Microtubules, the molecular tracks for motor proteins in intracellular transport, consist of typically 13 parallel filaments, which may be viewed as separate lanes. Experimentally,
2.3 Driven transport on multiple parallel lanes 19
Figure 2.4: Driven transport on two parallel lanes. Particles enter the upper (lower) lane at rate
α↑ (α↓), and leave at the right at rates β↑ (β↓). In bulk, they hop to the right (at unit rate) or change lanes at a low rate ω. All processes are constrained by hard-core on-site exclusion. The lanes’ weak coupling, stemming from the low lane change rate, induces involved behavior of the system’s non-equilibrium stationary state, see text.
molecular motors moving on them have been found to stay on one lane. However, the statistics of possible rare deviations, meaning occasional lane changes, is not know. Due to the intrinsic stochasticity of the motor’s steps as well as fluctuations in the cellular environment, it seems natural to assume the occurrence of rare “missteps”, where the motor, while stepping forward, also changes its lane. While experiments to investigate the abundance of such missteps seem too challenging to be undertaken nowadays, the implications for intracellular transport along microtubules are of genuine interest.
In this Thesis, we develop a paradigmatic model for driven transport on multiple parallel lanes. In the remainder of this Section, we introduce its dynamics, present unexpected effects arising from a small probability of particle lane changes, and discuss the system’s behavior in general terms.
Transport on multiple parallel lanes does not only occur within cells. Highway traffic [50, 51] often runs on several tracks, with vehicles moving parallel or antiparallel. The formation of traffic jams is crucially influenced by the different possible geometries of the underlying track [79]. Of course, in realistic models for highway traffic, several additional factors play important roles, such as different velocities of vehicles [80] or the existence of bottlenecks for the traffic flow [81, 82]. In our work, we ignore such additional complications and focus solely on the effects of multiple lanes with a (small) rate at which particles change between the lanes. Incorporating bottleneck effects or the existence of slow and fast moving particles represents an interesting issue and could motivate further studies.
The system’s dynamics, a driven exclusion process on parallel tracks, is depicted in Fig. 2.4. For simplicity, we consider two parallel lanes, each consisting of a large number of discrete lattice sites. The dynamics may be seen as a multiple-track generalization of the TASEP which has been introduced above. Particles enter the upper (lower) lane at rates α↑ resp.
20 2. Intracellular transport on parallel lanes
α↓, if the corresponding first site is empty. In bulk, they hop one site to the right, provided the neighboring site is unoccupied. The rates for these processes are set to one, and thereby fix the timescale. At a certain (low) rate ω, particles may change the lane, meaning they jump to the neighboring site on the other track if exclusion allows for it. Below, we explain the meaning of a “low” rate for this process. Exiting occurs at the right- most lattice site, at rates β↑ (β↓) on the upper resp. the lower lane. As in the TASEP, all processes are only allowed under the constraint of hard core on-site exclusion, which represents the only interaction of the particles. The two lanes are weakly coupled due to the lane change processes occurring at the low rate ω. In the following, we show how this weak coupling gives rise to a number of intriguing effects. Note that, in contrast to weak coupling, a vanishing lane change rate results in two uncoupled TASEPs, while a very high rate, meaning a strong coupling, induces behavior that is qualitatively the one of a one-lane TASEP [83]. The scenario of weak coupling is therefore the most interesting one. Moreover, it seems most relevant in intracellular transport where motors probably switch between different lanes on a microtubulus only very rarely.
Previous work on multiple lane exclusion processes has focussed on different regimes or types of interactions than the one introduced above. One branch of studies has dealt with indirect coupling of two lanes, see e.g. Refs. [74, 84]. There, the hopping rate of a particle in bulk depends on the occupation of the neighboring sites of the other lane. As an example, particles on adjacent sites on the two lanes can obstruct each other. Such effects are know from vehicular traffic [50, 51] and also plausible in the context of intracellular transport. There, molecular motors carrying large cargo particles are likely to obstruct each other when moving close to each other on parallel lanes. Recently, together with Anna Melbinger, Thomas Franosch and Erwin Frey, we have started to investigate the consequences of such obstruction on the transport properties [85, 86]. Here, we disregard such interactions and solely focus on direct coupling due to particles changing between the lanes. Such models have recently been investigated in the situation of large lane change rates [83, 87, 88]. Qualitatively behaving as a one-lane TASEP, on a quantitative level, deviations occur which can be treated by calculating correlations between the channels. Asymmetric lane change rates induce novel effects, and domain walls occur that perform coupled random walks. The situation of low, but non-vanishing lane change rate has first been considered by ourselves. The relevant articles have appeared in Physical Review Letters and the New Journal of Physics; they are reprinted at the end of this Chapter. Recently, a study of particles moving in opposite directions on two weakly coupled lanes has been published [89].