The Stochastic Evolution of Catalysts in Spatially Resolved Molecular Systems

Abstract A fully stochastic chemical modelling technique is derived which describes the influence of spatial separation and discrete population size on the evolutionary stability of coupled amplification in biopolymers. The model is analytically tractable for an dimensional space (simplex geometry), which also provides insight into evolution in normal Euclidean space. The results are compared with stochastic simulations describing the coevolution of combinatorial families of molecular sequences both in the simplex geometry and in lower (one, two and three) space dimensions. They demonstrate analytically the generic limits which exploitation place on coevolving multicomponent amplification systems. In particular, there is an optimal diffusion (or migration) coefficient for cooperative amplification and minimal and maximal threshold values for stable cooperation. Over a bounded range of diffusion rates, the model also exhibits stable limit cycles. Furthermore, the cooperatively coupled system has a maximum tolerable error rate at intermediate rates of diffusion. A tractable model is thereby established which demonstrates that spatial effects can stabilize catalytic biological information. The analytic behaviour in dimensional simplex space is seen to provide a reasonable guide to the spatial dependence of the error threshold in physical space. Nanoscale possibilities for the evolution of catalysis on the basis of the model are outlined. We denote the modelling technique by PRESS, Probability Reduced Evolution of Spatiallydiscrete Species.