Abstract
A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher’s work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. First, we consider the geometric case of fitness being isomorphic to distance from an optimum, and show how problems of optimal mutation rate control can be solved exactly or approximately depending on additional constraints of the problem. Then we consider the general case of fitness communicating only partial information about the distance. We define weak monotonicity of fitness landscapes and prove that this property holds in all landscapes that are continuous and open at the optimum. This theoretical result motivates our hypothesis that optimal mutation rate functions in such landscapes will increase when fitness decreases in some neighbourhood of an optimum, resembling the control functions derived in the geometric case. We test this hypothesis experimentally by analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. Our findings support the hypothesis and find that the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We discuss the relevance of these findings to living organisms.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-016-0995-3) contains supplementary material, which is available to authorized users.
Highlights
Mutation is one of the most important biological processes that influence evolutionary dynamics
Using the geometry of Euclidean space, Fisher showed that probability of adaptation decreases exponentially as a function of mutation size, and concluded, that adaptation is more likely to occur by small mutations
We shall apply the technique to the case when fitness is equivalent to negative distance from an optimum. The purpose of this exercise is to demonstrate that the functions μ(y) evolved by the Meta-genetic algorithms (GAs) have monotonic properties, similar to those possessed by the optimal mutation rate control function obtained analytically
Summary
Mutation is one of the most important biological processes that influence evolutionary dynamics. Subsequent geometric models based on Fisher’s, while explicitly modelling discrete mutational steps (e.g. Orr 2002), continue to assume that they occur within the same infinite Euclidean space This issue may contribute to the fact that the predictions of such models have at best only been partially verified in actual biological systems (McDonald et al 2011; Bataillon et al 2011; Kassen and Bataillon 2006; Rokyta et al 2008). The main theoretical result is a theorem about weak monotonicity of continuous landscapes, which establishes the condition for a similarity between fitness and distance to an optimum in a broad class of landscapes This suggests a similarity between fitness-based and distance-based optimal control functions for mutations rates.
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