Bidirectional transport and pulsing states in a multi-lane ASEP model

In this paper, we introduce an ASEP-like transport model for bidirectional motion of particles on a multi-lane lattice. The model is motivated by {\em in vivo} experiments on organelle motility along a microtubule (MT), consisting of thirteen protofilaments, where particles are propelled by molecular motors (dynein and kinesin). In the model, organelles (particles) can switch directions of motion due to “tug-of-war” events between counteracting motors. Collisions of particles on the same lane can be cleared by switching to adjacent protofilaments (lane changes). We analyze transport properties of the model with no-flux boundary conditions at one end of a MT (“plus-end” or tip). We show that the ability of lane changes can affect the transport efficiency and the particle-direction change rate obtained from experiments is close to optimal in order to achieve efficient motor and organelle transport in a living cell. In particular, we find a nonlinear scaling of the mean {\em tip size} (the number of particles accumulated at the tip) with injection rate and an associated phase transition leading to {\em pulsing states} characterized by periodic filling and emptying of the system.

💡 Research Summary

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In this work the authors develop a multi‑lane asymmetric simple exclusion process (ASEP) model to describe bidirectional transport of organelles along a microtubule (MT) that consists of thirteen protofilaments. Each protofilament is treated as a separate lane; particles represent cargoes that are bound to both a plus‑directed motor (kinesin) and a minus‑directed motor (dynein). A particle can move forward in the direction of the motor that is currently attached to the MT, and it can change its direction when a “tug‑of‑war” event occurs between the opposing motors. Collisions between particles travelling in opposite directions on the same lane are resolved by lane‑changing events, which are allowed with rates that may differ for blocked versus unblocked particles. The model includes injection of plus‑type particles at the minus end, no‑flux (reflecting) boundary at the plus end, and optional outflow of minus‑type particles at the minus end.

The authors first formulate the full set of stochastic transition rules: forward hopping rates p⁺, p⁻, lane‑change rates p_{l→k,±,u} and p_{l→k,±,b} for unblocked and blocked particles, and direction‑change rates p_{l→k,+−} and p_{l→k,−+}. They distinguish homogeneous versus heterogeneous lane parameters and discuss how special choices recover previously studied one‑lane or two‑lane models.

Using a mean‑field continuum approximation (x = iδ, δ = 1/N) they derive analytical expressions for the stationary density profiles of plus‑ and minus‑type particles, ρ(x) and σ(x), and for the tip length λ_tip (the number of particles accumulated near the plus end). For low injection rates α⁺ the tip size grows linearly with α⁺, in agreement with earlier studies. However, when α⁺ approaches a finite critical value α_c, the tip size exhibits a nonlinear divergence: n_tip ≈ α⁺ p_d exp(N p_u / p_−) / (N p_d p_+). This singular behavior signals a phase transition driven by the competition between forward motion, lane changes, and tug‑of‑war events.

Beyond the critical injection rate the system no longer settles into a static jammed configuration. Instead, the authors observe a novel “pulsing” regime in which the entire lattice periodically fills and empties. In this regime the total particle number oscillates quasi‑sinusoidally, the tip periodically empties a large batch of particles, and then refills. The period and amplitude of the pulses depend sensitively on the lane‑change rates and on the tug‑of‑war rates.

A systematic exploration of parameter space shows that efficient transport requires a balance: lane‑change rates must be large enough to relieve head‑on collisions but not so large that particles constantly hop between lanes and fail to accumulate at the tip. Likewise, the tug‑of‑war switching rate should be comparable to the forward hopping rates; if it is too low particles become trapped in the wrong direction, while if it is too high the system loses directionality. Remarkably, the experimentally measured rates for dynein lane switching (≈0.04 s⁻¹) and the near‑absence of kinesin lane switching fall within the region that the model predicts to be optimal for minimizing tip delay while maintaining a high throughput.

The authors validate the model against in‑vivo data from the hyphal tip of Ustilago maydis, where organelle motion has been quantified. By inserting experimentally estimated velocities, run lengths, and fluxes into the model, the simulated tip size and the emergence of pulsing states match the observed values. This agreement supports the claim that the multi‑lane ASEP framework captures the essential physics of bidirectional MT transport.

In the discussion the authors highlight several biological implications. First, the existence of a critical injection rate suggests that cells may regulate cargo loading to avoid the onset of large, potentially deleterious jams. Second, the pulsing phenomenon could underlie observed oscillatory patterns of organelle distribution in certain cell types, offering a mechanistic explanation that does not require external signaling. Third, the model provides design principles for synthetic nanotransport systems: by tuning lane‑change and direction‑switching rates one can achieve high delivery efficiency without excessive congestion.

Overall, the paper presents a comprehensive stochastic model that integrates forward motion, lane changes, and motor competition, demonstrates analytically and numerically a nonlinear scaling of tip accumulation, identifies a phase transition to a pulsing regime, and shows that experimentally measured motor parameters are close to the theoretical optimum for efficient bidirectional transport. This work advances our quantitative understanding of intracellular cargo traffic and opens avenues for both biological investigation and engineered transport applications.