Particle interactions and lattice dynamics: Scenarios for efficient bidirectional stochastic transport?

If the surroundings of the filament is an open space, motors can diffuse very far from the filament, and thus, most of the time, motors that attach to the filament have completely lost the memory of their previous detachment point. Hence there is no need to keep track of spatial positions of the diffusing motors.

Actually, the grand-canonical reservoir will be modeled by a single number, representing the density of motors in the reservoir, which is assumed to be constant and (in this paper) equal to $\rho_{\mathrm{u}}=2\rho_{\mathrm{u,+}}=2\rho_{\mathrm{u,-}}=0.1$. Attachment to empty sites on the filament occurs at rate $\rho_{\mathrm{u,\pm}}\cdot\omega_{\mathrm{a}}$.

This type of reservoir is relevant if the filament is embedded in a large volume and the number of attached motors is small compared to the number of motors in solution. This is the case for many in vitro experiments.

An infinite diffusion rate in the surroundings of the lattice is also considered in the canonical reservoir, and thus particles in the reservoir do not have any spatial coordinate and can attach to any empty site along the lattice.

However, in the canonical setup, we consider that the total number of particles in the system is finite. This implies, with our definition of the density, that $\rho_{\pm}=\rho_{\mathrm{b,\pm}}+\rho_{\mathrm{u,\pm}}$ is fixed. Again, attachment occurs at rate $\rho_{\mathrm{u,\pm}}\cdot\omega_{\mathrm{a}}$.

This type of reservoir is relevant if the filament is embedded in a large volume and the number of motors attached to the filament is of the same order as the number of motors in solution. This situation can also be encountered in in vitro experiments.

Actually, the axon is a rather crowded environment¹¹1 The importance of confinement in the axon was for example illustrated by the simulations in [22] which reproduce the experimental results of H. Erez et al [23] for transected axons. , where the size of the transported objects is of the same order as the distance between the microtubules [24, 25], and the previous reservoirs are not relevant to model the diffusive motion episodes. In order to study the effects of confinement, we shall introduce a reservoir with a memory effect: there must be a correlation between the locations where a motor detaches, and where it attaches again on the microtubule.

In order to do so, the model is extended by a second lane which represents the reservoir (see figure 2). Particles in the reservoir symmetrically hop to both neighboring sites with rate $D$ without any interaction between particles. Note that this one-dimensional reservoir actually represents a three-dimensional region around the microtubule, which allows for several particles at the same longitudinal position. Only the longitudinal diffusion, along the microtubule, is modelled explicitly. The transversal dimensions of the three-dimensional diffusion reservoir, which are not represented explicitly, can be tuned by taking a larger or weaker effective attachment rate (see [26] for an example). The larger the diffusive region is (in the direction transverse to the microtubule), the weaker the effective attachment rate is. Thus, under strong confinement, diffusion in the transverse direction should be more limited, the attachment rate in the model will be larger, and the memory effect will increase.

In this paper, we shall also consider that the underlying lattice itself can evolve dynamically. We shall consider the same type of lattice dynamics as in [19]: a site of the lattice is randomly eliminated at rate $k_{\mathrm{d}}$ and recreated at rate $k_{\mathrm{p}}$ (see figure 3). The indices refer to the processes of microtubule de-/polymerization, although the dynamics we consider here have little in common with realistic dynamics and must rather be considered as a minimal model. A particle cannot bind to an eliminated site and immediately detaches from the lattice if it steps onto an eliminated site or is bound to a site which is being eliminated. We fix the polymerization rate equal to $k_{\mathrm{p}}=1$ in order to reduce the number of free parameters (we saw indeed in [19] that transport features were more sensitive to $k_{\mathrm{d}}$ than to $k_{\mathrm{p}}$).

It has been shown that bidirectional transport models in which the particle species interact only via an exclusion interaction generically exhibit macroscopic clustering at any finite density $\rho$ [13]. This large cluster controls the current in the system and therefore limits the total transport capacity. In particular, for large systems, the current vanishes.

Considering the case of bidirectional transport on dynamical lattices, it has been shown that lattice dynamics can induce a transition from the clustered phase to a homogenous phase in which the particles are distributed over the whole system [19]. The homogenous states are density-dependent so that a finite (non-vanishing) current of particles is present even in very large systems, which would not be the case on a static lattice. The transition takes place because the lattice dynamics breaks up forming clusters and thus limits the reachable maximum cluster sizes. At a given strength of the lattice dynamics, the nascent clusters are dissolved more quickly than they can form and the system appears homogenous. The transition depends mostly on the rate at which binding sites disappear suggesting that the finite lifetime of individual binding sites is crucial for the observed transition.

In order to support this point of view, it has been verified that this effect is not simply due to the cutting of the filament into several segments of different lengths, by considering a static lattice with the same density of individual holes. In that case, the phase transition is not observed, thus indicating that the cluster dissolution is really an effect of the dynamics of the lattice.

Though this scenario was inspired by the fact that microtubules are themselves evolving through the so-called dynamic instability, i.e., events of polymerization and depolymerization, the dynamics of the filament as we have defined it in [19] or in section 2.3.1 do not aim at being realistic, but rather are the simplest kind of lattice dynamics that could be considered in order to illustrate the aforementioned mechanism.

In [15], it has been shown that for a grand-canonical setup, lane formation is observed if the particles’ attachment/detachment rates are modified according to figure 4 on a static lattice. In this section, we briefly present some new results obtained in the same frame, in order to discuss the efficiency of the lane formation mechanism.

In order to study the effect of confinement, we consider now a reservoir such as the one defined in section 2.2.3. Actually, we consider two identical subsystems (see figure 10) for the same reasons as explained above. Particles can hop from one reservoir to the other while conserving their longitudinal position at rate $c_{\mathrm{R}}$. This corresponds to diffusion in transverse direction. We choose the diffusion in both directions along the reservoir lattices to be equal to the attachment rate, i.e., $D=c_{\mathrm{R}}=\omega_{\mathrm{a}}=0.1$. The two filament lattices are not directly coupled in this model, an assumption that facilitates lane formation for this setup.

Without the modified attachment/detachment rates defined in section 3.1, this model has been shown to generically form a single big cluster which contains almost all of the particles in the system [13]. The existence of this big cluster is a clear indicator for an inefficient state since it blocks most of the transport. Furthermore, the cluster formation depends on the number of particles rather than density so that systems form larger and more stable clusters with increasing system size although the total density of particles is constant. As a consequence, the current vanishes in the thermodynamic limit.

If modified attachment/detachment rates are included, we see that the finite diffusion in the reservoir inhibits even partial demixing of the particles into the different subsystems (see figure 11(a)). Although lane formation is suppressed by the limited diffusion, the current (figure 11(b)) is enhanced by the interaction, though a single big cluster is still present in the system. The increase in the current is caused by the reduced time blocked particles spend in front of each other and the fact that, in low density regions, particles of a given species are more likely to have nearest neighbors of the same species.

Eliminating lattice sites reduces the effectivity of the particle-particle interaction since it is next-neighbor and does not reach across holes in the filament. As a consequence, the transition is shifted to higher values of $q$ with increasing depolymerization rate $k_{\mathrm{d}}$ as can be seen in figure 12(a). For a very dynamic lattice, there seems to be no transition toward lane formation at all. This effect is partly due to the fact that lattice dynamics cuts the filament in shorter segments on which lane formation is less stable, as has been verified by comparison with a static lattice with the same density of holes. However, the effect observed here is stronger if the lattice additionally is dynamic.

However, although lane formation is hindered by the lattice dynamics, it positively affects the current of the system (figure 12(b)). By pushing particles to the reservoir when a lattice site is eliminated, the lattice dynamics keeps the density on the filament low enough so that particles have enough space in front of them to move. Thus, extreme densities that were strongly limiting the current in the case of a static lattice (see section 3.2.1) are avoided here.

The results shown in figure 12 are obtained for the grand-canonical reservoir but are qualitatively the same for a canonical setup at high densities. For densities $\rho<1$, the current on the static lattice is already close to optimal (see section 3.2.1) and so there is little space for improvement. Actually, the lattice dynamics corrupts these states by shifting the transition to higher values of $q$ and making the phase separation less stable.

In the model with finite diffusion rates in the reservoir, there is no lane formation on the static lattice and the same remains true if a lattice dynamics at any strength is added. But as has already been reported in [19], a dynamic lattice is capable of dramatically increasing the current along the lattice and reaching an efficient state by dissolving large clusters. This effect is obviously also present in this system and the strength of the additional interaction only has very little impact on the current on the dynamic lattice.

The increase of current due to the modified attachment/detachment rates on the dynamic lattice is of the same order as the one observed on the static lattice (figure 11) and represents only a small fraction of the increase due to lattice dynamics: In a system of density $\rho=1$ and length $L=1000$, the maximum current increase caused by the lattice dynamics is almost ten-fold, whereas the modified attachment/detachment rates do not even double the current in the system.

On a static lattice, changing lanes in front of a single particle of opposite species does not lead to dramatic effects. In particular, no lane formation is observed although the absolute magnetization of the filament slightly increases. This effect competes with the reservoir exchange rate $c_{\mathrm{R}}$ which tends to mix the system and destroys any phase separation between the two subsystems. At $c_{\mathrm{R}}=D=0.1$, no effect of the additional particle-particle interaction is visible at any probability $\alpha$.

Nevertheless, there is a small effect for decoupled reservoirs at $c_{\mathrm{R}}=0$. The increase of the magnetization is stronger the fewer particles we have in the system, but even for densities as low as $\rho=0.1$, the magnetization does not exceed $|m|<0.3$. It is noteworthy that at the system size $L=1000$ considered here, there is no important clustering at this low density. In a clustered system with higher particle density, absolute increases in magnetization are almost negligible: For $\rho=1$, the magnetization passes from $|m|=0.047$ at $\alpha=0$ to $|m|=0.0784$ at $\alpha=1$.

Alike, the current does not improve much with this side-stepping interaction. The clustered systems remain clustered and exhibit a maximal gain of their current of $25\%$ while starting from an extremely low value being caused by the existence of the large cluster.

If we consider a stronger condition for filament changes as shown in figure 13(b), lane formation is obtained much more easily compared to the single obstacle model (section 4.2.1) and occurs at any density with decoupled reservoirs, i.e., $c_{\mathrm{R}}=0$. The current of particles along each filament is consequently very high and close to the optimal value found in a one-species TASEP. The interaction is asymmetric in the sense that a single particle changes the filament if opposed to two particles of opposite charge while the single particle cannot induce any lane change on the two particles. As a consequence, a particle species which is only slightly dominant on one filament can push more and more particles from the minority species to the other filament. Ultimately, complete lane formation is observed.

As discussed in the last section, the lane formation competes with the mixing through the reservoirs. If the reservoir coupling is increased, the magnetization is lowered at some point and lane formation disappears (figure 14(a)). The onset of remixing the separated lanes occurs at lower couplings $c_{\mathrm{R}}$ for higher densities as in these cases, more particles are continuously transported to reservoir belonging to the ‘wrong’ filament. For $c_{\mathrm{R}}=D=0.1$, the system is completely mixed at any density.

The current increases by one order of magnitude if lane formation occurs (figure 14(b)) and is very close to the optimal value in the one-species TASEP.

If the possibility to perform filament changes is combined with lattice dynamics, the system’s tendency toward lane formation is increased (see figure 15(a)). A system of density $\rho=0.4$ reaches complete lane formation at $k_{\mathrm{d}}=0.1$. However, at this value of the depolymerization rate $k_{\mathrm{d}}$, the basic model without filament changes does not exhibit a large cluster anymore, but is in a homogenous state with very small structures, and thus has already improved transport properties only due to lattice dynamics.

The enhanced lane formation can be understood in the following way: In order to form lanes, filament changes of particles are necessary which can only occur if particles of different species meet. Consequently, more lane changes are induced if there are more interfaces between positive and negative particles. In a clustered system, almost all the particles are accumulated in the cluster so that there are only very few interfaces. The lattice dynamics, though, inhibits the formation of large clusters and instead distributes the particles more homogeneously over the whole system leading to many very small clusters. This effectively increases the number of interfaces between positive and negative particles and thus favors lane changes and lane formation. The effect which has been mentioned in section 3.3.1 that holes in the lattice decrease the effectivity of a particle-particle interaction because the interaction is only next-neighbor seems to be overcompensated by the increase in interfaces due to the dissolution of the big cluster.

The strength of the lattice dynamics that is needed to form lanes depends on the particle density, e.g., a system of density $\rho=1$ does not exhibit complete lane formation at any lattice dynamics (see figure 15(c)) while a system of density $\rho=0.1$ forms lanes at much lower lattice dynamics (not shown).

Again, the current is more strongly (and positively) affected by the lattice dynamics than by the additional particle-particle interaction as can be seen in figure 15(b) and (d): The higher the lattice dynamics, the higher the overall position of the current curves. In particular, for the completely lane separated system at $\rho=0.4$ for $\alpha>0$, the current increases by $40\%$ due to the lane formation, but more than quadruples because of the lattice dynamics (figure 15(b)). This impression is strengthened by the observation that similar gains due to the dynamic lattice are obtained even if no lane formation occurs (figure 15(d)). The values of the current for $\rho=1$ are remarkable since they are rather close to the maximum current of $1/4$ that would be found in a one-species TASEP, although encounters between particles of different species are still frequent in our model even at important lattice dynamics.

These results have so far been obtained at decoupled reservoirs $c_{\mathrm{R}}=0$. The influence of the reservoir coupling $c_{\mathrm{R}}$ is shown in figure 16 for a dynamic lattice. Already an extremely small reservoir coupling destroys the lane formation (figure 16(a)). If diffusion in longitudinal and transverse direction are equal, i.e., $c_{\mathrm{R}}=D=0.1$, the lane formation effect disappears almost completely. However, the current is only marginally affected (figure 16(b)) and reaches almost identical values for all reservoir couplings at $\alpha=1$. This once more confirms that the lattice dynamics is dominant in determining the transport capacity of the system.

As for the the “single obstacle” interaction, the lattice dynamics also improve the lane formation mechanism by increasing the number of interfaces between oppositely moving particles for the “two obstacles” interaction (figure 17(a)). The influence of the lattice dynamics and the competition with the reservoir coupling $c_{\mathrm{R}}$ on the magnetization and therefore on the lane formation remains qualitatively unchanged compared to section 4.3.1.

There is nevertheless an interesting difference for the value of the current in the systems with lane formation: For the most dynamic lattices $k_{\mathrm{d}}=0.1$, the current in the separated system can sometimes be lower than in de-mixed or even mixed systems at lower lattice dynamics (blue and green lines in figure 17(b)). In these cases, the lattice dynamics actually inhibit any further improvement of the current by eliminating too many sites of the filament along which no transport can happen. This effect is the more pronounced, the lower the density of particles is.