Friday, June 06, 2008

Notes on Sewall Wright: Population Size   posted by DavidB @ 6/06/2008 05:30:00 AM

Continuing my series of notes on the work of Sewall Wright, I come to the question of population size. This is important in Wright's formulation of population genetics and his evolutionary theory generally. One of the major differences between Wright and R. A. Fisher is that Fisher believed that, in general, evolutionary processes could be treated as if they took place in a very large random-mating population. He did not believe, contrary to some caricatures, that species were literally random-mating across their entire range (which is obviously false), but rather that there was usually enough migration between different parts of that range that for most purposes the departures from random mating did not matter. Wright, on the other hand, believed that in many cases local populations were sufficiently isolated from each other that they could be treated as populations evolving separately. This difference of views had a major impact on Wright's and Fisher's assessment of the relative importance of selection and genetic drift.

In his treatment of genetic drift Wright showed that in the absence of mutation and migration, genetic diversity, as measured by the proportion of heterozygotes in the population, will decline at a rate of 1/2N per generation, where N is the relevant population size. The larger the size, the slower the loss of diversity. This raises the question what is the 'relevant' size of N. As Wright explained in his great 1931 paper 'Evolution in Mendelian Populations', 'The conception is that of two random samples of gametes, N sperms and N eggs, drawn from the total gametes produced by the generation in question (N/2 males and N/2 females each with a double representation from each series of allelomorphs). Obviously N applies only to the breeding population and not to the total number of individuals of all ages' (p.111, 'Evolution: Selected Papers' (ESP). Unless otherwise stated, all citations are from this source.)

Wright immediately goes on to say that this idealised model of the population is often an oversimplification. The effective size of the population is often different from the current actual number of breeding adults. If the effective size is smaller than the apparent size (the current number of breeding adults), genetic drift will be faster than expected. We may say that the effective size of the population is the size of an idealised population, meeting the criteria outlined in the quotation from p.111 given above, which would give rise to genetic drift at the same rate as actually observed. I am not sure that Wright ever formally defines effective size, but the definition I have suggested seems to be implied in various references, e.g. ESP pp.111, 157, 251, 354.

Wright repeatedly specifies three factors which tend to reduce the effective size of the population below its apparent size:

1) different numbers of breeding males and females (ESP, pp.112, 251, 299, 354, 370). The effective population size is closer to that of the rarer sex.

2) where variance in reproductive success greater than that assumed in the idealised model (ESP pp. 112, 251, 300, 354, 270), genetic drift will be faster.

3) Occasional or cyclical reductions in population size (ESP pp.112, 157, 251, 300, 354, 370). The effect of (non-selective) reductions in population size is to take a random sample out of the gene pool. Such samples will have a variance in gene frequencies proportional to 1/n, where n is the size of the sample. The smaller the number n, the larger the variance due to 'sampling error'. If n is small relative to N (the usual population size), the effect is equivalent to concentrating many generations of slow genetic drift into a single event. In the absence of mutation and selection the effect is irreversible. A subsequent expansion of population, however large, does not reverse the loss of genetic diversity. (But note that if there is mutation and selection, an expansion of population gives an opportunity for rare advantageous mutations to appear and be selected. An expansion of population is also often associated with a relaxation of natural selection, which means that slightly disadvantageous mutations, which would normally be weeded out, may survive. This could help shift the population across a 'valley' in the adaptive landscape, if such things exist).

These three factors all tend to reduce the effective population size below the current observed number of adult males and females. Wright repeatedly claims that the effective size is usually less than the apparent size, for example, 'The effective size (N) of the theory may, however, differ much from the apparent size, being usually much less' (ESP p.251). So far as I know, Wright only once mentions a factor that might increase the effective number above the apparent level: on ESP p.300 he mentions that the variance in reproductive success could be less than in the idealised model, in which case the effective population number could be up to twice the apparent size. But he comments that this improbable except in planned breeding experiments.

So far so good. But so far as I am aware, Wright never mentions another factor which may raise the effective population size above the current number of breeding adults. This is where there is a large reserve of juvenile or dormant individuals with the ability to replace the current adults in the event of a population reduction. Such a reserve population would contain a greater amount of genetic diversity than the reduced number of current adults. This is probably a minor factor in the case of vertebrate animals, but could be important among some small invertebrates, where the number of eggs or larvae may be many times the current 'crop' of adults. It is even more important in the case of plants. Most species of plants produce resistant seeds, bulbs, etc, which are orders of magnitude more numerous than the mature plants. In some cases they can survive for years or decades in a dormant state. The genetic effect of sharp reductions in adult population numbers (e.g. due to drought) may therefore be much less among plants than among animals. This oversight vitiated one of Wright's own major empirical studies (see Provine p.485).

Another major complication is migration. Wright's idealised model of genetic drift assumes that the population is completely self-contained, that is, reproductively isolated from other populations. If the population is an entire biological species, this is true by definition, since a biological species is defined by reproductive isolation. But if the population is a subdivision of a species, there is in principle the possibility that genes will enter the population from outside. My next note will examine how Wright dealt with this complication.

William B. Provine: Sewall Wright and Evolutionary Biology, 1986.

Sewall Wright: Evolution: Selected Papers, edited and with Introductory Materials by William B. Provine, 1986.

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