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Fisher's Fundamental Theorem of Natural Selection Steven A TREE vol. 7, no. 3, March 1992 Fisher's Fundamental Theorem of Natural Selection Steven A. Frank and Montgomery Slatkin relationship to Adaptive Land- scapes. We focus on three ques- tions: What did Fisher really mean by the Fundamental Theorem? is Fishes Fundamental Theorem of natural though it were governed by the Fisher's interpretation of the Fun- selection is one of the most widely cited average condition of the species or damental Theorem useful? Why was theories in evolutionary biology. Yet it has inter-breeding group' (Ref. 3, p. 58). Fisher misinterpreted, even though been argued that the standard interpret- Fisher also pointed out that he stated on many occasions that he ation of the theorem is very different from average fitness, measured by the was not talking about the average what Fisher meant to say. What Fisher intrinsic (malthusian) rate of in- fitness of a population? really meant can be illustrated by look- crease of a species, must fluctuate ing in a new way at a recent model for about zero (Ref. 4, pp. 41-45). What did Fisher really mean? the evolution of clutch size. Why Fisher Otherwise, if a species' rate of The standard interpretation of was misunderstood depends, in part, on the increase were consistently positive, the Fundamental Theorem is that contrasting views of evolution promoted by it would soon overrun the world, or natural selection increases the Fisher and Wright. if a species' rate of increase were average fitness of a population at a consistently negative, it would rate equal to the genetic variance in R.A. Fisher and Sewall Wright, two quickly become extinct. fitness. Thus, the average fitness of of the founders of modem evol- These comments about average a population is a nondecreasing utionary theory, fought bitterly for 30 fitness, repeated in different ways quantity. This interpretation is in years. Angry words were exchanged by Fisher, seem to contradict his accord with the intuitively appeal- on a variety of topics, but at the heart own proud claims for his Fundamen- ing idea that natural selection will of the controversy was a clash tal Theorem of natural selection: make species better adapted to between each scientist's vision of 'The rate of increase in fitness of their environments. natural selection and evolution: any organism at any time is equal One problem with the standard on one side, Fisher's Fundamental to its genetic variance in fitness interpretation of the Fundamental Theorem of natural selection, and at that time' (Ref. 4, p. 37), or Theorem is that, although Fisher on the other side, Wright's Adaptive 'The rate of increase of fitness of claimed that his theorem is exact, Landscape. any species is equal to its genetic realistic models show that average Fisher detested the Adaptive variance in fitness' (Ref. 4, p. 50). fitness does not always increase. Landscape formulation of natural If 'fitness of any species' is inter- For example, Moran 8 showed that selection, an opinion that he re- preted as 'average fitness', which average fitness can decrease when peated on many occasions. Al- is the usual interpretation, these selection acts on two linked loci that though Wright's original formu- statements certainly suggest that have epistatic effects on fitness (see lation' of the Adaptive Landscape the average fitness of a species Box 1 for definition of genetical in 1932 is obscure', the Adaptive increases steadily over time, just as terms). in general, the average Landscape soon came to mean that in Wright's Adaptive Landscape. fitness of a population increases at gene frequencies change as if pro- Everyone, including Wright, in- a rate equal to the additive variance ceeding up a hill of increasing terpreted Fisher as saying that the in fitness only when certain restric- average fitness. Wright justified this Fundamental Theorem was about tive conditions are met. For ex- metaphor by showing that under the average fitness of a species. In ample, there must be no epistasis, some conditions the rate of change fact, Wright often quoted the Fun- no linkage disequilibrium and no in gene frequency depends on the damental Theorem in support of his frequency dependence, among a steepness of the gradient in average gradient formulation of the Adapt- variety of other conditions about fitness, d W/dq, where W is the ive Landscape, dWIdq. As pointed mating and nonadditive genetic average fitness of the population out by Ewens5, Wright said in interactionsw.". and q is gene frequency. his last paper: 'The effects Ion Given that the standard interpret- Fisher objected to the idea that gene frequencies in an Adaptive ation was generally believed to be natural selection alone would have Landscape' may be calculated using what the Fundamental Theorem was any simple effect on the average Fisher's fundamental theorem of about, several theoreticians begin- fitness of the population. For natural selection' (Ref. 6, p. 118), ning with Kimura' 2 have focused on example: 'In regard to selection in spite of the fact that Fisher had the question of when the average theory, objection should be taken to already rejected this interpretation: fitness of a population increases Wright's equation (the expression Thave never, indeed, written about at a rate approximately equal to dW/dql principally because it repre- W114/ l and its relationships . the the additive variance in fitness. sents natural selection, which in existence of such a potential func- Recently, Nagylaki n has obtained reality acts upon individuals, as tion I i.e. a function nondecreasing in upper bounds on the error made in time' W is not a general property assuming that the average fitness of natural populations, but arises increases at a rate equal to the only in the special and restricted additive variance in fitness. Steven Frank is at the Dept of Ecology and Evol- utionary Biology, University of California, Irvine, cases Wright . considers' (Ref. 7, However, Price14 and Ewens5 CA 92717, USA; Montgomery Slatkin is at the p. 285). have shown convincingly that the Dept of Integrative Biology, University of California, Here, we reconsider Fisher's standard interpretation is very dif- Berkeley, CA 94720, USA. Fundamental Theorem and its ferent from what Fisher had in mind, 92 © 1992. Elsevier Science Publishers Ltd (UK) 0169-5347/92/S05.00 TREE vol. 7, no. 3, March 1992 Box 1. Glossary The following definitions provide a sense of how these concepts are typically used. Formal definitions for many of the following and they propose an alternative stress that this term is often negative terms can be found in Falconer9 or in other introductory texts of genetics. explanation that fits all the facts. because natural selection increases Average effect of a gene substitution can Price and Ewens provide formal fitness but the total change in fitness be illustrated by considering a locus that has derivations of the Fundamental is usually close to zero. two alleles, A and a. Let one randomly chosen . Theorem consistent with Fisher's a in the population be changed to A, and. Is Fisher's interpretation useful? measure the phenotypic difference caused by interpretation: the theorem is exact the change. The average change, measured and general in every way that Price" and Ewens5 both ex- over all a's taken one at a time, is the aver- Fisher claimed. Our purpose here pressed deep disappointment in age effect of a substitution by a. The average is to provide a sense of what the Fundamental Theorem, and effect of a substitution at this locus is the Fisher meant rather than to repeat Nagylaki" has recently echoed difference between the average effect of a substitution by a and a substitution by A. The the mathematical and historical that disappointment. Price states average effect is the foundation for the con- analyses provided by Price and (Ref. 14, p. 139): 'A ... !gravel cepts and mathematics of the Fundamental Ewens. defect is the matter of the shifting Theorem, and is also the basis for many of Fisher realized that the average standard of 'fitness' that gives the Fisher's criticisms of Wright. paradox of M 'WI tending always to Environment is an important aspect of the fitness of a group is a useful Fundamental Theorem. The average offect quantity only in the wider context increase and yet staying generally of an allele takes into account all poss- of other groups and the environ- close to zero. Much more interesting ible genetic and environmental effects by ment: a species in the context would be a theorem telling of measuring the phenotypic change of a sub- of competitors, diseases and food increase in 'fitness' defined in terms stitution in the context of the current popu- lation. When discussing his Fundamental availability; a genotype in the con- of some fixed standard.' Theorem, Fisher chose to lump genetic ef- text of a particular mix of competing It is a matter of taste whether a fects such as gene frequency and dominance genotypes within the population; particular partition of evolutionary with physical effects such as weather into a or an allele in the context of the change is useful; all of the com- single 'environmental' term. ponent parts must ultimately sum to Additive genetic variance is a measure for frequency of competing alleles at a the potential amount of evolutionary change locus. the same total change. We believe caused by natural selection. For a single To explain the notion of fac- that Fisher's partition is useful locus, the measure is obtained by multiplying tors extrinsic to the group being – indeed fundamental. To sup- the square of the average effect of a substi- analysed, Fisher referred to all port this conclusion, we analyse an tution by the variance in gene frequency – essentially, the amount of change caused by extrinsic forces as the environment.
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