Traffic of molecular motors: from theory to experiments

arXiv:0710.1384v1 [physics.bio-ph] 6 Oct 2007 Traffic of molecular motors: from theory to experiments∗ Paolo Pierobon Institut Curie, CNRS UMR 168, 11 Rue P. et M. Curie, F-75231 Cedex 05, France † Intracellular transport along microtubules or actin filaments, powered by molecular motors such as kinesins, dyneins or myosins, has been recently modeled using one-dimensional driven lattice gases. We discuss some generalizations of these models, that include extended particles and defects. We investigate the feasibility of single molecule experiments aiming to measure the average motor density and to locate the position of traffic jams by mean of a tracer particle. Finally, we comment on preliminary single molecule experiments performed in living cells. I. INTRODUCTION Living cell is a highly organised structure that constantly needs to move its constituent parts from one place to another. It is therefore provided with a complex and accurate distribution system: the cytoskeleton, the network of biopolymers (actin, microtubules and intermediate filaments) that gives the cell its structural and mechanical features, functions as road system for the transport of organelles and vesicles; the motion of these objects is entrusted to motor proteins moving along these filaments. These motors are enzymes that convert the energy obtained by hydrolysis of an ATP (adenosin-triphosphate) molecule into a work (displacement of their cargo). Myosin V (on actin filament) and kinesin or dynein (on microtubules) are well known examples of these so called processive motors [1, 2]. It has been observed that these motors can act cooperatively or interact with each other giving rise to collective phenomena. In particular, in some phase of the cell cycle, the motors can be expressed in high concentrations: it seems therefore natural to investigate analogies and differences with the traffic observed in a city. Experimental techniques as single molecule and fluorescence imaging have just started giving some hints on the complex behaviour of this system, but are still far from giving quantitative description of traffic situations. While waiting for experimental data some theoretical models have been developed to physically describe intracellular transport. II. DRIVEN LATTICE GASES: MODELS FOR INTRACELLULAR TRANSPORT A simple model to capture the behavior of many motors on a filaments needs to include three fundamental features: (i) the motors move in a step-like fashion on a one dimensional tracks and bind specifically to the monomer constituting the cytoskeletal filaments; (ii) cytoskeleton filaments present a specific polarization: the chemical properties of the track guarantee that the motion is always directed towards one of the two ends of the filament; (iii) the particles move according to a stochastic rule, i.e. they move upon a chemical reaction, the hydrolysis of ATP, which occurs randomly with a typical rate. The Total Asymmetric Simple Exclusion Process (TASEP) is a stochastic process first introduced to describe the motion of ribosomes on mRNA substrate [3] and encodes all these features. It rapidly became a paradigm of non- equilibrium statistical mechanics and one of the few example of exactly solvable systems [4, 5]. In this model each particle occupies a site on a one dimensional lattice and advances stochastically and in one direction. The most obvious observable is the average density profile of particles along the lattice. The system with open boundaries where particles enter the lattice with rate α at one end and leave with rate β at the other, shows a non trivial phase diagram where three distinct non-equilibrium steady states appear: a low density phase controlled by the left boundary, a high density phase controlled by the right boundary and a maximal current phase, independent of the boundaries. In a first attempt to construct a minimal model for molecular intracellular transport, one needs to add to the TASEP the fact that the tracks are embedded in the cytosol with a reservoir of motors in solution. This allows the motors to attach to or detach from the track. This led to the construction of the TASEP with Langmuir (i.e. attachment/detachment) kinetics or TASEP/LK model [6, 7] depicted in Fig. 1a. According to this model, in addition to the TASEP properties, particles enter (leave) the system with rate ωA (ωD) also in the bulk. All over the lattice they obey exclusion: two particles cannot occupy the same site. ∗Based on the oral contribution given at the Traffic and Granular Flow meeting 2007 † Email adress: pierobon@curie.fr 2 α α β β i = 1 i = 1 i = N i = N τ = 1 τ = 1 ωA ωA ωD ωD (a) (b) FIG. 1: Schematic representation of the TASEP with Langmuir kinetics in the case of (a) monomers and (b) dimers. The allowed moves are: forward jump (with rate τ = 1), entrance at the left boundary (with rate α), exit at the right boundary (with rate β), attachment (with rate ωA), and detachment (with rate ωD) in the bulk. According to the rules de

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