Mathematical modelling of driven diffusive transport processes: analyses and simulations

Abstract

Transportation is the center of the world! It is the glue of our daily lives. Prominent examples of complex transport systems include several physical as well as biological processes. The most commonly observed physical transport phenomenon are vehicular traffic, traffic noise on communication networks, noisy signals in driven disordered and self-organized systems, granular flow and many others. Likewise, transport mechanisms inside a biological cell play a very crucial role in maintaining and supporting various processes in a living organism. The tiny biological machines controlling multi cellular processes such as muscle functioning, cell motility, intracellular transportation, transfer of genetic information are motor proteins. These motors move along macromolecular highways, namely microtubules escorting cargo (e.g., vesicles, organelles, other proteins) to different compartments of the cell. The continuous supply of energy obtained by the hydrolysis of adenosine triphosphate triggers the motion of motor proteins. This categorizes the collective transport of motor proteins to fall under the peculiar class of non-equilibrium systems, driven diffusive systems. A unified framework of exclusion model, specifically, Totally Asymmetric Simple Exclusion Process (TASEP) has gained much attention in last few decades to study the non-equilibrium stochastic motion of physical and biological processes. Motivated by the complex dynamics of various stochastic transport phenomenon, we attempt to study physical properties of several generalizations of TASEP model. In the framework of TASEP, the moving entities are considered as physical particles and pathways as discrete lanes. Based on this structure we majorally contribute in understanding collective behavior of particles in variants of single as well as different topology of lanebased models. Instigated by intermolecular interactions, we investigate the role of particleparticle interactions in a TASEP model with a dynamic defect (another type of particle) introducing an inhomogeneity in the lattice. We explore non-trivial effects of interactions in terms of phase diagrams, density profiles in allowed local reversible association of particles in the system. In recent studies, the more realistic scenario that biological filaments are flexible polymers in three-dimensional environment has been considered in TASEP model that significantly affects the properties of the system. In this direction, we study the effect of bidirectional movement of two different species of particles on a flexible TASEP in a three-dimensional environment portraying the emergence of new phases. Stepping ahead, a bidirectional TASEP model with rigid lattice is generalized to a inhomogeneous variant by allowing local irreversible dissociation of particles. The interesting point of discussion is the occurrence of spontaneous symmetry breaking phenomenon arising due to bidirectional movement of two-distinct species of particles. Further, we study a topology of twointersecting lanes in the vicinity of limited supply of particles. Besides, we investigate the effect of local irreversible dissociation of particles from a bridge model coupled to inputoutput lanes represented by one-dimensional TASEPs considering two different species of driven diffusive particles. The properties of the proposed models are thoroughly investigated utilizing theoretical and computational techniques. We use different versions of mean-field theory to analyse novel features in terms of new phases and phase transitions that are extensively validated through Monte Carlo simulations. It is expected that the obtained results can provide a deeper insight into the study of non-equilibrium stochastic transport systems.