TREE vol. 7, no. 3, March 1992 Fisher's Fundamental Theorem of 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 , 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 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 , 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 , 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 , 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 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. interesting biological problem, the a rise or fall in gene frequency multiplied by The total change in fitness overtime, evolution of clutch size, and show the amount of variability in genetic material in the context of the environment, that Fisher's view leads to a sig- presently available. E, can be defined as nificantly deeper understanding of Epistasis is the genetic interaction between the problem. In discussing clutch different loci. Epistatic interactions are non- zero if the average effect of a substitution ow = e E size it is not our intention to resolve at one locus depends on the genotype fre- differences among various models, quencies at a seccnd locus. where primes denote one time step but instead to illustrate an appli- Linkage disequilibrium is the statistical or instant into the future, and i W cation of Fisher's theorem. correlation between alleles at different loci. Linkage disequilibrium is nonzero if multi- is the total change in fitness which The problem, presented in a locus gamete frequencies are different from everyone had assumed was the recent paper by Cooke et al.'', is the product of the allele frequencies at each object of Fisher's analysis. that many bird populations have locus. Fisher's theorem, however, was considerable amounts of additive Frequency dependence occurs when the genetic variance for clutch size, and absolute fitness (number of successful off- not concerned with the total evol- spring) of a genotype depends on the utionary change, which depends field studies typically show that frequency of genotypes in the population. at least as much on changes in larger clutch sizes would confer Density dependence occurs when the ab- the environment as it does on higher fitness. The paradox is often solute fitness of a genotype depends on the natural selection. Instead, Fisher stated in terms of the standard number of individuals in the population. partitioned the total change into interpretation of the Fundamen- tal Theorem: average clutch size ATV =1W E– El is expected to increase because + ( 'I Ei I El ( 1 1 there is significant additive genetic application of the theorem. They variance in fitness attributable to cite the standard interpretation of Fisher called the first term the differences in clutch size, but clutch the theorem, including Wright'6 and change in fitness caused by natural sizes in natural populations are Crow and Kimura' 7, to support their selection because there is a con- apparently not changing. conclusion. stant frame of reference, the initial Cooke et al. suggest that the In fact, Cooke et al.'s model is not environmental state E. The Fun- average clutch size may remain fixed an extension but a specific example damental Theorem states that the because of the balance between two of Fisher's partition of evolutionary change in fitness caused by natural forces: natural selection acts to change into a component caused by selection is equal to the additive increase clutch size, but a simul- natural selection and a component variance in fitness. Fisher referred to taneous increase in competition for caused by the environment. Fisher the second term as the change scarce resources reduces clutch size. often referred to the environmental caused by the environment, or more These authors stated that their component as a 'deterioration in the often, as the change caused by the model is an extension of the Fun- environment', and, as one example, deterioration of the environment, to damental Theorem rather than an he discussed this deterioration in 93 terms of an increase in ecological deterioration may occur. First, sup- when from the rate of increase pressure on population growth. We pose that P, is fixed and that only in the mean value of m 114/ I recast Cooke et al.'s argument in the level of aggressiveness, P2, is produced by Natural Selection, is terms of Fisher's analysis in order variable. Then, from Eqn 2, the deducted the rate of decrease to show the relationship between change in P is zero because due to deterioration in environ- ment, results not in an increase the clutch-size example and Fisher's in the average value of m, for ecological view of evolutionary c" (P2/152 P2/P2) this average value cannot greatly change. (P'2/P'2 — P'2/P2) (3) exceed zero, but principally in a Cooke et a!. consider clutch size, steady increase in population. P, which can be equated to fitness. where the first term is proportional (P is used to denote phenotype.) to the increase in the average level Another interesting aspect of P depends on two factors, P = P,T2, of aggressiveness in the population Fisher's view is that gene-frequency where P, is the number of eggs caused by natural selection, and the change and evolution may be per unit area of territory held by second term is proportional to the occurring continually even though the bird, and T2 is territory size. decrease in clutch size caused by there is no net change in mean Territory, in turn, depends on the increase in the average level of fitness: 'Intense selective activity is competition for space. shown to be compatible with an T2 = (AIN)PIP, The second type of environmen- entire absence of change in the tal deterioration occurs when P2 is average survival value of the where A is the total area avail- fixed and the number of eggs per population' (Ref. 3, p. 63). able for territories, N is population unit area, PI , varies. In this case, the size, and P2 summarizes the physical proportional change in Pis given by Why was Fisher misunderstood? . and behavioral traits that determine We think that Fisher was aggressiveness and thus relative AP oc PIM — PI/N) +(i5',IN' — P',/N) misunderstood because most evol- territory size. (All capital letters are utionary biologists sought a simple random variables; overbars denote If one makes the reasonable as- dynamic theory of evolution. That is population averages.) sumption that the carrying capacity, what Wright's Adaptive Landscape With these definitions, the clutch N, is proportional to the number provided. By contrast, Fisherdid not size, P, and the environmental influ- of eggs that can be raised per seek a dynamic theory and, in fact, ence on clutch size, E, are given by unit of territory, PI , then AP = 0. he believed that changes in fitness This balance occurs because the caused by natural selection could P = P1P2E increase in efficiency caused by never be equated with the total natural selection (the first term) is evolutionary change in fitness. The E = A/NP2 exactly balanced by the increased first sentence in the preface of his competition for space (the second book4, The Genetical Theory of From Eqn 1 above, the total change term). Natural Selection, is: 'Natural in the population mean for P is In this last case, clutch size Selection is not Evolution.' and the intrinsic rate of increase When Fisher discussed the evol- = E — 11E) + (PIE' — PIE) (mean fitness) of the population do ution of a trait, he generally (2) not change, but the population considered the direct effect of size does increase (see Ref. 18, natural selection, and the feedback The first term is the expected p. 201). This is exactly the situation caused by ecological changes due to response to selection when that Fisher had in mind for the only natural selection. Instead of provid- measured in the context of the case in which he explicitly discussed ing a precise description for the total original environmental state; this the evolution of the total change in change expected, he discussed at response is equal to the additive population mean fitness (Ref. 4, pp. some length the difficulties in- variance in fitness. The second term 45-46): volved in estimating all of the is the contribution of changing relevant selective and ecologi- environment to the total change in An increase in numbers of any organism will impair its environ- cal parameters involved (Ref. 4, clutch size. This equation for the ment in a manner analogous to, pp. 45-49). total change, AP, corresponds to and more surely than, an increase Fisher's ecological, holistic view, the traditional quantitative genetic in the numbers or efficiency of and the very reasonable interpret- formulation (see Eqn 2 in Ref. 15). its competitors. It is a patent ation of clutch size that follows from The puzzle ).o solve is: why oversimplification . to assert that this view, apparently leaves Fisher doesn't clutch size increase even the environment determines the in a safe and comfortable position. though the first term, the additive numbers of each sort of organism But our discussion so far has not variance in fitness for clutch size, is which it will support. The numbers addressed two important issues. often quite large? Cooke et a!. must indeed be determined by The first concerns Fisher's use of suggest that a decrease in the the elastic quality of the resist- ance offered to increase in num- changing gene frequencies as an environmental term can strike a bers, so that life is made some- ecological factor subsumed within balance so that no evolutionary what harder to each individual the environmental component of change in clutch size occurs. when the population is larger, and evolutionary change". The second In Cooke et al.'s model, there are easier when the population is issue concerns Fisher's criticisms of two ways in which environmental smaller. The balance left over the Adaptive Landscape. Fisher's peculiar treatment of mean fitness, or of a character corre- history or to predict future changes changing gene frequency as part of lated with fitness, is interesting only in traits correlated with fitness. the environment can be seen by in the context of population dy- an analogy with the clutch-size namics. Similarly, the evolutionary Acknowledgements We thank N. Barton, R. Bush, F. Cooke, example of the previous section. In significance of changing gene fre- I. Crow, W. Ewens, 0. Kempthome, M. that example, the level of aggress- quency can only be understood in Kirkpatrick, T. Nagylaki, T. Price, P. Taylor, iveness, P2, may evolve by the direct the context of the genomic environ- M. Turelli and M. Wade for their comments. action of natural selection, as shown ment. SAF's research is supported in part by NIH grants GM42403 and BRSG-S07-RR07008 and in the first term of Eqn 3, but the NSF grant BSR-9057331. increase in the average value of Fitness gradients and ecology aggressiveness, Pr is treated as Fisher did not question the math- an environmental variable, in the ematical validity of Wright's equa- References second term of Eqn 3. tions. He did, however, question 1 Wright, S. (1932) Proc. Sixth Int. Congr. Genet. 1, 356-366 Fisher followed an analogous par- the evolutionary significance of the 2 Provine, W.B. (1986) Sewall Wright and tition at the gene-frequency level. mean-fitness gradient because it Evolutionary Biology, University of Chicago By his view, natural selection in- ignores the ecological aspects of Press creases the frequency of a particular evolutionary change. 3 Fisher, R.A. (1941) Ann. Eugen. I I, 53-63 allele according to its average effect The gradient approach can in- 4 Fisher, R.A. (19581 The Genetical Theory of Natural Selection (2nd edn), Dover • on fitness, which depends on the corporate 'genomic ecology' (e.g. 5 Ewens, (1989) Theor. Popul. Biol. 36, frequencies and combinations of epistasis, frequency dependence) 167-180 other alleles – the genotypic 'en- and population ecology (density 6 Wright, S. (1988) Am. Nat. 131, 115-123 vironment'. Here allele frequency is dependence)°, but the primary use 7 Fisher, R.A. (1958)1. Ecol. 46, 289-293 8 Moran, P.A.P. (1964) Ann. Hum. Genet. 27, a character correlated with fitness, of the gradient for predicting or- 383-393 and natural selection affects mean interpreting the evolutionary conse- 9 Falconer, D.S. (1981) Introduction to fitness according to the additive quences of selection has been (2nd edn), Longman variance in fitness associated with based on the uncorrected Adaptive 10 Crow, I.F. and Nagylaki, T. (1976) Am. allele frequency. Landscape that Fisher criticized. Nat. 110, 207-213 II Kimura, M. (1958) Heredity 12, 145-167 The peculiar part of Fisher's The tension continues between 12 Kimura, M. (1965) Genetics 52, 875-890 argument is that any change in fitness-gradient and ecological 13 Nagylaki, T. (1991) Proc. Natl Acad. Sci. fitness caused by a change in the views of evolution. On one side, USA 88, 2402-2406 average effect of an allele is an much of evolutionary quantitative 14 Price, G.R. (1972) Ann. Hum. Genet. 36, 129-140 environmental effect. Because the genetic theory and the inferences 15 Cooke, F., Taylor, P.D., Francis, C.M. and average effect of an allele often drawn from measurements of Rockwell, R.F. (1990) Am. Nat. 136, 261-267 depends on its frequency, a change selection in the wild have been 16 Wright, S. (1949) in Genetics, in allele frequency caused by based on the mean-fitness gradient. Paleontology and Evolution (Jepson, G.L., natural selection is also a change On the other side, Cooke et al.'s Simpson, G.G. and Mayr, E., eds), pp. 365-389, Princeton University Press in the environment. By this formu- model shows clearly that measure- ;7 Crow, I.F. and Kimura, M. (1970i An lation, fitness, or a trait correlated ments of selection on traits like Introduction to Theory with fitness, increases by an exact clutch size may be misleading if Harper & Row amount because of natural selection interpreted outside of their eco- 18 Charlesworth, B. (1980) Evolution in Age-Structured Populations, Cambridge but simultaneously increases or logical context: a fitness gradient University Press decreases by an unpredictable (Adaptive Landscape) is not 19 Barton, N.H. and Turelli, M. (1987) amount because of the 'environ- sufficient to infer evolutionary Genet. Res. 49, 157-173 ment'. The Fundamental Theorem provides no general statement about evolutionary change. There is, naturally, a certain dis- appointment that the theorem says In the next issue of TREE: nothing about evolutionary dynam- ics. Indeed, since the goal of math- * Inverse modelling and the global carbon cycle, R.C. Dewar ematical population genetics is to * Marine speciation, S.R. Palumbi provide explicit dynamical models, * Muroid rodents: phylogeny and evolution, it is not surprising that most authors F.M. Catzeflis et al. have assumed that Fisher's theorem was intended to be part of a * Long-term population studies of seabirds, dynamical theory, or else have R.D. Wooller et al. expressed disappointment when * What is a quasispecies? M.A. Nowak they realized what Fisher had really * Resistance and virulence in plant–herbivore and plant– done. What one gains from the Fun- pathogen interactions, R.J. Marquis and H.M. Alexander damental Theorem, however, is a * Supply-side ecology and the history of larval biology, stronger ecological sense of the R.K. Grosberg and D.R. Levitan relationship between natural selec- tion and evolution: the evolution of 95