The purpose of science, says Herbert Simon, "is to find meaningful simplicity in the midst of disorderly complexity". A real population includes individuals of all ages, some dying, some being born, some choosing mates, some reproducing, some migrating, some simply growing old. And in a sexual population each individual has a different genotype. Keeping track of all these goings-on is impractical and not very interesting. Most questions of genetic or evolutionary importance do not require such detail.
The gene-pool model is a wonderful, simplifying convention. The analogy is close to game-playing. A dictionary definition of a pool is "an aggregated stake to which each player has contributed". In the population model, each individual contributes to a large (theoretically infinite) pool of gametes. An offspring is a sample of two gametes from this pool. In the simplest case, the parents contribute equally to the pool and the sample is drawn randomly. For large diploid populations, this leads immediately to the familiar Hardy–Weinberg law (see Box).
The simplicity and power of treating random sampling from a pool of genes as equivalent to random mating of diploid individuals is a bonus, especially appreciated by students in elementary genetics courses.
In a sexual population, each genotype is unique, never to recur. The life expectancy of a genotype is a single generation. In contrast, the population of genes endures. The quantities that are followed, in mathematical theories or in observations, are allele frequencies. The geneticist knows that at any desired time, the genotype frequencies can be obtained by the simple binomial rule.
Mutation replaces one allele in the pool by another. Selection is introduced when parents contribute unequally to the gene pool. Meiotic drive entails preferential contribution of certain alleles in the gametes from a parent. Inbreeding and assortative mating (mating of phenotypically similar individuals) can be treated by correlations between the chosen gametes. Random sampling from a small parental population leads to random gene-frequency drift, because the same allele may be drawn twice. In the draw some alleles are over-represented, others under-represented, or not at all. These models lead to the simple mathematical equations of which classical population genetics largely consists.
Departures from simple assumptions can be met by convenient artifices. The most widely used is Sewall Wright's effective population number, Ne. In an ideal population each allele in the pool has an equal probability of coming from any parent. Departures from this, caused by uneven sex ratio or unequal fertility so that alleles do not have an equal probability of coming from any parent, are adjusted in the equations by using Ne —the size of an ideal population that would produce the same random drift or consanguinity as the actual population. This can be calculated from demographic studies of sex ratios, survival probabilities and fertility distributions, in addition to population size.
Ernst Mayr compared this type of gene-pool modelling to sampling from a bag of coloured beans and dismissed it as "beanbag genetics". Others have also criticized such models, especially those with a holistic bent, as being simplistic and gene-centred.
J. B. S. Haldane counterattacked with his spirited and witty paper "A defence of beanbag genetics", giving numerous examples, often from his own work, of the value of beanbag theory. Gene-pool models are indeed simplified, but this is "meaningful simplicity" that sweeps away "disorderly complexity". Often it is such a simplified view that provides the most useful insights into evolutionary processes. No one doubts that gene interactions in development are complicated. Fortunately, these processes are increasingly amenable to molecular techniques. At the same time the simplified models have allowed striking advances in the fields of evolution and population genetics, especially dynamic aspects.
Of course one can use more realistic equations. There may be larger units, such as linked clusters of genes, which, if not permanent, are enduring. This is especially important in molecular studies, where the unit of interest is a haplotype, as well as a nucleotide. Variable selection and geographically structured populations do not lend themselves to simple gene pool methods and have been modelled more realistically in other ways, of course with greater complexity.
In classical population genetics — as developed by R. A. Fisher, Haldane, Wright and, more recently, Motoo Kimura — the specific model was often left unspecified. These days, thanks to the pioneering work of Gustave Malécot, models are stated much more precisely. The idea is to specify the model in enough detail for the equations, whether deterministic or stochastic, to follow unambiguously. A mathematical model is based on precisely stated assumptions as to population size, ploidy, extent of self- fertilization, rules of migration, time and nature of mutation, patterns of selection and mating combinations. Investigators using an equation then know that for at least one specified model the equation is correct, and they can judge how well this model mimics nature. In modern research, equations are very sophisticated, computers being available when the mathematics become prohibitive. As a means of bypassing distracting details to provide useful approximations and novel insights, gene-pool models are here to stay.
