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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.

Molecular motors
play a central role in many cellular
functions. For example, an

Due to the small size of molecular motors, the inertia
of the motor is negligible. As a result, the motor operation is dominated by
high viscous friction and large thermal fluctuations from the fluid environment
[

One of the main differences between molecular motors and macroscopic motors is manifested in the issue of efficiency. For a macroscopic motor, the efficiency is well defined no matter it is working against a conservative force or working against a friction force. For a molecular motor, we do need to distinguish these two cases. When a molecular motor is working against a conservative force, the thermodynamic efficiency is well defined and is the energy output to the external agent exerting the conservative force divided by the chemical free energy consumption in the motor. When a molecular motor is working against a viscous drag, the situation is completely different. It has no energy output at all. One way to define efficiency in this case is to proceed with the apparent energy output based on the average velocity. The efficiency defined this way, called Stokes efficiency, is different from the thermodynamic efficiency. First of all, these two efficiencies are for two different cases: the thermodynamic efficiency is for a motor working against a conservative force; and the Stokes efficiency is for a motor working against a viscous drag. But this is not the only difference. In single molecule experiments, a motor can be put to work against a conservative force. In a different experimental setup, the same motor can be put to work against a viscous drag. For a molecular motor, both the thermodynamic efficiency and the Stokes efficiency can be measured but in two different experimental setups. Before we can treat the Stokes efficiency as a valid efficiency measurement, we need to show that it is bounded by 100%. Since the apparent energy output used in the definition of the Stokes efficiency does not have a thermodynamic meaning, the Stokes efficiency being bounded by 100% cannot be argued simply from thermodynamical point of view. In this paper, we prove mathematically that the Stokes efficiency is bounded by 100%. The proof is based on the Fokker-Planck formulation of motor-cargo systems. Below we will first describe this mathematical formulation.

A molecular motor, in general, has many internal and
external degrees of freedom. One of these degrees of freedom is associated with
the motor's unidirectional motion, the main biological function of the motor.
For example, a Kinesin dimer walks along a microtubule filament toward the
positive end [

In a molecular motor, the potential is not static.
Instead, the potential changes with the current chemical occupancy state of the
motor. In this paper, we consider the case where a motor has only one catalytic
site (one reaction cycle). The extension of the analysis to the case of
multiple catalytic sites is straightforward but tedious. Let

The derivation in [

A motor-cargo system in single
molecule experiments, corresponding to the experimental setup
in [

Let

In [

In summary, for the general case where a motor is elastically linked to a cargo, we have proved mathematically that the Stokes efficiency is bounded 100%. Therefore, the Stokes efficiency is a justified efficiency measure for motor-cargo systems.

The authors thank Timothy Elston, George Oster, and Charles Peskin for many helpful discussions during the project. This work was partially supported by the National Science Foundation and the Air Force Office of Scientific Research grant F1ATA06313G003.