. 169.
[7] Peskin, C., and Oster, G., 1995. “Coordinated Hydrolysis
Explains the Mechanical Behavior of Kinesin”. Biophysical Journal, 68 (4 Suppl).
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c 2008 by ASME
Copyright