Dynamics of Evolutionary Optimization

General criteria of selection are derived from the kinetic equations of polynucleotide replication. As an illustrative example we discuss replication in the continuously stirred tank reactor (CSTR). The total rate of RNA synthesis is optimized during selection. The conjecture that the rate of approach towards the stable steady state is a maximum can be easily disproved. It is possible, nevertheless, to derive a potential function for polynucleotide replication in the CSTR. Following a method first introduced by Shahshahani we define a non Euclidean metric on the space of polynucleotide concentrations. In this space with a Riemannian metric the systems follows the corresponding generalized gradient during the process of selection and, therefore, the rate of ascent is now maximum. Potential functions can be derived also for some second order autocatalytic systems which are of interest in evolution, for a multidimensional Schloegl model in the CSTR and, as originally has been shown by Shahshahani, for the Fisher-Haldane-Wright equation of population genetics. In the general case, however, second order autocatalysis is not compatible with the existence of a potential. The elementary hypercycle is discussed as one simple example of a reaction network whose dynamics cannot be described by means of a generalized gradient system. Finite population size introduces a stochastic element into the selection process. Under certain conditions fluctuations in particle numbers become extremely important for the dynamics of selection. Two examples of this kind are: kinetic degeneracy of rate constants and low accuracy of replication.

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