Overdispersion of the molecular clock: temporal variation of gene-specific substitution rates in Drosophila

Simple models of molecular evolution assume that sequences evolve by a Poisson process in which nucleotide or amino acid substitutions occur as rare independent events. In these models, the expected ratio of the variance to the mean of substitution counts equals 1, and substitution processes with a ratio greater than 1 are called overdispersed. Comparing the genomes of 10 closely related species of Drosophila, we extend earlier evidence for overdispersion in amino acid replacements as well as in four-fold synonymous substitutions. The observed deviation from the Poisson expectation can be described as a linear function of the rate at which substitutions occur on a phylogeny, which implies that deviations from the Poisson expectation arise from gene-specific temporal variation in substitution rates. Amino acid sequences show greater temporal variation in substitution rates than do four-fold synonymous sequences. Our findings provide a general phenomenological framework for understanding overdispersion in the molecular clock. Also, the presence of substantial variation in gene-specific substitution rates has broad implications for work in phylogeny reconstruction and evolutionary rate estimation.     

On the Overdispersed Molecular Clock

Rates of molecular evolution at some loci are more irregular than described by simple Poisson processes. Three situations under which molecular evolution would not follow simple Poisson processes are reevaluated from the viewpoint of the neutrality hypothesis: concomitant or multiple substitutions in a gene, fluctuating substitution rates in time caused by coupled effects of deleterious mutations and bottlenecks, and changes in the degree of selective constraints against a gene (neutral space) caused by successive substitutions. The common underlying assumption that these causes are lineage nonspecific excludes the case where mutation rates themselves change systematically among lineages or taxonomic groups, and severely limits the extent of variation in the number of substitutions among lineages. Even under this stringent condition, however, the third hypothesis, the fluctuating neutral space model, can generate fairly large variation. This is described by a time-dependent renewal process, which does not exhibit any episodic nature of molecular evolution. It is argued that the observed elevated variances in the number of nucleotide or amino acid substitutions do not immediately call for positive Darwinian selection in molecular evolution.     

Amino acid substitution of proteins coded for in mitochondrial DNA during mammalian evolution.

Three Markov models (Dayhoff, Proportional and Poisson models; Hasegawa et al., 1992a) for amino acid substitution during evolution were used for maximum likelihood analyses of proteins coded for in mitochondrial DNA in estimating a phylogenetic tree among human, bovine and murids (mouse and rat) with chicken as an outgroup. It turned out that Dayhoff model is the most appropriate model among the alternatives in approximating the amino acid substitutions of proteins coded for in mitochondrial DNA. In spite of the presence of the complete sequence data of mitochondrial genomes, we could not resolve the trichotomy among human, bovine and murids, probably because the time length separating two branching events among these three lines was short and because chicken is too distant from mammals to be used as an outgroup. It was suggested that the average substitution rate of amino acids coded for in mitochondrial DNA is lower along the bovine line than those along the human or murid lines. Advantages of amino acid sequence analysis over nucleotide sequence analysis in phylogenetic study were discussed.     

Overdispersion of the molecular clock varies between yeast, Drosophila and mammals

Although protein evolution can be approximated as a "molecular evolutionary clock," it is well known that sequence change departs from a clock-like Poisson expectation. Through studying the deviations from a molecular clock, insight can be gained into the forces shaping evolution at the level of proteins. Generally, substitution patterns that show greater variance than the Poisson expectation are said to be "overdispersed." Overdispersion of sequence change may result from temporal variation in the rate at which amino acid substitutions occur on a phylogeny. By comparing the genomes of four species of yeast, five species of Drosophila, and five species of mammals, we show that the extent of overdispersion shows a strong negative correlation with the effective population size of these organisms. Yeast proteins show very little overdispersion, while mammalian proteins show substantial overdispersion. Additionally, X-linked genes, which have reduced effective population size, have gene products that show increased overdispersion in both Drosophila and mammals. Our research suggests that mutational robustness is more pervasive in organisms with large population sizes and that robustness acts to stabilize the molecular evolutionary clock of sequence change.     

The spatial distribution of fixed mutations within genes coding for proteins

We have examined the extensive amino acid sequence data now available for five protein families - the alpha crystallin A chain, myoglobin, alpha and beta hemoglobin, and the cytochromes c - with the goal of estimating the true spatial distribution of base substitutions within genes that code for proteins. In every case the commonly used Poisson density failed to even approximate the experimental pattern of base substitution. For the 87 species of beta hemoglobin examined, for example, the probability that the observed results were from a Poisson process was the minuscule 10(-44). Analogous results were obtained for the other functional families. All the data were reasonably, but not perfectly, described by the negative binomial density. In particular, most of the data were described by one of the very simple limiting forms of this density, the geometric density. The implications of this for evolutionary inference are discussed. It is evident that most estimates of total base substitutions between genes are badly in need of revision.     

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.     

Molecular evolution of hemagglutinin genes of H1N1 swine and human influenza A viruses

Summary The hemagglutinin (HA) genes of influenza type A (H1N1) viruses isolated from swine were cloned into plasmid vectors and their nucleotide sequences were determined. A phylogenetic tree for the HA genes of swine and human influenza viruses was constructed by the neighbor-joining method. It showed that the divergence between swine and human HA genes might have occurred around 1905. The estimated rates of synonymous (silent) substitutions for swine and human influenza viruses were almost the same. For both viruses, the rate of synonymous substitution was much higher than that of nonsynonymous (amino acid altering) substitution. It is the case even for only the antigenic sites of the HA. This feature is consistent with the neutral theory of molecular evolution. The rate of nonsynonymous substitution for human influenza viruses was three times the rate for swine influenza viruses. In particular, nonsynonymous substitutions at antigenic sites occurred less frequently in swine than in humans. The difference in the rate of nonsynonymous substitution between swine and human influenza viruses can be explained by the different degrees of functional constraint operating on the amino acid sequence of the HA in both hosts.     

Substitution rate variation among sites in hypervariable region 1 of human mitochondrial DNA

More than an order of magnitude difference in substitution rate exists among sites within hypervariable region 1 of the control region of human mitochondrial DNA. A two-rate Poisson mixture and a negative binomial distribution are used to describe the distribution of the inferred number of changes per nucleotide site in this region. When three data sets are pooled, however, the two-rate model cannot explain the data. The negative binomial distribution always fits, suggesting that substitution rates are approximately gamma distributed among sites. Simulations presented here provide support for the use of a biased, yet commonly employed, method of examining rate variation. The use of parsimony in the method to infer the number of changes at each site introduces systematic errors into the analysis. These errors preclude an unbiased quantification of variation in substitution rate but make the method conservative overall. The method can be used to distinguish sites with highly elevated rates, and 29 such sites are identified in hypervariable region 1. Variation does not appear to be clustered within this region. Simulations show that biases in rates of substitution among nucleotides and non-uniform base composition can mimic the effects of variation in rate among sites. However, these factors contribute little to the levels of rate variation observed in hypervariable region 1.     

Molecular clocks in reptiles: life history influences rate of molecular evolution

Life history has been implicated as a determinant of variation in rate of molecular evolution amongst vertebrate species because of a negative correlation between body size and substitution rate for many molecular data sets. Both the generality and the cause of the negative body size trend have been debated, and the validity of key studies has been questioned (particularly concerning the failure to account for phylogenetic bias). In this study, a comparative method has been used to test for an association between a range of life-history variables-such as body size, age at maturity, and clutch size-and DNA substitution rate for three genes (NADH4, cytochrome b, and c-mos). A negative relationship between body size and rate of molecular evolution was found for phylogenetically independent pairs of reptile species spanning turtles, lizards, snakes, crocodile, and tuatara. Although this study was limited by the number of comparisons for which both sequence and life-history data were available, the results suggest that a negative body size trend in rate of molecular evolution may be a general feature of reptile molecular evolution, consistent with similar studies of mammals and birds. This observation has important implications for uncovering the mechanisms of molecular evolution and warns against assuming that related lineages will share the same substitution rate (a local molecular clock) in order to date evolutionary divergences from DNA sequences.     

蕨类植物叶绿体rps12基因的分子进化研究

以68种蕨类植物和2种石松类植物的rps12基因为对象,在系统发育背景下,结合最大似然法,使用HyPhy和PAML软件对该基因进行进化速率和适应性进化研究。结果显示:位于IR区的外显子2~3,其替换率明显降低,rps12基因编码序列的替换率也随之降低,且rps12基因密码子第3位的GC含量明显升高;在蕨类植物的进化过程中,3′-rps12更倾向定位于IR区,以保持较低的替换率;rps12基因编码的123个氨基酸位点中,共检测到4个正选择位点和116个负选择位点。研究结果表明基因序列进入到IR区后,显示出降低的替换率;强烈的负选择压力表明RPS12蛋白的高度保守性以及rps12基因的功能和结构已经趋于稳定。     

Accuracy of estimated phylogenetic trees from molecular data

Summary The accuracies and efficiencies of four different methods for constructing phylogenetic trees from molecular data were examined by using computer simulation. The methods examined are UPGMA, Fitch and Margoliash's (1967) (F/M) method, Farris' (1972) method, and the modified Farris method (Tateno, Nei, and Tajima, this paper). In the computer simulation, eight OTUs (32 OTUs in one case) were assumed to evolve according to a given model tree, and the evolutionary change of a sequence of 300 nucleotides was followed. The nucleotide substitution in this sequence was assumed to occur following the Poisson distribution, negative binomial distribution or a model of temporally varying rate. Estimates of nucleotide substitutions (genetic distances) were then computed for all pairs of the nucleotide sequences that were generated at the end of the evolution considered, and from these estimates a phylogenetic tree was reconstructed and compared with the true model tree. The results of this comparison indicate that when the coefficient of variation of branch length is large the Farris and modified Farris methods tend to be better than UPGMA and the F/M method for obtaining a good topology. For estimating the number of nucleotide substitutions for each branch of the tree, however, the modified Farris method shows a better performance than the Farris method. When the coefficient of variation of branch length is small, however, UPGMA shows the best performance among the four methods examined. Nevertheless, any tree-making method is likely to make errors in obtaining the correct topology with a high probability, unless all branch lengths of the true tree are sufficiently long. It is also shown that the agreement between patristic and observed genetic distances is not a good indicator of the goodness of the tree obtained.     

Correlation between the substitution rate and rate variation among sites in protein evolution.

It is well known that the rate of amino acid substitution varies among different proteins and among different sites of a protein. It is, however, unclear whether the extent of rate variation among sites of a protein and the mean substitution rate of the protein are correlated. We used two approaches to analyze orthologous protein sequences of 51 nuclear genes of vertebrates and 13 mitochondrial genes of mammals. In the first approach, no assumptions of the distribution of the rate variation among sites were made, and in the second approach, the gamma distribution was assumed. Through both approaches, we found a negative correlation between the extent of among-site rate variation and the average substitution rate of a protein. That is, slowly evolving proteins tend to have a high level of rate variation among sites, and vice versa. We found this observation consistent with a simple model of the neutral theory where most sites are either invariable or neutral. We conclude that the correlation is a general feature of protein evolution and discuss its implications in statistical tests of positive Darwinian selection and molecular time estimation of deep divergences.     

Estimating the influence of selection on the variable amino acid sites of the cytochrome B protein functional domains

We evaluated the effects of selection on the molecular evolution of the functional domains of the mammalian cytochrome b gene as it relates to physicochemical properties shown to correlate with rates of amino acid replacement. Two groups of mammals were considered: pocket gophers of the rodent family Geomyidae, and cetaceans and ungulates of the monophyletic taxon Cetartopdactyla. Several characteristics of cytochrome b evolution were common to both mammal groups. The evolution of the matrix domain reflected the region's relative lack of function. Goodness of fit to neutral expectations indicated that external influences have had very little effect on the evolution of the matrix, although in some cases conservative and moderate changes have been favored. Although rates of synonymous nucleotide substitution have been relatively high, the transmembrane domain exhibited poor goodness of fit to neutral expectations. However, the evolution of the transmembrane domain has been constrained by negative selection, allowing a preponderance of conservative and moderate amino acid replacements. We hypothesize that a high rate of substitution is maintained in spite of negative selection because the codons of the transmembrane coding region are predisposed to conservative changes in all amino acid properties. The evolutionary patterns of the intermembrane domain in pocket gophers and cetartiodactyls, however, were very different. Changes inferred from the pocket gopher phylogenetic tree exhibited a significant fit to neutral expectations for each of the amino acid properties. Changes inferred from the cetartiodactyl tree exhibited significant fit to neutral expectations for polarity and isoelectric point, but not for composition, molecular volume, polar requirement, or hydropathy. In each case, lack of fit was due to selection that promoted conservative or moderate change, with the noteworthy exception of polar requirement. We detected an unexpectedly large change in polar requirement (from aspartic acid to threonine) in two separate lineages (Camelus bactrianus and all cetaceans) at amino acid position 159. This inferred change occurred in a region of the cyt-b protein that directly interacts with external surface proteins of the cytochrome bc(1) complex and resulted in a reversion to a more common character state in vertebrates.     

Empirical relationship between the number of nucleotide substitutions and interspecific identity of amino acid sequences in some proteins

There are three different methods of estimating the number of nucleotide substitutions between a pair of species from amino acid sequence data, i.e. the Poisson correction method, random evolutionary hit method, and counting the actual but minimum number of nucleotide substitutions. In this paper the relationships among the estimates obtained by these methods are studied empirically. The results obtained indicate that there is a high correlation among these estimates and in practice any of the three methods may be used for constructing evolutionary trees or relating nucleotide substitutions to evolutionary time. The effects of varying rates of nucleotide substition among different sites on the Poisson correction and random evolutionary hit methods are also studied mathematically. It is shown that these two methods are quite insensitive to the variation of the rate of nucleotide substitution.     

The Rates of Molecular Evolution in Rodent and Primate Mitochondrial DNA

A higher rate of molecular evolution in rodents than in primates at synonymous sites and, to a lesser extent, at amino acid replacement sites has been reported previously for most nuclear genes examined. Thus in these genes the average ratio of amino acid replacement to synonymous substitution rates in rodents is lower than in primates, an observation at odds with the neutral model of molecular evolution. Under Ohta's mildly deleterious model of molecular evolution, these observations are seen as the consequence of the combined effects of a shorter generation time (driving a higher mutation rate) and a larger effective population size (resulting in more effective selection against mildly deleterious mutations) in rodents. The present study reports the results of a maximum-likelihood analysis of the ratio of amino acid replacements to synonymous substitutions for genes encoded in mitochondrial DNA (mtDNA) in these two lineages. A similar pattern is observed: in rodents this ratio is significantly lower than in primates, again consistent only with the mildly deleterious model. Interestingly the lineage-specific difference is much more pronounced in mtDNA-encoded than in nuclear-encoded proteins, an observation which is shown to run counter to expectation under Ohta's model. Finally, accepting certain fossil divergence dates, the lineage-specific difference in amino acid replacement-to-synonymous substitution ratio in mtDNA can be partitioned and is found to be entirely the consequence of a higher mutation rate in rodents. This conclusion is consistent with a replication-dependent model of mutation in mtDNA. Received: 24 September 1999 / Accepted: 18 September 2000     

CodonTest: Modeling Amino Acid Substitution Preferences in Coding Sequences

Codon models of evolution have facilitated the interpretation of selective forces operating on genomes. These models, however, assume a single rate of non-synonymous substitution irrespective of the nature of amino acids being exchanged. Recent developments have shown that models which allow for amino acid pairs to have independent rates of substitution offer improved fit over single rate models. However, these approaches have been limited by the necessity for large alignments in their estimation. An alternative approach is to assume that substitution rates between amino acid pairs can be subdivided into rate classes, dependent on the information content of the alignment. However, given the combinatorially large number of such models, an efficient model search strategy is needed. Here we develop a Genetic Algorithm (GA) method for the estimation of such models. A GA is used to assign amino acid substitution pairs to a series of rate classes, where is estimated from the alignment. Other parameters of the phylogenetic Markov model, including substitution rates, character frequencies and branch lengths are estimated using standard maximum likelihood optimization procedures. We apply the GA to empirical alignments and show improved model fit over existing models of codon evolution. Our results suggest that current models are poor approximations of protein evolution and thus gene and organism specific multi-rate models that incorporate amino acid substitution biases are preferred. We further anticipate that the clustering of amino acid substitution rates into classes will be biologically informative, such that genes with similar functions exhibit similar clustering, and hence this clustering will be useful for the evolutionary fingerprinting of genes.     

Implications of Fluctuations in Substitution Rates: Impact on the Uncertainty of Branch Lengths and on Relative-Rate Tests

Many tests of the lineage dependence of substitution rates, computations of the error of evolutionary distances, and simulations of molecular evolution assume that the rate of evolution is constant in time within each lineage descended from a common ancestor. However, estimates of the index of dispersion of numbers of mammalian substitutions suggest that the rate has time-dependent variations consistent with a fractal-Gaussian-rate Poisson process, which assumes common descent without assuming rate constancy. While this model does not affect certain relative-rate tests, it substantially increases the uncertainty of branch lengths. Thus, fluctuations in the rate of substitution cannot be neglected in calculations that rely on evolutionary distances, such as the confidence intervals of divergence times and certain phylogenetic reconstructions. The fractal-Gaussian-rate Poisson process is compared and contrasted with previous models of molecular evolution, including other Poisson processes, the fractal renewal process, a Lévy-stable process, a fractional-difference process, and a log-Brownian process. The fractal models are more compatible with mammalian data than the nonfractal models considered, and they may also be better supported by Darwinian theory. Although the fractal-Gaussian-rate Poisson process has not been proven to have better agreement with data or theory than the other fractal models, its Gaussian nature simplifies the exploration of its impact on evolutionary distance errors and relative-rate tests. Received: 29 September 1999 / Accepted: 20 January 2000     

Bacterial Density in Water Determined by Poisson or Negative Binomial Distributions

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.