Saturday, September 27, 2008

R. A. Fisher and the Adaptive Landscape   posted by DavidB @ 9/27/2008 06:01:00 AM

In my note on Sewall Wright's concept of the Adaptive Landscape I said that I would later discuss R. A Fisher's views on the subject. Some commentators have claimed that Fisher held a definite view on the 'shape' of the landscape. For example, a book by Sergey Gavrilets includes a section on 'Fisher's single-peak fitness landscapes', with the claim that:

In contrast to Wright, Fisher... suggested that as the number of dimensions in a fitness landscape increases, local peaks in lower dimensions will tend to become saddle points in higher dimensions. In this case, according to Fisher, natural selection will be able to move the population without the need for genetic drift or other factors. A typical fitness landscape implied by Fisher's views has a single peak. - Gavrilets, p.36

I think this goes beyond anything that Fisher actually says about Wright's adaptive landscape. There is of course room for debate about what an author's views imply. My own interpretation is that Fisher was sceptical about the value of the landscape concept as such, because both environmental and genetic conditions were too changeable for the metaphor of a 'landscape' to be useful. For Fisher the question of the 'shape' of the landscape therefore did not arise as a major issue, and he had no need to take a firm view on it. I discuss this interpretation below the fold.


As I pointed out in my earlier note, Wright himself seldom if ever used the term 'landscape', so we should not expect to find the term in Fisher either. Wright usually referred to a 'field' of gene combinations, and a 'surface' of selective values. He used these concepts mainly to illustrate his shifting balance theory of evolution. Any comments by Fisher that are relevant to the shifting balance theory could therefore also be relevant to the landscape concept. Even with this broad scope, I can find few published comments by Fisher on the subject. The main ones are in his 1932 review of Wright's paper on 'Evolution in Mendelian Populations', reprinted in Bennett (ed.), his 1941 paper on 'Average excess and average effect of a gene substitution', his 1953 paper on 'Population genetics', and his 1958 paper on 'Polymorphism and natural selection', all available at the Fisher Archives here.

In addition to Fisher's published writings, his correspondence contains a few relevant remarks. Most of his correspondence is accessible at the Fisher Archives, and a good selection of his letters on evolution and genetics is published in Bennett (ed.) Two letters are especially relevant. In February 1931 Wright outlined his landscape concept in a letter to Fisher, quoted in Provine's biography of Wright (p.272). In a reply Fisher made some sceptical comments. Then in 1938 Fisher's colleague E. B. Ford described Wright's concept in a popular book on genetics. In a letter of 2 May 1938 to Ford, commenting on his book, Fisher gave what is probably his longest critique of the landscape concept. The letter is published in Bennett (ed.) (p.201-2) and available at the Fisher Archives, so I will not quote it in full, but it should certainly be read by anyone interested in this issue.

From Fisher's published and unpublished writings we can extract a number of criticisms of Wright's theory.

The interpretation of the dimensions of the landscape

In his biography of Wright, William B. Provine has pointed out that Wright in various places used two different interpretations of the genetic 'dimensions' of the landscape, which in Provine's view are inconsistent (Provine, p.313). In one interpretation the dimensions represent the number of alleles of a given type in an individual genome, while in the other interpretation they represent the frequency of those alleles in a population. Provine points out that in the first interpretation there is properly speaking no continuous surface, but only a lattice of discrete points. He also argues that there is no way of validly transferring conclusions from one interpretation to the other. I believe that these criticisms are somewhat overstated, but it is interesting to find that they are both anticipated by Fisher. In his letter to Ford, Fisher comments that either Ford's description of Wright's views, or the views themselves, are confused, and points out that 'so far as individuals are concerned, there is only a discontinuous aggregate of lattice points, each having its own selective value. There is no continuum of possible values in which we might speak of peaks or maxima.' In his article of 1941, Fisher also criticises one of Wright's own accounts, remarking that Wright 'confuses the number of genotypes, e.g. 3^1000, which may be distinguished among individuals, with the continuous field of variation of gene frequencies.... the large number of genotypes gives no reason for thinking that even one peak, maximal for variations of all gene ratios should occur in this field of variation' (1941, p.378). It is surprising that no-one else seems to have picked up on the apparent confusion in Wright's accounts until Provine's book in 1986.

The number of peaks

As discussed in my earlier post, Wright believed that there are usually a very large number of local fitness maxima in the landscape. Fisher, on the other hand, believed that this was unproven. As noted above, he thought that Wright's view was partly due to confusion between optimal genotypes and optimal frequencies. There is no easy transition from the existence of multiple optima among genotypes to multiple optima among frequencies. I have suggested in my earlier post that in some circumstances (notably where the optimal genotype is homozygous at all loci, and fitness is not frequency-dependent) there can be such a transition, but this is a special case. In general Fisher was correct to regard Wright's argument as inconclusive.

Fisher makes another criticism in his letters to Wright and Ford. In the letter to Wright he says:

In one dimension a curve gives a series of alternative maxima and minima, but in two dimensions two inequalities must be satisfied for a true maximum, and I suppose that only about one fourth of the stationary points will satisfy both. Roughly I would guess that with n factors only 2^-n of the stationary points would be stable for all types of displacement, and any new mutation will have a half chance of destroying the stability. This suggests that true stability in the case of many interacting genes may be of rare occurrence, though its consequence when it does occur is especially interesting and important.

In his letter to Ford, Fisher writes:

In one dimension, as in a road, we pass over an alternative series of hills and dips, so that half of the level points are maxima. In two dimensions, in addition to peaks and bottoms we have cols [i.e. saddle points], which may be regarded as the lowest points on ridges or the highest points on valleys, the curvature of the ground being positive in one direction and negative in another, and the peaks are only about a quarter of the level spots. In n dimensions only about one in 2^n can be expected to be surrounded by lower ground in all directions.

Disregarding for a moment the important comment in the first letter about new mutations, Fisher's thinking seems to be as follows. In each dimension of gene frequencies, only about half of the level points will be maxima. Assuming that the location of the maxima in each dimension is independent of the other dimensions, the probability that a level point will be simultaneously maximal in all dimensions will only be about (1/2)^n, or 1 in 2^n.

As these are just comments in private letters, it is difficult to know how much weight we should put on them. Fisher uses the words 'roughly', 'guess', and 'about', which do not suggest a dogmatic position. The validity of the two key assumptions - that about half of the level points in each dimension will be maxima, and that these will be independent of each other - could be discussed at length. But even at best, Fisher's argument only goes to show that the proportion of the level points which are all-round maxima will fall as the number of dimensions increases (which, incidentally, Wright himself accepted, e.g. at ESP p.226). It does not follow that the number of all-round maxima will remain small. If Fisher believed that this was necessarily the case (which is not clear), he was mistaken. It is quite possible that with an increasing number of dimensions the number of level points may increase faster than the proportion of all-round maxima declines. Indeed, it has been claimed that this is generally the case, but this is also unproven. (I will discuss this more fully in a separate post.)

I have not found any definite statement by Fisher either accepting or denying the existence of multiple optima. As I pointed out in my post on Fisher's views on epistasis, he accepted that there could be alternative stable allele frequencies at particular loci. As far as I can see, Fisher would not have denied in principle the possibility of multiple optima for the genome as a whole, and indeed his 1931 letter to Wright might be interpreted as accepting them as an important if rare phenomenon. But overall I think Fisher's position should be described as deeply sceptical. Wright himself said that Fisher 'did not accept the concept of multiple selective peaks' (Wright,1970, p.23), which is literally true, provided it is not taken as implying outright rejection either.

The mean fitness of the population

In Wright's theory, a population is expected to 'climb' up the slope of the fitness landscape under the influence of natural selection, implying that the mean fitness of the population increases. (Selection may however be offset by migration, recurrent mutation, or genetic drift.) In his publications from 1935 onwards (e.g. ESP p.239, 366) Wright uses a formula which may be expressed as delta-q = [q(1 - q)/2W][dW/dq], where q and (1 - q) are the frequencies of two alleles, delta-q is the single-generation change in q, W is the mean fitness of the population, and dW/dq is the partial derivative of W with respect to changes in q. The formula may be interpreted as saying that the effect of selection on the frequency of a particular allele is proportional to its effect on the mean fitness of the population (as well as to the current frequency distribution q(1 - q)).

In his 1941 paper Fisher strongly criticised this formulation, showing by a somewhat roundabout argument that it depends on the assumption of random mating, and claiming that any attempt to relate selection pressure to mean fitness is 'foredoomed to failure just so soon as the simplifying, but unrealistic, assumption of random mating is abandoned' (p.378). Wright's derivation of his formula, e.g. at ESP p.239, does indeed assume random mating. But Fisher's objection is not just technical: 'In regard to selection theory, objection should be taken to Wright's equation principally because it represents natural selection, which in reality acts upon individuals, as though it were governed by the average condition of the species or inter-breeding group. Early selectionists, following in this respect the language of the earlier theological writers on organic adaptation, often speak of selection as directed 'for the good of the species'. In reality it is always directed to the good, as measured by descendants, of the individual. Unless individual advantage can be shown, natural selection offers no explanation of structures or instincts which appear to be beneficial to the species. Yet in Wright's equation the whole evolutionary sequence would appear to be governed by the principle of increasing the 'general good'.' (p.378) I think this is somewhat unfair to Wright, who did not ascribe any causal efficacy to the fitness of the population as such, but Fisher's statement is important as his first general criticism of 'good of the species' thinking. He makes similar criticisms in his 1953 and 1958 papers. In the 1958 edition of GTNS a section on 'The Benefit of the Species' is added, which has become highly influential on modern evolutionary thinking. Although this new section does not refer to Wright, it is plausible that Fisher's sharpening of his hostility to 'good of the species' thinking was stimulated by his objections to Wright's equation.

New Mutations

As already mentioned, in his 1931 letter to Wright, Fisher argues that 'any new mutation will have a half chance of destroying the stability' of an optimal gene frequency. He makes a similar point in his published review of Wright's 1931 paper on 'Evolution in Mendelian Populations', 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]'.

Their attitude towards new mutations is one of the fundamental dividing lines between Wright and Fisher. Wright repeatedly played down the importance of favourable new mutations, on the grounds that their chance of occurring would be negligible even over long periods (see e.g. ESP pp.150, 165, and 321). He seems to have believed that all possible mutations would already have occurred often enough to be selected if they were favourable, so that the possibility of improvement through new mutations would already have been exhausted. Fisher, in contrast, believed that in large populations even very low mutation rates (say, of one in a thousand million per generation) could not be neglected, and that on an evolutionary time-scale of hundreds or thousands of generations they would provide scope for continuing evolution. It may of course be thought that neither Wright nor Fisher, in the 1930s, knew enough about the nature of genes to have any good basis for their opinions.

Changing Environment

Wright's concept of the adaptive landscape is explicitly based on the assumption of constant environmental conditions. Any change in those conditions involves a change in the landscape itself. Wright was of course aware that environments could change, but he seems to have regarded the 'landscape' as having an underlying continuity of existence even if environmental fluctuations might temporarily change its shape. (I will consider Wright's views on this further in my final post on the shifting balance theory.)

Fisher, on the other hand, believed that environmental change was in one sense irreversible. In the section 'Deterioration of the Environment' in GTNS he emphasised especially the organic environment of competitors, etc:

For the majority of organisms... the physical environment may be regarded as constantly deteriorating... Probably more important than the changes in climate will be the evolutionary changes in progress in associated organisms. As each organism increases in fitness, so will its enemies and competitors increase in fitness; and this will have the same effect, perhaps in a much more important degree, in impairing the environment, from the point of view of each organism concerned. - The Genetical Theory of Natural Selection, Variorum Edition, ed. Henry Bennett, 1999 p.41-2

In his review of Wright's 'Evolution in Mendelian Populations' (reprinted in Bennett, ed.) Fisher again emphasised environmental change:

Professor Wright considers that: 'In too large a freely interbreeding population there is great variability, but such a close approximation to complete equilibrium of all gene frequencies that there is no evolution under static conditions'. He therefore argues that the subdivision of species into partially isolated local races of small size is an important condition not merely, as is obvious, for fission into distinct species, but for progressive evolution. This conclusion is much more debatable [Fisher then makes his point about the importance of new mutations even under static conditions]... Moreover, static conditions in the evolutionary sense certainly do not occur, for, apart from geological and climatological changes, the evolutionary progress of associated organisms ensures that the organic environment shall be continually changing

In short, as several recent commentators have noted, Fisher held a 'Red Queen' conception of evolution, in which organisms have to keep constantly running just to keep up with the competition. This is quite alien to Wright's conception, in which under the influence of selection alone the organic world would soon grind to an evolutionary halt. The extent to which either of these views is correct is a matter for empirical observation. Genetic studies of living populations tend to show continual change, at least at a microevolutionary level, which might seem to support Fisher's view, whereas paleontologists often claim to observe long-term stasis in morphological traits, which might support Wright. This is of course one of the points at issue in the debate over 'punctuated equilibrium', which seems to have petered out through boredom (and the death of some key participants) rather than being resolved. A possible explanation of the apparent conflict of evidence is that traits in hard body parts may be more tightly constrained by stabilising selection than biochemical and behavioural traits. For other suggestions see Williams, Chapter 9.

J. H. Bennett, ed.: Natural Selection, Heredity and Eugenics: Including selected correspondence of R. A. Fisher with Leonard Darwin and others, 1983.
Sergey Gavrilets, Fitness Landscapes and the Origin of Species, 2004.
William B. Provine, Sewall Wright and Evolutionary Biology, 1986.
Sewall Wright: Evolution: Selected Papers (ESP), ed. William B.Provine, 1986.
George C. Williams: Natural Selection: Domains, Levels, and Challenges, 1992.
Sewall Wright: 'Random drift and the shifting balance theory of evolution', in Mathematical Topics in Population Genetics, ed. Kojima, 1970.

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