Monday, September 01, 2008
My series of posts on the work of Sewall Wright is now approaching its (anti?)climax. The next post, on the shifting balance theory, should be the last. The present note deals with a closely related subject. Wright introduced the concept of the 'adaptive landscape' largely in order to illustrate the shifting balance theory. It does however have great interest in its own right, and there is a substantial literature on the concept of adaptive landscapes. [Note 1]
Wright's own treatment of the subject has attracted some controversy following the biography of Wright by William B. Provine. Provine pointed out that Wright used two different interpretations of the 'landscape', which in Provine's view were inconsistent with each other: 'One of Wright's two versions of the fitness surface is unintelligible, and even if one were to escape this problem and put the gene combinations on continuous axes, the two versions would be mathematically wholly incompatible and incommensurable, and there would be no way to transform one into the other' (Provine, p.313). I believe that Provine's criticisms are overstated, but he was right to point out that Wright's concept of the landscape is problematic. This note examines the issues. It is long.
The general concept of the adaptive landscape is that the genetic constitution of an individual or a population can be represented by a point in a space of many dimensions. The biological fitness associated with that genetic constitution can then be represented by a measurement along a further dimension. The fitness 'heights' of different genetic constitutions form a quasi-surface. Points or areas of high fitness can be described as 'peaks', points of low fitness as 'pits', 'troughs', etc, and more complex configurations as 'ridges', 'valleys', 'passes', etc. The genetic evolution of a population can be represented by the movement of points around the 'landscape'. Subject to certain provisos, under the influence of natural selection a population will move up the steepest available slope towards areas of higher fitness. If the population reaches a local peak - a point surrounded in all directions by lower ground - evolution will stop until circumstances change in some significant way.
Wright believed that in general there will be many local peaks of fitness in the landscape, often differing in height from each other. It is therefore likely that under the influence of natural selection alone, and under constant environmental conditions, a population will get 'stuck' on a peak which is not the highest in the landscape. Evolution would be quicker, and more beneficial to the species, if there were some means of shifting populations away from these suboptimal local peaks. According to the shifting balance theory in its original form, the only way of moving a population from a peak, other than a large and permanent change in environmental conditions, is by genetic drift, which enables a population to cross 'valleys' of relatively low fitness. This is most likely to occur if the species is divided into a large number of small, partially isolated, subpopulations. Some subpopulations will then by chance find themselves on higher peaks of fitness, and their greater reproductive success will result in a net gene flow into other subpopulations, raising the general fitness of the species and enabling evolution to continue. Wright later abandoned his original exclusive emphasis on genetic drift, but this has not always been sufficiently emphasised. I will deal with this more fully in the final post.
To consider the 'landscape' in more detail:
Wright's first known use of the landscape concept is in a letter of February 3 1931 to R. A. Fisher, quoted in Provine's biography (p.272). Wright's first published account came in a short paper in 1932. Thereafter he discussed the concept in most of his general surveys of population genetics and evolutionary theory. I cannot claim to have read all of Wright's scattered papers, and I have relied heavily on the collection 'Evolution: Selected Papers', (ESP) edited by Provine with Wright's co-operation. Unfortunately, by the operation of Sod's Law, probably the best account of the 'landscape' is not included in ESP (it is in a 1960 Darwin symposium volume edited by Sol Tax). Surprisingly, Wright's huge 4-volume treatise on Evolution and the Genetics of Populations has no systematic treatment of the landscape concept, though various of its component parts are discussed. Finally, a special interest attaches to a paper of 1988, since this came after the publication of Provine's biography. For details see the references.
Wright himself seldom if ever uses the term 'landscape'. In fact, I have not found a single example of it. He does on one occasion (ESP p.625) use the similar term 'topography', but in general he uses two other terms: the field of gene combinations, and the surface of selective values. For convenience I will continue to use the term 'landscape', but anyone searching in Wright's own works should look for 'fields' and 'surfaces', not 'landscapes'. The popularity of the term 'landscape' probably stems from its use in George Gaylord Simpson's Tempo and Mode in Evolution (p.89) and The Major Features of Evolution (p.155), which were more widely read by biologists than Wright's own works. For the same reason, the landscape concept is often given interpretations which derive from Simpson rather than Wright, in which the 'peaks' of the landscape represent either locally optimal phenotypes, or ecological niches. These interpretations are compatible with those of Wright, but not the same as Wright's own landscape, in which the dimensions other than fitness always represent genetic rather than phenotypic variables.
The Number of Dimensions
Wright's landscape has one dimension for fitness, and others representing the genetic constitution of an individual or a population, which I will call the genetic dimensions. At least one genetic dimension is required for each distinct locus at which more than one allele is present in the population. A position along a genetic dimension represents either the number of copies of an allele (in the case of an individual) or the frequency (proportion) of that allele in the population. Since the number of genes at a locus in an individual must add up to the relevant ploidy (one for a haploid, two for a diploid, etc), and the frequencies of different alleles at a locus in a population must add up to 100%, it is only necessary to specify the number or frequencies of (n - 1) alleles at each locus, since the number or frequency of the n'th allele will then be determined as a residual. It is therefore sufficient to have (n - 1) dimensions for each locus, where n is the number of alleles in the population at that locus. The total number of genetic dimensions is the sum of the (n - 1)'s for all loci. The gene pool of any species probably has at least 1,000 loci at which there are two or more alleles present in the population. The number of genetic dimensions is therefore at least 1,000, and usually much larger.
It might be supposed that the genetic dimensions would be represented diagrammatically by Cartesian axes at right angles to each other (orthogonal axes). For loci with more than two alleles this would however have the disadvantage that the alleles would not be treated symmetrically. For example, with 3 alleles (A, B and C) represented on two orthogonal axes, if one axis represented the balance between A and B, and the other axis the balance between A and C, the balance between B and C could be inferred but would not be directly shown in the diagram. Wright therefore suggests in several places (e.g. Tax p.431-2) that the axes need not be orthogonal, so that for example in the case just mentioned the pairs A-B, A-C, and B-C could be represented by the sides of an equilateral triangle. In practice, Wright usually illustrates his concept with diagrams showing two orthogonal axes for genetic dimensions and one axis (height) for fitness, which on a flat page can be indicated either by perspective or by contours on a map.
The Number of Genotypes
The number of possible genotypes is vast. With at least 1,000 loci, even if only two positions were possible at each locus, the total number of genotypes representable in the system would be at least 2^1000. Wright himself gives a more generous estimate of 10^1000. Either way, the number is super-astronomical. As Wright points out, it is larger than the number of elementary particles in the universe. It is certainly far greater than the number of individuals in any species. It follows that most of the positions in the genetic 'space' of any actual species will be empty. Even if for most loci a single allele has a high frequency in the population, the genotypes of individuals will be very sparsely scattered over the space. Apart from clones, it is unlikely that two individuals will ever have exactly the same genotype.
Genotypes or Frequencies?
As Provine showed clearly in his biography (pp.307-17), Wright used two different interpretations of his genetic dimensions. In one interpretation, which I will call the genotype version, a position along a genetic dimension represents the number of alleles of a certain type in an individual genotype. For example, if the dimension represents the allele pair A-a at a diploid locus, a position at one end of the axis would represent the homozygote AA, a position at the other end would represent the homozygote aa, and a position in the middle would represent the heterozygote Aa. [Note 2] The whole genotype of an individual would be represented by a single point in the many-dimensional genotype space, and the allele composition of the individual at a given locus could be 'read off' from the projection of that point onto the relevant axis. The genetic composition of a population could then be represented by a number of points, one for each member of the population, at appropriate positions in the 'space'.
In the alternative interpretation, which I will call the frequency version, a position along a genetic dimension represents the proportion of alleles of a certain type in a population. For example, if the dimension represents the allele pair A-a at a diploid locus, a position at one end of the axis would represent fixation (100% frequency) for the allele A, a position at the other end would represent fixation for the allele a, and a position in between would represent an intermediate frequency, e.g. 60% A and 40% a. The entire genetic composition of a population could be represented by a single point at an appropriate position in the 'space'. It must not be inferred that all members of the population would have the genotype represented by this point under the genotype version. In fact, unless most loci are fixed for a single allele, it is extremely unlikely that any individual in the history of the species would have exactly that genotype.
There is no doubt that Wright uses both of these interpretations. In his first known account (in the 1931 letter to Fisher) he uses only the frequency version, but in the first published account (1932) he uses only the genotype version. From 1935 onwards his publications most often use the frequency version, but the genotype version is never entirely lost, and the two interpretations may even appear in the same work. (See Note 3 for my own attempt at a chronological listing.)
But is there really any inconsistency in the two different interpretations? It is evidently quite possible for a position along an axis to represent either an allele number or an allele frequency, and there is no fundamental reason why the two interpretations should not be used at different times, or even at the same time, provided the differences between the two interpretations are properly noted. There is of course a danger that the use of two different interpretations will lead to confusion, or even to actual error if theorems or generalisations which are valid only for one interpretation are applied to the other one. I am not aware that Wright himself ever falls into definite error, but his explanations are often unclear. According to Provine (p.311) , when he first pointed out the different interpretations to Wright, the latter was somewhat taken aback, and did not realise that he had been switching between them. Wright's 1988 paper, which includes a response to Provine's critique, is surprisingly insouciant about the issue, effectively taking the line: 'Why worry, it's only a diagram.'
Provine does have other criticisms, but before discussing these it will be useful to look at the remaining dimension of the landscape, that of fitness.
The Dimension of Fitness
In view of its importance Wright says surprisingly little about the nature or definition of fitness. In his first presentation of the landscape concept he says only that the entire field of gene combinations can be 'graded with respect to adaptive value under a particular set of conditions' (ESP p.162) . The word 'graded' seems to imply a relative measure of fitness, which is consistent with Wright's general approach and that of many other population geneticists, including Haldane. For most purposes a relative measure is sufficient. Wright does however recognise that an absolute measure, such as Fisher's Malthusian parameter, may be useful or necessary for some purposes, for example in dealing with overlapping generations (Tax, p.433).
A more important issue is the question of the relevant 'set of conditions', on which Wright is again disappointingly vague. Clearly the fitness of a given genotype will depend in part on the environment. It appears that Wright intends fitness to be averaged over the usual range of environments in which a species finds itself. But it would be reasonable to object that conditions will be constantly changing, so that there is no such thing as an 'average' environment except at a moment in time. Even at a moment in time the environment will vary in different parts of a species' geographical range. The most important aspect of a species' environment is often not the inorganic factors (climate, etc) but the organic or biotic environment of competitors, food, predators, parasites, and pathogens. These differ fundamentally from the inorganic environment because they are themselves evolving by natural selection, sometimes in response to the species of interest. For example, a new mutation occurring among any of the pathogens affecting a species may dramatically change the fitness of all the genotypes of that species. Wright does in various places recognise that the organic and inorganic environment are liable to change, but he tends to present this as a factor leading to movement of the species around the 'landscape', when it could arguably be seen as invalidating the concept of the landscape altogether. One of the essential features of a landscape, in the ordinary sense, is that it has at least a modicum of persistence through time.
For an individual member of a species, the other members of the same species are an important part of its biotic environment. This raises the possibility that the absolute or relative fitness of different genotypes may vary according to the genetic composition of the species population. Notably, this would be the case with various forms of frequency-dependent selection, for example, if pathogens or predators attack the most common variants. I cannot find any discussion of the issue in Wright's early papers. Under the first published (1932) account, which presents only the genotype version, it seems to be assumed that each genotype can be assigned a fitness regardless of gene frequencies. In the first published account of the frequency version (1935), Wright deals mainly with certain special cases, which again seem to be independent of frequency. In two more general presentations (1939 and 1940), I still find no clear statement. Finally, in 1942 (in an article based on a lecture given in September 1941) we find an explicit assumption that 'the relative selective values of these genotypes are independent of their frequencies' (ESP p.472). It may be relevant that in 1941, in a paper referenced in Wright's 1942 article, R. A. Fisher had sharply criticised Wright's 1940 presentation. Whatever the reasons, in later discussions, notably Tax and EGP, Wright gave more attention to the issue of frequency-dependence (see especially Tax pp.443-49). Generally speaking, frequency-dependence can involve either positive or negative feedback, in the first case driving alleles to fixation, and in the second often leading to a balanced polymorphism. If the latter case is common in nature, it would tend to make the landscape concept more difficult to interpret (see further below).
Is there a fitness surface?
On many occasions Wright refers to the values of the fitness dimension as forming a 'surface'. This would normally imply at least an approximate continuity of values for fitness with respect to changes along the other dimensions. Provine has pointed out that under the genotype version, the fitness values cannot be continuous. The genotype values themselves form a lattice of discrete points, not a surface, so the associated fitness values must likewise be discontinuous.
I think this objection is somewhat overstated. First, as a matter of textual detail, Wright seldom uses the term 'surface' when he is referring to the genotype version; in particular, he does not use the term in his first (1932) published account. But on at least one occasion (in 1939, ESP p.318), he does unambiguously refer to a fitness surface with respect to genotypes; also, as Provine points out, even in the 1932 account Wright uses a diagram which seems to imply a continuous surface. Provine's criticism therefore needs to be met, but I think it is not as serious as Provine suggests. It is true that the genotype values form a lattice of points rather than a surface, but it is possible to define a 'distance' between these points by the number of gene substitutions needed to go from one point to another. We can reasonably describe some points as being closer than others. It would then also be reasonable, if not mathematically exact, to say that the associated fitness values approximate to a surface, provided that small differences in distance correspond to small differences in fitness. The real objection, it seems to me, is not that the surface is not strictly continuous, but that the necessary correspondence between fitness and distance does not exist. Genotypes which differ only in a single allele may differ widely in fitness, for example if the heterozygote at a given locus has above-average fitness, whereas the recessive homozygote is lethal. I do not see any basis for an assumption that differences in fitness correspond, even loosely, to the number of genetic differences between two genotypes.
I suggest that the following picture is more plausible. A very large part of the 'genotype space' must correspond to zero fitness, since it would involve combinations of rare disadvantageous alleles which are unlikely ever to be combined in reality. Only a small 'corner' of the space is inhabited by actual genotypes. Most of these will have rather similar average fitness, equivalent to producing around two surviving offspring (by sexual reproduction), since, on average, this is what most genotypes actually achieve under their normal circumstances. (If they did not, the population would soon die out.) Among these mediocre genotypes there will be a scattering of super-fit types, and a larger scattering of low-fitness types. The geometrical picture is that most of the landscape would be flat, with uniformly zero fitness, rising gently up to a small inhabited plateau of mediocre fitness, in which there are numerous 'holes' corresponding to genotypes with low fitness (e.g. lethal recessives) compared to their immediate neighbours. [Note 4] There will also be scattered pimples or wrinkles of modest height representing clusters of genotypes containing advantageous genes that are still in the process of selection, and shallow depressions representing mildly disadvantageous genes. But because it contains numerous 'holes' - isolated genotypes or groups of genotypes with fitness much lower than their neighbours - the landscape is not even approximately a continuous surface.
If now we turn to the frequency version, there are better grounds for regarding the fitness surface as continuous. In the frequency version each point in the genetic space corresponds to a certain set of allele frequencies at each locus. Provided we make certain assumptions about the mating system and linkage (usually random mating and zero linkage), each array of allele frequencies will be associated with an array of all possible genotypes, each with a definite probability of occurrence. The mean fitness associated with a given point in the frequency space will therefore also be defined. As the point moves around the space, the genotype probabilities will vary continuously, and so will the average fitness, since the value of ab + cd varies continuously if a and c vary continuously, for any fixed values of b and d. It is true that in a finite population the allele frequencies cannot vary with strict mathematical continuity, since they are ultimately fractions with the population size as a denominator, but unless the population size is very small, the fitness surface will approximate to continuity.
What is a fitness peak?
The idea of a fitness 'peak' is central to Wright's use of the 'landscape' concept. So what exactly is a fitness peak? Characteristically, in introducing the term (in 1932) Wright does not formally define it, and his meaning has to be inferred from what he says about it.
This is one issue where it is important to distinguish between the genotype and frequency versions of the landscape. With the genotype version, the definition of a fitness peak is relatively straightforward. If a genotype has higher fitness than any genotype which can be derived from it by substituting another allele at a single site (including e.g. substituting a homozygote for a heterozygote at a given locus), then it may be described as a local fitness peak. So far as I am aware, this is how Wright always uses the term 'peak' under the genotype version.
Under the frequency version matters are less clear. We could, of course, stipulate that a set of frequencies is a local peak if any small frequency change at a single locus would reduce the mean fitness of the population. But this would exclude the reasonable possibility that frequencies may change slightly but simultaneously at more than one locus, which might increase mean fitness even though no single-locus change would do so. The natural definition of local fitness peak implied by these considerations is that a set of frequencies is a local fitness peak if no combination of small simultaneous frequency changes, at any number of loci, would increase mean fitness. Geometrically, this is equivalent to stipulating that a local fitness peak is immediately surrounded by downward slopes of fitness in all 'directions' in the genetic space. Probably this intuitive concept could be defined more precisely in terms of the 'principal directions' of differential geometry, but I am not aware that Wright himself ever took this approach. [Note 5] In practice, Wright deals mainly with specific cases where the intuitive meaning of a fitness peak is sufficiently clear.
How many peaks?
One of Wright's fundamental claims about the landscape is that it has numerous local peaks. Moreover, many of these have a different fitness 'height'. To give some examples (all page references to ESP), he claims that the number of peaks is 'many' (9, 483), 'enormous' (163, 370), 'large' (226), 'inconceivably great' (230), 'multiple' (318), 'innumerable' (348, 554), and even 'virtually infinite' (535). He also insists that many of these peaks will have a different selective value (see the cited or nearby pages for examples). Without these claims, the landscape concept has little interest. The basis of the claims therefore needs to be examined.
In his original 1932 presentation Wright used a simple probabilistic argument for the existence of numerous peaks. The number of possible genotypes is vast, so even if only a tiny proportion of them are local optima, the number of local optima would still be very large: 'With something like 10^1000 possibilities it may be taken as certain that there will be an enormous number of widely separated harmonious combinations. The chance that a random combination is as adaptive as those characteristic of the species may be as low as 10^-100 and still leave room for 10^800 separate peaks....(ESP p.163)'.
This is a dubious argument. It may be compared to a common argument for the existence of intelligent life elsewhere in the universe. There are around 10,000 billion billion stars in the universe, so even if the proportion of stars with planets supporting intelligent life is tiny - say, 1 in 10,000 billion - there would still be an enormous number of such stars. But consider the following counter-argument. It is plausible that the emergence and survival of intelligent life requires a moderately large number of conditions - say, at least 100 - to be met. It is also plausible that these conditions are largely independent, and individually quite improbable - say, with a probability of only 1 in 100. But with these assumptions, the probability that all of the necessary conditions are met in any given case is less than 1 in 1/100^100. This is vastly less than 1 in 10,000 billion billion, so rather than expecting there to be a large number of stars with planets supporting intelligent life, it would be a miracle if there are any at all. In reality, neither argument goes much further than establishing the bare possibility of the conclusion. Similarly, in the case of selective peaks, the sheer number of possible genotypes is in itself not a strong argument for the existence, rather than the bare possibility, of numerous different peaks.
Wright does later present better arguments for the existence of multiple peaks. By far his most common example is that of a quantitative trait controlled by several loci where the selective optimum for the trait is at an intermediate value, i.e. neither the highest nor the lowest that can be produced by the various possible combinations of alleles. In this situation it is likely that the optimum intermediate value of the trait can be produced by different allele combinations. The effect of an allele on fitness (not necessarily on the quantitative trait itself) is epistatic, i.e. dependent on the combination of other genes in the genotype. Which of the relevant alleles are favoured by selection may then depend on the accident of which allele at a locus happens to be most frequent when selection begins, with all other alleles at the locus being driven to extinction. This example is used repeatedly: ESP pp.247, 310, 319, 370, 477, 626, Tax p. 450, EGP vol. 1 pp.59-60.
The theoretical possibility of multiple selective peaks in this situation has been generally recognised. As I pointed out in a post on R. A. Fisher and epistasis, it was recognised by Fisher in 1930. It was also noted by J. B. S. Haldane, who is sometimes mentioned by Wright in this context. Indeed, a diagram used repeatedly by Wright to illustrate the point (e.g. ESP pp. 310, 371) looks suspiciously like an adaptation of one used by Haldane (Causes of Evolution, p.107).
It should be noted that the example of an intermediate optimal phenotype applies to both the genotype and frequency versions of the landscape concept. Provine has claimed that the two versions are 'mathematically wholly incompatible and incommensurable, and there would be no way to transform one into the other' (Provine, p.313). Like his other criticisms, I think this one is overstated. In at least one important class of cases a local peak under the genotype version will be a local peak under the frequency version as well. This is where the local optimum genotype is homozygous at all loci (or where the organism is haploid). In this case, if all the alleles of the optimum genotype are fixed (i.e. have a frequency of 100%) in the gene pool, all genotypes produced from the gene pool will be identical, and will have the local optimum value. Any change in frequencies (including simultaneous changes in several frequencies) can then only occur by mutations, producing a small proportion of alternative alleles. Assuming random mating and zero linkage, the genotypes produced from the new gene pool will usually differ from the local optimum genotype at no more than a single locus. But by definition these are all less fit than the local optimum, so the change in frequencies will be selected against. Genotypes which differ from the local optimum at more than one locus are indeed possible, and may be fitter than the local optimum, but they will occur so rarely that they can usually be neglected. The frequency array in which all the alleles of a local optimum genotype are fixed in the population will therefore usually be a local peak under the frequency version.
If the optimum genotype is not homozygous at all loci, I think Provine is right that there is no easy transition from the genotype version to the frequency version. For any locus that is heterozygous in the local optimum genotype, the heterozygote is most likely to be produced by a 50:50 ratio of the relevant alleles in the population. Let us suppose that the population is fixed for all the homozygous alleles in the optimum genotype, and has a 50:50 ratio for all the heterozygous alleles. Unlike the case where all loci are fixed, this frequency set will produce a multiplicity of genotypes. If there are more than a few heterozygous loci in the optimum genotype, only a small proportion of the genotypes produced from the frequency set will actually have the optimum genotype. (At any heterozygous locus a 50:50 frequency will produce 50% heterozygotes, so if there are n independent heterozygous loci the proportion of genotypes that are heterozygous at all the relevant loci will be (1/2)^n, which rapidly becomes negligible as n increases.) There is no guarantee that this frequency set will be a local fitness optimum (as defined under the frequency version), since this will depend on the fitness of numerous different genotypes, whose mean fitness may well be higher at some other nearby point in the frequency space. It all gets very complicated. If we also take account of frequency-dependent fitness, it is even messier, since there may be no such thing as a local optimum genotype that remains optimal under all frequency arrays.
The case of optimum fitness of a trait with an intermediate value does however go some way towards vindicating Wright's confidence in the existence of numerous local peaks. Assuming that there are several such traits which are genetically independent of each other, and of other loci, this may lead to a very large number of local optimum genotypes. With at least two independent optima for each trait, the total number of local optimal genotypes will be at least 2^n, where n is the number of traits. This quickly leads to large numbers: over a thousand for n = 10, over a million for n = 20, over a billion for n = 30, and so on. But there is a snag. Selection for an intermediate value of a trait will, if it is successful, always produce much the same phenotype. For example, if the optimum length of a canine tooth is 1 inch, selection will tend to produce that length of tooth even if different combinations of alleles are involved. In this case there will be multiple peaks in the genetic landscape, but they will all be of much the same 'height' in the fitness dimension. This would take much of the interest out of the concept. Wright recognised this snag at least from 1935 onwards. His answer to the problem was to emphasise that most genes have multiple (pleiotropic) effects, and that the system of peaks relative to one character is therefore not independent of that relative to another (ESP p.230, 320, etc.) In some places Wright seems to imply that the allele frequencies may be fixed at an arbitrary peak by selection for the optimal value of one trait, leaving the effects on some other trait varying and often suboptimal (e.g. ESP p.595, but he is not explicit). But this is doubtful. Suppose for example that an allele combination which determines the length of the canine teeth also affects the incisors. If two such combinations produce the same optimum length of canines, but different lengths of incisors, there will be selective pressure to bring the latter towards its own optimum. In this situation there may well be genes at other loci that are capable of modifying the trait. If necessary, new mutations could be selected (not necessarily absolutely new, but newly advantageous.) It is not clear that significantly different (in fitness) multiple peaks will persist for any trait. In at least one place (Tax p.450) Wright himself may recognise this possibility, but it does not seem to have dented his confidence in the existence of multiple peaks with different fitness.
Although the case of intermediate optimum traits is by far the most common reason given by Wright for the existence of multiple peaks, it is not quite the only one. He does occasionally mention the possibility of multiple peaks at a single locus with two or more alleles, if the homozygotes are fitter than the heterozygotes. He also recognises the value of Simpson's concept of phenotypic and ecological peaks, distinguishing two cases: those where different phenotypes give alternative ways of adapting to the same selective conditions, and those where they give ways of adapting to different ecological niches within the same environment (ESP p.555).
Overall, it seems to me that Wright makes out a plausible case that there are likely to be multiple peaks of fitness, but the arguments are not conclusive. If the environment is changing, as it always is, the landscape itself becomes fluid. And if there is widespread genetic polymorphism and/or frequency-dependence in a population, much of Wright's original formulation is (by his own admission) not directly applicable. Provine's criticisms of the two different versions of the landscape concept seem to me overstated, but he is right to question its usefulness as a heuristic device. If several generations of biologists failed even to notice the existence of the two versions, the metaphor of the landscape can hardly be said to have encouraged clarity of thought.
The discussion so far has left some important issues untouched. What are the reasons for expecting a population to 'climb' up a fitness slope? Even if there are many fitness peaks in the landscape, are they all accessible to the population? Will a population get 'stuck' on a peak for any length of time? If so, what circumstances may shift it away from that peak? These questions all go to the heart of the shifting balance theory, so rather than discuss them now I will leave them for my intended note on the shifting balance theory. But before I get there I think it will be useful to cover two supplementary issues which are less directly concerned with Wright's own views. First, what did R. A. Fisher think about all this? And second, apart from Wright's own arguments, what other theoretical or empirical reasons are there for believing in multiple fitness peaks?
Note 1: I do not claim to be very familiar with this literature, which is often highly technical and has little to do with Wright's own formulation. See for example the book by Gavrilets and its extensive bibliography.
Note 2: Wright himself sometimes uses a notation in which only one of the two alleles at a locus is indicated, so that for example if there are three loci with alleles Aa, Bb, and Cc, the genotype AabbCc could be represented by small letters as abbc, and AABbcc as bcc, and so on. The single genotype in which there are no small letters at all is represented by +. Some of Wright's examples are very difficult to follow if these conventions are not understood.
Note 3: 1931 (letter to Fisher): frequency; 1932 (ESP p.163): genotype; 1935 (ESP p.226): frequency; 1937 (ESP p.248): frequency; 1939 (ESP pp.310, 318): both; 1940 (ESP p.347): genotype; 1940 (ESP p.370): frequency; 1941 (ESP p.472): frequency; 1948 (ESP p.535): genotype; 1948 (ESP p543): frequency; 1949 (ESP p. 552): frequency; 1960 (Tax): both; 1977 (ESP p.9): frequency; 1980 (ESP p.626): genotype.
Note 4: Terms like 'hole' and 'wrinkle' must be understood as the n-dimensional analogues of these terms in three dimensions. A 'hole' may itself be a figure with many dimensions.
Note 5: Even in 3 dimensions, containing 2-dimensional surfaces, differential geometry is a tough subject. For an introduction see Aleksandrov, ed, chapter 7.
Works by Sewall Wright
Evolution: Selected Papers (ESP), ed. William B.Provine, 1986
Evolution and the Genetics of Populations (EGP), 4 vols., 1968-1978
'Physiological genetics, ecology of populations, and natural selection', in Evolution After Darwin, vol. 1, ed. Sol Tax, 1960 (Tax)
'Surfaces of selective value revisited', American Naturalist, 131, 1988, 115-23.
A. Aleksandrov et al., eds., Mathematics: its content, methods, and meaning, vol. 2, 1963
R. A. Fisher, 'Average excess and average effect of a gene substitution', Annals of Eugenics, 11, 1941, 53-63.
Sergey Gavrilets, Fitness Landscapes and the Origin of Species, 2004
J. B. S. Haldane, The Causes of Evolution, 1932 (reprint ed. E. Leigh, 1990)
William B. Provine, Sewall Wright and Evolutionary Biology, 1986
G. G. Simpson: Tempo and Mode in Evolution, 1944 (reprint 1984)
The Major Features of Evolution, 1953