The Arrival of the Frequent: How Bias in Genotype-Phenotype Maps Can Steer Populations to Local Optima
Steffen Schaper, Ard A. Louis mail
Published: February 05, 2014DOI: 10.1371/journal.pone.0086635
An illustration of the possible mutations available to an RNA molecule.
Abstract
Genotype-phenotype (GP) maps specify how the random mutations that change genotypes generate variation by altering phenotypes, which, in turn, can trigger selection. Many GP maps share the following general properties: 1) The total number of genotypes 
is much larger than the number of selectable phenotypes; 2) Neutral exploration changes the variation that is accessible to the population; 3) The distribution of phenotype frequencies 
, with 
the number of genotypes mapping onto phenotype 
, is highly biased: the majority of genotypes map to only a small minority of the phenotypes. Here we explore how these properties affect the evolutionary dynamics of haploid Wright-Fisher models that are coupled to a random GP map or to a more complex RNA sequence to secondary structure map. For both maps the probability of a mutation leading to a phenotype 
scales to first order as 
, although for the RNA map there are further correlations as well. By using mean-field theory, supported by computer simulations, we show that the discovery time 
of a phenotype 
similarly scales to first order as 
for a wide range of population sizes and mutation rates in both the monomorphic and polymorphic regimes. These differences in the rate at which variation arises can vary over many orders of magnitude. Phenotypic variation with a larger 
is therefore be much more likely to arise than variation with a small 
. We show, using the RNA model, that frequent phenotypes (with larger 
) can fix in a population even when alternative, but less frequent, phenotypes with much higher fitness are potentially accessible. In other words, if the fittest never ‘arrive’ on the timescales of evolutionary change, then they can't fix. We call this highly non-ergodic effect the ‘arrival of the frequent’.
is much larger than the number of selectable phenotypes; 2) Neutral exploration changes the variation that is accessible to the population; 3) The distribution of phenotype frequencies , with the number of genotypes mapping onto phenotype , is highly biased: the majority of genotypes map to only a small minority of the phenotypes. Here we explore how these properties affect the evolutionary dynamics of haploid Wright-Fisher models that are coupled to a random GP map or to a more complex RNA sequence to secondary structure map. For both maps the probability of a mutation leading to a phenotype scales to first order as , although for the RNA map there are further correlations as well. By using mean-field theory, supported by computer simulations, we show that the discovery time of a phenotype similarly scales to first order as for a wide range of population sizes and mutation rates in both the monomorphic and polymorphic regimes. These differences in the rate at which variation arises can vary over many orders of magnitude. Phenotypic variation with a larger is therefore be much more likely to arise than variation with a small . We show, using the RNA model, that frequent phenotypes (with larger ) can fix in a population even when alternative, but less frequent, phenotypes with much higher fitness are potentially accessible. In other words, if the fittest never ‘arrive’ on the timescales of evolutionary change, then they can't fix. We call this highly non-ergodic effect the ‘arrival of the frequent’.
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