Biological Review 18:32-64
(rec'd 26 June 1942)
Polygenic inheritance and natural selection
K. Mather

V. The Properties of Polygenic Combinations

(1) Internal and relational balance

The two features which determine the relations between free and potential variability are the aggregate recombination frequency and the arrangements of the polygenes in the chromosomes. Darlington (1939) has shown how completely genetic systems and their evolution are interpretable in terms of the control of recombination. We can now see why recombination is so important. It is the tap controlling the flow of polygenic variability.

We can see also how the theory of polygenic inheritance aids us in understanding the selective control of recombinations. Fisher (1930a) has shown that there is a general selective advantage in tight linkage between two genes, provided that their recombinations enjoy constant advantages relative to each other. But we have seen that polygenic inheritance is typified by inconstancy of such advantage. Consequently the situation here is very different. Just as the relative advantage of the allelomorphs of a given polygene vary with the other polygenes present, in that these others can enhance or counteract the effect of the gene in question, the advantage of a given combination of any two genes will depend on the constitution of the individual for the other polygenes affecting the character in question. Though in one set of individuals AB and ab may be more favoured than Ab or aB, the reverse may be true elsewhere and, as recombination is the only means short of mutation of changing the arrangement, this inconstancy of advantage must favour some degree of recombination.

Secondly, Sturtevant & Mather (1938), in listing the conditions necessary for the selective favouring of recombinations between two genes, found it necessary to stress that the genes must be maintained heterozygous in order that the recombination be effective. They were forced to complicate the conditions to this end. It is clear, however, that with systems of many polygenes some of them, though not necessarily the same ones, will always be heterozygous at different times. Consequently, as recombination must affect all gene arrangements, Sturtevant & Mather's special condition is not essential. Its necessity vanishes when we cease to think in terms of minimal numbers of genes.

Fig. 6. Internal and relational b alance of polygenic combinations. Capital letters indicate dominant and small letters recessive allelomorphs. The indices, + and –,, allow the direction of action of the allelomorphs, the sum effect of the allelomorphs at any onelocus being shown by the + or –, below. The optimum is assumed to be two +'s two –,'s, any departure from this being taken as indicating unbalance. Dominance permits the two balances to be varied independently of one another. Note that the case of poor relational balance is also one of heterosis.

Turning to the second agency affecting storage and release, viz. the genic arrangement, two types of polygenic balance must be recognized (Mather, 1941). (It should be made clear that we are referring to the balance within each chromosome. As we have seen above, different chromosomes usually recombine freely and hence each must achieve its own balance independently of the rest, the control of variability being achieved almost entirely by intro-chromosome adjustment.) In an inbreeding organism, such as wheat (Mather, 1940) or the grass mite Pediculopsis (Cooper, 1937), individuals homozygous for one or more chromosomes must occur freely. If such individuals are fit, their polygenic combinations must, when homozygous, have an effect near to the optimum (Fig. 6). The combinations must each be internally balanced, and natural selection will in fact favour the occurence of such a balance in inbreeding organisms.

If, however, we consider an inbreeding organism, such as maize, it is clear that heterozygosity of chromosomes will be the rule. The internal balance of a combination is then of little importance for survival, being overshadowed by the relational balance existing between pairs of different homologous combinations. In such cases natural selection will tend to build up relationally balanced combinations. We can see the consequence of this in maize, which when inbred gives homozygous types poor and feeble in comparison with the highly heterozygous individuals of an open-pollinated variety (Fig. 6). The internal balance displayed in the homozygotes is poor, but the relational balance shown in heterozygotes is good. We may note that the existence of poor internal combined with good relational balance implies dominance of polygenes, as otherwise the heterozygote would always be intermediate between the homozygotes, and the types of balance would not be capable of separate adjustment.

Adaptive relational balance supplies the key to a number of problems of equilibrium. It shows us, for example, how polymorphism can be maintained (Mather, 1941). If the internal balance is poor, homozygotes are less fit than heterozygotes; and when each combination is completely linked, as by means of an inversion, with a major oligogenic mutant or marker gene the polymorphic system, found for example in grouse locusts, (Fisher, 1930b, 1939) is complete. The consistent advantage of all the many heterozygotes over all the homozygotes, as observed in these cases, is not easy to understand except in terms of polygenic combinations.

The relative advantage of any combination depends on those which accompany it, and so we might expect the relative frequency of occurence of any one to vary with external conditions. Any major mutant inseparably linked with a given combination would vary correspondingly in frequency and give a 'cline' (Huxley, 1939). Such a cline would be stable unless conditions are changing rapidly in any place. Polygenic theory allows us to see how equilibria are maintained, because it shows us how optimum phenotypes depend on harmonious combinations of polygenes rather than on the properties of the individual units of genotypic variation. The statistical, and hence relatively constant, properties of the aggregate hide the vagaries of the individual gene.

(2) Heterois and the origin of isolation mechanisms

Heterosis is the name usually applied to the phenomenon of increase in vigour sometimes observed when strains of an organism are intercrossed (Darwin, 1876). This extra vigour is more apparent in cross-breeding organisms if the parents are themselves highly inbred and hence poor types. Heterosis should, however, be measured in such cases by the excess of the hybrids over the average of the open breeding varieties from which the parental homozygotes had been derived. In this way the distracting effect of inbreeding depression is avoided.

In crop plants, especially maize, heterosis is advantageous from the cultivator's point of view, for it allows him to obtain heavier or earlier crops. In nature, however, the situation must be very different.

Fig. 7. Heterosis. In a wild organism the phenotype varies round the optimum, O, without any dominance bias. Heterosis in natural populations is shown by the phenotypic distribution of the progeny (dotted line) having a greater spread than that of the parents (solid line). Artificial selection may be regarded as moving the otimum outside the range of the phenotypes. The distribution then shows dominance bias and, in consequence heterosis causes the progeny distribution to approach the optimum more nearly than that of the parents (see text).

Intercrossing two strains results in bringing together unlike polygenic combinations in the hybrid. If the histories of the two strains are separate, the two combinations received by the hybrid will not previously have been together and hence will not have been selected for good relational balance. The phenotype of the hybrid will be likely to show a greater departure from the optimum than does either parental strain, because combinations within the same strain will have been selected for relational balance. The departure of the hybrids from the optimum, or optima, to which the parental strains are adapted, will, if in the direction of increased size, be hybrid vigour or heterosis. We can thus recognize that in nature heterosis is a sign of poor adaptation and must be selectively disadvantageous. Hence, inasmuch as all departures from the optimum will be disadvantageous, whatever their direction (§IV (1)), the concept of heterosis may properly be extended to include all examples of poor relational balance between combinations of different wild interbreeding groups (Fig. 7).

Now if two strains or populations breed less freely with each other than either does within itself, selection will have less opportunity of maintaining the relational balance between combinations of different strains than between those of like strain. The genotype being fluid, even though the phenotype is constant (§IV (4)), relational balance will deteriorate unless constantly maintained by selection. Hence heterosis in our new sense is an automatic consequence of any diminution in the freedom of interbreeding between strains. Now as heterotic individuals will be less fit than the parental types, their production represents wastage of reproductive effort by the parents, and any variants tending to reduce outbreeding between the strains will be favoured. In this way isolating mechanisms (Dobzhansky, 1941) and hybrid sterility originate. It is to be presumed that both isolation mechanisms and hybrid sterility are polygenic characters capable of developing from the species, store of variability. They will then be inevitable results of heterosis. Other special conditions may have the same result (Sturtevant, 1938), but the inevitability of this polygenic effect makes it seem likely that the avoidance of heterosis is the most widespread stimulant of isolating devices.

Such a system must be self-propagating, because the less the intercrossing which goes on between strains, the greater is the chance of unbalance and heterosis, and the stronger is the advantage of isolation. Thus a small decrease in mating freedom, such as may be brought about by natural obstacles, slightly different conditions between adjacent localities, or a change in the breeding system (§VI (3)) will suffice to start the chain of events, which once in progress cannot be reversed. Once a complete isolation mechanism is achieved and crossing prevented, the two strains, now species, can invade each other's territory and all evidence of how they came to be separated will be lost. This process now appears to be nearing completion in Drosophila pseudo-obscura.

Heterosis, like the simultaneous occurence of good relational and poor internal balance, implies dominance of polygenes (cf. Jones, 1917). In III (2) we have given evidence that in natural populations dominance is not directional. The + allelomorph is as likely, but no more likely, to be dominant as recessive to the – allelomorph. This lack of direction is a consequence of the variation of the phenotype round its optimum. Hence if the optimum were maintained outside the phenotypic range, dominance of that allelomorph making for a phenotype nearer to the optimum would become more frequent than the reverse. So polygenic characters, such as yield in crop plants, where the breeder always selects in one direction, should show dominance preponderantly in the direction of selection, with the result that disturbed relational balance should most often cause hybrids to depart from the strain means in the direction of previous selection. Dominance bias and heterosis in crops should be directional (Fig. 7).

* Studies in plant breeding technique. IV. The inheritance of agricultural characters in three inter-strain crosses in cotton. Indian J. Agric. Sci. 8: 757-775

Hutchinson, Panse & Govande (1939)* have found this to be the case in cotton, wherever the character was one of commercial importance, and hence presumably the subject of artificial selection; but with unselected characters of no commercial interest, no such directional departure was observed. In their sense of the word, no heterosis was found.

Evidence of the undirectional nature of heterosis, in the present sense, in organisms not artificially selected, is provided by Sveshnikova's (1935) observations on Vicia cracca where interstrain hybrids were often poor, by East (1935) in species hybrids of Nicotiana and Fragaria, and by unpublished observations which my colleague Mr W. J. C. Lawrence has allowed me to quote. He finds that species hybrids of Streptocarpus are of very variable vigour but do not regularly depart in one direction from their parents. Crosses of these species with the artificially selected garden forms, however, give hybrids which more often than not show increased vigour.

(3) Mutation and the maintenance of variability

The heritable variability of a population is constantly being reduced in two ways (Mather & Wigan, 1942). One of these is by the random elimination of allelomorphs. Such elimination is a consequence of the fact that more gametes are produced by the zygotes of one generation than will be represented in the zygotes of the next generation. Sampling leads to random changes in the allelomorph frequences and will eliminate allelomorphs with a frequency inversely proportional to population size (Fisher, 1930a; Wright, 1940).

The second way is by selection. Selection can act as a stabilizing agent of the allelomorph frequencies, as in cases of good relational balance combined with poor internal balance. But if selection produces a permanent adaptive or evolutionary change in the organism, it must do so by changing the genotype, i.e. by an allelomorph completely replacing its competitors at one or more loci. Heritable variability will thereby be reduced. Selection will destroy the very variability by virtue of which it was effective.

Clearly there must be an agent counterbalancing both random and selective extinction, and it is to be found in mutation. Polygenes mutate just as do oligogenes, but the rate of accumulation of polygenic variation is slow. Johannsen (1909) with beans, Lindstrom (1941) with tomatoes, and others, found little evidence of polygenic mutation, but Mather & Wigan (1942) were able to observe the results of mutation during fifty-three generations of selection in Drosophila melanogaster. The new heritable variability was not immediately available to selection, as its small effect was masked and protected by non-heritable fluctuation. In time, however, recombination brought together mutant genes to give variants large enough for selection to act on in spite of fluctuations. Selective advances then occurred.

Fig. 8. The states of polygenic variability. New variability arises by mutation, most of it going into store, though a small amount is immediately free. Most free variability is derived from the potential store by segregation, and most of it returns to store by crossing. Some free variability is however fixed by response to selection, and some, like part of the potential variability, is lost by random fluctuation of the allelomorph frequencies. The potential variability is in a continual state of change from heterozygotic to homozygotic and vice versa, and also from one heterozygotic state to another. The relative magnitudes of the various normal changes are indicated by thickness of the solid arrows, the actual magnitudes being governed by the recombination frequencies and the linkage organization. The dotted arrows mark changes due to random fluctuations dependent for magnitude on the effective breeding size of the population.

The rate of origin of new variability was so slow that after twenty generations the effect of selection was still small as compared with that obtained in ten generations' selection of hybrid material (Mather, 1941; Wigan, 1942). Thus, unless highly inbred, an organism's response to natural selection must depend almost entirely on the utilization of stored variability. Mutation mainly serves to replenish the store and so compensate for the small regular loss by random and selective extinction (Fig. 8). A little of the new variability will be, of course, free from its time of origin, for, mutation being an undirected process, some must arise in combinations which thereby become slightly unbalanced. Most, however, passes into store.

This steady flow of new variability into store, and the equally steady release from and return to store of free variability, is possible only with polygenic characters. The new variability may be due to mutation of polygenes other than those which are being fixed by selection, but, since all like polygenes have similar effects, replacement of variability can be effected by genes other than those which are lost. It is impossible to say which polygenes will exist in the population as two or more allelomorphs, and which will not be varying at any given moment; but it can be said that a characteristic proportion will be in each state. It is on the fact that such a proportion exists in the heterogeneous state that variability depends.

The genotype is constantly changing as new mutants arise and old ones are fixed or lost. Variation is passing into store and is also being released (Fig. 8). Even inside the polygenic combinations there is a constant reshuffling, as new ones are formed and old ones destroyed by recombination. Selection is causing some to increase and others to decrease in frequency (Mather, 1942b). Yet, in spite of all this, the phenotypic distribution can be nearly constant, unless the environment is changing rapidly. Just as polygenetics resolves the contradiction between discontinuous genotypic and apparently continuous phenotypic variation, it resolves the contradiction between genotypic flux and constancy, or near constancy of the phenotypic distribution.

The importance of recombination is that its frequency regulates the rate of all these transformations of variability (Fig. 8). Low recombination gives stability and high recombination gives flux. In Drosophila, where recombination is low (Darlington, 1934b), stability is upset only by rare, and hence very obvious, changes (Mather, 1941). The flux is small as might perhaps be expected in an organism whose successive generations encounter seasonally variable environmental conditions. In mice recombination is much higher (Crew & Koller, 1932). Here the release of variability, and hence the response to artificial selection, is easier and steadier (Goodale, 1938). Maize is intermediate in recombination frequency (Darlington, 1934a) and response to selection (Winter, 1929).

(4) Correlated response

So far selective response has been discussed, for simplicity, in terms of a single polygenic character, to whose departure from the optimum phenotype loss of fitness was equated. In truth, however, fitness must be dependent on, and compounded of, many such polygenic characters, and response to natural selection must also be compound. The response in one character will depend on the correlated selection of other characters (Anderson, 1939).

Correlated response may happen in two ways. First of all, each polygene may be pleiotropic and simultaneously affect several characters. In such cases the correlation cannot be broken, and the two characters may be treated as one.

The second kind of correlation is mechanical. Polygenic combinations affecting two characters may occur in the same chromosome, and their constituent polygenes will then be intermingled. Recombination which changes the balance of one will usually, though not of necessity always, unbalance the other. The two characters will generally show simultaneous release of variability. So even though the phenotypic variants for one character, made possible by recombination, might enjoy a selective advantage in appropriate conditions, the correlated unbalance of the second combination, intermingled with the first, must often tend to nullify this advantage, as conditions are not likely to favour both variants simultaneously. The result will be a slowing down of changes in the hereditary constitution of the population undergoing selection; but the changes will generally be possible in the end as, sooner or later, recombination will, step by step, re-order the intermingled combinations until something approaching the optimum is obtained. Response of the first character then becomes possible without a deleterious effect on the second.

In special circumstances, however, events may follow a different course. The rise in selective advantage of changed manifestation in the first character may be due to conditions which reduce the disadvantage of variation in the second one. The balance of advantage will then be towards change in both, and the second character will degenerate by virtue of the mechanical relations of its polygenes with those of the first. This has perhaps occurred for example in cave animals as Professor R. A. Fisher has pointed out to me. The advantage of good sight ceases on migration into the cave, while new adaptation of, for example, touch is favoured. The mechanical relationship of the two sets of controlling polygenes in the chromosome would be sufficient to make degeneration of the eyes an inevitable accompaniment of the rise of these new adaptations.