Biological Reviews 18:32-64 (Jan 1943)
Polygenic inheritance and natural selection
K. Mather

I. NATURAL SELECTION

Evolution is the occurrence of persistent changes in the hereditary constitution of a population of organisms. Two conditions must be fulfilled if such changes are to occur by natural selection.

The first is that the population shall be, or shall be capable of becoming, genetically heterogeneous; for if, except for non-heritable variation, all the individuals are alike, and are incapable of changing, their progeny must also be alike and evolutionary changes become impossible.

The second condition is that genetically unlike types leave different average numbers of mature progeny in the next generation. Otherwise, natural selection would be inoperative, and evolutionary changes would depend, of necessity, on new variation being directional and perponderantly adaptive. Genetical evidence (see Muller, 1939*) indicates that this is not the case, mutation being a process which, as Darlington (1937) has said, is chemically determinate but biologically at random. With new variation not always directly adaptive, selective breeding is the necessary condition for sorting out advantageous and disadvantageous variants.

Now if these two conditions are fulfilled natural selection must occur (Fisher, 1930a), but this does not imply that evolution proceeds by this process. Unless the variation subjected to selection is of an appropriate kind, and unless the selective breeding is itself suitable, the effects of natural selection may be trivial and incapable of supplying a mechanism of evolutionary change. Thus, for example, selection is known to favour heterozygotes at the expense of homozygotes in some cases (Fisher, 1939); but this can only lead to the maintenance of a special kind of polymorphism. Evolutionary change cannot be a consequence of this particular kind of selective breeding. Natural selection is a phenomenon separable from evolution and capable of being studied independently. It is on such study, attempted by Fisher (1930a), that the demonstration of evolution by natural selection depends.

Any investigation of the mechanism of natural selection or of evolution must ultimately be genetical, for it must involve the separation of hereditary from non-hereditary variation and the analysis of the former's changes in succeeding generations under the action of changing conditions. Genetics has already contributed materially to our understanding of these processes, notably in removing one of Darwin's greatest difficulties by the demonstration that inheritance is particulate and hence that the decay of heritable variability in a population is slow (Fisher, 1932).

During the last ten years more attention than ever before has been devoted to the genetical study of populations, and the consequent contribution to evolutionary theory has been great. The validity of the two fundamental theorems of natural selection has been fully verified. Evidence of genetical heterogeneity in populations will be reviewed later (§II (2)) and that of the differing selective advantage of various genotypes has been given by Fisher (1939) and Ford (1940b). It has also been possible to go further and show that, so far, nothing has been discovered about evolutionary change which is in conflict with, or demands an extension of, the known genetical principles of variation and natural selection (Timofeeff-Ressovsky, 1940; Muller, 1940; Darlington, 1940). The application of genetics to the detailed analysis of evolutionary change has commenced to bear fruit. In order to see how fully the genetical principles of variation and natural selection can account for evolutionary changes, we must examine the nature of specific differences in relation to intra-specific variation, the way in which variability is maintained in populations and how responses to selection occur. Many of the questions which arise will be capable of more critical analysis as new facts become available in the future; but we can recognize now that the success of such analyses must depend on their being treated as special aspects of the general problem raised by natural selection.

II. NATURAL VARIATION

(1) Specific differences

Specific differences cannot commonly be subjected to genetical analysis, because individuals of different species will seldom hybridize, and, even when obtained, species hybrids are very often infertile. Thus information on the genetical nature of specific differences is somewhat scarce. Some evidence has, however, been obtained, and it is clear that several types of difference are involved. The problem now is that of deciding which of these are characteristic of the distinction between species.

Polyploidy will be omitted from the discussion. Though in many plant genera this phenomenon has played a great evolutionary part (Darlington, 1937; Dobzhansky, 1941), polyploidy has been of very minor importance in animals, and in any case, results more in combining the features of two existing species, than in initiating entirely new forms. Structural changes, such as inversions, interchanges and duplications, will also be omitted, except for a few special cases mentioned later. Selection acts directly on the genic constitution of the organism, structural rearrangements being affected only in so far as they control or determine the behaviour of individual genes. Our main concern must clearly be that of finding out how new genotypic systems are brought into being. This will be approached using only the concepts of formal genetics, to which cytological theory is ancillary for our purpose.

Before proceeding to a consideration of existing data it is necessary to decide on the criterion by which the importance to specific distinction of various types of genetical differences are to be found in embryo in racial and varietal differences, genetical variation of the kind found between species must also be of common occurrence within species; but it is clear that this variation within species, though of the same kind, must be of smaller magnitude or less extensive than the variation between species. Otherwise species differences would be genetically no more significant than intra-specific variation and the problem would not exist. The application of this criterion will be clearer when the evidence has been discussed.

First of all, differences distinguishing species may be oligogenic, i.e. they may be controlled by a small number of genes having effects large when compared with non-neritable fluctuation, and hence leading to sharp segregation. (The familiar genetical variants of the laboratory are of this kind.) Examples of oligogenic variation between species are not uncommon. Green (1935), for example, has shown that Mus bactrianus has the character white-bellied agouti. Petunia integrifolia has red flowers, differing from the white flowers of P. axillaris by two genes (Mather, unpublished). Similar flower-colour differences are also involved in Streptocarpus species (Lawrence, Scott-Moncrieff & Sturgess, 1939) and Nemesia species (Mather, unpublished). Other characters may show the same behaviour, as, for example, habit in crosses between Phaseolus vulgaris and Ph. multiflorus (Lamprecht, 1941). All these cases and others like them were found as a result of species hybridization, but a case in Drosophila is of interest as having been detected in a different way. Gottschewski & Tan (1938) have rendered likely the existence of a monogenic difference in eye colour between D. melanogaster and D. pseudo-obscura by transplantation experiments, the two species being incapable of crossing.

Oligogenic differences do not, however, satisfy our criterion, for species are known which need not differ in characters of this kind. Furthermore, differences of exactly the same kind as those distinguishing two species may be found within either or both of them. Indeed, differences of this kind, whose study has formed the main occupation of geneticists, are more common within than between species. There is no relation between closeness of relationship and number of oligogenic differences. Particular oligogenic variation seems, rather, to be an outcome of the possession of particular hereditary material in the chromosomes, for similar variants are often found in related species, genera or even higher groups (see Haldane, 1927a; and Vavilov, 1922), where it has given rise to Vavilov's 'Law of Homologous Variation'.

A second type of heritable difference shown by species is dependent on the joint action of many genes, wach having an effect small in relation to the total non-heritable fluctuation of the character in question. Such differences are termed polygenic, and polygenic characters do not show sharp segregation. They may exhibit any degree of expression between wide limits and hence have often been called quantitative characters. Size and shape are typically polygenic in most cases. Polygenes are inherited in exactly the same way as other genes, in that they are situated in the chromosomes (Warren, 1924; Mather, 1942b). They differ only in the type and magnitude of phenotypic effect produced.

The absence of simple sharp segregation for polygenic characters has made their exact genetical study difficult, but cases of polygenic specific differences are well known. Perhaps the most familiar is that described by Baur (1932) in Antirrhinum, but almost every account of species hybridization refers to complex segregations for some characters. Evidence has been given in detail by Harland (1936*) for Gossypium, Anderson (1939) and Smith (1937) for Nicotiana, Honing (1923, 1928) for Canna, Green (1935) for Mus, Iljin (1941) for Canis and others. Although data are absent in many cases, it seems to be generally agreed by all who have experience of species hybrids that they are characterized by polygenic segregation of size and shape characters (see especially East, 1935; Timofeeff-Ressovsky, 1940; and Muller, 1940).

Polygenic variations are known within species, as we shall see in the next section, the differences being smaller than those between species. It thus appears that polygenic differences fulfil the criterion and may be regarded as essential to specific distinction.

A third type of specific variation is described by Harland (1936) in Gossypium. Related species have mutant forms which can be shown to be dependent on mutation of homologous genes. Individuals of the two species ofen differ slightly when heterozygous for the mutant, so showing that their wild type, or commonly occurring, allelomorphs are not quite the same. It is of course possible that the differences are due to other genes very closely linked to the one in question, but, in any case, the differences between homozygotes are too small to add appreciably to the specific distinction, except, perhaps, as part of a polygenic combination.

A final type of interspecific difference must be mentioned. In some plant genera, notably Epilobium (Michaelis, 1937), Streptocarpus (Oehlkers, 1938) and Oenothera (Renner, 1936), species differ cytoplasmically as well as genetically. Cytoplasmic variation is not, however, a regular and characteristic feature of species distinction. It is apparently absent in most cases.

The genera Oenothera and Triticum (and possibly some others) require speciel mention, for they seem to constitute exceptions to the rule that specific differences are essentially polygenic. In each genus particular species appear to differ only by a single major heritable factor (Watkins, 1930; Ellerton, 1939; Renner, 1925). Special reasons for this behaviour are, however, known in Oenothera. Many of the species, like Oe. Lamarckiana, are permanent hybrids, whose chromosomes behave in such a way that recombination in the differential segments is suppressed. In consequence, apparently unifactorial segregation is inevitable, no matter how many genes are really involved (Darlington, 1931, 1939). The evidence of aberrant cytological behaviour in Triticum is less strong, but it seems likely that here too the differentiating genes are inherited as a complex. The anomalous nature of the species differences in these genera is probably more apparent than real, the general rule still holding good.

(2) Variation within species

All the types of heritable variation by which species may differ are also found within species. Oligogenic variation of two types is to be observed. The first kind, whose existence has long been recognized, gives polymorphism. This is widespread among butterflies (Ford, 1940b) and is established in some of the Coleoptera (Timofeeff-Ressovsky, 1940; Dobzhansky, 1933), in Helix (Diver, 1940), Lebistes (Winge, 1927), the grouse locusts, Paratettix, Apotettix and Acrydium (Nabours, 1929; Nabours, Larson & Hartwig, 1933), the grasshopper, Chorthippus parallelus (Sansome & La Cour, 1935), Lotus species (Dawson, 1941) and elsewhere (Timofeeff-Ressovsky, 1940; Ford, 1940b). Many organisms, both plant and animal, most probably fall into this class, though conclusive genetical evidence is lacking. In some cases the same polymorphic variant that is found within a species also serves to distinguish two related species. It is not, however, clear whether the polymorphic gene is an essential part of the species difference in such cases (Timofeeff-Ressovsky, 1940). It may be noted that certain examples of Huxley's (1939) 'clines' and and the wold inversions in Drosophila pseudo-obscura (Dobzhansky & Sturtevant, 1938) are special examples of polymorphic variation. Polymorphism depends on a curiously balances set of selective advantages (Fisher, 1927; Ford, 1940b), and there is reason to believe that this balance, and hence polymorphism, may depend for its maintenance on polygenic differences (see §V (1)).

The existence of the second type of oligogenic variation has been established more recently. Mutant genes of the type found and used in laboratory studies have been detected in wild material of several species of Drosophila (reviewed by Dobzhansky, 1939*), Dermestes (Philip, 1938), Gammarus (Spooner, 1932), rats (Lloyd, 1912), Peromyscus (Sumner, 1932), Trifolium (Williams, 1935) and grasses (Jenkin, 1930), in fact wherever a serious search has been made. The variants, which include lethals, sub-lethals and various visible colour and structural aberrants, are rarely to be observed homozygous in wild individuals, though some examples have been seen. A very large proportion of wild organisms have, however, been found by adequate tests, to be heterozygous for the mutants. Many of these genes reduce the viability of the carrier, and in any case, as we have seen, there is no evidence of specific difference being dependent on this type of variatin. Hence we cannot regard this type of oligogenic variation as having any adaptive or evolutionary significance in the general case.

Naturally occurring polygenic variation within species has been less analysed than the oligogenic type. It is, however, clear that variation of this kind is widespread. The biometrical studies made in man by Galton, Pearson and their associates have disclosed a wealth of variation suggestive of the polygenic interpretation (cf. Fisher, 1918), and nearly all important characters such as yield, quality and disease resistance in domestic animals and plants are of this type (Emerson & East, 1933; Hammond, 1940).

In Peromyscus the characters distinguishing races appear to be polygenic (Summer, 1932). Homologous oligogenic variants were, on the other hand, found in many races, again showing how different in evolutionary importance the two types of variation are. The races of Drosophila pseudo-obscura also differ polygenically, even in respect of their interfertility (Dobzhansky, 1936). Polygenic morphological differences observed between the races of this species are greater than those found between strains of the same race (Mather & Dobzhansky, 1939) but less than those between this and related species such as D. miranda. Local groups of D. melanogaster are distinguished by their numbers of sternopleural chaetae (Dubinin et al. 1934), which Wigan (1941) has shown to be subject to polygenic control, a gread deal of variation being present in wild flies.

In many cases, though the genetical data are inconclusive, wild individuals or their offspring have been found to differ in those characters, such as size, shape and growth habit, which are commonly subject to polygenic variation. There is little doubt that intra- and interspecific variation is polygenic, though the degree of variation is not so great.

A case of allelic variation, such as Harland described between different species of cotton, has been found within D. melanogaster. Two strains, one American and the other Russian, had, at the white locus, wild-type allelomorphs distinguishable from one another by their mutation and dominance characteristics (Timofeeff-Ressovsky, 1932; Muller, 1935). This is, however, a unique example outside cotton.

Cytoplasmically inherited variants are known within a number of species (Sirks, 1938), but they are not so common as the more familiar nuclear kind. Racial differences may be cytoplasmic as in the case of Vicia faba major and minor (Sirks, 1931), but this is not characteristic of racial differentiation (cf. Drosophila pseudo-obscura), and, as in the case of species differences, cytoplasmic distinctions between races seem to be exceptional rather than regular.

Finally, there exists within species a kind of variation which cannot develop directly into interspecific difference, viz. polymorphism affecting the breeding system, or system of mating, within the species. Related species may differ in their breeding systems, but the variation on which the control of breeding rests is always, of necessity, intraspecific. The most familiar kind, indeed the most familiar kind of all genetic variation, is sex separation, but others, e.g. incompatibility of the Nicotiana type (East, 1929) and heterostyly (see Darwin, 1877), are quite common. In the cases of sex separation and heterostyly the differences between male and female or between pin and thrum, are morphologically obvious, though certain other mechanisms also play a part as we shall see later (§§VI (1) and (2)). Incompatibility, or, as it is often called, self-sterility, depends on a physiological reaction between pollen and style. It involves no morphological differentiation and so was not understood until relatively recently in Nicotiana. Since that time it has been found to be of widespread occurrence in plants.

Other mechanisms controlling the rates of outcrossing and inbreeding also exist (see Kerner & Oliver, 1894-5) and probably involve some genetical variation, though little accurate knowledge of their genetics is available. Lewis (1941) has discussed the genetical possibilities of gynodiocy and concluded that it must depend on cytoplasmic differentiation if it is successfully to function as a means of controlling the breeding system.

This type of polymorphism is of special interest because both its adaptive value and genetical stabilization are more obvious than those of the other types of polymorphism mentioned earlier. All the various systems play the part of controlling outbreeding or inbreeding, and hence of affecting the average heterozygosity of the organism. The precise system operating in any species is determined by the existing morphological and physiological peculiarities (Mather, 1940). The question of how breeding mechanisms affect heritable variability, of how they are adapted to fresh circumstances and of the effects of these adaptive changes, will be discussed after the properties of polygenic inheritance have been considered.

III. POLYGENIC INHERITANCE

(1) Phenotype and genotype

Evolutionary and adaptive changes are dependent on polygenic characters, ans so our attention must be directed towards polygenic variation and its behaviour under the action of natural selection. Polygenic characters are controlled by many genes having effects small in comparison with non-heritable fluctuations. In consequence, polygenic inheritance is marked by certain peculiar features which distinguish it from oligogenic behaviour and which throw a fresh light on the interrelations of variation and selection. Polygenetics represents a new level of integration by means of which a better understanding of natural selection and its action may be achieved.

The quantitative difference in number of operative genes between polygenic and oligogenic variation gives a qualitative difference in behaviour. Laboratory genetics has been almost solely concerned with oligogenic variation, and so has proved disappointing to the evolutionist. Both types of gene are, however, inherited in the same way and so the success of polygenic analysis depends on the utilization of the principles elucidated in laboratory studies.

Before proceeding to a discussion of polygenic inheritance, one point must be made clear. Any given character may be subject to both polygenic and oligogenic variation. Thus a Drosophila melanogaster may be wild type and have some 18 or 20 chaetae on the ventral surface of each abdominal segment, but it may, on the other hand, show the effects of the mutant gene 'scute', in which case the number of chaetae is very much smaller. The flies of each kind are sharply distinct, for, though the chaeta number is variable, the two classes, wild type and scute, do not overlap. This is characteristic of oligogenic variation. But the precise number of chaetae on a wild-type fly is subject to the control of many genes each of small effect, as well as being influenced by environmental conditions. The continuous phenotypic variation produced in this way is characteristically polygenic. The number of polygenes is not known, but small-scale experiments have shown them to lie in all the major chromosomes (Mather, 1942b). Wild flies commonly show polygenic variation of this character, but oligogenic variation is very rare.

When a single gene is involved, zygotes must fall into three genotypic classes, the two homozygotes and the heterozygote. With incomplete dominance this means that the maximum number of three phenotypes will be found. With dominance the number of phenotypes is reduced to two, one of which will include both the homozygote AA and the heterozygote Aa. In such cases one phenotype, usually that associated with the dominant allelomorph, will be selectively advantageous when compared with the other. The disadvantageous allelomorph must tend to become less and less frequent until it is only maintained by mutation as a rarity in the population. No finer adjustment is possible.

Fig. 1. Genotype and phenotype. The phenotypic frequency distribution of a character controlled by two genes of equal and independent effect, without dominance. The phenotypic expression is proportional to the number of capital letters in the genotype (see text).
Where, however, a number of polygenes is involved, the situation is very different, for many phenotypes are possible, a large proportion of which will be produced by a number of different genotypes. With only three polygenes of equal effect, the genotypes AABBcc, AAbbCC and aaBBCC will, for example, give the same phenotype. This phenotype would also characterize the genotypes AaBBcc, AABbcc, AaBbcc, etc., if dominance were the rule, or AABbCc, AaBBCc, and AaBbCC in the absence of dominance. With more genes the possibilities are increased.

Two important consequences then follow. First, neither allelomorph of a polygene will have an unconditional advantage over the other, for each may form part of a distinct genotype giving the same phenotype. Thus, in the example considered above, AAbbCC and aaBBCC gave the same phenotype as each other. So would AAbbcc and aaBBcc. No matter which of these two phenotypes is more advantageous, A will sometimes be favoured at the expense of a, while the reverse will hold in other cases. The same is true of B, b. The selective properties of one polygene will be conditioned by the other polygenes which are heterogeneous, i.e. exist as at least two allelomorphs, in the population in question. The second consequence is that a very fine adjustment of phenotype to environment becomes possible, for the chance of finding a phenotype closely adapted to the prevailing conditions increases with the number of phenotypes which can occur. When circumstances change, a different phenotype will show maximum adaptation; but whatever the conditions, close adjustment is possible. Polygenic variation gives great adaptability, the nearly continuous variation permitting more regular and more accurate adaptation.

The different phenotypes will not be equally frequent in a population for two reasons. First, the numbers of genotypes which give rise to particular phenotypes are not constant, and secondly, the various genotypes have different frequencies of occurrence. This can be well seen from the simple example of a character controlled by two incompletely dominant independent genes, A, a and B, b, the two allelomorphs at each locus being equally frequent (Fig. 1). The allelomorphs designated by small letters are assumed to add nothing to the expression of the character, while each allelomorph designated by a capital letter adds 1 unit. Fig. 1 shows the gametes and zygotes which will be produced by only one genotype each, viz. aabb and AABB. The central and most frequent phenotype is given by three genotypes, AaBb, AAbb and aaBB, the first of which is itself the commonest genotype. The remaining two phenotypes are each produced by two genotypes of intermediate frequency, Aabb and aaBb in one case and AABb and AaBB in the other.

This type of phenotypic frequency distribution is characteristic of polygenic inheritance. As the number of genes involved increases, more phenotypes are possible, and the distribution becomes more nearly continuous. With many genes, and with non-heritable influences playing a part, the distribution must often approximate to the normal as observer, for example, in human stature.

(2) Dominance and interaction of polygenes

The exact shape of the phenotypic frequency distribution will be conditioned by the properties of the individual polygenes, their dominance relations and interactions with each other. If, for example, in the simple digenic case discussed above A and B were completely dominant over their allelomorphs a and b, or if the B, b locus could only have an effect on the phenotypes of aa individuals, i.e. A, a was epistatic to B, b, the frequency distribution of phenotypes, as shown in Fig. 1, would no longer be symmetrical. It would be skew, with the mode to the left and long tail to the right of the figure. Skew distributions of this and the opposite kind, with its long tail to the left, are quite common, an example being afforded by plant height in barley (Fisher, Immer & Tedin, 1932).

Little, however, is known of those properties of polygenes which are responsible for skewness. Fisher (1930a) has advanced reasons for expecting polygenes often to show some dominance. Clearly dominance in polygenes, unlike dominance of the familiar laboratory oligogenes, cannot depend on an intrinsic advantage of one allelomorph over the other, for, as we have seen, neither allelomorph has such a regular advantage. Each allelomorph will have an advantage over the other in particular cases. Thus, if dominance is related to selective advantage, the -allelomorph, decreasing the degree of phenotypic manifestation of a character, is just as likely to be dominant as it is to be recessive to the +allelomorph which increases the manifestation. Fisher's argument leads to the conclusion that dominance of one allelomorph or the other will be determined solely by their relative frequencies in the population. We may then expect equal numbers of dominants in either direction. If, however, selection were to be so adjusted as steadily and regularly to favour a high or a low degree of expression, i.e. the + or -allelomorphs, the balance would be upset and dominance equality would no longer be encountered. In such cases the long tail of the phenotypic frequency distribution, which contains the recessive less frequent phenotypes, will point away from the direction of selection, and may be used to detect the direction of action of a selective force (Fisher et al. 1932).

These expectations are well supported by such data as are available at present. In man it has been shown that the frequency distribution for stature is symmetrical; yet, by the use of correlation methods, which will detect dominance no matter what its direction may be, Fisher (1918) has found evidence of dominance of the polygenes controlling this character. Symmetry must then depend on an equal number of polygenes being dominant in either direction. This approach has been developed in more detail by Fisher et al. (1932) who introduced the use of third degree statistics. Taking data from Emerson & East (1913) these authors showed that the polygenes controlling plant height in maize, a character which shows heterosis, displayed dominance.

They also analysed data from a selection experiment with mice (Fortuyn, 1931). In unselected material the phenotypic frequency distribution was symmetrical, but selection in either direction introduced skewness with the long tail of the distribution towards the pre-selection mean, exactly as Fisher's view would lead us to expect.

Interaction of polygenes has been even less extensively analysed than their dominance. Two kinds of interaction, having very different genetical significance, can, however, be recognized. One gives rise to what Fisher terms metrical bias, the other being interaction of the epistasis type.


Fig. 2. The phenotypic distribution and optimum. The mean of the phenotypic distribution is M1. When the optimum is at O1, departing from the mean by h1, the next generation has a mean lying between M1 and O1 at M2. If, however, the environment is showing fluctuating change, the new optimum may be on the other side of M1, at O2. Then h2, the departure of M2 from O2 will be greater than h1. The selective response of the first generation has lowered fitness in the second. (The departures h1 and h2 are magnified for the sake of clarity.)
The first kind, leading to metrical bias, can perhaps best be understood by a consideration of Fig. 2, which shows a symmetrical frequency distribution. Now suppose that the scale of the abscissa was made logarithmic. What were originally equal intervals in Fig. 2 now become unequal and, in particular, the farther to the right we proceed along the abscissa, the greater is the shortening of the intervals. The effect of this will be to destroy the symmetry of the curve and render it skew. A similar result could be obtained by other transformations of the abscissa scale, as, for example, the square roots or cube roots of the abscissa values in Fig. 2. Skewness of the opposite kind would be obtained by such scalar transformations as using antilogs or squares. Now if the scale we find most convenient for measuring the character of the organism happens to bear the same relation to the unknown 'scale', on which the organism's physiological work, that, say, logarithms do to antilogs, or square roots to their generating numbers, skewness will be observed when the frequency distribution of the measurements is plotted. These kinds of skewness are described as metrical bias because they depend on the type of scale, or metric, used, and can be removed entirely by appropriate transformation.

No general theoretical significance can be attributed to metrical bias because, in the absence of external evidence, we have no right to assume that the scale we find convenient shoud bear any special relation to the processes on which depends the degree of expression of the phenotype in question. An understanding of metrical bias is, however, important for the analysis of polygenic behaviour, because this bias can give rise to unexpected curves for inbreeding depression and other phenomena (Rasmusson, 1933).

Epistasis, and the related types of interaction shown, for example, by complementary genes, are theoretically quite distinct from metrical bias. The latter depends on the scale of the measurement and will give a constant skewness, whereas the former results from the interdependence of genes in development. Thus complementary genes can produce no effect unless all are present. With two such genes (A, a and B, b) aabb, Abb and aaB are all alike, the distinct type being AB. Similarly when one gene (A, a) is epistatic to another B, b, the latter will have no effect unless a given allelomorph of the former is present, e.g. AB and Abb will be alike though aaB and aabb differ. It is clear that no scalar transformation can regularly remove skewness of the phenotypic frequency distribution arising from this type of interaction, for such skewness will not be constant in magnitude. Nor is there any reason to expect this type of skewness to be characteristically in one direction. Rather, like dominance, it may give eigher positively or negatively skew distributions.

There is no certain evidence of the occurence of polygenic interaction. Rasmusson (1935) has recorded a case in Pisum, but this example could equally be one of metrical bias. The same is true of Currence's (1938) findings in the tomato. Smith (1937) failed to find any evidence of interaction in the inheritance of corolla shape in Nicotiana species cross. Unless, and until, the contrary is shown to be the case, however, it is obviously necessary for caution's sake to suppose that any type of interaction shown by major mutants, i.e. oligogenes, may also be shown by polygenes.

Dominance, metrical bias and interaction can all cause skew phenotypic frequency distributions, and very extensive experiments would be required to separate their effects on any character. Such analyses have yet to be made, and so it is impossible at present to say how far the effects of the various agents are shown by the frequency distributions of polygenic characters.

These three agents may also have effects similar to one another on features other than skewness. In particular, all three will play their part in determining the relations which exist between the phenotypes of a hybrid and its parents. If, on the chosen scale, the genes are simply additive and show neither dominance nor interaction the hybrid phenotype will fall on the arithmetic mean of its parents; but this would also be the case where either dominance or interaction occurred, provided that it was equal in each direction. Lack of such equality or metrical bias, however, would destroy this simple relation. The observable relations between hybrid and parents would destroy this simple relation. The observable relations between hybrid and parents would then depend on the degree to which dominance or interaction preponderated in one direction, or on the nature of the metrical bias.

Without extensive analyses of the features, and such analyses have never been made, no expectation can be formulated for the F1 phenotype. It follows, then, that the discussion as to whether fruit size in the hybrid Solanum lycopersicum x S. pimpinellifolium falls on the arithmetic or geometric means of the parents must be sterile (MacArthur & Butler, 1938). The F1 fruit size need not fall on either. Furthermore, even if one or other of these alternatives was observed in a particular case, this fact would by itself give no certain information about the nature of the polygenes involved. Any given relation may be obtained in several ways.

In just the same way the development formulae (Charles & Smith, 1939) for distinguishing between additive polygenes with dominance in one direction and a type of inheritance showing metrical bias without dominance is of very limited value. It leaves out of account many other possibilibies, all equally likely a priori.

IV. THE STORAGE OF VARIABILITY

(1) Fitness and flexibility

Inasmuch as individuals vary in their phenotypic manifestation of a polygenic character, they may be expected to vary in the fitness, or relative selective advantage, which they enjoy in any given environment. The phenotype is, of course, subject to non-heritable as well as to heritable variation, but provided that heritable variation is present the mean degree of expression of any character in a progeny will be correlated with that of the parental phenotypes.

The magnitude of the regression of progeny on parent will be conditioned by the ratio of heritable to non-heritable variation, by the dominance, interaction, number and linkage of the polygenes and by the scale on which the phenotypes are measured; but whatever its exact value the regression must be positive. Big parents will produce big offspring and small parents small offspring. Hence any variation in selective advantage, consequent on variation in phenotype, is capable of causing the phenotypic frequency distribution of the enxt generation to differ from that of the parental generation. Such changes need not occur, however, where selective advantages are balanced against one another in the sense that every individual deviating in one direction from the centre of the parental distribution is matched, with regard to selective advantage, by an individual deviating in the opposite direction. A stable population is possible where the relative selective advantage dimishes as the phenotype departs from the centre of distribution. The resulting tendency towards diminution in phenotypic variation is offset by the extra segregation from the centrally placed phenotypes, which, as will be seen from Fig. 1, are produced by the most highly heterozygous genotypes. Evidence of this correlation of selective advantage and departure from the mean phenotype has been obtained for recovery from rain damage and body form in sparrows by Bumpus and for longevity and shell radius in snails by Weldon (quoted by Baily, 1941).

Thus the mean expression of any phenotype, measured on a scale giving no metrical bias, must be the optimum phenotype if the population is to be stable. Otherwise selection would constantly be moving the mean towards the optimum (Mather, 1941). Actually, of course, the environment is itself never stable and hence mean and optimum will not exactly coincide. Every generation will give progeny whose mean is nearer to its own optimum; but by the time the progeny has developed, the optimum will have changed and the process will be repeated (Fig. 2). Response to selection must always be a generation in arrears, because selective advantage is determined by the phenotype while selective response is expressed as a change in the daughter genotypes.

If the environment, and hence optimum phenotype, is changing steadily in one direction, the change which selection brings about in mean phenotype will be directly adaptive, in spite of the lag of one generation. But if the environment and optimum are subject to changes of a fluctuating kind, response by the organism is not adaptive. It is wasted, because the next generation is as likely as not to have a mean departing from the optimum in the direction opposite to the parental deviation. Since a persistent response to selection must involve one allelomorph of at least one gene replacing a competing allelomorph (see §V (3)) selection destroys heritable variation; so unless reaction to these fluctuating changes in environment will, by dimishing heritable variability, reduce the chance of later changes by which the organism could adapt itself to environmental trends. Equally, however, if the heritable variability necessary for adaptation is present, response to fluctuating changes in the environment is inevitable. Adaptability and conservation of variability make conflicting calls on the organism.

There is a close parallel between this conflict and that between fitness and flexibility. Heritable variability is necessary for adaptive change, but, in that it implies some individuals departing from the optimum, it lowers present fitness; for, as we have seen above, departure from the optimum must be correlated with reduction in fitness. In the same way response to fluctuating environmental changes reduces the heritable variability upon which depends adaptation to environmental trends. The success of any organism in competition with its contemporaries, must depend on the extent to which it reconciles these needs. Failure to achieve an adequate balance spells either its own doom, on the one hand, or that of its descendants, on the other. Existing organisms must therefore have descended from those which had most adequately balanced the advantages of fitness and flexibility in the past. The organisms of the future will equally be descended from those which, to-day, best reconcile the needs of fitness and flexibility, the rest dying out sooner or later.

(2) Free and potential variability

Immediate fitness requires that, phenotypically, all the individuals fall as near to the optimum as possible, i.e. that the phenotypic frequency distribution has a minimum variance. (The mean phenotypic variance of the progeny of an individual is, of course, bound to be highly correlated with that of the population as a whole.) But a complete absence of heritable variablity would rule out future adaptation, except by mutation, and so doom the population to extinction. Such variability, which is the only adequate basis for future adaptation, need not, however, cause a marked lowering of present fitness, for it need not be phenotypic.

Except in special cases fitness is a property of the phenotype, which, apart from non-heritable causes, is controlled by the genotype acting as a whole. If a gene is completely dominant, the heterozygote Aa will be phenotypically like the homozygote AA, but, nevertheless, Aa individuals have a measure of heritable variability not possessed by the homozygotes. Such heterozygotes can give rise to aa offspring which, in general, will differ phenotypically from their parents and from their Aa and AA sibs. Some of the variability is not displayed phenotypically. It exists in the genotype in a hidden, or, as we shall call it, potential, state.

The free variability of a population is that which is manifested by the phenotypes. It will be open to the action of selection and, inasmuch as the various phenotypes differ in their degree of adaptation to the environment, it will inevitably be acted on by selection. The potential variation of a population is, on the other hand, not manifest in the phenotype and hence will be incapable of being affected in any direct way by selective forces. It will play its part in later generations when it has passed from the state of potential to free; just as the potential energy of a weight held in the air is freed when it is allowed to fall. Thus the conflict between fitness and flexibility can largely be removed, for fitness and flexibility can largely be removed, for fitness is a function of the free phenotypic variability while flexibility largely depends on potential genotypical variability. Flexibility obviously will also depend in part on the free variability, but only to an extent proportional to the magnitude of free as compared with potential variability. The most advantageous arrangement will thus involve a small amount of the former and a large store of the latter. Inasmuch as the species of to-day are descended from the successful species of the past, their genetical structure must betray the means by which this balance of free and potential variability is achieved and maintained.

Before passing to a discussion of how storage of variability can be effected it must be made clear that no amount of potential variability is of use unless it can be freed. Future adaptation depends on the interaction of free variability with selection. Furthermore, as trend changes in the environment must be presumed to be always in progress, even though masked at any moment by fluctuation changes, a quantity of free variability must be available in every generation. In other words, there must be a steady flow variability from potential to free. The storage mechanism must provide for a gradual leakage.

It is also clear that unfixed free variability must pass back into store sooner or later, for otherwise the amount of free variability would increase from generation to generation. This cycle of variability change, from potential to free and back, may be interrupted by selective forces which fix some of the free variability, so causing selective response. This fixation will only occur with such variability as it appropriate to the selective force, which may thus appear to create its own directional variability (see 'Student', 1934). In truth, however, the discriminative action of selection is solely one of fixing and so rendering visible variability which would otherwise pass back into store. Selection cannot create new variability.

(3) Linkage and the storage of variability

The variability of a population will be wholly free when all the individuals have completely homozygous genotypes giving the maximum phenotypic departure from the mean possible with the available genes. If this condition is not fulfilled some of the variability must be in store. Thus with a single gene the variability will be wholly free when all individuals are either AA or aa (except in the rare and complex case of Aa being more extreme than either homozygote). If Aa individuals are present some of the variation is potential, for interbreeding these heterozygotes will give phenotypically detectable segregation in the next generation. The progeny will then show phenotypic variation not possessed by the parents. So we may recognize one way in which variability can be stored, viz. by heterozygosity, and the way in which it is released, viz. by segregation. We may also note that, since intercrossing AA and aa individuals produces Aa heterozygotes, crossing is the means of converting free into potential variability.

When two or more genes affect the same character, a further type of storage becomes possible. Consider two non-interacting genes A, a and B, b where the effects of A and B differ in a similar way from those of a and b respectively. The extreme homozygotes are AABB and aabb, and providing no other gene is involved the variability will be wholy free when all individuals fall into one or other of these classes. The various heterozygotes, Aabb, aaBb, AaBb, AaBB and AABb will show some storage of variability, but we have also to consider the remaining homozygous types AAbb and aaBB. Here the two genes are acting in opposite directions, the phenotypes being, in consequence, intermediate between those of AABB and aabb individuals. Hence some variability must be stored by these homozygotes. Such homozygotic variability cannot be released directly. It must always first be converted, by intercrossing, into heterozygotic variabiity whereupon release by segregation becomes possible (Fig. 3).


Fig. 3. The release of potential variability. Potential variability is released by segregation. Homozygotic potential variability cannot be released directly. It must first be converted by crossing into heterozygotic variability.
With more than two genes affecting the character this type of homozygotic storage becomes increasingly important because homozygous genotypes giving phenotypes of various degrees of intermediacy may occur. Thus the ratio of free to potential variability of a polygenic character is very flexible. All values are possible.

When the character is affected by two or more genes of similar effect, the release of variation will depend on recombination, and hence will be affected by linkage (Mather, 1942b). Continuing the two-gene example of the preceding paragraph, let Aa and Bb be intermediate between AA and aa, and BB and bb respectively, and let the two genes have equal additive effects. On selfing or interbreeding double heterozygotes, AaBb, the F2 will contain the ten possible genotypes with the following frequencies, p being the recombination value, and q=1-p.

Coupling AB/ab   Repulsion Ab/aB
  AA Aa aa     AA Aa aa
BB q2 2pq p2   BB p2 2pq q2
Bb 2pq R 2p2
C 2q2
2pq   Bb 2pq R 2q2
C 2p2
2pq
bb p2 2pq q2   bb q2 2pq p2

R indicates the repulsion and C the coupling double heterozygotes, Ab/aB and AB/ab. The phenotypic expression of the character may be measured by the number of capital letters in the genotypes. AABB gives a phenotypic expression of 4, AaBB and AABb of 3, AAbb, AaBb (Both C and R classes) and aaBB 2, aaBb and Aabb 1 and aabb 0 (see Fig. 1).


Fig. 4. Recombination and variability release. Starting with individuals heterozygous for two genes (AaBb), all the variability being thus potential, interbreeding releases some variability by segregation. The actual amount released is governed by the linkage relations of the two genes. With low frequency of recombination (p), release is slow in repulsion, and fast in coupling. Loose linkage gives intermediate release. The parental phenotypic distribution is shown in the centre, those of the next generation, assuming various recombination frequencies, being immediately round it. h is the fraction of the variability which is released. Its maximum value of 0.5 is achieved in close coupling (see text).
Fig. 4 shows the phenotypic frequency distributions obtained with various values of p in coupling and repulsion, and it will be seen that the distribution depends on the linkage conditions. The quantity h is the amount of free variability in the progeny expressed as a fraction of the total variability. Now the parents, AaBb, were the same for every progeny, and show no free variability. Now the parents, AaBb, were the same for every progeny, and show no free variability. All the variability was potential, and hence the differences in free variability shown in the next generation depend on differences in the rate of release of this potential variability. Rate of release is thus clearly modified by the linkage conditions. Tight linkage in repulsion, i.e. where the two genes have allelomorphs of like effect are in the same chromosome, quick release, loose linkage in either phase giving intermediate results. As the frequency of recombination between two genes in the same chromosome is capable of modification by selection (Detlefsen & Roberts, 1921) and by inversion of chromosome segments (Darlington, 1937) the rate of release can be adapted to an optimum value. The freedom of recombination of whole chromosomes can also be changed by reciprocal translocation, leading to ring formation; but this method would appear to offer greater difficulty and be less widespread than control of recombination within chromosomes. Hence it is to intra-chromosome adjustment that we must look for the storage and control of variability.

Storage may be either homozygotic or heterozygotic, release depending ultimately on segregation in either case. When two or more genes affect the character, and with polygenic characters many genes are involved, recombination frequency controls the rate of release, the main control being exercised within chromosomes. Segregation of any gene is not itself influenced by that of the others, but linkage determines the frequency with which the phenotypic effect of segregation in one gene is reinforced or nullified by that of the others. It determines whether segregation will release the variability or merely recast it into slightly different potential form. Intercrossing controls the flow of homozygotic to heterozygotic store, and also of free back to the potential state.

(4) Balanced polygenic combinations

Linkage can, according to its phase, either speed up or slow down the release of polygenic variability, but if a slow rate of release is determined by close linkage in the repulsion phase, very rapid release must occur when the coupling phase supervenes. Such a system would show sharp alternation between slow and rapid release. This clearly has serious drawbacks for the organism, for if slow release is advantageous, the periods when the coupling phase dominated would be markedly disadvantageous. Two factors help, however, in mitigating this consequence.

In the first place, distinctly more than two of the polygenes affecting a given character may be expected to lie in any chromosome. A very complex series of linkage types will then be possible. With two genes only two fully heterozygous zygotes are possible, AB/ab and Ab/aB. The number rises to four with three genes, ABC/abc, ABc/abC, Abc/aBC, AbC/aBc, and to eight with four genes, ABCD/abcd, ABCd/abcD, ABcD/abCd, ABcd/abCD, AbCD/aBcd, AbcD/aBCd, AbCd/aBcD, Abcd/aBCD. It is increasing according to the rule 2n-1, where n is the number of linked genes. Now certain of these multiple heterozygotes are complex mixtures of coupling and repulsion. Thus in AbCd/aBcD, two pairs of genes are coupled internally but repulsed with respect to each other. The most that a single cross-over can do in such a case is to bring three genes into the coupling phase, the fourth being in repulsion. The change in phasic balance here is not great, and with more genes still less drastic changes may be expected to follow single recombination. Polygenic behaviour has a statistical stability not shown by single genes, for the same reason that a volume of gas has predictable properties though the behaviour of a single molecule is quite unpredictable.

This conclusion will be true even if the arrangements in the chromosomes occur with random frequencies. But the second agent making for stability enters in at this point. There must be a selection for the more stable arrangements, i.e. for those in which the phasic balance is least upset by recombination.

We have already seen that the rates of release in repulsion and coupling are negatively correlated. In other words, the greater the advantage of repulsion, the greater the disadvantage of coupling. So a drastic change in phasic balance must ultimately be followed by a marked lowering of fitness and, in consequence, stable phasic types will carry a selective advantage will tend to oust their competitors. The less advantageous types will not of course disappear, for they will continually arise as recombination products from the favoured arrangements. We can thus recognize still another characteristic of polygenic characters. An equilibrium will exist between selective increase and recombinant decrease in favoured types and between selective decrease and recombinant increase in deleterious arrangements. We can thus recognize still another characteristic of polygenic characters. A population may show a constant phenotypic frequency distribution and a constant mass breeding behaviour, but this is the outcome of a complex of balancing processes causing a steady reshuffling of the genetic material in the very individuals which in the aggregate show this statistical constancy. Constancy of the phenotypic distribution in a constant environment is produced by the very agents on which depends selective change in a changing environment. Selection is necessary for stability, and, as Haldane (1936) points out, this stabilizing action of selection makes much more plausible its role as the agent of evolutionary change.

In the general case we may expect the free variability of a population to be low and the potential variability high. Hence neighboring polygenes in a chromosome should be in repulsion, i.e. have opposing tendencies. Otherwise the rate of release will be high and variability will be diminished to a dangerous extent by response to fluctuating changes in the environment. The desirable balance of fitness and flexibility would then be lost. An arrangement which maintains the optimum, or near optimum, balance of fitness and flexibility will be referred to as a balanced polygenic combination, and is characterized by the twin properties of having a phenotypic effect near to the optimum for the constituent polygenes, and of releasing its variability only slowly by recombination with other homologous combinations of the same chromosome. The further the phenotypic effect is from the optimum and the more easily the combination is broken up, the poorer its balance. A well balanced combination causes little phenotypic variability while maintaining, in relation to available homologous combinations, the possibility of great ultimate change slowly produced.

Selection experiments have shown that such combinations exist in Drosophila melanogaster (Mather, 1941; Wigan, 1942; Sismanidis, 1942) and they can be inferred from Winter's (1929) results with maize. The great storage capacity is well brought out by Payne's (1918) selection for increase in the number of scutellar bristles in Drosophila. In both wild and stock flies this number seldom varies from 4, but a fly with 5 is occasionally found. Starting with a 5 bristled female and a 4 bristled male, Payne raised the mean number to about 6.5 in eleven generations, and to over 9 in thirty generations. Sismanidis (1942) has confirmed Payne's results and has been able to assign the sudden responses to selection, to recombinations in particular chromosomes (Fig. 5).

Fig. 5. Recombination and selective response. The result of a selection experiment for increased number of scutellar chaetae in Drosophila melanogaster. The response occurs in two rapid steps, with intervening stability. Tests show that these steps are separable as being due to changes, presumably recombinations, in distinct and recognizable chromosomes. (Reproduced by kind permission of Dr A. Sismanidis.)

CybeRose note: The neo-Darwinists insisted that all heredity must be particulate and restricted to chromosomes. They "explained" continuous variation by postulating the existence of arbitrarily large sets of polygenes that could not, by definition, be demonstrated to exist.

In practice it does not matter whether variation is actually continuous or only appears to be so. We can spend our time calculating around imaginary polygenes, as Mather did, or simply accept that inheritance is polygenic (multifactorial) and set about exploiting it. DeVries and Dubois chose the latter approach with notable success.

Polygenic inheritance does not imply the existence of polygenes. A gene which has a major effect on one trait may also have a tiny influence on another. For example, a gene that confers resistance to a specific fungal infection would be a major gene if we were studying the inheritance of resistance to that fungus. But if we were studying yield in grain, and if that particular fungus was exceedingly rare in the fields we were studying, the anti-fungus gene would have only a very slight influence. So, if many genes have similarly small influences on the trait that interests us, we are not obliged to postulate the existence of a large set of genes that do nothing but interfere with Mendelian calculations.