A method to infer positive selection from marker dynamics in an asexual population

MOTIVATION: The observation of positive selection acting on a mutant indicates that the corresponding mutation has some form of functional relevance. Determining the fitness effects of mutations thus has relevance to many interesting biological questions. One means of identifying beneficial mutations in an asexual population is to observe changes in the frequency of marked subsets of the population. We here describe a method to estimate the establishment times and fitnesses of beneficial mutations from neutral marker frequency data. RESULTS: The method accurately reproduces complex marker frequency trajectories. In simulations for which positive selection is close to 5% per generation, we obtain correlations upwards of 0.91 between correct and inferred haplotype establishment times. Where mutation selection coefficients are exponentially distributed, the inferred distribution of haplotype fitnesses is close to being correct. Applied to data from a bacterial evolution experiment, our method reproduces an observed correlation between evolvability and initial fitness defect.     

Theory of evolutionary molecular engineering through simultaneous accumulation of advantageous mutations

We examined the effectiveness of an "adaptive leap" strategy using the "mutation scrambling" method as an efficient optimization technique (Uchiyama, 2000;J. Biochem.128, 441-447) for cases where mutational (rough) additivity holds in fitness. The mutation scrambling method is composed of the following three processes: (1) preliminary selection of several advantageous single-point mutations introduced in a wild-type sequence; (2) preparation of various multiple-point mutants incorporating the advantageous mutant residue or wild-type residue at each of the selected sites, by scrambling the mutant residues and wild-type residues (this process is called mutation scrambling); and (3) selection of the fittest through screening of the mutant pool. The fitness distribution in the mutant pool is controlled by the mixing ratio of the mutant residues to the wild-type residues. We focused on the mutant fitness distribution and obtained the optimal mixing ratio which efficiently generates superior multiple-point mutants with high fitnesses. As a result, we found that the optimal ratio lies between 7/3 and 9/1 in realistic cases. Particularly, this strategy works well in cases where the number of component mutations is large and the size of the population to be screened is small. Analysis of the mutant fitness distributions with various mixing ratios is also useful to explore local fitness landscapes.     

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.     

A minimum on the mean number of steps taken in adaptive walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on "gradient" adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L = e-1, where e approximately 2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L = e-1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.     

ADAPTIVE WALKS AND DISTRIBUTION OF BENEFICIAL FITNESS EFFECTS

We study the adaptation dynamics of a maladapted asexual population on rugged fitness landscapes with many local fitness peaks. The distribution of beneficial fitness effects is assumed to belong to one of the three extreme value domains, viz. Weibull, Gumbel, and Fréchet. We work in the strong selection‐weak mutation regime in which beneficial mutations fix sequentially, and the population performs an uphill walk on the fitness landscape until a local fitness peak is reached. A striking prediction of our analysis is that the fitness difference between successive steps follows a pattern of diminishing returns in the Weibull domain and accelerating returns in the Fréchet domain, as the initial fitness of the population is increased. These trends are found to be robust with respect to fitness correlations. We believe that this result can be exploited in experiments to determine the extreme value domain of the distribution of beneficial fitness effects. Our work here differs significantly from the previous ones that assume the selection coefficient to be small. On taking large effect mutations into account, we find that the length of the walk shows different qualitative trends from those derived using small selection coefficient approximation.     

Analysis of a local fitness landscape with a model of the rough Mt. Fuji-type landscape: application to prolyl endopeptidase and thermolysin

A method of analysis of a local fitness landscape for a current biopolymer is presented. Based on the assumption of additivity of mutational effects in the biopolymer, we assigned a site-fitness to each residue at each site. The assigned values of site-fitnesses were obtained by the least-squares method to minimize discrepancies between experimental fitnesses and theoretical ones. As test cases, we analyzed a section of a local landscape for the thermostability of prolyl endopeptidase and that for the enzymatic activity of thermolysin. These sections were proved to be of the rough Mt. Fuji-type with straight theta values of larger than 1.0, where straight theta is defined as the ratio of the "mean slope" to the "degree of roughness" on the fitness surface. Furthermore, we theoretically explained discrepancies between the fitnesses of multiple mutants and those predicted based on strict additivity of the component mutations by using a model of the rough Mt. Fuji-type landscape. According to this model, the discrepancies depend on the local landscape property (such as the straight theta value) and the location of the wild type on the landscape and the mean change in fitness by the component mutations. Our results suggest that this model may provide a good approximation of real sections of local landscapes for current biopolymers phenomenologically.     

The phenotype-fitness map in experimental evolution of phages

Evolutionary biologists commonly interpret adaptations of organisms by reference to a phenotype-fitness map, a model of how different states of a phenotype affect fitness. Notwithstanding the popularity of this approach, it remains difficult to directly test these mappings, both because the map often describes only a small subset of phenotypes contributing to total fitness and because direct measures of fitness are difficult to obtain and compare to the map. Both limitations can be overcome for bacterial viruses (phages) grown in the experimental condition of unlimited hosts. A complete accounting of fitness requires 3 easily measured phenotypes, and total fitness is also directly measurable for arbitrary genotypes. Yet despite the presumed transparency of this system, directly estimated fitnesses often differ from fitnesses calculated from the phenotype-fitness map. This study attempts to resolve these discrepancies, both by developing a more exact analytical phenotype-fitness map and by exploring the empirical foundations of direct fitness estimates. We derive an equation (the phenotype-fitness map) for exponential phage growth that allows an arbitrary distribution of lysis times and burst sizes. We also show that direct estimates of fitness are, in many cases, plausibly in error because the population has not attained stable age distribution and thus violates the model underlying the phenotype-fitness map. In conjunction with data provided here, the new understanding appears to resolve a discrepancy between the reported fitness of phage T7 and the substantially lower value calculated from its phenotype-fitness map.     

A sharp minimum on the mean number of steps taken in adaptive walks

It was recently conjectured by H.A. Orr that from a random initial point on a random fitness landscape of alphabetic sequences with one-mutation adjacency, chosen from a larger class of landscapes, no adaptive algorithm can arrive at a local optimum in fewer than on average e-1 steps. Here, using an example in which the mean number of steps to a local optimum equals (A-1)/A, where A is the number of distinct "letters" in the "alphabet" from which sequences are constructed, it is shown that as originally stated, the conjecture does not hold. It is also demonstrated that (A-1)/A is a sharp minimum on the mean number of steps taken in adaptive walks on fitness landscapes of alphabetic sequences with one-mutation adjacency. As the example that achieves the new lower bound has properties that are not often considered as potential attributes for fitness landscapes-non-identically distributed fitnesses and negative fitness correlations for adjacent points-a weaker set of conditions characteristic of more commonly studied fitness landscapes is proposed under which the lower bound on the mean length of adaptive walks is conjectured to equal e-1.     

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%.     

On the evolutionary stability of sink populations

The evolution of adaptive behaviours can influence population dynamics. Conversely, population dynamics can affect both the rate and direction of adaptive evolution. This paper examines reasons why sink populations – populations maintained by immigration, preventing local extinction – might persist in the habitat repertoire of a species over evolutionary time-scales. Two such reasons correspond to standard explanations for deviations from an ideal free habitat distribution: organisms may not be free to settle in whichever habitat has the highest potential fitness, and may be constrained by costs, perceptual limitations, or mode of dispersal in the acuity of their habitat selectivity. Here, I argue that a third general reason for persistent sink populations is provided by unstable population dynamics in source habitats. I present a simple model illustrating how use of a sink habitat may be selectively advantageous, when a source population has unstable dynamics (which necessarily reflects temporal variation in local fitnesses). Species with unstable local dynamics in high-quality habitats should be selected to utilize a broader range of habitats than species with stable local dynamics, and in particular in some circumstances should utilize sink habitats. This observation has implications for the direction of niche evolution, and the likelihood of niche conservatism.     

Phenogenetic drift and the evolution of genotype-phenotype relationships

Maintenance of a stable two-locus polymorphism is analyzed statistically by fitting a logistic regression with a quadratic function of genotypic fitnesses to the probability for a fitness set to maintain a polymorphism. The regression is fitted using a data set containing information on stable equilibria maintained by 32,00 randomly generated fitness sets with three recombination values (0. 005, 0.05, 0.5). Fitted logistic regressions discriminate with 88 to 90% accuracy between fitness sets maintaining and not maintaining a stable internal equilibrium, which implies the existence of a fitness structure (balance of fitnesses) maintaining a two-locus polymorphism. Aspects of the balance of fitnesses revealed by logistic regressions are discussed. It is demonstrated that logistic regression also discriminates between types of a stable polymorphism: globally stable polymorphism, several simultaneously stable polymorphisms, and stable equilibria in addition to a polymorphic one, which implies that different balances of fitnesses are responsible for the maintenance of different types of polymorphism.     

Genetic variation for total fitness in Drosophila melanogaster: complex yet replicable patterns

The extent of genetic variation in fitness is a crucial issue in evolutionary biology and yet remains largely unresolved. In Drosophila melanogaster, we have devised a method that allows the net effects on fitness of heterozygous wild-type chromosomes to be measured, by competing them against two different "balancer" chromosomes. We have applied the method to a large sample of 40 wild-type third chromosomes and have measured fitnesses of nonlethal chromosomes as well as chromosomes bearing recessive lethals. The measurements were made in the environment to which the population was adapted and did not involve inbreeding. The results show an extraordinary similarity in the behavior of replicates of the same chromosome, indicating consistent genetic effects on total fitness. Some invading chromosomes increased rapidly and some slowly, and some rose to appreciable frequency after several months, but then declined again: in every case, the same pattern was seen in each replicate. We estimated relative fitnesses, rates of change of fitness, and relative viabilities, for each chromosome. There were significant fluctuations around the fitted model, which were also highly replicable. Wild-type chromosomes varied substantially in their effects on heterozygous fitness, and these effects vary through time, most likely as a result of genotype x environment interactions.     

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.     

A macroscopic approach to population genetics

Conventional population genetics uses as primitive variables the frequencies and fitnesses of individual genes. This paper develops a formalism whose primitive variables are the frequencies and fitnesses of genotypes and environmental histories in a population. From the mathematical relation that describes genetic variation and selection of genotypes and environmental histories we derive a sequence of more specialized equations, including those of the conventional theory. Some familiar formulas of the conventional theory (including Fisher's fundamental theorem, the formula relating the rate of change of a metric character to selection pressure, and the definitions of broad and narrow heritability) are shown to be special cases of simpler and more general formulas. It is shown that the “genotypic value” of a trait, together with its heritability, may depend strongly on genotype-environment correlations.A generalization of Fisher's fundamental theorem shows that the rate of evolution of a trait depends on the skewness of its fitness distribution. An equation relating the second derivative of the mean fitness to the skewness is derived.Finally, the formalism is applied in a preliminary way to a recent theory of genetic variation (Layzer,1978a), according to which the genetic variability of a trait is selected along with the trait itself. It is shown that there is positive feedback between the two kinds of selection.     

Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.     

The evolution of alternative mating strategies in variable environments

Summary We assessed the influence of phenotypic plasticity in age at maturity on the maintenance of alternative mating strategies in male Atlantic salmon,Salmo salar. We calculated the fitness,r, associated with the parr and the anadromous strategies, using age-specific survival data from the field and strategy-specific fertilization data from the laboratory. The fitness of each strategy depended largely on mate competition (numbers of parr per female, i.e. parr frequency) and on age at maturity. Fitness declined with increasing numbers of parr per female with equilibrium frequencies (at which the fitnesses of each strategy are equal) being within the range observed in the wild. Equilibrium parr frequencies declined with decreasing growth rate and increasing age at maturity. Within populations, the existence of multiple age-specific sets of fitness functions suggests that the fitnesses of alternative strategies are best represented as multidimensional surfaces. The points of intersection of these surfaces, whose boundaries encompass natural variation in age at maturity and mate competition, define an evolutionarily stable continuum (ESC) of strategy frequencies along which the fitnesses associated with each strategy are equal. We propose a simple model that incorporates polygenic thresholds of a largely environmentally-controlled trait (age at maturity) to provide a mechanism by which an ESC can be maintained within a population. An indirect test provides support for the prediction that growth-rate thresholds for parr maturation exist and are maintained by stabilizing selection. Evolutionarily stable continua, maintained by negative frequency-dependent selection on threshold traits, provide a theoretical basis for understanding how alternative life histories can evolve in variable environments.     

Chromosome Interactions in DROSOPHILA MELANOGASTER. I. Viability Studies

The nature of fitness interactions is an important, yet unsolved, question in population genetics. We compare the egg-to-adult viability of individuals homozygous for either a second or a third chromosome with the viability of individuals homozygous for both chromosomes simultaneously. On the average, the viability of the two-chromosome homozygotes is somewhat greater than expected assuming that the fitnesses of the single-chromosome homozygotes interact in a multiplicative fashion. This result differs from previous observations that indicate either no significant deviations from the expectation or lower-than-expected average fitnesses for the double homozygotes.     

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.     

Pleiotropic Effects on Fitness of Mutations Affecting Viability in DROSOPHILA MELANOGASTER

The relative viabilities and fitnesses of wild-type second chromosomes in heterozygous condition were determined. Joint analysis of these permitted an estimation of a parameter that relates the viability effect of a mutation to its effect on fitness as a whole. For newly arisen mutations, the estimate was slightly greater than one, indicating that the reductions in viability caused by these mutations are associated with reductions in other components of fitness. For mutations from an equilibrium population, the estimate of the parameter was near zero, implying that the deleterious viability effects of these mutations are compensated by improvements in other aspects of fitness.