Edward Pollak

Fixation probabilities and effective population numbers in diploid populations with overlapping generations

Abstract A finite diploid population, observed at times t = 0, 1, 2,…, is studied. An individual is said to be in age group i at time t if its age is between i and i + 1 units at that time, where i ⩾ 1. It is assumed that the number of individuals in a particular age-sex class is the same for every t and that the probability that a male offspring was produced by a mating of a male in age group i and a female in age group j is p ij m (with a corresponding probability p f ij for a female offspring), regardless of when the individual is born. The probability of ultimate fixation of an allele A 1 and the inbreeding effective number, for large populations, is calculated under the further assumptions that A 1 is neutral and that mating is random, given the ages of the mates.

Effective population numbers and mean times to extinction in dioecious populations with overlapping generations.

Abstract We consider a finite monoecious population that is observed at times 0, 1,… and is divided into age classes 0, 1,…, K . It is assumed that the numbers of individuals in the various age classes do not change with time and that there is no mutation, selection or migration. If the allele A 1 is initially rare and the population large, the underlying Markov chain can be approximated by a nonsingular positively regular and critical multitype branching process. Functional equations are given for the generating functions of the invariant measure and the Greens function of the branching process, and properties of the solutions of these equations are obtained. These properties lead to expressions for the effective population number and the mean time to extinction of a line of A 1 A 2 heterozygotes descended from a single A 1 A 2 individual in class i .

Malthusian parameters in genetic populations. I. Haploid and selfing models

Abstract Leslies discrete time theory of population growth is applied to genetic populations reproducing wholly by selfing. If there is selection, but no mutation, the number of individuals having a particular genotype usually has a long run steady rate of geometric increase, which we call the Malthusian parameter of the genotype. If the population size changes slowly, the rate of change in the mean of the Malthusian parameters tends, as time increases, to the variance of the Malthusian parameters. If, however, mutations of all kinds are possible, the population has only one Malthusian parameter.

ON THE SURVIVAL OF A GENE IN A SUBDIVIDED POPULATION

A classical type of problem in population genetics is that of calculating the probability that a line descended from a particular gene will become extinct. In one problem of this sort, dealt with by Fisher (1922) and Haldane (1927), it is assumed that the population being studied is very large and that initially the number of genes of a particular type, say type A, is small. These authors obtained the solution by the use of the theory of branching processes. In another problem of this type it is assumed that the population always has a constant number N1 of genes and A has a certain selective advantage or disadvantage with respect to another gene A. Such a population will eventually have only one or the other of these genes. It is then desired to calculate the probabilities that the population has only A, or only A, genes. Fisher (1930) and Wright (1931) gave an approximate solution. Improved results have been obtained in recent years by Malecot (1952), Kimura (1957) and Moran (1960). In this paper we generalize these problems to a situation in which there are K subpopulations that inhabit different ecological niches, between which there can be migration. First it is assumed that there is no limit to the size that a population can attain and that the genes of type A reproduce independently of each other. It is then possible, by using the theory of multitype branching processes, to calculate the probability that A survives in a population. This quantity is expressible in terms of the numbers

The effective population size of an age-structured population with a sex-linked locus.

Let a population have the same age distribution and age-specific sex ratios at times 0, 1, 2,..., and let M, F, and L, respectively, be the numbers of males and females in the youngest age group and the generation interval. It can then be shown that if there is a sex-linked locus the fixation probabilities of a neutral allele are respectively 1/3LM or 1/3LF if the allele first appears in one newborn male or in one newborn female. The effective population size can then be derived. It is the same as for a population with discrete generations having the same means, variances, and covariances of male and female progeny during a lifetime and the same number of individuals entering the population per generation.

Malthusian parameters in genetic populations part II. Random mating populations in infinite habitats

Abstract This paper gives a development of ideas presented in a previous paper on the dynamics of haploid genetic populations in an infinite habitat. A brief review of necessary ideas and a discussion of a concept of value for haploid populations are given. A general model for a diploid population, which incorporates life tables, mating of individuals, and fecundities is presented and general recursive relationships for frequencies of genetic types developed. Because of its intractability special cases are then considered in which there is equal fecundity of all matings in the population of matings that occurs. It is found that in some cases the numbers of the various types increase asymptotically at geometric rates, so that each type may be said to have a Malthusian parameter. It is then found that a version of Fishers Fundamental Theorem holds in that the asymptotic rate of increase of the average Malthusian parameter is equal to the genotypic variance in this parameter. Additionally, the Malthusian parameters are found to be additive. It does not seem possible, however, to extend the concept of value to diploid mating populations.

The inbreeding effective number and the effective number of alleles in a population that varies in size

Abstract Consider a population that does not change in size. If it is assumed that there are an infinite number of possible neutral alleles at a locus and u is the probability that a particular gene mutates to some other gene in one generation, the effective number of alleles n e is computed to be 4 N e u + 1, where N e is the inbreeding effective population number. It is assumed in this paper that the number of individuals in a monoecious population, or the numbers of males and females in a dioecious population, are states in a finite irreducible Markov chain. In general it is impossible to obtain a single value of n e . In some cases where the computation of n e is possible, the results are as follows. When the population is monoecious, N e is the reciprocal of the asymptotic average, over population sizes, of the probabilities that two gametes uniting to form an individual came from the same individual one generation earlier. In dioecious populations, N e is the reciprocal of the long-run average of the probabilities that two homologous genes in separate individuals of one generation came from the same individual one generation earlier. Special cases are discussed.

The effective population size of some age-structured populations☆

It was shown in a previous paper that if generations are discrete, then the effective population size of a large population can be derived from the theory of multitype branching processes. It turns out to be proportional to the reciprocal of a term that appears in the denominator of expressions for survival probabilities when there is a supercritical positively regular branching process for which the dominant positive eigenvalue of the first moment matrix is slightly larger than 1. If there is an age-structured population with unchanging proportions among sexes and age groups, then the effective population size is shown to be also obtainable from the theory of multitype branching processes. The expression for this parameter has the same form as in the corresponding model for discrete generations, multiplied by an appropriate measure of the average length of a generation. Results are obtained for dioecious random mating populations, populations reproducing partly by selfing, and populations reproducing partly by full-sib mating.

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).

On sojourn times at particular gene frequencies

The distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of b ij s, where b ij is the mean number of visits to frequency j /2 N , given initial frequency i /2 N . The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.