Mathematical modelling of driven stochastic transport systems

Abstract

What sustains life or how living things originate from the non-living things is among the most challenging problems before the scienti c community. A biological cell is assumed as the smallest unit of life. The multiple cellular processes are mainly controlled by tiny biological machines called motor proteins which move along macromolecular highways, namely microtubules to transfer cargoes at distinct locations inside a cell. These motors work in groups for e cient supply of cargoes. To get insight on the life-sustaining mechanism, it is necessary to understand the collective transport of motor proteins which fall into a speci c category of the non-equilibrium system due to the presence of non-zero current governed by the continuous supply of energy. In the last few decades, dynamics of a single motor has been well analyzed through in vitro and in vivo experiments as well as theoretical models whereas the collective behavior of motors movement is poorly understood. Therefore, we investigated the collective dynamics of motion of motor proteins utilizing a mathematical model in the parameter space of important variables. In this work, the unidirectional non-equilibrium motion of physical and/or biological motors along the highways are modeled by totally asymmetric simple exclusion process (TASEP) which falls in the class of driven di usive systems. To explore the collective dynamics, the physics behind the uni ed motion of the biological motors, in particular, kinesin, a motor protein have been thoroughly explored. The biological motors and intracellular tracks are mimicked by physical particles and one dimension discrete lanes, respectively in the framework of TASEP. We investigated single as well as multi-lane TASEP by incorporating vital intracellular features to explore how molecules behave collectively under the in uence of di erent processes and environments. Like any resource in nature, motor proteins are also not in nite, and they move along multiple microtubule proto laments with the possibility to switch their lane. These processes, including nite resources and lane changing phenomena affect the system dynamics signi cantly by producing various new phases in the phase diagram stimulating the need to examine their e ect on uni ed system properties. Microtubule, a dynamic and exible polymer, displays a twist in terms of a small preferred curvature due to the internal organization of the proto laments forming constrained entrances. Recent studies suggest that these properties of a microtubule lead to many considerable impacts on the system properties motivating us to incorporate these features in our model for a better description of motor movement. The studied models include these essential observations and investigate their non-trivial role in overall system dynamics. Motivated by the non-trivial e ect of nite resources in past studies, we examined its role on system dynamics in the presence of coupling between lanes, constrained entrances, and exibility in the lattice with a 3D environment. Besides, mathematically, we explored how cooperative motor action regulates the length of a dynamic microtubule while another study investigates the system dynamics in the presence of three lanes with the in nite number of particles. We used di erent theoretical approaches, including mean- eld theory and its variants, to calculate analytical results, including many novel features in terms of new phases and phase transitions which are validated by Monte Carlo simulations and available experimental results in the literature. The proposed work tries to provide a natural means to get a more in-depth insight into the properties of collective dynamics on intracellular transport by motors. We hope that our theoretical outcomes also might be observed by conducting in vitro and in vivo experiments under a well-controlled environment. It is expected that the obtained results not only might be useful for experimental biologists in understanding motor proteins in a more precise manner, but also can enhance one's insight into the study of non-equilibrium systems. Further, there exist many physical and biological systems, including vehicular transport, ant trails, pedestrian ow, protein synthesis, etc., which are analogous to intracellular transport carried out by motor proteins in the modelling framework of TASEP. Therefore the proposed models, theoretical analysis, and derived results can be utilized to understand other non-equilibrium stochastic transport problems too. In addition to that, there are various diseases associated with improper functioning of motor proteins such as HIV, Charcot-Marie- Tooth disease, kidney diseases, etc., which can be treated in a much better way by analyzing the collective behavior of these biological molecules. In short, we used mathematical modeling to understand unexplored biophysical complexities, which are also observed through simulations and might arise during motor transport too, thereby, making it essential to understand the collective dynamics of motor proteins.