# Mathematical properties of mutation-selection models

@article{Brger2004MathematicalPO, title={Mathematical properties of mutation-selection models}, author={Reinhard B{\"u}rger}, journal={Genetica}, year={2004}, volume={102-103}, pages={279-298} }

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…

## 82 Citations

### Mutation-selection models of sequence evolution in population genetics

- Biology, Mathematics
- 2005

The equilibrium properties of a number of deterministic mutation- selection models of sequence evolution are investigated and maximum principles to determine the population mean fitness in equilibrium are derived, i.e., the phenomenon that for certain fitness functions the population changes at a critical mutation rate.

### Mutation-Selection Balance with Recombination: Convergence to Equilibrium for Polynomial Selection Costs

- Biology, MathematicsSIAM J. Appl. Math.
- 2009

It is established that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions and that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.

### Mutation-selection balance: ancestry, load, and maximum principle.

- BiologyTheoretical population biology
- 2002

The results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds) and lead to a clarification of concepts, as well as criteria for the existence of error thresholds.

### The frequency of the perfect genotype in a population subject to pleiotropic mutation.

- BiologyTheoretical population biology
- 2006

### A MUTATION-SELECTION MODEL FOR GENERAL GENOTYPES WITH RECOMBINATION

- Biology
- 2006

It is shown that the new model arises from the haploid model when recombination is added, in the limit as generations per unit time go to infinity, and selection strength and mutation per generation go to 0.

### Nonequivalent Loci and the distribution of mutant effects.

- BiologyGenetics
- 2002

Concentrating on the maintenance of quantitative genetic variation by mutation-selection balance, it is concluded that typically nonequivalent loci yield a genetic variance that is of order 10% smaller than that obtained from the previous approaches.

### Kingman’s Model with Random Mutation Probabilities: Convergence and Condensation II

- MathematicsAdvances in Applied Probability
- 2022

A generalisation of Kingman’s model of selection and mutation has been made in a previous paper which assumes all mutation probabilities to be i.i.d.. The weak convergence of fitness distributions to…

### The non-stationary dynamics of fitness distributions: asexual model with epistasis and standing variation

- BiologybioRxiv
- 2016

A general deterministic approximation is proposed that can yield predictions over both short and long timescales, hopefully fostering the quantitative test of alternative models, using data from experimental evolution in asexuals.

### Mutation–Selection Balance

- Biology
- 2014

Equilibrium expectations under the neutral model of mutation–selection balance provide important null hypotheses for measuring selection in natural populations.

### Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models.

- BiologyTheoretical population biology
- 2006

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