A simple completion of Fisher’s fundamental theorem of natural selection

Fisher''s fundamental theorem of natural selection shows that the part of the rate of change of mean fitness that is due to natural selection equals the additive genetic variance in fitness. Fisher embedded this result in a model of total fitness, adding terms for deterioration of the environment and density dependence. Here, a quantitative genetic version of this neglected model is derived that relaxes its assumptions that the additive genetic variance in fitness and the rate of deterioration of the environment do not change over time, allows population size to vary, and includes an input of mutational variance. The resulting formula for total rate of change in mean fitness contains two terms more than Fisher''s original, representing the effects of stabilizing selection, on the one hand, and of mutational variance, on the other, making clear for the first time that the fundamental theorem deals only with natural selection that is directional (as opposed to stabilizing) on the underlying traits. In this model, the total (rather than just the additive) genetic variance increases mean fitness. The unstructured population allows an explanation of Fisher''s concept of fitness as simply birth rate minus mortality rate, and building up to the definition in structured populations.     

Niche construction and the environmental term of the price equation: How natural selection changes when organisms alter their environments

Organisms construct their own environments and phenotypes through the adaptive processes of habitat choice, habitat construction, and phenotypic plasticity. We examine how these processes affect the dynamics of mean fitness change through the environmental change term of the Price Equation. This tends to be ignored in evolutionary theory, owing to the emphasis on the first term describing the effect of natural selection on mean fitness (the additive genetic variance for fitness of Fisher's Fundamental Theorem). Using population genetic models and the Price Equation, we show how adaptive niche constructing traits favorably alter the distribution of environments that organisms encounter and thereby increase population mean fitness. Because niche-constructing traits increase the frequency of higher-fitness environments, selection favors their evolution. Furthermore, their alteration of the actual or experienced environmental distribution creates selective feedback between niche constructing traits and other traits, especially those with genotype-by-environment interaction for fitness. By altering the distribution of experienced environments, niche constructing traits can increase the additive genetic variance for such traits. This effect accelerates the process of overall adaption to the niche-constructed environmental distribution and can contribute to the rapid refinement of alternative phenotypic adaptations to different environments. Our findings suggest that evolutionary biologists revisit and reevaluate the environmental term of the Price Equation: owing to adaptive niche construction, it contributes directly to positive change in mean fitness; its magnitude can be comparable to that of natural selection; and, when there is fitness G × E, it increases the additive genetic variance for fitness, the much-celebrated first term.     

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.     

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.     

Joint phenotypes,evolutionary conflict and the fundamental theorem of natural selection

Multiple organisms can sometimes affect a common phenotype. For example, the portion of a leaf eaten by an insect is a joint phenotype of the plant and insect and the amount of food obtained by an offspring can be a joint trait with its mother. Here, I describe the evolution of joint phenotypes in quantitative genetic terms. A joint phenotype for multiple species evolves as the sum of additive genetic variances in each species, weighted by the selection on each species. Selective conflict between the interactants occurs when selection takes opposite signs on the joint phenotype. The mean fitness of a population changes not just through its own genetic variance but also through the genetic variance for its fitness that resides in other species, an update of Fisher''s fundamental theorem of natural selection. Some similar results, using inclusive fitness, apply to within-species interactions. The models provide a framework for understanding evolutionary conflicts at all levels.     

The fundamental theorem of natural selection

R. A. Fisher's Fundamental Theorem of Natural Selection states that the rate of increase in the mean fitness of a population ascribable to gene-frequency changes is exactly equal to the additive genetic variance in fitness. It has been widely misunderstood, though clarification has gradually come about particularly through the work of G. R. Price, W. J. Ewens, and S. Lessard. Building on their interpretations we here explain the approach adopted by Fisher (1941), devising a figure as an aid to understanding this important paper.     

The ecology and genetics of fitness in Chlamydomonas. XIII. Fitness of long-term sexual and asexual populations in benign environments

We measured the mean fitness of populations of Chlamydomonas reinhardtii maintained in the laboratory as obligately sexual or asexual populations for about 100 sexual cycles and about 1000 asexual generations. Sexuality (random gamete fusion followed by meiosis) is expected to reduce mutational load and increase mean fitness by combining deleterious mutations from different lines of descent. We found no evidence for this process of mutation clearance: the mean fitness of sexual populations did not exceed that of asexual populations, whether measured through competition or in pure culture. We found instead that sexual progeny suffer an immediate loss in fitness, and that sexual lines maintain genetic variance for fitness. We suggest that sexual populations at equilibrium with selection in a benign environment may be mixtures of several or many epistatic genotypes with nearly equal fitness. Recombination between these genotypes reduces mean fitness and creates genetic variance for fitness. This may provide fuel for continued selection should the environment change.     

Sexual selection and alternative mating behaviours generate demographic stochasticity in small populations.

Recent theory predicts that environmental variation and small population size facilitate the coexistence of alternative phenotypes despite unequal mean fitness. However, traditional studies of reproductive strategies often assume that the stability of alternative mating behaviours relies on equal male fitness. We present results from field observations and experimental manipulations of thermal resources on territories demonstrating the coexistence of alternative reproductive behaviours with unequal fitness. The side-blotched lizard Uta stansburiana exhibits two alternative strategies for territoriality: "usurp" and "defend". Paternity analysis revealed significantly greater mean fitness for "usurpers" than "defenders" in our study of natural variation. Moreover, variance in fitness was significantly higher for usurpers on both experimental and natural plots, implying that "usurp" is a risky strategy with potentially large pay-offs or none at all. We show theoretically that significantly higher variance in usurper fitness can allow for coexistence with defenders despite higher mean fitness of usurpers. This coexistence is facilitated by small population size. Our results have general implications for the evolution of alternative strategies and the maintenance of genetic diversity in small populations.     

Elevated temperature increases genome-wide selection on de novo mutations

Adaptation in new environments depends on the amount of genetic variation available for evolution, and the efficacy by which natural selection discriminates among this variation. However, whether some ecological factors reveal more genetic variation, or impose stronger selection pressures than others, is typically not known. Here, we apply the enzyme kinetic theory to show that rising global temperatures are predicted to intensify natural selection throughout the genome by increasing the effects of DNA sequence variation on protein stability. We test this prediction by (i) estimating temperature-dependent fitness effects of induced mutations in seed beetles adapted to ancestral or elevated temperature, and (ii) calculate 100 paired selection estimates on mutations in benign versus stressful environments from unicellular and multicellular organisms. Environmental stress per se did not increase mean selection on de novo mutation, suggesting that the cost of adaptation does not generally increase in new ecological settings to which the organism is maladapted. However, elevated temperature increased the mean strength of selection on genome-wide polymorphism, signified by increases in both mutation load and mutational variance in fitness. These results have important implications for genetic diversity gradients and the rate and repeatability of evolution under climate change.     

The ecology and genetics of fitness in Chlamydomonas. IX. The rate of accumulation of variation of fitness under selection

Populations of Chlamydomonas founded by single cells were cultured in chemostats for 50 days, representing about 125 generations. The mean and variance of division rate was measured daily by withdrawing cells from the effluent and culturing them for 24 h on filtered effluent medium solidified with agar. Mean fitness did not change during the period of culture, and the behavior of neutral markers indicated that no substitutions of novel beneficial mutations occurred. However, the variance of fitness increased markedly at about the same rate in two replicate populations. The standardized rate, or mutational heritability, was Vm/VE = 4-5 x 10(-3) per generation. This is substantially greater than most other estimates for characters closely correlated with fitness. Moreover, it seems difficult to reconcile with the absence of any change in mean fitness. We investigated the possibility that frequency-dependent selection was created by spatial heterogeneity within the culture vessel by testing cell populations with different phenotypes from the top, bottom, and surface of the chemostats. However, the differentiation of these populations seemed to be attributable to phenotypic plasticity, with no evidence that their characteristics were heritable. Finally, we report an experiment in which lines were selected for about 100 generations on solid or liquid medium. These lines became specifically adapted to the medium on which they were cultured, showing that liquid and solid media, even when chemically identical, provide different conditions of growth for Chlamydomonas. The genetic variance appearing in the cultures was therefore attributed to conditionally neutral mutations that were not expressed in the chemostat. This implies that rates of accumulation of mutational variance measured in the culture environment itself (where this can be done) may greatly underestimate the variation available for a response through selection to environmental change. Moreover, it suggests that chemostat populations may be more dynamic and more diverse than is usually thought.     

Sex‐specific selection under environmental stress in seed beetles

Sexual selection can increase rates of adaptation by imposing strong selection in males, thereby allowing efficient purging of the mutation load on population fitness at a low demographic cost. Indeed, sexual selection tends to be male‐biased throughout the animal kingdom, but little empirical work has explored the ecological sensitivity of this sex difference. In this study, we generated theoretical predictions of sex‐specific strengths of selection, environmental sensitivities and genotype‐by‐environment interactions and tested them in seed beetles by manipulating either larval host plant or rearing temperature. Using fourteen isofemale lines, we measured sex‐specific reductions in fitness components, genotype‐by‐environment interactions and the strength of selection (variance in fitness) in the juvenile and adult stage. As predicted, variance in fitness increased with stress, was consistently greater in males than females for adult reproductive success (implying strong sexual selection), but was similar in the sexes in terms of juvenile survival across all levels of stress. Although genetic variance in fitness increased in magnitude under severe stress, heritability decreased and particularly so in males. Moreover, genotype‐by‐environment interactions for fitness were common but specific to the type of stress, sex and life stage, suggesting that new environments may change the relative alignment and strength of selection in males and females. Our study thus exemplifies how environmental stress can influence the relative forces of natural and sexual selection, as well as concomitant changes in genetic variance in fitness, which are predicted to have consequences for rates of adaptation in sexual populations.     

Natural selection and quantitative genetics of life-history traits in Western women: a twin study

Whether contemporary human populations are still evolving as a result of natural selection has been hotly debated. For natural selection to cause evolutionary change in a trait, variation in the trait must be correlated with fitness and be genetically heritable and there must be no genetic constraints to evolution. These conditions have rarely been tested in human populations. In this study, data from a large twin cohort were used to assess whether selection will cause a change among women in a contemporary Western population for three life-history traits: age at menarche, age at first reproduction, and age at menopause. We control for temporal variation in fecundity (the "baby boom" phenomenon) and differences between women in educational background and religious affiliation. University-educated women have 35% lower fitness than those with less than seven years education, and Roman Catholic women have about 20% higher fitness than those of other religions. Although these differences were significant, education and religion only accounted for 2% and 1% of variance in fitness, respectively. Using structural equation modeling, we reveal significant genetic influences for all three life-history traits, with heritability estimates of 0.50, 0.23, and 0.45, respectively. However, strong genetic covariation with reproductive fitness could only be demonstrated for age at first reproduction, with much weaker covariation for age at menopause and no significant covariation for age at menarche. Selection may, therefore, lead to the evolution of earlier age at first reproduction in this population. We also estimate substantial heritable variation in fitness itself, with approximately 39% of the variance attributable to additive genetic effects, the remainder consisting of unique environmental effects and small effects from education and religion. We discuss mechanisms that could be maintaining such a high heritability for fitness. Most likely is that selection is now acting on different traits from which it did in pre-industrial human populations.     

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.     

The evolution of genetic canalization under fluctuating selection

Abstract.— If the direction of selection changes from generation to generation, the ability to respond to selection is maladaptive: the response to selection in one generation leads to reduced fitness in the next. Because the response is determined by the amount of genetic variance expressed at the phenotypic level, rapidly fluctuating selection should favor modifier genes that reduce the phenotypic effect of alleles segregating at structural loci underlying the trait. Such reduction in phenotypic expression of genetic variation has been named "genetic canalization." I support this argument with a series of two- and multilocus models with alternating linear selection and Gaussian selection with fluctuating optimum. A canalizing modifier gene affects the fitness of its carriers in three ways: (1) it reduces the phenotypic consequences of genetic response to previous selection; (2) it reduces the genetic response to selection, which is manifested as linkage disequilibrium between the modifier and structural loci; and (3) it reduces the phenotypic variance. The first two effects reduce fitness under directional selection sustained for several generations, but improve fitness when the direction of selection has just been reversed. The net effect tends to favor a canalizing modifier under rapidly fluctuating selection regimes (period of eight generations or less). The third effect improves fitness of the modifier allele if the fitness function is convex and reduces it if the function is concave. Under fluctuating Gaussian selection, the population is more likely to experience the concave portion of the fitness function when selection is stronger. Therefore, only weak to moderately strong fluctuating Gaussian selection favors genetic canalization. This paper considerably broadens the conditions that favor genetic canalization, which so far has only been postulated to evolve under long-term stabilizing selection.     

The effect of trait type and strength of selection on heritability and evolvability in an island bird population

The heritability (h²) of fitness traits is often low. Although this has been attributed to directional selection having eroded genetic variation in direct proportion to the strength of selection, heritability does not necessarily reflect a trait's additive genetic variance and evolutionary potential (“evolvability”). Recent studies suggest that the low h² of fitness traits in wild populations is caused not by a paucity of additive genetic variance (V_A) but by greater environmental or nonadditive genetic variance (V_R). We examined the relationship between h² and variance‐standardized selection intensities (i or β_σ), and between evolvability (I_A:V_A divided by squared phenotypic trait mean) and mean‐standardized selection gradients (β_μ). Using 24 years of data from an island population of Savannah sparrows, we show that, across diverse traits, h² declines with the strength of selection, whereas I_A and I_R (V_R divided by squared trait mean) are independent of the strength of selection. Within trait types (morphological, reproductive, life‐history), h², I_A, and I_R are all independent of the strength of selection. This indicates that certain traits have low heritability because of increased residual variance due to the age at which they are expressed or the multiple factors influencing their expression, rather than their association with fitness.     

A general definition of the heritable variation that determines the potential of a population to respond to selection

Genetic selection is a major force shaping life on earth. In classical genetic theory, response to selection is the product of the strength of selection and the additive genetic variance in a trait. The additive genetic variance reflects a population's intrinsic potential to respond to selection. The ordinary additive genetic variance, however, ignores the social organization of life. With social interactions among individuals, individual trait values may depend on genes in others, a phenomenon known as indirect genetic effects. Models accounting for indirect genetic effects, however, lack a general definition of heritable variation. Here I propose a general definition of the heritable variation that determines the potential of a population to respond to selection. This generalizes the concept of heritable variance to any inheritance model and level of organization. The result shows that heritable variance determining potential response to selection is the variance among individuals in the heritable quantity that determines the population mean trait value, rather than the usual additive genetic component of phenotypic variance. It follows, therefore, that heritable variance may exceed phenotypic variance among individuals, which is impossible in classical theory. This work also provides a measure of the utilization of heritable variation for response to selection and integrates two well-known models of maternal genetic effects. The result shows that relatedness between the focal individual and the individuals affecting its fitness is a key determinant of the utilization of heritable variance for response to selection.     

Fisher’s fundamental theorem of inclusive fitness and the change in fitness due to natural selection when conspecifics interact

Competition and cooperation is fundamental to evolution by natural selection, both in animals and plants. Here, I investigate the consequences of such interactions for response in fitness due to natural selection. I provide quantitative genetic expressions for heritable variance and response in fitness due to natural selection when conspecifics interact. Results show that interactions among conspecifics generate extra heritable variance in fitness, and that interacting with kin is the key to evolutionary success because it translates the extra heritable variance into response in fitness. This work also unifies Fisher’s fundamental theorem of natural selection (FTNS) and Hamilton’s inclusive fitness (IF). The FTNS implies that natural selection maximizes fitness, whereas Hamilton proposed maximization of IF. This work shows that the FTNS describes the increase in IF, rather than direct fitness, at a rate equal to the additive genetic variance in fitness. Thus, Hamilton’s IF and Fisher’s FTNS both describe the maximization of IF.     

On a genetic model of intraspecific competition and stabilizing selection

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

Environmental variance in male mating success modulates the positive versus negative impacts of sexual selection on genetic load

Sexual selection on males is predicted to increase population fitness, and delay population extinction, when mating success negatively covaries with genetic load across individuals. However, such benefits of sexual selection could be counteracted by simultaneous increases in genome-wide drift resulting from reduced effective population size caused by increased variance in fitness. Resulting fixation of deleterious mutations could be greatest in small populations, and when environmental variation in mating traits partially decouples sexual selection from underlying genetic variation. The net consequences of sexual selection for genetic load and population persistence are therefore likely to be context dependent, but such variation has not been examined. We use a genetically explicit individual-based model to show that weak sexual selection can increase population persistence time compared to random mating. However, for stronger sexual selection such positive effects can be overturned by the detrimental effects of increased genome-wide drift. Furthermore, the relative strengths of mutation-purging and drift critically depend on the environmental variance in the male mating trait. Specifically, increasing environmental variance caused stronger sexual selection to elevate deleterious mutation fixation rate and mean selection coefficient, driving rapid accumulation of drift load and decreasing population persistence times. These results highlight an intricate balance between conflicting positive and negative consequences of sexual selection on genetic load, even in the absence of sexually antagonistic selection. They imply that environmental variances in key mating traits, and intrinsic genetic drift, should be properly factored into future theoretical and empirical studies of the evolution of population fitness under sexual selection.     

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.