# Double-temperature ratchet model and current reversal of coupled Brownian
motors

Chen-Pu Li  Department of Physics, Beijing Normal University, Beijing 100875,
China  College of Science, Hebei University of Architecture, Zhang Jiakou
075000, China  Hong-Bin Chen  Institute of Systems Science (ISS), Huaqiao
University, Xiamen 361021, China  College of Information Science and
Engineering, Huaqiao University, Xiamen 361021, China  Zhi-Gang Zheng
<EMAIL_ADDRESS>Institute of Systems Science (ISS), Huaqiao University,
Xiamen 361021, China  College of Information Science and Engineering, Huaqiao
University, Xiamen 361021, China

###### Abstract

On the basis of the transport features and experimental phenomena observed in
studies of molecular motors, we propose a double-temperature ratchet model of
coupled motors to reveal the dynamical mechanism of cooperative transport of
motors with two heads, where the interactions and asynchrony between two motor
heads are taken into account. We investigate the collective unidirectional
transport of coupled system and find that the direction of motion can be
reversed under certain conditions. Reverse motion can be achieved by
modulating the coupling strength, coupling free length, and asymmetric
coefficient of the periodic potential, which is understood in terms of the
effective potential theory. The dependence of the directed current on various
parameters is studied systematically. Directed transport of coupled Brownian
motors can be manipulated and optimized by adjusting the pulsation period or
the phase shift of the pulsation temperature.

###### pacs:

05.45.Xt, 05.40.-a, 05.60.-k 02.50.Ey

##  I Introduction

The directed transport of Brownian motors in periodic structures with the help
of fluctuations with zero mean has long been an important problem and has been
widely studied ref1 ; ref2 ; ref3 ; ref4 . In recent years, much attention has
been given to theoretical studies of directed transport of coupled Brownian
motors in many different scientific areas, such as molecular motors in
biological systems, the divergence of two polymers on a surface, and Josephson
junction arrays, to name but a few ref5 ; ref6 ; ref7 ; ref8 ; ref9 ; ref10 ;
ref11 ; ref12 ; ref13 ; ref14 ; ref15 ; ref16 . A nonequilibrium environment
and broken symmetry of the system are indispensable conditions for directional
transport of particles in a ratchet model. Symmetry breaking generally
includes the symmetry breaking of the periodic potential field, the symmetry
breaking induced by nonequilibrium perturbations, and the breaking induced by
mutual coupling between elements of the system ref17 ; ref18 ; ref19 ; ref20 ;
ref21 ; ref22 ; ref23 ; ref24 ; ref25 ; ref26 ; ref27 .

In this work, we consider two interacting Brownian motors in a spatially
periodic potential. We are motivated by experimental observations of the
motion patterns of molecular motors (protein motors). It has been found that
most protein motors possess a dimer structure in which each motor protein is
composed of two interacting identical monomers, and each monomer experiences
cyclic ATP hydrolysis ref28 ; ref29 . It has been found experimentally that
the coupling between two motor proteins, which do not act independently but
alternate in a sequential manner such that their catalytic cycles are out of
phase, plays a significant role in achieving directed motion and even reversed
motion. We noticed an important fact related to the symmetry breaking of dimer
molecular motors; i.e., it was also found that the hydrolysis processes of two
heads are continuous while they are asynchronous ref30 ; ref31 . This gives us
a hint for establishing a double-temperature ratchet model of coupled motors
in an asymmetric potential field.

In this paper, we analyze the effects of several parameters, such as the
coupling strength, asymmetry coefficient of the potential, pulsation period,
and phase shift of the temperature, on the average velocity of coupled motors.
Furthermore, the dynamical mechanism and reverse behavior of the coupled motor
are reasonably explained using effective potential theory in the strong-
coupling case.

##  II  The Coupled Double-temperature Brownian Motor Model

###  II.1 Dynamical model

We consider the overdamped Brownian motion of two coupled Brownian motors in
contact with two reservoirs with different temperatures in asymmetric periodic
potentials. The equations of motion of two mutually coupled motors can be
written as

$\dot{x_{i}}=-\frac{\partial{V(x_{i})}}{\partial{x_{i}}}-\frac{\partial{U_{0}}}{\partial{x_{i}}}+\xi_{i}(t),\quad
i=1,2,$ (1)

where $x_{i}$ is the coordinate of the $i$-th motor, and $V(x)$ is the
asymmetric periodic potential originating from the interaction between the
motor and the track. The following simplified form of $V(x)$ is chosen:

$V(x)=-V_{0}[\sin{(\frac{2\pi}{L}x)}+\frac{\Delta}{4}\sin{(\frac{4\pi}{L}x)}],$
(2)

where $L$ is the spatial period of the potential field $V(x)$, which is shown
in Fig.1, and $\Delta$ is the asymmetry coefficient of the potential field
$V(x)$. $U_{0}(x_{1},x_{2})$ denotes the interaction potential between the two
motors, which is set to have the following simple harmonic form:

$U_{0}(x_{1},x_{2})=\frac{1}{2}k(x_{1}-x_{2}-a)^{2},$ (3)

where $k$ is the coupling strength, and $a$ is the coupling free length. The
influence of the two reservoirs on the Brownian motors is described in terms
of the noises $\xi_{1}(t)$ and $\xi_{2}(t)$, which are assumed to be
independent, unbiased Gaussian white noises with

$\begin{split}&\langle\xi_{i}(t)\rangle=0,\\\
&\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2k_{B}T_{i}(t)\delta_{ij}\delta(t-t^{\prime}),\quad
i,j=1,2,\end{split}$ (4)

where $k_{B}T$ is the thermal energy, and $T_{1}(t)$, $T_{2}(t)$ are the
following modulated harmonically varying functions:

$\begin{split}&T_{1}(t)=T_{0}[1+A\sin{(\frac{2\pi}{t_{0}}t)}]^{2},\\\
&T_{2}(t)=T_{0}[1+A\sin{(\frac{2\pi}{t_{0}}t+\Delta\theta)}]^{2},\end{split}$
(5)

where $t_{0}$ is the pulsation period of the temperature, and $\Delta\theta$
is the phase shift between two temperature fluctuations. The mismatch between
the two temperatures denotes biologically the different ATP hydrolysis states
of the two motors. Throughout this paper, we set the parameters $T_{0}$ = 0.5
and $A$ = 0.8.

Figure 1:  the asymmetric periodic potential $V(x)$ which is originated from
the interaction between the motor and the track.

###  II.2 Adiabatic elimination and mass-center dynamics in strong-coupling
case

It is difficult to directly analyze the cooperative ratcheting effect of two
coupled Brownian motors theoretically. An effective method for dealing with
the cooperative ratcheting effect of coupled motors is to decrease the degrees
of freedom of the system to obtain a low-dimensional description. By
introducing the mass-center coordinate $X$=$(x_{1}+x_{2})/2$ and the relative
coordinate $Y$=$x_{1}-x_{2}$, one can transform Eq. (1) to

$\begin{split}&\dot{X}=-\frac{1}{2}\frac{\partial[V(X+\frac{Y}{2})+V(X-\frac{Y}{2})]}{\partial
X}\\\ &-\frac{\partial[V(X+\frac{Y}{2})-V(X-\frac{Y}{2})]}{\partial
Y}+\frac{1}{2}(\xi_{1}(t)+\xi_{2}(t)),\end{split}$ (6)

$\begin{split}&\dot{Y}=-\frac{\partial[V(X+\frac{Y}{2})-V(X-\frac{Y}{2})]}{\partial
X}\\\ &-2\frac{\partial[V(X+\frac{Y}{2})+V(X-\frac{Y}{2})]}{\partial
Y}-2k(Y-a)+\xi_{1}(t)-\xi_{2}(t).\end{split}$ (7)

To understand the effect of the coupling between two Brownian motors on the
directed transport of the system, it is instructive to study the dynamics in
the limit of a large but finite stiffness $k$. In this situation, the term
$-2k(Y-a)$ in Eq. (7) implies a much faster decay of the coordinate $Y$
compared with the relaxation of the variable $X$. This analysis indicates that
$Y$ is a fast variable and can be adiabatically eliminated in terms of the
slaving principle proposed by Haken ref10 ; ref27 ; ref32 . Therefore, in
$k\rightarrow\infty$, the dynamical equation of the mass center of coupled
motors can be described as

$\begin{split}&\dot{X}=-\frac{1}{2}\frac{\partial[V(X+\frac{a}{2})+V(X-\frac{a}{2})]}{\partial
X}\\\ &+\frac{1}{2}(\xi_{1}(t)+\xi_{2}(t)),\end{split}$ (8)

with $Y\approx a$.

The dynamical equation (8) can be simply expressed as

$\dot{X}=f(X)+q(t),$ (9)

with

$\displaystyle f(X)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\partial[V(X+\frac{a}{2})+V(X-\frac{a}{2})]}{\partial
X},$ $\displaystyle q(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\xi_{1}(t)+\xi_{2}(t)).$

Theoretically, this enables one to qualitatively calculate the effect of
multiple parameters, such as the coupling free length $a$, the modulation
period $t_{0}$ of the two temperatures, and the phase shift $\Delta\theta$
between the two temperatures, on the ratchet motion of the coupled Brownian
motors in terms of the dynamics of a single Brownian motor (the mass-center
dynamics).

According to the analyses in reference [4] of the temperature ratchet of a
single Brownian motor, when the period $t_{0}$ of the temperature fluctuation
tends to infinity, the temperature can be regarded as an approximate constant
in a small time interval, where the average velocity of a single Brownian
motor is zero in the periodic potential with Gaussian white noise. When the
period $t_{0}\ll 1$, one can readily find that

$\langle\dot{X}\rangle=t_{0}^{2}B_{k}\int_{0}^{L}dX(V_{k}^{{}^{\prime}}(X)[V_{k}^{{}^{\prime\prime}}(X)]^{2})+o(t^{3}),$
(10)

with

$\displaystyle
B_{k}=\frac{4L\int_{0}^{1}dh[\int_{0}^{h}d\hat{h}(\frac{1-\hat{T}(\hat{h})}{\bar{T}})]}{\eta^{3}\int_{0}^{L}dX(e^{\frac{V_{k}(X)}{k_{B}\bar{T}}})\int_{0}^{L}dX(e^{-\frac{V_{k}(X)}{k_{B}\bar{T}}})}$

and

$\displaystyle\bar{T}=\frac{1}{t_{0}}\int_{0}^{t_{0}}dtT(t)=\int_{0}^{1}dh\hat{T}(h),$
$\displaystyle\hat{T}(h)=T(t)=T(t_{0}h).$

###  II.3 Effective potential theory in strong-coupling case

To explain the current reversal of the ratchet system, it is convenient to
introduce the effective potential ref10 ; ref24 ; ref27 . In the above
discussion, we know that in the strong-coupling situation, the dynamics of the
relative coordinate $Y$ occurs on a much faster time scale than that of the
mass-center coordinate $X$. One can obtain the effective potential
$V_{eff}(X)$ of $X$ by replacing the $X$\- and $Y$-dependent potential with
the potential averaged with respect to the fast relative coordinate $Y$ as

$V_{eff}(X)=-\frac{1}{2}k_{B}\bar{T}\ln(\int_{-\infty}^{\infty}dY\rho(X,Y)),$
(11)

and

$\rho(X,Y)=e^{-\frac{U(X,Y)}{k_{B}\bar{T}}},$ (12)
$U(X,Y)=\frac{1}{2}k(Y-a)^{2}+V(X+\frac{Y}{2})+V(X-\frac{Y}{2}).$ (13)

In the following discussions, we continue to study the collective directed
transport of coupled Brownian motors using numerical simulations and make
comparisons with theoretical discussions. In numerical simulations of the
stochastic dynamics, the second-order Runge–Kutta numerical simulation
algorithm is adopted ref33 ; ref34 ; the number of ensembles $N$ is 1000, and
the time step $dt$ is 0.001. Throughout the numerical simulations, we set the
parameter $V_{0}$ to 1 and $L$ to 1. The average velocity or the current can
be obtained by obtaining both ensemble and time averages of the instantaneous
velocity as

$<v_{c}>=<\dot{X_{c}}>=\lim_{t\rightarrow\infty}\frac{1}{Nt_{0}}\sum_{i=1}^{N}\int_{0}^{t_{0}}dt^{{}^{\prime}}\dot{X}_{ci}(t^{{}^{\prime}}).$
(14)

##  III  Collective Directed Transport and Current Reversal

In this section, we analyze the influence of the coupling strength $k$ of the
coupled Brownian motors, the coupling free length $a$, and the phase shift
$\Delta\theta$ of the two motor heads on the average velocity. Theoretically,
one can discuss the current reversal using the effective potential theory for
the strong-coupling case.

###  III.1 Current reversal induced by the coupling strength

The coupling strength $k$ plays a significant role in directed transport of
the ratchet system. Fig.2(a) shows three curves of the average velocity
against the coupling strength $k$ with phase shifts $\Delta\theta=\pi$,
$\pi/2$, and 0. Fig.2(b) presents the curves of the effective potential with
coupling strengths $k$=0, 300, and 1000 (corresponding to the strong-coupling
limit $k\rightarrow\infty$ in the theoretical analysis). For every curve in
Fig.2(a), the sign of the velocity, which corresponds to the direction of
motion of the coupled motors, can be reversed when the coupling strength $k$
reaches a certain value.

For weak coupling [e.g., $0.01<k<1$ in Fig.2(a)], the average velocity
$v\approx-0.22$. This is because in the weak-coupling case, the motion can be
considered as a simple combination of that of two single motors, where each
particle is immersed in a common periodic potential $V(x)$ and the Gaussian
white noise, and the effective potential $V_{eff}(X)$ [e.g., the curve with
$k$ = 0 in Fig.2(b)] of the mass center could be considered as the potential
$V(x)$ of a single motor; the motion of two weakly coupled motors is
consistent with the relatively large gradient of the potential $V(x)$ in the
temperature ratchet model of a single motor ref10 . With increasing coupling
strength $k$ [e.g., $k=1-40$ in Fig.2(a)], the negative mean velocity
gradually increases and tends to zero, which shows that the coupling between
the two coupled motors affects the symmetry breaking of the entire system.
When the coupling is large enough [e.g., $k=40-1000$ in Fig.2(a)], the
symmetry breaking is opposite to that when the coupling is weak [$k\sim$ 0 to
40 in Fig.2(a)], such that the effective potential $V_{eff}(X)$ in the strong-
coupling case exhibits the opposite tendency of that in the weak-coupling case
in Fig.2(b). Finally, the positive mean velocity reaches saturation with
increasing coupling strength $k$.

In addition, when $k<40$, the three curves in Fig.2(a) almost overlap, and the
effect of the phase shift $\Delta\theta$ on the average velocity is very
small. That is because of the single-particle behavior and the fact that the
mean velocity is independent of $\Delta\theta$ for the weak-coupling case.
When $k>40$, the mean velocity decreases with increasing $\Delta\theta$ in the
range of [0,$\pi$] and tends to zero for $\Delta\theta=\pi$. This can be
interpreted in terms of the formulas for the two temporally modulated
temperatures $T_{1}(t)$=$T_{0}[1+A\sin(2\pi t/t_{0})]^{2}$ and
$T_{2}(t)$=$T_{0}[1+A\sin(2\pi t/t_{0}+\Delta\theta)]^{2}$. In the temperature
ratchet model, the maximum temperature represents the minimum binding of the
potential well to the coupled motors, and the minimum temperature corresponds
to the strongest binding. According to the two formulas with
$\Delta\theta=\pi$, one temperature of the coupled motors reaches the maximum
value when the other one is at the minimum, indicating that the coupling
reduces the motion of the coupled motor in this case. For $\Delta\theta=0$,
the temperature fluctuation of both motors is synchronized, which means that
the average velocity reaches the maximum value because the coupling could
enhance the motion.

Figure 2:  the average velocity as a function of the coupling strength $k$.

###  III.2 Current reversal induced by changing the coupling free length

The presence of coupling between two motor heads not only provides cooperative
directed motion, but also significantly affects the current. The direction of
the current can even be determined by the free length of the coupled motors,
where the current reversal can be interpreted well in terms of the effective
potential theory.

In Fig.3(a), we give the average velocity versus the coupling free length $a$
with the other parameters given in the plot, where the four curves correspond
to different coupling strengths $k$ and phase shifts $\Delta\theta$ between
the two temperatures, i.e., ($k$, $\Delta\theta$)=(1, 0), (300, 0), (300,
$\pi/2$), and (300, $\pi$). It can be found from Fig.3(a) that when the
coupling strength $k$ is small (e.g., $k=1$), the coupling free length $a$
cannot greatly influence the average velocity, which has an almost constant
negative value (e.g., $v=-0.2$) for all values of $a$. This is because the
motion of coupled motors can be considered as a simple combination of that of
two single motors in weak-coupling cases. For the three curves with strong
coupling strengths $k$ (e.g., $k=300$) between the two motors in Fig.3(a), the
average velocity first increases and then decreases with $a$ in the range of 0
to $L/2$, and the velocity is reversed when $a=L/4$. In addition, these curves
are symmetric about $a=L/2$. In Fig.3(b), the average velocity versus the
coupling free length $a$ is theoretically computed according to Eq. (10) with
the parameters given in the plot. Although this result differs from the
simulation results in Fig.3(a), the theoretical results can still
qualitatively display the effects of the coupling free length $a$ on the
average velocity. Note that the theoretical results agree well with that of
the simulation only when $t_{0}$ is close to zero.

Figure 3:  the average velocity as function of the coupling free length $a$.

In Fig.4(a), (b), (c), and (d), we give the effective potential $V_{eff}(X)$
for coupling free lengths $a$ of 0.1, 0.25, 0.3, and 0.5, respectively, where
the other parameters are $k$ = 300, $\Delta$ = 1, $t_{0}$ = 0.1, and
$\Delta\theta$=0. We introduce an effective asymmetric coefficient
$\Delta_{eff}$ of the effective potential $V_{eff}(X)$, where the effective
asymmetric coefficient $\Delta_{eff}=(L2-L1)/L$, which is shown in Fig.4(a).
In the strong-coupling case, the asymmetric coefficient of the effective
potential $V_{eff}(X)$ periodically changes against the coupling free length
$a$ with $L$ in one cycle in Fig.4(a), (b), (c), and (d). Fig.4 shows that the
asymmetric coefficient $\Delta_{eff}$ is less than zero when $0<a<L/4$, is
greater than zero when $L/4<a<L/2$, and equals zero when $a=L/4$ and $L/2$.
The cyclic behavior of the average velocity agrees well with the change in the
asymmetric coefficient of the effective potential $V_{eff}(X)$ for the strong-
coupling situations shown in Fig.4, which naturally leads to the periodic
variation of the average velocity.

The symmetry of the average velocity about $a=L/2$ can be understood well by
analyzing the space-time transformation invariance of the dynamical Eq. (1)
ref25 ; ref27 ; ref35 ; ref36 . Rewriting Eq. (1) with a superscript
corresponding to the value of the coupling free length $a$ yields

Figure 4:  the curves of effective potential with different the coupling free
length $a$.

$\begin{split}&\dot{x}_{1}^{(a)}=-\frac{\partial V(x_{1})}{\partial
x_{1}}-k(x_{1}-x_{2}-a)+\xi_{1}(t),\\\ &\dot{x}_{2}^{(a)}=-\frac{\partial
V(x_{2})}{\partial x_{2}}+k(x_{1}-x_{2}-a)+\xi_{2}(t).\end{split}$ (15)

By replacing the free length $a$ with $L-a$, we obtain

$\begin{split}&\dot{x}_{1}^{(L-a)}=-\frac{\partial V(x_{1})}{\partial
x_{1}}-k(x_{1}-x_{2}-L+a)+\xi_{1}(t),\\\ &\dot{x}_{2}^{(L-a)}=-\frac{\partial
V(x_{2})}{\partial x_{2}}+k(x_{1}-x_{2}-L+a)+\xi_{2}(t).\end{split}$ (16)

By inserting $x_{11}=x_{1}-L$ into Eq. (16), we have

$\begin{split}&\dot{x}_{11}^{(L-a)}=-\frac{\partial V(x_{11})}{\partial
x_{11}}-k(x_{11}-x_{2}+a)+\xi_{1}(t),\\\ &\dot{x}_{2}^{(L-a)}=-\frac{\partial
V(x_{2})}{\partial x_{2}}+k(x_{11}-x_{2}+a)+\xi_{2}(t).\end{split}$ (17)

By further transforming Eq.(17) by first $x_{11}\rightarrow x_{1}$ and then
$x_{1}\leftrightarrow x_{2}$, we obtain

$\begin{split}&\dot{x}_{2}^{(L-a)}=-\frac{\partial V(x_{2})}{\partial
x_{2}}+k(x_{1}-x_{2}-a)+\xi_{1}(t),\\\ &\dot{x}_{1}^{(L-a)}=-\frac{\partial
V(x_{1})}{\partial x_{1}}-k(x_{1}-x_{2}-a)+\xi_{2}(t).\end{split}$ (18)

Taking into account that $\xi_{1}$ and $\xi_{2}$ have the same statistical
properties, we finally obtain that

$\begin{split}&\dot{x}_{2}^{(L-a)}=-\frac{\partial V(x_{2})}{\partial
x_{2}}+k(x_{1}-x_{2}-a)+\xi_{2}(t)\\\ &=\dot{x}_{2}^{(a)},\\\
&\dot{x}_{1}^{(L-a)}=-\frac{\partial V(x_{1})}{\partial
x_{1}}-k(x_{1}-x_{2}-a)+\xi_{1}(t)\\\ &=\dot{x}_{1}^{(a)}.\end{split}$ (19)

That is,

$\begin{split}&v^{(L-a)}=(\dot{x}_{1}^{(L-a)}+\dot{x}_{2}^{(L-a)})/2=v^{(a)}\\\
&=(\dot{x}_{1}^{(a)}+\dot{x}_{2}^{(a)})/2.\end{split}$ (20)

This result makes it possible to interpret the symmetry of the average
velocity against the coupling free length $a$, as shown in Fig.3.

###  III.3 Current reversal induced by the potential asymmetry coefficient

In studies of Brownian ratchets, the asymmetry coefficient $\Delta$ of the
potential $V(x)$ significantly affects the directional current. In the above
discussions, we observed reversal of the directed motion by modulating the
coupling strength $k$ and coupling free length $a$. In fact, the direction can
also be determined by the asymmetric coefficient $\Delta$ for a given $a$ and
$k$. In Fig.5(a), the average velocity against the asymmetric coefficient
$\Delta$ is plotted for ($k$, $\Delta\theta$) = (1, 0), (300, 0), and (300,
$\pi$), where the other parameters are $t_{0}$ = 0.1 and $a$ = 0.3.

Figure 5:  the average velocity varying with the asymmetric coefficient
$\Delta$ of $V(x)$.

In the weak-coupling case (e.g., $k=1$), the curve for $k=1$, $\Delta\theta=0$
in Fig.5(a) shows that $v>0$ when $\Delta<0$, $v<0$ when $\Delta>0$, and $v=0$
when $\Delta=0$. Thus, the size and direction of the average velocity are
determined by the asymmetric coefficient $\Delta$ in this case. In addition,
the average velocity decreases approximately linearly with increasing
asymmetric coefficient $\Delta$. According to the above analysis, the motion
of coupled Brownian motors can be approximately considered as the motion of a
single Brownian motor immersed in the periodic potential $V(x)$ and Gaussian
white noise when the coupling strength $k$ is small, where the direction of
the velocity is determined by $\Delta$ in the temperature ratchet model of a
single particle. In the strong-coupling case (e.g., $k=300$), the average
velocity, the size and direction of which are also determined by the
asymmetric coefficient $\Delta$, increases linearly with increasing asymmetric
coefficient $\Delta$. The curves of the average velocity are reversed compared
to those of the weak-coupling case (e.g., $k=1$). Moreover, the average
velocity for $\Delta\theta=0$ is larger than that for $\Delta\theta=\pi$ when
$k=300$. Similarly, Fig.5(b) gives the theoretical curve of the average
velocity versus the asymmetric coefficient $\Delta$ according to Eq. (10),
which agrees qualitatively with the simulation results shown in Fig.5(a).

Figure 6:  the effective potential $V_{eff}(X)$ with different asymmetric
coefficient $\Delta$.

The effects of the asymmetric coefficient $\Delta$ on the average velocity can
be explained by the effective potential theory for the strong-coupling case.
When $\Delta_{eff}>0$, the average velocity $v>0$, and when $\Delta_{eff}<0$,
the velocity $v<0$ according to the dynamical mechanism of the temperature
ratchet model. The curve of the effective asymmetric coefficient
$\Delta_{eff}$ versus the asymmetric coefficient $\Delta$ is plotted in
Fig.6(a). One can easily find that the effective asymmetric coefficient
increases linearly with the asymmetric coefficient $\Delta$, and when
$\Delta=0$, the effective asymmetric coefficient $\Delta_{eff}$ is also zero.
Fig.6(b), (c), and (d) show the effective potential for different asymmetric
coefficients $\Delta$ given in the plot. Fig.6 shows that the asymmetry of
$V_{eff}(X)$ when $\Delta<0$ is opposite to that when $\Delta>0$, and the
effective potential $V_{eff}(X)$ is symmetric when $\Delta=0$. The asymmetry
of the effective potential decreases and it is close to symmetric when
$\Delta$ approaches zero. These results are in agreement with the dependence
of the averaged velocity on the asymmetric coefficient $\Delta$ for the
strong-coupling case shown in Fig.5(a).

Moreover, it is noteworthy that every curve in Fig.5(a) is antisymmetric about
$\Delta=0$ for different coupling cases. This can be explained well through an
analysis of the space-time transformation invariance. As we did above, we
rewrite Eq. (1), this time with a superscript corresponding to the value of
the asymmetric coefficient $\Delta$, and obtain

$\begin{split}&\dot{x}_{1}^{(\Delta)}=-\frac{\partial
V(x_{1},\Delta)}{\partial x_{1}}-k(x_{1}-x_{2}-a)+\xi_{1}(t),\\\
&\dot{x}_{2}^{(\Delta)}=-\frac{\partial V(x_{2},\Delta)}{\partial
x_{2}}+k(x_{1}-x_{2}-a)+\xi_{2}(t).\end{split}$ (21)

Adding the two formulas in Eq. (21), one naturally gets

$\begin{split}&\dot{x}_{1}^{(\Delta)}+\dot{x}_{2}^{(\Delta)}=-[\frac{\partial
V(x_{1},\Delta)}{\partial x_{1}}+\frac{\partial V(x_{2},\Delta)}{\partial
x_{2}}]\\\ &+\xi_{1}(t)+\xi_{2}(t).\end{split}$ (22)

By replacing the free length $\Delta$ with $L-\Delta$, we have

$\begin{split}&\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)}=-[\frac{\partial
V(x_{1},-\Delta)}{\partial x_{1}}+\frac{\partial V(x_{2},-\Delta)}{\partial
x_{2}}]\\\ &+\xi_{1}(t)+\xi_{2}(t).\end{split}$ (23)

Then, by putting $V(x)=-V_{0}[\sin(2\pi x/L)+(\Delta/4)\sin(4\pi x/L)]$ into
Eq. (23), we obtain

$\begin{split}&\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)}=\frac{2\pi
V_{0}}{L}[\cos(\frac{2\pi x_{1}}{L})-\frac{\Delta}{2}\cos(\frac{4\pi
x_{1}}{L})\\\ &+\cos(\frac{2\pi x_{2}}{L})-\frac{\Delta}{2}\cos(\frac{4\pi
x_{2}}{L})]+\xi_{1}(t)+\xi_{2}(t),\end{split}$ (24)
$\begin{split}&\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)}=-\frac{2\pi
V_{0}}{L}[\cos(\frac{2\pi x_{1}}{L}-\pi)+\frac{\Delta}{2}\cos(\frac{4\pi
x_{1}}{L})\\\ &+\cos(\frac{2\pi x_{2}}{L}-\pi)+\frac{\Delta}{2}\cos(\frac{4\pi
x_{2}}{L})]+\xi_{1}(t)+\xi_{2}(t).\end{split}$ (25)

By further inserting $x_{1}=x_{11}+L/2$ and $x_{2}=x_{21}+L/2$ into Eq. (25),
we have

$\begin{split}&\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)}=-\frac{2\pi
V_{0}}{L}[\cos(\frac{2\pi x_{11}}{L})+\frac{\Delta}{2}\cos(\frac{4\pi
x_{11}}{L})\\\ &+\cos(\frac{2\pi x_{21}}{L})+\frac{\Delta}{2}\cos(\frac{4\pi
x_{21}}{L})]+\xi_{1}(t)+\xi_{2}(t).\end{split}$ (26)

By then replacing $x_{11}\rightarrow x_{1}$ and $x_{21}\rightarrow x_{2}$, we
find that

$\begin{split}&\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)}=-\frac{2\pi
V_{0}}{L}[\cos(\frac{2\pi x_{1}}{L})+\frac{\Delta}{2}\cos(\frac{4\pi
x_{1}}{L})\\\ &+\cos(\frac{2\pi x_{2}}{L})+\frac{\Delta}{2}\cos(\frac{4\pi
x_{2}}{L})]+\xi_{1}(t)+\xi_{2}(t)\\\ &=[\frac{\partial
V(x_{1},\Delta)}{\partial x_{1}}+\frac{\partial V(x_{2},\Delta)}{\partial
x_{2}}]+\xi_{1}(t)+\xi_{2}(t)\\\
&=-(\dot{x}_{1}^{(\Delta)}+\dot{x}_{2}^{(\Delta)}).\end{split}$ (27)

Taking into account that the Gaussian white noises $\xi_{1}$ and $\xi_{2}$
have the same statistical properties, and comparing with Eq. (22), eventually
we find that

$\begin{split}&v^{(-\Delta)}=(\dot{x}_{1}^{(-\Delta)}+\dot{x}_{2}^{(-\Delta)})/2=-(\dot{x}_{1}^{(\Delta)}+\dot{x}_{2}^{(\Delta)})/2\\\
&=-v^{(\Delta)}.\end{split}$ (28)

This implies that the relationship between the velocity of the coupled
Brownian motors and the asymmetric coefficient $\Delta$ is antisymmetric. For
the special value $\Delta=0$, this leads to unbiased motion of the coupled
motors, and one has $v^{(0)}=0$. These results agree well with the effective
potential analysis.

##  IV  Optimization and Manipulation of Collective Directed Transport

It is interesting to investigate the dependence of the directed transport of
the system on the pulsation period $t_{0}$ and the phase shift $\Delta\theta$
between the two motor heads. In this section, we systematically determine how
to optimize the motor motion by modulating the pulsation period and phase
shift.

###  IV.1 Effect of the pulsation period of the temperature

The average velocity of the coupled motors against $t_{0}$ is plotted in Fig.7
for different coupling strengths $k$ and phase shifts $\Delta\theta$, with
$\Delta=1$ and $a=0.3$. The averaged velocity is negative when $k$ is small
(e.g., $k=1$), whereas the direction of motion is reversed when the coupling
strength $k$ is large (e.g., $k=300$). These results are consistent with the
discussion above in terms of the effective potential analysis shown in
Fig.2(b). For the two curves with $k=300$, the average velocity is larger for
the phase shift $\Delta\theta=0$ than for $\Delta\theta=\pi$, and the velocity
is nearly zero for the latter.

Figure 7:  the average velocity as the function of the pulsating period
$t_{0}$ of the temperature $T$.

In Fig.7, we plot the average velocity as a function of the pulsation period
$t_{0}$ of the temperature $T$. All the curves of the mean velocity clearly
have an optimal pulsation period $t_{max}$ having the largest velocity, and
the averaged velocity approaches zero when $t_{0}$ tends to zero and infinity.
This interesting behavior can be interpreted as follows. For an infinitely
large pulsation period $t_{0}$, the temperature pulsates so slowly that it can
be regarded as a constant. In this case, the coupled Brownian motors are
immersed in a stationary periodic potential and a white Gaussian noise with a
constant intensity, where the system has a null directed current regardless of
the coupling strength $k$. In contrast, when $t_{0}$ tends to zero, the
temperature fluctuates too rapidly, and the change in the configuration of the
coupled Brownian motors always lags behind the rapid temperature fluctuation.
This also leads to an absence of directed motion. Moreover, the solid curve in
Fig.7 gives the theoretical result according to formula (10) with $t_{0}$ and
$k$ tending to zero and infinity. The theoretical results agree well with that
of the simulation when $t_{0}$ is close to zero.

###  IV.2 Effect of the phase shift between the two temperatures

The phase shift $\Delta\theta$ between the temperatures of the two motors
significantly affects the current. In Fig.8(a), the average velocity versus
the phase shift $\Delta\theta$ is plotted for coupling strengths of $k$ = 1
and 300, and the other parameters are $\Delta$ = 1, $a$ = 0.3, and $t_{0}$ =
0.1. For a very small coupling strength $k$ (e.g., $k$ = 1), the effect of the
phase shift $\Delta\theta$ on the average velocity is almost negligible, and
the velocity is approximately a negative constant, $v=-0.2$. For strong
coupling (e.g., $k$ = 300), the phase shift $\Delta\theta$ has important
effects on the average velocity, as clearly shown in Fig.8 for the curve with
$k$ = 300. Moreover, the average velocity versus the phase shift
$\Delta\theta$ is symmetric about $\Delta\theta=\pi$ owing to the temporal
periodicity of the temperature $T(t)$ about $2\pi$, where $v$ first decreases
and then increase as $\Delta\theta$ increases in the range of 0 to $2\pi$.
Fig.8(b) shows the theoretical average velocity versus the phase shift
$\Delta\theta$ according to Eq. (10) with the parameters given in the figure,
which agrees qualitatively with the simulated data shown in Fig.8(a).

Figure 8:  The average velocity versus the phase shift $\Delta\theta$ with
different the coupling strengths $k$.

##  V Concluding Remarks

In this paper, we studied the mechanism of collaborative directed transport of
elastically coupled Brownian motors in an asymmetric periodic potential under
the modulation of two reservoirs with different and asynchronous temperatures.
We applied both invariance analysis of the space-time transformation of the
coupled dynamical equations and the effective potential theory to study the
coupling-induced symmetry breaking and the consequent collective directed
transport and current reversal behavior. The dynamical analysis indicated that
$\Delta\neq 0$ is the precondition of directed motion of the coupled system in
this dynamical model, and the presence of coupling between the two motors
influences the symmetry breaking of the ratchet system. In the strong-coupling
case, the directional transport of the coupled motors can be reversed by
modulating the coupling strength, coupling free length, or asymmetry
coefficient, which can be effectively illustrated using the effective
potential theory and invariance analysis of the space-time transformation.
Moreover, the relationships between the average velocity and various
parameters such as the pulsation period and the phase shift between the
temperatures of the two reservoirs are investigated systematically, and the
results provide a valuable way of optimizing and manipulating the collective
directed transport by adjusting different parameters in practice.

This work is partially supported by the National Natural Science Foundation of
China (Grant Nos.11075016 and 11475022),the Scientific Research project of
Zhangjiakou city (Grant Nos.1611064B) and the Scientific Research Funds of
Huaqiao University.

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