Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Theoretical Study of near Neutrality. II. Effect of Subdivided Population Structure with Local Extinction and Recolonization

There are several unsolved problems concerning the model of nearly neutral mutations. One is the interaction of subdivided population structure and weak selection that spatially fluctuates. The model of nearly neutral mutations whose selection coefficient spatially fluctuates has been studied by adopting the island model with periodic extinction-recolonization. Both the number of colonies and the migration rate play significant roles in determining mutants' behavior, and selection is ineffective when the extinction-recolonization is frequent with low migration rate. In summary, the number of mutant substitutions decreases and the polymorphism increases by increasing the total population size, and/or decreasing the extinction-recolonization rate. However, by increasing the total size of the population, the mutant substitution rate does not become as low when compared with that in panmictic populations, because of the extinction-recolonization, especially when the migration rate is limited. It is also found that the model satisfactorily explains the contrasting patterns of molecular polymorphisms observed in sibling species of Drosophila, including heterozygosity, proportion of polymorphism and fixation index.     

Relationship between DNA Polymorphism and Fixation Time

When there is no recombination among nucleotide sites in DNA sequences, DNA polymorphism and fixation of mutants at nucleotide sites are mutually related. Using the method of gene genealogy, the relationship between the DNA polymorphism and the fixation of mutant nucleotide was quantitatively investigated under the assumption that mutants are selectively neutral, that there is no recombination among nucleotide sites, and that the population is a random mating population with N diploid individuals. The results obtained indicate that the expected number of nucleotide differences between two DNA sequences randomly sampled from the population is 42% less when a mutant at a particular nucleotide site reaches fixation than at a random time, and that heterozygosity is also expected to be less when fixation takes place than at a random time, but the amount of reduction depends on the value of 4Nv in this case, where v is the mutation rate per DNA sequence per generation. The formula for obtaining the expected number of nucleotide differences between the two DNA sequences for a given fixation time is also derived, and indicates that, even when it takes a large number of generations for a mutant to reach fixation, this number is 33% less than at a random time. The computer simulation conducted suggests that the expected number of nucleotide differences between the two DNA sequences at the time when an advantageous mutant becomes fixed is essentially the same as that of neutral mutant if the fixation time is the same. The effect of recombination on the amount of DNA polymorphism was also investigated by using computer simulation.     

Fixation of advantageous alleles in partially self-fertilizing populations. The effect of different selection modes

The expected fixation probability of an advantageous allele was examined in a partially self-fertilizing hermaphroditic plant species using the diffusion approximation. The selective advantage of the advantageous allele was assumed to be increased viability, increased fecundity, or an increase in male fitness. The mode of selection, as well as the selfing rate, the population size, and the dominance of the advantageous allele, affect the fixation probability of the allele. In general it was found that increases in selfing rate decrease the fixation probability under male sexual selection, increase fixation probability under fecundity selection, and increase when recessive and decrease when dominant under viability selection. In some cases the highest fixation probability of advantageous alleles under fecundity or under male sexual selection occurred at an intermediary selfing rate. The expected mean fixation times of the advantageous allele were also examined using the diffusion approximation.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

The evolutionary rate of duplicated genes under concerted evolution

The effect of directional selection on the fixation process of a single mutation that spreads in a multigene family by gene conversion is investigated. A simple two-locus model with two alleles, A and a, is first considered in a random-mating diploid population with size N. There are four haplotypes, AA, Aa, aA, and aa, and selection works on the number of alleles A in a diplod (i = 0, 1, 2, 3, 4). Because gene conversion is allowed between the two loci, when the mutation rate is very low, either AA or aa will fix in the population eventually. We consider a situation where a single mutant, A, arises in one locus when a is fixed in both loci. Then, we derive the fixation probability analytically, and the fixation time is investigated by simulations. It is found that gene conversion has an effect to increase the "effective" population size, so that weak selection works more efficiently in a multigene family. With these results, we discuss the effect of gene conversion on the rate of molecular evolution in a multigene family undergoing concerted evolution. We also argue about the applicability of the theoretical results to models of multigene families with more than two loci.     

The influence of local extinctions on the probability of fixation of chromosomal rearrangements

We investigated the influence of local extinctions in a subdivided population on the probability of fixation of an initially rare allele, for different migration rates. The selective regimes considered were strict underdominance, meiotic drive, and underdominance associated with meiotic drive. We show that local extinctions can increase the probability of fixation of initially rare alleles in underdominant loci for relatively high migration rates, even when both homozygotes have the same fitness. This increase is due to drift during founder events. On the contrary, local extinctions decrease the probability of fixation of alleles favoured by meiotic drive. For a locus where both meiotic drive and underdominance act, the effect of local extinctions depends on the relative strength of the two selective regimes and the initial frequency of the rare allele. For parameter values such that the rare allele is initially selected against, local extinctions decrease the probability of fixation for low migration rates while they cause an increase for moderate migration rates. When the parameter values are such that the rare allele should always be favoured by selection, local extinctions always decrease the probability of fixation. In this latter case, we show the existence of an optimal migration rate which maximizes the probability of fixation.     

Effects of Partial Inbreeding on Fixation Rates and Variation of Mutant Genes

Diffusion methods were used to investigate the fixation probability, average time until fixation and extinction, and cumulative heterozygosity and genetic variance for single mutant genes in finite populations with partial inbreeding. The critical parameters in the approximation are the coefficient of inbreeding due to nonrandom mating (F) and the effective population size (Ne), which also depends on F and the variance of family size. For large Ns, the fixation probability (u) is u = 2(Ne/N)s (F + h - Fh), where N is the population census, s is the coefficient of selection of the mutant homozygote and h is the coefficient of dominance. For Poisson family size (independent Poisson distributions of selfed and nonselfed offspring with partial selfing, and independent Poisson distributions of male and female numbers with partial sib mating), Ne = N/(1 + F), and the time until fixation is approximately equal to Ne/N times the time to fixation with random mating, but this relation does not hold, however, for other distributions of family size. The cumulative nonadditive variance until fixation or loss for dominant genes is reduced with increasing F while for recessive genes it is increased with intermediate values of F. The average time until extinction of deleterious mutations is reduced by increasing F. This reduction, when expressed as a proportion, is approximately independent of the initial gene frequency as well as the selective disadvantage if this is large.     

Mutation Rate and Dominance of Genes Affecting Viability in DROSOPHILA MELANOGASTER

Spontaneous mutations were allowed to accumulate in a second chromosome that was transmitted only through heterozygous males for 40 generations. At 10-generation intervals the chromosomes were assayed for homozygous effects of the accumulated mutants. From the regression of homozygous viability on the number of generations of mutant accumulation and from the increase in genetic variance between replicate chromosomes it is possible to estimate the mutation rate and average effect of the individual mutants. Lethal mutations arose at a rate of 0.0060 per chromosome per generation. The mutants having small effects on viability are estimated to arise with a frequency at least 10 times as high as lethals, more likely 20 times as high, and possibly many more times as high if there is a large class of very nearly neutral mutations.-The dominance of such mutants was measured for chromosomes extracted from a natural population. This was determined from the regression of heterozygous viability on that of the sum of the two constituent homozygotes. The average dominance for minor viability genes in an equilibrium population was estimated to be 0.21. This is lower than the value for new mutants, as expected since those with the greatest heterozygous effect are most quickly eliminated from the population. That these mutants have a disproportionately large heterozygous effect on total fitness (as well as on the viability component thereof) is shown by the low ratio of the genetic load in equilibrium homozygotes to that of new mutant homozygotes.     

A simple proof that certain quantities are independent of the geographical structure of population

This paper gives a proof that certain quantities are independent of the geographical structure of a population. The quantities are: (1) the fixation probability of a mutant; (2) the sum of the quantity x(1 ? x), where x is the mutant frequency, while the mutant is segregating; and (3) the quantity x(1 ? x) summed over the generations during which the gene frequency in the whole population assumes a specified value. The independence of geographical structure for the latter two quantities is not exact if there is selection, but is a close approximation.The model is a geographically structured version of Moran's haploid overlapping generation model. The population consists of colonies connected genetically by migration. Each individual has the same negative exponential lifetime distribution. When an individual dies, it is immediately replaced by an individual born in the same colony with a probability proportional to the frequency and fitness of the type giving birth. In a diploid population the quantity x(1 ? x) is proportional to the heterozygosity.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.     

Fixation of clonal lineages under Muller's ratchet

For clonal lineages of finite size that differ in their deleterious mutational effects, the probability of fixation is investigated by mathematical theory and Monte Carlo simulations. If these fitness effects are sufficiently small in one or both lineages, then the lineage with the less deleterious effects will become fixed with high probability. If, however, in both lineages the deleterious effects are larger than a threshold s(c), then the probability of fixation is independent of the fitness effects and depends only on the initial frequencies of the lineages. This threshold decreases with decreasing genomic mutation rate U and increases with population size N. (For N = 10(5), we have s(c) approximately = 0.1 if U = 1, and s(c) approximately = 0.015 if U = 0.1). Above the threshold, the competition is not driven by the ratio of mean fitnesses of the lineages, but by the relative sizes of the zero-mutation classes, which are independent of the fitness effects of the mutations. After the loss of the zero-mutation class of a lineage, the other lineage will spread to fixation with high probability and within a short time span. If the mutation rates of the lineages differ substantially, the lineage with the lower mutation rate is fixed with very high probability unless the lineage with the larger mutation rate has very slightly deleterious mutational effects. If the mutation rates differ by not more than a few percent, then the lineage with the higher mutation rate and the more deleterious effects can become fixed with appreciable probability for a certain range of parameters. The independence of the fixation probability on the fitness effects in a single population leads to dramatic effects in metapopulations: lineages with more deleterious effects have a much higher fixation probability. The critical value s(c), above which this phenomenon occurs, decreases as the migration rate between the subpopulations decreases.     

Effect of mutation on genetic differentiation among nonequilibrium populations

The usefulness of GST and similar measures of genetic differentiation has been questioned repeatedly because of their dependence on the amount of heterozygosity within populations, creating problems when comparing degrees of divergence at loci with different mutation rates. Although the effect of mutation on GST is expected to be small in the early phases of divergence, it is unclear for how long after separation from a common ancestral population that GST is largely unaffected by mutation and by the resulting effect on heterozygosity. We address this question through analysis of the recursion equations for gene identity under the infinite allele model of mutation, and derive conditions describing when the effect of mutation on GST can be ignored under mutation-migration-drift equilibrium conditions and during the preceding transition phase. An important result is that during the transition phase GST is not only affected by mutation, but also by the heterozygosity in the base population from which the subpopulations diverged. The effect of mutation on GST is significant from the very start of the divergence process when initial heterozygosity is low, whereas GST is only weakly affected by mutation in the early phases of differentiation when initial heterozygosity is high. Thus, differentiation following a severe bottleneck is strongly dependent on mutation. The standardized measure of differentiation, G'ST, suggested by Hedrick (2005), may be helpful when comparing amounts of divergence at loci with different mutation rates under steady-state conditions, provided that migration is very low. In many other situations the use of G'ST might be misleading, however, and its application should be exercised with caution.     

Effect of population structure on the amount of polymorphism and the fixation probability under overdominant selection

Under overdominant selection, mutants substantially contribute to increase the amount of polymorphism. It is also known that under neutrality as the migration rates among demes decrease in a subdivided population, the amount of polymorphism increases along with the increase of the effective population size, N(e). In this study, under overdominant selection the effect of population subdivision on the amount of polymorphism was investigated using the diffusion approximation and the low migration approximation. It was shown that if selection is medium or strong (e.g., N(T)s > 1, where N(T) is the population size and s is the selective advantage of heterozygotes), the nucleotide diversity, pi, decreases along with the decrease of Nm against the increase of N(e), where N is the size of demes and m is the migration rate per deme. In addition, the ratio of the nucleotide diversity to the evolutionary rate also decreases along with the decrease of Nm. In some cases the ratio becomes smaller than that expected under neutrality as Nm decreases.     

A Study on a Nearly Neutral Mutation Model in Finite Populations

As a nearly neutral mutation model, the house-of-cards model is studied in finite populations using computer simulations. The distribution of the mutant effect is assumed to be normal. The behavior is mainly determined by the product of the population size, N, and the standard deviation, sigma, of the distribution of the mutant effect. If 4N sigma is large compared to one, a few advantageous mutants are quickly fixed in early generations. Then most mutation becomes deleterious and very slow increase of the average selection coefficient follows. It takes very long for the population to reach the equilibrium state. Substitutions of alleles occur very infrequently in the later stage. If 4N sigma is the order of one or less, the behavior is qualitatively similar to that of the strict neutral case. Gradual increase of the average selection coefficient occurs and in generations of several times the inverse of the mutation rate the population almost reaches the equilibrium state. Both advantageous and neutral (including slightly deleterious) mutations are fixed. Except in the early stage, an increase of the standard deviation of the distribution of the mutant effect decreases the average heterozygosity. The substitution rate is reduced as 4N sigma is increased. Three tests of neutrality, one using the relationship between the average and the variance of heterozygosity, another using the relationship between the average heterozygosity and the average number of substitutions and Watterson's homozygosity test are applied to the consequences of the present model. It is found that deviation from the neutral expectation becomes apparent only when 4N sigma is more than two. Also a simple approximation for the model is developed which works well when the mutation rate is very small.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Evolution of coadaptation in a subdivided population

The interplay between population subdivision and epistasis is investigated by studying the fixation probability of a coadapted haplotype in a subdivided population. Analytical and simulation models are developed to study the evolutionary fate of two conditionally neutral mutations that interact epistatically to enhance fitness. We find that the fixation probability of a coadapted haplotype shows a marked increase when the population is genetically subdivided and subpopulations are loosely connected by migration. Moderate migration and isolation allow the propagation of the mutant alleles across subpopulations, while at the same time preserving the favorable allelic combination established within each subpopulation. Together they create the condition most favorable for the ultimate fixation of the coadapted haplotype. On the basis of the analytical and simulation results, we discuss the fundamental role of population subdivision and restricted gene flow in promoting the evolution of functionally integrated systems, with some implications for the shifting-balance theory of evolution.     

The effect of subdivision on variation at multi-allelic loci under balancing selection

Simulations are used to investigate the expected pattern of variation at loci under different forms of multi-allelic balancing selection in a finite island model of a subdivided population. The objective is to evaluate the effect of restricted migration among demes on the distribution of polymorphism at the selected loci at equilibrium, and to compare the results with those expected for a neutral locus. The results show that the expected number of alleles maintained, and numbers of nucleotide differences between alleles, are relatively insensitive to the migration rate, and differentiation remains low even under very restricted migration. However, nucleotide divergence between copies of functionally identical alleles increases sharply when migration decreases. These results are discussed in relation to published surveys of allelic diversity in MHC and plant self-incompatibility systems, and to the possibility of inferring ancient population genetic events and processes. In addition, it is shown that, for sporophytic self-incompatibility systems, it is not necessarily true in a subdivided population that recessive alleles are more frequent than dominant ones.     

Intracellular Selection, Conversion Bias, and the Expected Substitution Rate of Organelle Genes

A key step in the substitution of a new organelle mutant throughout a population is the generation of germ-line cells homoplasmic for that mutant. Given that each cell typically contains multiple copies of organelles, each of which in turn contains multiple copies of the organelle genome, processes akin to drift and selection in a population are responsible for producing homoplasmic cells. This paper examines the expected substitution rate of new mutants by obtaining the probability that a new mutant is fixed throughout a cell, allowing for arbitrary rates of genome turnover within an organelle and organelle turnover within the cell, as well as (possibly biased) gene conversion and genetic differences in genome and/or organelle replication rates. Analysis is based on a variation of Moran's model for drift in a haploid population. One interesting result is that if the rate of unbiased conversion is sufficiently strong, it creates enough intracellular drift to overcome even strong differences in the replication rates of wild-type and mutant genomes. Thus, organelles with very high conversion rates are more resistant to intracellular selection based on differences in genome replication and/or degradation rates. It is found that the amount of genetic exchange between organelles within the cell greatly influences the probability of fixation. In the absence of exchange, biased gene conversion and/or differences in genome replication rates do not influence the probability of fixation beyond the initial fixation within a single organelle. With exchange, both these processes influence the probability of fixation throughout the entire cell. Generally speaking, exchange between organelles accentuates the effects of directional intracellular forces.(ABSTRACT TRUNCATED AT 250 WORDS)