Cooperative instability phenomena in arrays of catalytic sites

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Abstract
Cooperative instability phenomena are shown to arise in arrays of localized catalytic sites immersed in a bulk system. Nonlinear reaction mechanisms may occur on the catalytic sites; the reactions in the bulk are stable. From the partial differential reaction–diffusion equations for the total system we derive ordinary integral equations and obtain from these, by linearization, stability conditions as a function of catalytic site density. The theory is applied to one‐dimensional lattices of catalytic sites with several model reaction mechanisms. For a product activated enzyme mechanism occurring on each catalytic site we show that there exist cooperative effects among sites due to the intersite coupling via the bulk reactions and diffusion. For given constraints, the number of stationary states available (one or three) depends on the density of catalytic sites, such that as the density increases, first three, then one, then three stationary states exist. For the Prigogine–Lefever mechanism critical concentrations necessary to produce chemical oscillations are shown to depend on the catalytic site density; the functional dependence has a maximum due to cooperative interactions among the sites. We study next a linear array of two types of alternating catalytic sites on which occur reactions of mutual activation of two species (a generalization of a model of Shymko and Glass). Multiple stationary states are found, and again the number of such states available depends on the site density.

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