# Stokes Efficiency of Molecular Motor-Cargo Systems

Authors: Hongyun Wang; Hong Zhou
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

A molecular motor utilizes chemical free energy to generate a unidirectional motion
through the viscous fluid. In many experimental settings and biological settings, a
molecular motor is elastically linked to a cargo. The stochastic motion of a molecular
motor-cargo system is governed by a set of Langevin equations, each corresponding to
an individual chemical occupancy state. The change of chemical occupancy state is
modeled by a continuous time discrete space Markov process. The probability density
of a motor-cargo system is governed by a two-dimensional Fokker-Planck equation. The
operation of a molecular motor is dominated by high viscous friction and large thermal
fluctuations from surrounding fluid. The instantaneous velocity of a molecular motor
is highly stochastic: the past velocity is quickly damped by the viscous friction and
the new velocity is quickly excited by bombardments of surrounding fluid molecules.
Thus, the theory for macroscopic motors should not be applied directly to molecular
motors without close examination. In particular, a molecular motor behaves differently
working against a viscous drag than working against a conservative force. The Stokes
efficiency was introduced to measure how efficiently a motor uses chemical free energy
to drive against viscous drag. For a motor without cargo, it was proved that the Stokes
efficiency is bounded by 100% [H. Wang and G. Oster, (2002)].
Here, we present a proof for the general motor-cargo system.


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## Body

## 1. Introduction and Mathematical Formulation

Molecular motors
play a central role in many cellular
functions. For example, anFo motor utilizes the transmembrane proton gradient to power the ATP
synthesis in the F1 portion of the ATP synthase; a Kinesin dimer hydrolyzes ATP to drive
intracellular vesicle transportation; and a V-ATPase hydrolyzes ATP to pump
protons against a large proton gradient to regulate intracellular acidity.
Understanding the operating principles of molecular motors is crucial to
comprehending intracellular protein transport and cell motility.Due to the small size of molecular motors, the inertia
of the motor is negligible. As a result, the motor operation is dominated by
high viscous friction and large thermal fluctuations from the fluid environment
[1]. Because of these
properties of molecular motors, the results for macroscopic motors, in general,
do not necessarily extend to molecular motors. In both macroscopic motors and
molecular motors, a chemical reaction can be used to generate a conformational
change and an active force at the reaction site. The conformational change
along with the active force is then mechanically delivered to drive the motor
motion. This mechanism of generating directed motion is called a power stroke
motor [2, 3]. However, in a molecular
motor, a unidirectional motion can be generated by a completely different
mechanism. For simplicity, we consider the case where a motor is restricted to
moving in one spatial dimension, such as moving along a polymer track. Suppose
we use the free energy released from the chemical reaction to block thermal
fluctuations toward one direction. Then, the motor will move toward the
opposite direction. Here the unidirectional motion is not directly driven by an
active force produced at the reaction site. Rather it is directly driven by the
bombardments of surrounding fluid molecules. Of course, drawing thermal energy
from an isothermal environment to drive motor motion is not sustainable without
a free energy input. The free energy for blocking the backward fluctuations and
rectifying the forward fluctuations comes from the chemical reaction. This
mechanism of generating a unidirectional motion is called a Brownian ratchet
[4–7].One of the main differences between molecular motors
and macroscopic motors is manifested in the issue of efficiency. For a
macroscopic motor, the efficiency is well defined no matter it is working
against a conservative force or working against a friction force. For a
molecular motor, we do need to distinguish these two cases. When a molecular
motor is working against a conservative force, the thermodynamic efficiency is
well defined and is the energy output to the external agent exerting the
conservative force divided by the chemical free energy consumption in the
motor. When a molecular motor is working against a viscous drag, the situation
is completely different. It has no energy output at all. One way to define
efficiency in this case is to proceed with the apparent energy output based on
the average velocity. The efficiency defined this way, called Stokes
efficiency, is different from the thermodynamic efficiency. First of all, these
two efficiencies are for two different cases: the thermodynamic efficiency is
for a motor working against a conservative force; and the Stokes efficiency is
for a motor working against a viscous drag. But this is not the only
difference. In single molecule experiments, a motor can be put to work against
a conservative force. In a different experimental setup, the same motor can be
put to work against a viscous drag. For a molecular motor, both the
thermodynamic efficiency and the Stokes efficiency can be measured but in two
different experimental setups. Before we can treat the Stokes efficiency as a
valid efficiency measurement, we need to show that it is bounded by 100%. Since
the apparent energy output used in the definition of the Stokes efficiency does
not have a thermodynamic meaning, the Stokes efficiency being bounded by 100%
cannot be argued simply from thermodynamical point of view. In this paper, we
prove mathematically that the Stokes efficiency is bounded by 100%. The proof
is based on the Fokker-Planck formulation of motor-cargo systems. Below we will
first describe this mathematical formulation.A molecular motor, in general, has many internal and
external degrees of freedom. One of these degrees of freedom is associated with
the motor's unidirectional motion, the main biological function of the motor.
For example, a Kinesin dimer walks along a microtubule filament toward the
positive end [8, 9]. There are many levels of models for molecular motors,
from simple kinetic models with a few states to all atom molecular dynamics. We
adopt a modeling approach of intermediate level in which the unidirectional
motion is followed explicitly and the effects of other degrees of freedom are
modeled in the mean field potential affecting the unidirectional motion
[2, 5, 7, 10]. To introduce this modeling approach of intermediate
level, we start with the simple case of a small particle in a fluid
environment, restricted to moving in one spatial dimension, and subject to a
static potential, ϕ(x),
where x is the coordinate along the spatial dimension.
This situation is actually very close to that of a molecular motor. The main
difference is that a molecular motor is driven by switching among a set of
static potentials, each corresponding to one chemical occupancy state. The
particle experiences are (1) the force derived
from  the
potential, (2) the viscous drag force from the fluid environment, and (3) the
Brownian force also from the fluid environment. Both the drag force and the
Brownian force arise from collisions of the particle with the surrounding fluid
molecules. The drag force is the mean of this stochastic bombarding force and
the Brownian force is the fluctuating part of this bombarding force. The drag
force on the particle is always opposing the motion and is proportional to the
velocity: drag=−ζu,
where u is the velocity and ζ is the drag coefficient of the particle. Here,
we adopt the convention that stochastic processes are denoted by boldface
letters. The Brownian force on the particle has zero mean and is modeled as a
Gaussian white noise, which is the formal derivative of the Weiner process (the
Brownian motion). The magnitude of the Brownian force is related to the drag
coefficient as(1.1)Brownianforce=2kBTζdW(t)dt,where kB is the Boltzmann constant, T is the absolute
temperature [11], and W(t) is the Weiner process [12]. This is a result of the
fluctuation-dissipation theorem [13–15]. The stochastic motion of the particle is governed by
the Langevin equation(1.2)mdudt=−ζu−ϕ′(x)+2kBTζdW(t)dt,dxdt=u, where m is the mass of the particle. Equation (1.2) has
two very different time scales: the time scale of the motor forgetting about
its current velocity (the time scale of inertia), and the time scale of moving
over one period of potential ϕ(x) (the time scale of reaction cycle). Dividing
both sides of  (1.2) by m and noticing that quantity t0≡m/ζ has the dimension of time, we rewrite  (1.2)
as(1.3)dudt=−1t0[u−g(t)].Here g(t) is a stochastic process given
by(1.4)g(t)=−Dϕ′(x)kBT+2DdW(t)dt,where D is the diffusion coefficient of the particle
and is related to the drag coefficient by the Einstein relation: D≡kBT/ζ [1]. The solution of  (1.3) has the form(1.5)u(t)=g(t)+exp⁡(−tt0)(u(0)−g(0))+1t0∫0texp⁡(−(t−s)t0)(g(s)−g(t))ds.Note that  (1.5) does not provide
an explicit expression for the solution of  (1.3) since g(t) is unknown. Nevertheless, it is clear from  (1.5)
that the effect of the initial velocity u(0) decays exponentially with time scale t0≡m/ζ.
For that reason, we call t0 the time scale of inertia. In general, the
drag coefficient is proportional to the linear size of the particle [1], and the mass is
proportional to the volume of the particle. For a spherical particle of radius r,
the mass, the drag coefficient, and the time scale of inertia are,
respectively,(1.6)m=43πρr3,ζ=6πηt,t0=mζ=2ρ9ηr2,where ρ is the density of the particle and η is the viscosity of the surrounding fluid. In
general, the time scale of inertia t0 is proportional to the square of the linear
size of particle whereas the coefficient varies with the shape of particle.
Thus, for small particles, the time scale of inertia t0 is very small. For example, for a latex bead
of diameter 1⁢μm in water, we have t0=56×10−9⁢s=56ps.
A time period of 56ps is much smaller than the typical time scales
of chemical reaction cycles in molecular motors, which are of the order of
milliseconds. The short time dynamics of the particle is full of fast decaying
of the instantaneous velocity in time scale of t0 and fast increasing of the instantaneous
velocity excited by the stochastic Brownian force. These short-time dynamical
behaviors present significant difficulties for analysis and computations. It is
desirable to capture relatively long-time behaviors of the motor in the time
scales of reaction cycles, without resolving these short-time details. For that
purpose, we consider the mathematical limit of  (1.3) as time scale t0 goes to zero. As time scale t0 goes to zero, the limit of the solution of  (1.3)
satisfies(1.7)dxdt=−Dϕ′(x)kBT+2DdW(t)dt.The reduction from  (1.3) to  (1.7) in
the limit of small t0 is called the Einstein-Smoluchowski limit
[16].In a molecular motor, the potential is not static.
Instead, the potential changes with the current chemical occupancy state of the
motor. In this paper, we consider the case where a motor has only one catalytic
site (one reaction cycle). The extension of the analysis to the case of
multiple catalytic sites is straightforward but tedious. LetN be the number of chemical occupancy states of
the motor in consideration. Let {1,2,…,N} denote the set of N occupancy states, and
let {ϕ1(x),ϕ2(x),…,ϕN(x)} denote the corresponding set of N periodic potentials. Here ϕj(x) is the periodic motor potential when the motor
is in chemical occupancy state j.
Mathematically, ϕj(x) is a periodic function of period L where the period L is either one motor step or a multiple of
motor steps. For a Kinesin dimer walking on a microtubule filament, one motor
step is about 8 nm. For the rotary motor of F1  ATPase, one motor step is one third of one revolution (2π/3). The mechanical motion of the motor is
governed by Langevin equation:(1.8)dxdt=−DϕS(t)′(x)kBT+2DdW(t)dt,where S(t) denotes the current chemical occupancy state,
and ϕS(t)(x) is the periodic motor potential corresponding
to S(t).
Equation  (1.8) can either be viewed as one Langevin equation that varies with the
current occupancy state S(t) or be viewed as a collection of N Langevin equations, each corresponding to an
individual chemical occupancy state. Equation  (1.8) describes only the spatial
motion of the motor. Equation  (1.8) is not closed since it depends on the current
occupancy state S(t).
To close  (1.8), we need to follow the time evolution of the chemical occupancy
state. In a transition from occupancy state A to occupancy state B, the
commitment time is generally much smaller than the residence time. The
commitment time is the time of the actual transition process. More precisely,
the commitment time is the time period during which the system has left state A
but has not yet arrived at state B. If we define a transition state between
state A and state B, the commitment time can also be viewed as the residence time
of the transition state. Because the commitment time of transition is small in
comparison with the residence time, we can approximate the transition as
instantaneous. Thus, we model the stochastic evolution of the motor's chemical
state as a continuous time discrete space Markov process, also called a jump
process:(1.9)Pr[S(t+δt)=j∣S(t)=i]={(δt)ki→j(x(t))+o(δt),j⁢≠i,1−(δt)∑l⁢≠iki→l(x(t))+o(δt),j=i,where ki→j(x(t)) is the transition rate from state i to state j.
Notice that the transition rate depends on the motor position x(t).
In other words, in Markov process  (1.9), the evolution of chemical state S(t) is affected by the motor position x(t).
On the other hand, in Langevin equation  (1.8), the evolution of motor position x(t) is affected by the chemical state S(t).
Thus, Langevin equation  (1.8) and Markov process  (1.9) are coupled, and together
they govern the stochastic evolution of both the mechanical motion and the
chemical reaction of the motor. In most of chemical reactions, for example, in
the ATP hydrolysis cycle, the system goes through the set of N occupancy states sequentially. In this case,
the N occupancy states form a loop, and the only
allowed transitions are either forward to the next state or backward to the
previous state along the loop. Here, by convention, we represent the transition N⁢→1 by N⁢→N+1 and represent the transition 1⁢→N by N+1⁢→N.
With this convention, we have that ki→j(x(t))≠0 only when j=i+1 or j=i−1.The derivation in [17] was for motor systems without cargos. In many single
molecule experiments, the motor is not observed/recorded directly. Instead a
large latex bead is attached to the motor, the position of the bead is
observed/controlled with the help of a laser trap or a force clamp [9, 18–20]. Motor-cargo systems also occur in biological
settings. For example, a Kinesin dimer walks on a microtubule filament, towing
a vesicle toward the membrane for exporting. To accommodate these experimental
and biological settings, we need to study the behaviors of motor-cargo systems.
Figure 1 shows a motor-cargo system corresponding to the experimental setups in
[9, 19]. A Kinesin dimer walks on a
microtubule toward the plus end and a latex bead is linked to the Kinesin dimer
to visualize the motion  of the Kinesin dimer and to exert a force on the Kinesin
dimer. The force exerted on the Kinesin dimer is
varied either by changing the drag coefficient of the latex bead [19] or by applying a
conservative force on the bead using a laser trap [9]. In the recently developed
2D laser trap [20], a
force perpendicular to the direction of motor motion can also be exerted on the
Kinesin dimer.Figure 1
A motor-cargo system in single
molecule experiments, corresponding to the experimental setup
in [9, 19]. A Kinesin dimer walks on a
microtubule filament toward the plus end, and a latex bead is linked to the
Kinesin dimer to visualize the motion of the Kinesin
dimer and to exert a force on the Kinesin dimer.
The latex bead is either loaded by the force from a laser trap or loaded by the
viscous drag from the fluid media.Letx(t) be the coordinate of the motor and y(t) the coordinate of the cargo along the
direction of motion as illustrated in Figure 1. As we discussed above, the
motor is driven by switching among a set of N potentials, each corresponding to
an individual occupancy state. The stochastic motion of the motor is governed
by an overdamped Langevin equation(1.10)dxdt=DM[−E′(x−y−R)−ϕS(t)′(x)]kBT+2DMdW(t)dt,where DM is the diffusion coefficient of the motor, and −E′(x−y−R) is the elastic force on the motor exerted by
the elastic link between the motor and the cargo. Here E(x−y−R) is the elastic potential and R the rest length of the elastic link. The rest
length R can be eliminated using a change of variable y=y+R.
So without loss of generality, we assume R=0.
The stochastic evolution of the occupancy state is governed by jump process
(1.9). In the motor-cargo system, the external loading force no longer acts
directly on the motor. The external loading force acts directly on the cargo.
The stochastic motion of the cargo is governed by(1.11)dydt=DC[f+E′(x−y−R)]kBT+2DCdW(t)dt,where DC is the diffusion coefficient of the cargo, and f is the external loading force acting on the
cargo. Note that in the motor-cargo system, the motor does not directly feel
the external loading force on the cargo; the motor sees the elastic force from
the cargo. On the other hand, the cargo does not directly feel the internal
motor force; the cargo sees the elastic force from the motor. All
communications between the motor and the cargo are done via the elastic link.
In experiments, only average quantities can be measured repeatedly and
reliably. All average quantities can be calculated by following the probability
density of the motor-cargo system. Let ρj(x,y,t) be the probability density that the motor
has at position x⁢,
the cargo is at position y,
and the chemical reaction is in occupancy state j at time t.
Mathematically, ρj(x,y,t) is defined as(1.12)ρj(x,y,t)=lim⁡δx→0δy→0Pr[x≤x(t)<x+δxy≤y(t)<y+δy,S(t)=j]δxδy.The set of probability densities {ρj(x,y,t)} is governed by the two-dimensional
Fokker-Planck equation corresponding to Langevin equations  (1.10) and  (1.11) and
jump process  (1.9) [16].
The Fokker-Planck equation is based on the conservation of
probability(1.13)∂ρj∂t=−∂∂xJj(x)−∂∂yJj(y)−(pj+1/2−pj−1/2).In the above, Jj(x)(x,y,t) is the probability flux along the x-direction (the dimension of the motor
position) in occupancy state j and it has the expression(1.14)Jj(x)(x,y,t)=−DM(1kBT∂Φj(x,y)∂xρj+∂ρj∂x).Φj(x,y) is the total potential of the motor-cargo
system in occupancy state j,
which includes both the internal motor potential caused by the chemical
reaction and the elastic energy in the link connecting the motor and cargo. Φj(x,y) does not contain the effect of the external
loading force, which is accounted for separately. Φj(x,y) has the expression(1.15)Φj(x,y)=ϕj(x)+E(x−y).Jj(y)(x,y,t) is the probability flux along the y-direction (the dimension of the cargo
position) in occupancy state j and it has the expression(1.16)Jj(y)(x,y,t)=−DC(1kBT[−f+∂Φj(x,y)∂y]ρj+∂ρj∂y).pj+1/2(x,y,t) is the net probability flux density (net
probability flux per unit area in the x−y plane) along the direction of reaction from
occupancy state j to occupancy state j+1.
It has the expression(1.17)pj+1/2(x,y,t)=kj→j+1(x)ρj(x,y,t)−kj+1→j(x)ρj+1(x,y,t).Note that the transition rate kj→j+1(x) does not depend explicitly on the cargo
position (y).
The cargo can affect the chemical reaction indirectly by changing the motor
position. In the chemical reaction governed by jump process  (1.9), the transition
rates ki→j(x) cannot be arbitrarily specified. These rates
are constrained by detailed balance-like condition,
which ensures that if the system is brought to an equilibrium, then the
equilibrium solution is given by the Boltzmann distribution and the probability
flux vanishes everywhere [11]. Specifically, suppose state A and state B have well-defined
free energy GA and GB.
Then the transition rates between these two states must satisfy(1.18)kA→BkB→A=exp⁡(GA−GBkBT).It follows that the transition
rates used in  (1.17) are constrained by(1.19)kj→j+1(x)kj+1→j(x)=exp⁡(Φj(x,y)−Φj+1(x,y)kBT)=exp⁡(ϕj(x)−ϕj+1(x)kBT).The boundary conditions for  (1.13)
in the x-direction are pseudoperiodic:(1.20)ρj(x+L,y+L,t)=ρj(x,y,t),ϕj(x+L)=ϕj(x),ki→j(x+L)=ki→j(x).In the y-direction,  (1.13) extends from negative
infinity to positive infinity. For fixed x and large y,
the elastic energy E(x−y) is large and the probability density ρj(x,y,t) is exponentially small. Thus, at infinity, we
have(1.21)ρj(x,±∞,t)=0,∂ρj(x,±∞,t)∂x=0,∂ρj(x,±∞,t)∂y=0.The boundary conditions along
the reaction direction are also
pseudoperiodic:(1.22)ρN+j(x,y,t)=ρj(x,y,t),ϕN+j(x)=ϕj(x)+ΔG,kN+i→N+j(x)=ki→j(x),where ΔG<0 is the free energy change in one reaction
cycle.


## 2. Stokes Efficiency, Previous Result, and a General Proof

In [17], the Stokes efficiency was
proposed to measure how efficiently the motor utilizes the chemical free energy
to drive through the viscous fluid. Let 〈u〉 denote the average velocity and let r denote the chemical reaction rate (average
number of reaction cycles per unit time) of the motor system The Stokes
efficiency for a motor system without cargo is defined as(2.1)ηStokes≡ζ〈u〉2r(−ΔG)+f〈u〉,where ζ is the drag coefficient of the motor. In a
motor-cargo system, the motor part has drag coefficient ζM=kBT/DM and the cargo part has drag coefficient ζC=kBT/DC.
Since both the motor and the cargo move through the viscous fluid with the same
average velocity, the Stokes efficiency for a motor-cargo system is defined
as(2.2)ηStokes≡(ζM+ζC)〈u〉2r(−ΔG)+f〈u〉.The denominator of the Stokes
efficiency is the total free energy consumed per unit time: r(−ΔG) is the chemical free energy consumed in the
chemical reaction per unit time and f〈u〉 is the free energy input to the motor-cargo
system from the external agent exerting the conservative force. If the external
conservative force is against the motor motion, then f〈u〉 is negative, which means the free energy is
actually output to the external agent. The numerator of the Stokes efficiency, (ζM+ζC)〈u〉2,
has the dimension of energy per unit time. But it does not have a clear
thermodynamic meaning. As a result, ηStokes≤1 cannot be derived from simple thermodynamic
arguments. In [17], we
proved ηStokes≤1 for motor systems without cargos. Below we
present a proof for the general motor-cargo system described by Fokker-Planck
equation  (1.13) with detailed balance constraint  (1.19) and boundary conditions
(1.20),  (1.21), and  (1.22). Specifically, we want to prove(2.3)(ζM+ζC)〈u〉2≤r(−ΔG)+f〈u〉.Since all average quantities can
be determined from the steady-state solution, we
consider the steady state of  (1.13) as
follows:(2.4)∂∂xJj(x)+∂∂yJj(y)=−(pj+1/2−pj−1/2).To facilitate the analysis
below, we introduce a new function Hj(x,y):(2.5)Hj(x,y)≡Φj(x,y)kBT+ln⁡(ρj(x,y)).Note that in the steady state,
everything is independent of t.
From the pseudoperiodic boundary condition in the reaction direction, we obtain
immediately that(2.6)HN+j(x,y)=Hj(x,y)+ΔGkBT.With the help of function Hj(x,y),
we rewrite the fluxes Jj(x) and Jj(y) as (2.7)Jj(x)=−DM(1kBT∂Φj∂xρj+∂ρj∂x)=−DMρj∂Hj∂x,(2.8)Jj(y)=−DC(1kBT[−f+∂Φj∂y]ρj+∂ρj∂y)=−DCρj(∂Hj∂y−fkBT).The average velocity and the
chemical reaction rate have the expressions:
(2.9)〈u〉=∫0L∫−∞∞∑j=1NJj(x)dy⁢dx,(2.10)〈u〉=∫0L∫−∞∞∑j=1NJj(y)dy⁢dx,(2.11)r=∫0L∫−∞∞pj+1/2dy⁢dx. In the above, we have two
expressions for the average velocity 〈u〉.
The first expression  (2.9) is the average velocity of the motor, and the second
expression  (2.10) is the average velocity of the cargo. Since the motor and the
cargo are elastically linked, they must have the same average velocity.
Substituting flux  (2.7) into the average velocity for the motor given in  (2.9)
and applying the Cauchy-Schwarz inequality yield(2.12)〈u〉2=[∫0L∫−∞∞∑j=1N−DMρj∂Hj∂xdy⁢dx]2=[∫0L∫−∞∞∑j=1Nρj1/2(−DMρj1/2∂Hj∂x)dy⁢dx]2≤[∫0L∫−∞∞∑j=1Nρjdy⁢dx]⋅[∫0L∫−∞∞∑j=1NDM2ρj(∂Hj∂x)2dy⁢dx].Using the fact that the total
probability is one and converting back to flux Jj(x),
we get(2.13)〈u〉2≤−DM∫0L∫−∞∞∑j=1NJj(x)∂Hj∂xdy⁢dx.Pseudoperiodic boundary
condition  (1.20) implies(2.14)∫−∞∞∑j=1NJj(x)Hjdy|x=0=∫−∞∞∑j=1NJj(x)Hjdy|x=L.Integrating by parts the
right-hand side of  (2.13) and using  (2.14), we have(2.15)〈u〉2≤DM∫0L∫−∞∞∑j=1NHj∂Jj(x)∂xdy⁢dx.Equation  (2.15) is an inequality
for 〈u〉 based on the average velocity of the motor.
Now using the average velocity for the cargo given in  (2.10) with flux  (2.8),
applying the Cauchy-Schwarz inequality, converting back to flux Jj(y),
and then converting back to 〈u〉,
we obtain(2.16)〈u〉2=[∫0L∫−∞∞∑j=1Nρj1/2(−DCρj1/2(∂Hj∂y−fkBT))dy⁢dx]2≤∫0L∫−∞∞∑j=1NDC2ρj(∂Hj∂y−fkBT)2dy⁢dx=−DC∫0L∫−∞∞∑j=1NJj(y)(∂Hj∂y−fkBT)dy⁢dx=−DC∫0L∫−∞∞∑j=1NJj(y)∂Hj∂ydy⁢dx+DCfkBT〈u〉.Integrating by parts and using
boundary condition  (1.21) lead to another inequality for 〈u〉:(2.17)〈u〉2≤DC∫0L∫−∞∞∑j=1NHj∂Jj(y)∂ydy⁢dx+DCfkBT〈u〉.Combining inequalities  (2.15) and
(2.17) gives us(2.18)〈u〉2=DCDM+DC〈u〉2+DMDM+DC〈u〉2≤DCDMDM+DC∫0L∫−∞∞∑j=1NHj(∂Jj(x)∂x+∂Jj(y)∂y)dy⁢dx+DCDMDM+DC⋅fkBT〈u〉.Up to now, we have not yet used
the steady-state Fokker-Planck equation  (2.4). Using  (2.4), applying summation by
parts, and using relation  (2.6), we have(2.19)〈u〉2≤−DCDMDM+DC∫0L∫−∞∞∑j=1NHj(pj+1/2−pj−1/2)dy⁢dx+DCDMDM+DC⋅f〈u〉kBT=DCDMDM+DC∫0L∫−∞∞∑j=1Npj+1/2(Hj+1−Hj)dy⁢dx+DCDMDM+DC⋅(−ΔG)kBT∫0L∫−∞∞p1/2dy⁢dx+DCDMDM+DC⋅f〈u〉kBT≡Q1+Q2+Q3.Let us examine the three terms Q1, Q2,
and Q3 defined above one by one. For Q3,
we use the relation (DCDM/(DM+DC))⋅(1/kBT)=1/(ζM+ζC) to write it as(2.20)Q3=1ζM+ζC⋅f〈u〉.In Q2,
converting the integral back to the chemical reaction rate gives
us(2.21)Q2=1ζM+ζC⋅r(−ΔG).For Q1,
we first use detailed balance  (1.19) to write flux density pj+1/2 as(2.22)pj+1/2=kj→j+1ρj(1−kj+1→jkj→j+1⋅ρj+1ρj)=kj→j+1ρj(1−exp⁡(ϕj+1(x)−ϕj(x)kBT+ln⁡ρj+1ρj)).Then we use definition  (2.5) to
write Hj+1−Hj as(2.23)Hj+1−Hj=ϕj+1(x)−ϕj(x)kBT+ln⁡ρj+1ρj.Using these two results, we
write Q1 as(2.24)Q1=DCDMDM+DC∫0L∫−∞∞∑j=1Nkj→j+1ρj(1−exp⁡(ΔHj+1/2))ΔHj+1/2dy⁢dx,where ΔHj+1/2≡Hj+1−Hj. Since the expression (1−exp⁡(α))α is always nonpositive for any value of α,
it follows that Q1 is always nonpositive: Q1≤0.
Finally, substituting Q1, Q2,
and Q3 into  (2.19), we arrive at(2.25)〈u〉2≤1ζM+ζC(r(−ΔG)+f〈u〉)which leads directly to the
desired result  (2.3). Thus, the Stokes efficiency is indeed bounded by 100% for
a motor-cargo system.In summary, for the general case where a motor is
elastically linked to a cargo, we have proved mathematically that the Stokes
efficiency is bounded 100%. Therefore, the Stokes efficiency is a justified
efficiency measure for motor-cargo systems.



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*Source: 241736-2008-05-19.xml*