Variances and covariances of squared linkage disequilibria in finite populations

Analysis of linkage disequilibrium D among restriction sites or bases in DNA sequences, arising from mutations in finite populations, depends on a knowledge of the variance-covariance structure of measures such as D2 between different pairs of sites. This requires evaluation of the eighth moments of gene frequencies among two, three, and four loci, and the necessary methodology is derived here and results are computed. While primary emphasis is placed on disequilibrium arising from mutation or gene conversion, the methodology also allows for the joint effects of only drift and recombination. Numerical results confirm that squared linkage disequilibria can have high variances and covariances.     

Recombination, gene conversion, and identity-by-descent at three loci

We investigate the probabilities of identity-by-descent at three loci in order to find a signature which differentiates between the two types of crossing over events: recombination and gene conversion. We use a Markov chain to model coalescence, recombination, gene conversion and mutation in a sample of size two. Using numerical analysis, we calculate the total probability of identity-by-descent at the three loci, and partition these probabilities based on a partial ordering of coalescent events at the three loci. We use these results to compute the probabilities of four different patterns of conditional identity and non-identity at the three loci under recombination and gene conversion. Although recombination and gene conversion do make different predictions, the differences are not likely to be useful in distinguishing between them using three locus patterns between pairs of DNA sequences. This implies that measures of genetic identity in larger samples will be needed to distinguish between gene conversion and recombination.     

An Extension of a Model for the Evolution of Multigene Families by Unequal Crossing over

Evolution of a multigene family is studied from the standpoint of population genetics. It is assumed that the multigene family is undergoing continuous interchromosomal unequal crossing over, mutation and random frequency drift. The equilibrium properties of the probability of gene identity (clonality) are investigated, using two measures: identity probability within and between chromosomes. The measures represent homogeneity of genes within a family in one chromosome and similarity of gene families between two homologous chromosomes. The means, the variances and the covariance of these two measures of identity probability are obtained by using the diffusion equation method. It is shown that the means and the variances are generally smaller than those predicted in the previous model assuming intrachromosomal (sister chromatid) unequal crossing over (Ohta 1978a,b).     

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.     

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.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

Gene Conversion, Linkage, and the Evolution of Repeated Genes Dispersed among Multiple Chromosomes

The evolution of the probabilities of genetic identity within and between the loci of a multigene family dispersed among multiple chromosomes 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, but the same for every chromosome; the dependence of the probabilities of identity on the location on each chromosome is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. Under the assumption of loose linkage (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), explicit approximations are obtained for the equilibrium values of the probabilities of identity and of the linkage of disequilibria. The probabilities of identity are of order one [i.e., O(1)] and do not depend on location; the linkage disequilibria are of O(epsilon) and, within each chromosome, depend on location through the crossover rates. It is demonstrated also that the ultimate rate and pattern of convergence to equilibrium are close to that of a much simpler, location-independent model. If intrachromosomal conversion is absent, the above results hold even without the assumption of loose linkage. In all cases, the relative errors are of O(epsilon). Even if the conversion rate between genes on nonhomologous chromosomes is considerably less than between genes on the same chromosome or homologous chromosomes, the probabilities of identity between the former genes are still almost as high as those between the latter, and the rate of convergence is still not much less than with equal conversion rates. If the crossover rates are much less than 1/2, then most of the linkage disequilibrium is due to intrachromosomal conversion. If linkage is loose, the reduction of the linkage disequilibria to O(epsilon) requires only O(-ln epsilon) generations.     

A Defect in Mismatch Repair in Saccharomyces Cerevisiae Stimulates Ectopic Recombination between Homeologous Genes by an Excision Repair Dependent Process

Null mutations in three recombination and DNA repair genes were studied to determine their effects on mitotic recombination between the duplicate AdoMet (S-adenosylmethionine) synthetase genes (SAM1 and SAM2) in Saccharomyces cerevisiae. SAM1 and SAM2, located on chromosomes XII and IV, respectively, encode functionally equivalent although differentially regulated AdoMet synthetases. These similar but not identical (homeologous) genes are 83% homologous at the nucleotide level and this identity is limited solely to the coding regions of the genes. Single frameshift mutations were introduced into the 5' end of SAM1 and the 3' end of SAM2 by restriction site ablation. The sequences surrounding these mutations differ significantly in their degree of homology to the corresponding area of the other gene. Mitotic ectopic recombination between the mutant sam genes occurs at a rate of 8.4 x 10(-9) in a wild-type genetic background. Gene conversion of the marker within the region of greater sequence homology occurs 20-fold more frequently than conversion of the marker within the region of relative sequence diversity. The relative orientation of the two genes prevents the recovery of translocations. Mitotic recombination between the sam genes is completely dependent on the DNA repair and recombination gene RAD52. A mutation in PMS1, a mismatch repair gene, causes a 4.5-fold increase in the rate of ectopic recombination. RAD1, an excision repair gene, is required to observe this increased rate of ectopic conversion. In addition, RAD1 is involved in modulating the pattern of coconversion during recombination between the homeologous sam genes. These results suggest that interactions between mismatch repair, excision repair and recombinational repair functions are involved in determining the ectopic gene conversion frequency between the sam genes.     

On the divergence of genes in multigene families

Statistical properties of the amount of divergence of genes in multigene families are studied. The model considered is an infinite-site neutral model with unbiased intrachromosomal conversion, unbiased interchromosomal conversion, and recombination. By considering the time back to the most recent common ancestor of two genes, both the probability of identity and the moments of S, the number of sites that differ between two sampled genes, are obtained. We find that if recombination rates are large or conversion is always interchromosomal, then the expectation of S is 4N mu n where N is the population size, mu is the rate of mutation per generation per gene and n is the number of genes in the gene family, as the conversion rates approach zero, the moments of divergence do not approach the moments of divergence with conversion rates equal to zero, and it is possible for a decrease in the rate of intrachromosomal conversion to result in a higher probability of identity, but a greater mean divergence of the two genes.     

Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

High-frequency germ line gene conversion in transgenic mice.

Gene conversion is the nonreciprocal transfer of genetic information between two related genes or DNA sequences. It can influence the evolution of gene families, having the capacity to generate both diversity and homogeneity. The potential evolutionary significance of this process is directly related to its frequency in the germ line. While measurement of meiotic inter- and intrachromosomal gene conversion frequency is routine in fungal systems, it has hitherto been impractical in mammals. We have designed a system for identifying and quantitating germ line gene conversion in mice by analyzing transgenic male gametes for a contrived recombination event. Spermatids which undergo the designed intrachromosomal gene conversion produce functional beta-galactosidase (encoded by the lacZ gene), which is visualized by histochemical staining. We observed a high incidence of lacZ-positive spermatids (approximately 2%), which were produced by a combination of meiotic and mitotic conversion events. These results demonstrate that gene conversion in mice is an active recombinational process leading to nonparental gametic haplotypes. This high frequency of intrachromosomal gene conversion seems incompatible with the evolutionary divergence of newly duplicated genes. Hence, a process may exist to uncouple gene pairs from frequent conversion-mediated homogenization.     

Neutral evolution of multiple quantitative characters: a genealogical approach

The G matrix measures the components of phenotypic variation that are genetically heritable. The structure of G, that is, its principal components and their associated variances, determines, in part, the direction and speed of multivariate trait evolution. In this article we present a framework and results that give the structure of G under the assumption of neutrality. We suggest that a neutral expectation of the structure of G is important because it gives a null expectation for the structure of G from which the unique consequences of selection can be determined. We demonstrate how the processes of mutation, recombination, and drift shape the structure of G. Furthermore, we demonstrate how shared common ancestry between segregating alleles shapes the structure of G. Our results show that shared common ancestry, which manifests itself in the form of a gene genealogy, causes the structure of G to be nonuniform in that the variances associated with the principal components of G decline at an approximately exponential rate. Furthermore we show that the extent of the nonuniformity in the structure of G is enhanced with declines in mutation rates, recombination rates, and numbers of loci and is dependent on the pattern and modality of mutation.     

Tension versus ecological zones in a two-locus system

Previous theories show that tension and ecological zones are indistinguishable in terms of gene frequency clines. Here I analytically show that these two types of zones can be distinguished in terms of genetic statistics other than gene frequency. A two-locus cline model is examined with the assumptions of random mating, weak selection, no drift, no mutation, and multiplicative viabilities. The genetic statistics for distinguishing the two types of zones are the deviations of one- or two-locus genotypic frequencies from Hardy-Weinberg equilibrium (HWE) or from random association of gametes (RAG), and the deviations of additive and dominance variances from the values at HWE. These deviations have a discontinuous distribution in space and different extents of interruptions in the ecological zone with a sharp boundary, but exhibit a continuous distribution in the tension zone. Linkage disequilibrium enhances the difference between the deviations from HWE and from RAG for any two-locus genotypic frequency.     

The Role of DNA Repair Genes in Recombination between Repeated Sequences in Yeast

The presence of repeated sequences in the genome represents a potential source of karyotypic instability. Genetic control of recombination is thus important to preserve the integrity of the genome. To investigate the genetic control of recombination between repeated sequences, we have created a series of isogenic strains in which we could assess the role of genes involved in DNA repair in two types of recombination: direct repeat recombination and ectopic gene conversion. Naturally occurring (Ty elements) and artificially constructed repeats could be compared in the same cell population. We have found that direct repeat recombination and gene conversion have different genetic requirements. The role of the RAD51, RAD52, RAD54, RAD55, and RAD57 genes, which are involved in recombinational repair, was investigated. Based on the phenotypes of single and double mutants, these genes can be divided into three functional subgroups: one composed of RAD52, a second one composed of RAD51 and RAD54, and a third one that includes the RAD55 and RAD57 genes. Among seven genes involved in excision repair tested, only RAD1 and RAD10 played a role in the types of recombination studied. We did not detect a differential effect of any rad mutation on Ty elements as compared to artificially constructed repeats.     

Sequence Identity in an Early Chorion Multigene Family Is the Result of Localized Gene Conversion

The multigene families that encode the chorion (eggshell) of the silk moth, Bombyx mori, are closely linked on one chromosome. We report here the isolation and characterization of two segments, totaling 102 kb of genomic DNA, containing the genes expressed during the early period of choriogenesis. Most of these early genes can be divided into two multigene families, ErA and ErB, organized into five divergently transcribed ErA/ErB gene pairs. Nucleotide sequence identity in the major coding regions of the ErA genes was 96%, while nucleotide sequence identity for the ErB major coding regions was only 63%. Selection pressure on the encoded proteins cannot explain this difference in the level of sequence conservation between the ErA and ErB gene families, since when only fourfold redundant codon positions are considered, the divergence within the ErA genes is 8%, while the divergence within the ErB genes (corrected for multiple substitutions at the same site) is 110%. The high sequence identity of the ErA major exons can be explained by sequence exchange events similar to gene conversion localized to the major exon of the ErA genes. These gene conversions are correlated with the presence of clustered copies of the nucleotide sequence GGXGGX, encoding paired glycine residues. This sequence has previously been correlated with gradients of gene conversion that extend throughout the coding and noncoding regions of the High-cysteine (Hc) chorion genes of B. mori. We suggest that the difference in the extent of the conversion tracts in these gene families reflects a tendency for these recombination events to become localized over time to the protein encoding regions of the major exons.     

Analysis of linkage disequilibrium in an island model

Linkage disequilibria for two loci in a finite island model were parameterized. The total linkage disequilibrium was decomposed into three components, gametic, demic, and population, for which corresponding unbiased estimators were established. Other statistics encountered provided measures of differentiation corresponding to the hierarchical structure of the ecological model. Under the assumption of linkage equilibrium, the variances and covariances of these estimators and statistics were formulated in terms of descent measures, functions of gene frequencies, and the numbers of individuals, demes, and populations sampled. The functions of gene frequencies fall into two classes, one representing the differentiation of genes at each locus, and the other representing the association of genes between the loci. For a neutral model with extinction, migration, and linkage, transition equations were derived for the descent measures which also take into account deme size and numbers of dems within the population. With the addition of unequal mutation rates for a finite number of alleles at each locus, the transition equations were solved for the descent measures in the equilibrium state. This permitted the exact numerical evaluation of the effects of the sampling and ecological dimensions and of extinction, migration, and mutation rates in any parameter range. Some numerical results were presented for the effects of linkage, extinction, migration, and sampling on the variances of various measures of linkage disequilibrium and genetic differentiation. Also, some results were compared with the approximate numerical results of Ohta which agreed fairly well in the parameter ranges she considered, but not so well in other ranges.     

Gene Conversions and Crossing over during Homologous and Homeologous Ectopic Recombination in Saccharomyces Cerevisiae

The pma1-105 mutation reduces the activity of the yeast plasma membrane H(+)-ATPase and causes cells to be both low pH and ammonium ion sensitive and resistant to the antibiotic hygromycin B. Revertants that can grow at pH 3.0 and on ammonium-containing plates frequently arise by ectopic recombination between pma1-105 and PMA2, a diverged gene that shares 85% DNA sequence identity with PMA1. The gene conversion tracts of revertants of pma1-105 were determined by DNA sequencing the hybrid PMA1::PMA2 genes. Gene conversion tracts ranged from 18-774 bp. The boundaries of these replacements were short (3-26 bp) regions of sequences that were identical between PMA1 and PMA2. These boundaries were not located at the regions of greatest shared identity between the two PMA genes. Similar results were obtained among low pH-resistant revertants of another mutation, pma1-147. One gene conversion was obtained in which the resulting PMA1::PMA2 hybrid was low pH-resistant but still hygromycin B-resistant. This partially active gene differs from a wild-type revertant only by the presence of two PMA2-encoded amino acid substitutions. Thus, some regions of PMA2 are not fully interchangeable with PMA1. We have also compared the efficiency of recombination between pma1-105 and either homeologous PMA2 sequence or homologous PMA1 donor sequences inserted at the same location. PMA2 X pma1-105 recombination occurred at a rate approximately 75-fold less than PMA1 X pma1-105 events. The difference in homology between the interacting sequences did not affect the proportion of gene conversion events associated with a cross-over, as in both cases approximately 5% of the Pma(+) recombinants had undergone reciprocal translocations between the two chromosomes carrying pma1-105 and the donor PMA sequences. Reciprocal translocations were identified by a simple and generally useful nutritional test.     

Factors affecting ectopic gene conversion in mice

Duplicated genes and repetitive sequences are distributed throughout the genomes of complex organisms. The homology between related sequences can promote nonallelic (ectopic) recombination, including gene conversion and reciprocal exchange. Resolution of these events can result in translocations, deletions, or other harmful rearrangements. In yeast, ectopic recombination between sequences on nonhomologous chromosomes occurs at high frequency. Because the mammalian genome is replete with duplicated sequences and repetitive elements, high levels of ectopic exchange would cause aneuploidy and genome instability. To understand the factors regulating ectopic recombination in mice, we evaluated the effects of homology length on gene conversion between unlinked sequences in the male germline. Previously, we found high levels of gene conversion between lacZ transgenes containing 2557 bp of homology. We report here that genetic background can play a major role in ectopic recombination; frequency of gene conversion was reduced by more than an order of magnitude by transferring the transgenes from a CF1 strain background to C57BL/6J. Additionally, conversion rates decreased as the homology length decreased. Sequences sharing 1214 bp of sequence identity underwent ectopic conversion less frequently than a pair sharing 2557 bp of identity, while 624 bp was insufficient to catalyze gene conversion at significant levels. These results suggest that the germline recombination machinery in mammals has evolved in a way that prevents high levels of ectopic recombination between smaller classes of repetitive sequences, such as the Alu family. Additionally, genomic location appeared to influence the availability of sequences for ectopic recombination. Received: 12 September 1997 / Accepted: 29 December 1997     

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.