Monoecious Population Research Articles

Linkage disequilibria in finite populations

Four-locus recombination frequencies are summarized into two-locus pair frequencies and three-locus frequencies, and further, into two-locus frequencies such that higher-order frequencies are linear functions of lower-order frequencies. Frequencies of gene combinations are defined according to their position on the same or distinct gametes, and linear functions of these provide the measures of linkage disequilibria. These concepts are utilized to derive the transitional behavior of the gene combination frequencies and the linkage disequilibria in a finite monoecious population with random union of gametes for up to four loci. The transitions of lower-order disequilibria in a higher-order (more loci) setting involve the higher-order disequilibria which must be taken into account in arriving at the final (fixation) frequencies. The methods allow different initial conditions. Since corresponding data functions of the gene combination frequencies provide unbiased estimates of the parameters, estimators follow naturally.

Allele protectedness in frequency dependent biallelic selection models with separated generations

Gametic (Darwinian) and the pertaining fractional fitnesses are proposed as a natural means of analysing and characterizing population genetic problems on a more general level. This is demonstrated by treating the problem of protectedness of an allele in biallelic, frequency dependent selection models. Both types of fitnesses are given for the “classical” viability selection model for random mating monoecious populations, and it is pointed out that the gametic fitnesses are proportional to the viabilities. The “classical” principle of protectedness, namely that a mutant allele may or may not become established according to whether its heterozygote viability does or does not exceed that of the wild type homozygote, served as the basis for generalization. It is shown that this principle, when applied to the gametic fitnesses, can indeed be generalized to cover almost arbitrary frequency depend selection, provided the carriers of the mutant allele do not mate positively assortatively or self-fertilize (show no “reproductive self-preferences”). Herewith, “reproductive self-preference” is measured by means of the fractional fitnesses. If the mutant homozygote or the heterozygote show reproductive self-preferences, then, irrespective of their intensities, the mutant allele may become established if, at the outset, the heterozygote has twice the gametic fitness of the wild type homozygote. Otherwise, the fractional fitnesses and the gametic fitness of the mutant homozygote become decisive. Some of the results are illustrated with the help of a known selection model involving positive assortative mating.

Evolution of multigene families under interchromosomal 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 interchromosomal 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. Formulae for the equilibrium values of the probabilities of identity and for the rate of convergence are deduced. At equilibrium, the amount of intralocus homology, f, always exceeds the amount of interlocus homology, ĝ. The equilibrium homologies f and ĝ and the characteristic convergence time T are independent of the crossover rate. As the population size and the number of repeats increase, f and ĝ decrease and T increases; as the rate of gene conversion increases, f and T decrease whereas ĝ increases. The time T can be sufficiently short to imply that interchromosomal gene conversion may be an important mechanism for maintaining sequence homogeneity among repeated genes.

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.

Drift and mutation with a finite number of allelic states

The frequencies with which alleles are alike within (Q) and between (q) populations are formulated for monoecious populations undergoing drift and mutation with unequal mutation rates among alleles for a finite number (k) of allelic states. The effective number of alleles and an application to Nei's measure of genetic distance [Nei, M. (1972) Am. Nat. 106, 283-292] are also considered for this model. The equilibrium values of Q and q increase as k decreases. Unequal mutation rates further increase the equilibrium values and reduce the rates of approach of Q and q to these values. The transitional values of Q and q are very dependent on the initial population frequency composition when mutation rates are unequal. Reducing k, of course, reduces the effective number of alleles, which is further reduced by unequal mutation rates. Complications introduced by initial population composition, unequal mutation rates, and number of allelic states, coupled with data limitations for long-term measures of genetic distance or population differentiation, with mutation as the main driving force, are discussed.

ESTIMATION OF THE COANCESTRY COEFFICIENT: BASIS FOR A SHORT-TERM GENETIC DISTANCE

A distance measure for populations diverging by drift only is based on the coancestry coefficient theta, and three estimators of the distance D = -ln(1 - theta) are constructed for multiallelic, multilocus data. Simulations of a monoecious population mating at random showed that a weighted ratio of single-locus estimators performed better than an unweighted average or a least squares estimator. Jackknifing over loci provided satisfactory variance estimates of distance values. In the drift situation, in which mutation is excluded, the weighted estimator of D appears to be a better measure of distance than others that have appeared in the literature.

Evolution of a finite population under gene conversion.

Evolution at a multiallelic locus under the joint action of gene conversion, mutation, selection, and random genetic drift is studied. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Under the assumption that all four evolutionary forces are weak, a diffusion approximation is established for the dynamics of the gene frequencies. For two alleles, the inclusion of gene conversion merely alters one of the two selection parameters of the thoroughly investigated diffusion process without conversion. Therefore, all results for this classical process, some of which are reviewed and extended here, are immediately applicable to the biologically more general problem. Small conversional disparities can dramatically affect the fixation probability (and hence the rate of gene substitution) and can greatly reduce the mean conditional fixation time of a new mutant. The mean absorption and fixation times are often sufficiently short to imply that biased gene conversion can be an important mechanism for the loss of genetic variability in and the genetic divergence of isolated populations.

Evolution of a large population under gene conversion.

The dynamics of allelic frequencies at a single multiallelic locus under gene conversion is studied. Generations are discrete and nonoverlapping; the diploid monoecious population mates at random; selection, mutation, and random drift are negligible. Analytical and numerical investigation indicates the following. (i) If gene conversion is biased within at least one pair of alleles, then the frequency of at least one allele must become arbitrarily small. (ii) If conversion is biased within every pair of alleles, then the frequency of at most one allele can fail to become arbitrarily small. Although allelic frequencies may become repeatedly small instead of remaining small, the biological ubiquity of small random perturbations (due, e.g., to random genetic drift) guarantees the ultimate loss of at least one allele in case i and of all alleles but one in case ii. The decay of genetic variability is often sufficiently rapid to imply that biased gene conversion can be an important mechanism for the genetic divergence of isolated populations.

Geographical invariance in population genetics

Various quantities of evolutionary interest are shown to be independent of geographical structure. A diploid, monoecious population is subdivided into a finite number of panmictic colonies that exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and non-overlapping; the analysis is restricted to a single locus. Previous results are generalized in the neutral multiallelic case. With selection, it is assumed that there are only two alleles, dominance is absent, selection has the same intensity in all demes, migration does not change the subpopulation numbers, and all evolutionary forces are weak. A diffusion approximation is established for the gene frequencies, and the invariance of the fixation probability and of the moments of the conditional and unconditional total heterozygosities before absorption is demonstrated by a martingale argument.

The rate of mutant substitution in populations with overlapping generations.

SUMMARYWe consider an age-structured population that is observed at timest= 0, 1, 2,…. It is assumed that for eachtthere is the same number of individuals of a particular sex and age group. Another assumption we make is that an offspring of a specified sex has at all times the same probability of having a parent of a particular age and sex. It is shown that the rate of substitution of neutral mutants is/L, whereandLare respectively equal to the mean fraction of mutants among gametes succeeding in forming newborn individuals and the mean age of reproduction. This result also applies to monoecious populations. The substitution rate is also derived for advantageous mutants in a monoecious population. Once again, the mutation rate in the usual expression is replaced by/L. Implications of these results are discussed.

Is the between-population variance negligible in the total variance of heterozygosity? Case of a finite number of loci subject to the infinite-allele model in finite monoecious populations

The decomposition of the variance of the average heterozygosity into variances between and within populations is studied in the general case of a finite number of loci. These loci are assumed randomly distributed over chromosome pairs having a non-interference recombination scheme, and independently subject to mutation according to the infinite-allele model. The equilibrium behavior of that decomposition is discussed in the monoecious mating case with regard to each parameter of the model: mutation rate per gene per generation ( u), population size ( N), number of loci ( n), map length of chomosome pairs ( L). It is shown that the proportion Q of the between-population variability in the total variance of the average heterozygosity is decreasing as either the mean heterozygosity ( θ = 4 Nu/(1 + 4 Nu)) or the mean number of mutations per gamete per generation ( v = nu) is increasing. Moreover, even if Q is always smaller than 1 3 for this model, it is not negligible unless θ is close to one or v is much larger than one for L long enough.

Effect of mating structure on variation in inbreeding

An analysis is made of the variation among individuals in finite populations of the proportion of their genes which are identical by descent. There are two causes of this variability: variation in pedigree among individuals, and linkage which causes whole blocks of genes to be identical or nonidentical by descent. The variation between and within populations is analyzed in detail for several mating systems: monoecious populations with and without random selfing, and dioecious populations with and without a hierarchical mating structure. Transition matrices for two-locus descent measures are given for each system. Total variability is obtained by integrating these measures over the distribution of map distances over whole chromosomes. Approximate methods are also developed for unlinked loci. Unless populations have a very small effective size ( N e ) there is little variation in inbreeding between populations. For unlinked loci, the coefficient of variation in nonidentity within populations approaches about 1 √3N e for random selfing, 1 √6N e for monogamous matings and 1 √12N e for monoecy with selfing excluded or dioecy with random pairing. If there is no association between map distance and initial heterozygosity or effect on quantitative traits, the coefficient of variation in mean heterozygosity over the genome is related to that in nonidentity, and the additional variation in a quantitative trait due to dominant genes equals the product of the square of the initial inbreeding depression and the squared coefficient of variation of nonidentity.

Diffusion model for genotype-dependent migration

A diffusion model for the joint action of genotype-dependent migration and genotype-independent population regulation is derived and investigated. The diploid, monoecious population occupies a finite chain of equally spaced colonies; generations are discrete and nonoverlapping; mating is random in each colony. A boundary-value problem is deduced for the gene frequencies at a multiallelic locus in the absence of mutation, selection, and random drift. This problem is studied for two alleles and constant and uniform population density and drift and diffusion coefficients. All equilibria are shown to be monotone, and explicit conditions for a protected polymorphism are established. Two examples of asymptotically stable clines are presented. It is demonstrated that genotype dependence of the mean displacements is necessary to produce or maintain spatial differentiation of gene frequencies.

Effect of mating structure on variation in linkage disequilibrium.

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (Ne) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium. --Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N3, mating structure, n, and linkage or recombination parameter. --The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.

The strong-migration limit in geographically structured populations.

Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number Ne (less than or equal to NT) appears instead of the actual total population number NT. 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers Ni are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the Ni are proportional to 1/u, as NT leads to infinity, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection of coefficients proportional to 1/NT, let Pj be the frequency of the allele Aj in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As NT leads to infinity, the deviations of the allelic frequencies in each of the subpopulations from Pj converge to zero; the usual panmictic mutation-selection diffusion is obtained for Pj, with the selection intensities averaged with respect to the stationary distribution of M. In both models, Ne = NT and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.

The distribution of heterogeneity in finite random mating populations

The probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation both the lengths of heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of ‘external junctions’ (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's ‘theory of junctions’. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.

The distribution of heterogeneity in finite random mating populations

The probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation both the lengths of heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of ‘external junctions’ (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's ‘theory of junctions’. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.

Diffusion approximations of Markov chains with two time scales and applications to population genetics

ForN= 1, 2, ···, let {(XN(k),YN(k)),k= 0, 1, ···} be a homogeneous Markov chain in ℝmx ℝn. Suppose that, asymptotically asN→ ∞, the ‘infinitesimal’ covariances and means ofXN([·/∊N]) areaij(x, y) andbi(x, y), and those ofYN([·/δN]) are 0 andcl(x, y). Assume limN→∞δN= limN→∞∊N/δN= 0 and the zero solution ofý=c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i)XN([·/∊N]) converges weakly to a diffusion process with coefficientsaij(x, 0) andbi(x, 0), and (ii)YN([t/∊N]) → 0 in probability for everyt> 0. (The case limN→∞δN= δ∞> 0 = limN→∞∊Nis also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.

Diffusion approximations of Markov chains with two time scales and applications to population genetics

ForN= 1, 2, ···, let {(XN(k),YN(k)),k= 0, 1, ···} be a homogeneous Markov chain in ℝmx ℝn. Suppose that, asymptotically asN→ ∞, the ‘infinitesimal’ covariances and means ofXN([·/∊N]) areaij(x, y) andbi(x, y), and those ofYN([·/δN]) are 0 andcl(x, y). Assume limN→∞δN= limN→∞∊N/δN= 0 and the zero solution ofý=c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i)XN([·/∊N]) converges weakly to a diffusion process with coefficientsaij(x, 0) andbi(x, 0), and (ii)YN([t/∊N]) → 0 in probability for everyt> 0. (The case limN→∞δN= δ∞> 0 = limN→∞∊Nis also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.

Selection limits and strategies

Strategies are analyzed for maximizing the ultimate limit to truncation selection for a quantitative trait in finite populations. By using a formulation of Kimura and Crow [Kimura, M. & Crow, J. F. (1978) Proc. Natl. Acad. Sci. USA 75, 6168-6171], it is shown that the limit is maximized by truncating at the mode with the highest ordinate of the phenotypic distribution. This implies 50% selection for the normal or any unimodal symmetric distribution and, for skewed distributions, selection of more than one-half if desired phenotypes are in the long tail of the distribution, less than one-half if in the short tail. For dioecious populations, the optimal procedure requires, in addition, equal numbers recorded and selected of each sex. For monoecious populations the limit can be improved by using the best individuals mated to the next-best individuals. Limitations of the results are discussed.