2211076a0Nature221518519690315107610760028-0836196910.1038/2211076a0ukNatureNatureNATUREnatureNature is a weekly international journal publishing the finest peer-reviewed research in all fields of science and technology on the basis of its originality, importance, interdisciplinary interest, timeliness, accessibility, elegance and surprising conclusions. Nature also provides rapid, authoritative, insightful and arresting news and interpretation of topical and coming trends affecting science, scientists and the wider public./nature/journal/v221/n5185issueJournal homeArchiveCurrent issueAdvance online publicationPrivacy policySubscribeNature Publishing GroupCurrent issue2211076a0Mean Fitness Increases when Fitnesses are Additive
AU  - EWENS, W. J.Department of Mathematics, La Trobe University, Melbourne.FISHER'S fundamental theorem of natural selection1 states that for a random mating population with fitnesses dependent on the genetic constitution at a single locus, the mean fitness of the population increases with time. This is true for an arbitrary number of possible alleles at the locus and for arbitrary fitness values.Moran2 has shown that if fitness depends on the genetic constitution at two loci, then the mean fitness of the population can decrease monotonically with time. Two questions arise from this disturbing result: (i) can any restrictions be placed on the fitnesses to secure an analogue of Fisher's theorem; (ii) does any function other than the mean fitness increase monotonically with time? I should like to give an answer to (i) which is applicable in a wide range of important cases by proving the following theorem.
My theorem is that if fitnesses are assumed additive over loci, then the mean fitness of the population increases monotonically in time, irrespective of the number of loci on which fitness depends, of the number of possible alleles at each locus and of the linkage arrangement between the loci. (Fitnesses which are "additive over loci" are those for which, if fitness is supposed to depend on the genotypic constitution at k loci, the fitness of any individual can be expressed as a sum of k terms, the ith term in the sum being characteristic of the individual's genotype at the ith such locus.)
We consider the proof in detail only for the case of two loci A and B, admitting two alleles each: A1 A2 and B1 B2. Because fitnesses are assumed additive over loci, we can write the fitness matrix in the form
 	 A1A1 	 A1A2 	 A2,A2 
 B1Bl 	U1 + V1 	 U1 + V2 	 Ul + V3 
 B1B2 	U2 + Vl 	U2 + V2 	 U2 + V3 
 	B2B2 U3 + V1	U3 + V2 	 U3 + V3 
If the frequencies of the gametes A1B2, A2B1 A1B2 and A2B2 which make up the zygotes in generation t are denoted cl c2, c3, c4, then the mean fitness W of the population in generation t is given, according to Moran3, by
W = Ui (Cj+Ca)2 + 2U2 (C! + C2) (C3 + C4) + U3 (C3 + C4)2
+ vl (C! + C3)2 + 2v2 (Ci + Cs) (C2 + C4) + v, (c2 + c4)2 (1)
W depends on the frequencies c1 + c2 and c1+c3, only, for we can write c3 + c4= 1  (c1 + c3), c2 + c4= 1  (c1 + c2). Note that c1 + c2 is the frequency of the gene B1 and c1 + c3 is the frequency of the gene A1. The frequencies cl, c2, c3, c4 of the gametes which make up the zygotes in generation t + 1 are found from the recurrence relations (76)(79) in ref. 3. Given c1 c2, c3 and c4, the frequencies c1+c2 and c1+c3 are independent of the degree of linkage between A and B loci. Thus the mean fitness W of the population in generation t + 1-given by equation (1) with W replacing W and ci replacing ci, and therefore depending on c1+c2 and c1+c3 only-will itself be independent of the degree of linkage between A and B loci, once the values ci are given. In particular, it will be equal to the mean fitness in generation t+1 for the particular case R = 0. But it is a classical result that when R = 0 the population behaves as one where the fitness depends on one locus with four alleles, and Fisher's theorem asserts that for this latter situation, W > W. Thus, because W is independent of R, W>W for all values of R.
The key point in this proof is that W depends only on c1+c2 and c1+c3 (that is, only on gene frequencies). This makes its generalization to an arbitrary number of alleles and an arbitrary number of loci almost immediate, for in all such cases W again depends only on gene frequencies. Given the frequencies of the gametes making the zygotes of generation t, the frequencies of the genes making the zygotes of generation t+1 are independent of any linkage arrangement between the loci. It follows immediately that the mean fitness in generation t + 1 is again identical to that which obtains when all crossing-over frequencies are zero, and, because this latter case can again be viewed as one where fitnesses depend on the (very large number of) alleles at a single locus, we again have W'>W in the completely general case. Kingman's results4 show further that W > W unless the population is at an equilibrium point.
This result opens up a range of problems which will be most usefully associated with the forthcoming results of Moran5 concerning stability points and convergence behaviour in the same model.
I thank Professor P. A. P. Moran for his assistance in discussing this problem.Fisher, , R. A., The Genetical Theory of Natural Selection (Oxford University Press, 1930).Moran, , P. A. P., Ann. Hum. Genet., 27, 383 (1964).PubMedISIChemPortMoran, , P. A. P., Proc. Fifth Berk. Symp. Math. Stat. and Prob. (Univ. California Press, 1965).Kingman, , J. F. C., Quart. J. Math., 12, 78 (1961).ISIMoran, , P. A. P., Ann. Hum. Genet, (in the press).