Driven lattice gases: models for intracellular transport Paolo Pierobon1,2, Thomas Franosch2 , Mauro Mobilia2, Roger Kouyos3 and Erwin Frey2 1

Institut Curie, Paris; 2Arnold Sommerfeld Center & CeNS, Dept.of Physics LMU München; 3ETH, Zurich

Abstract

Intracellular transport and statistical mechanics

Intracellular transport driven by molecular motors along microtubules closely resembles the dynamics of a driven lattice gas without conservation of particles. The motion of the motors is unidirectional, asymmetric and stochastic: its properties are encoded in the well studied totally asymmetric simple exclusion process (TASEP). We extend this model by including the possibility of attachment/detachment kinetics (Langmuir kinetics, LK), the fact that as dimers they occupy two lattice sites and the presence of a bottleneck. We study the stationary phase diagrams by means of Monte Carlo simulations combined with a continuum description (based on an extended mean field theory). Finally we consider the dynamics of a tracer particle and we show how to reconstruct the density profiles from the statistical properties of its motion.

Intracellular transport of organelles and large proteins in the cytoplasm cannot rely on simple diffusion and is performed by active elements, the molecular motors, which move along microtubules always in the same direction powered by ATP. Anterograde motors (e.g. Kinesins) move from the – end to the + end while retrograde (e.g. Dyneins) move in the opposite direction. This work aims to describe some general properties of many molecular motors moving on a microtubule from a physical perspective. The most relevant quantity will be the average density profile in the lattice ρ=. Four features are essential to describe this system:

Unidirectional and stochastic motion (rate τ=1) Mutual exclusion Control at the boundaries (left-entrance α, right-exit β) Binding and unbinding (LK) ωA and ωD

Recently a driven lattice gas model has been proposed, that include the fourth feature, namely the finite processivity of motors [5]. It couples the usual TASEP to a reservoir (modelling the cytosol) via Langmuir kinetics. This new feature triggers new unexpected phenomena:

Competition between the reservoir and the driven kinetics The current and the density are now space dependent A localised shock shows up! there is phase coexistence! The position of the shock is tunable through α and β Richer phase diagram! Mean field (MF) theory works very well!

The Totally Asymmetric Simple Exclusion Process (TASEP) is a one dimensional driven lattice gas that describes a system with the first three properties. Depending on the parameter α and β it exhibits a phase diagram with three distinct phases: high density, low density and maximal current. It is a paradigm for motion on nucleic acid template and for non-equilibrium statistical mechanics

Dimers

Bottleneck

Tracer Motivations  Experiments with many label particles are difficult (bleaching).  More feasible to observe the motion of single marked tracer particle.

Motivations

Motivations

 Can we reconstruct the density profile from tracer particle properties?

Kinesins and Dyneins are dimers

Microtubules have structural defects or microtubule associated proteins

Chemically bind to two sites but step only one site.

The motion of the motor proteins can be affected by defects

Measured quantities

Is the TASEP with on-off kinetics robust upon the introduction of dimers?

What is the effect of disorder on non-equilibrium models?

 P(x,t|0,0) = prob. to find a particle at site x at time t started at x=0 at t=0.

The easiest case: pointlike disorder in TASEP+LK (monomer).

 We assume it is experimentally feasible to repeatedly collect the position as a function of the time for a large number of particles.

The bottleneck is a site xd with lower hopping rate q<1.

Contruction of the MF equation Naive MF fails → use prob. theory w/ three states model MF for the 2-TASEP (continuum approximation) Particle interaction → non linear current density relation! MF for the ON-OFF kinetics (cross-checked with grand-canonical results) MESOSCOPIC LIMIT: the interesting regime requires competition between boundaries and bulk, i.e. a scaling like ΩA=ωAN and ΩD=ωDN Eqn. for the stationary state in form of a continuity eq.:

 Estimate P(x,t|0,0) by constructing a histogram for the fraction of total number of particles that reached site x at time t.

Known results TASEP with a single inhomogeneity  usual phase diagram The maximal current is limited by the defect. What does LK change? the current is now space dependent!  If the system is diluted (dense) a bottleneck is negligible  local spike (dip).  Above critical current the defect has a macroscopic effect  discontinuity. Data analysis  First two moments of the P(x,t|0,0)  velocity and diffusion constant

Shock and phase diagram K=2 ΩD=0.1

Coexistence ρ

x

J

The DW is localized!!!

 Convolute the data obtained with a resolution function of ~50time steps

α=β=0.1

MF theory: splitting the system

The position of the DW (and therefore the phase boundaries) is determined using conservation of the left and right currents at the DW position (i.e. local conservation of the particle number). When the DW is outside the system (i.e. in the Low or High density phase jα or jβ, dominates. Note that the TASEP maximal current phase is replace by a “Meissner” phase where the boundaries do not influence the bulk density and the solution is given always by the same curve.

There is critical current above which the boundaries become irrelevant and the defect determines the maximal current. When the defect is relevant we can split the system in two and apply the MF analysis of simple TASEP+LK.

 Use rigorous results of TASEP on a ring and assume those results are not affected locally by the on-off kinetics:  Reconstruction works much worse using the second moment (diffusion constant) because of non-trivial correlations between the particles. Reconstruction from experimentally relevant parameters  Parameters experimentally testable: N=512, α=0, β=1 (free end) or β=1 (blocked end), ΩD=1, K=1.5 and K=0.5.  Very low statistic: 128 injected particles.

Phase diagram Match the currents imposed by the boundaries with the carrying capacity

 Deviations are observed in the last part of the lattice where the loss of particles affects the probability function.

Appearance of bottleneck phases.  Depending on the shape of the carrying capacity we identify phase diagrams with 4, 6 and 9 different bottleneck phases.

x The phase diagram shows (at fixed ΩA,D) four phases:

In some phases multiple shocks arise.

Low density (LD) phase (controlled by the left boundary); High density (HD) phase (controlled by the right boundary); Coexistence phase (two phases LD-HD and LD-MC coexist separated by a shock); “Meissner” (MC) phases (so called because independent of the boundaries) The results are qualitatively (though not quantitative) the same as for the TASEP+LK of monomers.

Conclusions [3]  We investigate theoretically the properties of a tracer particle.  Simulations show that density profile could be experimentally measured using a tracer particle! Conclusions [1]  Phase coexistence (shock) appears  The monomer model is ROBUST upon introducing larger particles!  OUTLOOK: what about dynamics? (see e.g. [4])

Conclusions [2]  A novel phase behavior emerges even with a simple defect!  Multiple shocks appear.  OUTLOOK: Effects of other kind of disorders (quenched, on the on-off rates, particle-wise…)

 OUTLOOK: a better theory for the variance. Bibliography [1] P. Pierobon, T. Franosch, and E. Frey, to appear in PRE 74 (2006), arXiv:cond-mat/0603385 [2] P. Pierobon, M. Mobilia, R. Kouyos and E. Frey, to appear in PRE 74 (2006), arXiv:cond-mat/0604356 [3] P. Pierobon, T. Franosch, and E. Frey, in preparation [4] P. Pierobon, A. Parmeggiani, F. von Oppen, and E. Frey, PRE 72, 036123 (2005) [5] A. Parmeggiani, T. Franosch, and E. Frey, Phys. Rev. Lett. 90, 086601, (2003), Phys. Rev. E 70, 046101 (2004). Email: [email protected]