Several Issues in Modeling Molecular Motors

Abstract

Protein motors play a central role in many cell functions. Understanding the operating principles of molecular motors is crucial to comprehending intracellular protein transport and cell motility. Due to the small size and the negligible inertia of molecular motors, the motor motion is damped by high viscous friction and is subject to large thermal excitations form the surrounding fluid environment. These properties determine that molecular motors behave drastically different from macroscopic motors. There are many levels of modeling for molecular motors: from simple chemical kinetic models with a few discrete states to all atom molecular dynamics simulations. Models at each level represent a compromise between being able to accommodate all aspects of motor operation and being easy to analyze/compute. All these models of different levels are valuable and are contributing to the understanding of the operating mechanism of molecular motors. In most of our studies of molecular motors, we adopt a mathematical framework of an intermediate level. In this framework, the major conformational changes of the motor are treated as continuous motions and changes in the occupancy state at the catalytic sites are modeled as discrete Markov transitions. In this framework, a motor system is described by a system of coupled Fokker-Planck equations, in which each equation corresponds to a chemical occupancy state. In this review, we discuss several issues in modeling molecular motors under this mathematical framework: (1) detailed balance and breaking of detailed balance; (2) motor potential profile and reconstruction of motor potential profile; (3) thermodynamic efficiency, Stokes efficiency and other efficiencies; and (4) numerical solutions of motor systems. In this review, we also examine the derivation of randomness parameter. We like to make it clear that this review is far from being comprehensive. The issues discussed are simply the ones we encounter in our modeling of molecular motors.

Asymptotic analysis of microtubule-based transport by multiple identical molecular motors

“…We say the motors are "processive" because they take many steps during each encounter with a microtubule. All told, we have substantial information about the operation and properties of individual molecular motors bound to cargo through work done by biologists [3,33,25,24], physicists [2,57,20,42], and mathematicians [44,69,53,8,56,30].…”

Influence of boundary conditions on stochastic movement of biological molecular motors

“…Myosins, due to their size, are influenced by their own Brownian motion and also by bombardment of surrounding fluids molecules [18]. For this reason, the most suitable description of muscle at the molecular level is a statistical one.…”

“…The others can be approximated by a mean field potential because their timescales are much shorter than the time set by the average motor velocity. Another reason, to create a mathematical model only in one spatial dimension, is to make the best connection with an experiment, where the motor is observed along one spatial dimension [18].…”

A modified active Brownian dynamics model using asymmetric energy conversion and its application to the molecular motor system

“…An overdamped limit-Since the very small mass (∼ 10 −22 kg ) of kinesin makes a negligible inertial contribution [49], we confine ourselves to an overdamped limit. Then velocity is not well-defined in the presence of thermal noise and hence the energy conversion rate cannot be determined.…”

“…Due to a negligible inertia effect in the nano-world, we consider the overdamped dynamics of the motor in contrast to the standard energy depot model. We introduce a modified energy depot model where the energy conversion function is defined in terms of the instantaneous drift velocity to avoid the difficulty of defining velocity in the overdamped Langevin equation of motion in the presence of thermal noise [49]. Based on our modified model, effects of the thermal fluctuations as well as of the active force, external load, and energy-supply rate associated with the molecular motor are studied and analyzed in detail.…”