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Mutation–selection models solved exactly with methods of statistical mechanics

Published online by Cambridge University Press:  29 August 2001

ELLEN BAAKE
Affiliation:
Zoologisches Institut, Universität München, Luisenstrasse 14, D-80333 Munich, Germany
HOLGER WAGNER
Affiliation:
Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg 11, D-37077 Göttingen, Germany
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Abstract

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We reconsider deterministic models of mutation and selection acting on populations of sequences, or, equivalently, multilocus systems with complete linkage. Exact analytical results concerning such systems are few, and we present recent and new ones obtained with the help of methods from quantum statistical mechanics. We consider a continuous-time model for an infinite population of haploids (or diploids without dominance), with N sites each, two states per site, symmetric mutation and arbitrary fitness function. We show that this model is exactly equivalent to a so-called Ising quantum chain. In this picture, fitness corresponds to the interaction energy of spins, and mutation to a temperature-like parameter. The highly elaborate methods of statistical mechanics allow one to find exact solutions for non-trivial examples. These include quadratic fitness functions, as well as ‘Onsager's landscape’. The latter is a fitness function which captures some essential features of molecular evolution, such as neutrality, compensatory mutations and flat ridges. We investigate the mean number of mutations, the mutation load, and the variance in fitness under mutation–selection balance. This also yields some insight into the ‘error threshold’ phenomenon, which occurs in some, but not all, examples.

Type
Research Article
Copyright
2001 Cambridge University Press
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