NULL MODELS FOR THE NUMBER OF EVOLUTIONARY STEPS IN A CHARACTER ON A PHYLOGENETIC TREE

Random trees and random characters can be used in null models for testing phylogenetic hypothesis. We consider three interpretations of random trees: first, that trees are selected from the set of all possible trees with equal probability; second, that trees are formed by random speciation or coalescence (equivalent); and third, that trees are formed by a series of random partitions of the taxa. We consider two interpretations of random characters: first, that the number of taxa with each state is held constant, but the states are randomly reshuffled among the taxa; and second, that the probability each taxon is assigned a particular state is constant from one taxon to the next. Under null models representing various combinations of randomizations of trees and characters, exact recursion equations are given to calculate the probability distribution of the number of character state changes required by a phylogenetic tree. Possible applications of these probability distributions are discussed. They can be used, for example, to test for a panmictic population structure within a species or to test phylogenetic inertia in a character's evolution. Whether and how a null model incorporates tree randomness makes little difference to the probability distribution in many but not all circumstances. The null model's sense of character randomness appears more critical. The difficult issue of choosing a null model is discussed.     

Mutational order: a major stochastic process in evolution

Computer simulations in which selection acts on a quantitative character show that the randomness of mutations can contribute significantly to evolutionary divergence between populations. In different populations, different advantageous mutations occur, and are selected to fixation, so that the populations diverge even when they are initially identical, and are subject to identical selection. This stochastic process is distinct from random genetic drift. In some circumstances (large populations or strong selection, or both) mutational order can be greatly more important than random drift in bringing about divergence. It can generate a 'disconnection' between evolution at the phenotypic and genotypic levels, and can give rise to a rough 'molecular clock', albeit episodic, that is driven by selection. In the absence of selection, mutational order has little or no effect.     

Evolutionary dynamics in frequency-dependent two-phenotype models

General frequency-dependent selection models based on two phenotypic classes are analyzed with underlying one-locus multiallele phenotypic determination systems in diploid populations. It is proved that the mean phenotypic fitnesses tend to equality over discrete generations and genetic mutations if a phenotypic polymorphism is to be maintained. The exact conditions are examined. The present results are valid for a wide class of models whenever random groupings or assortative patterns based on phenotype and affecting fitness, linearly or not, are independent of sex, mating preferences, or kinship. They can also be applied to two-sex haploid models.     

Polygenic inheritance is not necessary for sympatric speciation by sexual selection

 The theoretical possibility of sympatric speciation by sexual selection has been demonstrated by a number of mathematical models. Although these models assumed that sexually selected traits are additively determined by many genes, recent empirical studies suggest that many sexually selected traits are determined by major gene inheritance. Thus, using a mathematical simulation model, this article examines whether sympatric speciation by sexual selection can occur when sexually selected traits are determined by major gene inheritance. The model reveals that speciation can occur with major gene inheritance of sexually selected traits. Simulations show that speciation from an initially monomorphic population occurs via two successive Fisher's runaway processes of sexual selection. The first runaway causes the unidirectional evolution of male secondary sexual character toward one extreme in the trait space and of female mate preference for such a character. The second runaway then drives the male character and female preference of a part of the population toward the other extreme in the trait space, splitting the population into two reproductively isolated subgroups. The current results reinforce the plausibility of sympatric speciation by sexual selection. Received: January 30, 2002 / Accepted: June 27, 2002     

The genetics of adaptation: the roles of pleiotropy, stabilizing selection and drift in shaping the distribution of bidirectional fixed mutational effects

Pleiotropy allows for the deterministic fixation of bidirectional mutations: mutations with effects both in the direction of selection and opposite to selection for the same character. Mutations with deleterious effects on some characters can fix because of beneficial effects on other characters. This study analytically quantifies the expected frequency of mutations that fix with negative and positive effects on a character and the average size of a fixed effect on a character when a mutation pleiotropically affects from very few to many characters. The analysis allows for mutational distributions that vary in shape and provides a framework that would allow for varying the frequency at which mutations arise with deleterious and positive effects on characters. The results show that a large fraction of fixed mutations will have deleterious pleiotropic effects even when mutation affects as little as two characters and only directional selection is occurring, and, not surprisingly, as the degree of pleiotropy increases the frequency of fixed deleterious effects increases. As a point of comparison, we show how stabilizing selection and random genetic drift affect the bidirectional distribution of fixed mutational effects. The results are then applied to QTL studies that seek to find loci that contribute to phenotypic differences between populations or species. It is shown that QTL studies are biased against detecting chromosome regions that have deleterious pleiotropic effects on characters.     

Selection for altruism through random drift in variable size populations

ABSTRACT: BACKGROUND: Altruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have 'hidden' advantages if the 'common good' produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness. RESULTS: Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of "selfish" alleles. In other terms, the fixation probability of altruistic genes can be higher than those of a selfish ones, even though altruists have a smaller fitness. Moreover when populations are geographically structured, the altruists advantage can be highly amplified and the fixation probability of selfish genes can tend toward zero. The above results are obtained both by numerical and analytical calculations. Analytical results are obtained in the limit of large populations. CONCLUSIONS: The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists.     

Sexual selection and the evolution of costly female preferences: spatial effects

Abstract.— Models of Fisher's runaway process show that if there is a cost to female preference, no preference or male trait exaggeration will evolve. Surprisingly, this is true no matter how small the cost, which reveals that these models of Fisher's process are structurally unstable (Bulmer 1989). Here a model of Fisher's runaway process is presented to demonstrate that costly female preference evolves very easily when space is explicitly included in the model. The only requirement is that the optimal male phenotype changes across the species' range. The model shows that the spatial average of the female preference and male trait reach an evolutionary equilibrium that is identical to those of nonspatial models, but that the preference and male trait can deviate greatly from these averages at any point in space. For example, if random mating results in the lowest cost to females, then at equilibrium the spatial average preference will be zero. Nevertheless, there will be some locations at which females prefer males with larger ornaments and others where they prefer males with smaller ornaments. Results also show that the structural instability of nonspatial models of Fisher's process is less of a problem in spatial models. In particular, many of the main qualitative features of cost-free spatial models of Fisher's process remain valid even when there are small costs of female preference. Finally, the model shows that abrupt changes in the optimal male phenotype across space can result in an amplification of this pattern when preference has a small cost, but it can also result in a pattern similar to reproductive character displacement. Which of these occurs depends on the magnitude of the cost of female preference. This suggests that some patterns of reproductive character displacement in nature might be explained simply by sexual selection rather than by hybrid dysgenesis and reinforcement.     

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.     

Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.     

Models of clines in sexual selection by female choice

Most models of sexual selection by female choice have considered discrete, homogenous populations. This paper studies the evolution of a female preference along a cline in the frequency of a preferred male trait. Single alleles control both the preference and preferred character.
Fisher's process initially causes the preference to spread to some 'maximum' frequency with a corresponding rise in character gene frequency. At this stage, the cline in preference does not necessarily mirror that of the trait. Then, however, preference frequency usually decreases, albeit very slowly, and the preference cline always comes to mirror that of the preferred character. Eventually, preference is completely lost from the cline and the character cline decays to that seen under random mating. This loss can only be prevented if the preference is initially frequent enough to push the character to fixation throughout the cline.
Consequently, a preference that arises after the preferred trait may increase very little in frequency itself and have a negligible effect on trait frequency before being lost from the population. Special conditions, such as cyclical natural selection, may be necessary to explain the spread of a preference in a cline from a low initial frequency to frequencies as high as those observed. A preference that predates the preferred trait can enter the population at a high frequency and radically alter a cline in the frequency of a preferred male trait, albeit often transiently.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Not so different after all: a comparison of methods for detecting amino acid sites under selection

We consider three approaches for estimating the rates of nonsynonymous and synonymous changes at each site in a sequence alignment in order to identify sites under positive or negative selection: (1) a suite of fast likelihood-based "counting methods" that employ either a single most likely ancestral reconstruction, weighting across all possible ancestral reconstructions, or sampling from ancestral reconstructions; (2) a random effects likelihood (REL) approach, which models variation in nonsynonymous and synonymous rates across sites according to a predefined distribution, with the selection pressure at an individual site inferred using an empirical Bayes approach; and (3) a fixed effects likelihood (FEL) method that directly estimates nonsynonymous and synonymous substitution rates at each site. All three methods incorporate flexible models of nucleotide substitution bias and variation in both nonsynonymous and synonymous substitution rates across sites, facilitating the comparison between the methods. We demonstrate that the results obtained using these approaches show broad agreement in levels of Type I and Type II error and in estimates of substitution rates. Counting methods are well suited for large alignments, for which there is high power to detect positive and negative selection, but appear to underestimate the substitution rate. A REL approach, which is more computationally intensive than counting methods, has higher power than counting methods to detect selection in data sets of intermediate size but may suffer from higher rates of false positives for small data sets. A FEL approach appears to capture the pattern of rate variation better than counting methods or random effects models, does not suffer from as many false positives as random effects models for data sets comprising few sequences, and can be efficiently parallelized. Our results suggest that previously reported differences between results obtained by counting methods and random effects models arise due to a combination of the conservative nature of counting-based methods, the failure of current random effects models to allow for variation in synonymous substitution rates, and the naive application of random effects models to extremely sparse data sets. We demonstrate our methods on sequence data from the human immunodeficiency virus type 1 env and pol genes and simulated alignments.     

The Site-Frequency Spectrum of Linked Sites

The site-frequency spectrum, representing the distribution of allele frequencies at a set of polymorphic sites, is a commonly used summary statistic in population genetics. Explicit forms of the spectrum are known for both models with and without selection if independence among sites is assumed. The availability of these explicit forms has allowed for maximum likelihood estimation of selection, developed first in the Poisson random field model of Sawyer and Hartl, which is now the primary method for estimating selection directly from DNA sequence data. The independence assumption, which amounts to assume free recombination between sites, is, however, a limiting case for many population genetics models. Here, we extend the site-frequency spectrum theory to consider the case where the sites are completely linked. We use diffusion approximation to calculate the joint distribution of the allele frequencies of linked sites for models without selection and for models with equal coefficient selection. The joint distribution is derived by first constructing Green’s functions corresponding to multiallele diffusion equations. We show that the site-frequency spectrum is highly correlated between frequencies that are complementary (i.e., sum to 1), and the correlation is significantly elevated by positive selection. The results presented here can be used to extend the Poisson random field to allow for estimating selection for correlated sites. More generally, the Green’s function construction should be able to aid in studying the genetic drift of multiple alleles in other cases.     

Polymorphism with migration and selection

Summary Two single-locus, deterministic models with discrete nonoverlapping generations are formulated for the maintenance of genetic variation in each of two distinct biological situations. The first two models are applicable to an autosomal locus in an hermaphroditic plant population with mixed selfing and random mating. They describe the interaction of migration and viability selection for, respectively, an island migration model and for a subdivided population. Pollen as well as seed may disperse. Sufficient conditions are derived and discussed for the existence of protected polymorphism in the diallelic case. The remaining two models are pertinent to migration and selection at a single X-linked locus. An island model is again considered as well as that of a subdivided population. Mating is at random, selection occurs only through viability differences, and the migration structure for males and females may differ. For a diallelic population, protection conditions are derived and discussed vis-à-vis the autosomal case.M.M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Effects of Differential Selection in the Sexes on Cytonuclear Polymorphism and Disequilibria

We develop a series of models that examine the effects of differential selection between the sexes on cytonuclear polymorphism and disequilibria. A detailed analysis is provided for populations under constant fertility or viability selection censused at life stages without frequency differences in the sexes. We show analytically that cytonuclear disequilibria can be generated de novo if the cytoplasmic and nuclear loci each affect female fitness and there is a nonmultiplicative fitness interaction between them. While computer simulations demonstrate that the majority of disequilibria produced by random selection are transient and small in magnitude, measurable permanent disequilibria can result from selective differences both within and between the two sexes. We derive analytic conditions for a protected cytonuclear polymorphism and use numerical simulations to quantitate the likelihood of obtaining permanent nuclear, cytoplasmic, and cytonuclear variation under various patterns of selection. The numerical analysis identifies special selection regimes more likely to generate disequilibria and maintain cytonuclear polymorphism and reveals a direct correlation to the strength of selection. As a byproduct, our models also provide the first decomposition of the different parental contributions to cytonuclear dynamics and the analytic conditions under which selection can cause cytoplasmic frequency changes or a cytonuclear hitchhiking effect.     

Selection versus random drift: long-term polymorphism persistence in small populations (evidence and modelling)

Our data on a subterranean mammal, Spalax ehrenbergi, and other evidence, indicate that appreciable polymorphism can be preserved in small isolated populations consisting of several dozens of, or a hundred, individuals. Current theoretical models predict fast gene fixation in small panmictic populations without selection, mutation, or gene inflow. Using simple multilocus models, we demonstrate here that moderate stabilizing selection (with stable or fluctuating optimum) for traits controlled by additive genes could oppose random fixation in such isolates during thousands of generations. We also show that in selection-free models polymorphism persists only for a few hundred generations even under high mutation rates. Our multi-chromosome models challenge the hitchhiking hypothesis of polymorphism maintenance for many neutral loci due to close linkage with few selected loci.     

Multiplicative Genotype-Environment Interaction as a Cause of Reversed Response to Directional Selection

In experiments with directional selection on a quantitative character a "reversed response" to selection is occasionally observed, when selection of individuals for a higher (lower) value of the character results in a lower (higher) value of the character among their offspring. A sudden change in environments or random drift is often assumed to be responsible for this. It is demonstrated in this paper that these two causes cannot account for the reversed response at least in some of the experiments. Multiplicative genotype-environment interaction is discussed as a possible cause of a reversed response to directional selection. Such interaction entails either disruptive or stabilizing genotypic selection, even when the phenotypic selection is directional.     

Spatial patterns in the distributions of polygenic characters

The spatial patterns in the mean and variance of a quantitative character that result from the interaction of spatially varying, optimizing selection and gene flow are considered. The model analyzed is an extension of those of Kimura (1965) and Lande (1976) for the distribution of a quantitative character maintained in a population by independent mutations. For weak selection, it is shown that there is only a small effect of gene flow on the variance of the character and that the mean value changes on a length scale that is large compared to the average dispersal distance. As in models of clines in allele frequencies, it is possible to define a “characteristic length” in terms of the average dispersal distance and strength of selection. The characteristic length is the smallest length scale environmental change to which the mean value of the character can significantly respond. It is also shown that, for weak selection, an asymmetry in dispersal can result in a significant shift in location of a cline. By considering an infinite linear cline in optimal values, it is shown that gene flow can increase the variance only when there is sufficient mixing in each generation of individuals from locations with different means. A model of selection in different niches is also considered. There is an increase in variance due to the effective weakening of the intensity of selection because of the differences in optimal values in different niches.The implications of the different models for maintenance of genetic polymorphism are discussed. Under some conditions gene flow can produce a significant increase in heterozygosity. It is also argued that spatial variation in selection on a polygenic character can be much more effective in increasing heterozygosity than temporal variation because of the potentially greater increase in phenotypic variance. The difference between some of the results for polygenic characters from those of similar models of one and two locus systems is accounted for by the fact that for normally distributed polygenic characters, changes in the variance are effectively decoupled from changes in the mean.     

Riding across the selection landscape: fitness consequences of annual variation in reproductive characteristics

Evolutionary models estimating phenotypic selection in character size usually assume that the character is invariant across reproductive bouts. We show that variation in the size of reproductive traits may be large over multiple events and can influence fitness in organisms where these traits are produced anew each season. With data from populations of two orchid species, Caladenia valida and Tolumnia variegata, we used Bayesian statistics to investigate the effect on the distribution in fitness of individuals when the fitness landscape is not flat and when characters vary across reproductive bouts. Inconsistency in character size across reproductive periods within an individual increases the uncertainty of mean fitness and, consequently, the uncertainty in individual fitness. The trajectory of selection is likely to be muddled as a consequence of variation in morphology of individuals across reproductive bouts. The frequency and amplitude of such changes will certainly affect the dynamics between selection and genetic drift.