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.

Numerical methods for computing effective transport properties of flashing Brownian motors

“…We begin by describing how the ideas from the Wang-Peskin-Elston (WPE) method [21,22] can be adapted in order to compute the effective drift and diffusivity of the flashing Brownian ratchet (1) modulated by the continuous Markov process F (t). First, F (t) is approximated by a finite state, continuous-time Markov chain F ♯ (t) with state space {f n } n∈S ♯ F and transi-tion rate matrix K satisfying the property that all the row sums are zero n ′ ∈S ♯ F K nn ′ = 0 and all non-diagonal entries are nonnegative.…”

“…These last two supermatrices will have a block diagonal form since they do not couple across different modulation states. Wang et al [21] and Wang and Elston [22] show that the effective drift and diffusivity are obtained as the unique solutions to the following equations:…”

Molecular Motors: Design, Mechanism, and Control

“…Such models can be developed at different levels of molecular details [21]. Even at a given level, the dynamics of the system can be formulated using different types of formalisms or updating rules for computer simulations.…”

Interacting molecular motors: Efficiency and work fluctuations

“…Molecular motors [1][2][3][4][5][6][7][8][9][10] are subject to intense study both from biological and technological point of view. They are paradigmatic examples of machines operating at nanometer scale.…”