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