Dynamics of Evolutionary Optimization

Abstract

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.

Replicator Dynamics in Protocells

“…In the case of homogeneous interaction functions, F k (λ c) = h(λ)F k ( c), one can show that the CSTR and the constant organization model are the same up to a rescaling of the time axis (Schuster & Sigmund, 1985). An analogous result can be shown for the limit of small flux rates r in the CSTR and arbitrary interaction functions F k ( c) (Happel & Stadler, 1999).…”

“…Again, as demonstrated in (Schuster & Sigmund, 1985), the total concentration c only amounts to a re-scaling of the time axis in the case of homogeneous interaction functions F k ( . ).…”

Evolution in Systems of Ligation-Based Replicators

“…≥ α n , then there is an index m such thatx is of the formx k > 0 if k ≤ m andx k = 0 for k > m. In other words, m species survive while the n − m least efficient replicators die out. This behavior is completely analogous to the reversible exponential competition case [18] where the rate constants a k play the role of the Darwinian fitness parameters α k . The concentration dependent values β k , on the other hand, collectively influence the flux term and hence set the extinction threshold α * = 2Φ(x).…”

Some mechanistic requirements for major transitions

“…Cooperative networks of replicators with hypercyclic organization in the flow reactor were considered previously in some detail [31]. The mechanism is given by equations (2.1a-e) where according to (2.1c) we use k ij ¼ k i for j ¼ ði þ 1Þ mod n and k ij ¼ 0 otherwise.…”