Biased Brownian motion as the operating principle for microscopic engines

The hydrolysis of adenosine triphosphate (ATP) has been shown to drive the motion of motor proteins along a biopolymer. These motor proteins are the smallest engines known and, in the absence of an ATP-to-adenosine diphosphate chemical potential, they execute Brownian motion. Therefore, it is reasonable to imagine that the energy released in ATP hydrolysis is used to bias, or rectify, Brownian motion in one direction. In this paper, we show, in terms of Fokker-Planck equations that we solve analytically, how a net flow can occur along a periodic potential, provided that this potential has an anisotropy and that there is an energy input. We work out two cases: one case where the energy input comes from a fluctuation of the periodic potential in time and one case where a variation of temperature within a period is maintained. An interesting feature of these systems is that they need ''the correct amount'' of thermal noise. Without thermal noise or with too much thermal noise, no net flow occurs and, in this sense, the systems we discuss are one more example of the lately much discussed phenomenon of stochastic resonance.