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     

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     

分子进化研究：Ⅱ.5SrRNA序列的分形与分子进化的关系

本文根据分形理论的原理和方法,在对现行的计算核酸序列分维的方法进行修改的基础上,对各类生物的80余种5SrRNA序列的分维进行了计算,并结合耗散结构理论就其分维与分子进化的关系问题进行了研究和探讨。作者认为,5SrRNA序列的分维与其分子进化间的关系是一种复杂的非线性关系,在分子进化的过程中,序列的分维表现为随机涨落。     

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.     

The impact of single substitutions on multiple sequence alignments

We introduce another view of sequence evolution. Contrary to other approaches, we model the substitution process in two steps. First we assume (arbitrary) scaled branch lengths on a given phylogenetic tree. Second we allocate a Poisson distributed number of substitutions on the branches. The probability to place a mutation on a branch is proportional to its relative branch length. More importantly, the action of a single mutation on an alignment column is described by a doubly stochastic matrix, the so-called one-step mutation matrix. This matrix leads to analytical formulae for the posterior probability distribution of the number of substitutions for an alignment column.     

A compound poisson process for relaxing the molecular clock

The molecular clock hypothesis remains an important conceptual and analytical tool in evolutionary biology despite the repeated observation that the clock hypothesis does not perfectly explain observed DNA sequence variation. We introduce a parametric model that relaxes the molecular clock by allowing rates to vary across lineages according to a compound Poisson process. Events of substitution rate change are placed onto a phylogenetic tree according to a Poisson process. When an event of substitution rate change occurs, the current rate of substitution is modified by a gamma-distributed random variable. Parameters of the model can be estimated using Bayesian inference. We use Markov chain Monte Carlo integration to evaluate the posterior probability distribution because the posterior probability involves high dimensional integrals and summations. Specifically, we use the Metropolis-Hastings-Green algorithm with 11 different move types to evaluate the posterior distribution. We demonstrate the method by analyzing a complete mtDNA sequence data set from 23 mammals. The model presented here has several potential advantages over other models that have been proposed to relax the clock because it is parametric and does not assume that rates change only at speciation events. This model should prove useful for estimating divergence times when substitution rates vary across lineages.     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Rates of ribosomal RNA evolution are uniquely accelerated in eukaryotes

A novel procedure for testing the relative rates of evolution is described. The procedure, the distance-matrix rate test, consists of creating a graph that displays two complete distance matrices for two different genes derived from the same group of species, an approach made practical by numerous whole genomic sequences. The results in this paper show that the molecular clock of ribosomal RNA from Eukaryotes is uniquely accelerated and highly variable while those of Archaea and Bacteria are not. This idiosyncratic eukaryotic rRNA evolution is not observed with four different protein genes. The distance matrix rate test consists of plotting the distance of one gene (from two different species) against the distance of a second gene (from the same pair of species) in the form of a simple X-Y plot. Because it is not possible to compute variances (or co-variances in this case) that can be meaningfully compared to expectations from a Poisson process, the test does not permit calculations of an index of dispersion. In place of this, equations are given for the 95% confidence limits expected for a Poisson process. The test was applied to the proteins rpsl1 and rp114, as one example, and to rps11 and ssu rRNA as a second example. In addition, the cytochrome c and cytochrome c oxidase evolution from a larger group of Eukaryotes are compared to each other and that of the ssu rRNA. This graphical test shows that the evolution of the four proteins and the archael and bacterial ssu rRNA's are consistent with a Poisson process since last common ancestor. The distance-matrix rate test that is introduced in this study needs to make no assumptions regarding evolutionary rates, divergence times, or phylogenetic relationships.     

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.     

The index of dispersion of molecular evolution: slow fluctuations

The most simple neutral model of molecular evolution predicts that the number of substitutions within a lineage in T generations ought to be Poisson distributed. Therefore, the variance in the number of substitutions ought to equal the mean number. The ratio of the variance to the mean number of substitutions is called the index of dispersion, R(T). Assuming infinite sites, no recombination model of the gene, and a haploid, Moran population structure, R(T) is derived for a general stationary model of molecular evolution. R(T) is shown to be affected by fluctuations in parameters only when they occur on a very slow time scale. In order for parameter fluctuations to cause R(T) to deviate significantly from one, the time between parameter changes must be roughly as large, or larger, than the time between substitutions.     

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.     

Influence of the exclusion distance among trees on gap fraction and foliage clumping index of forest plantations

Key message

A hypergeometric model is proposed explicitly instead of two previous stochastic models (the Poisson model and Neyman-A model) to describe the topological relationship of trees and the influence of the exclusion distance on gap fraction and clumping index of forest plantation canopies.

Abstract

Gap fraction (GF) and clumping index (CI) play key roles in plant light interception, and therefore they have strong impacts on plant growth and canopy radiative transfer processes. Trees are usually assumed to be randomly distributed in natural forests in many previous studies. However, few studies have shown how trees are distributed in forest plantations and how these distribution patterns affect GF and CI in these forests. In this paper, a simple and general distance factor defined as relative allowable shortest distance between centers of two adjacent crowns divided by the mean diameter of the crowns (RASD) is proposed to describe quantitatively the degree of mutual exclusion among trees in forest plantations of various tree distribution patterns. A hypergeometric model is proposed instead of two previous stochastic tree distribution models (the Poisson model and Neyman-A model) to describe the topological relationship of trees and the influences of the exclusion distance on the GF and CI of the forest plantation canopies. The results show that: (1) the hypergeometric model is more suitable than the Poisson model and Neyman-A model for describing the topological relationship of trees in forest plantations; (2) the exclusion distance has strong impacts on GF and CI: there are significant differences between the results of the hypergeometric model and the Poisson model. Larger RASD causes lower GF and larger CI. The simulations are verified by field measurements in four forest plantation stands. Similarly, impacts of RASD on GF and CI are also found for other two crown shapes (prolate and oblate ellipsoids).

     

The Rate of Cu,Zn Superoxide Dismutase Evolution

The rate of amino acid replacement in Cu, Zn SOD greatly departs from the expectations of the molecular clock. We examine 27 Cu, Zn SOD sequences available and conclude that: (I) the SOD enzymes from different mammal families differ from each other by roughly the same number of replacements, which is consistent with a simultaneous mammalian radiation; (2) over the most recent 60 million years (MY) the rate of SOD evolution is fairly high (15aa/100aa/100MYR) and may be considered constant; (3) the rate of accumulation of amino acid replacements since the divergence of fungi. plants and animals to the present is inconstant along different branches of the evolutionary tree; moreover it steadily decreases with time, to the same extent in all lineages; (4) some comparisons exhibit divergences that are in any case greater than expected from a Poisson process on the assumption of a molecular clock; (5) plant chloroplast enzymes display fewer differences from each other than cytoplasmic ones; (6) bacteriocuprein (from Photobacterium leiognathi), fluke and human extracellular SOD are all three extremely remotely related to one another and to the SOD of other eukaryotes.

The process of consistent decline of the rate of evolution of Cu. Zn SOD can be described by a number of mathematical functions. We explore simple models that assume constant rates and might be applicable to other proteins or genes that apparently evolve at disparate rates.     

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.     

Stochastic modeling of the neuronal activity in the subthalamic nucleus and model parameter identification from Parkinson patient data

Several stochastic models, with various degrees of complexity, have been proposed to model the neuronal activity from different parts of the human brain. In this article, we use a simple Ornstein–Uhlenbeck process (OUP) to model the spike activity recorded from the subthalamic nucleus of patients suffering from Parkinson’s disease at the time of implantation of the electrodes for deep brain stimulation. From the recorded data, which contains information about the spike times of a single neuron, we identify and extract the model parameters of the OUP. We then use these parameters to numerically simulate the inter-spike intervals and the voltage across the neuron membrane. We finally assess how well the proposed mathematical model fits to the measured data and compare it with other commonly adopted stochastic models. We show an excellent agreement between the computer-generated data according to the OUP model and the measured one, as well as the superiority of the OUP model when compared to the Poisson process model and the random walk model; thus, establishing the validity of the OUP as a simple yet biologically plausible model of the neuronal activity recorded from the subthalamic nucleus of Parkinson’s disease patients.     

Determinants of rate variation in mammalian DNA sequence evolution

Attempts to analyze variation in the rates of molecular evolution among mammalian lineages have been hampered by paucity of data and by nonindependent comparisons. Using phylogenetically independent comparisons, we test three explanations for rate variation which predict correlations between rate variation and generation time, metabolic rate, and body size. Mitochondrial and nuclear genes, protein coding, rRNA, and nontranslated sequences from 61 mammal species representing 14 orders are used to compare the relative rates of sequence evolution. Correlation analyses performed on differences in genetic distance since common origin of each pair against differences in body mass, generation time, and metabolic rate reveal that substitution rate at fourfold degenerate sites in two out of three protein sequences is negatively correlated with generation time. In addition, there is a relationship between the rate of molecular evolution and body size for two nuclear-encoded sequences. No evidence is found for an effect of metabolic rate on rate of sequence evolution. Possible causes of variation in substitution rate between species are discussed.     

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.     

Evaluating scaling models in biology using hierarchical Bayesian approaches

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.