https://www.ucl.ac.uk/~ucackmp/Publications_files/unifying.pdf

Unifying evolutionary dynamics.

Fisher's Fundamental Theorem of Natural Selection is extended to the selection mutation model with mutation rates epsilon ij = epsilon i, i.e. depending only on the target gene, by constructing a simple Lyapunov function. For other mutation rates stable limit cycles are possible.

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The selection mutation equation.

This article provides an overview of the mathematical properties of various deterministic mutation-selection models. Many, even basic, results about models with more than two alleles per locus have only been obtained in the past two decades and are scattered throughout the literature. I review the fundamental results about existence and stability of equilibria for classical mutation-selection models with a finite number of alleles, for models like the stepwise-mutation model, and for the continuum-of-alleles model. For all these models, first- and second-order approximations for the equilibrium mean fitness and the mutation load are presented. Then I examine mutation-stabilizing selection balance. As an application of the results on the mutation load, second-order approximations for the equilibrium genetic variance can be derived. Various approximations for this equilibrium variance, which have been obtained in the literature, are concisely reviewed and their ranges of validity are compared. In particular, the dependence of the equilibrium variance on the parameters of the underlying genetic system, such as the form of the mutation distribution or the linkage map, is investigated.

Mathematical properties of mutation-selection models

Motivated by present activities in (statistical) physics directed towards biological evolution, we review the interplay of three evolutionary forces: mutation, selection, and genetic drift. The review addresses itself to physicists and intends to bridge the gap between the biological and the physical literature. We first clarify the terminology and recapitulate the basic models of population genetics, which describe the evolution of the composition of a population under the joint action of the various evolutionary forces. Building on these foundations, we specify the ingredients explicitly, namely, the various mutation models and fitness landscapes. We then review recent developments concerning models of mutational degradation. These predict upper limits for the mutation rate above which mutation can no longer be controlled by selection, the most important phenomena being error thresholds, Muller's ratchet, and mutational meltdowns. Error thresholds are deterministic phenomena, whereas Muller's ratchet requires the stochastic component brought about by finite population size. Mutational meltdowns additionally rely on an explicit model of population dynamics, and describe the extinction of populations. Special emphasis is put on the mutual relationship between these phenomena. Finally, a few connections with the process of molecular evolution are established.

Biological evolution through mutation, selection, and drift: An introductory review

A class of kinetic equations describing catalysed and template induced replication, and mutation is introduced. This ODE in its most general form is split into two vector fields, a replication and a mutation field. The mutation field is considered as a perturbation of the replicator equation. The perturbation expansion is a Taylor series in a mutation parameter lambda. First, second and higher order contributions are computed by means of the conventional Rayleigh-Schrodinger approach. Qualitative shift in the positions of rest points and limit cycles on the boundary of the physically meaningful part of concentration space are predicted from flow topologies. The results of the topological analysis are summarized in two theorems which turned out to be useful in applications: the rest point migration theorem (RPM) and the limit cycle migration theorem (LCM). Quantitative expressions for the shifts of rest points are computed directly from the perturbation expansion. The concept is applied to a collection of selected examples from biophysical chemistry and biology.

Mutation in autocatalytic reaction networks. An analysis based on perturbation theory.

In the continuous-time selection model for a single locus with several alleles, a stable polymorphism persists under mutation. Global stability can be shown for two and three alleles.For the discrete-time case there follows existence and uniqueness of a polymorphism.

K. P. Hadeler

Papers

Stable Polymorphisms in a Selection Model with Mutation