Analytical aspects of the Brownian motor effect in randomly flashing\n ratchets

The muscle contraction, operation of ATP synthase, maintaining the shape of a\ncell are believed to be secured by motor proteins, which can be modelled using\nthe Brownian ratchet mechanism. We consider the randomly flashing ratchet model\nof a Brownian motor, where the particles can be in two states, only one of\nwhich is sensitive the applied spatially periodic potential (the mathematical\nsetting is a pair of weakly coupled reaction-diffusion and Fokker-Planck\nequations). We prove that this mechanism indeed generates unidirectional\ntransport by showing that the amount of mass in the wells of the potential\ndecreases/increases from left to right. The direction of transport is\nunambiguously determined by the location of each minimum of the potential with\nrespect to the so-called diffusive mean of its adjacent maxima. The transport\ncan be generated not only by an asymmetric potential, but also by a symmetric\npotential and asymmetric transition rates, and as a consequence of the general\nresult we derive explicit conditions when the latter happens. When the\ntransitions are localized on narrow active sites in the protein conformation\nspace, we find a more explicit characterization of the bulk transport\ndirection, and infer that some common preconditions of the motor effect are\nredundant.\n