Stochastic models for a number of reversible chemical reactions involving one or two second-order steps are examined and exact solutions which involve standard orthogonal polynomials are found for the equilibrium states. The expected values and variances of the numbers of molecules present in the systems at equilibrium are expressed in terms of these polynomials. Comparison of the stochastic mean value and the classical deterministic expression, shows that they are equivalent for most systems of physical interest, that is when large numbers of molecules are present. The relative fluctuation is found to be O (N−½), where N is of the order of the number of molecules present in the system, in agreement with the results of equilibrium statistical mechanics.
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Data from two assay systems show that the kinetics of the hydrolysis of cytidine 2': 3'-cyclic monophosphate by bovine pancreatic RNAase (ribonuclease) is not consistent with conventional models. An allosteric model involving a substrate-dependent change in the equilibrium between two enzyme conformations is proposed. Such a model gives rise to a calculated curve ofvelocity versus substrate concentration which fits the experimental data. The model is also consistent with the results of an examination of the tryptic digestion of RNAase. Substrate analogues are able to protect RNAase against hydrolysis by trypsin and the percentage of RNAase activity which remains after digestion increases sigmoidally as the analogue concentration is increased. The model also explains the pattern seen in the Km values quoted in the literature and is consistent with strong physical evidence for a ligand-induced conformational change for RNAase reported in the literature.
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