Sunday, November 09, 2008

Notes on Sewall Wright: The Shifting Balance Theory (Part 2)   posted by DavidB @ 11/09/2008 01:32:00 AM

Part 1 of this note dealt with Sewall Wright's Shifting Balance theory of evolution (the SBT) in its original form, as propounded between 1929 and 1931. This final part deals with subsequent developments in the theory. These include refinements and elaborations, some changes of emphasis, one major addition, and one major change of substance. In particular I will cover:
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1. The role of new mutations
2. The concept of selective peaks
3. The effect of changes in environment
4. The adaptiveness of evolution
5. The process of intergroup selection
6. The three phases of the shifting balance.

I will throw in a few remarks about Fisher and Haldane as well.

NB: all page references are to Evolution: Selected Papers unless otherwise stated. Spelling and punctuation of quotations are as printed (some use American and some use British spelling). Square brackets indicate comments of my own.

1. The role of new mutations

First, a few words are necessary about the meaning of 'mutation'. In the 1930s very little was known about the physical and chemical nature of genes and therefore about the nature of changes to genes, in other words 'mutations'. In 1939 Wright gave a useful statement of current assumptions at that time: 'Presumably any particular gene can arise at a single step from only certain of the others and in turn mutate only to certain ones but the latter may be capable of producing mutations which could not have arisen from the former at one step and so on through a branching network of potentially unlimited extent' (306). This implies a 'step-by-step' evolution of genes themselves. Each gene may be said to have a first appearance in time, though recurrence of the same gene at different times is not excluded. The occurrence of mutations depends on the prior existence of the genes of which they are variants, so a particular type of mutation itself has an origin in time. The opportunity for mutations of a particular type will also depend on the frequency of the relevant genes in the population. If a gene is changing in frequency, the opportunity for new mutations of that gene will also be changing. We may therefore expect the rate of specific mutations to increase or decrease over time. This may explain some otherwise obscure comments in Fisher's Genetical Theory of Natural Selection (GTNS). In several places Fisher assumes that any new mutation will initially have a low rate of occurrence, but that this rate will increase over time (see especially GTNS p.78). This assumption makes sense if Fisher held the same view as Wright on the nature of mutations.

Wright's original formulation of the SBT said little about the role of beneficial new mutations in evolution. In 'Evolution in Mendelian populations' (EMP) (1931) Wright said only that in very large populations 'there is little scope for evolution. There would be complete equilibrium under uniform conditions if the number of allelomorphs at each locus were limited. With an unlimited chain of possible gene transformations, new favorable mutations should arise from time to time and gradually displace the hitherto more favored genes but with the most extreme slowness even in terms of geologic time' (150). This negative assessment of the prospects for evolution in large undivided populations conflicted with that of Fisher in GTNS, which appeared in 1930 after Wright's 'Evolution in Mendelian populations' (EMP) (1931) had been sent for printing. (A few short notes were added to take account of Fisher's work, but major changes were not possible.) Whereas Wright had concluded that large freely interbreeding populations were unfavourable to progressive evolution, Fisher believed that large populations (without strong barriers to gene flow) were favourable to evolution because of the greater scope they offered to new mutations. Fisher reinforced this in his published review of EMP, saying that 'even under static conditions, unless it is postulated that the organism is as well adapted as it could possibly be (in which case, obviously, evolutionary improvement is impossible), the equilibrium will be broken by the occurrence of any favourable mutation, of which a steady stream will doubtless occur in one or other of the very numerous individuals produced in each generation. The advantage of the large populations in picking up mutations of excessively low mutation rate seems to be overlooked [by Wright]... ' (Natural Selection, Heredity and Eugenics, p.288). Here, then, we find one of the major differences in the evolutionary theories of Wright and Fisher.

Wright elaborated and defended his position on this issue on several occasions, beginning with his own review of Fisher's GTNS in 1930. He notes that Fisher's 'scheme appears to depend on an inexhaustible flow of new favorable mutations. Dr. Fisher does not go into this matter of inexhaustibility but presumably it may be obtained by supposing that each locus is capable of an indefinitely extended series of multiple allelomorphs, each new gene becoming a potential source of genes which could not have appeared previously. The greatest difficulty seems to be in the posited favorable character of the mutations. Dr. Fisher, elsewhere presents cogent reasons as to why the great majority of all mutations should be deleterious. He shows that all mutations affecting a metrical character 'unless they possess countervailing advantages in other respects will be initially disadvantageous' [see Note 1]. He shows that in any case the greater the effect, the less the chance of being adaptive. [See Note 2] Add to this the point that mutations as a rule probably have multiple effects, and that the sign of the net selection pressure is determined by the greater effects, and it will be seen that the chances of occurrence of new mutations advantageous from the first are small indeed' (85).

There is a risk of ambiguity in this conclusion. If Wright means to say that only a small proportion of new mutations will be initially advantageous, his arguments are plausible, though not conclusive. If on the other hand he means to say that the 'chances of occurrence' of any such mutations, even in a large population, are small, the arguments are quite insufficient. It would be like confusing the probability that John Smith will die tomorrow, which is small, with the probability that someone will die tomorrow, which in a large population is virtually certain. Suppose that in a population of one billion, one in 100,000 individuals in each generation show some new mutation or other. There would then be 10,000 such new mutations in the population in each generation. Evidently, even if only a very small proportion of these mutations are advantageous, there might still be (in Fisher's terms) a 'steady stream' of them. Whether or not this is the case is an empirical matter.

Wright made similarly negative comments about new mutations on various occasions when defending the SBT:

1932: [under constant conditions] 'further evolution can only occur by the appearance of wholly new (instead of recurrent) mutations, and ones which happen to be favorable from the first. [Comment: this is valid only if 'new' means 'new under the same conditions'. Evolution might also occur through recurrence of mutations previously unfavourable but now favourable under new conditions.] Such mutations would change the character of the field [the 'adaptive landscape'] itself, increasing the elevation of the peak occupied by the species. Evolutionary progress through this mechanism is excessively slow since the chance of occurrence of such mutations is very small [comment: note the same ambiguity as in Wright's review of GTNS] and, after occurrence, the time required for attainment of sufficient frequency to be subject to selection to an appreciable extent is enormous' 165). [The last remark is puzzling. Any favourable new mutation is subject to selection from the outset, but it is at risk of being lost by random drift before it becomes safely established. It is not 'safe' until it has recurred a few hundred times. But in a large population, even with very low mutation rates this should only take a few hundred generations, which is not long in evolutionary time. This is one of Fisher's main arguments for the evolutionary advantage of large population size: see GTNS p.78. Once a mutation has reached a level of a hundred or so copies - say, a frequency of 1 in 10,000,000 in a population of a billion - the rate of advance will depend on the selective advantage of the gene. If the selective advantage is such as to double its frequency in 1,000 generations - equivalent to an advantage of rather less than 1 in 1,000 - the gene will go from first appearance to fixation (or equilibrium against back-mutation) in less than 30,000 generations. [See Note 3] This is not very long in geological time, though it would be imperceptibly slow to human observers, and until the later stages the gene would still be rare.]

1939: 'there is very little chance of occurrence of wholly new alleles in a large freely interbreeding population. There is also very little chance that any new mutation will be favorable at its first occurrence and even if favorable very little chance that it will attain sufficient frequency to be subject to selection to an appreciable extent' (321) [The italics for 'large' are Wright's own. The implicit assumption seems to be that in a large population every good mutation will already have been found. But note my previous comment that the advantageousness of a mutation is relative to conditions.]

1948: 'Presumably all mutations that are likely to arise at one or two steps from the more abundant genes present in the population have been tried by natural selection and found wanting, and thus are found at negligibly low frequencies if at all. There may be very valuable mutations which could only arise through a succession of unfavourable ones but these will have very little chance of occurring' (535) [see the previous comments]

1959: 'A genetic system can take the step from one selective peak to another one only by some non-selective process. A novel mutation may do this by creating a new peak, but this must be an excessively rare event' (Tax, p.451)

Wright maintained his opposition to the importance of new mutations to the end of his career. But his arguments are always brief and unquantified. There is a recurring ambiguity, as noted above, between the probability that a given new mutation will be advantageous, and the probability that any advantageous new mutation will occur. Fisher's view (GTNS p.78), was that in large populations, of the order of a billion (which includes most plant and invertebrate animal species), such mutations would occur often enough to be important in evolution. Wright opposed this conclusion, but it is difficult to avoid the feeling that in doing so he was trying to shore up a position which he had adopted without first considering mutation. It should at once be said that Fisher was equally stubborn (and more intemperate) in defending his own positions.

2. The concept of selective peaks

As noted in my post on Wright and the adaptive landscape, in 1932 Wright introduced the metaphor of a multidimensional field of gene combinations. I have discussed Wright's adaptive landscapes at length (see also here), so I will not repeat those discussions now. The point I wish to emphasize here is that the concept of selective peaks, valleys, etc, as introduced in 1932 was not just a new metaphor adopted for purposes of exposition, but an important addition of substance to the SBT.

From 1932 onwards it is a fundamental part of the SBT that there is a multiplicity of selective peaks in the field of possibilities available to a population. Many of these peaks are of different height (fitness). Under the influence of selection alone, and under constant conditions, a population cannot move from one peak to another. Under selection a population will tend to move towards one of the peaks, but usually the closest, which will seldom be the highest. It is therefore very likely that a population will be 'trapped' on an inferior peak, from which it cannot move purely by selection under constant conditions.

This aspect of the SBT is so important, and so familiar from Wright's later writings, that it is tempting to assume that in substance it was already there in the original version of the theory, even if the analogy of 'peaks' and 'valleys' was missing. In purely genetic terms, the meaning of a 'peak' in the landscape is that there is some set of gene frequencies such that any small departure from that set is opposed by selection. If there is more than one such set, there are multiple peaks. But the terminology of 'peaks', etc, is inessential. The substance of the theory could be stated quite well without it. It is therefore natural to expect some such equivalent statement in EMP, but I have not found one. It is true that, when discussing evolution in large populations in his 1929 summary, Wright does say that 'changed conditions cause a usually slight and reversible shift of the gene frequencies to new equilibrium points' (78), but in the context of his discussion in EMP (150) it appears that Wright was thinking only of a shift in the equilibrium between selection and mutation. His repeated claims that such shifts are essentially reversible would be difficult to reconcile with the concept of multiple peaks, and indeed, once Wright had clearly formulated that concept, he abandoned the claim of irreversibility.

The concept of multiple selective peaks is closely related to Wright's emphasis on epistatic fitness interactions, but this familiar feature of Wright's philosophy of evolution is also lacking from EMP. The beginnings of a new emphasis on epistasis can be found in 'Statistical theory of evolution' (1931), written after EMP but published slightly earlier. In discussing populations of intermediate size, Wright points out that 'it is the organism as a whole that is selected, not the individual genes, and a gene favored in one combination may be unfavorable in another' (95). And in subdivided populations 'exceptionally favorable combinations of genes may come to predominate in some of the subgroups' (95). But there is still, as far as I can see, no indication that even large populations may have alternative stable states, as proposed by Wright in 1932.

It is natural to wonder how Wright arrived at his 1932 conception of multiple selective peaks. It is possible that his reading of the section on 'Simple metrical characters' in GTNS had planted the seed. We know from Wright's correspondence that he was encouraged by receiving an offprint from Haldane in which the latter outlined similar ideas (Provine 275). It is also possible that Wright had privately reached his conception (without the geometrical analogy) much earlier, as Provine seems to think (Provine 275). But if Wright did indeed have the concept in mind when writing the paper which became EMP it is odd that he did not incorporate it in that work. I can only leave this as an unsolved puzzle.

3. The effect of changes in environment

As I have mentioned in previous posts (and as is also pointed out by Provine), until 1931 Wright considered that the evolutionary effects of temporary changes in environment would 'usually' or 'essentially' be reversible (78, 85, 150). But in 1932, with his paper on 'The roles of mutation, inbreeding, crossbreeding and selection in evolution', he took a new position. After introducing his concept of the multidimensional field of gene combinations, and the associated diagrams, he notes that 'the environment, living and non-living, of any species is actually in continual change. In terms of our diagram this means that certain of the high places are gradually being depressed and certain of the low places are becoming higher... Here we undoubtedly have an important evolutionary process and one which has been generally recognised. It consists largely of change without advance in adaptation. The mechanism is, however, one which shuffles the species around in the general field. Since the species will be shuffled out of low peaks more easily than high ones, it should gradually find its way to the higher general regions of the field as a whole' (167). This formulation is repeated, usually in similar words, in most of Wright's subsequent general surveys of evolutionary theory, e.g. 323, 374, 535, and 562.

It is perhaps not immediately clear (and Wright does not explain) why 'the species will be shuffled out of low peaks more easily than high ones'. Presumably it is partly because higher peaks may have stronger selection coefficients, and will therefore resist drift more strongly, but mainly because, other things being equal, higher peaks will have wider zones of attraction. A population may therefore drift further from the peak but still be pulled back towards it by selection. In geometrical terms, if two solid figures have the same shape, the taller figure will have the larger base. In genetic terms, the higher the fitness of a genotype relative to the average fitness of the population, the wider will be the range of gene frequencies within which the genes making up that genotype will be positively selected. But this is not an absolute rule. If a peak of fitness depends on very specific epistatic interactions of several genes, the peak may be high but narrow, like a spike. In this case a population may be easily jolted out of a high peak by environmental change, and never return to it. Changing environments may therefore be expected to promote mainly genes that are advantageous in a wide range of genetic combinations.

We are bound to ask why Wright changed his mind about the effects of environmental change. Wright himself gives no help on this point, because he never (I think) admitted that he had changed his mind. The change in 1932 goes together with Wright's formulation of the adaptive landscape concept, and in one sense goes very naturally with it. If we accept that there are multiple peaks of fitness in the landscape, and that it is largely a matter of chance which peak is most accessible to a population, then any factor which causes populations to move in a quasi-random way around the landscape could have the effect of 'shuffling' the population from one zone of attraction to another. But in another sense there is a tension between the landscape concept and environmental change, since the effect of environmental change is not so much to move the population around a fixed underlying landscape as to modify the landscape itself. As several commentators have suggested, in a changing environment the proper analogy is not so much with a solid landscape as with a choppy sea.

It is quite possible that Wright's change of mind in 1932 resulted simply from his own reflection on the issues. But he may also have been influenced by the positions already taken by Fisher and Haldane. As I mentioned in my post on Fisher and epistasis, in the section on 'Simple metrical characters' in GTNS Fisher had pointed out that metrical traits under stabilising selection could lead to multiple stable equilibrium gene frequencies, and that changes in selection coefficients due to environmental change could produce a lasting shift from one equilibrium to another. Wright had certainly read this section of GTNS, since he quotes from it in his review of the book. At that time (1930) he still thought that the effects of environmental change would usually be reversible, but he qualifies that position, saying: 'It may be granted that an irregular sequence of environmental conditions would result occasionally in irreversible changes (because of epistatic relationships), thus giving a real, if very slow, evolutionary process... ' (85). Over the next year Wright may have come to reconsider whether the process would only be 'occasional'. Haldane's The Causes of Evolution (1932, p.56) also contains a highly relevant passage: 'the change from one stable equilibrium to another may take place as the result of the isolation of a small unrepresentative group of the population, a temporary change in the environment which alters the relative viability of different types, or in several other ways...'. Unfortunately I do not know the exact dates of publication of Haldane's book and Wright's article of the same year, so it is not clear whether Wright could have seen it before writing his article. Wright had certainly read an article by Haldane of 1931 on 'Metastable Populations', which also discusses the theory of multiple equilibria, but this article refers only to chance fluctuations in the composition of populations, and not to environmental change, as possible reasons for a switch between alternative equilibria.

Whatever the reasons for Wright's new position on environmental fluctuation, he cannot be accused of playing down its importance. Several times he emphasised it: 'here we undoubtedly have an evolutionary process of major importance' (322), 'it can hardly be doubted that this has been one of the most important causes of evolution' (374), and 'there can be no doubt that a large part, perhaps the major portion, of evolutionary change, is of this character' (562). Nevertheless, it has often escaped the notice of later biologists, who assume that Wright continued to see genetic drift as the only way out of evolutionary stagnation.

Despite Wright's acceptance of, and even emphasis on, environmental change as a possible cause of 'peak shift', in some respects the implications of this new position were not fully assimilated into Wright's evolutionary philosophy. First, Wright might have been expected to rethink his position on the importance of population size and structure. On the face of it, a population of any size - large, small, or medium - may equally be affected by environmental change, and equally likely to shift from one peak to another. If this is so, Wright's belief in the ineffectiveness of evolution in large populations would need to be reconsidered. I am not aware that Wright did so. Second, if environmental change is capable of upsetting the equilibrium, perhaps other factors might also do so. One such factor is migration. If different gene frequencies are able to evolve in subpopulations, through genetic drift or local selective pressures, then migration between subpopulations may upset the equilibrium in some or all of them. Wright's SBT does allow for one particular effect of migration: if one subpopulation happens to have reached a higher selective peak than others, migration from that subpopulation may shift others towards the higher peak. But my point is that any migration between subpopulations with different gene frequencies may break up the existing equilibria and give the opportunity for new, and often higher, equilibria to be attained. It therefore seems that even if new favourable mutations are too rare, and mutation pressure is too weak, shifts between equilibria might occur in three ways: genetic drift in small subpopulations, environmental changes (biotic or nonbiotic) which might in principle affect populations of any size, and migration between subpopulations of any size.

4. The adaptiveness of evolution

I can deal more briefly with this topic because it has been dealt with thoroughly by Provine, who traces the change in emphasis from nonadaptive evolution, even at the level of differences between species, in Wright's early work, to a much stronger emphasis on adaptation in the post-war writings.

The only point I would add is that even in his later writings Wright saw adaptation as occurring mainly through intergroup selection. Selection within a single population, large or small, is in Wright's view ineffective in producing continuing adaptation because any single population will soon become stuck on a suboptimal selective peak. Evolution within subpopulations leads to divergence between them, either through genetic drift or fluctuating environmental factors. Neither of these is adaptive with respect to long term trends. This is obvious in the case of genetic drift, but even selection under fluctuating environment may be regarded as a quasi-random factor. It contributes to long-term adaptation only by providing the variation between subpopulations on which intergroup selection can work: 'In this theory [the SBT], the joint effects of random drift and intrademic selection merely supply raw material for interdemic selection' (618). Some subpopulations will, by chance, have combinations of genes which have the potential to increase fitness in the species as a whole, and these are spread by intergroup (interdemic) selection. The processes which generate diversity between subpopulations may be seen as analogous to mutation in the conventional neo-Darwinian framework: each mutation may have some underlying cause, and is not strictly random in the sense that mutations in all directions are equally probable, but it is random with respect to the long-term adaptiveness of the species as a whole.

It should be evident by now that Wright's SBT is a radical departure from the neo-Darwinism of Fisher, Haldane, and most other theorists of the 'evolutionary synthesis', and it should not be surprising that it has found admirers among such rebels against the synthesis as the punctuationists and the group selectionists of the last few decades.

5. The process of intergroup selection

Despite its importance in the SBT, Wright says little about the process of intergroup (or interdemic) selection. In principle one can envisage three different ways in which groups with higher average fitness could influence the properties of the wider population:

a) one group may become extinct, and a fitter group may then expand into the unoccupied territory

b) one group may move into the territory occupied by another group and displace it without interbreeding

c) members of one group may migrate into the territory of another, and influence its gene pool by interbreeding.

I do not think that Wright ever mentions process (a). In various places he seems to favour either process (b) or (c). In 1931 he says that 'exceptionally favorable combinations of genes may come to predominate in some of the sub-groups. These may be expected to expand their range while others dwindle' (95, see also 152). Since there is no mention of interbreeding, this seems to be closest to process (b). In 1932, on the other hand, he says that successful local races 'will expand in numbers and by crossbreeding will pull the whole species toward the new position' (168). This is closer to process (c). In 1939 he combines both (b) and (c), saying successful races 'by cross breeding with other races, as well as by actual displacement of these, will pull the species as a whole toward the new position' (324). In 1940 he says that successful local races may 'tend to displace all other local strains by intergroup selection (excess migration)' (351). The word 'displace' tends to suggest process (b). Also in 1940 he refers to some groups 'supplying more than [their] share of migrants to other regions, thus grading them up to the same type' (375, see also 423). The reference to 'grading up' may seem to imply a mingling of populations and interbreeding (process (c)). There is of course no reason why both processes should not play a part, as explicitly suggested in 1939. But both face some obvious difficulties. With process (b) it is necessary to explain why there is no interbreeding between the different types. This would be surprising unless some degree of reproductive isolation - i.e. speciation - had already evolved. With process (c) the problem is to explain why interbreeding does not break up the advantageous gene combinations on which the superiority of one group is supposed to rest. The problem is expecially severe if the successful group is initially small in relation to the whole population, as assumed at least in the original version of the SBT, with its reliance on genetic drift. This issue has been studied in several recent assessments of the SBT, the general conclusion being that the process is possible but, like the SBT as a whole, requires rather a lot of quantitative conditions to be met if it is to succeed.

As I mentioned in Part 1 of this note, 'intergroup selection' as envisaged by Wright has little to do with 'group selection' as envisaged by most of its recent advocates. Wright does not suggest that successful groups have evolved adaptations for group living, or that their members behave 'altruistically' towards each other. His claim is rather that the subdivided population structure allows some groups, by chance, to form combinations of genes that are advantageous to individual fitness. The higher mean fitness of the groups is the resultant of these individual fitness advantages.

However, in some of his later writings Wright does mention the possibility of the evolution of altruistic social traits through intergroup selection, for example: 'characters may be fixed [through random drift in small subpopulations] that are favourable to the group as a whole even though disadvantageous in individual competition' (536, see also Tax p.466). The problem, of course, is that this requires migration from other groups to be near zero if the 'altruistic' groups are to survive for more than a brief period without being undermined by freeloaders.

6. The three phases of the shifting balance

Finally, in his later writings on the SBT Wright often refers to three 'phases' of the shifting balance. Like the term 'shifting balance' itself, the 3-phase formulation seems to have been first used in the article of 1970 on 'Random drift and the shifting balance theory of evolution'. The phases are described as the 'phase of random drift', in which gene frequencies in each deme drift around the current selective peak; the 'phase of mass selection', in which a deme has drifted into the zone of attraction of a new selective peak, and moves rapidly towards it under the influence of selection; and the 'phase of interdemic selection'.

The explicit distinction between three phases seems to be new in 1970, but it is essentially a clarification of the process which had been implicit in various writings at least since 1932. I won't comment further on the substance of the three phases, which have already been discussed under various headings.


The purpose of this Note has been mainly to analyse the various aspects of the SBT in their chronological development, and not to assess its credibility. A few years ago I drew attention to some recent controversy, mainly in the journal 'Evolution', by biologists pro and con the SBT. These discussions still seem to be relevant, but I note that some aspects of the SBT (or of Wright's philosophy of evolution more generally) have not been sufficiently recognised. One is the important change in 1932 when Wright recognised that environmental fluctuations, as well as genetic drift, could have lasting effects on the genetic equilibrium of a population. Despite Wright repeating this point on several occasions, it has been widely overlooked (Dobzhansky being a notable exception, and Provine a more recent one). There is some excuse for this if, as I have argued, the implications of the change were never sufficiently absorbed by Wright himself. The second point is that Wright was consistently negative towards the prospects for new favourable mutations. I have suggested that his comments involve an ambiguity between the rarity of new favourable mutations among all mutations, which is not disputed, and the rarity of occurrence of any such mutations, even in a large population and over a timescale of many generations. Wright's negative conclusions are only valid if such mutations are rare in both senses. His position implies that the differences between populations, whether closely related species or subpopulations of the same species, will arise mainly by different epistatic combinations of existing genes, rather than by the selection of new variants. This is in principle testable.

This is the last of my planned notes on Sewall Wright, and it is a relief to get to the end of the journey. I will not attempt any overall assessment at this stage, but I will probably prepare a post giving links to all the notes in the series, as well as to related notes on Fisher and Haldane.

Note 1. See GTNS p.107, but note that according to Fisher, if the effect of the mutation is small (say, no more than 1 percent of the standard deviation of the trait), even mutation rates as low as one in a million may be sufficient to overcome the initial selective disadvantage and eventually push the mutation into a frequency where it is favoured by selection.

Note 2. The reference is evidently to the section in GTNS on 'The nature of adaptation'. What Fisher shows, given his assumptions, is that:

a) other things being equal, a smaller mutation is always more likely to be advantageous than a larger one. (As Kimura pointed out much later, this is partially offset by the consideration that the size of any advantage is likely to be greater for a larger mutation, and this affects the probability that it will survive in the population. Overall, mutations with effects somewhat above the minimum size have the highest probability of survival.)

b) for any given size of mutation, the probability of being advantageous is lower the more aspects of fitness are affected by it.

Using a very schematic geometrical model, Fisher quantifies the probability that mutations of a given size will be advantageous. It is assumed in the model that the present position of the organism is at some distance from a local optimum. The probability that a mutation will be advantageous is inversely related both to the size of the mutation and to the square root of the number of dimensions of fitness affected. For very small mutations the probability is close to 1/2, declining to zero for mutations with an effect more than twice the distance between the starting point and the local optimum (this zero probability being an assumption built into the model, rather than proved by it). But note that the probabilities are not always very small, even for mutations with an effect quite substantial relative to the present distance between the organism and the optimum. Also, since the probability declines in proportion only to the square root of the number of dimensions of fitness affected, not to that number itself, the decline is not as rapid as might be feared. Contrary to some popularisations, Fisher does not claim that mutations with very large or complex effects are impossible, or even highly improbable, only that they are less likely to be advantageous than those with smaller and/or simpler effects.

Note 3: Some readers may wonder how this can be reconciled with Haldane's rule of thumb that up to one mutation can go to fixation, on average, in every 300 generations - see my post on Haldane's Dilemma. I think the explanation has two parts. First, Haldane's '300 generations' estimate assumes that a gene under selection starts from a position of balance between adverse selection and mutation pressure, and then becomes favourable due to a change in environment. On this assumption the gene will already have a small but not negligible frequency in the population. Second, the '300 generations' figure does not mean that a single gene under selection goes from rarity to fixation in 300 generations, but rather that, on average, one gene could be fixed in every 300 generations. There is a difference between these two claims. Under typical selection intensities of 1 in 1000, or even 1 in 100, the process of fixation for a single initially rare gene would obviously take longer than 300 generations. Haldane's model assumes that there are a number of genes undergoing selection simultaneously or overlapping with each other. If we imagine, say, 100 genes starting the process of selection at the same time, and all taking 30,000 generations to reach fixation, the average number of genes fixed per generation over the period of 30,000 generations would be 100/30,000 = 1/300, but these would all reach fixation in a bunch at the end of the period. More realistically, if the periods of selection are overlapping in a more-or-less random way, and selection has been in progress for long enough, we would expect any period of, say, a thousand generations to see a few genes reaching fixation, with an average of about 1 per 300 generations.


R. A. Fisher, The Genetical Theory of Natural Selection, 1931, variorum edition ed. J. H. Bennett, 1999.
R. A. Fisher: Natural Selection, Heredity and Eugenics: Including selected correspondence of R. A. Fisher with Leonard Darwin and others, edited by J. H. Bennett (1983).
J. B. S. Haldane, 'Metastable populations', Proceedings of the Cambridge Philosophical Society, 27, 1931, 137-142.
J. B. S. Haldane, The Causes of Evolution, 1932 (reprint ed. E. Leigh, 1990)
William B. Provine, Sewall Wright and Evolutionary Biology, 1986.
Sewall Wright: 'Physiological genetics, ecology of populations, and natural selection', in Evolution After Darwin, vol. 1, ed. Sol Tax, 1960 (Tax). (Article first published in 1959.)
Sewall Wright: Evolution: Selected Papers (ESP), ed. William B.Provine, 1986.
Sewall Wright: 'Random drift and the shifting balance theory of evolution', in Mathematical Topics in Population Genetics, ed. Kojima, 1970.

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