Equilibrium Genetic Variance Research Articles

The Maintenance of Polygenic Variation in Finite Populations

Models of the maintenance of genetic variance in a polygenic trait have usually assumed that population size is infinite and that selection is weak. Consequently, they will overestimate the amount of variation maintained in finite populations. I derive approximations for the equilibrium genetic variance, VG in finite populations under weak stabilizing selection for triallelic loci and for an infinite rare model. These are compared to results for neutral characters, to the Gaussian allelic model, and to Wright's approximation for a biallelic locus under arbitrary selection pressures. For a variety of parameter values, the three-allele, Gaussian, and Wrightian approximations all converge on the neutral model when population size is small. As expected, far less equilibrium genetic variance can be maintained if effective population size, N, is on the order of a few hundred than if Nis infinite. All of the models predict that comparisons among populations with N less than about 104 should show substantial differences in VG. While it is easier to maintain absolute VG Awhen alleles interact to yield dominance or overdominance for fitness, less additivity also makes VG more susceptible to differences in N. I argue that experimental data do not seem to reflect the predicted degree of relationship between N and VG. This calls into question the ability of mutation-selection balance or simple balancing selection to explain observed VG. The dependence of VG on N could be used to test the adequacy of mutation-selection balance models.

HOW MUCH HERITABLE VARIATION CAN BE MAINTAINED IN FINITE POPULATIONS BY MUTATION-SELECTION BALANCE?

The joint effects of stabilizing selection, mutation, recombination, and random drift on the genetic variability of a polygenic character in a finite population are investigated. A simulation study is performed to test the validity of various analytical predictions on the equilibrium genetic variance. A new formula for the expected equilibrium variance is derived that approximates the observed equilibrium variance very closely for all parameter combinations we have tested. The computer model simulates the continuum-of-alleles model of Crow and Kimura. However, it is completely stochastic in the sense that it models evolution as a Markov process and does not use any deterministic evolution equations. The theoretical results are compared with heritability estimates from laboratory and natural populations. Heritabilities ranging from 20% to 50%, as observed even in lab populations under a constant environment, can only be explained by a mutation-selection balance if the phenotypic character is neutral or the number of genes contributing to the trait is sufficiently high, typically several hundred, or if there are a few highly variable loci that influence quantitative traits.

Quantitative genetic variability maintained by mutation-stabilizing selection balance in finite populations.

SummaryModels of variability in quantitative traits maintained by a balance between mutation and stabilizing selection are investigated. The effects of mutant alleles are assumed to be additive and to be randomly sampled from a stationary distribution. With a two-allele model the equilibrium genetic variance in an infinite population is independent of the distribution of mutant effects, and dependent only on the total number of mutants appearing per generation. In a finite population, however, both the shape and standard deviation of the distribution of mutant effects are important. The equilibrium variance is lower when most of the mutational variance is contributed by few genes of large effect. Genes of small effect can eventually contribute substantially to the variance with increasing population size (N.) The equilibrium variance can be higher in a finite than an infinite population since near-neutral alleles can drift to intermediate frequencies where selection is weakest. Linkage leads to a reduction in the maintained variance which is small unless linkage is very tight and selection is strong, but the reduction becomes greater with increasingNsince more mutants segregate. A multi-allele model is simulated and it is concluded that the two-allele model gives a good approximation of its behaviour. It is argued that the total number of loci capable of influencing most quantitative traits is large, and that the distribution of mutant effects is highly leptokurtic with the effects of most mutants very small, and such mutants are important in contributing to the maintained variance since selection against them is slight. The weakness of the simple optimum model is discussed in relation to the likely consequences of pleiotropy.

Directional selection and variation in finite populations.

Predictions are made of the equilibrium genetic variances and responses in a metric trait under the joint effects of directional selection, mutation and linkage in a finite population. The "infinitesimal model" is analyzed as the limiting case of many mutants of very small effect, otherwise Monte Carlo simulation is used. If the effects of mutant genes on the trait are symmetrically distributed and they are unlinked, the variance of mutant effects is not an important parameter. If the distribution is skewed, unless effects or the population size is small, the proportion of mutants that have increasing effect is the critical parameter. With linkage the distribution of genotypic values in the population becomes skewed downward and the equilibrium genetic variance and response are smaller as disequilibrium becomes important. Linkage effects are greater when the mutational variance is contributed by many genes of small effect than few of large effect, and are greater when the majority of mutants increase rather than decrease the trait because genes that are of large effect or are deleterious do not segregate for long. The most likely conditions for "Muller's ratchet" are investigated.

The Effect of Post-Metamorphic Dispersal on the Population Genetic Structure of Fowler's Toad, Bufo woodhousei fowleri

A 4 yr mark and recapture study of Fowler's toad, Bufo woodhousei fowleri, determined the rate of post-metamorphic dispersal among seven breeding areas distributed linearly within a 2 km section of the Indiana Dunes National Lakeshore. Of more than 17,000 1 wk old tadpoles carrying a unique electromorph and introduced into the population and 8539 toads marked at metamorphosis, 37 animals were recaptured as breeding adults. Of these, 73% bred for the first time in natal ponds, most migrants moved only one or two breeding areas from the natal pond (24%), and one individual was observed migrating the maximum distance across the study site. These data combined with estimates of further dispersal in the adult stage and adult mortality yielded a per-generation migration rate among breeding areas of 49%. A Lincoln-Petersen estimate of the number of breeding adults and a direct count of egg masses produced upper and lower bounds on effective population size in each area of 152 and 38. These population sizes and migration rates were used in a computer simulation to estimate the standardized equilibrium genetic variance among populations. This analysis showed that local breeding aggregations of amphibians following similar patterns of migration are not closed genetic units and would not be expected to exhibit much genetic differentiation.

Adaptive landscapes, genetic distance and the evolution of quantitative characters.

SummaryThe maintenance of polygenic variability by a balance between mutation and stabilizing selection has been analysed using two approximations: the ‘Gaussian’ and the ‘house of cards’. These lead to qualitatively different relationships between the equilibrium genetic variance and the parameters describing selection and mutation. Here we generalize these approximations to describe the dynamics of genetic means and variances under arbitrary patterns of selection and mutation. We incorporate genetic drift into the same mathematical framework.The effects of frequency-independent selection and genetic drift can be determined from the gradient of log mean fitness and a covariance matrix that depends on genotype frequencies. These equations describe an ‘adaptive landscape’, with a natural metric of genetic distance set by the covariance matrix. From this representation we can change coordinates to derive equations describing the dynamics of an additive polygenic character in terms of the moments (means, variances, …) of allelic effects at individual loci. Only under certain simplifying conditions, such as those derived from the Gaussian and house-of-cards approximations, do these general recursions lead to tractable equations for the first few phenotypic moments. The alternative approximations differ in the constraints they impose on the distributions of allelic effects at individual loci. The Gaussian-based prediction that evolution of the phenotypic mean does not change the genetic variance is shown to be a consequence of the assumption that the allelic distributions are never skewed. We present both analytical and numerical results delimiting the parameter values consistent with our approximations.

Evolution of genetic variability in a spatially heterogeneous environment: effects of genotype-environment interaction.

SummaryClassical population genetic models show that disruptive selection in a spatially variable environment can maintain genetic variation. We present quantitative genetic models for the effects of disruptive selection between environments on the genetic covariance structure of a polygenic trait. Our models suggest that disruptive selection usually does not alter the equilibrium genetic variance, although transient changes are predicted. We view a quantitative character as a set ofcharacter states, each expressed in one environment. The genetic correlation between character states expressed in different environments strongly affects the evolution of the genetic variability. (1) If the genetic correlation between character states is not ± 1, then the mean phenotype expressed in each environment will eventually attain the optimum value for that environment; this is the evolution of phenotypic plasticity (Via & Lande, 1985). At the joint phenotypic optimum, there is no disruptive selection between environments and thus no increase in the equilibrium genetic variability over that maintained by a balance between mutation and stabilizing selection within each environment. (2) If, however, the genetic correlation between character states is ± 1, the mean phenotype will not evolve to the joint phenotypic optimum and a persistent force of disruptive selection between environments will increase the equilibrium genetic variance. (3) Numerical analyses of the dynamic equations indicate that the mean phenotype can usually be perturbed several phenotypic standard deviations from the optimum without producing transient changes of more than a few per cent in the genetic variances or correlations. It may thus be reasonable to assume a roughly constant covariance structure during phenotypic evolution unless genetic correlations among character states are extremely high or populations are frequently perturbed. (4) Transient changes in the genetic correlations between character states resulting from disruptive selection act to constrain the evolution of the mean phenotype rather than to facilitate it.

The selection limit due to the conflict between truncation and stabilizing selection with mutation.

Long-term selection response could slow down from a decline in genetic variance or in selection differential or both. A model of conflict between truncation and stabilizing selection in infinite population size is analysed in terms of the reduction in selection differential. Under the assumption of a normal phenotypic distribution, the limit to selection is found to be a function of kappa, the intensity of truncation selection, omega 2, a measure of the intensity of stabilizing selection, and sigma 2, the phenotypic variance of the character. The maintenance of genetic variation at this limit is also analyzed in terms of mutation-selection balance by the use of the "House-of-cards" approximation. It is found that truncation selection can substantially reduce the equilibrium genetic variance below that when only stabilizing selection is acting, and the proportional reduction in variance is greatest when the selection is very weak. When truncation selection is strong, any further increase in the strength of selection has little further influence on the variance. It appears that this mutation-selection balance is insufficient to account for the high levels of genetic variation observed in many long-term selection experiments.

Effects of pleiotropy on predictions concerning mutation-selection balance for polygenic traits.

Previous mathematical analyses of mutation-selection balance for metric traits assume that selection acts on the relevant loci only through the character(s) under study. Thus, they implicitly assume that all of the relevant mutation and selection parameters are estimable. A more realistic analysis must recognize that many of the pleiotropic effects of loci contributing variation to a given character are not known. To explore the consequences of these hidden effects, I analyze models of two pleiotropically connected polygenic traits, denoted P1 and P2. The actual equilibrium genetic variance for P1, based on complete knowledge of all mutation and selection parameters for both P1 and P2, can be compared to a prediction based solely on observations of P1. This extrapolation mimics empirically obtainable predictions because of the inevitability of unknown pleiotropic effects. The mutation parameters relevant to P1 are assumed to be known, but selection intensity is estimated from the within-generation reduction of phenotypic variance for P1. The extrapolated prediction is obtained by substituting these parameters into formulas based on single-character analyses. Approximate analytical and numerical results show that the level of agreement between these univariate extrapolations and the actual equilibrium variance depends critically on both the genetic model assumed and the relative magnitudes of the mutation and selection parameters. Unless per locus mutation rates are extremely high, i.e., generally greater than 10(-4), the widely used gaussian approximation for genetic effects at individual loci is not applicable. Nevertheless, the gaussian approximations predict that the true and extrapolated equilibria are in reasonable agreement, i.e., within a factor of two, over a wide range of parameter values. In contrast, an alternative approximation that applies for moderate and low per locus mutation rates predicts that the extrapolation will generally overestimate the true equilibrium variance unless there is little selection associated with hidden effects. The tendency to overestimate is understandable because selection acts on all of the pleiotropic manifestations of a new mutation, but equilibrium covariances among the characters affected may not reveal all of this selection. This casts doubt on the proposal that much of the additive polygenic variance observed in natural populations can be explained by mutation-selection balance. It also indicates the difficulty of critically evaluating this hypothesis.

Pleiotropic overdominance and the maintenance of genetic variation in polygenic characters.

A model of selection is described in which optimizing phenotypic selection is combined with pleiotropic overdominance. Thus, the role that mutation commonly plays in models of phenotypic evolution is replaced by balancing selection. Expressions are provided for the equilibrium genetic variance in phenotype and for the heterozygosity. An approximate analysis of the transient properties of the model shows that, in certain circumstances, the behavior is quite similar to that of models based on the interaction of mutation and selection.

Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

Most quantitative traits in most populations exhibit heritable genetic variation. Lande proposed that high levels of heritable variation may be maintained by mutation in the face of stabilizing selection. Several analyses have appeared of two distinct models with n additive polygenic loci subject to mutation and stabilizing selection. Each is reviewed and a new analysis and model are presented. Lande and Fleming analyzed extensions of a model originally treated by Kimura which assumes a continuum of possible allelic effects at each locus. Latter and Bulmer analyzed a model with diallelic loci. The published analyses of these models lead to qualitatively different predictions concerning the dependence of the equilibrium genetic variance on the underlying biological parameters. A new asymptotic analysis of the Kimura model shows that the different predictions are not consequences of the number of alleles assumed but rather are attributable to assumptions concerning the relative magnitudes of per locus mutation rates, the phenotypic effects of mutation, and the intensity of selection. This conclusion is reinforced by analysis of a model with triallelic loci. None of the approximate analyses presented are mathematically rigorous. To quantify their accuracy and display the domains of validity for alternative approximations, numerically determined equilibria are presented. In addition, empirical estimates of mutation rates and selection intensity are reviewed, revealing weaknesses in both the data and its connection to the models. Although the mathematical results and underlying biological requirements of my analyses are quite different from those of Lande, the results do not refute his hypothesis that considerable additive genetic variance may be maintained by mutation-selection balance. However, I argue that the validity of this hypothesis can only be determined with additional data and mathematics.