Measuring fitness by means of balancer chromosomes

We present the theoretical background to a new method for measuring genetic variation for total fitness in Drosophila. The method allows heterozygous effects on total fitness of whole wild-type chromosomes to be measured under normal demography with overlapping generations. The wild-type chromosomes are competed against two balancer chromosomes (B1, B2, say), providing a standard genotype B1/B2 against which variation in the fitness effects of the wild-type chromosomes can be assessed. Fitness can be assessed in two ways: (i) at equilibrium of all three chromosomes under heterozygote advantage, and (ii) during displacement of one balancer by the other. Equilibrium with all three chromosomes present will be achieved only if the wild-type homozygote is not too fit, and if the fitnesses of the three heterozygotes are not too unequal. These conditions were not satisfied for any of a sample of 12 lethal-bearing chromosomes isolated from a random-bred laboratory population of Drosophila. At equilibrium, genotypic frequencies show low sensitivity to changes in genotypic fitness. Furthermore, where all four genotypes are viable and fertile, supplementary information from cages with only two chromosomes present and from direct measurements of pre-adult viability are required to estimate fitnesses from frequencies. The invasion method has the advantages of a greater sensitivity and of not requiring further data to estimate fitnesses if the wild-type homozygote is fertile. However, it requires that multiple samples be taken as the invasion progresses. In a discrete generation model, generation time influences fitness estimates from this method and is difficult to estimate accurately from the data. A full age-structured model can also be applied to the data from both types of experiment. For the invasion method, this gives fitness estimates close to those from the discrete generation model.     

Fitness relationships under different temperature regimes in Drosophila melanogaster: the eyeless/shaven-naked genetic system

Genotypic fitnesses were estimated over the temperature range 15°C to 29°C for genotypes of the eyeless/shaven-naked system. Total fitness was determined directly from estimates of mating ability, fecundity and egg-to-adult development time and viability, by gene frequency changes in discrete generation populations and in a single generation population experiment involving culture on a rotational basis at 29°C and 15°C. Genotypic differences were detected for mating ability and egg to adult development time and survival. Heterozygote advantage was observed for total fitness and this effect was greatest at 15°C and for culture on a rotational basis at 29°C and 15°C. There was evidence for genetic associations among some fitness components. The tendency for heterozygote advantage in extreme environments supports the general observation of high expressed genetic variation for fitness under extreme stresses. The results suggest an approach to the understanding of the genetic basis of fitness variation in natural populations based upon direct assessments of environmental stresses of ecological importance.     

Testing the extreme value domain of attraction for distributions of beneficial fitness effects

In modeling evolutionary genetics, it is often assumed that mutational effects are assigned according to a continuous probability distribution, and multiple distributions have been used with varying degrees of justification. For mutations with beneficial effects, the distribution currently favored is the exponential distribution, in part because it can be justified in terms of extreme value theory, since beneficial mutations should have fitnesses in the extreme right tail of the fitness distribution. While the appeal to extreme value theory seems justified, the exponential distribution is but one of three possible limiting forms for tail distributions, with the other two loosely corresponding to distributions with right-truncated tails and those with heavy tails. We describe a likelihood-ratio framework for analyzing the fitness effects of beneficial mutations, focusing on testing the null hypothesis that the distribution is exponential. We also describe how to account for missing the smallest-effect mutations, which are often difficult to identify experimentally. This technique makes it possible to apply the test to gain-of-function mutations, where the ancestral genotype is unable to grow under the selective conditions. We also describe how to pool data across experiments, since we expect few possible beneficial mutations in any particular experiment.     

On the evolution of epistasis I: diploids under selection

One interpretation of recent literature on the evolution of phenotypic modularity is that evolution should act to decrease the degree of interaction between genes that contribute to different phenotypes. This issue is addressed directly here using a fitness scheme determined by two genetic loci and a third locus which modifies a measure of statistical interaction between the fitnesses due to the first two. The equilibrium structure of such an epistasis-modifying locus is studied. It is shown that under well-specified conditions a modifying allele that increases epistasis succeeds. In other words, genetic interactions tend to become stronger. It is speculated that this occurs because the mean fitness in such models is locally increasing as a function of the degree of epistasis.     

Frequency-dependent selection with dominance: a window onto the behavior of the mean fitness

Selection in which fitnesses vary with the changing genetic composition of the population may facilitate the maintenance of genetic diversity in a wide range of organisms. Here, a detailed theoretical investigation is made of a frequency-dependent selection model, in which fitnesses are based on pairwise interactions between the two phenotypes at a diploid, diallelic, autosomal locus with complete dominance. The allele frequency dynamics are fully delimited analytically, along with all possible shapes of the mean fitness function in terms of where it increases or decreases as a function of the current allele frequency in the population. These results in turn allow possibly the first complete characterization of the dynamical behavior by the mean fitness through time under frequency-dependent selection. Here the mean fitness (i) monotonically increases, (ii) monotonically decreases, (iii) initially increases and then decreases, or (iv) initially decreases and then increases as equilibrium is approached. We analytically derive the exact initial and fitness conditions that produce each dynamic and how often each arises. Computer simulations with random initial conditions and fitnesses reveal that the potential decline in mean fitness is not negligible; on average a net decrease occurs 20% of the time and reduces the mean fitness by >17%.     

Variable pleiotropic effects from mutations at the same locus hamper prediction of fitness from a fitness component

The relationship of genotype, fitness components, and fitness can be complicated by genetic effects such as pleiotropy and epistasis and by heterogeneous environments. However, because it is often difficult to measure genotype and fitness directly, fitness components are commonly used to estimate fitness without regard to genetic architecture. The small bacteriophage X174 enables direct evaluation of genetic and environmental effects on fitness components and fitness. We used 15 mutants to study mutation effects on attachment rate and fitness in six hosts. The mutants differed from our lab strain of X174 by only one or two amino acids in the major capsid protein (gpF, sites 101 and 102). The sites are variable in natural and experimentally evolved X174 populations and affect phage attachment rate. Within the limits of detection of our assays, all mutations were neutral or deleterious relative to the wild type; 11 mutants had decreased host range. While fitness was predictable from attachment rate in most cases, 3 mutants had rapid attachment but low fitness on most hosts. Thus, some mutations had a pleiotropic effect on a fitness component other than attachment rate. In addition, on one host most mutants had high attachment rate but decreased fitness, suggesting that pleiotropic effects also depended on host. The data highlight that even in this simple, well-characterized system, prediction of fitness from a fitness component depends on genetic architecture and environment.     

Theories of social selection in human populations.

Using Huntington disease, mental retardation, and schizophrenia, it has been shown that two individuals with identical genotypes or phenotypes have different fitnesses because of affected nuclear family members. Such fitness interaction seems to occur because of cultural and social reactions due to the presence of affected individuals, and the interaction has been termed "social selection." Without assuming any specific genetic control for the social behavior, we can study the effect of social behavior on the incidence of a genetic disease.     

Fluctuation Analysis: Can Estimates Be Trusted?

The estimation of mutation rates and relative fitnesses in fluctuation analysis is based on the unrealistic hypothesis that the single-cell times to division are exponentially distributed. Using the classical Luria-Delbrück distribution outside its modelling hypotheses induces an important bias on the estimation of the relative fitness. The model is extended here to any division time distribution. Mutant counts follow a generalization of the Luria-Delbrück distribution, which depends on the mean number of mutations, the relative fitness of normal cells compared to mutants, and the division time distribution of mutant cells. Empirical probability generating function techniques yield precise estimates both of the mean number of mutations and the relative fitness of normal cells compared to mutants. In the case where no information is available on the division time distribution, it is shown that the estimation procedure using constant division times yields more reliable results. Numerical results both on observed and simulated data are reported.     

Multiple adaptive substitutions during evolution in novel environments

We consider an asexual population under strong selection-weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

Genetic and phenotypic models of natural selection

The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.     

Models of general frequency-dependent selection and mating-interaction effects and the analysis of selection patterns in Drosophila inversion polymorphisms

We investigate mechanisms of balancing selection by extending two deterministic models of selection in a one-locus two-allele genetic system to allow for frequency-dependent fitnesses. Specifically we extend models of constant selection to allow for general frequency-dependent fitness functions for sex-dependent viabilities and multiplicative fertilities, while non-multiplicative mating-dependent components remain constant. We compute protected polymorphism conditions that take the form of harmonic means involving both the frequency- and the mating-dependent parameters. This allows for a direct comparison of the equilibrium properties of the frequency-dependent models with those of the constant models and for an analysis of equilibrium of the general model of constant fertility. We then apply the theory to analyze the maintenance of inversion polymorphisms in Drosophila subobscura and D. pseudoobscura, for which data on empirical fitness component estimates are available in the literature. Regression on fitness estimates obtained at different starting frequencies enables us to implement explicit fitness functions in the models and therefore to perform complete studies of equilibrium and stability for particular sets of data. The results point to frequency dependence of fitness components as the main mechanism responsible for the maintenance of the inversion polymorphisms considered, particularly in relation to heterosis, although we also discuss the contribution of other selection mechanisms.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

The estimation of epistasis in components of fitness in experimental populations of Drosophila melanogaster I. A two-stage maximum likelihood model

Laboratory populations of Drosophila melanogaster bearing the Curly and Plum marked second chromosome inversions were observed in selection experiments for ten discrete generations. Maximum likelihood estimates of the relative fitnesses of Curly, Plum, Curly-Plum, and wild phenotypes were obtained from selection trajectories. Using these estimates, measures of multiplicative and additive epistasis were calculated. These were partitioned into pre-sampling and post-sampling components, and both were found to be significant. In several cases the sign of the epistasis of the two components was reversed, and the direction of net epistasis depended on the particular inversion. The significance of partitioning epistasis into components is discussed in the light of two locus population genetic theory.     

Selfishness and altruism can coexist when help is subject to diminishing returns

Altruism and selfishness are 30-50% heritable in man in both Western and non-Western populations. This genetically based variation in altruism and selfishness requires explanation. In non-human animals, altruism is generally directed towards relatives, and satisfies the condition known as Hamilton's rule. This nepotistic altruism evolves under natural selection only if the ratio of the benefit of receiving help to the cost of giving it exceeds a value that depends on the relatedness of the individuals involved. Standard analyses assume that the benefit provided by each individual is the same but it is plausible in some cases that as more individuals contribute, help is subject to diminishing returns. We analyse this situation using a single-locus two-allele model of selection in a diploid population with the altruistic allele dominant to the selfish allele. The analysis requires calculation of the relationship between the fitnesses of the genotypes and the frequencies of the genes. The fitnesses vary not only with the genotype of the individual but also with the distribution of phenotypes amongst the sibs of the individual and this depends on the genotypes of his parents. These calculations are not possible by direct fitness or ESS methods but are possible using population genetics. Our analysis shows that diminishing returns change the operation of natural selection and the outcome can now be a stable equilibrium between altruistic and selfish alleles rather than the elimination of one allele or the other. We thus provide a plausible genetic model of kin selection that leads to the stable coexistence in the same population of both altruistic and selfish individuals. This may explain reported genetic variation in altruism in man.     

THE EVOLUTION OF FITNESS

In every generation, the mean fitness of populations increases because of natural selection and decreases because of mutations and changes in the environment. The magnitudes of these effects can be measured in two ways: either directly, by comparing the fitnesses of selected and unselected populations, or indirectly, by measuring the additive variance of fitness and making use of the fundamental theorem of natural selection. The available data suggest that the amount by which natural selection increases mean fitness each generation (or degradation decreases mean fitness) will usually be between 0.1% and 30%; more tentatively, it is suggested that values will typically fall between 1% and 10%. These values can be used to set an upper limit of 5%–10% on the genetic advantage of mate choice.     

Fitness conflicts and the costs of sociality in communal egg layers: a theoretical model and empirical tests

Individuals within complex social groups often experience reduced reproduction owing to coercive or suppressive actions of other group members. However, the nature of social and ecological environments that favour individual acceptance of such costs of sociality is not well understood. Taxa with short periods of direct social interaction, such as some communal egg layers, are interesting models for study of the cost of social interaction because opportunities to control reproduction of others are limited to brief periods of reproduction. To understand the conditions under which communal egg layers are in fitness conflict and thus likely to influence each other's reproduction, we develop an optimality model involving a brood guarding 'host' and a nonguarding disperser, or 'egg dumper'. The model shows that when, where intermediate-sized broods have highest survival, lifetime inclusive fitnesses of hosts and dumpers are often optimized with different numbers of dumped eggs. We hypothesize that resolution of this conflict may involve attempts by one party to manipulate the other's reproduction. To test model predictions we used a lace bug (Heteroptera: Tingidae) that shows both hosts and egg dumpers as well as increased offspring survival in response to communal egg laying. We found that egg-dumping lace bugs oviposit a number of eggs that very closely matches predicted fitness optimum for hosts rather than predicted optimum of dumpers. This result suggests that dumpers pay a social cost for communal egg laying, a cost that may occur through host suppression of dumper reproduction. Although dumper allocation of eggs is thus sub-optimal for dumpers, previous models show that the decision to egg dump is nevertheless evolutionarily stable, possibly because hosts permit just enough dumper oviposition to encourage commitment to the behaviour.     

Beneficial fitness effects are not exponential for two viruses

The distribution of fitness effects for beneficial mutations is of paramount importance in determining the outcome of adaptation. It is generally assumed that fitness effects of beneficial mutations follow an exponential distribution, for example, in theoretical treatments of quantitative genetics, clonal interference, experimental evolution, and the adaptation of DNA sequences. This assumption has been justified by the statistical theory of extreme values, because the fitnesses conferred by beneficial mutations should represent samples from the extreme right tail of the fitness distribution. Yet in extreme value theory, there are three different limiting forms for right tails of distributions, and the exponential describes only those of distributions in the Gumbel domain of attraction. Using beneficial mutations from two viruses, we show for the first time that the Gumbel domain can be rejected in favor of a distribution with a right-truncated tail, thus providing evidence for an upper bound on fitness effects. Our data also violate the common assumption that small-effect beneficial mutations greatly outnumber those of large effect, as they are consistent with a uniform distribution of beneficial effects.     

Frequency-dependent viability selection (a single-locus, multi-phenotype model)

Frequency-dependent natural selection models are examined where the viability of an individual in the diploid population is determined by its phenotype and the frequency of other phenotypes present. The equilibria of the multi-phenotypic system are characterized through local mean fitness functions. It is shown that stability can best be analyzed by combining the principles of maximization of population mean fitness with the evolutionary stability conditions that apply when phenotypic fitnesses relative to the genetic constraints are equal.