Simulation Study of a Multigene Family, with Special Reference to the Evolution of Compensatory Advantageous Mutations

We investigate the evolution of a multigene family incorporating the forces of drift, mutation, gene conversion, unequal crossing over and selection. The use of simulation studies is required due to the complexity of the model. Selection is modeled in two modes: positive selection as a function of the number of different beneficial alleles and negative selection against deleterious alleles. We assume that gene conversion is unbiased, and that all mutations are initially deleterious. Compensation between mutants creates beneficial and neutral alleles, and allowances are made for compensatory mutations either within or between the members of a multigene family. We find that gene conversion can enhance the rate of acquisition of compensatory advantageous mutations when genes are redundant.     

Interchromosomal Biased Gene Conversion, Mutation and Selection in a Multigene Family

A mathematical model of the effects of interchromosomal biased gene conversion, mutation and natural selection on a multigene family is developed and analyzed. The model assumes two allelic states at each of n loci. The effects of genetic drift are ignored. The model is developed under the assumption of no recombination, but the analysis shows that, at equilibrium, there is no linkage disequilibrium, which implies that the conclusions are valid for arbitrary recombination among loci. At equilibrium, the balance between mutation, gene conversion and selection depends on the ratio of the mutation rates to the quantity [s + g(2α - 1)/ n], where s is the increment or decrement in relative fitness with each additional copy of one of the alleles, g is the conversion rate, and α is a measure of the bias in favor of one of the alleles. When this quantity is large relative to the mutation rates, the allele that has the net advantage, combining the effects of selection and conversion, will be nearly fixed in the multigene family. A comparison of these results with those from a comparable model of intrachromosomal biased conversion shows that biased interchromosomal conversion leads to approximately the same equilibrium copy number as does intrachromosomal conversion of the same strength. Interchromosomal conversion is much more effective in causing the substitution of one allele by another. The relative frequencies of interchromosomal and intrachromosomal conversion is indicated by the extent of the linkage disequilibrium among the loci in a multigene family.     

The Evolution of Multigene Families under Intrachromosomal Gene Conversion

A model for the evolution of the probabilities of genetic identity within and between loci of a multigene family in a finite population is formulated and investigated. Unbiased intrachromosomal gene conversion, equal crossing over between tandemly repeated genes, random genetic drift and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Formulas for the equilibrium values of the probabilities of identity and a cubic equation for the rate of convergence are deduced. Numerical examples indicate the following. The amount of homology at equilibrium generally decreases as the mutation rate, the population size and the number of repeats increase; it may increase or decrease with increasing crossover rate. The intralocus homology has an intermediate minimum, whereas the interlocus homology increases, as the rate of gene conversion increases. The intralocus homology decreases, whereas the interlocus homology increases, as the proportion of symmetric heteroduplexes increases. The characteristic convergence time can be sufficiently short to imply that intrachromosomal gene conversion may be an important mechanism for maintaining sequence homogeneity among repeated genes. The convergence time decreases as the conversion rate and the proportion of symmetric heteroduplexes increase; although exceptions occur, it generally increases as the population size and the number of repeats increase; it may increase or decrease with increasing crossover rate.     

Intrachromosomal gene conversion, linkage, and the evolution of multigene families

The evolution of the probabilities of genetic identity within and between tandemly repeated loci of a multigene family is investigated analytically and numerically. Unbiased intrachromosomal 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. Under the restriction that there is at most one crossover in the multigene family per individual per generation, the dependence on location of the probabilities of identity is treated exactly. In the "homogeneous" approximation to this "exact" model, end effects are disregarded; in the "exchangeable" approximation, to which all previous work was confined, all position dependence is neglected. Numerical results indicate that the exchangeable and homogeneous models are both qualitatively correct, the exchangeable model is sometimes too inaccurate for quantitative conclusions, and the homogeneous model is always more accurate than the exchangeable one and is always sufficiently accurate for quantitative conclusions.     

DIFFERENTIATION OF A MULTIGENE FAMILY BETWEEN POPULATIONS

Differentiation of a multigene family between populations is investigated under the forces of intrachromosomal unbiased gene conversion, neutral mutation, recombination, and random drift. The infinite-site model without recombination within the gene is employed, and the average numbers of the different sites between pairs of genes at the same locus and at different loci from two populations that divided some time ago are calculated as functions of time. Since differentiation is a relative concept in which differences between populations are expressed in comparison with those within a population and since there is most likely a reduction in population size after the division of populations, the time-dependent behavior of the corresponding quantities within a population is also investigated using approximate and numerical methods. It is found that intrachromosomal unbiased gene conversion promotes differentiation at the same locus but promotes or retards that between different loci, depending on the parameters in a recently divided population.     

A method for estimating the mutation,gene conversion and recombination parameters in small multigene families

A simple two-locus gene conversion model is considered to investigate the amounts of DNA variation and linkage disequilibrium in small multigene families. The exact solutions for the expectations and variances of the amounts of variation within and between two loci are obtained. It is shown that gene conversion increases the amount of variation within each locus and decreases the amount of variation between two loci. The expectation and variance of the amount of linkage disequilibrium are also obtained. Gene conversion generates positive linkage disequilibrium and the degree of linkage disequilibrium decreases as the recombination rate is increased. Using the theoretical results, a method for estimating the mutation, gene conversion, and recombination parameters is developed and applied to the data of the Amy multigene family in Drosophila melanogaster. The gene conversion rate is estimated to be approximately 60-165 times higher than the mutation rate for synonymous sites.     

Role of biased gene conversion in one-locus neutral theory and genome evolution

The implications of biased gene conversion acting on selectively neutral alleles are investigated for a single diallelic locus in a finite population. Even a very slight conversion bias can significantly alter fixation probabilities. We argue that most newly arising mutants will be at a conversion disadvantage, resulting in a potentially greatly decreased substitution rate of new alleles compared with predictions from strict neutral theory. Thus, conversion bias potential allows for conservation of particular alleles without having to invoke selection. Conversely, we also show that bias can be important in the maintenance of repeated gene families without altering the substitution rate at other loci that experience the same amount of conversion bias, provided that the number of genes in the family is sufficiently large. Bias can, therefore, be important at the genomic level and yet be unimportant at the populational level. Finally, we discuss the role of biased gene conversion in speciation events, concluding that this type of molecular turnover acting independently at many individual loci is very unlikely to decrease the time required for two allopatric populations to speciate.     

Variation after a selective sweep in a subdivided population

The effect of genetic hitchhiking on neutral variation is analyzed in subdivided populations with differentiated demes. After fixation of a favorable mutation, the consequences on particular subpopulations can be radically different. In the subpopulation where the mutation first appeared by mutation, variation at linked neutral loci is expected to be reduced, as predicted by the classical theory. However, the effect in the other subpopulations, where the mutation is introduced by migration, can be the opposite. This effect depends on the level of genetic differentiation of the subpopulations, the selective advantage of the mutation, the recombination frequency, and the population size, as stated by analytical derivations and computer simulations. The characteristic outcomes of the effect are three. First, the genomic region of reduced variation around the selected locus is smaller than that predicted in a panmictic population. Second, for more distant neutral loci, the amount of variation increases over the level they had before the hitchhiking event. Third, for these loci, the spectrum of gene frequencies is dominated by an excess of alleles at intermediate frequencies when compared with the neutral theory. At these loci, hitchhiking works like a system that takes variation from the between-subpopulation component and introduces it into the subpopulations. The mechanism can also operate in other systems in which the genetic variation is distributed in clusters with limited exchange of variation, such as chromosome arrangements or genomic regions closely linked to targets of balancing selection.     

Allele frequencies in multigene families. I. Diffusion equation approach

Simulation studies by Ohta, and theoretical studies by Shimizu, indicated that the alleles in one chromosome multigene family follow Ewens' sampling distribution at stationarity. The reasons for this are explored, and the theory is extended to obtain transient and stationary results for the sampling distribution. Expressions are obtained for the amount of variation of numbers of alleles across the chromosomes in a population. The population model includes mutation, and gene conversion, but not recombination.     

Germ line maintenance of the pseudogene donor pool for somatic immunoglobulin gene conversion in chickens.

Somatic immunoglobulin diversity is generated in avian species by sequential gene conversion of variable (V) gene segments of the immunoglobulin heavy- and light-chain loci during B-cell development. The germ line pools of donor sequence information for somatic V-region gene conversion are found in families of V pseudogenes, located 5' of the single functional V gene of each locus. The sequence relationships among the pseudogenes (psi VL) and functional VL1 gene of the chicken light-chain alleles in three inbred strains were compared to determine the extent of diversity within the germ line pseudogene cluster. Numerous differences were observed. For example, compared with the previously reported CB allele and the G4 allele, the S3 allele contains two intact pseudogenes between psi VL16 and psi VL18. These two adjacent psi VL gene segments (psi VL17a and psi VL17b) could have given rise to the psi VL17 segment of the G4 and CB alleles by homologous recombination. The majority of other sequence polymorphisms among the psi VL alleles appear to be the result of meiotic gene conversion. The incidence of untemplated mutations within psi VL segments is significantly lower than the incidence of mutation within the pseudogene flanking regions. Together with the observations that most psi VL segments have open reading frames and lack stop codons, these data support the hypothesis that the psi VL cluster resembles a functional multigene family maintained by evolutionary selection for its functional role in generating somatic antibody diversity. Meiotic gene conversion events within the psi VL cluster serve both to introduce diversity by the exchange of short segments between family members and to prevent the accumulation of random mutations.     

Selection and Biased Gene Conversion in a Multigene Family: Consequences of Interallelic Bias and Threshold Selection

In a previous paper, I investigated the interactions in a gene family of additive selection and biased gene conversion in a finite population when conversion events are rare. Here I extend my "weak-conversion limit" model by allowing biased interallelic conversion (conversion between alleles at the same locus) of arbitrary frequency and various threshold selection schemes for rare interlocus conversion events. I suggest that it is not unreasonable for gene families to experience threshold fitness functions, and show that certain types of thresholds can greatly constrain the rate at which advantageous alleles are fixed as compared to other fitness schemes, such as additive selection. It is also shown that the double sampling process operating on a gene family in a finite population (sampling over the number of genes in the gene family and over the number of individuals in the population) can have interesting consequences. For selectively neutral alleles that experience interallelic bias, the probability of fixation at each single locus may be essentially neutral, but the cumulative effects on the entire gene family of small departures from neutrality can be significant, especially if the gene family is large. Thus, in some situations, gene families can respond to directional forces that are weak in comparison to drift at single loci.     

Theoretical population genetics of repeated genes forming a multigene family

The evolution of repeated genes forming a multigene family in a finite population is studied with special reference to the probability of gene identity, i.e., the identity probability of two gene units chosen from the gene family. This quantity is called clonality and is defined as the sum of squares of the frequencies of gene lineages in the family. The multigene family is undergoing continuous unequal somatic crossing over, ordinary interchromosomal crossing over, mutation and random frequency drift. Two measures of clonality are used: clonality within one chromosome and that between two different chromosomes. The equilibrium properties of the means, the variances and the covariance of the two measures of clonality are investigated by using the diffusion equation method under the assumption of constant number of gene units in the multigene family. Some models of natural selection based on clonality are considered. The possible significance of the variance and covariance of clonality among the chromosomes on the adaptive differentiation of gene families such as those producing antibodies is discussed.     

Genetic Differentiation of Transposable Elements under Mutation and Unbiased Gene Conversion

A model is developed to predict the extent of genetic differentiation in a family of transposable elements under the combined effects of genetic drift, transposition, mutation and unbiased gene conversion. The model is based on simplifying assumptions that are valid when transposition is always to new sites and copy number per site is low. In the absence of gene conversion, the degree of differentiation as measured by the probability of identity of different elements is the same as at a single locus with the same mutation rate but in a population of effective size Nc/2, where N is the population size and c is the number of copies per individual. The inclusion of unbiased gene conversion does not significantly change this result. If, as seems to be the case, families of transposable elements are relatively homogeneous, then the model implies either that mutation rates for transposable elements are much lower than at comparable single-copy loci or that some other force, such as natural selection or biased gene conversion, is at work. Transposition is a very ineffective force for homogenizing a family of transposable elements.     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

Two patterns of variation among MHC class I loci in Tuatara (Sphenodon punctatus)

The genes of the major histocompatibility complex (MHC) are a central component of the immune system in vertebrates and have become important markers of functional, fitness-related genetic variation. We have investigated the evolutionary processes that generate diversity at MHC class I genes in a large population of an archaic reptile species, the tuatara (Sphenodon punctatus), found on Stephens Island, Cook Strait, New Zealand. We identified at least 2 highly polymorphic (UA type) loci and one locus (UZ) exhibiting low polymorphism. The UZ locus is characterized by low nucleotide diversity and weak balancing selection and may be either a nonclassical class I gene or a pseudogene. In contrast, the UA-type alleles have high nucleotide diversity and show evidence of balancing selection at putative peptide-binding sites. Twenty-one different UA-type genotypes were identified among 26 individuals, suggesting that the Stephens Island population has high levels of MHC class I variation. UA-type allelic diversity is generated by a mixture of point mutation and gene conversion. As has been found in birds and fish, gene conversion obscures the genealogical relationships among alleles and prevents the assignment of alleles to loci. Our results suggest that the molecular mechanisms that underpin MHC evolution in nonmammals make locus-specific amplification impossible in some species.     

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.     

Transition stages of molecular drive in multiple-copy DNA families in Drosophila

Multigene and non-genic DNA families are in a state of turnover and hence are continually being replaced throughout a population by new variant repeats. To quantify such molecular processes, in the absence of selection, it is necessary to find and compare stages of transistion during the homogenization of at least two non-genic families evolving in parallel in a closely related group of species. Detailed sequence analysis of patterns of variation, at each nucleotide position considered independently, amongst repeats of two tandem DNA families from seven related Drosophila species, reveals all stages of transition during the spread of randomly produced variant repeats. Variant repeats are found at different stages of homogenization and fixation in a population, irrespective of the loci, chromosomes or individuals from which they were cloned. Differences between the families in the relatively small number of variants at each transition stage and the greater number of fully homogenized and fixed variants between species of greater divergence indicate that the process of spread (molecular drive) is rapid relative to the mutation rate and occurs at seemingly different constant rates for each family. Occasional gene conversions, in addition to unequal exchanges, have contributed to family turnover. The significance of these results to the evolution of functional multigene families and divergence and conservation of sequences is discussed.     

Allele frequencies in multigene families. II. Coalescent approach

Shimizu's model for the evolution of multigene families, subject to gene conversion and mutation, is studied. The probability distribution for the allelic configuration of two chromosomes is found, by using "coalescent" methods. For the two chromosomes, moments are found for the number of alleles they have in common and for their total number of alleles, together with the probability that there is only one allele present. Some numerical examples are given.     

Neutral mutation as the source of genetic variation in life history traits

The mechanism underlying the maintenance of adaptive genetic variation is a long-standing question in evolutionary genetics. There are two concepts (mutation-selection balance and balancing selection) which are based on the phenotypic differences between alleles. Mutation - selection balance and balancing selection cannot properly explain the process of gene substitution, i.e. the molecular evolution of quantitative trait loci affecting fitness. I assume that such loci have non-essential functions (small effects on fitness), and that they have the potential to evolve into new functions and acquire new adaptations. Here I show that a high amount of neutral polymorphism at these loci can exist in real populations. Consistent with this, I propose a hypothesis for the maintenance of genetic variation in life history traits which can be efficient for the fixation of alleles with very small selective advantage. The hypothesis is based on neutral polymorphism at quantitative trait loci and both neutral and adaptive gene substitutions. The model of neutral - adaptive conversion (NAC) assumes that neutral alleles are not neutral indefinitely, and that in specific and very rare situations phenotypic (relative fitness) differences between them can appear. In this paper I focus on NAC due to phenotypic plasticity of neutral alleles. The important evolutionary consequence of NAC could be the increased adaptive potential of a population. Loci responsible for adaptation should be fast evolving genes with minimally discernible phenotypic effects, and the recent discovery of genes with such characteristics implicates them as suitable candidates for loci involved in adaptation.     

Linkage and the Limits to Natural Selection

The probability of fixation of a favorable mutation is reduced if selection at other loci causes inherited variation in fitness. A general method for calculating the fixation probability of an allele that can find itself in a variety of genetic backgrounds is applied to find the effect of substitutions, fluctuating polymorphisms, and deleterious mutations in a large population. With loose linkage, r, the effects depend on the additive genetic variance in relative fitness, var (W), and act by reducing effective population size by (N/N(e)) = 1 + var (W)/2r(2). However, tightly linked loci can have a substantial effect not predictable from N(e). Linked deleterious mutations reduce the fixation probability of weakly favored alleles by exp(-2U/R), where U is the total mutation rate and R is the map length in Morgans. Substitutions can cause a greater reduction: an allele with advantage s < s(crit) = (π(2)/6) log(e) (S/s)[var(W)/R] is very unlikely to be fixed. (S is the advantage of the substitution impeding fixation.) Fluctuating polymorphisms at many (n) linked loci can also have a substantial effect, reducing fixation probability by exp [ &2Kn var(W)/R] [K = -1/E((u - u)(2)/uv) depending on the frequencies (u,v) at the selected polymorphisms]. Hitchhiking due to all three kinds of selection may substantially impede adaptation that depends on weakly favored alleles.