Pleiotropic Models of Polygenic Variation, Stabilizing Selection, and Epistasis

We show that in polymorphic populations many polygenic traits pleiotropically related to fitness are expected to be under apparent ``stabilizing selection' independently of the real selection acting on the population. This occurs, for example, if the genetic system is at a stable polymorphic equilibrium determined by selection and the nonadditive contributions of the loci to the trait value either are absent, or are random and independent of those to fitness. Stabilizing selection is also observed if the polygenic system is at an equilibrium determined by a balance between selection and mutation (or migration) when both additive and nonadditive contributions of the loci to the trait value are random and independent of those to fitness. We also compare different viability models that can maintain genetic variability at many loci with respect to their ability to account for the strong stabilizing selection on an additive trait. Let V(m) be the genetic variance supplied by mutation (or migration) each generation, V(g) be the genotypic variance maintained in the population, and n be the number of the loci influencing fitness. We demonstrate that in mutation (migration)-selection balance models the strength of apparent stabilizing selection is order V(m)/V(g). In the overdominant model and in the symmetric viability model the strength of apparent stabilizing selection is approximately 1/(2n) that of total selection on the whole phenotype. We show that a selection system that involves pairwise additive by additive epistasis in maintaining variability can lead to a lower genetic load and genetic variance in fitness (approximately 1/(2n) times) than an equivalent selection system that involves overdominance. We show that, in the epistatic model, the apparent stabilizing selection on an additive trait can be as strong as the total selection on the whole phenotype.     

Multilocus models of sympatric speciation: Bush versus Rice versus Felsenstein

Abstract. In populations of phytophagous insects that use the host plant as a rendezvous for mating, divergence in host preference could lead to sympatric speciation. Speciation requires the elimination of "generalist" genotypes, that is, those with intermediate host preference. This could occur because such genotypes have an inherent fitness disadvantage, or because preference alleles become associated with alleles that are oppositely selected on the two hosts. Although the former mechanism has been shown to be plausible, the latter mechanism has not been studied in detail. I consider a multilocus model (the "Bush model") in which one set of biallelic loci affects host preference, and a second set affects viability on the hosts once chosen. Alleles that increase viability on one host decrease viability on the other, and all loci are assumed to be unlinked. With moderately strong selection on the viability loci, preference alleles rapidly become associated with viability alleles, and the population splits into two reproductively isolated host specialist populations. The conditions for speciation to occur in this model, as measured by the strength of selection required, are somewhat more stringent than in a model in which preference and viability are controlled by the same loci (one-trait model). In contrast, the conditions are much less stringent than in a model in which speciation requires buildup of associations between viability loci and loci controlling a host-independent assortative mating trait (canonical two-trait model). Moreover, in the one-trait model, and to a lesser extent the Bush model, the strength of selection needed to initiate speciation is only slightly greater than that needed to complete it. This indicates that documenting instances of sympatric species that are reproductively isolated only by host or habitat preference would provide evidence for the plausibility of sympatric speciation in nature.     

A population genetics model for multiple quantitative traits exhibiting pleiotropy and epistasis

We study a population genetics model of an organism with a genome of L(tot)loci that determine the values of T quantitative traits. Each trait is controlled by a subset of L loci assigned randomly from the genome. There is an optimum value for each trait, and stabilizing selection acts on the phenotype as a whole to maintain actual trait values close to their optima. The model contains pleiotropic effects (loci can affect more than one trait) and epistasis in fitness. We use adaptive walk simulations to find high-fitness genotypes and to study the way these genotypes are distributed in sequence space. We then simulate the evolution of haploid and diploid populations on these fitness landscapes and show that the genotypes of populations are able to drift through sequence space despite stabilizing selection on the phenotype. We study the way the rate of drift and the extent of the accessible region of sequence space is affected by mutation rate, selection strength, population size, recombination rate, and the parameters L and T that control the landscape shape. There are three regimes of the model. If LTL(tot), there are many small peaks that can be spread over a wide region of sequence space. Compensatory neutral mutations are important in the population dynamics in this case.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

INBREEDING DEPRESSION WITH HETEROZYGOTE ADVANTAGE AND ITS EFFECT ON SELECTION FOR MODIFIERS CHANGING THE OUTCROSSING RATE

The equilibrium level of inbreeding depression in populations with different selfing rates is studied for models with symmetrical or asymmetrical heterozygous advantage at several loci with partial linkage. As for the case of a single locus, the inbreeding depression caused by loci with heterozygous advantage can be higher for partially selfing populations than for complete outcrossing. The spread of modifier alleles at another locus that affects the selfing rate is studied. The stability of outcrossing populations to invasion by alleles that give increased selfing is found to depend on levels of inbreeding depression being greater than one-half, in accordance with earlier models that assumed a fixed level of inbreeding depression. However, in partially selfing populations the spread of such alleles can be checked by smaller levels of inbreeding depression than one-half, so that they do not always spread to fixation. This is interpreted as being due to associations between the genotypes at the modifier locus and the selected loci, together with increasing inbreeding depression as selfing increases, and does not occur if the inbreeding depression is due to mutation-selection balance.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

The genetic architecture of adaptation under migration-selection balance

Many ecologically important traits have a complex genetic basis, with the potential for mutations at many different genes to shape the phenotype. Even so, studies of local adaptation in heterogeneous environments sometimes find that just a few quantitative trait loci (QTL) of large effect can explain a large percentage of observed differences between phenotypically divergent populations. As high levels of gene flow can swamp divergence at weakly selected alleles, migration-selection-drift balance may play an important role in shaping the genetic architecture of local adaptation. Here, we use analytical approximations and individual-based simulations to explore how genetic architecture evolves when two populations connected by migration experience stabilizing selection toward different optima. In contrast to the exponential distribution of allele effect sizes expected under adaptation without migration (Orr 1998), we find that adaptation with migration tends to result in concentrated genetic architectures with fewer, larger, and more tightly linked divergent alleles. Even if many small alleles contribute to adaptation at the outset, they tend to be replaced by a few large alleles under prolonged bouts of stabilizing selection with migration. All else being equal, we also find that stronger selection can maintain linked clusters of locally adapted alleles over much greater map distances than weaker selection. The common empirical finding of QTL of large effect is shown to be expected with migration in a heterogeneous landscape, and these QTL may often be composed of several tightly linked alleles of smaller effect.     

Systematics of Mielichhoferia (Bryaceae: Musci). III. Hybridization between M. elongata and M. mielichhoferiana

Allozyme variation in mixed populations of Mielichhoferia elongata and M. mielichhoferiana was investigated to determine if interspecific hybridization occurs when these two closely related species grow together. Previous research has shown that M. elongata and M. mielichhoferiana can be distinguished by three diagnostic isozyme loci (Gpi-1, Mdh-2, and Mdh-3) at which the two species do not share alleles in 32 allopatric populations from North America and Europe. The present study shows that in five populations from Colorado, Norway, and Sweden, gametophytes resulting from interspecific hybridization can be recognized by recombinant genotypes combining alleles of the otherwise diagnostic loci. A total of 32 multilocus genotypes was found among the 111 individuals sampled, of which 13 were recombinants. The frequency of recombinants ranged from 12% to 35% within populations, and all but one population contained both parental species. Moreover, recombinant genotypes could be accounted for by the allelic constitution of sympatric parents. In two of the populations, more than one hybridization event was necessary to account for the diversity of recombinant genotypes. Twenty-nine of the 32 genotypes detected in this study were restricted to one population each, two occurred in two Swedish populations separated by approximately 14 km, and one occurred in both Sweden and Norway.     

Mutant alleles of small effect are primarily responsible for the loss of fitness with slow inbreeding in Drosophila melanogaster.

Multilocus simulation is used to identify genetic models that can account for the observed rates of inbreeding and fitness decline in laboratory populations of Drosophila melanogaster. The experimental populations were maintained under crowded conditions for approximately 200 generations at a harmonic mean population size of Nh approximately 65-70. With a simulated population size of N = 50, and a mean selective disadvantage of homozygotes at individual loci approximately 1-2% or less, it is demonstrated that the mean effective population size over a 200-generation period may be considerably greater than N, with a ratio matching the experimental estimate of Ne/Nh approximately 1.4. The buildup of associative overdominance at electrophoretic marker loci is largely responsible for the stability of gene frequencies and the observed reduction in the rate of inbreeding, with apparent selection coefficients in favor of the heterozygote at neutral marker loci increasing rapidly over the first N generations of inbreeding to values approximately 5-10%. The observed decline in fitness under competitive conditions in populations of size approximately 50 in D. melanogaster therefore primarily results from mutant alleles with mean effects on fitness as homozygotes of sm < or = 0.02. Models with deleterious recessive mutants at the background loci require that the mean selection coefficient against heterozygotes is at most hsm approximately 0.002, with a minimum mutation rate for a single Drosophila autosome 100 cM in length estimated to be in the range 0.05-0.25, assuming an exponential distribution of s. A typical chromosome would be expected to carry at least 100-200 such mutant alleles contributing to the decline in competitive fitness with slow inbreeding.     

Genome-wide association study for agronomic and physiological traits in spring wheat evaluated in a range of heat prone environments

Key message

We identified 27 stable loci associated with agronomic traits in spring wheat using genome-wide association analysis, some of which confirmed previously reported studies. GWAS peaks identified in regions where no QTL for grain yield per se has been mapped to date, provide new opportunities for gene discovery and creation of new cultivars with desirable alleles for improving yield and yield stability in wheat.

Abstract

We undertook large-scale genetic analysis to determine marker-trait associations (MTAs) underlying agronomic and physiological performance in spring wheat using genome-wide association studies (GWAS). Field trials were conducted at seven sites in three countries (Sudan, Egypt, and Syria) over 2–3 years in each country. Twenty-five agronomic and physiological traits were measured on 188 wheat genotypes. After correcting for population structure and relatedness, a total of 245 MTAs distributed over 66 loci were associated with agronomic traits in individual and mean performance across environments respectively; some of which confirmed previously reported loci. Of these, 27 loci were significantly associated with days to heading, thousand kernel weight, grain yield, spike length, and leaf rolling for mean performance across environments. Despite strong QTL by environment interactions, eight of the loci on chromosomes 1A, 1D, 5A, 5D, 6B, 7A, and 7B had pleiotropic effects on days to heading and yield components (TKW, SM^?2, and SNS). The winter-type alleles at the homoeologous VRN1 loci significantly increased days to heading and grain yield in optimal environments, but decreased grain yield in heat prone environments. Top 20 high-yielding genotypes, ranked by additive main effects and multiplicative interaction (AMMI), had low kinship relationship and possessed 4–5 favorable alleles for GY MTAs except two genotypes, Shadi-4 and Qafzah-11/Bashiq-1–2. This indicated different yield stability mechanisms due to potentially favorable rare alleles that are uncharacterized. Our results will enable wheat breeders to effectively introgress several desirable alleles into locally adapted germplasm in developing wheat varieties with high yield stability and enhanced heat tolerance.

     

Silencing of duplicate genes: a null allele polymorphism for lactate dehydrogenase in brown trout (Salmo trutta)

A previously described isozyme polymorphism at one of two skeletal muscle LdhA loci in brown trout is due to a null allele, Ldh1(n), producing no detectable catalytic activity. Homozygotes for this allele have approximately only 56% of the LDH activity in skeletal muscle relative to homozygotes for the active allele. The remaining activity results from enzyme subunits produced by other LDH loci. The Ldh1(n) allele is common and widespread throughout brown trout populations in Sweden and is also found in populations from Ireland. The persistence of duplicate gene expression for the LdhA loci in almost all salmonid species is best explained by natural selection against individuals containing null alleles. However, there is no indication of natural selection against brown trout with the Ldh1(n) allele: We suggest that the selection against individuals containing null alleles that is apparently responsible for the persistence of duplicate LdhA loci in salmonids occurs only under certain environmental conditions.      

Multiplex PCR identification of wheat HMW glutenin subunit genes by allele-specific markers

Wheat bread-making quality is closely correlated with composition and quantity of gluten proteins, in particular with high-molecular weight (HMW) glutenin subunits encoded by the Glu-1 genes. A multiplex polymerase chain reaction (PCR) method was developed to identify the allele composition of HMW glutenin complex Glu-1 loci (Glu-A1, Glu-B1 and Glu-D1) in common wheat genotypes. The study of multiplex PCR to obtain a well-balanced set of amplicons involved examination of various combinations of selected primer sets and/or thermal cycling conditions. One to three simultaneously amplified DNA fragments of HMW glutenin Glu-1 genes were separated by agarose slab-gel electrophoresis and differences between Ax1, Ax2* and Axnull genes of Glu-A1 loci, Bx6, Bx7 and Bx17 of Glu-B1, and Dx2, Dx5 and Dy10 genes of Glu-D1 loci were revealed. A complete agreement was found in identification of HMW glutenin subunits by both multiplex PCR analysis and SDS-PAGE for seventy-six Polish cultivars/strains of both spring and winter common wheat. Rapid identification of molecular markers of Glu-1 alleles by multiplex PCR can be an efficient alternative to the standard separation procedure for early selection of useful wheat genotypes with good bread-making quality.     

Persistence time of loss-of-function mutations at nonessential loci affecting eye color in Drosophila melanogaster

Persistence time of a mutant allele, the expected number of generations before its elimination from the population, can be estimated as the ratio of the number of segregating mutations per individual over the mutation rate per generation. We screened two natural populations of Drosophila melanogaster for mutations causing clear-cut eye phenotypes and detected 25 mutant alleles, falling into 19 complementation groups, in 1164 haploid genomes, which implies 0.021 eye mutations/genome. The de novo haploid mutation rate for the same set of loci was estimated as 2 x 10(-4) in a 10-generation mutation-accumulation experiment. Thus, the average persistence time of all mutations causing clear-cut eye phenotypes is approximately 100 generations (95% confidence interval: 61-219). This estimate shows that the strength of selection against phenotypically drastic alleles of nonessential loci is close to that against recessive lethals. In both cases, deleterious alleles are apparently eliminated by selection against heterozygous individuals, which show no visible phenotypic differences from wild type.     

Substitution Rates under Stabilizing Selection

Allelic substitutions under stabilizing phenotypic selection on quantitative traits are studied in Monte Carlo simulations of 8 and 16 loci. The results are compared and contrasted to analytical models based on work of M. Kimura for two and "infinite" loci. Selection strengths of S = 4Nes approximately four (which correspond to reasonable strengths of selection for quantitative characters) can retard substitution rates tenfold relative to rates under neutrality. An important finding is a strong dependence of per locus substitution rates on the number of loci.     

Marker-Based Inferences about Epistasis for Genes Influencing Inbreeding Depression

We describe a multilocus, marker-based regression method for inferring interactions between genes controlling inbreeding depression in self-fertile organisms. It is based upon selfing a parent heterozygous for several unlinked codominant markers, then analyzing the fitness of progeny marker genotypes. If loci causing inbreeding depression are linked to marker loci, then viability selection is manifested by distorted segregation of markers, and fecundity selection by dependence of the fecundity character upon the marker genotype. To characterize this selection, fitness is regressed on the proportion of loci homozygous for markers linked to deleterious alleles, and epistasis is detected by nonlinearity of the regression. Alternatively, fitness can be regressed on the proportion of heterozygous loci. Other modes of selection can be incorporated with a bivariate regression involving both homozygote and heterozygote marker genotypes. The advantage of this marker-based approach is that ``purging' is minimized and specific chromosomal segments are identified; its disadvantage lies in low statistical power when linkage is not strong and/or the linkage phase between marker and selected loci is uncertain. Using this method, in the wildflower Mimulus guttatus, we found predominant multiplicative gene interaction determining fecundity and some negative synergistic (nonmultiplicative) interaction for viability.     

The use of the expectation-maximization (EM) algorithm for maximum likelihood estimation of gametic frequencies of multilocus polymorphic codominant systems based on sampled population data

Estimation of gametic frequencies in multilocus polymorphic systems based on the numerical distribution of multilocus genotypes in a population sample ("analysis without pedigrees") is difficult because some gametes are not recognized in the data obtained. Even in the case of codominant systems, where all alleles can be recognized by genotypes, so that direct estimation of the frequencies of genes (alleles) is possible ("complete data"), estimation of the frequencies of multilocus gametes based on the data on multilocus genotypes is sometimes impossible, whether population data or even family data are used for studying genotypic segregation or analysis of linkage ("incomplete data"). Such "incomplete data" are analyzed based on the corresponding genetic models using the expectation-maximization (EM) algorithm. In this study, the EM algorithm based on the random-marriage model for a nonsubdivided population was used to estimate gametic frequencies. The EM algorithm used in the study does not set any limitations on the number of loci and the number of alleles of each locus. Locus and alleles are identified by numeration making possible to arrange loops. In each combination of alleles for a given combination of m out of L loci (L is the total number of loci studied), all alleles are assigned value 1, and the remaining alleles are assigned value 0. The sum of zeros and unities for each gamete is its gametic value (h), and the sum of the gametic values of the gametes that form a given genotype is the genotypic value (g) of this genotype. Then, gametes with the same h are united into a single class, which reduces the number of the estimated parameters. In a general case of m loci, this procedure yields m + 1 classes of gametes and 2m + 1 classes of genotypes with genotypic values g = 0, 1, 2, ..., 2m. The unknown frequencies of the m + 1 classes of gametes can be represented as functions of the gametic frequencies whose maximum likelihood estimations (MLEs) have been obtained in all previous EM procedures and the only unknown frequency (Pm(m)) that is to be estimated in the given EM procedure. At the expectation step, the expected frequencies (Fm(g) of the genotypes with genotypic values g are expressed in terms of the products of the frequencies of m + 1 classes of gametes. The data on genotypes are the numbers (ng) of individuals with genotypic values g = 0, 1, 2, 3, ..., 2m. The maximization step is the maximization of the logarithm of the likelihood function (LLF) for ng values. Thus, the EM algorithm is reduced, in each case, to solution of only one equation with one unknown parameter with the use of the ng values, i.e., the numbers of individuals after the corresponding regrouping of the data on the individuals' genotypes. Treatment of the data obtained by Kurbatova on the MNSs and Rhesus systems with alleles C, Cw, c, D, d, E, e with the use of Weir's EM algorithm and the EM algorithm suggested in this study yielded similar results. However, the MLEs of the parameters obtained with the use of either algorithm often converged to a wrong solution: the sum of the frequencies of all gametes (4 and 12 gametes for MNSs and Rhesus, respectively) was not equal to 1.0 even if the global maximum of LLF was reached for each of them (as it was for MNSs with the use of Weir's EM algorithm), with each parameter falling within admissible limits (e.g., [0, min(PN,Ps)] for PNs). The chi 2 function is suggested to be used as a goodness-of-fit function for the distribution of genotypes in a sample in order to select acceptable solutions. However, the minimum of this function only guarantee the acceptability of solutions if all limitations on the parameters are met: the sum of estimations of gametic frequencies is 1.0, each frequency falls within the admissible limits, and the "gametic algebra" is complied with (none of the frequencies is negative).     

标记辅助回交育种中所需最小样本容量的近似估计

回交育种是把有利基因从供体亲本向受体亲本转移的一种有效方法,标记辅助选择可加速其进程。为了制定合理的标记辅助选择计划,育种家必须知道所需的后代群体大小。该文提出了一种估算在标记辅助回交育种中同时进行前景选择和背景选择所需群体大小的方法。在假定所需转移的目标基因座与遗传背景之间为相互独立的简化假设下,可以通过将解析方法（针对前景选择）与基于回交亲本图示基因型的模拟方法（针对背景选择）相结合,近似地估计出在每一世代中选到所需基因型的概率,进而估算出在一定概率水平下至少获得一个符合要求的个体所需的最小样本容量,用假想的例子演示了该方法的使用情况。该方法可以很方便地应用于实际的回交育种。     

The use of microsatellites for detecting DNA polymorphism, genotype identification and genetic diversity in wheat

A set of 20 wheat microsatellite markers was used with 55 elite wheat genotypes to examine their utility (1) in detecting DNA polymorphism, (2)in the identifying genotypes and (3) in estimating genetic diversity among wheat genotypes. The 55 elite genotypes of wheat used in this study originated in 29 countries representing six continents. A total of 155 alleles were detected at 21 loci using the above microsatellite primer pairs (only 1 primer amplified 2 loci; all other primers amplified 1 locus each). Of the 20 primers amplifying 21 loci, 17 primers and their corresponding 18 loci were assigned to 13 different chromosomes (6 chromosomes of the A genome, 5 chromosomes of the B genome and 2 chromosomes of the D genome). The number of alleles per locus ranged from 1 to 13, with an average of 7.4 alleles per locus. The values of average polymorphic information content (PIC) and the marker index (MI) for these markers were estimated to be 0.71 and 0.70, respectively. The (GT)_n microsatellites were found to be the most polymorphic. The genetic similarity (GS) coefficient for all possible 1485 pairs of genotypes ranged from 0.05 to 0.88 with an average of 0.23. The dendrogram, prepared on the basis of similarity matrix using the UPGMA algorithm, delineated the above genotypes into two major clusters (I and II), each with two subclusters (Ia, Ib and IIa, IIb). One of these subclusters (Ib) consisted of a solitary genotype (E3111) from Portugal, so that it was unique and diverse with respect to all other genotypes belonging to cluster I and placed in subcluster Ia. Using a set of only 12 primer pairs, we were able to distinguish a maximum of 48 of the above 55 wheat genotypes. The results demonstrate the utility of microsatellite markers for detecting polymorphism leading to genotype identification and for estimating genetic diversity. Received: 15 May 1999 / Accepted: 27 July 1999     

Altruism, sex, and inbreeding when the genotype-phenotype map is additive

Recently published theoretical results suggest that, in a sexual population, when genotypes code for phenotypes in a complex manner, it is possible for altruistic genotypes to spread through a metapopulation (i.e. through a collection of subpopulations). This spread tends to occur during periods when the environment deteriorates throughout the metapopulation. By contrast, under asexual reproduction, non-altruistic genotypes seem to be favoured, at least when subpopulations are substantial in size. The most relevant previous study makes use of Kauffman and Levin's "NK model" as a way to relate genotypes to fitness. Unfortunately, there are both conceptual and technical problems with the application of the NK model to populations that contain many different genotypes (e.g. polymorphic diploid populations with more than a few loci under selection). The present study presents a more tractable and biologically plausible model to study the causal relationship between sexual reproduction and altruism. In particular, phenotypes are determined by additive interactions among alleles at different loci in a diploid genome, with up to 200 loci under selection. In addition, subpopulations are substantially larger than those considered in the most relevant previous work. The results show that, so long as there are multiple "fitness peaks" in "phenotype space", the additive genotype-phenotype map leads to results that are similar to those from the NK model. Various parameters are manipulated in an effort to discover the determinants of altruistic and non-altruistic outcomes. The findings should facilitate further investigations, and they should help to establish the plausibility of the suggested relationship between sexual reproduction and altruism. The results also suggest that inbreeding can lead to a similar result as asexuality. That is, inbreeding seems to enhance the probability that altruistic phenotypes will be eliminated.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.