Uniqueness of polymorphic equilibria under soft selection

THEORETICAL

POPULATION

BIOLOGY

24, 295-301 (1983)

Uniqueness of Polymorphic Equilibria under Soft Selection * R. B. CAMPBELL Department of Mathematics and Computer Science, University of Northern Iowa, Cedar Falls, Iowa 50614

Received May 5, 1983

Any two allele polymorphic equilibrium of a subdivided haploid population subject to soft selection is stable. This provides that for a two allele system in a subdivided haploid population there is a globally attracting equilibrium which is polymorphic if a polymorphic equilibrium exists, otherwise monomorphic. These results extend to diploid populations if within each habitat the heterozygote viability is greater than or equal to the geometric mean of the homozygote viabilities.

Wright’s concept of an adaptive landscape (1932; or see Dobzhansky, 1970, pp. 24 ff.) describes the behavior of a population or population system if the shape of the landscape is known. However, the structure of the landscape is in general not known. In particular, a question with pervasive import is whether there is a single adaptive peak toward which populations are always climbing, or whether there are many hillocks upon which populations may settle. This is a question of chance versus necessity. If there are many adaptive peaks natural history and random perturbations will determine at which peak the population is and perhaps shift the population from one peak to another. The presence of a single adaptive peak mandates where the population will be; if two populations have different genetic configurations they must be upon different adaptive landscapes. Mathematicians may prefer to call adaptive peaks stable equilibria and movement on the landscape the dynamics of allele frequencies.The dynamics of the frequencies of two alleles in a single habitat are well known (overdominance provides a stable polymorphism, directional selection provides a stable monomorphism, underdominance provides stability to both fixation states with domains of attraction demarcated by an unstable polymorphic equilibrium). The case of several alleles at a locus in a single habitat is not * Supported in part by NSF Grant MCS 8206837 while the author was at Purdue University.

295 0040-5809183$3.00 Copyright 0 1983 by Academtc Press. Inc. All rights of reproductmn in any form reserved.

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known, but insight into this problem is provided by Karlin (1981) and Karlin and Feldman (198 1). The present work extends investigations into the problem of two alleles at a single locus in several habitats by Karlin and Campbell (1980) (and obviates Campbell, 1981). Some preliminary investigations with several alleles in several environments have been made by Campbell (1982). The result of this paper is that any polymorphic equilibrium for two alleles in a subdivided haploid population is stable. This completes the proof of Principle III, hence Principles II and I, in Karlin and Campbell (1980). Therefore, a one locus two allele diploid system in which the heterozygote viability is greater than or equal to the geometric mean of the two homozygote viabilities in each habitat (i.e., submultiplicative viabilities prevail) has a unique globally attracting stable equilibrium.

THE MODEL

We employ the soft selection model for a subdivided population employed by Christiansen (1974). Within the ith habitat, in the haploid case,one allele has relative viability si and the other allele has relative viability 1. This provides the change in the frequency, xi, of the first allele in the ith habitat due to selection: Ti =

SiXi

sixi + 1 -xi

*

Migration then completes the generation as X’==Mi

(2)

where the prime (‘) denotes the next generation and A4 is a constant (independent of x) stochastic (nonnegative; row elements sum to one) matrix which is called the backward migration matrix (Bodmer and Cavalli-Sforza, 1968). We may express the equilibrium condition in components as SjXj

xi=plij .i

sjxj

+ 1 -xj

(3)

where the mij are the elements of M and the summation is taken over all demes (habitats). It is also useful to express the equilibrium condition in terms of 2 for which inversion of (1) provides

zq xi=&+si(l-.q

(4)

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hence (3) becomes ii + Si(1 - ai)

= 1 mijij ,j

i

=Jfi

i+so(l-i)

(5)

(6)

where 0 denotes the Schur (component) product and, as an abuse of notation which we shall employ in the sequel, division of two vectors is interpreted as division of their corresponding components.

THE THEOREM THEOREM. Subject to the action of soft selection as employed by Christiansen (1974); if a haploid system has a polymorphic equilibrium, it is a stable polymorphism. This equilibrium is asymptotically stable (attracting) unless there is no selection (s = I), which circumstance provides (neutral) stability.

We shall employ stable (unmodified) to refer to asymptotically stable (attracting) in the sequel. Proof: A polymorphic equilibrium for (2) incorporating (1) is stable if S

where p designatesthe spectral radius (magnitude of the largest eigenvalue), diag[ .] is the diagonal matrix with O’s off the diagonal and the ith component of the vector argument as its iith entry, and the abuse of notation illustrated in the juxtaposition of (5) and (6) is also employed with respect to squaring (Campbell 1981, Eq. (A.4)). Solving (5) or (6) for s provides Zi( 1 - C mijij) si = (1 - ai)(C mijij)

(8)

i 0 (1 -Mir) s=(l-+(M+

(9)

or

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Substituting (4) into (7) replaces x by rZ and subsequent use of (8) or (9) provides that (7) is equivalent to i 0 (1 -i) (Mi)o (1 -&f%) 1) < la

i I

p Mdiag

(10)

This inequality follows from the equality i 0 (1 - 2) M(i 0 (1 - i)) 1 ]Mdiag[io M(i 0 (1 - i))

(1 -i)])

= 1

(11)

where the first equality ensues because the spectral radius of a matrix product is invariant under cyclic permutations of the factors and the second equality acknowledges that the second matrix product is a stochastic matrix. The discrepancy between (10) and (11) lies in the inequality (Mi) 0 (1 - Mi) = Mi - (MC?)0 (MZ) > Mi - M(i 0 i) =M(i

0 (1 -2))

where the equalities rely on the distributive property of multiplication over addition and the inequality is a version of the Tchebychev rearrangement inequality. By the Frobenious theory of positive matrices, inequality in (12) and equality in (11) entails that inequality holds in (10). If equality holds in (12) then equality holds in (lo), and the stability analysis is indeterminate. However, equality holds in (12) only if all Zi are equal, which provides s = 1 by (8). This is the only circumstance in which a polymorphic equilibrium is not attracting. It is well known that if s = 1, all demes will approach the same frequency (which frequency depends on the original frequency distribution). Small perturbations in allele frequencies can only result in small perturbations in equilibrium allele frequencies, i.e., neutral stability. In summary, if s # 1 there is a globally attracting stable equilibrium which does not have the same allele frequencies in each deme if it is polymorphic. Conversely, if there is a polymorphic equilibrium which does not have the same allele frequencies in each deme, it is a stable globally attracting equilibrium. If all demeshave the same allele frequencies at equilibrium, it is a neutrally stable equilibrium. Remark. Equation (8) is valid for all i interior to the unit cube 0 & 2 Q 1. Thus given the migration structure M and i, there is a unique selection regime s which provides i as a polymorphic equilibrium. This is

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because i censusesthe population before migration. If we wished to census the population after migration (i.e., use x), there would be a smaller set of admissable equilibrium (x) values. Furthermore, it is important that (8) is not defined for the fixation states 0 and 1. Indeed 0 and 1 are equilibria for any selection regime s and are stable equilibria (not simultaneously) for many s. But our proof entails generating a unique s which provides the polymorphic equilibrium and then showing that the equilibrium is stable; the fixation states are not associated with a unique s nor are they stable whenever they exist.

IMPLICATIONS OF THE THEOREM

The import of the preceding theorem lies in the context of Karlin and Campbell (1980). This rests upon two previous observations. The first, which is valid for (2) incorporating (1) or its diploid analog, is that if every polymorphic equilibrium is stable, then there is a single globally attracting stable equilibrium. The latter extends the haploid result of this paper to submultiplicative viabilities by proving that showing that all polymorphic equilibria subject to haploid viabilities are stable suffices to show that all polymorphic equilibria subject to submultiplicative viabilities are stable. Sketches of the proofs of these facts follow. For (2) incorporating (1) or its diploid analog it can be shown that one stable equilibrium (which will be 0 if 0 is stable, and in some circumstances will be 1) has allele frequencies (xi or ii may be used) in each deme which are less than the allele frequencies associated with any other stable equilibrium. The closure of the intersection of the boundary of its domain of attraction with the interior of the unit cube is a closed contractable set which (2) maps into itself. Hence by the Brouwer fixed point theorem it contains a fixed point with respect to the mapping (2). This fixed point cannot be a stable equilibrium becauseit lies on the boundary of the domain of attraction of a stable equilibrium. It is readily verified that 0 and 1 are the only equilibria on the boundary of the unit cube. Hence the hypothesis that all polymorphic equilibria are stable entails that the domain of attraction of the stable equilibrium must include the entire interior of the unit cube, i.e., it is a globally attracting stable equilibrium. This suffices to prove uniqueness of the stable equilibrium for haploid systems in conjunction with the theorem of this paper. Extension to submultiplicative diploid viabilities rests upon identification of a polymorphic equilibrium under diploid viabilities with the same equilibrium under a corresponding set of haploid viabilities, and demonstration that if the diploid viabilities are submultiplicative, then the diploid equilibrium is stable if the corresponding haploid equilibrium is stable. Since stability of the

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haploid equilibrium is proved in this paper, there is also a unique globally attracting equilibrium (which may be monomorphic on polymorphic) when submultiplicative viabilities prevail under soft selection.

DISCUSSION

This paper completes the proof of Principle III, therewith Principles II and I, in Karlin and Campbell (1980); i.e., under the action of soft selection, overdominance in each deme assures a unique globally attracting polymorphic equilibrium, and submultiplicative viabilities (which include overdominance as a special case) in each deme assure a unique stable equilibrium which may be monomorphic or polymorphic. It is not appropriate to iterate arguments for the importance of knowing whether stable equilibrium configurations are unique. Essentially they entail the circumstance where the gene frequency configuration reflects only the selection regime rather than the selection regime and past history of the system. But it is always necessary to clarify exactly what has been shown. Random sampling of gametes has not been incorporated into the deterministic model of this paper. Steady ascents of adaptive peaks become capricious gambols on the adaptive landscape with random drift. Natural selection’s ascent of the adaptive peak is Sisyphean both in the sensethat it is unobtainable since random drift pushes the population off the peak and in the sensethat only the one peak is sought. This latter sense is the result of this paper: the population will always be near the only adaptive peak. The results of this paper are achieved under the hypothesis of soft selection. There are many other models for selection-migration interaction. Indeed, soft selection is one of the canonical models of selection migration interaction; it has been studied for at least 30 years (Levene, 1953). But it is neither more nor less appropriate than other models in all circumstances. We have proved an important result for a fundamental model of selectionmigration interaction: it provides useful insight into the nature of selectionmigration interaction, but it is not a definitive statement for (all) selectionmigration interaction. The concept of submultiplicative viabilities is a mathematical construct. It arose in the study of additive nonepistasis between alleles. But its biological implications are much broader. In particular, it subsumes selection regimes characterized by a phenotypic additive scale and concave fitness function which Gillespie (1976) has assertedshould govern most electrophoretic loci. Extension of these results to several alleles segregating at a locus is not immediate. Any multiple allele concept of submultiplicative viabilities will be more restrictive than Gillespie’s (1977) multiple allele additive scale concave

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fitness function (Campbell 1982) hence probably have less biological relevance. Nothing has been said about multiple locus selection regimes. Indeed it would be nice if Wright’s adaptive landscape always had a single adaptive peak. Yet we can construct models involving two alleles at a single locus in a single deme for which this is not the case, hence multiple peaks may often occur in nature which allows multiple alleles, multiple loci, and multiple demes. The breadth of selection-migration structures which soft selection with submultiplicative viabilities encompassesshould not be interpreted as implying that most adaptive landscapes should have a unique adaptive peak. Rather, the generality renders soft selection with submultiplicative viabilities a suitable paradigmatic class of selectionmigration structures to contrast with other selection migration structures which may or may not provide unique adaptive peaks.

REFERENCES BODMER,W. F., AND CAVALLI-SFORZA,L. L. (1968). A migration matrix model for the study of random genetic drift, Genetics 59, 565-592. CAMPBELL, R. B. (1981). Some circumstances assuring monomorphism in subdivided population, Theor. Pop. Biol. 20, 118-125. CAMPBELL,R. B. (1982). The effect of variable environments on polymorphism at loci with several alleles. II. Submultiplicative viabilities, J. Mofh. Biology IS, 293-303. CHRISTIANSEN,F. B. (1974). Sufficient conditions for protected polymorphism in a subdivided population, Amer. Narur. 108, 157-166. DOBZHANSKY,TH. (1970). Genetics of the Euolufionarv Process, Columbia Univ. Press, New York. GILLESPIE, J. H. (1976). A general model to account for enzyme variation in natural populations. II. Characterization of the litness function, Amer. Natur. 110, 809-82 I. GILLESPIE, J. H. (1977). A general model to account for enzyme variation in natural populations. III. Multiple alleles. Evolution 31, 85-90. KARLIN, S. (1981). Some natural viability systems for a multiallelic locus; A theoretical study, Genetics 91, 457-473. KARLIN, S., AND CAMPBELL, R. B. (1980). Selection-migration regimes characterized by a globally stable equilibrium, Genetics 94, 1065-1084. KARLIN, S., AND FELDMAN, M. W. (1981). A theoretical and numerical assessmentof genetic variability, Genetics 97, 475493. LEVENE, H. (1953). Genetic equilibrium when more than one ecological niche is available, Amer. Natur. 81, 333-333.

WRIGHT. S. (1932). The roles of mutation, imbreeding, and selection in evolution, “Proceedings VI Intl. Cong. Genet. I,” pp. 356-366.