J. B. S. Haldane

A Mathematical Theory of Natural and Artificial Selection, Part V: Selection and Mutation

New factors arise in a species by the process of mutation. The frequency of mutation is generally small, but it seems probable that it can sometimes be increased by changes in the environment (1,2). On the whole mutants recessive to the normal type occur more commonly than dominants. The frequency of a given type of mutation varies, but for some factors in Drosophila it must be less than 10 −6 , and is much less in some human cases. We shall first consider initial conditions, when only a few of the new type exist as the result of a single mutation; and then the course of events in a population where the new factor is present in such numbers as to be in no danger of extinction by mere bad luck. In the first section the treatment of Fisher (3) is followed.

THE ESTIMATION AND SIGNIFICANCE OF THE LOGARITHM OF A RATIO OF FREQUENCIES

It is occasionally useful to estimate the logarithm of the ratio of two classes in a sample. But a much more important case is discussed by Woolf (1955). Out of n cases of a disease, h had the character a and k the character p. Out of N unaffected persons, H had the character a and hthe character /3. For example, of 1490 cases of peptic ulcer in London investigated by Aird, Bentall, Mehigan & Roberts (1954), h=Y11, k=579, while among 8797 controls, H =4578, K = 4219, where the characters a and are membership of blood groups 0 and A . Hence Woolf calculated hK/kH = 1.4500, In (hK/kH) = 0.3716, where In x= logp. Suppose that the frequencies of groups 0 and A in all cases of peptic ulcer in London were p andq, among allother Londoners P and Q, then ln ( h K / k H ) is an efficient estimate of In (pQ/qP) . But it has a bias which is not always negligible. The estimate hK/kH has formally an infinite expectation, and all the moments of its distribution are infinite. For there is a finite, though very small, probability that either k or H should be zero. Even if such cases are neglected it has a bias of order n-1. As pointed out by Haldane & Maynard Smith (1056), t.he bias of h K / { ( k + 1) ( H + 1)) tends to zero more rapidly than any negative power of n or X . Similarly, all the moments of the distribution of the logarithm are formally infinite, since any of h, k , H and K may be zero with a finite probability. This fact by itself is unimportant, but i t is an indication that a less biased expression can be found. Since the two samples were taken independently, their sampling errors are uncorrelated, and we can therefore consider the estimation of In ( p l y ) . Let h = p n +a, k =qn -01. I am assuming sampling from an infinite population, and shall neglect the errors due to the fact that this is not so. Then

The cost of natural selection

SummaryUnless selection is very intense, the number of deaths needed to secure the substitution, by natural selection, of one gene for another at a locus, is independent of the intensity of selection. It is often about 30 times the number of organisms in a generation. It is suggested that, in horotelic evolution, the mean time taken for each gene substitution is . about 300 generations. This accords with the observed slowness of evolution.

Disease and Evolution

RIASSUNTO. – Negli esempi addotti dai biologi per mostrare come la selezione naturale agisce, la struttura o la funzione presa in esame è per lo più collegata con la protezione contro forze naturali a vverse, contro predatori, oppure con la conquista di alimento o dell’altro sesso. L’A. mostra che la lotta contro le malattie, e in particolare contro le malattie infettive, ha rappresentato un fattore evolutivo molto importante e che alcuni dei suoi risultati sono diversi da quelli raggiunti attraverso le forme consuete della lotta per l’esistenza. RÉSUMÈ. – Les exemples portés par les biologistes pour montrer comment la séléction naturelle opère tiennent compte d’ordinaire de structures ou de fonctions liées à la protection contre des forces naturelles hostiles, contre des prédateurs, ou bien liées à la conquéte de la nourriture ou du sexe opposé. L’A. montre que la lutte contre les maladies, et en particulier contre les maladies infectieuses, a représenté un facteur évolutif très important et que qualques-uns parmi ses résultats diffèrent bien de ceux qui ont été atteints par les formes ordinaires de la lutte pour la vie. SUMMARY. – Examples quoted by biologists, in order to show how natural selection is working, almost present structures or functions concerned either with protection against natural forces or against predators, or with purchase of food or mates. The Author suggests that the struggle against diseases, and especially infectious diseases, has been a very important evolutionary agent and that some of its results have been rather unlike those of the struggle for life in its common meaning. It is generally believed by biologists that natural selection has played an important part in evolution. When however an attempt is made to show how natural selection acts, the structure or function considered is almost always one concerned either with protection against natural «forces» such as cold or against predators, or one which helps the organism to obtain

The rate of spontaneous mutation of a human gene

SummaryThe rate of mutation at which the gene for haemophilia appears in the population of London is estimated at about once in 50,000 human life cycles. There are probably two distinct allelomorphs at the same locus, the milder type arising less frequently by mutation than the severe type.I have to thank Dr Julia Bell and Dr C. V. Green for most generously placing at my disposal data collected on behalf of the Medical Research Council and the Research Committee of the American Medical Association.

An exact test for randomness of mating

SummaryAn exact test is given for randomness of mating, and it is shown that in a population ofPanaxia dominula there is no evidence that mating was not random in any of 15 years.

The Relation between Density Regulation and Natural Selection

Darwin (1859) introduced the notion of natural selection by showing that if the density of a species is to remain steady, most of the individuals in each generation must die prematurely. Nicholson (1954) has analyzed the factors determining the density of a species in great detail. Haldane (1953) did so more summarily. It is simplest to consider annual plants or animals. Their increase or decrease from one year to another is a function, among other things, of their density. A factor which leads to increase as the density increases is called density-disturbing (Nicholson), or positive density-dependent (Haldane). Such factors include all forms of mutual aid, from the mutual protection of trees from storms and the greater ease of finding mates if the population is not too sparse, to various forms of social behaviour. Density-regulating (Nicholson) or negative density-dependent (Haldane) factors include all forms of competition, including competition for food and space, and disease facilitated by overcrowding. Other factors, which Nicholson calls density-legislative and density-inactive, are independent of density. Examples are the effects of heat and cold, in so far as they are not modified by competition or co-operation. Clearly at high densities some negative density-dependent factor must come into action, or density would increase indefinitely. Positive density-dependent factors maybe important at low densities. If so there may be an unstable equilibrium, and populations which fall below it decrease further and die out, for example, sessile bisexual or self-sterile organisms with juvenile dispersal. They always make for instability of equilibrium. Negative density-dependent factors may do so if their effect increases very sharply with density, or if it is delayed (Nicholson & Bailey 1935).

The estimation of viabilities

SummaryIf a sample containsa of a phenotype A andb of a phenotype B, where equal numbers of the two types are expected, thenb/a is a biased estimate of the relative viability of B; andb/(a+ I) is an almost unbiased estimate.

The Theory of Selection for Melanism in Lepidoptera

Dr Kettlewells proof that the dark form carbonaria of Biston betularia has replaced the type, at least in part as the result of selection by bird predators, gives me the right to bring my calculation (Haldane 1924) on this matter up to date. Assuming random mating, if the relative fitnesses of the genotypes CC, Cc and cc are as 1:1 - k:1 - K, and the gene frequencies are pC + qc, then the annual change in the value of p is Δp = (pq(kp - kq + Kq)) / (1 - 2kpq - Kq2) . But K and k are not constants, even in a constant environment. They change with the frequency of other genes (or possibly plasmons). At present in industrial areas, K is positive and k nearly zero, since C is apparently fully dominant. However, on the genetic background of unpolluted areas, Cc is intermediate and k may be about 1/2K (Ford 1955).

The theory of inbreeding with forced heterozygosis

SummaryWhen a foreign gene is being bred into a pure line, or when a population with a lethal gene, incomplete sex linkage, heterostylism, or self-sterility is being inbred, the population is kept heterozygous for a particular gene or chromosome segment, but otherwise inbred. In these cases expressions are found for the probability of introducing a gene linked with the given gene, or for finding such a linked gene still heterozygous after inbreeding. Expressions are also found for the mean length of foreign chromosome introduced, or the length of chromosome remaining heterozygous (Table VI). The difficulty of introducing a single gene without any linked genes is emphasised.