Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

Most quantitative traits in most populations exhibit heritable genetic variation. Lande proposed that high levels of heritable variation may be maintained by mutation in the face of stabilizing selection. Several analyses have appeared of two distinct models with n additive polygenic loci subject to mutation and stabilizing selection. Each is reviewed and a new analysis and model are presented. Lande and Fleming analyzed extensions of a model originally treated by Kimura which assumes a continuum of possible allelic effects at each locus. Latter and Bulmer analyzed a model with diallelic loci. The published analyses of these models lead to qualitatively different predictions concerning the dependence of the equilibrium genetic variance on the underlying biological parameters. A new asymptotic analysis of the Kimura model shows that the different predictions are not consequences of the number of alleles assumed but rather are attributable to assumptions concerning the relative magnitudes of per locus mutation rates, the phenotypic effects of mutation, and the intensity of selection. This conclusion is reinforced by analysis of a model with triallelic loci. None of the approximate analyses presented are mathematically rigorous. To quantify their accuracy and display the domains of validity for alternative approximations, numerically determined equilibria are presented. In addition, empirical estimates of mutation rates and selection intensity are reviewed, revealing weaknesses in both the data and its connection to the models. Although the mathematical results and underlying biological requirements of my analyses are quite different from those of Lande, the results do not refute his hypothesis that considerable additive genetic variance may be maintained by mutation-selection balance. However, I argue that the validity of this hypothesis can only be determined with additional data and mathematics.     

The molecular origins of evolutionary innovations

The history of life is a history of evolutionary innovations, qualitatively new phenotypic traits that endow their bearers with new, often game-changing abilities. We know many individual examples of innovations and their natural history, but we know little about the fundamental principles of phenotypic variability that permit new phenotypes to arise. Most phenotypic innovations result from changes in three classes of systems: metabolic networks, regulatory circuits, and macromolecules. I here highlight two important features that these classes of systems share. The first is the ubiquity of vast genotype networks - connected sets of genotypes with the same phenotype. The second is the great phenotypic diversity of small neighborhoods around different genotypes in genotype space. I here explain that both features are essential for the phenotypic variability that can bring forth qualitatively new phenotypes. Both features emerge from a common cause, the robustness of phenotypes to perturbations, whose origins are linked to life in changing environments.     

THE EVOLUTION OF PHENOTYPIC PLASTICITY IN LIFE-HISTORY TRAITS: PREDICTIONS OF REACTION NORMS FOR AGE AND SIZE AT MATURITY

We used life-history theory to predict reaction norms for age and size at maturation. We assumed that fecundity increases with size and that juvenile mortality rates of offspring decrease as ages-at-maturity of parents increase, then calculated the reaction norm by varying growth rate and calculating an optimal age at maturity for each growth rate. The reaction norm for maturation should take one of at least four shapes that depend on specific relations between changes in growth rates and changes in adult mortality rates, juvenile mortality rates, or both. Most organisms should mature neither at a fixed size nor at a fixed age, but along an age-size trajectory. The model makes possible a clear distinction between the genetic and phenotypic components of variation. The evolved response to selection is reflected in the shape and position of the reaction norm. The phenotypic response of a single organism to rapid or slow growth is defined by the location of its maturation event as a point on the reaction norm. A quantitative test with data from 19 populations and species of fish showed that predictions were in good agreement with observations (r = 0.93, P < 0.0001). The predictions of the model also agreed qualitatively with observed phenotypic variation in age and size at maturity in humans, platyfish, fruit flies, and red deer. This preliminary success suggests that experiments designed to test the predictions directly will be worthwhile.     

Comparative methods with sampling error and within-species variation: contrasts revisited and revised

Comparative methods analyses have usually assumed that the species phenotypes are the true means for those species. In most analyses, the actual values used are means of samples of modest size. The covariances of contrasts then involve both the covariance of evolutionary changes and a fraction of the within-species phenotypic covariance, the fraction depending on the sample size for that species. Ives et al. have shown how to analyze data in this case when the within-species phenotypic covariances are known. The present model allows them to be unknown and to be estimated from the data. A multivariate normal statistical model is used for multiple characters in samples of finite size from species related by a known phylogeny, under the usual Brownian motion model of change and with equal within-species phenotypic covariances. Contrasts in each character can be obtained both between individuals within a species and between species. Each contrast can be taken for all of the characters. These sets of contrasts, each the same contrast taken for different characters, are independent. The within-set covariances are unequal and depend on the unknown true covariance matrices. An expectation-maximization algorithm is derived for making a reduced maximum likelihood estimate of the covariances of evolutionary change and the within-species phenotypic covariances. It is available in the Contrast program of the PHYLIP package. Computer simulations show that the covariances are biased when the finiteness of sample size is not taken into account and that using the present model corrects the bias. Sampling variation reduces the power of inference of covariation in evolution of different characters. An extension of this method to incorporate estimates of additive genetic covariances from a simple genetic experiment is also discussed.     

The evolutionary significance of cis-regulatory mutations

For decades, evolutionary biologists have argued that changes in cis-regulatory sequences constitute an important part of the genetic basis for adaptation. Although originally based on first principles, this claim is now empirically well supported: numerous studies have identified cis-regulatory mutations with functionally significant consequences for morphology, physiology and behaviour. The focus has now shifted to considering whether cis-regulatory and coding mutations make qualitatively different contributions to phenotypic evolution. Cases in which parallel mutations have produced parallel trait modifications in particular suggest that some phenotypic changes are more likely to result from cis-regulatory mutations than from coding mutations.     

Adaptive dynamics in diploid, sexual populations and the evolution of reproductive isolation

Evolutionary branching is the process whereby an initially monomorphic population evolves to a point where it undergoes disruptive selection and splits up into two phenotypically diverging lineages. We studied evolutionary branching in three models that are ecologically identical but that have different genetic systems. The first model is clonal, the second is sexual diploid with additive genetics on a single locus and the third is like the second but with an additional locus for mate choice. Evolutionary branching occurred under exactly the same ecological circumstances in all three models. After branching the evolutionary dynamics may be qualitatively different. In particular, in the diploid, sexual models there can be multiple evolutionary outcomes whereas in the corresponding clonal model there is only one. We showed that evolutionary branching favours the evolution of (partial) assortative mating and that this in turn effectively restores the results from the clonal model by rendering the alternative outcomes unreachable except for the one that also occurs in the clonal model. The evolution of assortative mating during evolutionary branching can be interpreted as the initial phase of sympatric speciation with phenotypic divergence and partial reproductive isolation.     

Rate heterogeneity, ancestral character state reconstruction, and the evolution of limb morphology in Lerista (Scincidae, Squamata)

Rates of phenotypic evolution derive from numerous interrelated processes acting at varying spatial and temporal scales and frequently differ substantially among lineages. Although current models employed in reconstructing ancestral character states permit independent rates for distinct types of transition (forward and reverse transitions and transitions between different states), these rates are typically assumed to be identical for all branches in a phylogeny. In this paper, I present a general model of character evolution enabling rate heterogeneity among branches. This model is employed in assessing the extent to which the assumption of uniform transition rates affects reconstructions of ancestral limb morphology in the scincid lizard clade Lerista and, accordingly, the potential for rate variability to mislead inferences of evolutionary patterns. Permitting rate variation among branches significantly improves model fit for both the manus and the pes. A constrained model in which the rate of digit acquisition is assumed to be effectively zero is strongly supported in each case; when compared with a model assuming unconstrained transition rates, this model provides a substantially better fit for the manus and a nearly identical fit for the pes. Ancestral states reconstructed assuming the constrained model imply patterns of limb evolution differing significantly from those implied by reconstructions for uniform-rate models, particularly for the pes; whereas ancestral states for the uniform-rate models consistently entail the reacquisition of pedal digits, those for the model incorporating among-lineage rate heterogeneity imply repeated, unreversed digit loss. These results indicate that the assumption of identical transition rates for all branches in a phylogeny may be inappropriate in modeling the evolution of phenotypic traits and emphasize the need for careful evaluation of phylogenetic tests of Dollo's law.     

The evolution of human phenotypic plasticity: age and nutritional status at maturity

Several evolutionary optimal models of human plasticity in age and nutritional status at reproductive maturation are proposed and their dynamics examined. These models differ from previously published models because fertility is not assumed to be a function of body size or nutritional status. Further, the models are based on explicitly human demographic patterns, that is, model human life-tables, model human fertility tables, and, a nutrient flow-based model of maternal nutritional status. Infant survival (instead of fertility as in previous models) is assumed to be a function of maternal nutritional status. Two basic models are examined. In the first the cost of reproduction is assumed to be a constant proportion of total nutrient flow. In the second the cost of reproduction is constant for each birth. The constant proportion model predicts a negative slope of age and nutritional status at maturation. The constant cost per birth model predicts a positive slope of age and nutritional status at maturation. Either model can account for the secular decline in menarche observed over the last several centuries in Europe. A search of the growth literature failed to find definitive empirical documentation of human phenotypic plasticity in age and nutritional status at maturation. Most research strategies confound genetics with phenotypic plasticity. The one study that reports secular trends suggests a marginally insignificant, but positive slope. This view tends to support the constant cost per birth model.     

Phenotypic switching and its implications for the pathogenesis of Cryptococcus neoformans

Phenotypic switching has been described in several strains of Cryptococcus neoformans. It occurs in vivo during chronic infection and is associated with differential gene expression and changes in virulence. The switch involves changes in the polysaccharide capsule and cell wall that affect the yeast's ability to resist phagocytosis. In addition, the phenotypic switch variants elicit qualitatively different inflammatory responses in the host. The host's immune response ultimately affects selection of the switch variants in animal models of chronic cryptococcosis. The biological relevance of phenotypic switching is demonstrated in several murine infection models and further underlines the importance of phenotypic switching in the setting of human disease. This includes the association of switching and poor outcome in chronic infection, the ability of the mucoid variant of strain RC-2 (RC-2 MC) but not the smooth variant (RC-2 SM) to promote increased intracranial pressure in a rat model, and lastly the observation that antifungal interventions can promote the selection of more virulent switch variants during chronic murine infection.     

Patterns of variation of a common fern (Athyrium filix-femina; Woodsiaceae): population structure along and between altitudinal gradients

Genetic variability of Athyrium filix-femina populations was evaluated with regard to phenotypic, allozyme, and RAPD variation in 20 Swiss populations along five altitudinal gradients at four different elevations in the northern Swiss Alps. Additionally, allozyme and phenotypic variations in one Italian and two Spanish populations were compared with the variation in the Swiss populations. We hypothesized that there will be statistically significant genetic differences among populations of different altitudes and sites. The results showed no substantial correlation between genetic variation and phenotypic variation among Swiss populations. These results imply that outbreeding and effective gene exchange (long-distance spore dispersal) are the keys to population structure in this fern species, and as a consequence, phenotypic plasticity is assumed to be favored. This contrasts with results found in similar studies of herbaceous flowering plants where genetic adaptation to gradients like altitude is common. However, when data from the more distant Italian and Spanish populations of A. filix-femina were included, significant variation was detected.     

The onion model, a simple neutral model for the evolution of diversity in bacterial biofilms

Bacterial biofilms are particularly resistant to a wide variety of antimicrobial compounds. Their persistence in the face of antibiotic therapies causes significant problems in the treatment of infectious diseases. Seldom have evolutionary processes like genetic drift and mutation been invoked to explain how resistance to antibiotics emerges in biofilms, and we lack a simple and tractable model for the genetic and phenotypic diversification that occurs in bacterial biofilms. Here, we introduce the 'onion model', a simple neutral evolutionary model for phenotypic diversification in biofilms. We explore its properties and show that the model produces patterns of diversity that are qualitatively similar to observed patterns of phenotypic diversity in biofilms. We suggest that models like our onion model, which explicitly invoke evolutionary process, are key to understanding biofilm resistance to bactericidal and bacteriostatic agents. Elevated phenotypic variance provides an insurance effect that increases the likelihood that some proportion of the population will be resistant to imposed selective agents and may thus enhance persistence of the biofilm. Accounting for evolutionary change in biofilms will improve our ability to understand and counter diseases that are caused by biofilm persistence.     

Costly plastic morphological responses to predator specific odour cues in three-spined sticklebacks (<Emphasis Type="Italic">Gasterosteus aculeatus</Emphasis>)

Predation risk is one of the major forces affecting phenotypic variation among and within animal populations. While fixed anti-predator morphologies are favoured when predation level is consistently high, plastic morphological responses are advantageous when predation risk is changing temporarily, spatially, or qualitatively. Three-spined sticklebacks (Gasterosteus aculeatus) are well known for their substantial variability in morphology, including defensive traits. Part of this variation might be due to phenotypic plasticity. However, little is known about sticklebacks’ plastic ability to react morphologically to changing risks of predation and about the proximate cues involved. Using a split-clutch design we show that odour of a predatory fish induces morphological changes in sticklebacks. Under predation risk, i.e., when exposed to odour of a predator, fish grew faster and developed a different morphology, compared to fish reared under low predation risk, i.e., exposed to odour of a non-predatory fish, or in a fish-free environment. However, fast growing comes at cost of increased body asymmetries suggesting developmental constraints. The results indicate that sticklebacks are able to distinguish between predatory and non-predatory fishes by olfactory cues alone. As fishes were fed on invertebrates, this reaction was not induced by chemical cues of digested conspecifics, but rather by predator cues themselves. Further, the results show that variation in body morphology in sticklebacks has not only a strong genetical component, but is also based on plastic responses to different environments, in our case different predation pressures, thus opening new questions for this model species in ecology and evolution.     

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.     

Evolution and maintenance of the environmental component of the phenotypic variance: benefit of plastic traits under changing environments

How environmental variances in quantitative traits are influenced by variable environments is an important problem in evolutionary biology. In this study, the evolution and maintenance of phenotypic variance in a plastic trait under stabilizing selection are investigated. The mapping from genotypic value to phenotypic value of the quantitative trait is approximated by a linear reaction norm, with genotypic effects on its phenotypic mean and sensitivity to environment. The environmental deviation is assumed to be decomposed into environmental quality, which interacts with genotypic value, and residual developmental noise, which is independent of genotype. Environmental quality and the optimal phenotype of stabilizing selection are allowed to randomly fluctuate in both space and time, and individuals migrate equally before development and reproduction among different niches. Analyses show that phenotypic plasticity is adaptive within variable environments if correlations have become established between the optimal phenotype and environmental quality in space and/or time. The evolved plasticity increases with variances in optimal phenotypes and correlations between optimal phenotype and environmental quality; this further induces increases in mean fitness and the environmental variance in the trait. Under certain circumstances, however, the environmental variance may decrease with increase in variation in environmental quality.     

An Emerging Allee Effect Is Critical for Tumor Initiation and Persistence

Tumor cells develop different strategies to cope with changing microenvironmental conditions. A prominent example is the adaptive phenotypic switching between cell migration and proliferation. While it has been shown that the migration-proliferation plasticity influences tumor spread, it remains unclear how this particular phenotypic plasticity affects overall tumor growth, in particular initiation and persistence. To address this problem, we formulate and study a mathematical model of spatio-temporal tumor dynamics which incorporates the microenvironmental influence through a local cell density dependence. Our analysis reveals that two dynamic regimes can be distinguished. If cell motility is allowed to increase with local cell density, any tumor cell population will persist in time, irrespective of its initial size. On the contrary, if cell motility is assumed to decrease with respect to local cell density, any tumor population below a certain size threshold will eventually extinguish, a fact usually termed as Allee effect in ecology. These results suggest that strategies aimed at modulating migration are worth to be explored as alternatives to those mainly focused at keeping tumor proliferation under control.     

Population dynamics of mutant alleles under maternal effect in plant populations

Fitness interactions via maternal effects were analyzed by constructing a mathematical model. Here, the fitness of an individual is determined by its own phenotype and the maternal phenotype. It is assumed that two alternative phenotypic states are determined by two alleles at a single locus. Under this assumption, the model developed here reduces to a social selection model of human populations that considers that the fitness of an individual is determined by the individual as well as the parental phenotypes. The effect on genetic variability of random fluctuations in the magnitude of maternal effects was studied by considering the fixation probability and the equilibrium frequency of mutant genes. It is shown that genetic variability decreases as the magnitude of variation in maternal effect intensity increases.     

The probability of parallel genetic evolution from standing genetic variation

Parallel evolution is often assumed to result from repeated adaptation to novel, yet ecologically similar, environments. Here, we develop and analyse a mathematical model that predicts the probability of parallel genetic evolution from standing genetic variation as a function of the strength of phenotypic selection and constraints imposed by genetic architecture. Our results show that the probability of parallel genetic evolution increases with the strength of natural selection and effective population size and is particularly likely to occur for genes with large phenotypic effects. Building on these results, we develop a Bayesian framework for estimating the strength of parallel phenotypic selection from genetic data. Using extensive individual‐based simulations, we show that our estimator is robust across a wide range of genetic and evolutionary scenarios and provides a useful tool for rigorously testing the hypothesis that parallel genetic evolution is the result of adaptive evolution. An important result that emerges from our analyses is that existing studies of parallel genetic evolution frequently rely on data that is insufficient for distinguishing between adaptive evolution and neutral evolution driven by random genetic drift. Overcoming this challenge will require sampling more populations and the inclusion of larger numbers of loci.     

CFD simulation of non-Newtonian fluid flow in anaerobic digesters

A general mathematical model that predicts the flow fields in a mixed-flow anaerobic digester was developed. In this model, the liquid manure was assumed to be a non-Newtonian fluid, and the flow governed by the continuity, momentum, and k-epsilon standard turbulence equations, and non-Newtonian power law model. The commercial computational fluid dynamics (CFD) software, Fluent, was applied to simulate the flow fields of lab-scale, scale-up, and pilot-scale anaerobic digesters. The simulation results were validated against the experimental data from literature. The flow patterns were qualitatively compared for Newtonian and non-Newtonian fluids flow in a lab-scale digester. Numerical simulations were performed to predict the flow fields in scale-up and pilot-scale anaerobic digesters with different water pump power inputs and different total solid concentration (TS) in the liquid manure. The optimal power inputs were determined for the pilot-scale anaerobic digester. Some measures for reducing dead and low velocity zones were proposed based upon the CFD simulation results.     

A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies

We consider the movement and viability of individual cells in cell colonies. Cell movement is assumed to take place as a result of sensing the strain energy density as a mechanical stimulus. The model is based on tracking the displacement and viability of each individual cell in a cell colony. Several applications are shown, such as the dynamics of filling a gap within a fibroblast colony and the invasion of a cell colony. Though based on simple principles, the model is qualitatively validated by experiments on living fibroblasts on a flat substrate.