John H. Gillespie

A simple stochastic gene substitution model

If the fitnesses of n haploid alleles in a finite population are assigned at random and if the alleles can mutate to one another, and if the population is initially fixed for the kth most fit allele, then the mean number of substitutions that will occur before the most fit allele is fixed is shown to be (formula; see text) when selection is strong and mutation is weak. This result is independent of the parameters that went into the model. The result is used to provide a partial explanation for the large variance observed in the rates of molecular evolution.

Some Properties of Finite Populations Experiencing Strong Selection and Weak Mutation

Some general properties of populations experiencing strong selection (α = 2NS ≫ 1) and weak mutation (Θ = 4NU ≪ 1) with k ≪ ∞ alleles are described. Under these assumptions a true boundary layer dynamics emerges with rare alleles remaining near zero for an exponentially distributed length of time; thereafter they enter the interior of the allelic frequency space where natural selection alone operates at a much faster time scale than occurs in the boundary layer. This structure allows a much simpler description of the evolutionary process than by the conventional diffusion analysis. Particular models examined include overdominant selection and selection in a randomly fluctuating environment. For the latter model it is shown that drift and mutation can have a profound effect on the number of polymorphic alleles as suggested earlier by Nei and Takahata. The analysis indicates that a fundamental parameter is the product αΘ. If this quantity is very small, evolution effectively stagnates. If αΘ is moderate, the above described boundary layer dynamics emerge, whereas if αΘ is very large, drift does not effectively inhibit the progress of evolution. In the time scale of the process the rate of evolution is directly proportional to the population size.

The neutral theory in an infinite population.

Selective substitutions at one locus induce stochastic dynamics at a linked neutral locus that resemble genetic drift even when the population size is infinite. This new stochastic force, which is called genetic draft, causes genetic variation at the neutral locus to decrease with population size and the rate of deleterious substitution to increase with population size. The fact that heterozygosities in natural populations are only weakly dependent on population size suggests that genetic draft may be a much more important stochastic force than genetic drift in natural populations. Some of the mathematical properties of genetic draft are explored.

SUBSTITUTION PROCESSES IN MOLECULAR EVOLUTION. II. EXCHANGEABLE MODELS FROM POPULATION GENETICS

Substitution processes are of two sorts: origination processes record the times at which nucleotide mutations that ultimately fix in the population first appear, and fixation processes record the times at which they actually fix. Substitution processes may be generated by combining models of population genetics—here the symmetrical‐neutral, overdominance, underdominance, TIM, and SAS‐CFF models—with the infinite‐sites, no‐recombination model of the gene. This paper is mainly concerned with a computer simulation study of these substitution processes. The rate of substitution is shown to be remarkably insensitive to the strength of selection for models with strong balancing selection caused by the genealogical drift of mutations through alleles held in the population by selection. The origination process is shown to be more regular than Poisson for the overdominance, TIM, and SAS‐CFF models but more clustered for the underdominance model. A class of point processes called Sawyer processes is introduced to help explain these observations as well as the observation that the times between successive originations are nearly uncorrelated. Fixation processes are shown to be more complex than origination processes, with regularly spaced bursts of multiple fixations. An approximation to the fixation process is described. One important conclusion is that protein evolution is not easily reconciled with any of these models without adding perturbations that recur on a time scale that is commensurate with that of molecular evolution.

A GENERAL MODEL TO ACCOUNT FOR ENZYME VARIATION IN NATURAL POPULATIONS. III. MULTIPLE ALLELES

Recent biochemical explorations into electrophoretic classes of enzyme variants have revealed a wealth of previously suspected but undetected variation (Bernstein et al., 1973; Singh et al., 1975). The existence of this variation poses new problems for the hypothesis that natural selection is maintaining the variation by heterotic selection since the conditions for polymorphism under heterosis become incredibly restrictive as the number of alleles increases (Dr. Ken-ichi Kojima, pers. comm.). One way to quantify this idea would be to ask: Of the total set of all possible fitnesses for the n(n + 1)72 genotypes at a locus with n alleles, what fraction leads to a stable polymorphism with all alleles segregating? This fraction can be easily calculated by a Monte-Carlo simulation on the computer by assigning each of the n(n + 1)72 genotypes a fitness value uniformly and independently distributed on the unit interval and applying the usual criterion for the existence and stability of an internal equilibrium in a population with n alleles (see, e.g. Mandel, 1959). When this experiment is repeated a large number of times, an estimate of the proportion of the fitness space which yields stable internal equilibria results. Figure 1 gives the results of such a simulation and illustrates well the truth of Kojimas statement about the restrictive nature of the heterosis assumption. There is, however, a very unrealistic aspect of this argument. In biological sysstems, certain laws govern the relationships of fitnesses in related genotypes. This aspect is amply born out in the work on the quantitative genetics of fitness components (e.g., Mukai et al., 1972) which gives estimates of the correlation of the heterozygote fitness to the mean of the two parental homozygotes. Although the laws governing the relationships of fitnesses of related genotypes are not known, it is clear they can radically change the impression given by Figure 1. For example, if heterozygotes were always intermediate in fitness between the parental homozygotes, and if the environment were constant, no stable polymorphism would be possible and the subset of stable fitnesses would be empty. On the other hand, if some biological principle dictated that heterozygotes are invariably more fit than their parental homozygotes, the fraction of the space of fitnesses yielding stable points would be much larger than given in Figure 1. Finding constraints on the assignment of fitnesses which are biologically meaningful and which will allow the maintenance of large numbers of alleles should be a major area of investigation now that new genetic variation is being uncovered in natural populations. In this paper, the constraints suggested in the previous papers in this series (Gillespie and Langley, 1974; Gillespie, 1976), will be examined and shown to allow readily the stable existence of large numbers of alleles by balancing selection in a fluctuating environment.

ON OHTA'S HYPOTHESIS: MOST AMINO ACID SUBSTITUTIONS ARE DELETERIOUS

Ohtas hypothesis that most amino acid substitutions are deleterious grew out of a class of population-genetics models called shift models. Recently, shift models have been shown to be biologically unreasonable and have been replaced by a more plausible house-of-cards model. In this paper, the simplest form of the house-of-cards models is shown to be incompatible with most of the major features of protein evolution. Moreover, this model is shown to not be a model of exclusively deleterious-allele evolution, but rather to be a model with an equal mix of deleterious and advantageous substitutions.

Codon usage divergence of homologous vertebrate genes and codon usage clock

SummaryThis paper is concerned with the divergence of synonymous codon usage and its bias in three homologous genes within vertebrate species. Genetic distances among species are described in terms of synonymous codon usage divergence and the correlation is found between the genetic distances and taxonomic distances among species under study. A codon usage clock is reported in alphaglobin and beta-globin. A method is developed to define the synonymous codon preference bias and it is observed that the bias changes considerably among species.

The interaction of genetic drift and mutation with selection in a fluctuating environment.

The interaction of genetic drift, mutation, and selection in a random environment is investigated using an asymptotic analysis based on assumptions of weak mutation and strong selection. It is shown that genetic drift can be a potent force for removing variation from the population when the random environment tends to occasionally push alleles down to low frequencies.

Molecular Evolution and Polymorphism: SAS-CFF Meets the Mutational Landscape

A Markov model of molecular evolution resulting from natural selection in a fluctuating environment that appears to be in substantial agreement with many of the observations on genetic variation within and between species is described. Although the Markov model is based on a particular model of selection, the Stochastic Additive Scale-Concave Fitness Function model, the approach-which uses strong-selection, weak-mutation limits--suggests a general strategy for approximating the dynamics of a much broader class of models. The main dynamic features of the model are a rapid buildup phase that introduces new alleles into the population, an allelic-exchange phase that slowly replaces polymorphic alleles, and allelic constrictions, rare events that cause a complete turnover of alleles. The model exhibits many properties commonly observed in sequence data, among which are the variability of rates of evolution, the frequency spectrum of segregating sites, and the generation-time effect

The transient properties of balancing selection in large finite populations

The transient properties of balancing selection in large, but finite, populations are described by means of an asymptotic analysis. Heterotic selection is shown to retard the rate of loss of genetic variation while random environment selection is shown to retard the rate of loss of variation when the initial variant is a rare mutant. Otherwise random environment selection can speed up the loss of variation for certain parametric cases. The asymptotic analysis leads to a particularly simple conceptualization of the selection process which allows the computation of asymptotic forms of the dominant eigenvalue of the process.