Stochastic modeling of driven transport processes: analyses and simulations

Abstract

Transportation is a fundamental aspect of human and biological systems, serving as the mechanism for the movement of people, goods, and organisms from one place to another. Its significance lies in its ability to connect diverse locations, assist in economic growth, cultural exchange, and ecological balance. In biological systems, intricate mechanisms such as motor proteins traveling along microtubules play a crucial role in cellular transport, ensuring the proper distribution of essential molecules within cells. Onamacroscopic scale, vehicular traffic and pedestrian flow are essential components of urban life, influencing the efficiency and livability of cities. The coordinated movement of ants, exemplifying collective behavior, showcases how transportation is vital for the survival and thriving of social organisms. Therefore, it is important to investigate the collective motion of the entities involved, by employing a mathematical model, with the objective of explaining the complex dynamics that characterize their interactions. Over the years, a discrete lattice gas model, namely totally asymmetric simple exclusion process (TASEP), has emerged as a paradigmatic model, which is often adopted to study the non-equilibrium stochastic motion of various physical and biological transport processes. Treating entities such as vehicles, motor proteins, and ants as particles, this model aptly captures their unidirectional movement along the respective track by representing them with discrete lattice structures. Considering the numerous scenarios involving systems with open boundaries, the terminal points of the lattice are linked to boundary reservoirs and the dynamics of the particles are governed by simple Poisson process. To provide a sound theoretical framework, mean-field theory and its variants are adopted which neglects all the correlations between neighbouring particles. A notable consistency between mean-field predictions and Monte Carlo simulations is observed, affirming the theoretical predictions of the stationary properties. In this thesis, we contribute to a thorough understanding of collective behaviour of particles by incorporating various realistic features. The first part is inspired by the motion of a limited number of motor proteins like kinesin and dynein in opposite directions on a microtubules. In particular, we examine the influence of individual particle constraints on both species in a one-dimensional bidirectional transport model. The stationary properties of the system is regulated by the entry-exit rates of the species as well as the occupancy of the respective reservoir. Continuing our exploration in bidirectional transport, further we analyze an exclusion model resembling the structure of a roundabout. The second part broadens a single-lane model to a two-lane TASEP by introducing diverse elements such as stochastic blockages at each site, the existence of narrow entrances, reservoir crowding, or inter-lane coupling. These incorporated features emulate various scenarios found in both intracellular transport and vehicular traffic. Initially, our focus is on a system characterized by narrow two-lane entrances, where defects can stochastically bind and unbind within a resource-constrained environment. Despite the inherent complexity, we are able to derive theoretical expressions for stationary properties through the usage of mean-field theory. Furthermore, we explore a resource-limited, two-lane coupled model where simple mean-field methods prove inadequate for obtaining explicit solutions. To conduct a thorough analysis of this challenging problem, we utilize vertical cluster mean-field in conjunction with singular-perturbation technique. The last part of the thesis centers around the attachment and detachment dynamics of motor proteins on complex structure of microtubules, known as the Langmuir kinetics. Initially, a network junction model is investigated under conditions of infinite resources, followed by an examination under finite resource constraints. Overall, the objective of this work is to utilize mathematical modeling as a tool to attain a deep comprehension of the complexities that may emerge in a transport system where particles undergo random motion.