Driven lattice gases

This thesis has been motivated by intracellular transport phenomena, such as kinesins and myosins moving along cytoskeletal filaments or ribosomes along messenger RNA. In both cases molecular motors move uni-directionally along a one-dimensional track. This motion is driven by the free energy released in the chemical hydrolysis reaction of ATP (adenosine-triphosphate). It drives conformational changes of the protein and at the same time switches the affinity to the cytoskeletal filaments. The detailed mechanisms are still a matter of debate and intensive research. They do not concern us here, since we are not interested in the principles governing the chemo-mechanics of individual enzymes, but in the possible cooperativity in the intracellular transport resulting from the interplay between externally driving the system and the interaction between the motors. We are interested in the emergent properties of the system, such as the density and current profiles along the track in the ensuing non-equilibrium steady state. We idealize the dynamics in terms of driven lattice gases, which model the motors as particles occupying one or more lattice sites whose dynamics is given by a simple Poisson process. The track is represented by a one-dimensional periodic lattice with open boundaries, where particles may enter or leave the system stochastically. Interaction between the particles is restricted to hard core repulsion such that each lattice site can at most be occupied by one particle. As such the system is known as the Totally Asymmetric Simple Exclusion Process (TASEP), which shows a non-trivial phase diagram as a function of the entrance and exit rates. Though describing some key features of intracellular transport this simple model misses several important aspects such as the exchange of particles between molecular track and the cytoplasm, the extended molecular structure of each motor, and the interaction of motors with microtubule associated proteins which may act as road blocks for intracellular traffic. The goal of this thesis is to account for some of these additional features and explore their possible relevance for the nature of the non-equilibrium steady state and correlations in the single particle and collective dynamics. To achieve this goal we use both analytical approaches (mean field theories, Langevin equations) and numerical methods (kinetic Monte Carlo simulations). In the first part of the thesis we build on recent advances in the field, which have been achieved by taking into account particle exchange between the track and bulk solution (Langmuir kinetics). It was found that this violation of current conservation along the track leads to phase coexistence regions in the phase diagram not present in the TASEP. We