The nearly neutral and selection theories of molecular evolution under the fisher geometrical framework: substitution rate, population size, and complexity

The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model in which evolutionary rate does not depend on population size, but is critically dependent on the complexity of organisms and mutation size. For SR as well as VR we found that key parameters of molecular evolution are linked by biological factors, and we showed that they cannot be fixed independently by arbitrary criteria, as has usually been assumed in previous molecular evolutionary models.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

The Effects of Background and Interference Selection on Patterns of Genetic Variation in Subdivided Populations

It is well known that most new mutations that affect fitness exert deleterious effects and that natural populations are often composed of subpopulations (demes) connected by gene flow. To gain a better understanding of the joint effects of purifying selection and population structure, we focus on a scenario where an ancestral population splits into multiple demes and study neutral diversity patterns in regions linked to selected sites. In the background selection regime of strong selection, we first derive analytic equations for pairwise coalescent times and F_ST as a function of time after the ancestral population splits into two demes and then construct a flexible coalescent simulator that can generate samples under complex models such as those involving multiple demes or nonconservative migration. We have carried out extensive forward simulations to show that the new methods can accurately predict diversity patterns both in the nonequilibrium phase following the split of the ancestral population and in the equilibrium between mutation, migration, drift, and selection. In the interference selection regime of many tightly linked selected sites, forward simulations provide evidence that neutral diversity patterns obtained from both the nonequilibrium and equilibrium phases may be virtually indistinguishable for models that have identical variance in fitness, but are nonetheless different with respect to the number of selected sites and the strength of purifying selection. This equivalence in neutral diversity patterns suggests that data collected from subdivided populations may have limited power for differentiating among the selective pressures to which closely linked selected sites are subject.     

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.     

The Properties of Adaptive Walks in Evolving Populations of Fungus

The rarity of beneficial mutations has frustrated efforts to develop a quantitative theory of adaptation. Recent models of adaptive walks, the sequential substitution of beneficial mutations by selection, make two compelling predictions: adaptive walks should be short, and fitness increases should become exponentially smaller as successive mutations fix. We estimated the number and fitness effects of beneficial mutations in each of 118 replicate lineages of Aspergillus nidulans evolving for approximately 800 generations at two population sizes using a novel maximum likelihood framework, the results of which were confirmed experimentally using sexual crosses. We find that adaptive walks do indeed tend to be short, and fitness increases become smaller as successive mutations fix. Moreover, we show that these patterns are associated with a decreasing supply of beneficial mutations as the population adapts. We also provide empirical distributions of fitness effects among mutations fixed at each step. Our results provide a first glimpse into the properties of multiple steps in an adaptive walk in asexual populations and lend empirical support to models of adaptation involving selection towards a single optimum phenotype. In practical terms, our results suggest that the bulk of adaptation is likely to be accomplished within the first few steps.     

Rapid evolutionary escape by large populations from local fitness peaks is likely in nature

Fitness interactions between loci in the genome, or epistasis, can result in mutations that are individually deleterious but jointly beneficial. Such epistasis gives rise to multiple peaks on the genotypic fitness landscape. The problem of evolutionary escape from such local peaks has been a central problem of evolutionary genetics for at least 75 years. Much attention has focused on models of small populations, in which the sequential fixation of valley genotypes carrying individually deleterious mutations operates most quickly owing to genetic drift. However, valley genotypes can also be subject to mutation while transiently segregating, giving rise to copies of the high fitness escape genotype carrying the jointly beneficial mutations. In the absence of genetic recombination, these mutations may then fix simultaneously. The time for this process declines sharply with increasing population size, and it eventually comes to dominate evolutionary behavior. Here we develop an analytic expression for N(crit), the critical population size that defines the boundary between these regimes, which shows that both are likely to operate in nature. Frequent recombination may disrupt high-fitness escape genotypes produced in populations larger than N(crit) before they reach fixation, defining a third regime whose rate again slows with increasing population size. We develop a novel expression for this critical recombination rate, which shows that in large populations the simultaneous fixation of mutations that are beneficial only jointly is unlikely to be disrupted by genetic recombination if their map distance is on the order of the size of single genes. Thus, counterintuitively, mass selection alone offers a biologically realistic resolution to the problem of evolutionary escape from local fitness peaks in natural populations.     

Riding across the selection landscape: fitness consequences of annual variation in reproductive characteristics

Evolutionary models estimating phenotypic selection in character size usually assume that the character is invariant across reproductive bouts. We show that variation in the size of reproductive traits may be large over multiple events and can influence fitness in organisms where these traits are produced anew each season. With data from populations of two orchid species, Caladenia valida and Tolumnia variegata, we used Bayesian statistics to investigate the effect on the distribution in fitness of individuals when the fitness landscape is not flat and when characters vary across reproductive bouts. Inconsistency in character size across reproductive periods within an individual increases the uncertainty of mean fitness and, consequently, the uncertainty in individual fitness. The trajectory of selection is likely to be muddled as a consequence of variation in morphology of individuals across reproductive bouts. The frequency and amplitude of such changes will certainly affect the dynamics between selection and genetic drift.     

Relative Fitness, Teleology, and the Adaptive Landscape

The metaphor of the adaptive landscape, introduced by Sewall Wright in 1932, has played, and continues to play, a central role in much evolutionary thought. I argue that the use of this metaphor is tied to a teleological view of the evolutionary process, in which natural selection directs evolution toward an improved future state. I argue further that the use of “relative fitnesses” standardized to an arbitrary value, which is closely connected with the metaphor of an adaptive landscape, produces a disconnect between the mean fitness of a population and any real property of that population. This allows for a vague and ill-defined improvement to occur under the influence of selection. Instead, I suggest that relative fitnesses should be standardized by the mean absolute fitness (expected population growth rate), so that they express the expected rate of increase in frequency, rather than number. Under this definition, the mean relative fitness of all populations is always 1.0, and never changes as long as the population continues to exist.     

Population dynamics in a spin-glass model of chemical evolution

Summary We introduce a simple model describing the evolution of a population of information-carrying macromolecules. We discuss the asymptotic dependence of the variability of the population on different parameters, representing the severity or the fluctuations of the environment. We show the existence of a transition separating a neutralist evolutionary regime from a trapped one. We investigate the dependence of the evolutionary behavior of the population on the correlation properties of the fitness landscape.     

A model for habitat selection and species distribution derived from central place foraging theory

We have developed a habitat selection model based on central place foraging theory. An individual’s decision to include a patch in its habitat depends on the marginal fitness contribution of that patch, which is characterized by its quality and distance to the central place. The essence of the model we have developed is a fitness isocline which is a function of patch quality and travel time to the patch. It has two parameters: the maximum travel distance to a patch of infinite quality and a coefficient that appropriately scales quality by travel time. Patches falling below the isocline will have positive marginal fitness values and should be included in the habitat. The maximum travel distance depends on the availability and quality of patches, as well as on the forager’s life history, whereas the scaling parameter mostly depends on life history properties. Using the model, we derived a landscape quality metric (which can be thought of as a connectivity measure) that sums the values of available habitat in the landscape around a central place. We then fitted the two parameters to foraging data on breeding white storks (Ciconia ciconia) and estimated landscape quality, which correlated strongly with reproductive success. Landscape quality was then calculated for a larger region where re-introduction of the species is currently going on in order to demonstrate how this model can also be regarded as a species distribution model. In conclusion, we have built a general habitat selection model for central place foragers and a novel way of estimating landscape quality based on a behaviorally scaled connectivity metric.     

Simulating natural selection in landscape genetics

Linking landscape effects to key evolutionary processes through individual organism movement and natural selection is essential to provide a foundation for evolutionary landscape genetics. Of particular importance is determining how spatially-explicit, individual-based models differ from classic population genetics and evolutionary ecology models based on ideal panmictic populations in an allopatric setting in their predictions of population structure and frequency of fixation of adaptive alleles. We explore initial applications of a spatially-explicit, individual-based evolutionary landscape genetics program that incorporates all factors--mutation, gene flow, genetic drift and selection--that affect the frequency of an allele in a population. We incorporate natural selection by imposing differential survival rates defined by local relative fitness values on a landscape. Selection coefficients thus can vary not only for genotypes, but also in space as functions of local environmental variability. This simulator enables coupling of gene flow (governed by resistance surfaces), with natural selection (governed by selection surfaces). We validate the individual-based simulations under Wright-Fisher assumptions. We show that under isolation-by-distance processes, there are deviations in the rate of change and equilibrium values of allele frequency. The program provides a valuable tool (cdpop v1.0; http://cel.dbs.umt.edu/software/CDPOP/) for the study of evolutionary landscape genetics that allows explicit evaluation of the interactions between gene flow and selection in complex landscapes.     

An experimental assessment of artificial within-family selection for fitness in conservation programs

Equalizing familiar contributions is the simplest recommended strategy to maintain genetic diversity in conservation programs. However, this method implies a relaxation of natural selection and the possibility of accumulation of deleterious mutations. Computer simulations have shown that performing selection within families for fitness traits in a conservation program can be useful to alleviate such problems. We thus carried out an experiment with the model species Drosophila melanogaster in order to assess whether or not selection for fitness traits can be useful. We considered a fitness trait (pupa productivity) that was first checked to perform as a typical fitness component. The trait showed an inbreeding depression of 1.2 per 1 % increase in inbreeding and an asymmetrical response to selection with average realized heritabilities of about 0.04 in the upward direction and an order of magnitude larger (0.36) in the downward direction. The management experiment indicated that artificial within-family selection for fitness had only a marginal success for two reasons. First, there was not an appreciable decline in fitness across the experiment despite the low population sizes assumed (N = 10 or 20), even in the population not subjected to selection. This result is compatible with fitness models which imply the segregation of few deleterious mutations of large effect. Second, artificial selection within families had a limited impact on the trait, as one expects for a typical fitness component with very low heritability.     

Genealogical process on a correlated fitness landscape

We study with extensive numerical simulation the genealogical process of 2N haploid genetic sequences. The sequences are under selective pressure, and fitness values are assigned at random, but with a tunable degree of correlation to the fitness values of closely related sequences. The genealogies that we observe can be classified into three different categories, corresponding to different regimes of the mutation rate. At low mutation rates, the sequences remain localized around a small number of central sequences, which leads to trees with short pairwise distances and slow turnover of the most recent common ancestor of the population. At high mutation rates, we observe trees similar (but not identical) to those of neutral evolution. In this regime, the population drifts rapidly, and selection does not influence the distribution of fitness values in the population. The third regime, for intermediate mutation rates, is only found in strongly correlated landscapes. It resembles the one for high mutation rates in that the population drifts rapidly, but nevertheless selection still shapes the distribution of fitness values.     

Finite size scaling in the local abundances of geographic populations

We analyzed the statistical distribution of intra-specific local abundances for a set North American breeding bird species. We constructed frequency plots for every species and found that they showed long-tail power-law behavior, truncated at an upper abundance cut-off value. Based on finite size scaling arguments, we investigated whether frequency curves may be considered scaled copies of each other. Data collapse was possible after taking powers of the total abundance of each species, in order to correct deviations from the underlying universal finite size scaling function (UFSS). The UFSS power law exponent oscillated in time within the regime of unbounded variance, which is consistent with the wild fluctuations that characterize ecological phenomena. We speculate that our results may eventually be linked to other law-like macroecological phenomena, such as energetic constraints reported in allometric scaling.     

Are There Laws of Genome Evolution?

Research in quantitative evolutionary genomics and systems biology led to the discovery of several universal regularities connecting genomic and molecular phenomic variables. These universals include the log-normal distribution of the evolutionary rates of orthologous genes; the power law-like distributions of paralogous family size and node degree in various biological networks; the negative correlation between a gene's sequence evolution rate and expression level; and differential scaling of functional classes of genes with genome size. The universals of genome evolution can be accounted for by simple mathematical models similar to those used in statistical physics, such as the birth-death-innovation model. These models do not explicitly incorporate selection; therefore, the observed universal regularities do not appear to be shaped by selection but rather are emergent properties of gene ensembles. Although a complete physical theory of evolutionary biology is inconceivable, the universals of genome evolution might qualify as "laws of evolutionary genomics" in the same sense "law" is understood in modern physics.     

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.