Friday, July 18, 2008

R. A. Fisher and Epistasis   posted by DavidB @ 7/18/2008 05:21:00 AM

My next note on Sewall Wright will cover the exciting subject of the adaptive landscape. As every schoolboy knows, Wright considered epistatic gene interactions very important in determining the 'peaks' of the landscape. A sharp contrast is sometimes drawn between Wright and R. A. Fisher in this respect. For example:

Fisher believed that the process of genetical evolution occurred through selection that acts on the additive effects of genes in large populations. Although Fisher formally considered gene interactions, he was also dismissive of them, likening epistatic genetic variation to nonheritable (i.e. nontransmissible) environmental variations of phenotype. In contrast, Wright believed that nonadditive, or epistatic, effects were of primary importance, particularly in subdivided populations.
- from the editors' Preface to Epistasis and the Evolutionary Process

What is said here about Wright seems broadly correct, but what is said about Fisher is seriously misleading. Before continuing with my notes on Wright, I will therefore try to clarify Fisher's views on epistasis.[Note: due to formatting problems, italics and other refinements may be omitted.]

First, it is necessary to say something about the meaning of epistasis. The term 'epistasis' itself seems to have emerged around 1917. The first use cited in the OED is from the index to the 1917 volume of the journal Genetics. Around the same time Fisher, in writing his 1918 paper on the Correlation of Relatives, coined the term 'epistacy', but this never caught on. Both terms were derived from the adjective 'epistatic'. Like much of the terminology of genetics (including the word 'genetics' itself) this was coined by William Bateson, in 1907. Bateson used it with a relatively limited meaning to describe cases where a gene at one locus masked or suppressed the action of genes at another locus. For example, genes at one locus might affect the pigmentation of an animal's fur, but a gene at another locus might suppress the production of pigment entirely, causing albinism. In this case the trait of albinism (or the gene producing it) would be called epistatic (literally 'standing over'), while the traits that were masked would be called 'hypostatic' (literally 'standing under'). This limited usage of 'epistatic' is still sometimes found in medical genetics, but in evolutionary genetics a wider usage is more common. In the wider usage, epistasis is any kind of interaction between genes at different loci. Of course, many traits are affected by genes at more than one locus, but this does not necessarily imply interaction. The meaning of 'interaction' is that the genes at different loci do not act independently. For qualitative traits, the usual test of this is that the traits of the offspring do not show the expected Mendelian ratios (which is how epistasis in Bateson's sense was originally discovered). For quantitative traits, the usual criterion is that the value of the trait is not simply the sum of the values attributable to the individual genes concerned. If it is simply the sum, the genes are often said to have a purely 'additive' effect. If not, the trait either shows dominance (if the interaction is between genes at the same locus) or epistasis (if at different loci).

Assuming that epistasis can be identified (which in practice is often very difficult for small effects), it may be asked how the effects of epistatic interaction on a quantitative trait can be measured. One answer to this would be to decide that where interaction is involved, the entire effect of the interacting genes should be counted as epistatic. But this seems unreasonable if the same genes would still have some effect even if there were no interaction. An ideal solution might be to find cases in which the genes concerned are not involved in any epistatic relations, and measure their effect in these circumstances, then subtract this from the effect in the case of epistasis. But if epistasis is a widespread phenomenon, it would be difficult to find these non-epistatic cases, since most genes would show some effects of interaction. In any event, a different approach is generally taken.

The usual approach to measuring the effects of epistasis is roughly as follows. Each gene is assigned a value (the 'average effect' of the gene) based on the average value for the trait concerned among those members of the population who carry that gene, expressed as a deviation from the population mean. Each genotype (gene combination) is then assigned a value based simply on the sum of these average values. This is called the 'breeding value', since it is the part of the genetic makeup of the individual which enables the traits of its offspring to be predicted for breeding purposes. These breeding values will have a certain variance, relative to the population mean, usually called the additive genetic variance. The actual observed values will have a greater variance than this, due to the effects of environment, dominance, epistasis, and various other complications. The portion of the observed variance attributable to epistasis is estimated after the effects of environment and dominance have been subtracted. Genes with epistatic effects are not excluded from the analysis, and they may contribute to both additive and (in a more complicated way) to dominance variance as well as to the specific epistatic or 'genetic interaction' variance. All this is explained more fully, and no doubt more clearly, in Falconer. For a simple worked example of my own see Note 1.

The standard terminology is unfortunate. It cannot be stressed too strongly that 'additive' variance is not the same as the variance due to genes with purely additive effects. The additive variance takes account of the average effects of all genes, including those that may show strong dominance or epistasis. These average effects depend in part on the gene frequencies present in the population in question, and assume that all possible genotypes occur in the proportions expected under a given system of mating (usually assumed to be random). Part of the average effect is therefore due to the effects of gene interactions. Conversely, the so-called 'epistatic variance' covers only a part - usually the minority - of the effects that might intuitively be ascribed to interaction. Enthusiasts for epistasis (as in the volume already cited) sometimes complain that the standard method of apportioning variance tends to understate the effects of epistasis, and makes it difficult to detect. For example, James Cheverud comments that 'most tests for epistasis rely on the epistatic variance alone and ignore its contribution to additive and dominance variance' (p.65) and Edmund Brodie says that 'under a wide range of allele frequencies and strengths of interaction, the majority of variance produced by gene interaction is actually additive' (p.10). It would be possible in principle to use alternative measures which assign more of the observed variance to epistasis. But the standard method does have the advantage that it is possible to estimate the additive variance from the observed correlation between parents and offspring, and conversely to estimate the value of offspring from that of parents. This is particularly important if we wish to predict the effects of natural or artificial selection. Whatever we call it, the 'additive' variance is a useful concept and is not going to go away.

It is also desirable to distinguish between epistasis for fitness and for other traits of the organism. Fitness itself (whether measured simply by number of offspring or otherwise) shows epistasis if the effects on fitness of genes at different loci are not purely additive. If fitness is measured in relation to some particular trait, the fitness may show epistasis even if the trait as such does not. (And presumably vice versa, though I cannot think of a plausible scenario for this.) For example, a trait such as body size might be influenced by several genes acting purely additively in their effects on body size, but epistatically in their effect on fitness. This will often be the case if fitness is highest for some intermediate value of the trait. The fitness effects of genes tending to raise (or lower) the value of the trait will then depend crucially on the other genes they happen to be combined with. In the simplest case, if there are two haploid loci, with alleles H and L (for High and Low) at one locus, and h and l at the other, the combinations Hl and hL, which give intermediate size, may be favoured by selection, while the combinations Hh and Ll, which give high and low size respectively, are selected against. In this case the fitness is epistatic even though the direct effect of the genes on the phenotype is additive.

After all these preliminaries, I turn to discuss what Fisher actually said about epistasis.

Correlation of relatives

As already mentioned, Fisher's great 1918 paper on the 'Correlation of Relatives' proposed the term 'epistacy' to allow for the interaction of genes at different loci, and devised the standard method for apportioning variance. Fisher introduces his definition of 'epistacy' as follows: 'There is in dominance a certain latency. We may say that the somatic [phenotypic] effects of identical genetic changes are not additive, and for this reason the genetic similarity of relations is partly obscured in the statistical aggregate [see Note 2]. A similar deviation from the addition of superimposed effects may occur between different Mendelian factors [genes at different loci]. We may use the term Epistacy to describe such deviation, which although potentially more complicated, has similar statistical effects to dominance. If the two sexes are considered as Mendelian alternatives, the fact that other Mendelian factors affect them to different extents may be regarded as an example of epistacy. The contributions of imperfectly additive genetic factors divide themselves for statistical purposes into two parts: an additive part which reflects the genetic nature without distortion, and gives rise to the correlations which one obtains, and a residue which acts in much the same way as an arbitrary error introduced into the measurements. ' (p.404) Note that Fisher says here quite explicitly that part of the contribution of 'imperfectly additive' genes is itself additive, or as we would say, falls within the additive variance. Fisher does not say a great deal more about 'epistacy' in this paper (but see p.408-9 for the mathematical treatment of epistatic variance), and one of the contributors to the volume cited earlier claims that in his 1918 paper Fisher 'dismissed gene interactions as being of only minor importance in the evolutionary process, analogous to nonheritable modifications of the phenotype' (p.125). This goes beyond anything Fisher says. What he does say is that 'Throughout this work it has been necessary not to introduce any avoidable complications, and for this reason the possibilities of Epistacy have only been touched upon...' (p.432). For Fisher's specific purpose in this paper, which was to explain the correlation between relatives on Mendelian principles, and not to discuss evolutionary theory in general, his brief treatment of 'epistacy' seems sufficient. Fisher finds that with his methods the existing data on the correlation of relatives (mainly the data of Karl Pearson on humans) can be explained satisfactorily by additive variance, dominance, and assortative mating, without much influence of other factors, which by implication include epistatic variance. Fisher is more explicit about this in his 1922 paper on the Dominance Ratio, where he says that 'special causes, such as epistacy, may produce departures [from the expected correlations], which may in general be expected to be very small from the general simplicity of the results'. But before interpreting this as a general pronouncement on the insignificant role of epistasis in evolution, we should note that (a) the additive variance includes much of the effect of 'epistatic' genes, and (b), the discussion was concerned with ordinary traits such as height, and not with fitness. As emphasised earlier, there may be epistasis for fitness even if the underlying traits are purely additive.

The evolution of dominance

One of Fisher's best-known, and most controversial, theories is that of the evolution of dominance. Noting that harmful mutations are usually (though not always), recessive in their effects, Fisher sought to explain this by the action of modifier genes at other loci, which would be gradually selected to minimise the harmful effects of common recurring mutations by making them recessive. The theory has not been generally accepted, and Wright in particular opposed it, mainly on the grounds that the selective advantage of modifier genes would be so weak that it would usually be overpowered by their other, more direct, effects. Regardless of whether Fisher was right or wrong on this issue, the point to note here is that his theory depends entirely on epistatic effects! In this respect, at least, Fisher was more enthusiastic about epistasis than Wright himself.


A whole chapter of the Genetical Theory of Natural Selection is concerned with Mimicry. In discussing the underlying genetics of mimicry, Fisher emphasises the role of modifier genes, including those that act as 'switches' for other genes. For example, discussing the 'hooded' gene in rats, he says 'The gene, then, may be taken to be uninfluenced by selection, but its external effect may be influenced, apparently to any extent, by means of the selection of modifying factors' (p.185). And in discussing another case he goes on to say 'The gradual evolution of such mimetic resemblances is just what we should expect if the modifying factors, which always seem to be available in abundance, were subjected to the selection of birds or other predators' (p.185). While modifiers might in principle be purely additive in effect, they are more likely to be epistatic. This is presumably always the case with 'switch' genes.


Chapter 6 of GTNS deals with a variety of issues concerning sex, sexual selection, sex-limited traits, and speciation. Some of these could well involve epistasis - indeed, 'sex-limited' traits (those which are only manifested in one sex) do so almost by definition, if sex is genetically determined. (As mentioned in Fisher's paper on 'Correlation of Relatives', quoted above, differences between the sexes can be regarded as a case of 'epistacy'.) However, I find only one definite reference in the chapter to epistatic effects. In his discussion of speciation, Fisher points out that the adaptiveness of genes will vary in the different parts of a species's range, and says that 'In addition to those genes which are selected differentially by the contrasted environments, we must moreover add those, the selective advantage or disadvantage of which is conditioned by the genotype in which they occur, and which will therefore possess differential survival value, owing not directly to the contrast in environments, but indirectly to the genotypic contrast which these environments induce' (p.141). A difference in the selective advantage of a gene according to the genotypic background implies epistatic fitness. What Fisher is describing here is actually what is often called a 'co-adapted gene complex', much beloved of Wrightians.

The Fundamental Theorem of Natural Selection

The Fundamental Theorem of Natural Selection states that 'The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time' (GTNS p.37). The FTNS is notoriously difficult to interpret, and I do not intend to say much about it here. It is however now generally accepted, following the interpretations by George Price and A. W. F. Edwards, that when Fisher refers to 'genetic variance' he means the 'additive' genetic variance. The additive variance takes account of the average effect of genes in all the various environmental circumstances and genetic combinations in which they are found, in the proportions to be expected under a given system of mating. (See expecially p.31 of GTNS, where Fisher defines 'average excess' and 'average effect'.) It therefore incorporates the effects of dominance and epistasis to the extent that these contribute to the additive value of the genes. There is no reason at all to suppose that genes with epistatic effects are excluded from the FTNS. What is excluded is only that part of the total variance that is not covered by the contribution of those genes to additive variance. This can be justified on the grounds that the non-additive variance does not predictably change gene frequencies in the next generation and therefore has little effect on evolution. As Cheverud admits, 'the rate of evolution is determined by the additive genic [sic] variance alone' (p.65).

Selection at two loci

Before 1930 neither Fisher nor Wright had treated selection at more than one locus. As so often, the pioneer of the subject was J. B. S. Haldane, in 1926. In 1930 Fisher did however give the subject a short section in Chapter 5 of GTNS, under the heading 'Equilibrium involving two factors'. (This chapter is one of several that appear to be invisible to some readers.) The interesting situation, as Fisher recognises, is where two different combinations of alleles (e.g. AB and ab) are both favoured by selection, while the same genes are disadvantageous in other possible combinations (e.g. Ab and aB). Fitness in this case is therefore clearly epistatic. In his chapter summary Fisher says that stable equilibria may be established, but he is rather vague about the conditions for stability. But his main point is that there will be selection in favour of closer linkage between favourable gene combinations on the same chromosomes, and it is therefore a puzzle why recombination is as frequent as it is. I think this remains a problem. In any event, it is a case where Fisher clearly recognised the role of epistasis.

Selection of metrical characters

One of the most intriguing, but difficult, sections of GTNS is the one (also in the 'invisible' Chapter 5) on 'Simple metrical characters'. (I sometimes wonder if Fisher's use of the word 'simple' was a sly joke.) The case of interest is where a quantitative character, such as the size of a tooth, is regulated by genes at more than one locus, and subject to stabilising selection in favour of an intermediate size. Egbert Leigh has described this (in his 'Afterword' to the 1990 reprint of Haldane's 'The Causes of Evolution') as 'a topic still replete with mysteries and surprises'. Fisher's account is even more tangled than most, because he attempts to explain simultaneously selection of the metrical trait itself and selection for dominance of the genes controlling it. I cannot pretend to understand everything he says on the subject, but what is clear for the present purpose is that fitness in this case is epistatic, and that there may be more than one outcome of selection, depending on the initial frequencies of the genes concerned: 'the conditions of equilibrium are always unstable. Whichever gene is at less than its equilibrium frequency will tend to be further diminished by selection' (p.121). This is precisely the situation which Wright often emphasised as leading to alternative 'selective peaks'. But unlike Wright, Fisher did not believe a species was likely to get 'stuck' permanently on a selective peak (not that Fisher had much time for the adaptive landscape anyway). Fisher believed that following any change in the optimum phenotypic value due to environmental change there would be sufficient genetic variation (in a large population) for selection to shift organisms quickly towards the new optimum. His confidence in this was based mainly on the results of artificial selection, as he referred to 'the extreme rapidity with which such measurements are modified when selection is directed to this end' (p.119). The effects of such changes on gene frequencies might be lasting, even if the initiating circumstances were temporary. In Fisher's analogy, which may be more illuminating to physicists than to me, 'the system resembles one in which a tensile force is capable of producing both elastic and permanent strain, and in which the permanent deformations always tend to relieve the elastic forces which are set up' (p. 125).

This section of GTNS raises a rather intriguing historical possibility. As Provine has noted in his biography of Wright (Provine p.285-6), there was an unexplained change in Wright's account of the 'shifting balance' theory between his exposition in 'Evolution in Mendelian Populations' (1931), and his next major account in 1932. In 1931 he had asserted that temporary changes in the environment would only have temporary effects on the gene pool, being essentially reversible. Hence his emphasis on genetic drift in small subpopulations, as the only possible means of shifting from one peak to another. In 1932, on the other hand, he accepted that environmental changes could also shift a population from one stable peak to another, so that their effects might be lasting even after the change in environment had reversed. Unfortunately Wright did not explain the reasons for his change of mind, nor did he draw attention to the change, which is really very important, since it greatly weakens Wright's argument for the importance of genetic drift in small local subpopulations. Provine speculates, plausibly enough, that Wright's correspondence with Fisher, his reading of GTNS, and Fisher's own published review of 'Evolution in Mendelian Populations', had something to do with the change. My own suggestion, to build on this, is that Fisher's discussion of metrical characters in Chapter 5 of GTNS was a particular influence. But I have no direct evidence of this, so it will probably remain a mere speculation.


The main purpose of this note has been to identify and document what R. A. Fisher himself, as opposed to the straw man 'Fisher', actually said and believed about epistasis. Readers will be able to draw their own conclusions, but I will briefly indicate my own.

a) Fisher did not deny the existence of epistasis, in the broad sense, and in some specific cases - including the evolution of dominance, selection at two loci, and quantitative (metrical) traits under stabilising selection - he gave it an important role.

b) Fisher agreed with Wright (and Haldane) that in some circumstances, including stabilising selection, there could be more than one outcome of selection in terms of the resulting gene frequencies. Unlike Wright (in 1931), but like Wright (in 1932), he believed that temporary environmental change could shift a population durably from one equilibrium set of gene frequencies to another. Fisher's treatment of the problem in GTNS may have influenced Wright's unexplained volte-face on this important issue.

c) Fisher did not believe populations were likely to get stuck on a local peak in the selective landscape, but this was not because he did not believe in epistatic effects, but because he did not believe in the validity of the selective landscape concept at all. I will probably say more about Fisher's thinking on this in another post.

d) Fisher's general concept of evolutionary change, as expressed in the Fundamental Theorem of Natural Selection, does not exclude epistatic effects. The FTNS takes account of epistasis (and dominance) precisely to the extent that they do affect the rate of evolutionary change. The FTNS is neutral with respect to the importance of epistasis: whether it is important or unimportant cannot be inferred from the theorem, which takes account of additive variance in fitness whatever its source. Unfortunately much confusion has arisen about the meaning of 'additive' and 'epistatic' variance. If it is not understood that 'additive' variance includes much of the effect of epistatic genes, while 'epistatic' variance excludes much of that effect, the scope of the FTNS will be seriously misconstrued. It would be better to call additive variance something like 'heritable variance', while the non-additive effects of dominance and epistasis are clearly labelled in such a way as to make it clear that they are only part of the total effect of gene interactions.

e) Unlike Wright, Fisher did not, at least in his published works, put any emphasis on epistasis as a major factor in evolution. It is necessary to read GTNS quite carefully (or at least to look at all the chapters!) to find the references I have gathered together here. It is an empirical matter whether epistasis plays the central role that Wright gave it. Or it might have an important role that neither Wright nor Fisher had thought of, as suggested in Kondrashov's theory of sex.

I have not dealt here with another aspect of Fisher's views, namely his rejection of the importance in evolution of large single mutations. I have no doubt that Fisher believed that evolution occurred mainly through the selection of a large number of genes with individually small effects. I have not discussed this because (a) it was not a point of disagreement between Fisher and Wright, and (b) it does not seem relevant to the issue of epistasis. As far as I can see, large mutations are no more or less likely to have epistatic effects than small ones.


After writing the above, I came across a further reference to epistasis in Fisher's correspondence. Writing to Leonard Darwin in 1928, Fisher said 'I am inclining to the idea that the main work of evolution lies in the discovery by trial of perhaps rare combinations of its existing variants, which work better than the commoner combinations. A slight increase in the number of individuals bearing such a favourable combination will then set up selection in favour of all the genes in the combination, with marked evolutionary results. Many of these genes would have been previously rare mutant types (not necessarily rare mutations) unfavourable to survival. I think of the species not as dragged along laboriously by selection like a barge in treacle, but as responding extremely sensitively whenever a perceptible selective difference is established. All simple characters, like body size, must be always very near the optimum, so much so that the average body sizes of two alternative genes must be balanced on either side of the optimum, selection always tending to eliminate the rarer because it is further from the optimum...' (Correspondence p.88). In his Introduction to the correspondence, J. H. Bennett draws attention to this letter, and remarks that 'It is interesting, and perhaps needs emphasizing, that both Fisher and Wright considered systems of interacting genes to be of critical importance in evolution. A fundamental difference in their views of the evolutionary process concerned the means by which interaction systems could be exploited' (p.47) While I agree with Bennett that Fisher took some account of 'interaction systems' , in other words epistasis in the broad sense, this letter of 1928 seems a good deal more positive on the subject than anything I have noticed in his published works. I take this opportunity to say that Bennett's Introduction is one of the most useful things yet written on Fisher's work and ideas, and deserves repeated reading.

Note 1

Consider the simplest case of a haploid organism with a quantitative trait determined by genes at two loci. I assume complete genetic determination. Let the alleles in the population be A and a at one locus, and B and b at the other, each with a frequency of 50% in the population. Under random mating the four genotypes AB, Ab, aB and ab will therefore all have the frequency 25%. (In a diploid there would be nine genotypes to consider, and the possible complication of dominance, which is why I have chosen the haploid case.)

Let us suppose that the measurements of the trait for the four genotypes are as follows, where c and d are any numerical values:

AB........c + d

I have chosen these values to dramatise the situation. Intuitively, one would say that all of the variation in the trait was due to the epistatic interaction of A and B, since all other genotypes than AB have the identical value c. So let us see how the variance comes out under the standard method.

The mean value of the trait in the population is evidently .75c + .25(c + d) = c + .25d. The mean values for each gene considered separately, measured by the average value of the individuals who possess that gene, are:

A........ .5(c + d) + .5c = c + .5d
a......... c
B........ .5(c + d) + .5c = c + .5d
b........ c

Expressed as deviations from the population mean, c + .25d, these values come out as:

A........ + .25d
a......... - .25d
B........ + .25d
b........ - .25d

These are known as the 'average effects' of the genes in question.

The so-called 'breeding value' of a genotype is simply the sum of the average effects of its component genes, so for the four genotypes we have the breeding values:

AB.......... + .5d
Ab.......... 0
aB.......... 0
ab.......... - .5d

It may be noted that the combination ab has a substantial (negative) breeding value, even though there is, intuitively, no interaction between a and b. This reflects the fact that the interaction of A and B pulls up the population mean, and therefore affects the deviation values of other alleles and genotypes. The combination ab falls as far below the resulting mean as the combination AB rises above it. The symmetry is of course a consequence of the symmetry of the chosen assumptions about gene frequencies, etc.

The breeding values are already deviations from the population mean, so for the variance of breeding values (the so-called additive genetic variance) we have:

.25(.5d)^2 + .25(0)^2 + .25(0)^2 + .25(.5d)^2 = .125d^2.

It is already apparent that although the variance is intuitively entirely due to epistasis, the 'additive' variance is not zero. For comparison, we can measure the total variance of the values of the genotypes. The deviation values are as follows:

AB.......... c + d - (c + .25d) = .75d
Ab, aB, and ab.......... c - (c + .25d) = - .25d

Taking account of the proportions of the genotypes in the population we therefore have the variance of genotypic values as follows:

.25(.75d)^2 + .75(- .25d)^2 = .1875d^2

Subtracting the 'additive' variance from the total genotypic variance we find only .0625d^2 left for the 'epistatic' variance. So even where we have rigged the example to give a strong influence to epistasis, 2/3 of the resulting variance is 'additive', and only 1/3 'epistatic'!

Note 2: I think that by 'genetic changes' in this sentence Fisher means not just mutations, but any gene substitution, such as may occur through the normal processes of sexual reproduction. So, for example, if at a single locus the combination aa is replaced by the combination Aa, there will be a certain measurable effect of the change. If the effect of substituting two As is twice the effect of substituting just one A, the effect is additive. Otherwise the locus shows some degree of dominance.


D. S. Falconer: Introduction to Quantitative Genetics, 3rd. edn., 1989

R. A Fisher: The Genetical Theory of Natural Selection, 1930. I have given page references to the revised Dover edition of 1958, but the quoted passages are all unchanged from the first edition. For scholarly purposes the best edition is now the Variorum edition of 1999, edited by Henry Bennett.

Fisher's papers are cited from the online copies available from the archives at Adelaide (see link on sidebar)

Natural Selection, Heredity and Eugenics: Including selected correspondence of R. A. Fisher with Leonard Darwin and others, edited by J. H. Bennett (1983). Much of the correspondence is also available online from the archives at Adelaide.

Epistasis and the Evolutionary Process, ed. J. B. Wolf, E. D. Brodie, and M. J. Wade. 2000

William B. Provine: Sewall Wright and Evolutionary Biology, 1986. (Paperback edn. 1989)

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