A general multilocus two-allele model for epistatic selection

Observations show that evolutionary processes often relate to multilocus epistatic selection. Here we develop further the approach suggested earlier by Zhivotovsky and Gavrilets to admit arbitrary multilocus epistasis. The obtained dynamic equations for allelic frequencies and gametic disequilibria are represented in a simple form. If selection is weak, this result extends Wright’s evolutionary equation to the case of cis-trans effects and sex differences in both recombination rates and genotypic fitnesses. Additionally to Wright’s equations for allelic frequencies, we derive equations for the gametic disequilibrium terms. We also give a general expression for the gametic disequilibria in a quasi-linkage state.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

Definition and Properties of Disequilibrium Statistics for Associations between Nuclear and Cytoplasmic Genotypes

We define and establish the interrelationships of four components of statistical association between a diploid nuclear gene and a uniparentally transmitted, haploid cytoplasmic gene: an allelic (gametic) disequilibrium (D), which measures associations between alleles at the two loci; and three genotypic disequilibria (D1, D2, D3), which measure associations between two cytotypes and the three respective nuclear backgrounds. We also consider an alternative set of measures, including D and the residual disequilibrium (d). The dynamics of these disequilibria are then examined under three conventional models of the mating system: (1) random mating; (2a) assortative mating without dominance (the "mixed-mating model"); and (2b) assortative mating with dominance ("O'Donald's model"). The trajectories of gametic disequilibria are similar to those for pairs of unlinked nuclear loci. The dynamics of genotypic disequilibria exhibit a variety of behaviors depending on the model and the initial conditions. Procedures for statistical estimation of cytonuclear disequilibria are developed and applied to several real and hypothetical data sets. Special attention is paid to the biological interpretations of various categories of allelic and genotypic disequilibria in hybrid zones. Genetic systems for which these statistics might be appropriate include nuclear genotype frequencies in conjunction with those for mitochondrial DNA, chloroplast DNA, or cytoplasmically inherited microorganisms.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

Dynamics of correlated genetic systems. II. Simulation studies of chromosomal segments under selection

The dynamics of chromosomal segments under selection are investigated by comparing experimental data to simulations of simple models of selection. The simulations assume 93 loci distributed evenly along an entire chromosome. The two issues addressed in this paper concern rates of decay of linkage disequilibria for chromosomes under selection and rates of gene frequency change after perturbation of gametic frequencies to states near the edge of the gametic frequency simplex. The findings are: (1) If reasonable values of inbreeding depression are assumed, linkage disequilibria decays to zero but at a rate nearly twice that expected from neutral theory. Experimental results also show accelerated decay rates. The acceleration of decay seems to be a simple consequence of the increased heterozygosity produced by selection. It is, therefore, argued that massive linkage disequilibria, of the kind found by Franklin and Lewontin (1970) in their simulations, are unlikely to characterize the genetic structure of natural populations of random mating organisms. (2) It is possible to distinguish between two time-honored models of multilocus selection, known as the symmetric overdominant and classical models, on the basis of gene frequency change near the edge of the gametic frequency simplex assuming linkage disequilibria is intense. (3) Examination of experimental data from perturbation experiments shows that neither of these elementary models adequately account for observed rates of gene frequency change, although the symmetric overdominant model does provide the better fit. Instead the experimental data suggest a markedly nonuniform distribution of selective effects along the chromosome. The data also suggest that these selective effects combine in markedly nonadditive ways in determining joint fitness.     

Evolution under Fertility and Viability Selection

Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s2)], then to O(s2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

Modeling extent and distribution of zygotic disequilibrium: implications for a multigenerational canine pedigree

Unlike gametic linkage disequilibrium defined for a random-mating population, zygotic disequilibrium describes the nonrandom association between different loci in a nonequilibrium population that deviates from Hardy-Weinberg equilibrium. Zygotic disequilibrium specifies five different types of disequilibria simultaneously that are (1) Hardy-Weinberg disequilibria at each locus, (2) gametic disequilibrium (including two alleles in the same gamete, each from a different locus), (3) nongametic disequilibrium (including two alleles in different gametes, each from a different locus), (4) trigenic disequilibrium (including a zygote at one locus and an allele at the other), and (5) quadrigenic disequilibrium (including two zygotes each from a different locus). However, because of the uncertainty on the phase of the double heterozygote, gametic and nongametic disequilibria need to be combined into a composite digenic disequilibrium and further define a composite quadrigenic disequilibrium together with the quadrigenic disequilibrium. To investigate the extent and distribution of zygotic disequilibrium across the canine genome, a total of 148 dogs were genotyped at 247 microsatellite markers located on 39 pairs of chromosomes for an outbred multigenerational pedigree, initiated with a limited number of unrelated founders. A major portion of zygotic disequilibrium was contributed by the composite digenic and quadrigenic disequilibrium whose values and numbers of significant marker pairs are both greater than those of trigenic disequilibrium. All types of disequilibrium are extensive in the canine genome, although their values tend to decrease with extended map distances, but with a greater slope for trigenic disequilibrium than for the other types of disequilibrium. Considerable variation in the pattern of disequilibrium reduction was observed among different chromosomes. The results from this study provide scientific guidance about the determination of the number of markers used for whole-genome association studies.     

Gene Conversion, Linkage, and the Evolution of Multigene Families

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

Selection Response in Finite Populations

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (σ(AW0)(2)) and mutational (σ(M)(2)) variance. The cumulative response to selection until generation t(CR(t)) can be approximated as & where N(e) is the effective population size, σ(AW &)(2) = N(e)σ(M)(2) is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R(0) is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. β is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R(0) is the major determinant of the short term selection response, but σ(M)(2), N(e) and β are also important for the long term. A selection method of high accuracy using family information gives a small N(e) and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.     

A macroscopic approach to population genetics

Conventional population genetics uses as primitive variables the frequencies and fitnesses of individual genes. This paper develops a formalism whose primitive variables are the frequencies and fitnesses of genotypes and environmental histories in a population. From the mathematical relation that describes genetic variation and selection of genotypes and environmental histories we derive a sequence of more specialized equations, including those of the conventional theory. Some familiar formulas of the conventional theory (including Fisher's fundamental theorem, the formula relating the rate of change of a metric character to selection pressure, and the definitions of broad and narrow heritability) are shown to be special cases of simpler and more general formulas. It is shown that the “genotypic value” of a trait, together with its heritability, may depend strongly on genotype-environment correlations.A generalization of Fisher's fundamental theorem shows that the rate of evolution of a trait depends on the skewness of its fitness distribution. An equation relating the second derivative of the mean fitness to the skewness is derived.Finally, the formalism is applied in a preliminary way to a recent theory of genetic variation (Layzer,1978a), according to which the genetic variability of a trait is selected along with the trait itself. It is shown that there is positive feedback between the two kinds of selection.     

Further analysis of complex allozyme polymorphisms in a barley population

New theory has recently been developed for two-locus models. In the light of this theory, an earlier analysis of esterase allozyme data from an experimental barley population has been modified to take proper account of initial gametic phase (linkage) disequilibria. The results show that the directions in which two-locus genotypic frequencies deviated from products of one-locus frequencies in this population followed those predicted by neutral descent theory. The observed departures were, however, much larger in size than predicted by the new descent measure theory, indicating that selection is operating in the population.     

Analysis of multilocus zygotic associations

While nonrandom associations between zygotes at different loci (zygotic associations) frequently occur in Hardy-Weinberg disequilibrium populations, statistical analysis of such associations has received little attention. In this article, we describe the joint distributions of zygotes at multiple loci, which are completely characterized by heterozygosities at individual loci and various multilocus zygotic associations. These zygotic associations are defined in the same fashion as the usual multilocus linkage (gametic) disequilibria on the basis of gametic and allelic frequencies. The estimation and test procedures are described with details being given for three loci. The sampling properties of the estimates are examined through Monte Carlo simulation. The estimates of three-locus associations are not free of bias due to the presence of two-locus associations and vice versa. The power of detecting the zygotic associations is small unless different loci are strongly associated and/or sample sizes are large (>100). The analysis of zygotic associations not only offers an effective means of packaging numerous genic disequilibria required for a complete characterization of multilocus structure, but also provides opportunities for making inference about evolutionary and demographic processes through a comparative assessment of zygotic association vs. gametic disequilibrium for the same set of loci in nonequilibrium populations.     

Fitness of allozyme variants in Drosophila pseudoobscura III. Possible factors contributing to the maintenance of polymorphisms in nature

Experiments have been performed to investigate the mechanisms maintaining enzyme polymorphisms in natural populations. We have measured effects on fitness of genotypic variants at three loci, Est-5, Odh, and Mdh-2, in D. pseudoobscura. Significant differences exist among the genotypes in the rate of development from egg-to-adult; there is also indication of differences in larval survival. In a population segregating for allelic variants at all three loci, there is indication that segregation distortion at meiosis or some form of gametic selection might be involved. The relative fitnesses of alternative genotypes are reversed when either different fitness components are considered, or the genotypic frequencies are changed, or the larval density is increased. These fitness reversals may contribute to the maintenance of the polymorphisms, and may account for cyclical oscillations of allozyme frequencies observed in natural populations.Research supported by U.S. Public Health Service Research Fellowship (1F05 TWO 1991-01) to D.M. and by contract AT(04-3)34 with the U.S. Atomic Energy Commission. Adress reprint requests from Europe to D.M.; from elsewhere to F.J.A.     

Inconsistencies in standard approximations for selection coefficients at loci affecting a polygenic character

Inconsistencies exist in the standard expansions used to approximate selection coefficients for alleles at a locus underlying a quantitative character. Allelic (marginal) fitnesses obtained from expansions based on average excesses differ from allelic fitnesses obtained from expansions based on genotypic values. Similarly, the mean population fitness based on summing over either allelic or genotypic fitnesses usually differs mean population fitness obtained by averaging over the unrestricted phenotypic distribution. A consistent value of requires no variation in genotypic values. If, as suggested by Nagylaki (1984), expansions are corrected for the decrease in phenotypic variance resulting from conditioning on the presence of a particular allele or genotype, inconsistencies still exist. Unless W(z)[V _z p(z) + zp(z) + p(z)] dz = 0, where p(z) is the phenotypic probability density function, V _z the phenotypic variance, W( _z ) the fitness of phenotypic value z, the primes denote differentiation with respect to z, allelic fitnesses based on average effects differ from allelic fitnesses based on genotypic values. This condition must also be satisfied in order for either expansion to give a consistent , as first shown by Nagylaki. For arbitrary W(z), this is satisfied if and only if phenotypes are normally distributed.     

The effect of positive assortative mating at one locus on a second linked locus

Summary Considerations proceed from a model of positive assortative mating based on genotype at one locus, with an arbitrary number of alleles, assuming no selection, mutation, or migration, hypothetically infinite population size, and discrete non-overlapping generations. From these conditions, inferences are made about the genotypic structure at a linked locus, as well as about the corresponding 2-locus gametic structure.The following main results are presented: in the course of the generations, the genotypic structure at the second locus and the 2-locus gametic structure always tend to a limit responsive to the initial conditions concerning the joint genotypic structure at the two loci and the degree of assortativity and linkage. A complete, analytical representation of the limits is given. In particular, if assortative mating is only partial and at the same time linkage is not complete, a population is not able to maintain a permanent deviation of the gametic structure from linkage equilibrium, and thus the genotypic structure at the second locus tends to Hardy-Weinberg proportions. On the other hand, if initial linkage disequilibrium is combined with partial assortative mating and complete linkage (or with complete assortative mating and unlinked loci) the population maintains this disequilibrium and thus the genotypic structure at the second locus need not tend to Hardy-Weinberg proportions. It turns out that the conditions not only of complete linkage, but also of unlinked loci together with complete assortativity, imply no change in gametic structure from the initial structure.In order to demonstrate the influence of several parameters on the speed of convergence to and the magnitude of the respective limits, several graphs are included.     

Bottleneck Effect on Evolutionary Rate in the Nearly Neutral Mutation Model

Variances of evolutionary rates among lineages in some proteins are larger than those expected from simple Poisson processes. This phenomenon is called overdispersion of the molecular clock. If population size N is constant, the overdispersion is observed only in a limited range of 2Nσ under the nearly neutral mutation model, where σ represents the standard deviation of selection coefficients of new mutants. In this paper, we investigated effects of changing population size on the evolutionary rate by computer simulations assuming the nearly neutral mutation model. The size was changed cyclically between two numbers, N(1) and N(2) (N(1) > N(2)), in the simulations. The overdispersion is observed if 2N(2)σ is less than two and the state of reduced size (bottleneck state) continues for more than ~0.1/u generations, where u is the mutation rate. The overdispersion results mainly because the average fitnesses of only a portion of populations go down when the population size is reduced and only in these populations subsequent advantageous substitutions occur after the population size becomes large. Since the fitness reduction after the bottleneck is stochastic, acceleration of the evolutionary rate does not necessarily occur uniformly among loci. From these results, we argue that the nearly neutral mutation model is a candidate mechanism to explain the overdispersed molecular clock.     

Inference from Clines Stabilized by Frequency-Dependent Selection

Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between square root of s/square root of 8 sigma and square root of s/square root of 12 delta, where sigma is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations ("linkage disequilibria"), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequencies and of gametic correlations from a hybrid zone.     

The Direction of Linkage Disequilibrium

The previous paper (Langley, Tobari and Kojima 1974) reports that the directional linkage disequilibria, D_ω = P_ABP_ab-P_AbP_aB, tend to be negative for data between allozymes and linked to inversions. A and B stand for the two alleles with the greatest frequency in the population. In this paper we show that linkage disequilibrium in this direction is produced at equilibrium when double homozygotes have fitnesses that are a constant fraction of the product of the two component single homozygote fitnesses, a pattern that is frequently observed in experimental data.     

The impact of COVID-19 lockdown on professional soccer players’ body composition and physical fitness

During the COVID-19 lockdown, professional soccer players ceased their regular team training sessions and were provided with exercise programs to follow independently. This investigation assessed the impact of a 7-week COVID-19 lockdown and home-based individual physical training on professional soccer players’ body composition and physical fitness. The study consisted of nineteen division 1 elite soccer players (age 27.68 ± 5.99 years, height 178.47 ± 5.44 cm) and compared the anthropometric and physical fitness parameters obtained post-transition period to those obtained post-COVID-19 lockdown. The statistical analysis indicated that body fat percentage was significantly higher after the lockdown period [t(18) = -5.59, p < 0.01, d = 0.56]. Furthermore, VO_2max [t(17) = -11.54, p < 0.01, d = 0.57] and running time [t(17) = 3.94, p < 0.01, d = 0.76] values were significantly higher after the COVID-19 lockdown than those obtained after the transition period. In addition, significantly higher level of performance was demonstrated on squat jump [t(18) = -4.10, p < 0.01, d = 0.30], countermovement jump [t(18) = -7.43, p < 0.01, d = 1.11] and sit and reach tests [t(19) = -5.33, p < 0.01, d = 0.32]. Concurrently, lower body strength was indicated to be significantly greater (p < 0.01) following the COVID-19 lockdown. The training protocol provided during the confinement, due to the COVID-19 outbreak, was effective in keeping physical fitness at a significantly higher level compared to the transition period. Coaches and trainers are encouraged to examine the effectiveness of this protocol, as it may help them develop effective periodization programs during the transition period. This protocol may aid in the development of effective periodization programs that require minimal equipment and can be followed in similar situations.