Stochastic free energy transduction.

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Abstract
Theoretical free-energy coupling systems in which the free energy coupling intermediate (e.g., the proton) occurs only in small numbers of molecules per coupling unit are shown to exhibit a number of peculiar properties: (i) the reactions involving the intermediates do not follow conventional kinetic (or nonequilibrium thermodynamic) rate laws in terms of the average concentration or chemical potential of the intermediate, (ii) the variation of the output reaction rate with the average intermediate concentration (or apparent chemical potential) is not unequivocal but depends on whether the input reaction or the leak is varied to alter that concentration, and (iii) when the apparent free energy contained in the average concentration of the intermediate is compared with the average free energy recovered in the output reaction, apparent violations of the second law of thermodynamics can occur. These phenomena are reminiscent of experimental observations in proton-linked free-energy transducing systems that suggest a more direct coupling between electron transfer chains and H+-ATPases than only through a bulk proton gradient, delta muH. Consequently, the chemiosmotic coupling theory can account for those observations if it limits the number of free energy coupling protons per chemiosmotic coupling unit to small values.