Traffic by multiple species of molecular motors

Intracellular transport by molecular motor proteins is essential for many cellular functions such as shuttling vesicles or organelles to their proper destinations [1, 2]. In eukaryotic cells, polar cytoskeletal filaments serve as tracks for three families of cytoskeletal motors: kinesins and dyneins that move along microtubules, and myosins that walk along actin filaments [1, 2].

The movements of a molecular motor along its filamentous track can be described as a biased random walk. However, after a certain number of steps the motor unbinds from the track because the thermal noise of the environment overcomes the finite motor–filament binding energy. The motor then diffuses in the surrounding fluid as a Brownian particle until it comes close to the track to rebind to it. On large time scales, motors therefore perform peculiar ’motor walks’ that consist of alternating periods of directed motion or biased random walks on the tracks and undirected diffusive motion in the fluid environment [3].

Intracellular transport is typically achieved by several types of motors rather than just by one type [4, 5]. Different types of motors may either move independently along the same filament or work together to power bi-directional transport of a cargo that carries two types of motors [6]. Here we consider the first case, where different species of motors move on the same filament, and interact only via their mutual exclusion. These motors species might move into different directions, or they might move into the same direction but have different velocities or different affinities to the filament. We are interested in the traffic behavior of the latter systems.

A simple model for motor traffic is based on the asymmetric simple exclusion process (ASEP) [7]. In the ASEP, particles that interact through mutual exclusion from lattice sites perform biased random walks along a one dimensional lattice. This paradigmatic model for non-equilibrium transport exhibits interesting phenomena such as boundary induced phase transitions and traffic jams [7]. To describe the traffic of motor proteins, ASEP-like models have to be supplemented with the dynamics of binding to and unbinding from the track to account for the finite run length of the motors [8, 9, 10, 11, 12].

ASEPs with a single species of particles can be generalized to models with multiple species [13, 14]. Even without attachment and detachment of particles, multi-species ASEPs exhibit a variety of cooperative phenomena, such as spontaneous symmetry breaking [15] and phase separation [16, 17].

To model cellular traffic with different types of motors, one has to consider multi-species ASEPs with attachment and detachment of particles. Apart from investigations on conditions for the exact solvability [18, 19], most work on multi-species ASEPs with detachment and attachment has focused on the traffic of two species of particles moving into opposite directions [20, 21, 22, 23, 24]. These models exhibit continuous phase transitions [20, 21, 23], hysteresis of the total current [21], traffic jams [24], spontaneous symmetry breaking and localized shocks [22]. Here we focus on the case where two species of particles move in the same direction, but do so either with different velocities or with different binding affinities to the filament. A few such systems have been studied experimentally in vitro, in particular the traffic of mixtures of fast-moving and slow- or non-moving kinesin motors [25].

The paper is organized as follows: In section II we will present the theoretical model for multi-species motor traffic. We will focus on the situation that two motor species differ in only one parameter. In section III we will discuss the traffic of two types of motors which differ only in their unbinding parameters. In section IV we will discuss the traffic of two species with different stepping parameters, specifically one species with zero velocity. We will end with a summary and discussion in section V.

To investigate cellular transport or in vitro experiments with multiple species of molecular motors, we study multi-species ASEP models with attachment and detachment of particles. As mentioned in the introduction section, motors perform random walks, which are biased towards the same direction for all motor species, along a cytoskeletal filament. Such filaments are polymers which have periodic motor binding sites with repeat distance $\ell$, and motors walk along the filament with steps of the same size $\ell$. For microtubules, the lattice constant is $\ell=8$ nm. We therefore represent the filament by a one-dimensional lattice of $L$ binding sites with spacing $\ell$. Per unit time $\tau$, a bound motor of type $i$ on the filament makes a forward step with probability $\alpha_{\rm i}$ if the target site is free, unbinds with probability $\epsilon_{\rm i}$ and remains at the same site otherwise [8]. Unbound motors are treated as a motor reservoir [9, 10] with unbound density $\rho_{\rm ub,i}$ for the $i$-th type of motors. An unbound motor binds to the filament with the sticking probability $\pi_{\rm ad,i}$ when the binding site is empty. The treatment of unbound motors as a reservoir is slightly different from that in a previous lattice model for motor traffic in Ref. [8] where the diffusion of unbound motors has been treated explicitly as Brownian movement. The simplification to treat the unbound motors as a reservoir is appropriate if the density of unbound motors is homogeneous, as is the case for sufficiently fast diffusion and large motor numbers in the bulk solution. This is typically the case in in vitro experiments, where the buffer is an aqueous solution with motor concentrations in the nano- or micro-molar range. The typical length of a microtubule is of the order of $10\mu\rm{m}$ or even larger, which is one order of magnitude larger than the typical motor run length, which is of the order of $1\mu\rm{m}$ for kinesins. Therefore, we use periodic boundary conditions throughout this paper. This simplification is appropriate as most motors do not feel the ends of the filament, so that effects of the filament ends can be neglected. ¹¹1Note also that we do not scale the binding and unbinding probabilities of motors with the system size $L$. The latter scaling, which has been used in Ref. [10], leads to a motor run length that is comparable to the filament length $L$ in the limit of large $L$. In our simulations, the length $L$ of the filaments is chosen sufficiently large so that finite-size effects can be ignored, as we show explicitly in Sec. IV below. ²²2In most of our simulations, the filament length $L$ was chosen to be $L=200$ corresponding to about $2\mu\rm{m}$. In the presence of slow motors, the actual run length of the fast motors is strongly reduced and small compared to $L=200$. In the absence of slow motors, as in Sec. III, the length of the filament is irrelevant as well since our system then exhibits mean-field behavior for all $L$.

To sum up, as shown in Fig. 1, during each time step $\tau$ one of the $L$ sites is randomly chosen and updated according to the following stochastic dynamical rules:

where $0$ denotes a vacancy site and $i$ represents a site occupied by a motor particle of species $i$. All probabilities for other transitions are zero.

We performed Monte Carlo simulations on the the one-dimensional periodic lattice with $L$ sites. In the simulations, we control the number of motors in the system by changing the value of the unbound motor densities. This mimics in vitro experiments where the motor concentration in the buffer is an easily accessible control parameter. Cells can also control the motor concentration in the cytoplasm by regulating gene expression. Once the unbound motor densities have been settled, we use a random-sequential update. Each simulation step, which corresponds to a unit of the basic time scale $\tau$, consists of $L$ Monte Carlo moves. At each such move, a lattice site is chosen randomly and updated according to the dynamical rules (1) to (3). The simulations are performed long enough ($10^{10}$ steps) to ensure that the steady state of the system has been reached. The current and bound density are calculated by averaging over all the simulation steps and over all lattice sites.

In this article, we consider the traffic of two species of motors for long times, i.e. in the stationary state. Generally speaking, all the parameters for different motor species might be different. Here, as a simple starting point of theoretical work, we focus on the situation that the two motor species differ in only one parameter. We consider different unbinding probabilities, i.e. $\epsilon_{1}\neq\epsilon_{2}$, in section III and different forward stepping probabilities, i.e. $\alpha_{1}\neq\alpha_{2}$, in section IV. We will not explicitly consider the case of different binding probabilities for different motor species, i.e. $\pi_{\rm ad,1}\neq\pi_{\rm ad,2}$, because this case is equivalent to that of different species with only different unbound densities, since the attachment probability is the product of binding probability for single motor $\pi_{\rm ad,i}$ and unbound motor density $\rho_{\rm ub,i}$, see the dynamical rule (3).

Since we are interested in modeling molecular motor traffic, we choose the probabilities for motor movement as found experimentally for some specific molecular motors. We focus on the well-studied motor kinesin-1 for which all necessary parameters have been measured in single molecule experiments. Kinesin takes about 100 steps before unbinding from the filament [1], so that we can estimate the ratio of stepping and unbinding probabilities to be $\alpha/\epsilon\approx 100$. Kinesin’s microtubule desorption constant $K_{\rm D,MT}$ is of the order of $0.1-1\ \mu$M [26, 25], which fixes the ratio of the unbinding and binding probabilities $\epsilon/\pi_{\rm ad}=N_{\rm A}\ell^{3}K_{\rm D,MT}\approx 10^{-4}$ with the Avogadro constant $N_{\rm A}$, see Ref. [27]. We choose the discretization time unit $\tau$ so that the binding probability $\pi_{\rm ad}\equiv 1$, which means that we have unbinding probability $\epsilon=10^{-4}$ and stepping probability $\alpha=10^{-2}$. Considering kinesin’s velocity $v_{0}=\alpha\ell/\tau\approx 1\,\mu$m/s [1], this means that our discretization time is $\tau\approx 10^{-4}$ s, which is sufficiently small to avoid discretization artifacts. The typical values of the probabilities used in this article are summarized in Table 1.

In this section we consider the case of two species of motors which have different unbinding probabilities $\epsilon_{1}$ and $\epsilon_{2}$. The stepping probability $\alpha$ and the binding probability $\pi_{\rm ad}$ for both species are the same so that the indices of the corresponding symbols can be omitted.

Since the detachment and attachment rules (2) and (3) do not involve neighboring sites, there is no correlation of motors in the processes of binding and unbinding. Therefore the balances of attachment and detachment for both species of motors in the stationary state can be described by mean field theory as follows:

where $\rho_{\rm b,1}$ and $\rho_{\rm b,2}$ are the bound densities of type-1 and type-2 motors, respectively, and $\rho_{\rm b}=\rho_{\rm b,1}+\rho_{\rm b,2}$ is the total bound density. The terms on the left-hand side of Eqs. (4) and (5) represent the attachment of motors, with the mean field factor $(1-\rho_{\rm b})$ for the exclusion interaction between motors. The terms on the right-hand side represent the detachment of bound motors. Here we ignore the exclusion interaction when motors try to detach from the filament since the total unbound density of motors $\rho_{\rm ub}=\rho_{\rm ub,1}+\rho_{\rm ub,2}$ is very low, typically below $10^{-3}$, which corresponds to a bulk concentration of unbound motor proteins below $\rho_{\rm ub}/\ell^{3}N_{\rm A}\approx 1$ M.

For convenience, we define the fractions

of the $i$-th type of motors in the unbound and bound states, respectively, with $\Psi_{\rm ub,1}+\Psi_{\rm ub,2}=\Psi_{\rm b,1}+\Psi_{\rm b,2}=1$. Because the two species of motors have different unbinding probabilities, the fractions of these two types of motors in the bound states are not equal to those in the unbound states, or more precisely,

From equations (4) and (5) we obtain the bound densities of both types of motors and the total bound density

with an effective unbinding probability

Eq. (8) has the same form as the relation of binding and unbinding probabilities in the 1-species ASEP with attachment and detachment, except that the unbinding probability of the single motor species is replaced by the weighted average effective unbinding probability $\epsilon_{\rm eff}$. In the limit of $\Psi_{\rm ub,1}=0$ or $\Psi_{\rm ub,2}=0$, the effective unbinding probability becomes $\epsilon_{\rm eff}=\epsilon_{2}$ or $\epsilon_{\rm eff}=\epsilon_{1}$, respectively, and Eq. (8) reduces to the corresponding relation of the single-species ASEP.

The total current $J$ of both species can be determined by mean field theory as the product of forward stepping probability $\alpha$, total bound density $\rho_{\rm b}$ of the two types of motors and density of vacancies $1-\rho_{\rm b}$,

The total motor current has a maximum at $\rho_{\rm b}=0.5$. It decreases for bound densities larger than $\rho_{\rm b}=0.5$, because overcrowding of bound motors leads to traffic jams.

Simulation results of the ASEP with two species of particles with different unbinding probabilities are in excellent agreement with mean field theory, see the example with $\epsilon_{2}=8\epsilon_{1}$ in Fig. 2. This observation strongly suggests that mean-field theory is exact in the case of two species of motors which differ only in the unbinding probabilities.

Results of the mean-field theory for the ASEP with two species of particles with different unbinding probabilities are in excellent agreement with simulation results (see the example with $\epsilon_{2}=8\epsilon_{1}$ in Fig. 2) and with the numerical solution of the associated stochastic process for small system sizes (not shown). This observation indicates that mean-field theory is exact in the case of two species of motors which differ only in the unbinding probabilities.

As shown in Fig. 2, the total bound density $\rho_{\rm b}$ increases with the total unbound density $\rho_{\rm ub}$, as to be expected. The total current $J$ as function of unbound density $\rho_{\rm ub}$ has a maximum because overcrowding by motors causes traffic jams. The maximum shifts to higher unbound density with increasing fraction of type-2 motors, because type-2 motors have a larger unbinding probability than type-1 motors. Therefore a higher fraction of type-2 motors leads to smaller bound density and less traffic jam.

We next consider two motor species that differ only in the forward stepping probabilities, i.e. $\alpha_{1}\neq\alpha_{2}$. The binding probability $\pi_{\rm ad}$ and unbinding probability $\epsilon$ for these two species are the same so that the indices of the corresponding symbols can be omitted. Without loss of generality, we assume that type-1 motors have a larger stepping probability than type-2 motors. We will focus on the case that the second motor species has zero forward stepping probability $\alpha_{2}=0$, because we expect that this largest difference between $\alpha_{1}$ and $\alpha_{2}$ leads to the strongest effect. This choice is also motivated by in vitro experiments, which addressed the traffic of mixtures of moving motors and non-moving mutant motors [25].

Motivated by cellular traffic, which is powered by several different species of molecular motors, we have studied the traffic of two motor species along the same track. For this purpose we have used a two-species ASEP model with periodic boundary conditions with attachment and detachment kinetics, in which the two motor species move into the same direction but have different velocities or unbinding rates. We have determined characteristic properties of the movements, such as motor density and total motor current, especially the current-density relationship, by computer simulations and by analytical calculations. As in the cell or in in vitro experiments, our control parameter is provided by the densities of unbound motors.

We have considered two motor species that differ in only one property. If the two motor species only differ in the unbinding probabilities, increasing the fraction of motors with larger unbinding probability leads to a decrease of the total bound density of motors. As a consequence, upon increasing the fraction of motors with larger unbinding probability, the total current decreases in the low density region and increases in the high density region, see Fig. 2(b). For the case of two motor species with different velocities we have focused on the extreme case of non-zero and zero stepping probability for the two motor types. As to be expected, the total motor current decreases with increasing fraction of non-moving motors. Since particle-hole symmetry is broken for two motor types with different stepping parameters, the current-density relationship becomes asymmetric with respect to the bound density $\rho_{\rm b}=0.5$. As shown in Fig. 3, its maximum shifts towards lower bound densities because non-moving motors hinder the stepping of moving motors and hence enhance traffic jams.

To obtain these results, we have employed computer simulations and analytical calculations. We have first used mean field theory to analyze our model. The attachment and detachment processes only involve single filament sites and therefore do not introduce new correlations into the model, so that mean field theory works well for the case of two species which differ only in their unbinding parameters. However, mean field theory fails to estimate the total current of two motor species with different stepping probabilities, as illustrated in Fig. 3. In this case, the exclusion interaction leads to strong interference of the motors moving at different velocities and therefore to correlations and failure of mean field theory. In order to obtain analytical results in this latter case, we go beyond mean field and use finite-size scaling of the total current to extrapolate the total current value of small systems to the thermodynamic limit of large system size. This is possible because the total current decreases exponentially with the system size, as shown in Fig. 4.

This work presents a simple model for traffic by multiple motor species, which can be regarded as a starting point towards the understanding of intracellular transport with different kinds of molecular motors. Our model could be extended in many ways, such as by considering more than two different motor species which differ in multiple properties. Also, the model for the motors can be refined to include internal states of the stepping process or interactions between the motors which go beyond spatial exclusion. In the simple form presented here, our model gives a qualitative picture of the effects of traffic by several motor species. It makes predictions for the current-density relationship (Fig. 2 and Fig. 5) and for the velocity of a single motor in the presence of immobile obstacles (Fig. 6). It also suggests that the movements of different species of motors, and therefore of different kinds of cargo, can be cross-regulated, so that the movement of one type of motor is controlled indirectly by the presence or absence of other types of motors or by changes in the parameters of these motors. As an example, in the case of two species with different stepping probabilities, a small fraction of slow moving motors can strongly suppress the movement of fast moving motors and lead to a significant reduction of the current. This may be useful for the regulation of transport of different types of cargo along the same filament, both for in vivo traffic and for in vitro biomimetic applications.

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