Probability of fixation of a mutant gene in a finite population when selective advantage decreases with time

A mutation-selection model of protein evolution under persistent positive selection

We use first principles of population genetics to model the evolution of proteins under persistent positive selection (PPS). PPS may occur when organisms are subjected to persistent environmental change, during adaptive radiations, or in host–pathogen interactions. Our mutation–selection model indicates protein evolution under PPS is an irreversible Markov process, and thus proteins under PPS show a strongly asymmetrical distribution of selection coefficients among amino acid substitutions. Our model shows the criteria ω>1 (where ω is the ratio of nonsynonymous over synonymous codon substitution rates) to detect positive selection is conservative and indeed arbitrary, because in real proteins many mutations are highly deleterious and are removed by selection even at positively selected sites. We use a penalized-likelihood implementation of the PPS model to successfully detect PPS in plant RuBisCO and influenza HA proteins. By directly estimating selection coefficients at protein sites, our inference procedure bypasses the need for using ω as a surrogate measure of selection and improves our ability to detect molecular adaptation in proteins.

Natural selection and the distribution of chromosomal inversion lengths

Chromosomal inversions contribute substantially to genome evolution, yet the processes governing their evolutionary dynamics remain poorly understood. Theory suggests that a readily measurable property of inversions-their length-can potentially affect their evolutionary fates. Emerging data on the lengths of polymorphic and fixed inversions may therefore provide clues to the evolutionary processes promoting inversion establishment. However, formal predictions for the distribution of inversion lengths remain incomplete, making empirical patterns difficult to interpret. We model the relation between inversion length and establishment probability for four inversion types: (1) neutral, (2) underdominant, (3) directly beneficial, and (4) indirectly beneficial, with selection favouring the latter because they capture locally adapted alleles at migration-selection balance and suppress recombination between them. We also consider how deleterious mutations affect the lengths of established inversions. We show that length distributions of common polymorphic and fixed inversions systematically differ among inversion types. Small rearrangements contribute the most to genome evolution under neutral and underdominant scenarios of selection, with the lengths of neutral inversion substitutions increasing, and those of underdominant substitutions decreasing, with effective population size. Among directly beneficial inversions, small rearrangements are preferentially fixed, whereas intermediate-to-large inversions are maintained as balanced polymorphisms via associative overdominance. Finally, inversions established under the local adaptation scenario are predominantly intermediate-to-large. Such inversions remain polymorphic or approach fixation within the local populations where they are favoured. Our models clarify how inversion length distributions relate to processes of inversion establishment, providing a platform for testing how natural selection shapes the evolution of genome structure.

Inversions are bigger on the X chromosome

In many insects, X-linked inversions fix at a higher rate and are much less polymorphic than autosomal inversions. Here, we report that in Drosophila, X-linked inversions also capture 67% more genes. We estimated the number of genes captured through an approximate Bayesian computational analysis of gene orders in nine species of Drosophila. X-linked inversions fixed with a significantly larger gene content. Further, X-linked inversions of intermediate size enjoy highest fixation rate, while the fixation rate of autosomal inversions decreases with size. A less detailed analysis in Anopheles suggests a similar pattern holds in mosquitoes. We develop a population genetic model that assumes the fitness effects of inversions scale with the number of genes captured. We show that the same conditions that lead to a higher fixation rate also produce a larger size for inversions on the X.

Fixation probability of a beneficial mutation conferring decreased generation time in changing environments

BACKGROUND

One central building block of population genetics is the fixation probability. It is a probabilistic understanding of the eventual fate of new mutations. Moreover, the fixation probability of new beneficial mutations plays an important effect on the adaptation of populations to environmental challenges. Great progress has been made in the study of the beneficial mutations that increases offspring number. However, the fixation probability of beneficial mutations with a shorter generation time under various genetic and ecological conditions has not been explored.

RESULTS

Here we extend the classical result of the fixation probability of beneficial mutations obtained by Haldane, and estimate the fixation probability of a beneficial mutation with a reduced generation time in a changing environment. Assuming that the selective advantage is very small, we concentrate all the changing factors of environment on a single quantity: effective selective advantage. Using a time-dependent branching process, we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time. Then, we apply this approximation to four interesting biological cases.

CONCLUSIONS

In these instances, we show the comparison of the approximation with the accurate values. We find that they are consistent, demonstrating the effectiveness of our result for the fixation probability of beneficial mutations conferring a reduced replication time.

Hitchhiking of deleterious alleles and the cost of adaptation in partially selfing species

Self-fertilization is generally seen to be disadvantageous in the long term. It increases genetic drift, which subsequently reduces polymorphism and the efficiency of selection, which also challenges adaptation. However, high selfing rates can increase the fixation probability of recessive beneficial mutations, but existing theory has generally not accounted for the effect of linked sites. Here, we analyze a model for the fixation probability of deleterious mutants that hitchhike with selective sweeps in diploid, partially selfing populations. Approximate analytical solutions show that, conditional on the sweep not being lost by drift, higher inbreeding rates increase the fixation probability of the deleterious allele, due to the resulting reduction in polymorphism and effective recombination. When extending the analysis to consider a distribution of deleterious alleles, as well as the average fitness increase after a sweep, we find that beneficial alleles generally need to be more recessive than the previously assumed dominance threshold (h < 1/2) for selfing to be beneficial from one-locus theory. Our results highlight that recombination aiding the efficiency of selection on multiple loci amplifies the fitness benefits of outcrossing over selfing, compared to results obtained from one-locus theory. This effect additionally increases the parameter range under which obligate outcrossing is beneficial over partial selfing.

Establishment of new mutations in changing environments

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.

Asexuals, polyploids, evolutionary opportunists...: the population genetics of positive but deteriorating mutations

Some genetic phenomena originate as mutations that are initially advantageous but decline in fitness until they become distinctly deleterious. Here I give the condition for a mutation–selection balance to form and describe some of the properties of the resulting equilibrium population. A characterization is also given of the fixation probabilities for such mutations.

A unified treatment of the probability of fixation when population size and the strength of selection change over time

The fixation probability is determined when population size and selection change over time and differs from Kimura's result, with long-term implications for a population. It is found that changes in population size are not equivalent to the corresponding changes in selection and can result in less drift than anticipated.

Evolution in heterogeneous populations: from migration models to fixation probabilities

Although dispersal is recognized as a key issue in several fields of population biology (such as behavioral ecology, population genetics, metapopulation dynamics or evolutionary modeling), these disciplines focus on different aspects of the concept and often make different implicit assumptions regarding migration models. Using simulations, we investigate how such assumptions translate into effective gene flow and fixation probability of selected alleles. Assumptions regarding migration type (e.g. source-sink, resident pre-emption, or balanced dispersal) and patterns (e.g. stepping-stone versus island dispersal) have large impacts when demes differ in sizes or selective pressures. The effects of fragmentation, as well as the spatial localization of newly arising mutations, also strongly depend on migration type and patterns. Migration rate also matters: depending on the migration type, fixation probabilities at an intermediate migration rate may lie outside the range defined by the low- and high-migration limits when demes differ in sizes. Given the extreme sensitivity of fixation probability to characteristics of dispersal, we underline the importance of making explicit (and documenting empirically) the crucial ecological/ behavioral assumptions underlying migration models.

The fixation probability of beneficial mutations

The fixation probability, the probability that the frequency of a particular allele in a population will ultimately reach unity, is one of the cornerstones of population genetics. In this review, we give a brief historical overview of mathematical approaches used to estimate the fixation probability of beneficial alleles. We then focus on more recent work that has relaxed some of the key assumptions in these early papers, providing estimates that have wider applicability to both natural and laboratory settings. In the final section, we address the possibility of future work that might bridge the gap between theoretical results to date and results that might realistically be applied to the experimental evolution of microbial populations. Our aim is to highlight the concrete, testable predictions that have arisen from the theoretical literature, with the intention of further motivating the invaluable interplay between theory and experiment.

The Effect of Deleterious Alleles on Adaptation in Asexual Populations

We calculate the fixation probability of a beneficial allele that arises as the result of a unique mutation in an asexual population that is subject to recurrent deleterious mutation at rate U. Our analysis is an extension of previous works, which make a biologically restrictive assumption that selection against deleterious alleles is stronger than that on the beneficial allele of interest. We show that when selection against deleterious alleles is weak, beneficial alleles that confer a selective advantage that is small relative to U have greatly reduced probabilities of fixation. We discuss the consequences of this effect for the distribution of effects of alleles fixed during adaptation. We show that a selective sweep will increase the fixation probabilities of other beneficial mutations arising during some short interval afterward. We use the calculated fixation probabilities to estimate the expected rate of fitness improvement in an asexual population when beneficial alleles arise continually at some low rate proportional to U. We estimate the rate of mutation that is optimal in the sense that it maximizes this rate of fitness improvement. Again, this analysis relaxes the assumption made previously that selection against deleterious alleles is stronger than on beneficial alleles.

Probability of gene fixation in an expanding finite population

A mathematical theory was developed, based on diffusion models, that enables us to compute the probability of a rare mutant allele eventually spreading through a population when the population size changes with time. In particular, we elaborated the case in which the mutant allele has a definite selective advantage and the population expands following the logistic law. In this case, the probability of ultimate fixation of a single mutant is given by u = 2s(Z/N), where s is the selective advantage and Z/N is a factor by which the probability of fixation is modified through population expansion. Analytical expression was obtained for Z/N, and the validity of the formula for u was checked by Monte Carlo experiments.

[Simulation of populations with randomly fluctuating fitness values]

The behaviour of populations due to fluctuating fitness values has been investigated using a computer. Six model systems were considered. The results have been illustrated by figures showing the composition of the populations as a function of the number of generations. The curves obtained for the genotype frequencies of the zygotes and adults (Z i and f i) from repeated runs have been analysed statistically by characteristic parameters as means, relative variances RV zi and RV fi and fluctuation frequencies. The study gave the following results: 1. The value of the parameters is determined by the selection coefficient s, the relative variance of the fitness values RV wi, the position of the deterministic equilibrium and the fertilization system. 2. In the systems without selfing the extent of the fluctuations, measured by RV zi and RV fi, primarily is caused by RV wi; the fluctuation frequency, as a measure for the length of time of the deviation from the mean, on the contrary is determined by s. 3. Equal rates of selfing (S i) of the three genotypes and lesser rates of selfing of the heterozygotes (S 2 < S 1 = S 3) extend, whereas greater rates of selfing of the heterozygotes (S 2 > S 1 = S 3) reduce, the fluctuations. 4. In all systems the mean of the frequency of the heterozygotes was smaller than the value of the deterministic equilibrium. The size of the difference, called heterozygote decrease A z2 resp. A f2, showing dependencies similar to RV zi and RV fi. 5. The relative variances of the heterozygote frequencies ordinarily were lower than those of the homozygote frequencies. This means that the heterozygote frequencies are less influenced by fluctuations of the fitness values.

Dynamics of finite populations : II. A time-homogeneous stochastic process describing the ultimate probability of and the expected time to fixation or loss of an allele or type in a population of variable size

A time-homogeneous stochastic process was used to derive exact expressions for the ultimate probability of fixation or loss and the expected time to fixation or loss of an allele in a haploid population whose size is a Poisson random variable. The treatment included selection and multiple alleles.