Mutation Rate Evolution in Replicator Dynamics

The mutation rate of an organism is itself evolvable. In stable environments, if faithful replication is costless, theory predicts that mutation rates will evolve to zero. However, positive mutation rates can evolve in novel or fluctuating environments, as analytical and empirical studies have shown. Previous work on this question has focused on environments that fluctuate independently of the evolving population. Here we consider fluctuations that arise from frequency-dependent selection in the evolving population itself. We investigate how the dynamics of competing traits can induce selective pressure on the rates of mutation between these traits. To address this question, we introduce a theoretical framework combining replicator dynamics and adaptive dynamics. We suppose that changes in mutation rates are rare, compared to changes in the traits under direct selection, so that the expected evolutionary trajectories of mutation rates can be obtained from analysis of pairwise competition between strains of different rates. Depending on the nature of frequency-dependent trait dynamics, we demonstrate three possible outcomes of this competition. First, if trait frequencies are at a mutation–selection equilibrium, lower mutation rates can displace higher ones. Second, if trait dynamics converge to a heteroclinic cycle—arising, for example, from “rock-paper-scissors” interactions—mutator strains succeed against non-mutators. Third, in cases where selection alone maintains all traits at positive frequencies, zero and nonzero mutation rates can coexist indefinitely. Our second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics.     

Ascertainment of the effect of differential growth rates of mutants on observed mutant frequencies in X-irradiated mammalian cells An indication for the occurrence of X-ray-induced untargeted mutagenesis

As it is not known to what extent differential growth rates of induced mutants lead to over- and under-representation of mutants in treated populations and thereby affect the determination of mutant frequencies, the mutation induction in X-irradiated L5178Y mouse lymphoma cells was determined via two methods. The first method involves the standard protocol which may suffer from the effect of differential growth rates, while the second method is based upon the fluctuation test in which the differential growth rates can be actually measured. It appeared that the standard protocol led to a mutant frequency that was similar to the mutant frequency determined in the fluctuation test. Therefore, the standard protocol appears to lead to only a minor under-estimation if any. Substantial heterogeneity in growth rates of induced mutants was observed, but the mutants with a selective advantage appear largely to compensate for the mutants that are lost because of selective disadvantage. It was calculated that the chance for isolating the same mutant twice from a treated population had been increased 2.2-fold because of the observed differential growth rates. Therefore, our data indicate that the standard protocol does not lead to serious errors in the determination of mutant frequencies and in the sampling of mutants. The fluctuation tests were also used to determine the spontaneous mutation frequency per cell per generation. The mutation rate appeared more than 10-fold enhanced in X-irradiated cells which may be attributed to the induction of a process of untargeted mutagenesis in mammalian cells.     

Rates and fitness consequences of new mutations in humans

The human mutation rate per nucleotide site per generation (μ) can be estimated from data on mutation rates at loci causing Mendelian genetic disease, by comparing putatively neutrally evolving nucleotide sequences between humans and chimpanzees and by comparing the genome sequences of relatives. Direct estimates from genome sequencing of relatives suggest that μ is about 1.1 × 10(-8), which is about twofold lower than estimates based on the human-chimp divergence. This implies that an average of ~70 new mutations arise in the human diploid genome per generation. Most of these mutations are paternal in origin, but the male:female mutation rate ratio is currently uncertain and might vary even among individuals within a population. On the basis of a method proposed by Kondrashov and Crow, the genome-wide deleterious mutation rate (U) can be estimated from the product of the number of nucleotide sites in the genome, μ, and the mean selective constraint per site. Although the presence of many weakly selected mutations in human noncoding DNA makes this approach somewhat problematic, estimates are U ≈ 2.2 for the whole diploid genome per generation and 0.35 for mutations that change an amino acid of a protein-coding gene. A genome-wide deleterious mutation rate of 2.2 seems higher than humans could tolerate if natural selection is "hard," but could be tolerated if selection acts on relative fitness differences between individuals or if there is synergistic epistasis. I argue that in the foreseeable future, an accumulation of new deleterious mutations is unlikely to lead to a detectable decline in fitness of human populations.     

Heterozygosity, Heteromorphy, and Phylogenetic Trees in Asexual Eukaryotes

Little attention has been paid to the consequences of long-term asexual reproduction for sequence evolution in diploid or polyploid eukaryotic organisms. Some elementary theory shows that the amount of neutral sequence divergence between two alleles of a protein-coding gene in an asexual individual will be greater than that in a sexual species by a factor of 2tu, where t is the number of generations since sexual reproduction was lost and u is the mutation rate per generation in the asexual lineage. Phylogenetic trees based on only one allele from each of two or more species will show incorrect divergence times and, more often than not, incorrect topologies. This allele sequence divergence can be stopped temporarily by mitotic gene conversion, mitotic crossing-over, or ploidy reduction. If these convergence events are rare, ancient asexual lineages can be recognized by their high allele sequence divergence. At intermediate frequencies of convergence events, it will be impossible to reconstruct the correct phylogeny of an asexual clade from the sequences of protein coding genes. Convergence may be limited by allele sequence divergence and heterozygous chromosomal rearrangements which reduce the homology needed for recombination and result in aneuploidy after crossing-over or ploidy cycles.     

Life history and the male mutation bias

Abstract If DNA replication is a major cause of mutation, then those life-history characters, which are expected to affect the number of male germline cell divisions, should also affect the male to female mutation bias (a_m). We tested this hypothesis by comparing several clades of bird species, which show variation both in suitable life-history characters (generation time as measured by age at first breeding and sexual selection as measured by frequency of extrapair paternity) and in a_m, which was estimated by comparing Z-linked and W-linked substitution rates in gametologous introns. a_m differences between clades were found to positively covary with both generation time and sexual selection, as expected if DNA replication causes mutation. The effects of extrapair paternity frequency on a_m suggests that increased levels of sexual selection cause higher mutation rates, which offers an interesting solution to the paradox of the loss of genetic variance associated with strong directional sexual selection. We also used relative rate tests to examine whether the observed differences in a_m between clades were due to differences in W-linked or Z-linked substitution rates. In one case, a significant difference in a_m between two clades was shown to be due to W-linked rates and not Z-linked rates, a result that suggests that mutation rates are not determined by replication alone.     

Estimating the per-base-pair mutation rate in the yeast Saccharomyces cerevisiae

Although mutation rates are a key determinant of the rate of evolution they are difficult to measure precisely and global mutations rates (mutations per genome per generation) are often extrapolated from the per-base-pair mutation rate assuming that mutation rate is uniform across the genome. Using budding yeast, we describe an improved method for the accurate calculation of mutation rates based on the fluctuation assay. Our analysis suggests that the per-base-pair mutation rates at two genes differ significantly (3.80x10(-10) at URA3 and 6.44x10(-10) at CAN1) and we propose a definition for the effective target size of genes (the probability that a mutation inactivates the gene) that acknowledges that the mutation rate is nonuniform across the genome.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.     

Selection-mutation at a diallelic autosomal locus in a dioecious population

The population is assumed to be infinite dioecious with nonoverlapping discrete generations and random mating. It is assumed that the fitnesses and mutation rates are constant, heterozygotes are viable and the mutation rates are less than one-half. It is proved that the allelic frequencies converge to equilibria as the number of generations tends to infinity. The a priori types of phase portraits are determined. The method employed is elementary. The results extend those of [1, 2, 5, 8] to the case of selection-mutation rather than pure selection and those of [7] to the case of an autosomal rather than a sex-linked locus.     

Conditional Mutator Gene in Escherichia coli: Isolation, Mapping, and Effector Studies

A mutator gene, mutD5, whose phenotype is conditional, has been identified in Escherichia coli. By P1 transduction it has been shown to lie at about 5.7 min on the chromosome, being co-transduced with proA and argF. In rich medium, streptomycin- and nalidixic acid-resistant mutation frequencies are 50 to 100 times higher than those in minimal medium. In minimal medium, the mutD5-induced mutation frequencies are still 50 to 100 times above co-isogenic wild-type (mut(+)) levels. Similar results were obtained with all markers tested. Mutant frequencies can be raised by thymidine in the medium at concentrations as low as 0.04 muM, or by the endogenous generation of thymidine from thymine plus a deoxyribosyl donor. Deoxyadenosine, various ribonucleosides, thymine, and 2-deoxyribose do not stimulate mutation. None of these effects are related to growth rate, since growth rate and mutation rate can be decoupled completely.     

The Age of a Neutral Mutant Persisting in a Finite Population

Formulae for the mean and the mean square age of a neutral allele which is segregating with frequency x in a population of effective size N(e) have been obtained using the diffusion equation method, for the case of 4N(e)v<1 where v is the mutation rate. It has been shown that the average ages of neutral alleles, even if their frequencies are relatively low, are quite old. For example, a neutral mutant whose current frequency is 10% has the expected age roughly equal to the effective population size N(e) and the standard deviation 1.4N(e) (in generations), assuming that this mutant has increased by random drift from a very low frequency. Also, formulae for the mean "first arrival time" of a neutral mutant to a certain frequency x have been presented. In addition, a new, approximate method has been developed which enables us to obtain the condition under which frequencies of "rare" polymorphic alleles among local populations are expected to be uniform if the alleles are selectively neutral.-It was concluded that exchange of only a few individuals on the average between adjacent colonies per generation is enough to bring about such a uniformity of frequencies.     

Gene Conversion, Linkage, and the Evolution of Multigene Families

The evolution of the probabilities of genetic identity within and between the loci of a multigene family is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, and the location dependence of the probabilities of identity is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. For interchromosomal conversion, the equilibrium probabilities of identity are within order epsilon [i.e., O(epsilon)] of those in a simple model that has no location dependence and, at equilibrium, no linkage disequilibrium. At equilibrium, the linkage disequilibria are of O(epsilon); they are evaluated explicitly with an error of O(epsilon 2); they may be negative if symmetric heteroduplexes occur. The ultimate rate and pattern of convergence to equilibrium are within O(epsilon 2) and O(epsilon), respectively, of that of the same simple model. If linkage is loose (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), the linkage disequilibria are reduced to O(epsilon) in a time of O(-ln epsilon). If intrachromosomal conversion is incorporated, the same results hold for loose linkage, except that, if the crossover rates are much less than 1/2, then the linkage disequilibria generally exceed those for pure interchromosomal conversion.     

Evolution under multiallelic migration-selection models

The loss of a specified allele and the convergence of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection are investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Sufficient conditions are established for global fixation and for global loss of a particular allele. When migration is either sufficiently weak or sufficiently strong relative to selection, the equilibria are described, convergence of the gene frequencies is demonstrated, and sufficient conditions for the increase of a suitably defined mean fitness are offered. If the selection pattern is the same in every colony and such that in a panmictic population there is a globally asymptotically stable, internal (i.e., completely polymorphic) equilibrium point, then under certain weak assumptions on migration, the gene frequencies in the subdivided population converge globally to that equilibrium point. Thus, in this case, the ultimate state of the population is unaffected by geographical structure.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Inferring admixture proportions from molecular data

We derive here two new estimators of admixture proportions based on a coalescent approach that explicitly takes into account molecular information as well as gene frequencies. These estimators can be applied to any type of molecular data (such as DNA sequences, restriction fragment length polymorphisms [RFLPs], or microsatellite data) for which the extent of molecular diversity is related to coalescent times. Monte Carlo simulation studies are used to analyze the behavior of our estimators. We show that one of them (mY) appears suitable for estimating admixture from molecular data because of its absence of bias and relatively low variance. We then compare it to two conventional estimators that are based on gene frequencies. mY proves to be less biased than conventional estimators over a wide range of situations and especially for microsatellite data. However, its variance is larger than that of conventional estimators when parental populations are not very differentiated. The variance of mY becomes smaller than that of conventional estimators only if parental populations have been kept separated for about N generations and if the mutation rate is high. Simulations also show that several loci should always be studied to achieve a drastic reduction of variance and that, for microsatellite data, the mean square error of mY rapidly becomes smaller than that of conventional estimators if enough loci are surveyed. We apply our new estimator to the case of admixed wolflike Canid populations tested for microsatellite data.      

Selection at a diallelic autosomal locus in a dioecious population

The model used is that of an infinite dioecious population with nonoverlapping discrete generations and random mating. If the fitnesses are constant and heterozygotes are viable, it is proved that the allelic frequencies converge to equilibria as the number of generations tend to infinity. The results complement those of Karlin and Lessard [1] and Selgrade and Ziehe [5] in that hyperbolicity of equilibria is not assumed, use of index theory is avoided and it is determined how the number of equilibria and phase portraits depend on the fitnesses in the most general case. Lessard [2] gives, in the same situation, a condensed proof of convergence of allelic frequencies off the separatrix under the hypothesis that 1 is not an eigenvalue at any equilibrium. Our method of study is elementary.     

Genetic variance in the F2 generation of divergently selected parents

 Either by selective breeding for population divergence or by using natural population differences, F₂ and advanced generation hybrids can be developed with high variances. We relate the size of the genetic variance to the population divergence based on a forward and backward mutation model at a locus with two alleles with additive gene action. The effects of population size and initial gene frequency are also explored. Larger parental population sizes increase the F₂ genetic variance if the initial probability distribution is uniform or U-shaped. However, population size has the opposite effect if the initial distribution of gene frequencies is skewed such as it would be with newly arriving alleles. These alleles contribute to the genetic variance sooner when the selection pressure is higher or when the effective population size is smaller. Received: 5 April 1998 / Accepted: 22 April 1998     

An Approach to Population and Evolutionary Genetic Theory for Genes in Mitochondria and Chloroplasts, and Some Results

We developed population genetic theory for organelle genes, using an infinite alleles model appropriate for molecular genetic data, and considering the effects of mutation and random drift on the frequencies of selectively neutral alleles. The effects of maternal inheritance and vegetative segregation of organelle genes are dealt with by defining new effective gene numbers, and substituting these for 2N(e) in classical theory of nuclear genes for diploid organisms. We define three different effective gene numbers. The most general is N(lambda), defined as a function of population size, number of organelle genomes per cell, and proportions of genes contributed by male and female gametes to the zygote. In many organisms, vegetative segregation of organelle genomes and intracellular random drift of organelle gene frequencies combine to produce a predominance of homoplasmic cells within individuals in the population. Then, the effective number of organelle genes is N(eo), a simple function of the numbers of males and females and of the maternal and paternal contributions to the zygote. Finally, when the paternal contribution is very small, N( eo) is closely approximated by the number of females, N( f). Then if the sex ratio is 1, the mean time to fixation or loss of new mutations is approximately two times longer for nuclear genes than for organelle genes, and gene diversity is approximately four times greater. The difference between nuclear and organelle genes disappears or is reversed in animals in which males have large harems. The differences between nuclear and organelle gene behavior caused by maternal inheritance and vegetative segregation are generally small and may be overshadowed by differences in mutation rates to neutral alleles. For monoecious organisms, the effective number of organelle genes is approximately equal to the total population size N. We also show that a population can be effectively subdivided for organelle genes at migration rates which result in panmixis for nuclear genes, especially if males migrate more than females.     

A stochastic model for estimation of mutation rates in multiple-replication proliferation processes

In this paper we propose a stochastic model based on the branching process for estimation and comparison of the mutation rates in proliferation processes of cells or microbes. We assume in this model that cells or microbes (the elements of a population) are reproduced by generations and thus the model is more suitably applicable to situations in which the new elements in a population are produced by older elements from the previous generation rather than by newly created elements from the same current generation. Cells and bacteria proliferate by binary replication, whereas the RNA viruses proliferate by multiple replication. The model is in terms of multiple replications, which includes the special case of binary replication. We propose statistical procedures for estimation and comparison of the mutation rates from data of multiple cultures with divergent culture sizes. The mutation rate is defined as the probability of mutation per replication per genome and thus can be assumed constant in the entire proliferation process. We derive the number of cultures for planning experiments to achieve desired accuracy for estimation or desired statistical power for comparing the mutation rates of two strains of microbes. We establish the efficiency of the proposed method by demonstrating how the estimation of mutation rates would be affected when the culture sizes were assumed similar but actually diverge.      

Gamete frequencies at two sex-linked loci in random mating age-structured populations

A study is made of the change with time of frequencies of gametic types with one or two sex-linked loci in an infinite random mating age-structured population. Recurrence equations for these gamete frequencies are derived under the assumptions that all matings of adults are equally fertile and the number of matings at any time is proportional to the number of mature females at that time. These generalize others in the literature. It is shown that gamete frequencies approach their limiting values at geometric rates in the long run. This implies that the asymptotic behavior of the gamete frequencies is like what it is in populations with discrete generations if the unit of time is replaced by an appropriately chosen generation interval. With either one locus or two loci, the generation interval is bounded below by an analogous measure from standard demographic theory. This result also holds when there are two autosomal loci. In numerical examples from both this paper and a previous one by Pollak and Callanan, the lower bound is a good estimate of the generation interval.