Phenotypic noise and the cost of complexity

Experimental studies demonstrate the existence of phenotypic diversity despite constant genotype and environment. Theoretical models based on a single phenotypic character predict that during an adaptation event, phenotypic noise should be positively selected far from the fitness optimum because it increases the fitness of the genotype, and then be selected against when the population reaches the optimum. It is suggested that because of this fitness gain, phenotypic noise should promote adaptive evolution. However, it is unclear how the selective advantage of phenotypic noise is linked to the rate of evolution, and whether any advantage would hold for more realistic, multidimensional phenotypes. Indeed, complex organisms suffer a cost of complexity, where beneficial mutations become rarer as the number of phenotypic characters increases. Using a quantitative genetics approach, we first show that for a one-dimensional phenotype, phenotypic noise promotes adaptive evolution on plateaus of positive fitness, independently from the direct selective advantage on fitness. Second, we show that for multidimensional phenotypes, phenotypic noise evolves to a low-dimensional configuration, with elevated noise in the direction of the fitness optimum. Such a dimensionality reduction of the phenotypic noise promotes adaptive evolution and numerical simulations show that it reduces the cost of complexity.     

Proceedings of the SMBE Tri-National Young Investigators' Workshop 2005. Mutation rate and the cost of complexity

Two recent theoretical studies of adaptation suggest that more complex organisms tend to adapt more slowly. Specifically, in Fisher's "geometric" model of a finite population where multiple traits are under optimizing selection, the average progress ensuing from a single mutation decreases as the number of traits increases--the "cost of complexity." Here, I draw on molecular and histological data to assess the extent to which on a large phylogenetic scale, this predicted decrease in the rate of adaptation per mutation is mitigated by an increase in the number of mutations per generation as complexity increases. As an index of complexity for multicellular organisms, I use the number of visibly distinct types of cell in the body. Mutation rate is the product of mutational target size and population mutation rate per unit target. Despite much scatter, genome size appears to be positively correlated with complexity (as indexed by cell-type number), which along with other considerations suggests that mutational target size tends to increase with complexity. In contrast, effective population mutation rate per unit target appears to be negatively correlated with complexity. The net result is that mutation rate probably does tend to increase with complexity, although probably not fast enough to eliminate the cost of complexity.     

Fisher'S geometrical model of fitness landscape and variance in fitness within a changing environment

The fitness of an individual can be simply defined as the number of its offspring in the next generation. However, it is not well understood how selection on the phenotype determines fitness. In accordance with Fisher's fundamental theorem, fitness should have no or very little genetic variance, whereas empirical data suggest that is not the case. To bridge these knowledge gaps, we follow Fisher's geometrical model and assume that fitness is determined by multivariate stabilizing selection toward an optimum that may vary among generations. We assume random mating, free recombination, additive genes, and uncorrelated stabilizing selection and mutational effects on traits. In a constant environment, we find that genetic variance in fitness under mutation-selection balance is a U-shaped function of the number of traits (i.e., of the so-called "organismal complexity"). Because the variance can be high if the organism is of either low or high complexity, this suggests that complexity has little direct costs. Under a temporally varying optimum, genetic variance increases relative to a constant optimum and increasingly so when the mutation rate is small. Therefore, mutation and changing environment together can maintain high genetic variance. These results therefore lend support to Fisher's geometric model of a fitness landscape.     

The evolution of mutation rate in finite asexual populations

In this article, we model analytically the evolution of mutation rate in asexual organisms. Three selective forces are present. First, everything else being equal, individuals with higher mutation rate have a larger fitness, thanks to the energy and time saved by not replicating DNA accurately. Second, as a flip side, the genome of these individuals is replicated with errors that may negatively affect fitness. Third, and conversely, replication errors have a potential benefit if beneficial mutations are to be generated. Our model describes the fate of modifiers of mutation rate under the three forces and allows us to predict the long-term evolutionary trajectory of mutation rate. We obtain three major results. First, in asexuals, the needs for both adaptation and genome preservation are not evolutionary forces that can stabilize mutation rate at an intermediate optimum. When adaptation has a significant role, it primarily destabilizes mutation rate and yields the emergence of strong-effect mutators. Second, in contrast to what is usually believed, the appearance of modifiers with large mutation rate is more likely when the fitness cost of each deleterious mutation is weak, because the cost of replication errors is then paid after a delay. Third, in small populations, and even if adaptations are needed, mutation rate is always blocked at the minimum attainable level, because the rate of adaptation is too slow to play a significant role. Only populations whose size is above a critical mass see their mutation rate affected by the need for adaptation.     

The population genetics of adaptation: the adaptation of DNA sequences

I describe several patterns characterizing the genetics of adaptation at the DNA level. Following Gillespie (1983, 1984, 1991), I consider a population presently fixed for the ith best allele at a locus and study the sequential substitution of favorable mutations that results in fixation of the fittest DNA sequence locally available. Given a wild type sequence that is less than optimal, I derive the fitness rank of the next allele typically fixed by natural selection as well as the mean and variance of the jump in fitness that results when natural selection drives a substitution. Looking over the whole series of substitutions required to reach the best allele, I show that the mean fitness jumps occurring throughout an adaptive walk are constrained to a twofold window of values, assuming only that adaptation begins from a reasonably fit allele. I also show that the first substitution and the substitution of largest effect account for a large share of the total fitness increase during adaptation. I further show that the distribution of selection coefficients fixed throughout such an adaptive walk is exponential (ignoring mutations of small effect), a finding reminiscent of that seen in Fisher's geometric model of adaptation. Last, I show that adaptation by natural selection behaves in several respects as the average of two idealized forms of adaptation, perfect and random.     

Experimental adaptation of Salmonella typhimurium to mice

Experimental evolution is a powerful approach to study the dynamics and mechanisms of bacterial niche specialization. By serial passage in mice, we evolved 18 independent lineages of Salmonella typhimurium LT2 and examined the rate and extent of adaptation to a mainly reticuloendothelial host environment. Bacterial mutation rates and population sizes were varied by using wild-type and DNA repair-defective mutator (mutS) strains with normal and high mutation rates, respectively, and by varying the number of bacteria intraperitoneally injected into mice. After <200 generations of adaptation all lineages showed an increased fitness as measured by a faster growth rate in mice (selection coefficients 0.11-0.58). Using a generally applicable mathematical model we calculated the adaptive mutation rate for the wild-type bacterium to be >10(-6)/cell/generation, suggesting that the majority of adaptive mutations are not simple point mutations. For the mutator lineages, adaptation to mice was associated with a loss of fitness in secondary environments as seen by a reduced metabolic capability. During adaptation there was no indication that a high mutation rate was counterselected. These data show that S. typhimurium can rapidly and extensively increase its fitness in mice but this niche specialization is, at least in mutators, associated with a cost.     

On adaptive accuracy and precision in natural populations

Adaptation is usually conceived as the fit of a population mean to a fitness optimum. Natural selection, however, does not act only to optimize the population mean. Rather, selection normally acts on the fitness of individual organisms in the population. Furthermore, individual genotypes do not produce invariant phenotypes, and their fitness depends on how precisely they are able to realize their target phenotypes. For these reasons we suggest that it is better to conceptualize adaptation as accuracy rather than as optimality. The adaptive inaccuracy of a genotype can be measured as a function of the expected distance of its associated phenotype from a fitness optimum. The less the distance, the more accurate is the adaptation. Adaptive accuracy has two components: the deviance of the genotypically set target phenotype from the optimum and the precision with which this target phenotype can be realized. The second component, the adaptive precision, has rarely been quantified as such. We survey the literature to quantify how much of the phenotypic variation in wild populations is due to imprecise development. We find that this component is often substantial and highly variable across traits. We suggest that selection for improved precision may be important for many traits.     

Compensating for our load of mutations: freezing the meltdown of small populations

We have investigated the reduction of fitness caused by the fixation of new deleterious mutations in small populations within the framework of Fisher's geometrical model of adaptation. In Fisher's model, a population evolves in an n-dimensional character space with an adaptive optimum at the origin. The model allows us to investigate compensatory mutations, which restore fitness losses incurred by other mutations, in a context-dependent manner. We have conducted a moment analysis of the model, supplemented by the numerical results of computer simulations. The mean reduction of fitness (i.e., expected load) scaled to one is approximately n/(n+2Ne), where Ne is the effective population size. The reciprocal relationship between the load and Ne implies that the fixation of deleterious mutations is unlikely to cause extinction when there is a broad scope for compensatory mutations, except in very small populations. Furthermore, the dependence of load on n implies that pleiotropy plays a large role in determining the extinction risk of small populations. Differences and similarities between our results and those of a previous study on the effects of Ne and n are explored. That the predictions of this model are qualitatively different from studies ignoring compensatory mutations implies that we must be cautious in predicting the evolutionary fate of small populations and that additional data on the nature of mutations is of critical importance.     

Adaptation to slow environmental change, with apparent anticipation of selection

We investigate a genetic model of a large population of sexual organisms in a changing environment. The organisms are subject to stabilising selection on a quantitative trait, with environmental change causing the fitness optimum to move. When the fitness optimum moves slowly, adaptation to the changing environment occurs by means of reasonably well-separated substitutions at the loci controlling the trait. In this way, the trait generally tracks the moving optimum, but in such a case, the population may exhibit periods of time where the mean trait value overshoots the moving optimal trait value, thereby exhibiting an apparent anticipation of selection. The mechanism underlying this phenomenon is determined from consideration of a simpler model that correctly captures the observed dynamical behaviour. We note that very slow rates of changes of traits are seen in the fossil record and the present work may be relevant to this topic.     

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.     

FISHER'S GEOMETRIC MODEL WITH A MOVING OPTIMUM

Fisher's geometric model has been widely used to study the effects of pleiotropy and organismic complexity on phenotypic adaptation. Here, we study a version of Fisher's model in which a population adapts to a gradually moving optimum. Key parameters are the rate of environmental change, the dimensionality of phenotype space, and the patterns of mutational and selectional correlations. We focus on the distribution of adaptive substitutions, that is, the multivariate distribution of the phenotypic effects of fixed beneficial mutations. Our main results are based on an “adaptive‐walk approximation,” which is checked against individual‐based simulations. We find that (1) the distribution of adaptive substitutions is strongly affected by the ecological dynamics and largely depends on a single composite parameter γ, which scales the rate of environmental change by the “adaptive potential” of the population; (2) the distribution of adaptive substitution reflects the shape of the fitness landscape if the environment changes slowly, whereas it mirrors the distribution of new mutations if the environment changes fast; (3) in contrast to classical models of adaptation assuming a constant optimum, with a moving optimum, more complex organisms evolve via larger adaptive steps.     

Perspective: Here's to Fisher,additive genetic variance,and the fundamental theorem of natural selection

Fisher's fundamental theorem of natural selection, that the rate of change of fitness is given by the additive genetic variance of fitness, has generated much discussion since its appearance in 1930. Fisher tried to capture in the formula the change in population fitness attributable to changes of allele frequencies, when all else is not included. Lessard's formulation comes closest to Fisher's intention, as well as this can be judged. Additional terms can be added to account for other changes. The "theorem" as stated by Fisher is not exact, and therefore not a theorem, but it does encapsulate a great deal of evolutionary meaning in a simple statement. I also discuss the effectiveness of reproductive-value weighting and the theorem in integrated form. Finally, an optimum principle, analogous to least action and Hamilton's principle in physics, is discussed.     

Restrictions to RNA virus adaptation: an experimental approach

Some basic properties of RNA viruses are their high mutation rate, their enormous population sizes and their short generation time. These properties allow RNA virus populations to quickly explore fitness landscapes. A great adaptability has been amply demonstrated in experimental, as well as in natural, populations of RNA viruses. However, at least from a theoretical point of view, a limit to the extent of viral adaptation may exist as a consequence of adaptive trade-offs arising during evolution in changing environmental conditions. Here, I review previously published results searching for such fitness trade-offs. The following scenario has been explored: the cost of host-range expansion, the cost of resistance to antiviral drugs, and the adaptation to different population densities. Despite the environmental conditions tested, results show a common pattern: whenever a virus adapt to a simple environmental situation it pays a cost in terms of adaptation to alternative situations. However, in those cases where the virus has been simultaneously adapted to different environmental conditions, this cost disappears or, at least, is greatly reduced. Finally, and as another factor imposing a limit to their speed of adaptation, I review results showing that clonal interference also plays an important role during viral evolution. This revised version was published online in August 2006 with corrections to the Cover Date.     

The stochastic edge in adaptive evolution

In a recent article, Desai and Fisher proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time tau until a new class of mutants has been established and interpret 1/tau as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine tau: Desai and Fisher back extrapolate the value of tau from the best-fit class's exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows nonexponentially during the relevant time interval. Here, we substantially extend Desai and Fisher's analysis of the stochastic edge. We show that we can apply Desai and Fisher's method to high speeds by either exponentially back extrapolating from finite time or using a nonexponential back extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al.) and agree well with numerical simulations.     

The genetics of sex ratio distortion by cytoplasmic infection under maternal and contagious transmission: An epidemiological study

Nuclear and cytoplasmic determinants jointly influence the sex ratio in several organisms. A mathematical model of a maternally inherited extra-chromosomal agent that affects the fitness of its carriers and distorts the sex ratio in their broods is analyzed. The agent is transmitted through the cytoplasm from mother to daughter, or it may pass contagiously among females of the same generation. It is shown that under natural selection the deviation between the population sex ratio and Fisher's optimum value evolves to a minimum.     

A general multivariate extension of Fisher's geometrical model and the distribution of mutation fitness effects across species

The evolution of complex organisms is a puzzle for evolutionary theory because beneficial mutations should be less frequent in complex organisms, an effect termed "cost of complexity." However, little is known about how the distribution of mutation fitness effects (f(s)) varies across genomes. The main theoretical framework to address this issue is Fisher's geometric model and related phenotypic landscape models. However, it suffers from several restrictive assumptions. In this paper, we intend to show how several of these limitations may be overcome. We then propose a model of f(s) that extends Fisher's model to account for arbitrary mutational and selective interactions among n traits. We show that these interactions result in f(s) that would be predicted by a much smaller number of independent traits. We test our predictions by comparing empirical f(s) across species of various gene numbers as a surrogate to complexity. This survey reveals, as predicted, that mutations tend to be more deleterious, less variable, and less skewed in higher organisms. However, only limited difference in the shape of f(s) is observed from Escherichia coli to nematodes or fruit flies, a pattern consistent with a model of random phenotypic interactions across many traits. Overall, these results suggest that there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works. More generally, the model seems to qualitatively capture and possibly explain the variation of f(s) from lower to higher organisms, which opens a large array of potential applications in evolutionary genetics.     

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.     

Fitness cost from fluctuating ultraviolet radiation in Daphnia magna

Solar ultraviolet radiation (UVR) is an important environmental threat for organisms in aquatic systems, but its temporally variable nature makes the understanding of its effects ambiguous. The aim of our study was to assess potential fitness costs associated with fluctuating UVR in the aquatic zooplankter Daphnia magna. We investigated individual survival, reproduction and behaviour when exposed to different UVR treatments. Individuals exposed to fluctuating UVR, resembling natural variations in cloud cover, had the lowest fitness (measured as the number of offspring produced during their lifespan). By contrast, individuals exposed to the same, but constant UVR dose had similar fitness to control individuals (not exposed to UVR), but they showed a significant reduction in daily movement. The re-occurring threat response to the fluctuating UVR treatment thus had strong fitness costs for D. magna, and we found no evidence for plastic behavioural responses when continually being exposed to UVR, despite the regular, predictable exposure schedule. In a broader context, our results imply that depending on how variable a stressor is in nature, populations may respond with alternative strategies, a framework that could promote rapid population differentiation and local adaptation.     

Selective advantage for sexual reproduction with random haploid fusion

This article develops a simplified set of models describing asexual and sexual replication in unicellular diploid organisms. The models assume organisms whose genomes consist of two chromosomes, where each chromosome is assumed to be functional if it is equal to some master sequence σ₀, and non-functional otherwise. We review the previously studied case of selective mating, where it is assumed that only haploids with functional chromosomes can fuse, and also consider the case of random haploid fusion. When the cost for sex is small, as measured by the ratio of the characteristic haploid fusion time to the characteristic growth time, we find that sexual replication with random haploid fusion leads to a greater mean fitness for the population than a purely asexual strategy. However, independently of the cost for sex, we find that sexual replication with a selective mating strategy leads to a higher mean fitness than the random mating strategy. The results of this article are consistent with previous studies suggesting that sex is favored at intermediate mutation rates, for slowly replicating organisms, and at high population densities. Furthermore, the results of this article provide a basis for understanding sex as a stress response in unicellular organisms such as Saccharomyces cerevisiae (Baker’s yeast).