Properties of adaptive walks on uncorrelated landscapes under strong selection and weak mutation

We examine properties of adaptive walks on uncorrelated (i.e. random) fitness landscapes starting from moderately fit genotypes under strong selection weak mutation. As an extension of Orr's model for a single step in an adaptive walk under these conditions, we show that the fitness rank of the dominant genotype in a population after the fixation of a beneficial mutation is, on average, (i+6)/4, where i is the fitness rank of the starting genotype. This accounts for the change in rank due to acquiring a new set of single-mutation neighbors after fixing a new allele through natural selection. Under this scenario, adaptive walks can be modeled as a simple Markov chain on the space of possible fitness ranks with an absorbing state at i = 1, from which no beneficial mutations are accessible. We find that these walks are typically short and are often completed in a single step when starting from a moderately fit genotype. As in Orr's original model, these results are insensitive to both the distribution of fitness effects and most biological details of the system under consideration.     

Microevolution in an electronic microcosm

The evolution of microbial populations in simple environments such as chemostats is still not fully understood. The classical interpretation of adaptation involves a process of successive substitution whereby a new dominant genotype arises by mutation from the genotype previously dominant and spreads more or less rapidly through the population until it is nearly fixed. The population is, thus, nearly uniform most of the time. Some observations suggest that the process may be more complicated, but it remains formidably difficult to assemble the phylogeny of an evolving culture in sufficient detail to be sure. We report experiments with an electronic microcosm inhabited by self-replicating computer programs whose phylogeny can be rendered completely transparent. The physiology of these programs is different in many respects from that of organic creatures, but their population biology has many features in common, including a very extensive, if not unbounded, range of variation. Experimental populations evolved through point mutations (many of which were quasi-neutral when they were viable) and through rearrangements that led to a change in genome size and often had large effects on fitness. As a general rule, smaller genomes execute fewer instructions in order to replicate, the rate of replication increases as the number of instructions executed declines, and the rate of replication in pure culture is a good predictor of success in mixture. When cultured with CPU (central processing unit) time as the sole limiting resource, smaller genomes, therefore, evolve as a correlated response to natural selection for faster replication. The genetic basis of adaptation was highly contingent and always differed in replicate experiments. The pattern of evolution depends on mutation rate. At low mutation rates of 0.01 per genome per generation or less, we observed classic periodic selection, with each dominant genotype descending from the previous dominant and rising to a frequency of 0.8 or more. At higher mutation rates of about 0.1 per genome per generation, the most abundant genotypes rarely exceeded a frequency of about 0.4, and rare genotypes present in a few copies comprised a large part of the population. New dominant genotypes did not usually descend directly from previous dominants but, instead, from one of the many rare or moderately abundant genotypes. We suggest that the conventional chemostat paradigm may hold only as a special case at very low mutation rates and that the dynamics and diversity of evolving populations, even in the simplest conditions, may be more complex than is usually recognized. Artificial genetic autoadaptive systems are likely to be useful in constructing theory for situations that lie beyond the boundary of conventional population genetics.     

Visualizing fitness landscapes

Fitness landscapes are a classical concept for thinking about the relationship between genotype and fitness. However, because the space of genotypes is typically high-dimensional, the structure of fitness landscapes can be difficult to understand and the heuristic approach of thinking about fitness landscapes as low-dimensional, continuous surfaces may be misleading. Here, I present a rigorous method for creating low-dimensional representations of fitness landscapes. The basic idea is to plot the genotypes in a manner that reflects the ease or difficulty of evolving from one genotype to another. Such a layout can be constructed using the eigenvectors of the transition matrix describing the evolution of a population on the fitness landscape when mutation is weak. In addition, the eigendecomposition of this transition matrix provides a new, high-level view of evolution on a fitness landscape. I demonstrate these techniques by visualizing the fitness landscape for selection for the amino acid serine and by visualizing a neutral network derived from the RNA secondary structure genotype-phenotype map.     

An explicit model for the inbreeding load in the evolutionary analysis of selfing

I present analytical predictions for the equilibrium inbreeding load expected in a population under mutation, selection, and a regular mating system for any population size and for any magnitude and recessivity of the deleterious effects. Using this prediction, I deduce the relative fitness of mutant alleles with small effect on selfing to explore the situations where selfing or outcrossing are expected to evolve. The results obtained are in agreement with previous literature, showing that natural selection is expected to lead to stable equilibria where populations show either complete outcrossing or complete selfing, and that selfing is promoted by large deleterious mutation rates. I find that the evolution of selfing is favored by a large recessivity of deleterious effects, while the magnitude of homozygous deleterious effects only becomes relevant in relatively small populations. This result contradicts the standard assumption that purging in large populations will only promote selfing when homozygous deleterious effects are large, and implies that previously published results obtained assuming lethal mutations in large populations can be extrapolated to nonlethal alleles of similar recessivity. This conclusion and the general approach used in this analysis can be useful in the study of the evolution of mating systems.     

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.     

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.     

Choosing fitness-enhancing innovations can be detrimental under fluctuating environments

The ability to predict the consequences of one's behavior in a particular environment is a mechanism for adaptation. In the absence of any cost to this activity, we might expect agents to choose behaviors that maximize their fitness, an example of directed innovation. This is in contrast to blind mutation, where the probability of becoming a new genotype is independent of the fitness of the new genotypes. Here, we show that under environments punctuated by rapid reversals, a system with both genetic and cultural inheritance should not always maximize fitness through directed innovation. This is because populations highly accurate at selecting the fittest innovations tend to over-fit the environment during its stable phase, to the point that a rapid environmental reversal can cause extinction. A less accurate population, on the other hand, can track long term trends in environmental change, keeping closer to the time-average of the environment. We use both analytical and agent-based models to explore when this mechanism is expected to occur.     

Computer experiments on the evolution of sex: the haploid case

We have conducted a set of computer experiments that investigate the conditions under which sexual reproduction can evolve. In these experiments, haploid genomes are treated as integer strings able to undergo mutation and, in the case of sexual strings, recombination, with string sequence (the genotype) determining fitness. Our results indicate that the likelihood of a rare sexual mutant spreading through an otherwise asexual population is small when sexual reproduction entails the classical two-fold cost of males. However, our results indicate that when mutation rates are not excessively high, sexual reproduction constitutes a more efficient fitness optimization algorithm, allowing a sexual population to adapt more quickly to its environment than an asexual population.     

Epistasis and the adaptability of an RNA virus

We have explored the patterns of fitness recovery in the vesicular stomatitis RNA virus. We show that, in our experimental setting, reversions to the wild-type genotype were rare and fitness recovery was at least partially driven by compensatory mutations. We compared compensatory adaptation for genotypes carrying (1) mutations with varying deleterious fitness effects, (2) one or two deleterious mutations, and (3) pairs of mutations showing differences in the strength and sign of epistasis. In all cases, we found that the rate of fitness recovery and the proportion of reversions were positively affected by population size. Additionally, we observed that mutations with large fitness effect were always compensated faster than mutations with small fitness effect. Similarly, compensatory evolution was faster for genotypes carrying a single deleterious mutation than for those carrying pairs of mutations. Finally, for genotypes carrying two deleterious mutations, we found evidence of a negative correlation between the epistastic effect and the rate of compensatory evolution.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.     

Density dependent selection in parthenogenetic and self-mating populations

The consequences of density dependent selection on genetically heterogeneous, diploid populations reproducing by self-mating or various parthenogenetic mechanisms is investigated. A logistic fitness function that depends upon both the genotype of an individual and the density of the population is used. Such a fitness function simultaneously determines the population size and the genotype frequencies. The equilibrium solutions to a one locus and two locus model are given as well as some generalizations to n loci and nonlogistic fitness functions. Conditions are found that maintain several different genotypes simultaneously in the equilibrium population. The interaction of such selection with the genetic mechanisms which determine mode of reproduction in parthenogenetic populations is also discussed.     

Population dependent Fourier decomposition of fitness landscapes over recombination spaces: Evolvability of complex characters

The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes. Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p < n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal. Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.     

Is beauty in the eye of the beholder?

Theories of beauty were evaluated by requiring subjects to “evolve” a beautiful female face using a Genetic Algorithm. In this procedure, a computer program generated a small population of faces (first generation of phenotypes) from a set of random binary strings (genotypes). Genotypes specified the shapes and soft tissue anthropometrics of facial features. Each of the first generation of faces was rated by a subject (relative fitness measure) for beauty. The fittest genotypes then bred in proportion to their fitness, with crossover and mutation of the binary strings, to produce offspring which were again rated by the subject. This process continued until the most beautiful face, for that subject, was evolved. Forty Caucasian subjects (20 M, 20 F) were required to evolve their idealized beautiful female face using this procedure. The features and soft tissue anthropometrics of their final composites were compared to population norms. Also, the final composites, and different faces generated from the same data base, were rated for beauty by independent judges. The results support the conclusion that the concept of facial beauty is the result of sexual selection, and a beautiful female face has features and proportions indicative on high fertility.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.     

Population genetics from an information perspective

Some basic effects of population genetics are derived governing the occurrences of alleles A(i)and genotypes A(i)A(j)among its members. A principle of extreme physical information (EPI) is used. These effects are (1) the equation of genetic change, (2) Fisher's theorem of partial change, (3) a new uncertainty principle, and (4) the monotonic decrease of Fisher information with time, indicating increased disorder for the population. General conditions of population change are allowed: fitness coefficients w(ij)generally changing with time [except in effect (2)], population randomly or non-randomly mating, and a general number of loci present within each chromosome. EPI is a practical tool for deriving probability laws. It is an outgrowth of a physical process that occurs during any act of measurement. Here the measurement is the random observation of a genotype A(i)A(j). This observation is to be used to estimate the time of the observation, called "evolutionary time". The measurement activity incurs errors in the estimated observation time and fitness value of the observed genotype. By the Cramer-Rao inequality, the product of the two uncertainties must exceed unity [effect (3)]. The Fisher information I in data space is postulated to originate in the space of the genotype where it had some generally larger value J. The EPI principle extremizes the loss of information (I--J) with I=1/2 J. The solution gives rise to effects (1) and (2). Finally, it is shown that effect (4) holds when the population approaches an equilibrium state, e.g. for time values greater than a threshold if fitness coefficients w(ij)are constant. EPI provides a common framework for deriving physical laws and laws of population genetics. The new effects (3) and (4) are confirmed through computer simulation.     

Mutation-selection balance: ancestry,load, and maximum principle

We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.     

Adaptation to the cost of resistance in a haploid clonally reproducing organism: the role of mutation, migration and selection

A model of compensatory evolution with respect to fungicide resistance in a haploid clonally reproducing fungus is developed in which compensatory mutations mitigate fitness costs associated with resistance. The role of mutation, migration and selection in invasion of rare genotypes when the environment changes from unsprayed to sprayed and from sprayed to unsprayed is analysed in detail. In some circumstances (ignoring back mutations) stable internal steady-state values for multiple genotypes can be obtained. In these cases a threshold value (f*) for the fraction of the population exposed to the fungicide can be derived for the transition between different steady-state conditions. Conditions are derived for invasion-when-rare of resistant genotypes at boundary equilibria established sometime after the onset of spraying and conversely of sensitive genotypes sometime after the cessation of spraying are derived. In these cases conditions are presented for (a) the invasion of a resistant genotype with a compensatory mutation (resistant-compensated) into a sensitive-uncompensated population that has re-equilibrated following the onset of spraying and (b) the invasion of a susceptible-uncompensated genotype into a resistant-compensated population that has re-equilibrated following the cessation of spraying, provided certain conditions are met. A resistant-compensated genotype may be fixed (or at near-fixation) in the population following a period of spraying, provided the mean intrinsic growth rate of the resistant-compensated genotype in a sprayed environment (over exposed and non-exposed parts of the population) is greater than that of the susceptible-uncompensated genotype. The fraction of the population exposed (the efficiency of spraying) is critical in this respect. However, it is possible for a sensitive-uncompensated genotype to invade provided there is no fitness gain associated with the resistant-compensated genotype, introduction by migration occurs following equilibration of the population to the new environment, and competitive effects are re-imposed when spraying ceases. We further derive a threshold level for the resident resistant-compensated population to reduce to following the cessation of spraying, such that the introduced susceptible-uncompensated genotype will invade. These results will be of use in determining the long-term persistence of resistance in a pathogen population once a fungicide is no longer effective and removed from use.     

Genetic loads under fitness‐dependent mutation rates

Abstract Although much theory depends on the genome‐wide rate of deleterious mutations, good estimates of the mutation rate are scarce and remain controversial. Furthermore, mutation rate may not be constant, and a recent study suggests that mutation rates are higher in mildly stressful environments. If mutation rate is a function of condition, then individuals carrying more mutations will tend to be in worse condition and therefore produce more mutations. Here I examine the mean fitnesses of sexual and asexual populations evolving under such condition‐dependent mutation rates. The equilibrium mean fitness of a sexual population depends on the shape of the curve relating fitness to mutation rate. If mutation rate declines synergistically with increasing condition the mean fitness will be much lower than if mutation rate declines at a diminishing rate. In contrast, asexual populations are less affected by condition‐dependent mutation rates. The equilibrium mean fitness of an asexual population only depends on the mutation rate of the individuals in the least loaded class. Because such individuals have high fitness and therefore a low mutation rate, asexual populations experience less genetic load than sexual populations, thus increasing the twofold cost of sex.     

Inbreeding tolerance and fitness costs in wild bottlenose dolphins

In wild populations, inbreeding tolerance is expected to evolve where the cost of avoidance exceeds that of tolerance. We show that in a wild population of bottlenose dolphins found in East Shark Bay, Western Australia, levels of inbreeding are higher than expected by chance alone, and demonstrate that inbreeding is deleterious to female fitness in two independent ways. We found that inbred females, and females with inbred calves, have reduced fitness (lower calving success). We further show that one of the costs of inbreeding is extended weaning age, and that females'' earlier calves are more likely to be inbred. While the exact causes of inbreeding remain obscure, our results indicate that one factor is female age, and thus experience. Any inbreeding avoidance mechanisms such as female evasion of kin, or male dispersal, do not seem to be completely effective in this population, which supports the view that inbreeding avoidance does not always evolve wherever inbreeding incurs a cost.     

A population genetics model for multiple quantitative traits exhibiting pleiotropy and epistasis

We study a population genetics model of an organism with a genome of L(tot)loci that determine the values of T quantitative traits. Each trait is controlled by a subset of L loci assigned randomly from the genome. There is an optimum value for each trait, and stabilizing selection acts on the phenotype as a whole to maintain actual trait values close to their optima. The model contains pleiotropic effects (loci can affect more than one trait) and epistasis in fitness. We use adaptive walk simulations to find high-fitness genotypes and to study the way these genotypes are distributed in sequence space. We then simulate the evolution of haploid and diploid populations on these fitness landscapes and show that the genotypes of populations are able to drift through sequence space despite stabilizing selection on the phenotype. We study the way the rate of drift and the extent of the accessible region of sequence space is affected by mutation rate, selection strength, population size, recombination rate, and the parameters L and T that control the landscape shape. There are three regimes of the model. If LTL(tot), there are many small peaks that can be spread over a wide region of sequence space. Compensatory neutral mutations are important in the population dynamics in this case.