Mathematical and Computational Methods for Studying Energy Transduction in Protein Motors

Abstract

Protein motors play a central role in many cellular functions. Due to the small size of these molecular motors, their motion is dominated by high viscous friction and large thermal fluctuations. There are many levels of modeling molecular motors: from simple chemical kinetic models with a small number of discrete states to all atom molecular dynamics simulations. Here we describe a mathematical framework for an intermediate level of description. In this approach the major conformational changes of the motor protein are treated as continuous motions and changes in the chemical state of the motor are modeled as discrete Markov transitions. We discuss a numerical method for solving the Fokker-Planck equations that result from this mathematical framework and describe its extension to solving motor-cargo systems. We show that when the potential is discontinuous, detailed balance is a necessary condition for numerical convergence. We study the behavior of a motor-cargo system where the motor is driven by a tilted periodic potential. In particular, we derive a formula for the effective diffusion coefficient in the weak spring limit and analytically show that if the ratio of the motor size to that of the cargo is sufficiently large, then the velocity does not obtain its maximum value in the weak spring limit.