Thomas Nagylaki

A model for the evolution of self-fertilization and vegetative reproduction.

Abstract It was shown in a simple model that a gene for selfing will be established in a population if and only if at least one of the selfing genotypes also contributes to random fertilization. The same conclusion is valid for vegetative reproduction and ameiotic parthenogenesis. This connection between selfing and the mating system is consistent with the much greater frequency of uniparental reproduction among plants than animals.

The Rate of Change of a Character Correlated with Fitness.

An extended form of Fishers Fundamental Theorem of Natural Selection gives the rate of change of the mean value, \( \bar{C} \) , of a measured character. For a character determined by multiple alleles at two loci, this is \( \dot{\bar{C}} = {\mathrm{cov}_{g}} (m, \gamma) + \overline{\dot{C}} + \sum\limits^{2}_{n=1} \overline{\Delta^{(n)}\mathring{\theta}^{(n)}}+\overline{\varepsilon\mathring{\theta}} \) where the Newtonian superior dot means the time derivative and the circle is the time derivative of the logarithm. Covg (m, γ) is the genic (additive genetic) covariance of the character and fitness. Specifically, it is the covariance of the average excess of an allele for fitness and its average effect on the character. \( \overline{\dot{C}} \) is the average rate of change of the value of the character for individual genotypes, weighted by their frequencies. The value could be nonzero because of changing environments or change in the age distribution of the population. The third term on the right is the average over all pairs of alleles at both loci of the product of the dominance deviation and the rate of change of ln θ(n), where θ(n) is a measure of departure from random proportions. The last term is a similar expression for epistatic interactions. If selection is much weaker than recombination, after several generations, the last two terms are much smaller than the first. When the measured character is fitness, our result reduces to Kimuras generalization of Fishers Fundamental Theorem of Natural Selection.

Continuous selective models

Abstract Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.

The decay of genetic variability in geographically structured populations. II

Abstract The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that f( 0, x ) = O(x −2−η ), η > 0, as x ≡ ¦ x ¦ → ∞ . In the absence of mutation, f(t, x) tends to unity at the rate t −1 2 in one dimension and (ln t)−1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate (1 − u) 2t t −3 2 in one dimension and (1 − u)2tt−1(lnt)−2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malecot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.

Polymorphisms in cyclically-varying environments.

SummaryWe analysed both continuous and discrete two-allele models of cyclically-varying environments with an arbitrary degree of dominance. In continuous models, the gene frequency fluctuates with the period of the environmental oscillation. For the discrete case, the calculations were carried out to second order in selection. In contrast to the continuous models, and depending on the amount of dominance and the initial gene frequency, fixation is possible as well as polymorphism.

Continuous selective models with mutation and migration

Abstract The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.

The relation between distant individuals in geographically structured populations

Abstract The equilibrium structure of a population distributed continously and homogeneously in an infinite habitat is investigated. The analysis is confined to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Asymptotic expressions are derived for the probability that two homologous genes separated by a given distance are the same allele for a migration function which decays at least exponentially in three dimensions and for one with an infinite variance in one dimension. In the second case, the heterozygosity in the population is also calculated.

The Distribution of Sojourn Times in Finite Absorbing Markov Chains.

Abstract The distribution of the time spent before absorption in various states of finite Markov chains with time-independent transition probabilities is derived. All the moments of this distribution are calculated. The problem is of interest in population genetics.

A continuous selective model for an X-linked locus.

SummaryNeglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, a continuous selective model is formulated for a single X-linked locus with an arbitrary number of alleles. Without restricting the mating system, differential equations are derived for the genotypic and allelic frequencies. Assuming random mating, no selection, and constant fertilities and mortalities, these differential equations are solved explicitly. For this case, in contrast to the corresponding phenomenon in the usual model with discrete, non-overlapping generations, the difference between the frequencies of any allele in males and females approaches zero without oscillation.

Sampling truncated distributions

It is pointed out that the difference between the means of two populations is alwaysgreater than the difference between the sample means derived from the two distributions truncated at a lower limit, upper limit, or both. The form of the distributions is assumed to be identical but is otherwise arbitrary.