In modeling the interior of cells by simulating a reaction–diffusion master equation over a grid of compartments, one employs the assumption that the copy numbers of various chemical species are small, discrete quantities. We show that, in this case, textbook expressions for the change in Gibbs free energy accompanying a chemical reaction or diffusion between adjacent compartments are inaccurate. We derive exact expressions for these free energy changes for the case of discrete copy numbers and show how these expressions reduce to traditional expressions under a series of successive approximations leveraging the relative sizes of the stoichiometric coefficients and the copy numbers of the solutes and solvent. Numerical results are presented to corroborate the claim that if the copy numbers are treated as discrete quantities, then only these more accurate expressions lead to correct behavior. Thus, the newly derived expressions are critical for correctly computing entropy production in mesoscopic simulations based on the reaction–diffusion master equation formalism.
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The product rule can be derived as d(fg) = (f + df) (g + dg) − fg = fdg + gdf + dfdg = fdg + gdf since the term dfdg is assumed to be negligible. We give this reminder to emphasize the presence of the cross term, which we do not neglect in the discrete case.
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