Gene-culture coevolution and sex ratios: II. Sex-chromosomal distorters and cultural preferences for offspring sex

Cultural preferences for the sex of offspring may produce behavior, such as female infanticide, sex-selective abortion and sex-selective parental investment, which alter the sex ratio in a population. Empirical evidence suggests that some genetic sex-ratio distorters are located on the sex chromosomes. Interactions between cultural preferences and sex-linked sex-ratio distorters are examined. Criteria for the spread of cultural preferences and sex-chromosomal distorter alleles are derived analytically, and the coevolution of preferences and distorters is examined through numerical iteration. Evolutionary equilibria and trajectories of gene-culture interactions involving sex-chromosomal distorter alleles may produce severely male- or female-biased primary sex ratios and adult sex ratios in populations. Adult sex ratios, primary sex ratios, allele frequencies and the prevalence of cultural preferences in the population are sensitive to initial conditions and cultural transmission parameters. During the coevolutionary process phenoallelic association is observed in many cases and is associated with unusual dynamics.

Deleterious mutations, variable epistatic interactions, and the evolution of recombination

In this paper, we examine the conditions that allow increased recombination to evolve in the presence of recurrent deleterious mutation. We focus on a three-locus model first studied by Feldman et al. (1980), which follows the dynamics of a modifier locus that alters the recombination rate between two loci subject to deleterious mutation. Although Feldman et al. (1980) indicated that increased recombination might be favored if there is diminishing-returns epistasis, we show that alleles that increase the recombination rate can only invade if there is synergistic epistasis between the loci under selection. Even with synergistic epistasis, evolution at the modifier locus will lead to decreased recombination if the modifier locus is loosely linked and epistasis is strong. Using the multi-locus analysis of Barton (1995), we show that variability among loci in the sign and strength of epistasis further decreases the parameter space over which increased recombination may evolve. We conclude that, even with negative epistasis, increased recombination may only be favored when linkage is tight, especially if, as seems likely, epistatic interactions are highly variable among loci.

Microsatellite genetic distances with range constraints: analytic description and problems of estimation

Statistical properties of the symmetric stepwise-mutation model for microsatellite evolution are studied under the assumption that the number of repeats is strictly bounded above and below. An exact analytic expression is found for the expected products of the frequencies of alleles separated by k repeats. This permits characterization of the asymptotic behavior of our distances D1 and (delta mu)2 under range constraints. Based on this characterization we develop transformations that partially restore linearity when allele size is restricted. We show that the appropriate transformation cannot be applied in the case of varying mutation rates (beta) and range constraints (R) because of statistical difficulties. In the special case of no variation in beta and R across loci, however, the transformation simplifies to a usable form and results in a distance much more linear with time than distances developed for an infinite range. Although analytically incorrect in the case of variation in beta and R, the simpler transformation is surprisingly insensitive to variation in these parameters, suggesting that it may have considerable utility in phylogenetic studies.

Population genetic perspectives on the evolution of recombination

Optimality arguments and modifier theory are reviewed as paradigms for the study of the evolution of recombination. Optimality criteria (such as maximization of mean fitness) may agree with results from models developed in terms of the evolution of recombination at modifier loci. Modifier models demonstrate, however, that equilibrium mean fitness can decrease during the evolution of recombination rates and is not always maximized. Therefore, optimality arguments do not successfully predict the conditions under which increased or decreased recombination will evolve. The results from modifier models indicate that decreased recombination rates are usually favored when the population is initially near a polymorphic equilibrium with linkage disequilibrium. When the population is subject to directional selection or to deleterious mutations, increased recombination may be favored under certain conditions, provided that there is negative epistasis among alleles.

Fitness patterns and phenotypic plasticity in a spatially heterogeneous environment

We analyse patterns of the means and variances of genotypic fitnesses across different niches in a randomly mating haploid population. The population inhabits a spatially heterogeneous environment where it is subject to mutation and weak multilocus additive selection, with different election coefficients in different niches. Approximate analytical expressions are derived for the stationary mean and variance of genotypic fitnesses among the niches in terms of environmental and genetic parameters. As a special case, we analyse an environment described by a variable t, distributed among the niches with mean t(star) and variance D(star) and quadratic decrease in correlation between environments as a function of the difference in values of t. If the niches have the same qualities, the mean and variance of genotypic fitnesses evolve to be quadratic functions of t that achieve their maximum and minimum, respectively, at t(star). With unequal niche qualities, these are non-polynomial functions that attain their extrema at different, usually intermediate values of t, although the coefficient of variation of the genotypic fitnesses still attains its minimum near t(star). The functions involve the total mutation rate, the combination of the loci to genotypic fitnesses, and the frequency and quality distributions of the niches. Thus, for this relatively simple model the norms of reaction may be calculated in terms of the detailed properties of the environmental heterogeneity, and the genetic system.

Statistics for microsatellite variation based on coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate x population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci-effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation-drift equilibrium.

Statistical properties of the variation at linked microsatellite loci: implications for the history of human Y chromosomes

It has recently been suggested that observed levels of variation at microsatellite loci can be used to infer patterns of selection in genomes and to assess demographic history. In order to evaluate the feasibility of these suggestions it is necessary to know something about how levels of variation at microsatellite loci are expected to fluctuate due simply to stochasticity in the processes of mutation and inheritance (genetic sampling). Here we use recently derived properties of the stepwise mutation model to place confidence intervals around the variance in repeat score that is expected at mutation-drift equilibrium and outline a statistical test for whether an observed value differs significantly from expectation. We also develop confidence intervals for the time course of the buildup of variation following a complete elimination of variation, such as might be caused by a selective sweep or an extreme population bottleneck. We apply these methods to the variation observed at human Y-specific microsatellites. Although a number of authors have suggested the possibility of a very recent sweep, our analyses suggest that a sweep or extreme bottleneck is unlikely to have occurred anytime during the last approximately 74,000 years. To generate this result we use a recently estimated mutation rate for microsatellite loci of 5.6 x 10(-4) along with the variation observed at autosomal microsatellite loci to estimate the human effective population size. This estimate is 18,000, implying an effective number of 4,500 Y chromosomes. One important general conclusion to emerge from this study is that in order to reject mutation-drift equilibrium at a set of linked microsatellite loci it is necessary to have an unreasonably large number of loci unless the observed variance is far below that expected at mutation-drift equilibrium.

Geographic clustering of human Y-chromosome haplotypes

Five polymorphic markers on the Y-chromosome (mostly microsatellites) were typed in 121 individuals from 13 populations around the world. With these markers 78 different haplotypes were detected. Haplotypes present more than once tend to be shared by individuals from the same population or continent. A reconstruction of haplotype phylogeny also indicates significant geographic structure in the data. Based on the similarity of the haplotypes, population relationships were examined and found to be largely concordant with those obtained with other markers. Even though the sample size and the number of markers are small, there is very signficant clustering of the haplotypes by continent of origin.

On the modification of recombination with sex-dependent fitnesses and linkage

According to the Reduction Principle, when a recombination-reducing allele is introduced near an equilibrium that depends on recombination, that allele will increase in frequency. If the allele increases the recombination rate, it will be expelled from the population. There are known cases where this principle fails. In this respect, an interesting question is what kind of two-sex viability regimes support a general Reduction Principle. In this paper, we construct a model of viabilities, due to two autosomal linked genes, which differ between the sexes, such that recombination is different in the sexes. A complete analysis is provided for the case where recombination is absent in one sex. It is proved that the Reduction Principle is still valid for recombination in the other sex.

Microsatellite variability and genetic distances

We analyze the within- and between-population dynamics of the distribution of the number of repeats at multiple microsatellite DNA loci subject to stepwise mutation. Analytical expressions for moments up to the fourth order within a locus and the variance of between-locus variance at mutation-drift equilibrium have been obtained. These statistics may be used to test the appropriateness of the one-step mutation model and to detect between-locus variation in the mutation rate. Published data are compatible with the one-step mutation model, although they do not reject the two-step model. Using both multinomial sampling and diffusion approximations for the analysis of the genetic distance introduced by Goldstein et al. [Goldstein, D. B., Linares, A. R., Cavalli-Sforza, L. L. & Feldman, M. W. (1995) Proc. Natl. Acad. Sci. USA 92, 6723-6727], we show that this distance follows a chi 2 distribution with degrees of freedom equal to the number of loci when there is no variation in mutation rates among the loci. In the presence of such variation, the variance of the distance is obtained. We conclude that the number of microsatellite loci required for the construction of phylogenetic trees with reliable branch lengths may be several hundred. Also, mutations that change repeat scores by several units, even though extremely rare, may dramatically influence estimates of population parameters.

A gene-culture model of human handedness

A model of handedness incorporating both genetic and cultural processes is proposed, based on an evolutionary analysis, and maximum-likelihood estimates of its parameters are generated. This model has the characteristics that (i) no genetic variation underlies variation in handedness, and (ii) variation in handedness among humans is the result of a combination of cultural and developmental factors, but (iii) a genetic influence remains since handedness is a facultative trait. The model fits the data from 17 studies of handedness in families and 14 studies of handedness in monozygotic and dizygotic twins. This model has the additional advantages that it can explain why monozygotic and dizygotic twins and siblings have similar concordance rates, and no hypothetical selection regimes are required to explain the persistence of left handedness.

Population structure, fitness surfaces, and linkage in the shifting balance process

Wright first introduced the idea that random genetic drift and classical mass-action selection might combine in such a way as to allow populations to find the highest peak in complicated adaptive surfaces. His theory assumes large but structured populations, in which mating is spatially local. If gene flow is sufficiently low, and the subpopulations (demes) are small enough, they will be subject to genetic drift. Distant demes drift independently, allowing many independent searches of the adaptive surface to take place. A deme that has shifted to a higher peak can, by emigration, cause the rest of the demes to shift to the higher peak. The probability of this shift depends on the migration rate. Previous studies have concluded that very little migration is necessary to effect the shift in adaptive peaks that characterizes the last phase of Wright's Shifting Balance Process (SBP). Here we present the results of a computer study that investigates the roles of dispersal distance, the degree of epistasis in the fitness surface, and recombination on the shifting balance process. In particular, we measure their effect on the population's mean fitness. We show that over a range of dispersal distances the advantage of the SBP is a monotonically increasing function of the amount of epistasis. Our results show that the extent of dispersal that results in the greatest effect of the SBP in increasing mean fitness depends on the extent of epistasis. Finally, for low levels of epistasis, higher recombination performs better, while for intermediate levels, lower recombination results in a greater advantage of the SBP.

Genetic absolute dating based on microsatellites and the origin of modern humans

We introduce a new genetic distance for microsatellite loci, incorporating features of the stepwise mutation model, and test its performance on microsatellite polymorphisms in humans, chimpanzees, and gorillas. We find that it performs well in determining the relations among the primates, but less well than other distance measures (not based on the stepwise mutation model) in determining the relations among closely related human populations. However, the deepest split in the human phylogeny seems to be accurately reconstructed by the new distance and separates African and non-African populations. The new distance is independent of population size and therefore allows direct estimation of divergence times if the mutation rate is known. Based on 30 microsatellite polymorphisms and a recently reported average mutation rate of 5.6 x 10(-4) at 15 dinucleotide microsatellites, we estimate that the deepest split in the human phylogeny occurred about 156,000 years ago. Unlike most previous estimates, ours requires no external calibration of the rate of molecular evolution. We can use such calibrations, however, to test our estimate.

Effects of cis-trans viability selection on some two-locus models

Most analyses of two-locus viability models have assumed that the fitness of double heterozygotes are the same whether in the cis or trans configuration. This assumption is unlikely to hold for polymorphic sites within the same locus. We examine the quantitative and qualitative effects of incorporating cis-trans viability differences into a number of deterministic two-locus models. A new result is the finding that two asymmetric equilibria with similar levels of linkage disequilibrium, but different gene frequencies, may arise in quite realistic biological models. The general, and most important, conclusion is that high levels of linkage disequilibrium may be generated by very small selective differences in cis and trans. Polymorphic sites that interact selectively are expected to show high levels of linkage disequilibrium. Conversely, if two polymorphic sites within a gene are found to be in linkage equilibrium, it is likely that one or both are selectively neutral.

The reduction principle for recombination under density-dependent selection

In diploid random mating populations with constant viability selection, genetic modifiers of recombination, introduced near equilibria that exhibit genetic association, invade if they reduce recombination. In this study we combine ecological and standard populations genetics in a haploid multilocus model that includes density-dependent regulation of population size and weak density-dependent differential selection among the multilocus genotypes. An allele that affects recombination among the genes contributing to the ecological selection, introduced near a stable equilibrium of the ecological-genetic system, invades if it reduces a weighted average of the recombination rates among pairs of loci under selection. This generalizes the Reduction Principle for the evolution of recombination (M. W. Feldman and U. Liberman, Proc. Nat. Acad. Sci. USA 83, 4824-4827, 1986; L. A. Zhivotovsky, W. M. Feldman, and F. B. Christiansen, Theor. Popul. Biol. 44, 225-245, 1993). It is also shown that the stronger the extent of density-dependence, the weaker the selection for reduced recombination.

High sex ratios in China's future

In China in recent years, male live births have exceeded those of females by amounts far greater than those that occur naturally in human populations, a trend with significant demographic consequences. The resulting imbalance in the first-marriage market is estimated to be about 1 million males per year after 2010. These "excess" males were not easily accommodated in models with substantial changes in first-marriage patterns. The current sex ratio at birth has little effect on a couple's probability of having at least one son, so future increases in the sex ratio may well occur, especially given increasing access to sex-selective abortion.

Sex determination in a symmetric autosomal multi-locus model

A model of determination of sex by an individual's genotype at n loci is discussed. The parameters that determine the probability that an individual is male (or female) depend only on the loci at which the individual is heterozygous, extending the two-gene theory of Feldman et al. (1991, Genetics 129, 297-312). The analysis uses a set indexation technique that allows a compact expression of the recombination process, and a transformation of the chromosome frequencies that induces a natural subgroup structure on hyperplanes in the frequency simplex. It is shown that the polymorphic equilibria correspond to these subgroups. Conditions are determined under which equilibria that exhibit the even sex ratio are stable. How recombination affects the stability of other equilibria, including those with linkage disequilibrium, are also explored.

An evaluation of genetic distances for use with microsatellite loci

Mutations of alleles at microsatellite loci tend to result in alleles with repeat scores similar to those of the alleles from which they were derived. Therefore the difference in repeat score between alleles carries information about the amount of time that has passed since they shared a common ancestral allele. This information is ignored by genetic distances based on the infinite alleles model. Here we develop a genetic distance based on the stepwise mutation model that includes allelic repeat score. We adapt earlier treatments of the stepwise mutation model to show analytically that the expectation of this distance is a linear function of time. We then use computer simulations to evaluate the overall reliability of this distance and to compare it with allele sharing and Nei's distance. We find that no distance is uniformly superior for all purposes, but that for phylogenetic reconstruction of taxa that are sufficiently diverged, our new distance is preferable.

Gene-culture coevolution and sex ratios: the effects of infanticide, sex-selective abortion, sex selection, and sex-biased parental investment on the evolution of sex ratios

The evolutionary consequences of culturally transmitted practices that cause differential mortality between the sexes, thereby distorting the sex ratio (e.g., female infanticide and sex-selective abortion), are explored using dynamic models of gene-culture coevolution. We investigate how a preference for the sex of offspring may affect the selection of genes distorting the primary sex ratio. Sex-dependent differences in mortality have been predicted to select for a male- or female-biased primary sex ratio, to have no effect, or to favor either under different circumstances. We find that when a mating pair's behavior modifies mortality rates in favor of one sex, but does not change the number of offspring produced in the mating, the primary sex ratio will evolve a bias against the favored sex. However, when the total number of offspring of a mating pair is significantly reduced as a consequence of their prejudice, the primary sex ratio will evolve to favor the preferred sex. These results hold irrespective of whether the sex ratio is distorted by the mother's, the father's or the individual's own autosomal genes. The use of dynamic models of gene-culture coevolution allows us to explore the evolution of alleles which distort the sex ratio, as well as the final equilibrium states of the system. Gene-culture interactions can provide equilibria different from those in purely genetic systems, slow the approach to these equilibria by orders of magnitude, and move the primary (PSR) and the adult sex ratio (ASR) away from any stable equilibrium for hundreds of generations.

Cultural transmission of a sign language when deafness is caused by recessive alleles at two independent loci

Two unlinked autosomal loci are assumed to affect the ability to hear in such a way that homozygosity for the recessive allele at either locus causes deafness. The five deaf genotypes are subject to the same negative selection due to a lower likelihood of marriage, but unmarried deaf persons remain socially active and participate in the cultural transmission of sign languages. Marriages are assortative for deafness or for hearing, and mutation occurs irreversibly from the dominant to recessive allele at each locus at the same rate. At mutation-selection balance, the fully polymorphic equilibrium is symmetrical. Based on this genetic model, we consider the relative importance of various forms of cultural transmission as they affect the persistence of sign languages. Horizontal transmission is shown to be effective when deaf children are able to interact with many peers. This observation is especially pertinent if assortative meeting of deaf children occurs, for example, at schools for the deaf. Oblique transmission can also be effective, but the literature suggests that this kind of transmission plays only a minor role. It is necessary, however, that some form of cultural transmission occur between generations. Thus, vertical transmission is a critical factor, despite the fact that parent-child transmission is often interrupted due to the recessive inheritance of deafness. In particular, the contribution of vertical transmission is enhanced by assortative mating for deafness.

Evolution of recombination among multiple selected loci: a generalized reduction principle

Conditions for invasion by a new allele that controls the recombination pattern among an arbitrary number of genes under viability selection are studied. The recombination pattern may include interference. The new allele increases if its appropriately averaged marginal fitness is greater than the mean fitness prior to its introduction. Under weak additive-by-additive epistatic selection, this condition involves a weighted average of the changes in pairwise recombination rates relative to those prior to the introduction of the modifier. The weights here are positive functions of the epistatic selection components. In particular, the modifier allele may succeed even if it increases recombination among some pairs of loci, provided the overall average effect is one of reduction.

The evolution of interference: reduction of recombination among three loci

Crossover events along chromosomes do not occur independently, but influence the probability of other nearby events. The most common interaction between nearby crossover events is inhibitory: a crossover event tends to reduce the probability of other such events nearby, and this is called positive interference. A crossover event may increase the probability of events nearby, and this rare phenomenon is called negative interference. In this paper, we use numerical methods to investigate how interference among three loci would evolve if it were under the genetic control of a fourth, selectively neutral locus. We first discuss the effect of interference on the overall rate of recombination among the three loci, and then show that, under a variety of conditions, interference evolves in the same way as would be predicted based upon its effect on the overall rate of recombination. That is, the overall rate evolves in the same direction as would the rate at a locus that controls recombination between two loci directly. We then check for the existence of viability-analogous Hardy-Weinberg equilibria in the four-locus model of interference modification.

Heterogeneous selection in subdivided populations

The dynamics of allele frequencies changing under migration and heterogeneous selection in a subdivided population are investigated. Using perturbation techniques, a stationary state is obtained when migration and selection are both small. Heterogeneous selection leads to a positive correlation between values of F-statistics and heterozygosities when these are compared among sets of subdivided populations. This contrast with a negative value of the correlation obtained under Wright's classical model of homogeneous selection, and with the absence of correlation in the completely neutral situation.

On the probability of loss of new mutations in the presence of linkage disequilibrium

A new selectively neutral mutation occurs in a multilocus genetic background that has achieved a stable equilibrium at which there is a linkage disequilibrium. Perturbation techniques are applied to an extension of the branching process formulation of Fisher in order to address the question of extinction probabilities. We show that under appropriate conditions the probability of extinction of the new mutant is increased by the existence of linkage disequilibrium in the genetic background.

Gene-culture coevolution: toward a general theory of vertical transmission

A general formulation of cultural and genetic transmission is developed. The cultural transmission is vertical and the genetics may involve multiple loci. Each individual is represented by a phenogenotype, and conditions are given under which the evolutionary dynamics of phenogenotype frequencies are reducible to phenogametic or phenoallelic frequencies. The interaction between genes and culture is specified by an association measure, and results on the order of magnitude of this association at equilibrium are presented.

Assortative mating and grandparental transmission facilitate the persistence of a sign language

Conditions for the persistence (i.e., protection from loss) of a sign language are investigated assuming monogenic recessive inheritance of deafness, assortative mating for deafness or hearing, and cultural transmission of the sign language to deaf individuals from their deaf parents and deaf maternal grandparents. A new method is introduced to deal with the problem of grandparental transmission in which the basic variables are the frequencies of triplets comprising a mother, a father, and their daughter of permissible phenogenotypes. Usual stability analysis is then done on the system of linear recursions in the frequencies of these triplets, derived on the assumption that signers (users of the sign language) are rare. It is shown that assortative mating is the most important factor contributing to persistence, but that grandparental transmission can also have a significant effect when assortment is as strong as observed in England and the United States.

On models of quantitative genetic variability: a stabilizing selection-balance model

A model of stabilizing selection on a multilocus character is proposed that allows the maintenance of stable allelic polymorphism and linkage disequilibrium. The model is a generalization of Lerner's model of homeostasis in which heterozygotes are less susceptible to environmental variation and hence are superior to homozygotes under phenotypic stabilizing selection. The analysis is carried out for weak selection with a quadratic-deviation model for the stabilizing selection. The stationary state is characterized by unequal allele frequencies, unequal proportions of complementary gametes, and a reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared with Mather's polygenic balance theory, with models that include mutation-selection balance, and others that have been proposed to study the role of linkage disequilibrium in quantitative inheritance.

Lewontin and Kojima meet Fisher: linkage in a symmetric model of sex determination

The effect of linkage and epistasis on the evolution of the sex-ratio is studied in a symmetric two-locus model of autosomal sex determination closely related to the symmetric viability model of R. C. Lewontin and K. Kojima. R. A. Fisher's expectation of an even sex ratio for autosomal sex determination by a single gene governs the dynamics when the loci are tightly linked. However, recombination may preclude optimization of the sex ratio just as occurs in viability selection models. Many of the evolutionary phenomena known for the symmetric viability model also occur here. In addition, we exhibit a series of new phenomena related to the presence of surfaces of even sex ratio.

Recessive hereditary deafness, assortative mating, and persistence of a sign language

We model the cultural transmission of sign language when there is one-locus genetic variation for deafness and hearing. Our premises are that the deaf are more motivated to learn sign language than the hearing, and that a vertically transmitted sign language, unlike recessive hereditary deafness, cannot "jump a generation." Conditions are obtained for persistence (i.e. protection from loss) of signers. These conditions are more easily satisfied the greater the fraction of the hearing who also learn sign language and as the frequency of the recessive gene for deafness increases. Persistence is also facilitated by assortative mating for deafness, but not by assortment for signing. With vertical transmission only, it is necessary that one signer parent be able to transmit sign language with greater than one-half the efficiency of two. Under the assumption that the hearing do not learn sign language, the following additional results are obtained. Persistence is more likely with dominant as opposed to recessive inheritance. When recessive hereditary and acquired deafness co-occur, increasing the frequency of the latter has opposite effects depending on the degree of assortment. Opportunities for the deaf to learn sign language outside the family seem not to affect the conditions for persistence.

Mutation modification with multiplicative fertility selection

Two diallelic loci in an infinite panmictic population of diploid individuals are modelled. The A/a locus is subject to unidirectional mutation and either multiplicative fertility selection or, equivalently, sex-asymmetric viability selection. The M/m locus acts as a selectively neutral modifier of the mutation rate at A/a. The loci recombine at rate R. If the M/m locus is initially monomorphic, and the A/a locus has reached equilibrium, the fate of a new modifier allele is found to depend not just on its relative effect on mutation but also upon the linkage, R. Each initial equilibrium may be characterized by a critical value of the recombination rate, R*. If 0 less than R* less than 0.5, a sufficiently small "down" modifier of the mutation rate will invade the population when R less than R* whereas a sufficiently small "up" modifier will succeed when R greater than R*. If R* less than 0 or R* greater than 0.5, only mutation reduction may occur. Numerical analysis of 56,000 sample equilibria indicates that mutation rates may be increased, but only when the selection regime is such that the A/a locus would remain polymorphic in the absence of mutation.

Two-locus autosomal sex determination: on the evolutionary genetic stability of the even sex ratio

In two-locus models of sex determination, there are two kinds of interior (polymorphic) equilibria. One class has the even sex ratio, and the other has equal allele frequencies in the two sexes. Equilibria of the second class may exhibit linkage disequilibrium. The condition for external stability of these second-class equilibria to invasion by a new allele is that the appropriately averaged sex ratio near the equilibrium be moved closer to the even sex ratio than the average among the resident genotypes. However, invasion by a new chromosome depends on the recombination fraction in a way that appears to preclude general results about the evolutionary genetic stability of the even sex ratio in this situation.

Spatial subdivision of populations and estimates of genetic variation

Measures of variation in space are strongly affected by correlations between subdivisions used for sampling. Here we consider variation in gene frequencies across populations. Usually the variance of gene frequencies is standardized by dividing it by the mean gene frequency times one minus the mean (FST). Under the model of isolation by distance (usually called the "stepping stone" model), at the stationary state the correlation between the gene frequencies of two populations falls exponentially with the geographic distance between them. Using this model, we derive formulas for variances of blocks of populations of different sizes in one- and two-dimensional space and suggest that the theoretical results may be useful for understanding real observations, some examples of which are presented. We demonstrate how FST increases with the degree of subdivision among populations. We also show the effect of gaps between the sampled populations. Our results are valid, however, for traits other than gene frequencies, as long as their correlation with geographic distance falls exponentially. In the extension to 2-dimensional spaces, we present in closed form the distributions of distances between nodes of a lattice or of two lattices. These distributions might have applications in ecology.

Pleiotropy and preadaptation in the evolution of human language capacity

The capacity for spoken language in the human is a genetic trait, but the information communicated by this means is to a large extent culturally determined. Using a gene-culture coevolutionary approach, we model the hypothesis that speech evolved as a channel for the communication of adaptive cultural traits from parent to offspring. The motivation for this paper is a condition obtained previously that initial increase of communication would require at least a two-fold advantage for the transmitted trait. Here, we show that under reasonable assumptions the invasion condition becomes less stringent. In Model 1, we assume that two adaptive cultural traits can be transmitted. A gene which permits communication of the second adaptive trait. In Model 2, we assume that a related function such as greater memory capacity is a prerequisite for speech, and that this function confers an advantage independent of its association with speech. In both models we assume haploid sexual genetics and a simple scheme of vertical transmission. The stability properties of all corner and edge equilibria of the models are analyzed. The two models taken together suggest a possible scenario for the initial stages of the evolution of speech.

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.

A general asymptotic property of two-locus selection models

It is shown that any two-locus, two-allele model of selection with constant fitnesses has at least one polymorphic equilibrium for which the linkage association measure, D, is arbitrarily close to zero for large enough recombination, R. As R----+/- infinity, D----0 in such a way that the product l = RD----a non-zero finite constant. There may be 1, 3, or 5 distinct asymptotic equilibria, depending upon fitness parameters.

Kin selection and the evolution of monogamy

A two-locus genetic model is studied in which one locus controls the tendency of individuals to act altruistically toward siblings and the other locus controls the mating habits of females. It is demonstrated that genetic variation at the altruism locus is often sufficient to induce an increase in the frequency of genes that cause females to produce all of their offspring with a single mate. This occurs because of nonrandom associations that develop between genes that cause altruism and those that affect female mating behavior. The results provide a new explanation for the evolution of monogamy, and they suggest a previously unexplored mechanism for the evolution of a variety of other behavioral traits as well.

Selection, generalized transmission and the evolution of modifier genes. I. The reduction principle

Modifier gene models are used to explore the evolution of features of organisms, such as the genetic system, that are not directly involved in the determination of fitness. Recent work has shown that a general "reduction principle" holds in models of selectively neutral modifiers of recombination, mutation, and migration. Here we present a framework for models of modifier genes that shows these reduction results to be part of a more general theory, for which recombination and mutation are special cases. The deterministic forces that affect the genetic composition of a population can be partitioned into two categories: selection and transmission. Selection includes differential viabilities, fertilities, and mating success. Imperfect transmission occurs as a result of such phenomena as recombination, mutation and migration, meiosis, gene conversion, and meiotic drive. Selectively neutral modifier genes affect transmission, and a neutral modifier gene can evolve only by generating association with selected genes whose transmission it affects. We show that, in randomly mating populations at equilibrium, imperfect transmission of selected genes allows a variance in their marginal fitnesses to be maintained. This variance in the marginal fitnesses of selected genes is what drives the evolution of neutral modifier genes. Populations with a variance in marginal fitnesses at equilibrium are always subject to invasion by modifier genes that bring about perfect transmission of the selected genes. It is also found, within certain constraints, that for modifier genes producing what we call "linear variation" in the transmission processes, a new modifier allele can invade a population at equilibrium if it reduces the level of imperfect transmission acting on the selected genes, and will be expelled if it increases the level of imperfect transmission. Moreover, the strength of the induced selection on the modifier gene is shown to range up to the order of the departure of the genetic system from perfect transmission.

Behavior-dependent contexts for repeated plays of the Prisoner's Dilemma: II. Dynamical aspects of the evolution of cooperation

Iterated Prisoner's Dilemma models are proposed in which, at any trial, the probability of staying in the game depends on the outcome of the previous trial. If a player's choice depends on its own play (cooperate or defect) at the previous trial, it becomes possible for cooperative strategies to increase when rare in a population of egoists. A dynamic analysis is used to demonstrate that stable polymorphisms may result, and may involve more strategies than just Tit-for-Tat and all-Defect. The tendency for clustering among like strategists to enhance their initial increase when rare is also explored dynamically.

Toward a theory for the evolution of cultural communication: coevolution of signal transmission and reception

A haploid sexual two-locus model of gene-culture coevolution is examined, in which a dichotomous phenotype subject to natural selection is transmitted vertically with probabilities dependent on the chosen parent's genotype and phenotype and the offspring's genotype. Stability conditions for the genetically monomorphic corner equilibria are obtained. In a specialization of this general model, one locus controls the transmission and the other controls the reception of adaptive information. The corner and edge equilibria of this doubly coevolutionary model are fully analyzed, and conditions for transmission and reception to coevolve are derived in terms of the efficiency of vertical transmission, the selective advantage gained from possessing the information, the costs of transmission and reception, and the recombination fraction between the two loci. Possible applications of the model are to the evolution of semantic alarm calls in vervet monkeys and the phonetic aspects of human language. In a third model with diploid genetics, we consider the initial increase of cultural transmission from a mutation-selection balance in which the adaptive phenotype is the consequence of a dominant gene at one locus. A second gene controls the transmission of the phenotype in such a way that a new mutant at this second locus permits learning of the adaptive phenotype from a parent who has it. This new mutant cannot increase when rare.

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.

An evolutionary reduction principle for genetic modifiers

The joint evolution of major genes under viability selection and a modifier locus that controls recombination between the major genes, mutation at the major gene, or migration between two demes is studied. The modifying locus is selectively neutral and may have an arbitrary number of alleles. For each case a class of polymorphic equilibria exists in which the frequencies of the modifying alleles are those computed by assuming that the recombination, mutation, or migration rates were viabilities and in which the major and modifier loci are not statistically associated. These are called viability-analogous Hardy-Weinberg (VAHW) equilibria. A new allele introduced near these equilibria will enter the population if its marginal average rate of recombination, mutation, or migration (whichever applies) is less than the population average prior to its introduction. Stability properties of these VAHW equilibria are also reported.

A numerical simulation of the one-locus, multiple-allele fertility model

Numerical simulations were performed to determine the equilibrium behavior of the one-locus fertility model in which fitness is considered as a property of a pair of mating diploids. A series of patterns of "fertility matrices" were considered for a single locus with two to six alleles. From these simulations, 19 different statistics were collected that characterize, at equilibrium, the heterozygosity, the mean fitness and the fate of populations begun at the allele-frequency centroid. For more than one-half of the trajectories produced by random fertility matrices, there was a decrease in the mean fitness at some time on the way to equilibrium. The mean number of alleles maintained at equilibrium increased only slightly with matrix dimension. Despite the potential for fertility models to display multiple stable equilibria, random fertility models maintain fewer distinct stable points than do random one-locus viability models. Pleiotropic models were also considered with fertility and viability selection operating sequentially within each generation. Most of the equilibrium statistics (with the exception of mean fertility) for the pleiotropic model were intermediate between the corresponding random viability and fertility models.

Selection for increased mutation rates with fertility differences between matings

Previous studies of mutation modification have considered models in which selection is a result of viability differences that are sex symmetric. The results of a numerical study of a model in which selection is a result of fertility differences between mated pairs demonstrate that the type of selection to which a population is subject can have a significant impact on the evolution of various parameters of the genetic system. When the fertility of matings between individuals with different genotypes exceeds the fertility of at least some of the matings between individuals with the same genotype, selection may favor increased rates of mutation, in contrast to the results from all existing constant viability models with random mating and infinite population size. Increased mutation rates are most frequently favored when forward and back mutation occur at approximately equal rates and when the modifying locus is loosely linked to the selected locus. We present one example in which selection favors increased rates of mutation even though the selection scheme is reducible to one of differential viability between the sexes.