Biological molecular machines are proteins that operate under isothermal conditions and 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 (input) reaction and the free energy-accepting (output) 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. This makes the description of the enzymatic reaction kinetics in terms of conventional rate constants insufficient. In the steady state, upon taking advantage of the assumption that each reaction proceeds through a single pair (the gate) of transition conformational substates of the enzyme-substrates complex, the degree of coupling between the output and the input reaction fluxes has been expressed in terms of the mean first-passage times on a conformational transition network between the distinguished substates. The theory is confronted with the results of random-walk simulations on the five-dimensional hypercube. The formal proof is given that, for single input and output gates, the output-input degree of coupling cannot exceed unity. As some experiments suggest such exceeding, looking for the conditions for increasing the degree of coupling value over unity challenges the theory. Performed simulations of random walks on several model networks involving more extended gates indicate that the case of the degree of coupling value higher than 1 is realized in a natural way on critical branching trees extended by long-range shortcuts. Such networks are scale-free and display the property of the small world. For short-range shortcuts, the networks are scale-free and fractal, representing a reasonable model for biomolecular machines displaying tight coupling, i.e., the degree of coupling equal exactly to unity. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality.

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Computer simulations of random walk on the Sierpiński gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser f lash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.Formulation of an advanced statistical theory of biochemical processes needs simple but realistic models of phenomena underlying microscopic dynamics of proteins. Many experiments performed since the mid-70s have indicated the existence of a slow activated dynamics of conformational transitions in the protein native phase apart from the usual much faster vibrations (1-6). The spectrum of relaxation times characterizing the conformational transition dynamics spreads over many orders of magnitude from 10 Ϫ11