Understanding the overdispersed molecular clock

Rates of molecular evolution at some protein-encoding loci are more irregular than expected under a simple neutral model of molecular evolution. This pattern of excessive irregularity in protein substitutions is often called the "overdispersed molecular clock" and is characterized by an index of dispersion, R(T) > 1. Assuming infinite sites, no recombination model of the gene R(T) is given for a general stationary model of molecular evolution. R(T) is shown to be affected by only three things: fluctuations that occur on a very slow time scale, advantageous or deleterious mutations, and interactions between mutations. In the absence of interactions, advantageous mutations are shown to lower R(T); deleterious mutations are shown to raise it. Previously described models for the overdispersed molecular clock are analyzed in terms of this work as are a few very simple new models. A model of deleterious mutations is shown to be sufficient to explain the observed values of R(T). Our current best estimates of R(T) suggest that either most mutations are deleterious or some key population parameter changes on a very slow time scale. No other interpretations seem plausible. Finally, a comment is made on how R(T) might be used to distinguish selective sweeps from background selection.     

Genetic evolution,plasticity, and bet‐hedging as adaptive responses to temporally autocorrelated fluctuating selection: A quantitative genetic model

Adaptive responses to autocorrelated environmental fluctuations through evolution in mean reaction norm elevation and slope and an independent component of the phenotypic variance are analyzed using a quantitative genetic model. Analytic approximations expressing the mutual dependencies between all three response modes are derived and solved for the joint evolutionary outcome. Both genetic evolution in reaction norm elevation and plasticity are favored by slow temporal fluctuations, with plasticity, in the absence of microenvironmental variability, being the dominant evolutionary outcome for reasonable parameter values. For fast fluctuations, tracking of the optimal phenotype through genetic evolution and plasticity is limited. If residual fluctuations in the optimal phenotype are large and stabilizing selection is strong, selection then acts to increase the phenotypic variance (bet‐hedging adaptive). Otherwise, canalizing selection occurs. If the phenotypic variance increases with plasticity through the effect of microenvironmental variability, this shifts the joint evolutionary balance away from plasticity in favor of genetic evolution. If microenvironmental deviations experienced by each individual at the time of development and selection are correlated, however, more plasticity evolves. The adaptive significance of evolutionary fluctuations in plasticity and the phenotypic variance, transient evolution, and the validity of the analytic approximations are investigated using simulations.     

Lineage effects and the index of dispersion of molecular evolution

Recent efforts to estimate the index of dispersion [R(t)] of molecular evolution-i.e., the ratio of the variance in the number of substitutions on a lineage to the mean number-have suffered from an inability to adjust the data for lineage effects. These effects may include the generation-time dependency of the rate of evolution or improper assumptions about the branching pattern of a phylogenetic tree. In the present paper a method for correcting for lineage effects in the estimation of R(t) is presented for trees made up of three species. The recent data published by Li et al. for 20 loci in three orders of mammals is examined, and the average R(t), corrected for lineage effects, is shown to be 7.75 for replacement substitutions and 3.3 for silent substitutions. Thus the high values reported earlier may not be dismissed as due to generation-time effects or improper assumptions about phylogenies. Computer simulations are presented to give confidence in the estimate for replacement substitutions but also to demonstrate that the estimate for silent substitutions is sensitive to corrections for multiple substitutions and is not as reliable. This work's implications for our understanding of the mechanism of molecular evolution are discussed, and the arguments in favor of the hypothesis that replacement substitutions are mostly selected while silent substitutions are mostly neutral is presented.     

Molecular Evolution Modeled as a Fractal Renewal Point Process in Agreement with the Dispersion of Substitutions in Mammalian Genes

A fractal renewal point process (FRPP) is used to model molecular evolution in agreement with the relationship between the variance and the mean numbers of nonsynonymous and synonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike certain other episodic models, it also accounts for the increase in the index of dispersion with the mean number of substitutions in Ohta's (1995) data. We find that this correlation is significant for nonsynonymous substitutions at the 1% level and for synonymous substitutions at the 10% level, even after removing lineage effects and when using Bulmer's (1989) unbiased estimator of the index of dispersion. This model is simpler than most other overdispersed models of evolution in the sense that it is fully specified by a single interevent probability distribution. Interpretations in terms of chaotic dynamics and in terms of chance and selection are discussed. Received: 12 January 1998 / Accepted: 19 May 1998     

Variability of Evolutionary Rates of DNA

A statistical analysis of DNA sequences from four nuclear loci and five mitochondrial loci from different orders of mammals is described. A major aim of the study is to describe the variation in the rate of molecular evolution of proteins and DNA. A measure of rate variability is the statistic R, the ratio of the variance in the number of substitutions to the mean number. For proteins, R is found to be in the range 0.16 less than R less than 35.55, thus extending in both directions the values seen in previous studies. An analysis of codons shows that there is a highly significant excess of double substitutions in the first and second positions, but not in the second and third or first and third positions. The analysis of the dynamics of nucleotide evolution showed that the ergodic Markov chain models that are the basis of most published formulas for correcting for multiple substitutions are incompatible with the data. A bootstrap procedure was used to show that the evolution of the individual nucleotides, even the third positions, show the same variation in rates as seen in the proteins. It is argued that protein and silent DNA evolution are uncoupled, with the evolution at both levels showing patterns that are better explained by the action of natural selection than by neutrality. This conclusion is based primarily on a comparison of the nuclear and mitochondrial results.     

Robustness of the Estimator of the Index of Dispersion for DNA Sequences

If substitutions in DNA sequences follow a Poisson process, the ratio of the variance in the number of substitutions to the mean number of substitutions (the index of dispersion) should equal 1. In this paper, the robustness of the commonly applied estimator of the index of dispersion in replacement sites and silent sites to various assumptions regarding DNA evolution is explored using simulation methods. The estimate of the index of dispersion may be strongly biased if the assumptions of the model of substitution are violated. However, the results of this study support the conclusions of studies by Gillespie and Ohta that the process of substitution in replacement sites is overdispersed. This result contradicts those of a recent study and shows that the high index of dispersion for replacement sites is not an artifact caused by the method of estimation.     

Statistical models of the overdispersed molecular clock

The most commonly used statistical model to describe the rate constancy of molecular evolution (molecular clock) is a simple Poisson process in which the variance of the number of amino acid or nucleotide substitutions in a particular gene should be equal to the mean and henceforth the dispersion index, the ratio of the variance to the mean, should be equal to one. Recent sequence data, however, have shown that the substitutional process in molecular evolution is often considerably overdispersed and have called into question the generality of using a simple Poisson process. Several efforts have been made to develop more realistic models of molecular evolution. In this paper, I will show that the spatial (site-specific) variation in the rate of molecular evolution is an improbable cause of the overdispersion and then review various statistical models which take the temporal variation into account. Although these models do not immediately specify what the mechanisms of molecular evolution might be, they do make qualitatively different predictions and give some insight into their inference. One way to distinguish them is suggested. In addition, effects of selected substitutions that presumably occur after a major change in a molecule are quasi-quantitatively examined. It is most likely that the overdispersion of molecular clock is due either to a major molecular reconfiguration (fluctuating neutral space) led by a series of subliminal neutral changes or to selected substitutions fine-tuning a molecule after a major molecular change. Although the latter possibility, of course, violates the simplest neutrality assumption, it would not impair the neutral theory as a whole.     

Synonymous and nonsynonymous substitutions in mammalian genes and the nearly neutral theory

The nearly neutral theory of molecular evolution predicts larger generation-time effects for synonymous than for nonsynonymous substitutions. This prediction is tested using the sequences of 49 single-copy genes by calculating the average and variance of synonymous and nonsynonymous substitutions in mammalian star phylogenies (rodentia, artiodactyla, and primates). The average pattern of the 49 genes supports the prediction of the nearly neutral theory, with some notable exceptions.The nearly neutral theory also predicts that the variance of the evolutionary rate is larger than the value predicted by the completely neutral theory. This prediction is tested by examining the dispersion index (ratio of the variance to the mean), which is positively correlated with the average substitution number. After weighting by the lineage effects, this correlation almost disappears for nonsynonymous substitutions, but not quite so for synonymous substitutions. After weighting, the dispersion indices of both synonymous and nonsynonymous substitutions still exceed values expected under the simple Poisson process. The results indicate that both the systematic bias in evolutionary rate among the lineages and the episodic type of rate variation are contributing to the large variance. The former is more significant to synonymous substitutions than to nonsynonymous substitutions. Isochore evolution may be similar to synonymous substitutions. The rate and pattern found here are consistent with the nearly neutral theory, such that the relative contributions of drift and selection differ between the two types of substitutions. The results are also consistent with Gillespie's episodic selection theory.     

Perspective: gene divergence, population divergence, and the variance in coalescence time in phylogeographic studies

Molecular methods as applied to the biogeography of single species (phylogeography) or multiple codistributed species (comparative phylogeography) have been productively and extensively used to elucidate common historical features in the diversification of the Earth's biota. However, only recently have methods for estimating population divergence times or their confidence limits while taking into account the critical effects of genetic polymorphism in ancestral species become available, and earlier methods for doing so are underutilized. We review models that address the crucial distinction between the gene divergence, the parameter that is typically recovered in molecular phylogeographic studies, and the population divergence, which is in most cases the parameter of interest and will almost always postdate the gene divergence. Assuming that population sizes of ancestral species are distributed similarly to those of extant species, we show that phylogeographic studies in vertebrates suggest that divergence of alleles in ancestral species can comprise from less than 10% to over 50% of the total divergence between sister species, suggesting that the problem of ancestral polymorphism in dating population divergence can be substantial. The variance in the number of substitutions (among loci for a given species or among species for a given gene) resulting from the stochastic nature of DNA change is generally smaller than the variance due to substitutions along allelic lines whose coalescence times vary due to genetic drift in the ancestral population. Whereas the former variance can be reduced by further DNA sequencing at a single locus, the latter cannot. Contrary to phylogeographic intuition, dating population divergence times when allelic lines have achieved reciprocal monophyly is in some ways more challenging than when allelic lines have not achieved monophyly, because in the former case critical data on ancestral population size provided by residual ancestral polymorphism is lost. In the former case differences in coalescence time between species pairs can in principle be explained entirely by differences in ancestral population size without resorting to explanations involving differences in divergence time. Furthermore, the confidence limits on population divergence times are severely underestimated when those for number of substitutions per site in the DNA sequences examined are used as a proxy. This uncertainty highlights the importance of multilocus data in estimating population divergence times; multilocus data can in principle distinguish differences in coalescence time (T) resulting from differences in population divergence time and differences in T due to differences in ancestral population sizes and will reduce the confidence limits on the estimates. We analyze the contribution of ancestral population size (theta) to T and the effect of uncertainty in theta on estimates of population divergence (tau) for single loci under reciprocal monophyly using a simple Bayesian extension of Takahata and Satta's and Yang's recent coalescent methods. The confidence limits on tau decrease when the range over which ancestral population size theta is assumed to be distributed decreases and when tau increases; they generally exclude zero when tau/(4Ne) > 1. We also apply a maximum-likelihood method to several single and multilocus data sets. With multilocus data, the criterion for excluding tau = 0 is roughly that l tau/(4Ne) > 1, where l is the number of loci. Our analyses corroborate recent suggestions that increasing the number of loci is critical to decreasing the uncertainty in estimates of population divergence time.     

Statistical properties of neutral evolution

Neutral evolution is the simplest model of molecular evolution and thus it is most amenable to a comprehensive theoretical investigation. In this paper, we characterize the statistical properties of neutral evolution of proteins under the requirement that the native state remains thermodynamically stable, and compare them to the ones of Kimura's model of neutral evolution. Our study is based on the Structurally Constrained Neutral (SCN) model which we recently proposed. We show that, in the SCN model, the substitution rate decreases as longer time intervals are considered. Fluctuations from one branch of the evolutionary tree to another are strong, leading to a non-Poissonian statistics for the substitution process. Such strong fluctuations are in part due to the fact that neutral substitution rates for individual residues are strongly correlated for most residue pairs. Interestingly, structurally conserved residues, characterized by a much below average substitution rate, are also much less correlated to other residues and evolve in a much more regular way. Our results can improve methods aimed at distinguishing between neutral and adaptive substitutions as well as methods for computing the expected number of substitutions occurred since the divergence of two protein sequences. In particular, we compute the minimal sequence similarity below which no information about the evolutionary divergence of the compared sequences can be obtained.     

The evolution of postzygotic isolation: accumulating Dobzhansky-Muller incompatibilities

Hybrid sterility and inviability often result from the accumulation of substitutions that, while functional on their normal genetic backgrounds, cause a loss of fitness when brought together in hybrids. Previous theory has shown that such Dobzhansky-Muller incompatibilities should accumulate at least as fast as the square of the number of substitutions separating two species, the so-called snowball effect. Here we explicitly describe the stochastic accumulation of these incompatibilities as a function of time. The accumulation of these incompatibilities involves three levels of stochasticity: (1) the number of substitutions separating two allopatric lineages at a given time; (2) the number of incompatibilities resulting from these substitutions; and (3) the fitness effects of individual incompatibilities. Previous analyses ignored the stochasticity of molecular evolution (level 1) as well as that due to the variable effects of incompatibilities (level 3). Here we approximate the full stochastic process characterizing the accumulation of hybrid incompatibilities between pairs of loci. We derive the distribution of the number of incompatibilities as a function of divergence time between allopatric taxa as well as the distribution of waiting times to speciation by postzygotic isolation. We provide simple approximations for the mean and variance of these waiting times. These results let us estimate. albeit crudely, the probability, p, that two diverged sites from different species will contribute to hybrid sterility or inviability. Our analyses of data from Drosophila and Bombina suggest that p is generally very small, on the order of 10(-6) or less.     

Adaptation to an extraordinary environment by evolution of phenotypic plasticity and genetic assimilation

Adaptation to a sudden extreme change in environment, beyond the usual range of background environmental fluctuations, is analysed using a quantitative genetic model of phenotypic plasticity. Generations are discrete, with time lag τ between a critical period for environmental influence on individual development and natural selection on adult phenotypes. The optimum phenotype, and genotypic norms of reaction, are linear functions of the environment. Reaction norm elevation and slope (plasticity) vary among genotypes. Initially, in the average background environment, the character is canalized with minimum genetic and phenotypic variance, and no correlation between reaction norm elevation and slope. The optimal plasticity is proportional to the predictability of environmental fluctuations over time lag τ. During the first generation in the new environment the mean fitness suddenly drops and the mean phenotype jumps towards the new optimum phenotype by plasticity. Subsequent adaptation occurs in two phases. Rapid evolution of increased plasticity allows the mean phenotype to closely approach the new optimum. The new phenotype then undergoes slow genetic assimilation, with reduction in plasticity compensated by genetic evolution of reaction norm elevation in the original environment.     

Standard error of immunological dating of evolutionary time

Summary The empirical variance of the immunological distance as measured by microcomplement fixation with albumin is determined. The variance obtained is at least two times larger than the mean when the mean is small and the ratio of the variance to the mean increases with increasing mean. Thus, the immunological dating of evolutionary time has a large standard error. It is shown that in bird lysozymes the relationship between immunological distance (y) and the number of amino acid substitutions per 100 sites (x) is given byy = 4.2x approximately.     

Goodman et al.'s method for augmenting the number of nucleotide substitutions

Summary Statistical properties of Goodman et al.'s (1974) method of compensating for undetected nucleotide substitutions in evolution are investigated by using computer simulation. It is found that the method tends to overcompensate when the stochastic error of the number of nucleotide substitutions is large. Furthermore, the estimate of the number of nucleotide substitutions obtained by this method has a large variance. However, in order to see whether this method gives overcompensation when applied together with the maximum parsimony method, a much larger scale of simulation seems to be necessary.     

The genetic consequences of fluctuating inbreeding depression and the evolution of plant selfing rates

The magnitude of inbreeding depression, a central parameter in the evolution of plant mating systems, can vary depending on environmental conditions. However, the underlying genetic mechanisms causing environmental fluctuations in inbreeding depression, and the consequences of this variation for the evolution of self‐fertilization, have been little studied. Here, we consider temporal fluctuations of the selection coefficient in an explicit genetic model of inbreeding depression. We show that substantial variance in inbreeding depression can be generated at equilibrium by fluctuating selection, although the simulated variance tends to be lower than has been measured in experimental studies. Our simulations also reveal that purging of deleterious mutations does not depend on the variance in their selection coefficient. Finally, an evolutionary analysis shows that, in contrast to previous theoretical approaches, intermediate selfing rates are never evolutionarily stable when the variation in inbreeding depression is due to fluctuations in the selection coefficient on deleterious mutations.     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

The Evolution of Canalization and Evolvability in Stable and Fluctuating Environments

Using a multilinear model of epistasis we explore the evolution of canalization (reduced mutational effects) and evolvability (levels of additive genetic variance) under different forms of stabilizing and fluctuating selection. We show that the total selection acting on an allele can be divided into a component deriving from adaptation of the trait mean, a component of canalizing selection favoring alleles that epistatically reduce the effects of other allele substitutions, and a component of conservative selection disfavoring rare alleles. While canalizing selection operates in both stable and fluctuating environments, it may not typically maximize canalization, because it gets less efficient with increasing canalization, and reaches a balance with drift, mutation and indirect selection. Fluctuating selection leads to less canalized equilibria than stabilizing selection of comparable strength, because canalization then becomes influenced by erratic correlated responses to shifting trait adaptation. We conclude that epistatic systems under bounded fluctuating selection will become less canalized than under stabilizing selection and may support moderately increased evolvability if the amplitude of fluctuations is large, but canalization is still stronger and evolvability lower than expected under neutral evolution or under patterns of selection that shift the trait in directions of positive (reinforcing) epistasis.     

Size scaling of mutation avalanches in a model for protein evolution

The tunably rugged NK-model is used to study avalanche-like events that occur when environmental change causes fitness optima to disappear. The probability of an event with Delta substitutions scales as exp(-c Delta) for smooth landscapes, and as exp(-c Delta(2)) for rugged landscapes. Increasing the ruggedness leads to two competing effects: (1) more possible routes by which single mutations can increase the fitness, which dominates at low ruggedness and acts to increase Delta; and (2) a higher density of fitness optima, which dominates at high ruggedness and acts to decrease Delta. Due to these competing effects, the largest average values of Delta occur at intermediate ruggedness. The effects of system size on the avalanche events are examined, and average values of Delta increase logarithmically with system size. The variance to mean ratios for the number of substitutions per unit time are consistent with experimental results for protein evolution.     

Mammalian nonsynonymous sites are not overdispersed: comparative genomic analysis of index of dispersion of mammalian proteins

It is often stated that patterns of nonsynonymous rate variation among mammalian lineages are more irregular than expected or overdispersed under the neutral model, whereas synonymous sites conform to the neutral model. Here we reexamined genome-wide patterns of the variance to mean ratio, or index of dispersion (R), of substitutions in proteins from human, mouse, and dog. Contrary to the prevailing notion, we found that the mean index of dispersion for nonsynonymous sites of mammalian proteins is not significantly different from 1. We propose that earlier analyses were biased because the data included disproportionately more protein hormones, which tend to be more dispersed than genes in other functional categories. Synonymous sites exhibit greater degree of dispersion than nonsynonymous sites, although similar to earlier estimates and potentially due to errors associated with correction for multiple hits. Overall, our analysis identifies strong genome-wide generation-time effect and natural selection as important determinants of among-lineage variation of protein evolutionary rates. Furthermore, patterns of lineage-specific selective constraint are consistent with the nearly neutral model of molecular evolution.     

Overdispersed molecular clock at the major histocompatibility complex loci.

The extent of amino acid differences of major histocompatibility complex molecules within species is unusually high, consistent with the finding that some pairs of alleles have persisted for more than ten million years and the view that the polymorphism has been maintained by natural selection. The disparity between synonymous and non-synonymous substitutions in the antigen recognition site, however, suggests that some non-synonymous sites have undergone a number of substitutions whereas others have little or none. To describe statistically such an overdispersed underlying process, commonly used Poisson processes are inadequate. An alternative process leads to the surprising conclusion that each non-synonymous site has accumulated as many as 2.6 substitutions, on the average, in the two lineages leading to humans and mice. The standard deviation is also very large (6.6) and the dispersion index (the ratio of the variance to the mean) is at least 17. The substitution process thus inferred qualitatively agrees with the disposition (a boomerang pattern) of substitutions between HLA-A2 and Aw68 alleles, and quantitatively agrees well with that expected where the evolution of major histocompatibility complex molecules has long been driven mostly by balancing selection.