Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

Populational heritability: empirical studies of evolution in metapopulations

Using demes from experimental metapopulations of the flour beetle, Tribolium castaneum, we investigated phase 3 of Wright's shifting balance process. Using parent demes of high, intermediate, and low mean fitness, we experimentally modeled migration of varying amounts from demes of high mean fitness into demes of lower mean fitness (like phase 3) as well as the reciprocal (the opposite of phase 3). In natural populations, some migration among demes occurs independently of deme fitness. In this case, demes of high mean fitness are likely to receive migrants from demes of lower mean fitness; these effects might limit the effectiveness of phase 3 but have not been studied experimentally. We estimated the populational heritability of mean fitness by the regression of offspring deme means on the weighted parental means and found moderate levels of demic heritability one (0.641-0.690) and two (0.518-0.552) generations after migration. We discuss our findings in relation to the role of interdemic migration in "adaptive peak shifts" in metapopulations and the controversies over group selection and the units of inheritance.     

Rate of adaptive peak shifts with partial genetic robustness

How adaptive evolution occurs with individually deleterious but jointly beneficial mutations has been one of the major problems in population genetics theory. Adaptation in this case is commonly described as a population's escape from a local peak to a higher peak on Sewall Wright's fitness landscape. Recent molecular genetic and computational studies have suggested that genetic robustness can facilitate such peak shifts. If phenotypic expressions of new mutations are suppressed under genetic robustness, mutations that are otherwise deleterious can accumulate in the population as neutral variants. When the robustness is perturbed by an environmental change or a major mutation, these variants become exposed to natural selection. It is argued that this process promotes adaptation because allelic combinations enriched under genetic robustness can then be positively selected. Here, I propose simple two- and three-locus models of adaptation with partial genetic robustness as suggested by recent studies. The waiting time until the fixation of an adaptive haplotype was observed in stochastic simulations and compared to the expectation without robustness. It is shown that peak shifts can be delayed or accelerated depending on the conditions of genetic robustness. The evolutionary significance of these processes is discussed.     

A quantitative genetic model of two-policy games between relatives

Equations are derived for the change per generation of the population mean of the probability that an individual adopts a policy 1 as opposed to a policy 2 in a behavioral interaction between two diploid individuals of the same generation in which two policies are possible. The probability is assumed to be a quantitative genetic trait determined by many additively acting genes of small effects and an independent environmental component. Equations are derived for the case that interactions occur at random between all members of the population and also for the case that interactions occur between relatives of the same average degree of relatedness. It is assumed that each group of relatives and the number of such groups is sufficiently large. For a quantitative genetic trait with the additional assumption of unlinked loci the latter equation can be heuristically derived from the first by substituting the corresponding inclusive fitness effects. When per locus selection coefficients are small and linkage equilibrium holds, the average degree of relatedness can be equated approximately with Wright's coefficient of relationship. Thus, the quantitative genetic model provides a genetic basis for the inclusive fitness approach toward games between relatives. By contrast, in a monogenic system with major gene effects we obtain substantially different results which contradict those obtained by the inclusive fitness approach in game theory. Applications are made to the hawk-dove game, and the simple and iterated forms of the prisoner's dilemma.     

Populational Heritability: Extending Punnett Square Concepts to Evolution at the Metapopulation Level

In a previous study, using experimental metapopulations of the flour beetle, Tribolium castaneum, we investigated phase III of Wright's shifting balance process (Wade and Griesemer 1998). We experimentally modeled migration of varying amounts from demes of high mean fitness into demes of lower mean fitness (as in Wright's characterization of phase III) as well as the reciprocal (the opposite of phase III). We estimated the meta-populational heritability for this level of selection by regression of offspring deme means on the weighted parental deme means.Here we develop a Punnett Square representation of the inheritance of the group mean to place our empirical findings in a conceptual context similar to Mendelian inheritance of individual traits. The comparison of Punnett Squares for individual and group inheritance shows how the latter concept can be rigorously defined and extended despite the lack of explicitly formulated, simple Mendelian laws of inheritance at the group level. Whereas Wright's phase III combines both interdemic selection and meta-populational inheritance, our formulation separates the issue of meta-populational heritability from that of interdemic selection. We use this conceptual context to discuss the controversies over the levels of selection and the units of inheritance.     

Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.     

Cultural transmission and the evolution of human behaviour: a general approach based on the Price equation

Transmitted culture can be viewed as an inheritance system somewhat independent of genes that is subject to processes of descent with modification in its own right. Although many authors have conceptualized cultural change as a Darwinian process, there is no generally agreed formal framework for defining key concepts such as natural selection, fitness, relatedness and altruism for the cultural case. Here, we present and explore such a framework using the Price equation. Assuming an isolated, independently measurable culturally transmitted trait, we show that cultural natural selection maximizes cultural fitness, a distinct quantity from genetic fitness, and also that cultural relatedness and cultural altruism are not reducible to or necessarily related to their genetic counterparts. We show that antagonistic coevolution will occur between genes and culture whenever cultural fitness is not perfectly aligned with genetic fitness, as genetic selection will shape psychological mechanisms to avoid susceptibility to cultural traits that bear a genetic fitness cost. We discuss the difficulties with conceptualizing cultural change using the framework of evolutionary theory, the degree to which cultural evolution is autonomous from genetic evolution, and the extent to which cultural change should be seen as a Darwinian process. We argue that the nonselection components of evolutionary change are much more important for culture than for genes, and that this and other important differences from the genetic case mean that different approaches and emphases are needed for cultural than genetic processes.     

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Darwinian aspects of molecular evolution at sublinear propagation rates

Mean fitness is non-decreasing in the symmetry sector of the frequency trajectory followed in competitive replication at sublinear propagation rates (parabolic time course). This sector contains the pairwise symmetric distribution of species frequencies and its neighboring states, and represents at least half the possible states of an evolving sublinear system. States in the non-symmetry sector produce a negative rate of change in mean fitness. The heterogeneous steady state attained in a finite sublinear system is destabilized by formation of a variant with above-threshold fitness. Evolution in the post-steady-state interval elevates the fitness threshold for coexistence. Contrary to the proposition that ‘parabolic growth invariably results in the survival of all competing species’, only species with sufficient fitness to avoid subthreshold fitness survive. An erratum to this article is available at .     

The rank ordering of genotypic fitness values predicts genetic constraint on natural selection on landscapes lacking sign epistasis

Sewall Wright's genotypic fitness landscape makes explicit one mechanism by which epistasis for fitness can constrain evolution by natural selection. Wright distinguished between landscapes possessing multiple fitness peaks and those with only a single peak and emphasized that the former class imposes substantially greater constraint on natural selection. Here I present novel formalism that more finely partitions the universe of possible fitness landscapes on the basis of the rank ordering of their genotypic fitness values. In this report I focus on fitness landscapes lacking sign epistasis (i.e., landscapes that lack mutations the sign of whose fitness effect varies epistatically), which constitute a subset of Wright's single peaked landscapes. More than one fitness rank ordering lacking sign epistasis exists for L > 2 (where L is the number of interacting loci), and I find that a highly statistically significant effect exists between landscape membership in fitness rank-ordering partition and two different proxies for genetic constraint, even within this subset of landscapes. This statistical association is robust to population size, permitting general inferences about some of the characteristics of fitness rank orderings responsible for genetic constraint on natural selection.     

Hts1 Encodes Both the Cytoplasmic and Mitochondrial Histidyl-Trna Synthetase of Saccharomyces Cerevisiae: Mutations Alter the Specificity of Compartmentation

Wright's method of estimating the number of genes contributing to the difference in a quantitative character between two populations involves observing the means and variances of the two parental populations and their hybrid populations. Although simple, Wright's method provides seriously biased estimates, largely due to linkage and unequal effects of alleles. A method is suggested to evaluate the bias of Wright's estimate, which relies on estimation of the mean recombination frequency between a pair of loci and a composite parameter of variability of allelic effects and frequencies among loci. Assuming that the loci are uniformly distributed in the genome, the mean recombination frequency can be calculated for some organisms. Theoretical analysis and an analysis of the Drosophila data on distributions of effects of P element inserts on bristle numbers indicate that the value of the composite parameter is likely to be about three or larger for many quantitative characters. There are, however, some serious problems with the current method, such as the irregular behavior of the statistic and large sampling variances of estimates. Because of that, the method is generally not recommended for use unless several favorable conditions are met. These conditions are: the two parental populations are many phenotypic standard deviations apart, linkage is not tight, and the sample size is very large. An example is given on the fruit weight of tomato from a cross with parental populations differing in means by more than 14 phenotypic standard deviations. It is estimated that the number of loci which account for 95% of the genic variance in the F2 population is 16, with a 95% confidence interval of 7-28, and the effect of the leading locus is 13% of the parental difference, with 95% confidence interval 8.5-25.7%.     

COMPARING STRENGTHS OF DIRECTIONAL SELECTION: HOW STRONG IS STRONG?

The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the variance of the selected trait yields the response in units of standard deviation as the product of the heritability and the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and biases in publication or choice of traits to study.     

On an extremum principle in the genetical theory of natural selection

Natural selection causes gene frequency changes in a large population leading to genetic evolution over evolutionary time scales. Such gene frequency changes, however, involve an optimizing principle. According to Kimura, such changes, over a short interval of time, occur in a manner such that the increase in population fitness is maximum for a given distance between parent and daughter generation gene frequencies. But according to Ewens, of all gene frequency changes, including those that lead to the same partial increase in mean fitness as the natural selection gene frequency changes, the natural selection values minimize the generalized distance measure between parent and daughter generation gene frequency values. These two optimality principles happen to be mirror images of each other. However, the optimality principles are restricted to the case where the increase in mean fitness is to thefirst order in natural selection gene frequency changes. I show in this paper that, instead of linear approximation to the increase in mean fitness, the treatment can be fairly general, and the exact increase in mean fitness can be considered so as to include the dominance effects of the genes.     

Early results of closed partial menisectomy.

Thirty patients underwent partial meniscectomy through two puncture incisions by a closed technique under arthroscopic control. "Bucket-handle" fragments were removed from 17 knees, and flaps or tags from the remainder. The mean stay in hospital after operation was 1.3 days and the mean time for return to working fitness was 10.5 days. There were no serious complications. Further study of the long-term results of this technique is needed.     

Nuclear–mitochondrial epistasis: a gene's eye view of genomic conflict

We use population genetic models to investigate the cooperative and conflicting synergistic fitness effects between genes from the nucleus and the mitochondrion. By varying fitness parameters, we examine the scope for conflict relative to cooperation among genomes and the utility of the “gene's eye view” analytical approach, which is based on the marginal average fitness of specific alleles. Because sexual conflict can maintain polymorphism of mitochondrial haplotypes, we can explore two types of evolutionary conflict (genomic and sexual) with one epistatic model. We find that the nuclear genetic architecture (autosomal, X‐linked, or Z‐linked) and the mating system change the regions of parameter space corresponding to the evolution by sexual and genomic conflict. For all models, regardless of conflict or cooperation, we find that population mean fitness increases monotonically as evolution proceeds. Moreover, we find that the process of gene frequency change with positive, synergistic fitnesses is self‐accelerating, as the success of an allele in one genome or in one sex increases the frequency of the interacting allele upon which its success depends. This results in runaway evolutionary dynamics caused by the positive intergenomic associations generated by selection. An inbreeding mating system tends to further accelerate these runaway dynamics because it maintains favorable host–symbiont or male–female gene combinations. In contrast, where conflict predominates, the success of an allele in one genome or in one sex diminishes the frequency of the corresponding allele in the other, resulting in considerably slower evolutionary dynamics. The rate of change of mean fitness is also much faster with positive, synergistic fitnesses and much slower where conflict is predominant. Consequently, selection rapidly fixes cooperative gene combinations, while leaving behind a slowing evolving residue of conflicting gene combinations at mutation–selection balance. We discuss how an emphasis on marginal fitness averages may obscure the interdependence of allelic fitness across genomes, making the evolutionary trajectories appear independent of one another when they are not.     

An optimizing principle of natural selection in evolutionary population genetics

This paper brings together two themes in evolutionary population genetics theory. The first concerns Fisher's Fundamental Theorem of Natural Selection: a recent interpretation of this theorem claims that it is an exact result, relating to the so-called "partial" increase in mean fitness. The second theme concerns the desire to find an optimality principle in genetic evolution. Such a principle is found here: of all gene frequency changes which lead to the same partial increase in mean fitness as the natural selection gene frequency changes, the natural selection values minimize a generalized distance measure between parent and daughter generation gene frequency values.     

Spatial mosaic and interfacial dynamics in a Müllerian mimicry system

Uncovering why spatial mosaics of mimetic morphs are maintained in a Müllerian mimicry system has been a challenging issue in evolutionary biology. In this article, we analyze the reaction diffusion system that describes two-species Müllerian mimicry in one- and two-dimensional habitats. Due to positive frequency-dependent selection, a local population first approaches the state where one of the comimicking patterns predominates, which is followed by slow movement of boundaries where different patterns meet. We then analyze the interfacial dynamics of the boundaries to find whether a stable cline is maintained and to obtain the wave speed if the cline is unstable. The results are: (1) In a spatially uniform habitat the morph with greater base fitness spreads both in one and two species system. (2) The strength of cross-species interaction determines whether the mimetic morph clines of model and mimic species coalesce into the same geographical region or pass through each other. The joint wave speed of clines decreases by increasing the number of comimicking species in the mimicry ring. (3) In spatial heterogeneous habitats, stable clines can be maintained due to the balance between the base fitness gradient and the biased gene flow by negative curvature of boundary. This allows the persistence of a spatial mosaic even if one of the morphs is in every place advantageous over the other. A balanced cline is also maintained if there is a gradient in the population density. (4) A new advantageous morph occurring at a local region is doomed to go to extinction in a finite time if the "radius" of initial distribution is below a threshold. Possible applications to the heliconiine butterfly mimicry ring, heterozygous disadvantage systems of chromosomal rearrangement and hybrid zone, the third phase of Wright's Shifting Balance theory, and cytoplasmic incompatibility are discussed.     

Genetics of Darwinian fitness. II. Pleiotropic selection on fertility and survival

A non-overlapping generation model is proposed which links Wright's adaptive topography concept to the rather inexact notion of Darwinian fitness as survival and reproduction. In general, evolution is seen to weight proportionate increases in survival twice as greatly as proportionate increases in fertility. Certain special cases are also delineated in which measurements of survival and fertility receive equal weight in the fitness equations. In addition, some implications of the present study for Wright's shifting balance theory and models based on Fisher's reproductive value are discussed.