Theoretical formalism for bead movement powered by single two-headed motors in a motility assay

Kinesins and dyneins are protein motors that can use the free energy of ATP hydrolysis to carry a cargo and move uni-directionally along a microtubule filament. The purpose of this paper is to derive the formalism connecting the ATP-driven translocation reactions of these motors on microtubule filaments and the movement of the bead carried by the motor in a motility assay in which the bead is clamped at an arbitrary constant force. The formalism is thus useful in elucidating the load-dependent kinetic mechanism of the free-energy transduction of the motor using the mechanical data obtained from the motility assay. The formalism is also useful in assessing the effect on the measured motility data of various physical and hydrodynamic parameters of the assay, such as the size of the bead, the viscosity of the medium, the stiffness of the elastic element connecting the motor and the bead, etc. In a previous paper [Biophys. J. 67 (2000) 313] (hereafter referred to as paper I), we have derived the formalism for the case that the motor in the assay has only one head. In this paper we extend the derivation to the case that the motor is two-headed. The formalism is derived based on a simple two-state hand-over-hand model for the movement of the motor on microtubule, but can be easily extended to more complicated kinetic models. Effects of various hydrodynamic parameters on the velocity of the bead are studied with numerical calculations of the model. The difference between the formalism presented in this paper and the widely used "chemical" formalism, in which the movement of the kinesin and the bead is described by pure chemical reactions, is discussed.