  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.
