Physical and mathematical models for biological machines
Life is a collection of molecules. It often seems to behave in a highly complex manner; nonetheless, we hope to dissect them using physics and mathematics. In this thesis, we explore some of the physics and mathematical models to describe the biological process in a cell, mainly focused on molecular machines that consume energy from chemical reactions and convert it into mechanical work. We also discuss an attempt to find general principles among various phases found in biomolecular condensates. First in chapter 1, we present the biological perspectives of some of the molecular motors, which will be the subjects of the analysis in this thesis. We enumerate the hydrolysis and ATPase cycle rate for various biological motors and surmise a connection to the underlying mechanism of the motors: Power stroke and Brownian ratchet. Next, an introduction to kinetic models and concomitant non-equilibrium thermodynamics for molecular machines are given. A formalism of Information thermodynamics, which we apply to allosteric communication for molecular motors (chapter 3), is introduced. Chapter 2 introduces the possibly simplest model, which can recapitulate the transport of the cargos by kinesin-1 (Kin1). We derive the expression of the velocity distribution for Kin1. We propose that examination of the randomness parameter is valuable information to infer the chemo-mechanical kinetics involving the stepping mechanism for Kin1. The subsequent chapter (chapter 3) analyzes the coordination in the chemo-mechanical cycle for Kin1, so-called gating, from non-equilibrium thermodynamics and information-theoretical perspective. The analysis illustrates that the stall load of Kin1 may be stemmed from the gating mechanism. In chapter 4, we turn the subject into the genome organization problem. We propose a simple theoretical model to explain the recently discovered loop extrusion process by a novel type of molecular motors: Structural Maintenance of Chromosomes (SMC) proteins. Theoretical work for SMC proteins is scarce. Thus, our work constitutes one of the crucial pieces in the field. Finally, in chapter 5, we explore the protein aggregation and liquid-liquid phase separation in a cell. We characterize liquid droplets and other form aggregates (crystal, amorphous solid, etc.) using molecular dynamics simulation. We see sequence and rigidity of the single polymer significantly affect the resulting phase and morphology of the aggregates. This attempts to elucidate the general instruction for bio-molecules resulting in various morphologies depending on the environmental conditions such as pH in the solution.