Output-input coupling in thermally fluctuating biomolecular machines
  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. 
Michal Kurzynski, Mieczyslaw Torchala, and Przemyslaw Chelminiak
physics.bio-ph q-bio.BM