A molecular motor utilizes chemical free energy to generate a unidirectional motion
through the viscous fluid. In many experimental settings and biological settings, a
molecular motor is elastically linked to a cargo. The stochastic motion of a molecular
motor-cargo system is governed by a set of Langevin equations, each corresponding to
an individual chemical occupancy state. The change of chemical occupancy state is
modeled by a continuous time discrete space Markov process. The probability density
of a motor-cargo system is governed by a two-dimensional Fokker-Planck equation. The
operation of a molecular motor is dominated by high viscous friction and large thermal
fluctuations from surrounding fluid. The instantaneous velocity of a molecular motor
is highly stochastic: the past velocity is quickly damped by the viscous friction and
the new velocity is quickly excited by bombardments of surrounding fluid molecules.
Thus, the theory for macroscopic motors should not be applied directly to molecular
motors without close examination. In particular, a molecular motor behaves differently
working against a viscous drag than working against a conservative force. The Stokes
efficiency was introduced to measure how efficiently a motor uses chemical free energy
to drive against viscous drag. For a motor without cargo, it was proved that the Stokes
efficiency is bounded by 100% [H. Wang and G. Oster, (2002)].
Here, we present a proof for the general motor-cargo system.
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