  Biological molecular machines are proteins that operate under isothermal
conditions hence are referred to as free energy transducers. They can be
formally considered as enzymes that simultaneously catalyze two chemical
reactions: the free energy-donating reaction and the free energy-accepting one.
Most if not all biologically active proteins display a slow stochastic dynamics
of transitions between a variety of conformational substates composing their
native state. In the steady state, this dynamics is characterized by mean
first-passage times between transition substates of the catalyzed reactions. On
taking advantage of the assumption that each reaction proceeds through a single
pair (the gate) of conformational transition substates of the enzyme-substrates
complex, analytical formulas were derived for the flux-force dependence of the
both reactions, the respective stalling forces and the degree of coupling
between the free energy-accepting (output) reaction flux and the free
energy-donating (input) one. The theory is confronted with the results of
random walk simulations on the 5-dimensional hypercube. The formal proof is
given that in the case of reactions proceeding through single gates, the degree
of coupling cannot exceed unity. As some experiments suggest such exceeding,
looking for conditions of increasing the degree of coupling over unity
challenges theory. Though no analytical formulas for models involving more
transition substates are available, study simulations of random walks on
several model networks indicate that the case of the degree of coupling value
higher than one occurs in a natural way for scale-free tree-like networks. This
supports a hypothesis that the protein conformational transition networks, like
higher level biological networks: the proteome and the metabolome, have evolved
in a process of self-organized criticality.
