Bidirectional transport in a multispecies totally asymmetric exclusion-process model

Coordination, cooperation, competition, crowding and congestion of molecular motors: Theoretical models and computer simulations

Cytoskeletal motor proteins are biological nanomachines that convert chemical energy into mechanical work to carry out various functions such as cell division, cell motility, cargo transport, muscle contraction, beating of cilia and flagella, and ciliogenesis. Most of these processes are driven by the collective operation of several motors in the crowded viscous intracellular environment. Imaging and manipulation of the motors with powerful experimental probes have been complemented by mathematical analysis and computer simulations of the corresponding theoretical models. In this article, we illustrate some of the key theoretical approaches used to understand how coordination, cooperation and competition of multiple motors in the crowded intra-cellular environment drive the processes that are essential for biological function of a cell. In spite of the focus on theory, experimentalists will also find this article as an useful summary of the progress made so far in understanding multiple motor systems.

Effect of contact inhibition locomotion on confined cellular organization

Experiments performed using micro-patterned one dimensional collision assays have allowed a precise quantitative analysis of the collective manifestation of contact inhibition locomotion (CIL) wherein, individual migrating cells reorient their direction of motion when they come in contact with other cells. Inspired by these experiments, we present a discrete, minimal 1D Active spin model that mimics the CIL interaction between cells in one dimensional channels. We analyze the emergent collective behaviour of migrating cells in such confined geometries, as well as the sensitivity of the emergent patterns to driving forces that couple to cell motion. In the absence of vacancies, akin to dense cell packing, the translation dynamics is arrested and the model reduces to an equilibrium spin model which can be solved exactly. In the presence of vacancies, the interplay of activity-driven translation, cell polarity switching, and CIL results in an exponential steady cluster size distribution. We define a dimensionless Péclet number Q-the ratio of the translation rate and directional switching rate of particles in the absence of CIL. While the average cluster size increases monotonically as a function of Q, it exhibits a non-monotonic dependence on CIL strength, when the Q is sufficiently high. In the high Q limit, an analytical form of average cluster size can be obtained approximately by effectively mapping the system to an equivalent equilibrium process involving clusters of different sizes wherein the cluster size distribution is obtained by minimizing an effective Helmholtz free energy for the system. The resultant prediction of exponential dependence on CIL strength of the average cluster size and [Formula: see text] dependence of the average cluster size is borne out to reasonable accuracy as long as the CIL strength is not very large. The consequent prediction of a single scaling function of Q, particle density and CIL interaction strength, characterizing the distribution function of the cluster sizes and resultant data collapse is observed for a range of parameters.

The interplay of active and passive mechanisms in slow axonal transport

A combination of intermittent active movement of transient aggregates and a paused state that intervenes between periods of active transport has been proposed to underlie the slow, directed transport of soluble proteins in axons. A component of passive diffusion in the axoplasm may also contribute to slow axonal transport, although quantitative estimates of the relative contributions of diffusive and active movement in the slow transport of a soluble protein, and in particular how they might vary across developmental stages, are lacking. Here, we propose and study a model for slow axonal transport, addressing data from bleach recovery measurements on a small, soluble, protein, choline acetyltransferase, in thin axons of the lateral chordotonal (lch5) sensory neurons of Drosophila. Choline acetyltransferase is mainly present in soluble form in the axon and catalyzes the acetylation of choline at the synapse. It does not form particulate structures in axons and moves at rates characteristic of slow component b (≈ 1-10 mm/day or 0.01-0.1 μm/s). Using our model, which incorporates active transport with paused and/or diffusive states, we predict bleach recovery, transport rates, and cargo trajectories obtained through kymographs, comparing these with experimental observations at different developmental stages. We show that changes in the diffusive fraction of cargo during these developmental stages dominate bleach recovery and that a combination of active motion with a paused state alone cannot reproduce the data. We compared predictions of the model with results from photoactivation experiments. The importance of the diffusive state in reproducing the bleach recovery signal in the slow axonal transport of small soluble proteins is our central result.

A closed-loop multi-scale model for intrinsic frequency-dependent regulation of axonal growth

This article develops a closed-loop multi-scale model for axon length regulation based on a frequency-dependent negative feedback mechanism. It builds on earlier models by linking molecular motor dynamics to signaling delays that then determine signal oscillation period. The signal oscillation is treated as a front end for a signaling pathway that modulates axonal length. This model is used to demonstrate the feasibility of such a mechanism and is tested against two previously published reports in which experimental manipulations were performed that resulted in axon growth. The model captures these observations and yields an expression for equilibrium axonal length. One major prediction of the model is that increasing motor density in the body of an axon results in axonal growth-this idea has not yet been explored experimentally.

Modelling collective cell migration: neural crest as a model paradigm

A huge variety of mathematical models have been used to investigate collective cell migration. The aim of this brief review is twofold: to present a number of modelling approaches that incorporate the key factors affecting cell migration, including cell-cell and cell-tissue interactions, as well as domain growth, and to showcase their application to model the migration of neural crest cells. We discuss the complementary strengths of microscale and macroscale models, and identify why it can be important to understand how these modelling approaches are related. We consider neural crest cell migration as a model paradigm to illustrate how the application of different mathematical modelling techniques, combined with experimental results, can provide new biological insights. We conclude by highlighting a number of future challenges for the mathematical modelling of neural crest cell migration.

Driven transport on open filaments with interfilament switching processes

We study a two-filament driven lattice gas model with oppositely directed species of particles moving on two parallel filaments with filament-switching processes and particle inflow and outflow at filament ends. The filament-switching process is correlated with the occupation number of the adjacent site such that particles switch filaments with finite probability only when oppositely directed particles meet on the same filament. This model mimics some of the coarse-grained features observed in context of microtubule-(MT) based intracellular transport, wherein cellular cargo loaded and off-loaded at filament ends are transported on multiple parallel MT filaments and can switch between the parallel microtubule filaments. We focus on a regime where the filaments are weakly coupled, such that filament-switching rate of particles scale inversely as the length of the filament. We find that the interplay of (off-) loading processes at the boundaries and the filament-switching process of particles leads to some distinctive features of the system. These features includes occurrence of a variety of phases in the system with inhomogeneous density profiles including localized density shocks, density difference across the filaments, and bidirectional current flows in the system. We analyze the system by developing a mean field (MF) theory and comparing the results obtained from the MF theory with the Monte Carlo (MC) simulations of the dynamics of the system. We find that the steady-state density and current profiles of particles and the phase diagram obtained within the MF picture matches quite well with MC simulation results. These findings maybe useful for studying multifilament intracellular transport.

Motor-mediated bidirectional transport along an antipolar microtubule bundle: a mathematical model

Long-distance bidirectional transport of organelles depends on the coordinated motion of various motor proteins on the cytoskeleton. Recent quantitative live cell imaging in the elongated hyphal cells of Ustilago maydis has demonstrated that long-range motility of motors and their endosomal cargo occurs on unipolar microtubules (MTs) near the extremities of the cell. These MTs are bundled into antipolar bundles within the central part of the cell. Dynein and kinesin-3 motors coordinate their activity to move early endosomes (EEs) in a bidirectional fashion where dynein drives motility towards MT minus ends and kinesin towards MT plus ends. Although this means that one can easily assign the drivers of bidirectional motion in the unipolar section, the bipolar orientations in the bundle mean that it is possible for either motor to drive motion in either direction. In this paper we use a multilane asymmetric simple exclusion process modeling approach to simulate and investigate phases of bidirectional motility in a minimal model of an antipolar MT bundle. In our model, EE cargos (particles) change direction on each MT with a turning rate Ω and there is switching between MTs in the bundle at the minus ends. At these ends, particles can hop between MTs with rate q(1) on passing from a unipolar to a bipolar section (the obstacle-induced switching rate) or q(2) on passing in the other direction (the end-induced switching rate). By a combination of numerical simulations and mean-field approximations, we investigate the distribution of particles along the MTs for different values of these parameters and of Θ, the overall density of particles within this closed system. We find that even if Θ is low, the system can exhibit a variety of phases with shocks in the density profiles near plus and minus ends caused by queuing of particles. We discuss how the parameters influence the type of particle that dominates active transport in the bundle.

Macroscopic limits of individual-based models for motile cell populations with volume exclusion

Partial differential equation models are ubiquitous in studies of motile cell populations, giving a phenomenological description of events which can be analyzed and simulated using a wide range of existing tools. However, these models are seldom derived from individual cell behaviors and so it is difficult to accurately include biological hypotheses on this spatial scale. Moreover, studies which do attempt to link individual- and population-level behavior generally employ lattice-based frameworks in which the artifacts of lattice choice at the population level are unclear. In this work we derive limiting population-level descriptions of a motile cell population from an off-lattice, individual-based model (IBM) and investigate the effects of volume exclusion on the population-level dynamics. While motility with excluded volume in on-lattice IBMs can be accurately described by Fickian diffusion, we demonstrate that this is not the case off lattice. We show that the balance between two key parameters in the IBM (the distance moved in one step and the radius of an individual) determines whether volume exclusion results in enhanced or slowed diffusion. The magnitude of this effect is shown to increase with the number of cells and the rate of their movement. The method we describe is extendable to higher-dimensional and more complex systems and thereby provides a framework for deriving biologically realistic, continuum descriptions of motile populations.