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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AAL SS. Performed the experiments: AAL SS. Analyzed the data: AAL SS. Contributed reagents/materials/analysis tools: AAL SS. Wrote the paper: AAL SS.

Current address: Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Aachen, Germany

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

Darwin's account of biological evolution

GP mappings have been studied at different levels of abstraction

Most basically, the number of possible genotypes

Even though

Perhaps the most striking commonality of these GP maps is a strong bias in assignment of genotypes to phenotypes: Most phenotypes are realised by a tiny proportion of all genotypes, while most genotypes map into a small fraction of all phenotypes. This property is shared by all the GP maps we noted before. Typically the number

In this paper we study the evolutionary dynamics of a classical Wright-Fisher model, but with explicit microscopic GP maps that capture the three generic properties of such maps introduced above. Motivated by the strong bias in the distribution of the

These approximations are then tested against extensive simulations of two models: firstly, a simple GP map where the genotypes are randomly assigned to phenotypes according to a pre-determined distribution for the frequencies

Our analytic expressions agree quantitatively with the simulations in the polymorphic limit where

Finally, we employ the RNA GP map to study the case where two phenotypes

We study the evolution of a population of

The expected number of individuals with phenotype

While exact, these dynamic expressions depend implicitly on time through stochastic changes in the set

If

Neutral spaces can be astronomically large

Between fixations, the population undergoes periods of genotypic stasis in which only the 1-mutant neighborhood of the current genotype

It is instructive to compare the ratio

In the opposite extreme

Actual discovery and neutral fixation times can show strong fluctuations. As our evolutionary process is a Markov process – the next set of mutants depends only on the parents, not on earlier mutants – the first discovery time of a neighbour genotype as well as the arrival time of the neutral mutant “destined” to be fixed, are distributed geometrically (or exponentially in a model with continuous time). Thus the mean of

Let

If fixations are the rate-limiting step (ie.

The dynamics in the monomorphic regime are thus relatively straightforward. But whether some new phenotype

Over a large number of generations (

Finally, let us compare our results for large populations in the monomorphic and polymorphic limits. Most importantly, in both cases

These results suggest that for intermediate

In order to test our mean-field theory we study two kinds of GP maps that both include the generic properties of GP maps that we introduced earlier.

In the random GP map, the total number of phenotypes

Studying this map has two motivations: First, ignoring some biophysical detail may help illuminate generic features shared by the systems described in the introduction. Second, a simple model may clarify which deviations from our theory arise from population dynamic effects rather than from detailed (and system-specific) structure in the GP map.

In this simple model, correlations between genotypes are absent, facilitating analysis of the resulting neutral spaces. For example,

Here we study a particular random GP map with

For all regimes the

Locally frequent phenotypes (i.e. those with high

A subset of the phenotypes with

In the fully polymorphic regime where each individual essentially explores independently, any phenotype with

In the intermediate regime

In summary then, our theory derived in the previous section accurately describes the median discovery time

One of the best studied GP mappings has RNA genotypes of length

We performed extensive simulations of the

Overall, the evolutionary dynamics of this rather complex RNA system resembles that of the much simpler random GP map. Most importantly, the discovery times vary over many orders of magnitude. More precisely, as long as

The many orders of magnitude difference in the arrival rate of variation between phenotypes should have many important implications for evolutionary dynamics. Consider for example the situation where the population has equilibrated to a phenotype

Consider also the situation where two phenotypes

To illustrate this effect, we study the

We simulated a population of

Results are shown in

Probability that phenotype

Overall, our simulations show how the more frequent phenotype

Finally, phenotype

We note that differences in neutral network size have traditionally also been taken into account in terms of free fitness

Mutations provide the fuel for natural selection. Based on this principle, we have presented a detailed model of evolutionary dynamics that focuses on a microscopic description of the outcome of mutations. The phenotypic effect of mutations is mediated by the genotype-phenotype (GP) map which is therefore a crucial ingredient. As outlined in the introduction, several generic features are shared by many different example maps, independent of model details. Here we mainly focussed on the fact that these mapping are highly

Our calculations for a simplified random mapping and for the more complex RNA secondary structure model predict that the large bias observed in the GP maps translates into a similar order of magnitude variation in the median discovery times

In light of the simplicity of our mean-field approximation, its success in predicting the first-discovery time

The large differences we observe in the rate with which potential variation appears should have many consequences for evolutionary dynamics. There is of course a long history of invoking processes that impose directionality on the pathways available for evolutionary exploration (see ref.

We explicitly showed how phenotypes with a high local frequency can fix at the expense of locally rare phenotypes, even if the latter have much higher fitness. Taken together, these arguments suggest that the vast majority of possible phenotypes may never be found, and thus never fix, even though they may globally be the most fit: Evolutionary search is deeply non-ergodic. When Hugo de Vries was advocating for the importance of mutations in evolution, he famously said “Natural selection may explain the survival of the fittest, but it cannot explain the arrival of the fittest”

In the dynamic simulations, all

Secondary structures for RNA were predicted from sequence using the Vienna package

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Static properties of the random GP map.

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