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\begin{document}
\begin{frontmatter}
\title{Movements of molecular motors: Ratchets, random walks, and
traffic phenomena}
\author[mpi]{Stefan Klumpp \thanksref{thank1}},
\author[amst]{Theo M. Nieuwenhuizen} and
\author[mpi]{Reinhard Lipowsky}
\address[mpi]{Max-Planck-Institut f\"ur Kolloid- und
Grenzfl\"achenforschung, 14424~Potsdam-Golm, Germany}
\address[amst]{Instituut voor Theoretische Fysica, Universiteit van
Amsterdam, Valckenierstraat~65, 1018 XE Amsterdam, The Netherlands}
\thanks[thank1]{
Corresponding author.
E-mail: klumpp@}
\begin{abstract}
Processive molecular motors which power the traffic of organelles in
cells move in a directed way along cytoskeletal filaments. On large
time scales, they perform random walks which arise from the repeated
unbinding from and rebinding to filaments. Unbound motors perform
Brownian motion in the surrounding fluid. In addition, traffic of
molecular motors faces similar problems as the traffic on streets
such as the occurrence of traffic jams and the coordination of
(two-way) traffic. These issues are studied theoretically using
lattice models.
\end{abstract}
\begin{keyword}
molecular motors \sep active movement \sep random walks \sep
lattice models
\sep traffic jams
\PACS 87.16.Nn \sep 05.40.-a \sep 05.60.-k
\end{keyword}
\end{frontmatter}
%%%%%%%%%%%%
\section{Introduction}
The idea of constructing nanometer-sized devices and machines has
created a lot of excitement during the last years. The functionality
of artificial nano-devices is, however, rather limited by now. At the
same time, more and more biomolecular nano-machines have been
identified in the cells of living beings where they accomplish a huge
variety of tasks. Many of these molecular motors are now rather well
studied and it turned out that these motors, as a result of
billions of years of evolution, work with an amazing degree of
precision and efficiency \cite{Howard2001,Schliwa2003}. In the
following, we will focus on one class of molecular motors which has
been studied quite extensively during the last decade, namely
processive cytoskeletal motors which power the traffic of vesicles and
organelles within cells. These motors hydrolyze adenosinetriphosphate
(ATP) and convert the free energy from this chemical reaction into
directed movements along filaments of the cytoskeleton. This class of
motors contains kinesins and dyneins which move along microtubules and
certain myosins which move along actin filaments. These motors walk
along the filaments by performing discrete steps with a step size
which corresponds to the repeat distance of the filament, 8~nm for
kinesins and dyneins and 36~nm for myosin V. They are called
processive if they make many steps while staying in contact with the
filament.
From a physical point of view, much of the interest in molecular
motors
is due to the fact that
the difference in size compared to macroscopic engines implies also
conceptual differences, despite many similarities such as the use of a
chemical 'fuel' which provides the (free) energy converted into
mechanical work. The difference in size is accompanied by a difference
in the typical energy scale of the engine. The typical energy of
macroscopic motors is much larger than the thermal energy, $k_{\rm
B}T$, while the typical energies of molecular motors are of the
order
of $k_{\rm B}T$.
On the one hand, this implies that molecular motors have to cope with
thermal fluctuations as a source of perturbation, and on the other
hand, the unavoidable presence of noise suggests that evolution has
come up with motor which make use of this noise in order to generate
work or movement.
From a technological viewpoint, the amazing properties of single
biological motor molecules and the complexity of the systems into
which they can be integrated in the cell provide inspiration for the
design of artificial active transport systems and suggest that
integrating biological motor molecules into synthetic devices is a
promising route towards a nanotechnology of molecular motors
\cite{Hess_Vogel2001}.
In this article, we discuss several theoretical aspects of the motor
movements.
In section
\ref{sec:Ratchets}, we start by summarizing some recent experimental
results and discuss the question whether noise-driven mechanisms are
used by these motors. In section \ref{sec:RW}, we discuss another
effect of noise, namely that processive linear motors detach from
their tracks due to thermal fluctuations, which leads to peculiar
random walks. We derive the asymptotic behavior of these random walks
using the statistical properties of the returns of motors to the
filament. Finally in section \ref{sec:traffic}, we summarize our
recent studies of traffic problems which arise in systems with many
molecular motors due to their mutual exclusion from binding sites of
the filaments.
\section{Active movements of molecular motors}
\label{sec:Ratchets}
Molecular motors can be studied outside cells using biomimetic model
systems. In these experiments, the biological complexity is reduced to
a minimal number of components, namely motors, filaments, and ATP.
These experiments allow one to observe movements of single motor
molecules and to measure transport properties such as velocities, step
sizes, and forces \cite{Howard2001,Schliwa2003}. This implies that,
on the one hand, these experiments provide insight into the motor
mechanisms and the dynamics of the single steps, and, on the other
hand, they allow to measure systematically the transport properties as
a function of certain external control parameters. In addition, they also
provide the basic setup for applications in nanotechnology, see
\bref{Hess_Vogel2001}.
With respect to the motor mechanisms, a main breakthrough was to
resolve the discrete steps of the motors and to measure the step size
of the motor walk which corresponds to the repeat distance of the
filament. This has first been achieved for kinesin
\cite{Svoboda__Block1993}. More recently, it has been shown that kinesin
\cite{Asbury__Block2003,Yildiz__Selvin2004} and also myosin~V
\cite{Yildiz__Selvin2003} move in a hand-over-hand way, i.e.\ the two
heads of the dimer alternate in stepping forward in a way that the rear head
always moves in front of the leading lead, similar to human walking.
With respect to the systematic measurement of the transport properties
as functions of external control parameters, the main focus has been
on the velocity as a function of the ATP concentration and of force
applied with, e.g., optical tweezers to oppose the movements, see,
e.g., \cite{Visscher__Block1999}. Other quantities that have been
measured are the one-dimensional diffusion coefficient of motors bound
to filaments or the randomness parameter and the walking distance
before unbinding from the filament. These measurements have stimulated
a large amount of theoretical work, see, e.g.\
\cite{Juelicher__Prost1997,Lipowsky2000b,Astumian_Haenggi2002},
modeling the walks of motors along filaments in order either to fit
the experimental data or to find out the generic properties of these
walks. For example, it turns out that the motor velocity as a function
of the ATP concentration is given by a universal relationship which
should be valid for many types of motors
\cite{Lipowsky2000a,Lipowsky_Jaster2003}.
Since nanometer-sized molecular motors have to live in a noisy
environment, it has soon been speculated whether these motors exploit
the noise to generate their directed movements and various variants of
ratchet models have been proposed as reviewed in
\cite{Juelicher__Prost1997,Lipowsky2000b,Astumian_Haenggi2002}. In the
simplest case, the conformational changes associated with the chemical
cycle of the motor rectify the one-dimensional Brownian motion along
the filament. While such a simple mechanism is not consistent with the
measurements for dimeric motors such as conventional kinesin or myosin
V, it can describe movements of processive monomeric motors such as
the monomeric kinesin KIF1A
\cite{Okada_Hirokawa1999,Tomishige__Vale2002,Okada__Hirokawa2003} (for
other processive monomeric motors, see the review
\cite{Schliwa_Woehlke2003}). These motors exhibit biased, but strongly
diffusive movements along the filaments similar to what one obtains in
the simplest ratchet models. Interestingly, coupling two such ratchets
by a spring \cite{Ajdari1994,Klumpp__Wald2001} leads to driving
mechanism which does not depend on the diffusion along the filament.
Similarly, dimerization of monomeric kinesin results in a higher
velocity and smaller diffusion coefficient of the filament-bound
motors \cite{Tomishige__Vale2002}. Many monomeric motors, however, do
not remain bound to the filament during the whole chemical cycle.
These results suggest that the dimerization of motors has two effects:
It allows the motor to stay bound to the filament for many chemical
cycles (which many monomeric motors do not), and in addition, it
allows the motors to move by a more efficient mechanism which, in
contrast to the basal motility of monomers, does not
rely on diffusion along the filament .
\section{Random walks arising from many diffusional encounters with filaments}
\label{sec:RW}
\subsection{Random walks in open compartments}
% Fig.1
\begin{figure}[tb]
\begin{center}
\includegraphics[angle=0,width=.5\textwidth]{fig1}
\caption{Random walk of a molecular motor: The motor performs directed
movement along a filament (grey rod) and unbinds from it after a
certain walking distance. The unbound motor diffuses in
the surrounding fluid until it rebinds to the filament and
resumes directed motion.}
\label{fig:randWalk}
\end{center}
\end{figure}
The fact that nanometer-sized motor molecules necessarily work in a
noisy environment also has another consequence: Even processive motors
do not move along a filament forever, but unbind from it after a
certain binding time (which corresponds to a typical walking
distance), because the binding energy is finite and can be overcome by
thermal fluctuations. For a single kinesin or myosin V motor, the
binding times and walking distances are of the order of 1 s and 1
$\mu$m, respectively. Much longer walking distances can be obtained if
several motors form a complex or if a cargo is transported by a larger
number of motors. The unbinding which arises as the effect of
unavoidable fluctuations may also have a biological function, since it
allows the motors to diffuse around obstacles such as other proteins
bound to the filament.
Unbound motors perform simple Brownian motion until they rebind to the
same or another filament. On large time scales, the combination of
active directed movements along filaments and non-directed Brownian
motion leads to peculiar random walks which consist of alternating
sequences the two types of movments
\cite{Ajdari1995,Lipowsky__Nieuwenhuizen2001}.
In order to determine the effective transport properties of these
random walks, we have studied several simple arrangements of filaments
embedded in compartments of various geometries. A particularly simple
but intriguing case consists of a single filament and a set of
confining walls, which restrict the diffusion of unbound motors. In
the simplest case, there are no confining walls and the unbound motors
can diffuse freely in the full three-dimensional space (similar
behavior is obtained for a half space geometry which is more easily
accessible to experiments). By placing the filament in a quasi
two-dimensional slab or in a cylindrical tube (geometries, which are
also accessible to {\it in vitro} experiments), diffusion can be
restricted along one or two dimensions perpendicular to the filament.
We have studied these random walks by mapping them to random walks on
a lattice \cite{Lipowsky__Nieuwenhuizen2001}. A line of lattice sites
represents the filament. Motors at these sites perform a biased random
walk and move predominantly into one direction, which we choose to be
the positive $x$ direction. With a small probability $\epsilon/2d$,
they move to each of the adjacent non-filament sites and thus unbind
from the filament. At the non-filament sites the motors perform simple
symmetric random walks and move to each neighbor site with probability
$1/2d$ ($d$ denotes the spatial dimension) and rebind to the filament
with probability $\piad$ when they reach again a filament site.
Confining walls are implemented as repulsive boundaries, at which all
movements into the walls are rejected.
We have used scaling arguments, computer simulations, and exact
solutions of the master equations to study these random walks
\cite{Lipowsky__Nieuwenhuizen2001,Nieuwenhuizen__Lipowsky2002,Nieuwenhuizen__Lipowsky2004}.
The solutions exhibit anomalous drift behavior and strongly enhanced
diffusion parallel to the filament due to the repeated binding and
unbinding. At large times, motors move with an effective velocity
given by $v_\bd P_\bd$, where $v_\bd$ is the velocity of the directed
walks while bound to the filament and $P_\bd$ is the probability that
the motor is bound to the filament. In the tube geometry, $P_\bd$ is
time-independent at large times and given by the equilibrium of
binding/unbinding and diffusion perpendicular to the filament. In the
slab and half-space geometries as well as in two- and
three-dimensional space without confining walls, equilibrium is not
reached at any time, because motors can rebind to the filament after
arbitrarily large excursions, and because the longer one observes the
motors, the larger the excursions that contribute to the average
behavior. Therefore, $P_\bd$ and the effective velocity are
time-dependent in these cases, namely $P_\bd(t)\sim t^{-d_\perp/2}$
for compartments with $d_\perp$ dimensions of unconfined diffusion
(for $d$-dimensional space without confining walls, we have
$d_\perp=d-1$). The time-dependent effective velocity implies that the
average displacement of the motors grows sublinearly. In the
effectively two-dimensional slab geometry (as well as on a
two-dimensional lattice without confining walls), the displacement
behaves as $x(t)\sim\sqrt{t}$ at large times, and in the half space
(or full three-dimensional space), it is given by $x(t)\sim \ln t$.
In the following, we will give a simplified description which explains
these features and relates them to known results from the theory of
random walks.
\subsection{Asymptotics and return to the filament}
For simplicity, we discuss the case of a single infinitely long
filament embedded into $d$-dimensional space with $d=2$ or $d=3$.
Motor particles binding to this filament walk along it with velocity
$v_\bd$ until they unbind. Unbinding occurs with a rate
$\sim\epsilon$, so that the motors perform straight movements or
effective 'active steps' of length $x_s\sim v_\bd/\epsilon$. The
diffusive excursions between two such steps bring the motors back to
the filament. (For simplicity, we take the sticking probability
$\piad$ to be one. If $\piad<1$, the effective step size is $x_s\sim
v_\bd\piad/\epsilon$ on average, because the motor has to return to
the filament $1/\piad$ times before rebinding.) The distribution
$\psi(\tau)$ of the excursion times $\tau$ is therefore given by the
distribution of return times of a random walker to a line in $d$
dimensions or, projected into the subspace perpendicular to the
filament, to the origin in $d-1$ dimensions. This is a classical
problem in the theory of random walks which was solved by Polya in
1921 and has lead to the remarkable result that the return of a random
walker to the origin is certain on one- and two-dimensional lattices,
but not in three dimensions \cite{Polya1921}. For the molecular
motors, this implies that they will return to the filament with
certainty.
If we are only interested in the movement parallel to the filament, we
can consider the excursions away from the filament as periods of rest.
The random walks of the motors are then described by continuous time
random walks with dwell time distribution $\psi(\tau)$. For this type
of random walks, solutions can be obtained using Fourier--Laplace
transforms \cite{Weiss1994}. A short summary of this method is given
in the appendix. In that way, one can finds that the Laplace
transforms of the first two moments of the position of this random
walker are given by
\begin{eqnarray}\label{asMoments_PSIabh}
x(s)&=&\frac{\langle x_s \rangle\psi(s)}{s[2-\psi(s)]} \nonumber \\
% \qquad {\rm and}
\qquad x^2(s)&=&\frac{\langle x_s\rangle^2 \psi(s)}{s[1-\psi(s)]}+\frac{2\langle x_s^2\rangle \psi^2(s)}{s[1-\psi(s)]^2}
\end{eqnarray}
where $\psi(s)$ is the Laplace transform of the waiting time
distribution $\psi(s)\equiv\int_{0}^{\infty} \d t\, \psi(t) e^{-st}$.
From these relations, the average displacement and the dispersion of
the random walks of molecular motors for large times can be obtained
by inverting the Laplace transform. In particular, these relations
imply that the asymptotic displacement of the motors is given by the
large-time (or small $s$) behavior of the distribution of return times
to the origin in $d-1$ dimensions.
Normal drift behavior with $x(t)\sim t$ is obtained, as long as
$\psi(\tau)$ has a finite mean value, $\bar\tau$, and thus
$\psi(s)\approx 1-\bar\tau s$ for small $s$. If, however,
$\psi(\tau)$ decays slower than $\sim\tau^{-2}$ at large $\tau$, the
mean waiting time diverges (which implies a divergence of
$[\psi(s)-1]/s$ for small $s$), and anomalous drift is obtained, i.e.,
the average displacement $x(t)$ grows slower than linearly with time.
The latter behavior occurs in our case (where the waiting times are
given by return times to the filament) if the diffusion away from the
filament is not restricted.
In the two-dimensional case, the return time distribution behaves as
$\psi(\tau)\approx 1/(2\sqrt{\pi} \tau^{3/2})$ for large $\tau$ or
$\psi(s)\approx 1-\sqrt{s}$ for small $s$. Inserting this into
\eq{asMoments_PSIabh} and inverting the Laplace transform, we obtain
\begin{eqnarray}
x(s)\approx \frac{x_s}{s^{3/2}} &=& \frac{v_\bd}{\epsilon s^{3/2}}
\nonumber \\
\qquad{\rm and}\qquad
x(t)\approx \frac{2 x_s\sqrt{t}}{\sqrt{\pi}} &=& \frac{2v_\bd\sqrt{t}}{\epsilon\sqrt{\pi}}
\end{eqnarray}
for small $s$ and large $t$, respectively. Similarly, in the
three-dimensional case, the return time distribution is
$\psi(\tau)\approx 2\pi/(3 \tau\ln^2 \tau)$ for large $\tau$ or
$\psi(s)\approx 1-2\pi/(3\ln s^{-1})$ for small $s$, which leads to
\begin{eqnarray}
x(s) &\approx& \frac{3 x_s \ln s^{-1}}{2\pi\, s} = \frac{3v_\bd \ln
s^{-1}}{2\pi\epsilon\, s} \nonumber \\
% \qquad{\rm and}\qquad
x(t) &\approx& \frac{3 x_s}{2\pi}\ln t = \frac{3v_\bd}{2\pi\epsilon}\ln t.
\end{eqnarray}
Likewise, we can obtain the dispersion $\Delta x^2(t)$ of the motors
from the second moment of the distribution arising from the encounters
with filaments. Using \eq{asMoments_PSIabh} leads to $\Delta
x^2(t)\approx \frac{2(\pi-2)}{\pi}(v_\bd/\epsilon)^2 t+\frac{1}{2}t$
and $\Delta x^2(t)\approx \frac{9}{4\pi^2}(v_\bd/\epsilon)^2 \ln t
+\frac{1}{3}t$ in two and three dimensions, respectively, where we
have added the contribution due to the diffusion of unbound motors
parallel to the filament. Note that the broadening of the distribution
of motors due to the encounters with the filament is characterized by
an anomalously high effective diffusion coefficient of the order of
$(v_\bd/\epsilon)^2$ in two dimensions, while in three dimensions, the
leading term is the unbound diffusion, but with a larger logarithmic
correction, again of the order $(v_\bd/\epsilon)^2$.
These results agree with the corresponding asymptotic results from the
exact solution of the full master equations
\cite{Nieuwenhuizen__Lipowsky2002,Nieuwenhuizen__Lipowsky2004}.
\section{Traffic phenomena in many-motor systems}
\label{sec:traffic}
\subsection{Traffic jams and density patterns}
Finally, we consider systems with many interacting molecular motors.
The simplest type of motor--motor interaction is simple exclusion
which arises from the fact that a motor occupies a certain volume and,
in particular, excludes other motors from the binding site of the
filament to which it is bound as observed in decoration experiments,
e.g.\ \cite{Song_Mandelkow1993}. Exclusion is most important if motors
accumulate in certain regions along the filaments, where it leads to
the formation of molecular traffic jams. These interactions are
easily incorporated into our lattice model by rejecting all movements
to occupied lattice sites. These models then represent new variants of
exclusion processes or driven lattice gas models, where the active or
driven movements are localized to the filaments. In driven exclusion
processes, the boundary conditions can determine the state of the
system.
We have studied tube systems with a single filament located along the
axis of a cylindrical tube and with several types of boundary
conditions at the left and right tube ends. We use the convention that
the active movements along the filament is directed to the right.
There are several possibilities to build such tube systems
artificially such as micropipette glass tubes or liquid microchannels,
but there are also several tubular compartments within cells for which
these tube systems provide simple descriptions. The most prominent
example for the latter is the axon of a nerve cell.
The simplest situation is given by {\it periodic boundary conditions}
which can be solved exactly \cite{Klumpp_Lipowsky2003}. In this case,
motor particles reaching the right end of the tube simply restart
their movements at the left end. This leads to constant density
profiles for both the bound and unbound motors in the stationary
state. Diffusive currents in the radial direction vanish and binding
to the filament is locally balanced by unbinding.
%In this situation,
%which we call radial equilibrium, the bound and unbound motors
%densities $\rho_\bd$ and $\rho_\ub$, respectively, fulfill the
%condition
%\begin{equation}
% \epsilon\rho_\bd(1-\rho_\ub)=\piad\rho_\ub(1-\rho_\bd).
%\end{equation}
The current of motors through the tube is given by
$J=v_\bd\rho_\bd(1-\rho_\bd)$ and the value of the bound density
$\rho_\bd$ is determined by the total number of motor particles in the
tube. If the number of motors within the tube is increased beyond an
optimal number (where $\rho_\bd=1/2$ and $J=v_\bd/4$), the current
through the tube decreases due to jamming of the motors.
In a {\it closed tube}, motors accumulate in front of the right end,
and a diffusive current of unbound motors to the left balances the
current along the filament in the stationary state
\cite{Lipowsky__Nieuwenhuizen2001}. In that case, motors are
essentially localized at the right tube end if the number of motors in
the tube is small. If the number of motors within the tube is
increased, a jammed region at the right tube end builds up, separated
from a low density region to its left by a rather sharp interface,
which provides probably the simplest example for active pattern
formation by molecular motors. The crowded domain spreads to the left
at higher numbers of motors, and, eventually, if there are many motors
in the tube, the filament is uniformly covered by motors and rather
crowded.
Tubes with {\it open boundaries} which are coupled to reservoirs of
motors at both ends, so that motors enter the tube at the left end and
leave it at the right end, exhibit boundary-induced phase transitions
\cite{Klumpp_Lipowsky2003}. The three phases that occur in these
systems are characterized by the bottleneck which determines the motor
current through the tube. This bottleneck can be the left boundary,
the right boundary or the interior of the tube. These three cases
corresponds to the low-density (LD), high-density (HD) and
maximal-current (MC) phase. If changing the motor densities in the
reservoirs at the boundaries leads to a change in the bottleneck
position, a phase transition occurs which can be either discontinuous
(LD--HD) or continuous (LD--MC and HD--MC). These types of phases and
transitions are known from the one-dimensional asymmetric simple
exclusion process (ASEP) \cite{Krug1991,Kolomeisky__Straley1998} which
corresponds to the dynamics along the filament in our model without
the binding and unbinding processes. The presence of the unbound state
of the motors, however, increases the number of ways how to chose
precisely the boundary conditions, and since the phase transitions are
boundary-induced, the location of the transition lines within the
phase diagram is very sensitive to that choice, which may be hard to
control in an experimental system, so that precise predictions will be
difficult. Nevertheless, systems of molecular motors are promising
candidates for the observation of boundary-induced phase transitions.
\subsection{Phase transitions in two-way traffic}
Another type of phase transitions occurs in systems with two-way
traffic of motors: Each motor species moves either towards the plus-
or towards the minus-end of the corresponding filament. However,
different types of motors move into opposite directions along the same
filament. Decoration experiments indicate that these motors interact
in such a way that a motor is more likely to bind to the filament if
another motor which belongs to the same species is already bound
there, and that motors have a tendency to unbind more easily if they
encounter a motor moving into opposite direction. This interaction is
probably mediated by conformational changes or deformations of the
filament upon binding of a motor. We have studied systems with two
species of motors moving into opposite directions along the same
filament and modeled these interactions using a model with a
single interaction parameter $q$ by which the binding and unbinding
rates are increased or reduced in order to enhance binding and reduce
unbinding if a motor of the same type is present at a neighbor site
and to enhance unbinding and reduce binding if a motor with opposite
directionality is bound at a neighbor site \cite{Klumpp_Lipowsky2004}.
It turns out that if there are no such effectively attractive and
repulsive interactions and motors interact only through their mutual
exclusion, the motor current will be very small because motors often
block each other and continue moving only after one of them detached
from the filament. For sufficiently strong interaction with $q>q_c$
where $q_c$ is a critical value of the interaction parameter which
depends on the overall motor concentration and for equal concentrations of
both motor species, however, there is spontaneous symmetry breaking,
so that one motor species occupies the filament, while the other one
is largely excluded from it. If several filament aligned in parallel
and with the same orientation are present, this symmetry breaking
leads to the spontaneous formation of traffic lanes for motor traffic
with opposite directionality. Varying the relative concentration of
the two motor species at $q>q_c$ leads to a discontinuous phase
transition with hysteresis, similar to the transitions induced by
changes of the external fields in magnetic systems. In contrast to
the boundary-induced phase transitions discussed in the preceding
section, these transitions do not depend on the boundaries, but are
induced by the binding and unbinding dynamics. Correspondingly, they
are robust against the choice of boundary conditions to a large
extent. They depend, however, on the active movements of the motors,
and are not found in an equilibrium situation where the motor velocity
is zero, $v_\bd=0$, e.g.\ because ATP is absent.
\section{Summary}
In summary, molecular motors exhibit interesting movements on several
length scales. Here we have addressed two of these length scales,
namely the walks along filaments which consist of typically of the
order of 100 steps with a step size of the order of 10nm, and random
walks which consist of many such walks along filament interrupted by
periods of diffusion after unbinding from the filament. In addition,
the presence of many motors leads to traffic phenomena such as traffic
jams and traffic lanes, similar to the macroscopic traffic on streets.
However, unbinding of motors from the filaments due to thermal
fluctuations (which is a consequence of their microscopic size) plays
an important role and can help to circumvent obstacles and to
regulate the traffic.
\appendix
\section{Continuous time random walks}
\label{sec:app}
In this appendix, we summarize some results for random walks with a dwell time distribution $\psi(\tau)$ and a step
distribution $\mathcal{P}(x_s)$ which are used in section
\ref{sec:RW}.
We consider random walkers in one dimension which make the first step at time $t=0$ starting from the origin, $x=0$. The probability distribution $p(x,t)$ of such a random
walk fulfills the recursion relation
\cite{Montroll_Weiss1965,Weiss1994}
\begin{equation}
p(x,t)=\sum_{n=0}^{\infty}p_n(x)\int_0^t \psi_n(t')\Psi(t-t')\d t',
\end{equation}
where $p_n(x)$ is the probability density that the walker is at
position x after the $n$'th step, $\psi_n(t)$ is the probability that the
$n$'th step occurs at time $t$, and $\Psi(t)\equiv\int_t^\infty
\psi(\tau)\d\tau$ is the probability that no step occured until time
$t$. The initial conditions are $p_0(x)=\delta(x)$ and
$\psi_0(t)=\delta(t)$. The solution of this recursion can be obtained using
Fourier--Laplace transforms, %see, e.g., \cite{Weiss1994},
which leads
to
\begin{equation}\label{CTRW_sq}
p(q,s)=\frac{1-\psi(s)}{s[1-\mathcal{P} (q)\psi(s)]}
\end{equation}
for the Fourier--Laplace transform of the probability distribution
$p(x,t)$ \cite{Montroll_Weiss1965} with the Laplace transform of the
waiting time distribution,
%\begin{equation}
$\psi(s)\equiv\int_{0}^{\infty} \d t\, \psi(t) e^{-st}$
%\end{equation}
and the Fourier transform of the step distribution
%\begin{equation}
$\mathcal{P}(q)\equiv\int_{-\infty}^{\infty}\d x_s \mathcal{P}(x_s) e^{iqx_s} \approx 1+i\langle x_s\rangle q -\langle x_s^2\rangle q^2/2$.
%\end{equation}
The latter expansion is valid for small $q$, provided that the moments $\langle x_s^n\rangle$ of the step distribution $\mathcal{P}(x_s)$ with $n=1,2$ are finite. Using this expansion,
we can derive expressions for the Laplace
transforms of the moments of our random walk by expanding $p(q,s)$ as given in \eq{CTRW_sq} in powers of $q$
\cite{Shlesinger1974} which leads to \eq{asMoments_PSIabh} from which
the asymptotic behavior of the time-dependent moments can by obtained
via the Tauberian theorems, see \cite{Weiss1994}.
%\bibliographystyle{unsrt}
%\bibliographystyle{elsart-num}
%\bibliography{../../Bibliographien/motoren,../../Bibliographien/DrivenLatticeGases,../../Bibliographien/meinePapiere,../../Bibliographien/Tools,../../Bibliographien/sonstiges,../../Bibliographien/Biology,../../Bibliographien/RandomWalks}
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\end{document}