A Mathematical Theory of Natural and Artificial Selection

JBS Haldane

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Because mutation is rare, double- and higher-order mutations can be neglected. Each of these sequences, including the wild-type, is assigned a fitness from some distribution. The key point, however, is that this overall distribution of fitness is unknown. Despite this, we do know two things.

Second, any beneficial mutations would be even fitter and so would fall even farther out in the tail of the fitness distribution. At some point in time, the environment changes and the wild-type allele slips slightly in fitness and one or more of the m mutations becomes beneficial.

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The question is: what is the size of the fitness gap between the wild-type and a beneficial sequence? To answer this, Gillespie assumed that only one beneficial mutation is available. This result was later generalized by Orr to any modest number of beneficial mutations i. Mutation should thus often yield beneficial alleles of small effect and rarely yield those of large effect. In retrospect, it is clear that this exponential distribution of beneficial effects is a simple consequence of a well-known result from so-called peaks-over-threshold models in EVT Leadbetter et al. A large set of results, concerning both the first substitution during adaptation and the properties of entire adaptive walks to local optima rest on this result Orr , , ; Rokyta et al.

While the first assumption seems reasonable, the second can be, and has been, questioned. But this need not be true Leadbetter et al. In the Weibull domain, tails are truncated, as with a uniform distribution.

To make matters complicated, the Gumbel domain also includes some truncated distributions. Recently, Joyce et al. In particular, they analysed the generalized Pareto distribution GPD , which describes the distribution of values above a high threshold e. The probability density of the GPD is given by.

On the change of population fitness by natural selection

By analysing adaptation using the GPD, Joyce et al. First, if departures from the Gumbel domain are mild, i. This is an important, and reassuring, result. Second, under the same conditions, results for entire adaptive walks involving several to many adaptive substitutions can show substantial departures from traditional Gumbel-based genetics of adaptation. Third, as departures from the Gumbel domain grow larger, traditional theory typically does not hold. The possibility that the distribution of fitness effects might often be approximately exponential has generated considerable interest among experimentalists.

Indeed, several have been motivated to attempt the daunting task of estimating the distribution of effects among beneficial mutations in actual organisms. While quantitative geneticists have shown some interest in back-calculating the desired distribution from quantitative trait locus data, here I discuss data from microbial experimental evolution studies as these analyses provide higher resolution data. In these studies, experimentalists typically allow a clonal population of microbes to adapt to a novel environmental challenge and then attempt to isolate new mutations that are advantageous under these changed conditions.

The fitness effects of these mutations can then be assessed, e. It should be emphasized that the distribution we hope to characterize experimentally is that among new beneficial mutations, not among those that are substituted during adaptation.

The latter distribution can differ from the former for two reasons. First, even when beneficial mutations are rare, the distribution of fitness effects among substituted mutations is affected by probabilities of fixation Kimura To make matters worse, the distribution of effects among substituted mutations given clonal interference also falls into the Gumbel domain Rozen et al.

Experimentalists must thus attempt to sidestep these problems, estimating the desired distribution of effects among newly arising mutations. Unfortunately, the data available thus far from the relevant experiments are mixed. Several experiments have found that the distribution of beneficial effects among new mutations is at least roughly exponential. Although they were unable to identify individual mutations at the DNA sequence level and so were unable to determine the number of distinct mutations sampled , they could not reject the null hypothesis of an exponential distribution of beneficial effects.

Similarly, Sanjuan et al. Although few distinct mutations were obtained, they concluded that the distribution of beneficial effects was gamma, with a shape parameter close to but statistically distinguishable from exponential. The gamma distribution belongs to the Gumbel domain.

Rokyta et al. Employing a statistical likelihood ratio test approach developed by Beisel et al. Instead, the distribution belonged to the Weibull domain, i. Given these mixed results, it is difficult, at present, to draw firm conclusions.

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As each experiment measures with error fitness in one taxon in one environment and generally from one wild-type sequence, we cannot yet have confidence in any generalities—or the lack thereof. It may be worth noting, however, one possible explanation of the difference between the exponential-like results seen in some experiments and the uniform-like ones seen by Rokyta et al.

When a mutation of given size is allowed to have random direction in Fisher's geometric model of adaptation, the distribution of fitness effects among beneficial mutations is more uniform-like at low dimensions i. This conclusion makes good intuitive sense: it is easier for a large mutation to be beneficial when it can affect only a few characters, not many. Though wholly speculative, it is possible that this finding has some bearing on the bacteriophage data discussed above as phages are presumably of low dimensionality. Just as we can ask about the additive effects on fitness of beneficial mutations, so we can ask about their dominance.

In diploids, are most beneficial mutations recessive or dominant for fitness? Once again, the rarity of beneficial mutations makes it difficult to answer this question directly. Nearly all modern literature on the dominance of beneficial mutations traces to a seminal paper by Charlesworth et al. In this paper, Charlesworth and colleagues presented a general analysis of the evolution of genes that reside in different regions of the genome and that, consequently, show different patterns of inheritance.

In particular, they considered the evolution of sex chromosomes versus autosomes, investigating both single-locus evolution and evolution at a quantitative character that has experienced a shift in optimum. For simplicity—and because it became the focus of subsequent research—I consider here only single-locus results. As we will see, the contrast between rates of evolution at X-linked versus autosomal genes often depends on the dominance of beneficial mutations. As Charlesworth et al. The problem was clearly not the mathematical difficulty of the problem for, as Charlesworth et al.

This result, a branching process approximation, is reasonably accurate unless h is close to zero Moran Turning to X-linked loci, both the number of beneficial mutations that appear per generation and their probability of fixation differ from the autosomal case. With weak selection, it can be shown that the probability of fixation of an X-linked mutation approximately equals the weighted average of the relevant probabilities for female-limited and male-limited mutations Nagylaki : 2 hs and 2 s , respectively, where the latter probability reflects the haploid-like probability of fixation of a hemizygous mutation.

We assume dosage compensation such that a single dose of the mutation in males provides the same fitness advantage as two doses in females.

Finally, then, we see that the ratio of substitution rates on autosomes to X chromosomes is. Charlesworth et al. In addition, they considered evolution due to the fixation of slightly deleterious mutations.

I do not consider this complication here. The resulting body of theory has experienced a somewhat surprising history. Though presented initially in the contexts of chromosomal evolution and speciation—much of Charlesworth et al. Equation 4. In particular, this equation shows that, if long-term evolution reflects positive natural selection and all else is equal, patterns of species divergence across the genome depend on the dominance of beneficial mutations.

In all cases, we can now speak of the average dominance of beneficial mutations: as substitution rates are approximately linear in h , expected substitution rates depend linearly on mean h.

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