C. C. Li

Similarity of DNA fingerprints due to chance and relatedness.

Given the DNA fingerprints of two individuals with some bands being shared by both individuals, we define a new measure of the degree of similarity between the DNA profiles of two individuals. We use this measure to calculate the expected DNA similarity of two unrelated individuals of a randomly mating population; this similarity is due to chance only. Then, the expected similarity between two related individuals is obtained; this similarity is due to chance and relatedness. From these results, the degree of similarity due to relatedness alone may be calculated.

The Stability of an Equilibrium and the Average Fitness of a Population

The effect of intra-population selection on gene frequency in a large random mating population has been examined. In the absence of other forces when the genotypic selective values are independent of gene frequencies, a stable equilibrium value of gene frequency yields a maximum value of the average fitness of the population and an unstable equilibrium yields a minimum average fitness. This relationship is best illustrated by plotting both the W̄ curve and the Δ q curve on parellel q axis of the same scale. In more involved cases, it is often possible to define a modified average fitness of the population so that the above conclusion holds. Only relative magnitudes of the intra-population selective values are relevant. It was emphasized that they cannot be used as a basis for comparison between two separate populations under two different environments. A comparison of absolute adaptiveness of two populations cannot be done until absolute adaptiveness is defined and measured.

THE CONCEPT OF PATH COEFFICIENT AND ITS IMPACT ON POPULATION GENETICS

The method of path coefficients was first published by Professor Sewall Wright thirty-five years ago. In 1921 there appeared in the Journal of Agricultural Research (Vol. 20) a general account of the method and the relationship between correlation and path coefficients, together with some examples of application; and in Genetics (Vol. 6) a series of five papers dealing exclusively with the application of path coefficients to genetic problems. Previously known results of various mating systems, obtained by laborious arithmetical procedures, were confirmed by the more elegant method, and many new results were reached, some of which were later corroborated by the method of matrix algebra while others are still difficult to obtain by any other method today. These classical papers, together with the pioneer work of Fisher (1918), still constitute the basic readings for students of population genetics, although the method has since taken a more sophisticated form and the field of application has been widened. However, one must admit that the method of path coefficients, as powerful and flexible as it is, was not immediately very popular among geneticists, still less so among professional statisticians. It was much later that its usefulness became gradually and generally appreciated. Path coefficients can be treated at various mathematical levels. The most important properties, however, can be deduced and studied by standard statistical tools. To understand the method requires little more than a knowledge of multiple and partial regression and correlation. It is a special type of multi-variate analysis a method of dealing with a closed system of variables that are linearly related. (For nonlinearly related variables, an appropriate transformation of scales may

Population subdivision with respect to multiple alleles

In view of the current interest in studying human isolated populations and the fact that many human gene markers have multiple alleles, the writer thought that it would be helpful to have the problem of population subdivision discussed in more detail and hence stimulate further investigation. Although the problem under consideration seems at first sight purely genetical, an appropriate change of a few technical terms will render the problem identical with those of epidemiologists and sociologists studying the association of certain traits or diseases in specific (homogeneous) groups and in the combined (heterogeneous) group. It may be said that these problems are ‘isomorphic’. Thus, an investigationin one area has similar implications in the others. We shall illustrate these ideas very briefly for the case of two alleles before introducing multiple alleles.

Fundamental Theorem of Natural Selection

FISHER1 in 1930 stated his “fundamental theorem of natural selection” in the form: “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.” Later, Fisher2 restated his theorem more clearly: “The rate of increase in the average fitness of a population is equal to the genetic variance of fitness of that population”. The “genetic variance” in the foregoing statements is the linear or additive component of the fitness variance in current literature. Fisher obtained his result on the basis of a continuous time model with logarithmic fitness. This communication gives a simple derivation for an appropriate corresponding expression for the discrete-generation model and points out when Fishers theorem still applies and when it does not.

CHROMOSOMAL ADAPTIVE POLYMORPHISM IN DROSOPHILA PERSIMILIS III. MATING PROPENSITY OF HOMOKARYOTYPES

The outcome of selection is determinate in laboratory populations of Drosophila containing variable proportions of gene arrangements of uniform geographic origin (Dobzhansky, 1954, 1957). Consequently such gene arrangements must carry genic complexes which confer adaptive functions upon their carriers. While the evidence for adaptive control by these complexes in populations of D. pseudoobscura and D. persimilis, for example, is overwhelming, the identification of those adaptive functions conferred by the genic complexes has remained somewhat cryptic. In the cases of chromosome III polymorphism in these species, fitness, characters like viability, fertility, longevity, rates of development and maturation have not been proved to be intrinsic to the gene arrangements adaptive control either in laboratory or natural populations, although many suggestive data have been accumulated (Spiess and Schuellein, 1956). When raised under competitive conditions as in a population cage, flies with specific karyotypes may differ in combinations of adaptive traits which when considered separately are weak and barely measurable but which might bring about selective differences of considerable strength by their net action. Such combinations of fitness traits could account for a portion of changes observed in populations cages (Moos, 1954; Spiess, 1958). Logically it is to be expected that constant adaptive control of certain karyotypes in particular environments results from a physiological action which is capable of being ascertained in spite of genic interactions of some complexity and the sensitivity of such control to environmental variables. The net fitness of any gene ar-

EQUILIBRIUM UNDER DIFFERENTIAL SELECTION IN THE SEXES

When selection operates with the same intensity in both females and males, the selectional equilibrium condition of a Mendelian population may be found easily and is familiar to population geneticists. If the selection coefficients against a certain genotype differ only slightly in the two sexes, the average value of the two selection coefficients may be taken as the common selection coefficient for both sexes. In all such cases the gene frequencies in the two sexes are the same. However, when the selection intensities differ widely between the two sexes, as was found in the experiments of Woolf and Church (1963), the equilibrium gene frequencies in the two sexes are no longer equal, and this is characteristic of differential selection in sexes. I would venture to say that differential selection in sexes is more common than is generally recognized. In the following, explicit solutions for the equilibrium gene frequencies under the conditions of Woolf and Church have been given.

Is Rh Facing a Crossroad? A Critique of the Compensation Effect

The compensation on the part of rh-negative mothers who tend to replace their lost (heterozygous) children by having more (recessive) births cannot lead to a stable equilibrium value of rh frequency. The compensation coefficient (t) has been estimated to be roughly 0.36 to 0.40 for American Whites, being much larger than selection intensity (.025 to .10). Although it seems to be on the increase at present, the future direction of change in rh frequency and its ultimate fate in the population is still indefinite, depending upon the future status of selection and compensation.

Correspondence between a Linear Restriction and a Generalized Inverse in Linear Model Analysis

Abstract In experimental statistics the usual method of estimating treatment effects is to introduce arbitrary linear restrictions among the treatment effects in order to obtain solutions of the normal equations. A comparatively recent approach is to dispense with the linear restriction and use a generalized inverse solution to the normal equations. The present note is an attempt to bring these two methods closer together and show their correspondence; namely, for each given linear restriction, there exists a corresponding generalized inverse that yields the same solution for the treatment effects as the linear restriction does.

Estimation and testing of a measure of non-random mating

The Hardy-Weinberg Law expresses the relationship between the expected relative frequencies of genotypes in a population subject to random mating. Let the relative frequencies of the three genotypes GG, Gg, and gg be D, H , and R, D + H + R = 1, so that the frequencies of the alleles G and g are p = D + aH and q = R + 3H respectively. Under random mating D = p2, H = 2pq, R = q2 so that DR-