Evolution. 38(2),1984, pp. 359-367
EVOLUTIONARY LANDSCAPES FOR COMPLEX SELECTION JAMES
W.
CURTSINGER
Department of Genetics and Cell Biology, University of Minnesota, St. Paul, Minnesota 55108 Received March 16, 1983.
Wright's (1932, 1969) "surface of selective value" is one of the most influential concepts in evolutionary genetics. The surface, also known as an "adaptive topography" or "evolutionary landscape," is an n-dimensional picture ofthe way that natural selection changes the genetic composition of a population. For one-locus models, the surface is defined by n - 1 independent allele frequency axes and one dimension for population "fitness." At any given time, a population can be represented as a point on the surface. Through time, the point moves on the surface in such a way that population "fitness" is always increasing, until equilibrium is attained at a maximal state of adaptation. For the well known deterministic model of viability selection at one locus, the notion of increasing population "fitness" until equilibrium is an exact mathematical theorem (Kingman, 1961), provided that certain assumptions are satisfied: mating occurs at random, mutation and immigration are negligible, genotypespecific viabilities are constant, segregation follows Mendelian ratios, and selection is independent of sex. The appropriate measure of population "fitness" under these circumstances is the average zygote viability, usually called "mean fitness." However, for many models that depart from the assumptions of the classical viability selection model, mean fitness is not necessarily an increasing quantity that is maximized at equilibrium. Following is a representative list of biologically significant situations in which the principle ofincreasing population fitness is known to be violated. Multiple Loci.-If an individual's viability depends on the genotype at two
Revised July 24, 1983
loci, then the population mean viability is not generally maximized at equilibrium (Moran, 1964). Mean viability can decrease from generation to generation, even though selection coefficients are constant (Karlin and Carmelli, 1975). The mean viability is nondecreasing if fitness is determined by additive interactions between loci (Ewens, 1969), but epistasis is often found when the appropriate experimental design is employed (e.g., Clark and Feldman, 1981a; see Barker, 1979, for a review). Wright (1969 p. 475) was aware of this limitation, noting that "Selection formulas for single loci can never be more than momentary approximations because ofthe practical universality of interactions with the other loci, ...." Meiotic Drive.-If heterozygotes produce a non-Mendelian ratio of gamete types, then mean viability at the polymorphic equilibrium is always less than maximal (Hiraizumi et al., 1960). There can exist a stable fixation equilibrium that is a minimum of the mean viability. Naturally occurring major meiotic drive elements have been documented in the rodents Myopus schisticolor (Fredga et al., 1976) and Mus musculus (Dunn, 1956), two lepidopteran species (Chanter and Owen, 1972; Smith, 1975), at least 12 species of Drosophila (Sturtevant and Dobzhansky, 1936; Stalker, 1961; Faulhaber, 1967; Hartl and Hiraizumi, 1976), two other dipterans (Hiroyoshi, 1964; Hickey and Craig, 1966), two species of Neurospora (Turner and Perkins, 1976), and several higher plants (Cameron and Moav, 1957; Loegering and Sears, 1963; Rectei, 1965; Carlson, 1969; Mimtzing, 1968). Further, there is evidence that populations of Drosophila carry minor elements that modify segregation (Hanks,
359
360
JAMES W. CURTSINGER
1965; Lyttle, 1979; Curtsinger, 1981; Hiraizumi and Gerstenberg, 1981; Curtsinger and Hiraizumi, pers. observ.). Gametic Selection. - Organisms that have extended phases in the haploid stage can be subject to selection on the basis of haplotype. Likely examples include some algae, Foraminifera, and many higher plants (Mulcahy, 1975). Theoretical analysis of combined haploid and diploid selection has been considered by Scudo (1967) who showed that the overall state of adaptation attained by the population is a compromise between haploid and diploid selection. Organismal Interaction.-If interactions between organisms influence viability in a nonsymmetrical way, then mean viability may not be maximized at equilibrium (Cockerham et aI., 1972). The idea that behavioral interactions constitute a major selective force is currently an active one, but its general significance in comparison with other modes of selection remains to be determined. The extensive literature of frequency dependent selection in experimental populations of Drosophila (Ayala and Campbell, 1974; Lewontin, 1974) is consistent with the idea that the genetic composition of a population is a significant part of an organism's environment, contributing to the biotic and abiotic factors that determine fitness. Fertility Selection.-If the number of progeny produced by a mating depends on the statistical interaction of parental genotypes, then the population mean fertility may not be maximized at equilibrium, may steadily decrease through time, and may even exhibit oscillations (Kempthorne and Pollak, 1970; Pollak, 1978). There is increasing evidence from experimental populations of Drosophila that fertility selection is often a primary source of net fitness differences between genotypes (Prout, 1971; Sved, 1971; Bungaard and Christiansen, 1972; Curtsinger and Feldman, 1980; Brittnacher, 1981; Clark and Feldman, 1981b). Selective Differences Between the Sexes. - Models incorporating sex-de-
pendent viability selection at an autosomal locus are formally equivalent to multiplicative fertility selection (Bodmer, 1965), and are thus subject to nonmaximization behavior. Various authors have argued for the likelihood of sex-dependent viability selection (Haldane, 1962; Li, 1963; Kidwell et aI., 1977; Wilson and Sved, 1979; Curtsinger, 1980; Lande, 1980). The prevalence of sexual dimorphism among higher animals can be taken as evidence for the general importance of sex-dependent selection and sexual selection. In light of the documented violations of the assumptions of the classical viability selection model, it is important to investigate more complex models, and to ask whether population "fitness" can be redefined in a meaningful way. In particular, a redefinition is sought such that population "fitness" increases through time to its maximal value at equilibrium. It is desirable that the newly defined "fitness" be interpretable as a measure of population adaptation. In this paper, one-locus deterministic models of meiotic drive and gametic selection are investigated. These modes of selection have been chosen because they are analytically the simplest cases in which the principle of maximized mean viability is violated. For meiotic drive and gametic selection models there is a clear problem in defining population "fitness," because selection can be thought of as operating in both haploid and diploid stages. The two modes of selection are analyzed together for conciseness and because of structural similarities in the models, but the biological distinctions are clear: meiotic drive means that heterozygotes produce non-Mendelian ratios of gamete types, as in the SD system of Drosophila melanogaster. In the gametic selection case, the fitness of haplonts depends only on haplotype, and is independent of the diploid genotype from which it is derived. The first step of the analysis involves an extension of Lewontin's (1958) method of weights, by which a general formula is obtained for deriving
361
EVOLUTIONARY LANDSCAPES
new maximization functions. A preliminary report of this method has appeared elsewhere (Curtsinger, 1983). EXTENDING THE METHOD OF WEIGHTS
Consider a large population ofdiploids having discrete generations and two alleles at an autosomal locus. In the parental generation, alleles A and a have relative frequencies p and q, respectively, where p + q = 1. For the moment, no assumptions are made about the mating system, zygotic selection, segregation parameters, mutation, or other factors that modify allele frequencies. In the offspring generation, the three genotypes occur in the following relative frequencies among reproductive-aged adults: AA
Aa
aa
p2U 11
2pqU12
q2U22 •
(1)
In general, the Us are not relative viabilities or Wrightian fitness values. They are weights, as defined by Lewontin (1958), taking on values such that relative genotype frequencies conform to the proportions indicated. The Us could be functions of allele frequencies, genotype frequencies, selection coefficients, and other parameters. Denoting allele frequencies in the offspring generation by pi and q', the allele frequency transformation equations are
,_ q2U22 + pqU I2 q - P2U11 + 2 pq U 12 + q2 U22' and pi = 1 - q'.
(2)
The change in allele frequency over one generation is b.q: !!.q = q' - q =
pq[V12
V" + q(V" - 2V12 + V 22 ) ] p2V" + 2pqV12 + q2V22
(3)
Equilibria of the system defined by (2), denoted q, are q = 0, q = 1, and A
q
=
U 11
U 11 - U 12 2U12 + U22 •
(4)
Now consider the special case of ran-
dom mating, Mendelian segregation, and constant viability selection. Then Uij = wij' where the wij are constant, and (3) reduces to Wright's well-known formulation: b.q = pq dW
(5)
2W dq ' where W =
p2 W I 1
+
+
2pqw I2
q2W22 >
o.
W is the mean fitness of the zygotes of the offspring generation; it is a function of the allele frequencies, but the argument is usually suppressed. Equation (5) defines an "adaptive topography" or "evolutionary landscape" upon which populations move through time. It shows that allele frequencies change in such a way that the mean fitness increases, except at equilibrium, justifying the notion that populations climb out of "adaptive troughs" and scale "adaptive peaks." Equation (5) also implies that stable equilibria lie at maxima of the W function while unstable equilibria lie at minima: W is a Liapunov function, or, in the jargon of evolutionary biology, a maximization function, for the special case of Mendelian segregation, random mating, and constant viability selection. The mean fitness function is not necessarily maximized or nondecreasing in cases that depart from the classical viability selection scheme. Suppose that there exists for those more complex cases some maximization function "T." (T is a function of allele frequencies, but the argument will be suppressed for notational simplicity.) Further suppose that T is defined such that b.q = pq dT 2[; dq' where D = p2U t l + 2pqU12
+
q2 U22 >
(6)
o.
Then, by comparison ofequations (3) and (6), the following expression is obtained for the hypothetical maximization function:
362
T
JAMES W. CURTSINGER
=
2
f
+
+
V 12 - VII
q(V t t - 2Vt2 + V 22) dq.
(7)
Equation (7) is the basic expression for defining new maximization function for complex selection at one locus. The integrand of(7) can be obtained in two ways: by solving explicitly for the weights satisfying (1) (see Lewontin, 1958), or by deriving equation (3). To illustrate the application of the extended method ofweights, consider a discrete generation model incorporating diploid viability selection, meiotic drive, and gametic selection. In a large population there exist haplotypes A and a with relative frequencies p and q, respectively, where p + q = 1. Haplonts fuse at random to form diploid zygotes AA, Aa, and aa with relative viabilities Wl l , w t2, and W 22' respectively. Following viability selection, surviving diploids produce haplonts by generalized segregation, heterozygotes producing (1 - m)/2 A-types and (1 + m)/2 a-types. If m > then meiotic drive favors the a allele, while m = corresponds to Mendelian segregation. Viability selection then discriminates among haplotypes A and a with relative survival probabilities equal to hand 1 respectively. The entire life cycle then repeats. Parameter constraints are wij 2: 0, h 2: 0, and -1 :5 m :5 + 1. Censusing the postselection haplont pool, the allele frequency transformation equations for an entire generation of the life cycle are
°
WHDq'
= q2W22 + pqw 12 (1 +
°
m)
and p' = 1 - q', where W HD = p2Wl l h + pqw t2 .(1 + h + m - hm) + q2W22.
(8)
The change in allele frequency from generation to generation is WHv!lq = pq[wd1
+
m) - WIth
q(W1lh -
.(1
+
+
W22)].
m
+
W l2
h - hm) (9)
There are at most three equilibria: q = 0, q = 1, and •
q=
wllh - w.,(l + rn) wllh - w.2 ( 1 + rn + h - hrn)
+ W2 2
.
(10)
The equilibrium structure is governed by the following inequalities: a. If w l2h(1 - m) > W22 and wdl + m) > WIth then the polymorphic equilibrium exists, is stable, and the boundaries are unstable. If the inequality signs are reversed then the polymorphic equilibrium is unstable and the boundaries are stable. b. If w 12 h(1 - m) < W 22 and wdl + m) > WIth then the polymorphic equilibrium does not exist, q = 1 is stable, and q = is unstable. If the inequality signs are reversed then q = 1 is unstable and q = is stable.
°
°
In all cases allele frequencies converge monotonically to equilibrium (see Appendix). The maximization function is obtained by integrating the term in brackets on the right side of (9):
T
=
q2[wt t h - w12 ·(1 + m + h - hm)
+ +
2q[wdl
+
+
W 22]
m) - Wtth]
(11)
const
where "const" is a constant of integration. Note that when it exists the polymorphic equilibrium (10) corresponds to the critical point of T; in particular, the critical point is a maximum if q is stable and is a minimum if q is unstable. Critical points of W do not coincide with q of (10) if m or h 1. Given that T is a quadratic function of q, that it is locally maximized at stable
*' °
*'
363
EVOLUTIONARY LANDSCAPES
equilibria and locally minimized at unstable equilibria, and that allele frequencies converge monotonically to equilibria, it follows that T is nondecreasing. The change in T from generation to generation is t:J.T
=
pqWHD-ZZZ ·[WHD + pwllh
+
qwzz]'
(12)
where Z = wdl + m) - wl1h + q[wllh - w12 (1 + m + h - hm) + wn ]. t:J.T:::: 0, with equality holding only at q = 0, q = 1, or q = q of (10), the polymorphic equilibrium. Three special cases are of particular interest: a. If m = 0 and h = I then we have the classical viability selection model. In this case T differs from W (the usual diploid mean viability) by a constant of integration. b. If h = I and m *- 0 then we have the autosomal meiotic drive model of Hiraizumi et al. (1960). c. If h *- I and m = 0 then we have the gametic selection model of Scudo (1967). The extended method of weights unites the three cases by providing a single maximization function. Now consider the application of this method to the case of n alleles A I' Az, , An having relative frequencies PI' Pz, , Pn (~Pi = 1). For simplicity only the case h = 1 will be considered, though the conclusion applies to other cases. Individuals of the genotype A;Aj produce Ai-bearing and Arbearing gametes in the proportions (1 + m;)l2 and (1 + mj;)/2, respectively, where -1 -s m =::; + 1, mij = - m ji, and m; = O. The allele frequency transformation equations are Wp;'
=
PiWi
(i
=
1,2, ... ,n)
where Wi = ~ pjwij(1
+ m i)
j
W
= ~ ~ PiPjWij. j
,
(13)
One-generation changes in allele frequencyare Wt:J.Pi = Pi(Wi -
W).
(14)
By analogy with the Mendelian n-allele model, we seek a maximization function T such that t:J.Pi
=
Pi(1 2W
pJ bT bpi·
(15)
Functions satisfying (15) are (16)
For n > 2, equation (16) generates a number of different T, functions that are not generally well behaved. Functions are not constrained to finite values and can be shown to exhibit nonmonotone behavior. There might exist a maximization function for the multiple-allele model, but it is not generated by the method described here. It is conjectured that the extended method of weights fails where allele frequency change is nonmonotonic. DISCUSSION
How can one best describe the way that natural selection changes the genetic composition of a population? The common notion is that natural selection improves "population fitness," but that quantity is not readily defined with either precision or generality. The maximization ofmean viability is an appealing idea, governing the behavior of the classical viability selection model, but it is not robust. In particular, relaxing the assumptions concerning the mode of selection, the segregation pattern, the number ofloci, the constancy of the selection parameters, or the structure of the life cycle can result in nonmaximization behavior. Meiotic drive, gametic selection, epistasis, frequency dependence, and other complex modes of selection are real phenomena; their existence challenges the common notion of adaptation. In this paper a new approach to the
364
JAMES W. CURTSINGER
study of complex modes of selection has been developed. By extending Lewontin's (1958) method of weights, it is possible to define a quantity that is maximized at stable equilibria, minimized at unstable equilibria, and always increasing except at equilibrium. The strength of this approach lies in its generality: the derivation entails no specific assumptions other than a discrete generation treatment of a one-locus, two-allele deterministic model. Because of the generality of equation (7), it is possible to obtain maximization principles for a variety of deterministic models, including frequency dependent selection, nonrandom mating, inbreeding, and other cases. Equations (11) and (6) define evolutionary landscapes for meiotic drive and gametic selection in the same way that Wright's mean viability principle defines an adaptive topography for the classical viability selection model. In fact, the usual mean fitness principle is a special case of equations (11) and (6). While the generality of the extended method of weights has appeal, it also has limitations. The method fails ifthere are more than two alleles, though the advantages of being able to describe maximization principles for many modes of selection may outweigh the limitation associated with the number of alleles. A second limitation is that the method has no obvious application to an important class of complex models: selection at two or more loci. Third, some modes of selection may generate nonintegrable functions of weights. Finally, the maximization principles for meiotic drive and gametic selection are remarkably simple, but defy simple explanation. The quantities succeed in combining the effects of selection in haploid and diploid stages, but contain no simple combination ofintuitively defined haploid and diploid "fitness." Nor are the quantities obviously related to gene "transmissibility." The question of interpreting these maximization principles is an important one that will be taken up in a later paper; the T functions will be shown to be related to Fisher's "average effect."
No work in theoretical population genetics is totally new. There is a scattered literature on the derivation of maximization principles for various models that depart from the assumptions of the classical viability selection model. Li has employed heuristic methods to derive maximization principles for the cases of spatial heterogeneity in selection coefficients, inbreeding, compensation (1955), and selection at a sex-linked locus (1967). Levins and MacArthur (1966) and Cannings (197 1) have furthered the analysis of selection in multiple niches. Wright (1955, 1969) has defined a maximized "fitness function," distinct from the usual mean fitness or mean viability, that always exists in the case of a character determined by a single pair of alleles. A survey of standard population genetics texts suggests that the "fitness function" concept has had minimal impact (but see Arnold and Anderson, 1983). The fitness function seems to be related to the ideas developed in this paper, and deserves further analysis. Another related approach is the extension of Fisher's Fundamental Theorem of Natural Selection to take into account variable fitness and nonrandom mating (Kimura, 1958; Crow and Kimura, 1970), multiple loci (Turner, 1981), meiotic drive (Hartl, 1970), and selection on a sex-linked locus (Hartl, 1972). The outcome of those analyses is an expression for the rate of change in mean fitness, in the form of additive genetic variance plus a correction term, rather than the direct definition ofa maximization function. It can be concluded that, in spite of its limitations, the extended method of weights has much potential as a general description of the effect of natural selection. The application of the method to other complex modes ofselection and the interpretation ofthe maximization function will be taken up in subsequent papers. SUMMARY
Wright's principle of maximized mean viability is known to be violated in a number of biologically significant situa-
EVOLUTIONARY LANDSCAPES
tions involving complex modes of selection, including epistasis, meiotic drive, organismal interaction, fertility selection, selective differences between the sexes, and gametic selection. In this paper, Lewontin's (1958) method of weights is extended to derive a very general maximization principle for general one-locus, two-allele deterministic selection models. New maximization principles are developed for the cases of meiotic drive and gametic selection, thereby defining evolutionary landscapes. The extended method of weights has considerable potential as a general descriptor of complex selective processes. ACKNOWLEDGMENTS
I thank Dr. Richard Lewontin for his helpful comments in the early stages of this work. I also thank Drs. Lisa Brooks, Chris Cannings, Andrew Clark, and Larry Jacobson for their criticisms of the manuscript. Drs. Jonathan Arnold and Richard Lewontin made helpful comments following the presentation of this material at the 1983 Annual Meetings of the Genetics Society of America. Dr. Cannings suggested a concise proof of convergence. Ms. Kae Ebling provided expert secretarial assistance. Research is supported by NSF Grant No. BSR8211667. LITERATURE CITED ARNOLD, J., AND W. W. ANDERSON. 1983. Density-regulated selection in a heterogeneous environment. Amer. Natur. 121:656-668. AYALA, F. J., AND C. A. CAMPBELL. 1974. Frequency-dependent selection. Ann. Rev. Ecol. Syst.5:115-138. BARKER, J. S. F. 1979. Inter-locus interactions: a review of experimental evidence. Theoret. Pop. BioI. 16:323-346. BODMER, W. F. 1965. Differential fertility in population genetic models. Genetics 51:411-424. BRITTNACHER, J. G. 1981. Genetic variation and genetic load due to the male reproductive component of fitness in Drosophila. Genetics 97: 719-730. BUNGAARD, J., AND F. B. CHRISTIANSEN. 1972. Dynamics of polymorphisms. 1. Selection components in an experimental population of Drosophila melanogaster. Genetics 71:439-460. CAMERON, D. R., AND R. MOAV. 1957. Inheritance in Nicotiana tabacum. XXVIII. Pollen killer, an alien genetic locus inducing abortion
365
of microspores not carrying it. Genetics 42:326335. CANNINGS, C. 1971. Natural selection at a multiallelic autosomal locus with multiple niches. J. Genet. 60:255-259. CARLSON, W. R. 1969. Factors affecting preferential fertilization in maize. Genetics 62:543554. CHANTER, D.O., AND D. F. OWEN. 1972. The inheritance and population genetics of sex ratio in the butterfly Acraea encedon. J. Zool. 166: 363-383. CLARK, A. G., AND M. W. FELDMAN. 1981a. Estimation ofepistasis in components offitness in experimental populations of D. melanogaster. II. Assessment of meiotic drive, viability, fecundity, and sexual selection. Heredity 46:347377. - - . 1981b. Density dependent fertility selection in experimental populations of Drosophila melanogaster. Genetics 98:849-869. COCKERHAM, C. c., P. M. BURROWS, S. S. YOUNG, AND T. PROUT. 1972. Frequency dependent selection in randomly mating populations. Amer. Natur. 106:493-515. CROW, J. F., AND M. KIMURA. 1970. An Introduction to Population Genetics Theory. Harper and Row, N.Y. CURTSINGER, J. W. 1980. On the opportunity for polymorphism with sex-linkage or haplodiploidy. Genetics 96:995-1006. - - . 1981. Artificial selection on the sex ratio in Drosophilapseudoobscura. J. Hered. 72:377381. - - . 1983. Evolutionary landscapes for complex selection. Genetics 104:s20. CURTSINGER, J. W., AND M. W. FELDMAN. 1980. Experimental and theoretical analysis ofthe "sexratio" polymorphism in Drosophila pseudoobscura. Genetics 94:445-466. DUNN, L. C. 1956. Analysis ofa complex gene in the house mouse. Cold Spring Harbor Symp. Quant. BioI. 21:187-195. EDWARDS, A. W. F. 1972. Foundations of Mathematical Genetics. Cambridge Univ. Press, Cambridge. EWENS, W. J. 1969. Mean fitness increases when fitnesses are additive. Nature 221:1076. FAULHABER, S. H. 1967. An abnormal sex ratio in Drosophila simulans. Genetics 56: 189-213. FREDGA, K., A. GROPP, H. WINKING, AND F. FRANK. 1976. Fertile XX- and XY-type females in the wood lemming Myopus schisticolor. Nature 261 :225-227. HALDANE, J. B. S. 1962. Conditions for stable polymorphism at an autosomal locus. Nature 193:1108. HANKS, G. D. 1965. Are deviant sex ratios in normal strains of Drosophila caused by aberrant segregation? Genetics 52:259-266. HARTL, D. L. 1970. Population consequences of non-Mendelian segregation among multiple alleles. Evolution 24:415-423. - - . 1972. A fundamental theorem of natural
366
JAMES W. CURTSINGER
selection for sex linkage or arrhenotoky. Amer. Natur. 106:516-524. HARTL, D. L., AND Y. HIRAIZUMI. 1976. Segregation distortion, p. 615-666. In M. Ashburner and E. Novitski (eds.), Genetics and Biology of Drosophila, Vol. lb. Academic Press, N.Y. HICKEY, W. A., AND G. B. CRAIG, JR. 1966. Genetic distortion ofsex ratio in a mosquito, Aedes aegypti. Genetics 53:1177-1196. HIRAIZUMI, Y., AND M. V. GERSTENBERG. 1981. Gametic frequency of second chromosomes of the T-007 type in a natural population of Drosophila melanogaster. Genetics 98:303-316. HIRAIZUMI, Y., L. SANDLER, ANDJ. F. CROW. 1960. Meiotic drive in natural populations of Drosophila melanogaster. III. Population implications of the segregation-distorter locus. Evolution 14:433-444. HIROYOSHI, T. 1964. Sex-limited inheritance and abnormal sex-ratio in strains of the housefly. Genetics 50:373-385. KARLIN, S., AND D. CARMELLI. 1975. Numerical studies on two-loci selection models with general viabilities. Theoret. Pop. BioI. 7:399-421. KEMPTHORNE, 0., AND E. POLLAK. 1970. Concepts of fitness in Mendelian populations. Genetics 64:125-145. KIDWELL, J. F., M. T. CLEGG, F. M. STEWART, AND T. PROUT. 1977. Regions of stable equilibria for models of differential selection in the two sexes under random mating. Genetics 85: I 71183. KIMURA, M. 1958. On the change of population fitness by natural selection. Heredity 12:145167. KINGMAN, J. F. C. 1961. A mathematical problem in population genetics. Proc. Cambridge Philos. Soc. 57:574-582. LANDE, R. 1980. Sexual dimorphism, sexual selection, and adaptation in polygenic characters. Evolution 34:292-305. LEVINS, R., AND R. MACARTHUR. 1966. The maintenance ofgenetic polymorphism in a spatially heterogeneous environment: variations on a theme by Howard Levene. Amer. Natur. 100: 585-589. LEWONTIN, R. C. 1958. A general method for investigating the equilibrium ofgene frequency in a population. Genetics 43:421-433. - - - . 1974. The Genetic Basis of Evolutionary Change. Columbia Univ. Press, N.Y. LI, C. C. 1955. The stability of an equilibrium and the average fitness of a population. Amer. Natur. 89:281-296. - - - . 1963. Equilibrium under differential selection in the sexes. Evolution 17:493-496. - - - . 1967. The maximization of average fitness by natural selection for a sex-linked locus. Proc. Nat. Acad. Sci. USA 57:1260-1261. LOEGERING, W. Q., AND E. R. SEARS. 1963. Distorted inheritance ofstem-rust resistance oftimstein wheat caused by a pollen-killing gene. Can. J. Genet. Cytol. 5:65-72.
LYTTLE, T. W. 1979. Experimenta1populationgenetics of meiotic drive systems. II. Accumulation of genetic modifiers of Segregation Distorter (SD) in laboratory populations. Genetics 91:339-357. MORAN, P. A. P. 1964. On the nonexistence of adaptive topographies. Ann. Hum. Genet. 27: 383-393. MULCAHY, D. L. (ed.). 1975. Gamete Competition in Plants and Animals. North-Holland, Amsterdam. MONTZING, A. 1958. A new category of chromosomes. Proc. X Int. Genet. Congr. 1:453467. POLLAK, E. 1978. With selection for fecundity the mean fitness does not necessarily increase. Genetics 90:383-389. PROUT, T. 1971. The relation between fitness components and population prediction in Drosophila. II. Population prediction. Genetics 68: 151-168. REDEl, G. P. 1965. Non-Mendelian megagametogenesis in Arabidopsis. Genetics 51:857-872. SCUDO, F. M. 1967. Selection on both haplo and diplophase. Genetics 56:693-704. SMITH, D. A. S. 1975. All-female broods in the polymorphic butterfly Danaus chrysippus L. and their ecological significance. Heredity 34:363371. STALKER, H. D. 1961. The genetic systems modifying meiotic drive in Drosophila paramelanica. Genetics 46: 177-202. STURTEVANT, A. H., AND TH. DOBZHANSKY. 1936. Geographical distribution and cytology of sexratio in Drosophila pseudoobscura and related species. Genetics 21:473-490. SVED, J. A. 1971. An estimate ofheterosis in Drosophila melanogaster. Genet. Res. 18:97-105. TURNER, B. c., AND D. D. PERKINS. 1976. Spore killer genes in Neurospora. Genetics 83:suppl. s77. TURNER, J. R. G. 1981. "Fundamental theorem" for two loci. Genetics 99:365-369. WILTON, A., AND J. SVED. 1979. X-chromosomal heterosis in Drosophila melanogaster. Genet. Res. 34:303-315. WRIGHT, S. 1932. The roles of mutation, inbreeding, crossbreeding, and selection in evolution. Proc. VI Int. Congr. Genet. 1:356-366. - - - . 1955. Classification of the factors ofevolution. Cold Spring Harbor Symp. Quant. BioI. 20:16-24. - - - . 1969. Evolution and the Genetics of Populations, Vol. II. Univ. Chicago Press, Chicago. Corresponding Editor: J. Felsenstein
ApPENDIX Proof of monotone convergence to equilibrium by an extension of Edwards' (1972) Theorem 2.2.1 : Define r = p/q. Then
EVOLUTIONARY LANDSCAPES r'
=
r.
p2Wllh + pqW12h(1 - m). q2W22 + pqW 12(l + m)
=
q'
Equilibria are f
=
r = r*
0, f =
and
c.
W12h(l - m) - wn . wdl + m) - wllh
d.
= 00,
Some algebraic manipulation leads to r' - r*
= (r -
r*)Q,
where Q = (rwllh + wn)/(rw 12(l + m) + W22). There are six cases to be analyzed. a. If wllh < w12(l + m) and W22 < w 12h(l - m), then r* > and Q < 1. It follows that r' lies between rand r*, indicating monotonic convergence to the polymorphic equilibrium. b. If wllh > W 12 (1 + m) and W22 > w 12h(l - m),
°
e. f.
367
then r* > 0, Q > I, and rlies between r' and r*, indicating monotonic divergence from the polymorphic equilibrium. If wllh < wdl + m) and W22 2:: w 12h(1 - m), then r* ::'0 0, Q < I, and r decreases monotonically to r = 0. If wllh > wdl + m) and W22 ::'0 w 12h(1 - m), then r* ::'0 0, Q> I, and r increases monotonically to r = 00. If wllh = w12(l + m) and Wn #' w 12h(1 - m), then we redefine r = qlp and proceed as above. If wllh = wdl + m) and W22 = w12h(1 - m), then r' = r.
Thus r is either at equilibrium or monotonically converging to equilibrium. Because zsrand!:>.p must have the same sign, allele frequencies are also monotonically convergent.
EVOLUTIONARY LANDSCAPES FOR COMPLEX SELECTION JAMES
W.
CURTSINGER
Department of Genetics and Cell Biology, University of Minnesota, St. Paul, Minnesota 55108 Received March 16, 1983.
Wright's (1932, 1969) "surface of selective value" is one of the most influential concepts in evolutionary genetics. The surface, also known as an "adaptive topography" or "evolutionary landscape," is an n-dimensional picture ofthe way that natural selection changes the genetic composition of a population. For one-locus models, the surface is defined by n - 1 independent allele frequency axes and one dimension for population "fitness." At any given time, a population can be represented as a point on the surface. Through time, the point moves on the surface in such a way that population "fitness" is always increasing, until equilibrium is attained at a maximal state of adaptation. For the well known deterministic model of viability selection at one locus, the notion of increasing population "fitness" until equilibrium is an exact mathematical theorem (Kingman, 1961), provided that certain assumptions are satisfied: mating occurs at random, mutation and immigration are negligible, genotypespecific viabilities are constant, segregation follows Mendelian ratios, and selection is independent of sex. The appropriate measure of population "fitness" under these circumstances is the average zygote viability, usually called "mean fitness." However, for many models that depart from the assumptions of the classical viability selection model, mean fitness is not necessarily an increasing quantity that is maximized at equilibrium. Following is a representative list of biologically significant situations in which the principle ofincreasing population fitness is known to be violated. Multiple Loci.-If an individual's viability depends on the genotype at two
Revised July 24, 1983
loci, then the population mean viability is not generally maximized at equilibrium (Moran, 1964). Mean viability can decrease from generation to generation, even though selection coefficients are constant (Karlin and Carmelli, 1975). The mean viability is nondecreasing if fitness is determined by additive interactions between loci (Ewens, 1969), but epistasis is often found when the appropriate experimental design is employed (e.g., Clark and Feldman, 1981a; see Barker, 1979, for a review). Wright (1969 p. 475) was aware of this limitation, noting that "Selection formulas for single loci can never be more than momentary approximations because ofthe practical universality of interactions with the other loci, ...." Meiotic Drive.-If heterozygotes produce a non-Mendelian ratio of gamete types, then mean viability at the polymorphic equilibrium is always less than maximal (Hiraizumi et al., 1960). There can exist a stable fixation equilibrium that is a minimum of the mean viability. Naturally occurring major meiotic drive elements have been documented in the rodents Myopus schisticolor (Fredga et al., 1976) and Mus musculus (Dunn, 1956), two lepidopteran species (Chanter and Owen, 1972; Smith, 1975), at least 12 species of Drosophila (Sturtevant and Dobzhansky, 1936; Stalker, 1961; Faulhaber, 1967; Hartl and Hiraizumi, 1976), two other dipterans (Hiroyoshi, 1964; Hickey and Craig, 1966), two species of Neurospora (Turner and Perkins, 1976), and several higher plants (Cameron and Moav, 1957; Loegering and Sears, 1963; Rectei, 1965; Carlson, 1969; Mimtzing, 1968). Further, there is evidence that populations of Drosophila carry minor elements that modify segregation (Hanks,
359
360
JAMES W. CURTSINGER
1965; Lyttle, 1979; Curtsinger, 1981; Hiraizumi and Gerstenberg, 1981; Curtsinger and Hiraizumi, pers. observ.). Gametic Selection. - Organisms that have extended phases in the haploid stage can be subject to selection on the basis of haplotype. Likely examples include some algae, Foraminifera, and many higher plants (Mulcahy, 1975). Theoretical analysis of combined haploid and diploid selection has been considered by Scudo (1967) who showed that the overall state of adaptation attained by the population is a compromise between haploid and diploid selection. Organismal Interaction.-If interactions between organisms influence viability in a nonsymmetrical way, then mean viability may not be maximized at equilibrium (Cockerham et aI., 1972). The idea that behavioral interactions constitute a major selective force is currently an active one, but its general significance in comparison with other modes of selection remains to be determined. The extensive literature of frequency dependent selection in experimental populations of Drosophila (Ayala and Campbell, 1974; Lewontin, 1974) is consistent with the idea that the genetic composition of a population is a significant part of an organism's environment, contributing to the biotic and abiotic factors that determine fitness. Fertility Selection.-If the number of progeny produced by a mating depends on the statistical interaction of parental genotypes, then the population mean fertility may not be maximized at equilibrium, may steadily decrease through time, and may even exhibit oscillations (Kempthorne and Pollak, 1970; Pollak, 1978). There is increasing evidence from experimental populations of Drosophila that fertility selection is often a primary source of net fitness differences between genotypes (Prout, 1971; Sved, 1971; Bungaard and Christiansen, 1972; Curtsinger and Feldman, 1980; Brittnacher, 1981; Clark and Feldman, 1981b). Selective Differences Between the Sexes. - Models incorporating sex-de-
pendent viability selection at an autosomal locus are formally equivalent to multiplicative fertility selection (Bodmer, 1965), and are thus subject to nonmaximization behavior. Various authors have argued for the likelihood of sex-dependent viability selection (Haldane, 1962; Li, 1963; Kidwell et aI., 1977; Wilson and Sved, 1979; Curtsinger, 1980; Lande, 1980). The prevalence of sexual dimorphism among higher animals can be taken as evidence for the general importance of sex-dependent selection and sexual selection. In light of the documented violations of the assumptions of the classical viability selection model, it is important to investigate more complex models, and to ask whether population "fitness" can be redefined in a meaningful way. In particular, a redefinition is sought such that population "fitness" increases through time to its maximal value at equilibrium. It is desirable that the newly defined "fitness" be interpretable as a measure of population adaptation. In this paper, one-locus deterministic models of meiotic drive and gametic selection are investigated. These modes of selection have been chosen because they are analytically the simplest cases in which the principle of maximized mean viability is violated. For meiotic drive and gametic selection models there is a clear problem in defining population "fitness," because selection can be thought of as operating in both haploid and diploid stages. The two modes of selection are analyzed together for conciseness and because of structural similarities in the models, but the biological distinctions are clear: meiotic drive means that heterozygotes produce non-Mendelian ratios of gamete types, as in the SD system of Drosophila melanogaster. In the gametic selection case, the fitness of haplonts depends only on haplotype, and is independent of the diploid genotype from which it is derived. The first step of the analysis involves an extension of Lewontin's (1958) method of weights, by which a general formula is obtained for deriving
361
EVOLUTIONARY LANDSCAPES
new maximization functions. A preliminary report of this method has appeared elsewhere (Curtsinger, 1983). EXTENDING THE METHOD OF WEIGHTS
Consider a large population ofdiploids having discrete generations and two alleles at an autosomal locus. In the parental generation, alleles A and a have relative frequencies p and q, respectively, where p + q = 1. For the moment, no assumptions are made about the mating system, zygotic selection, segregation parameters, mutation, or other factors that modify allele frequencies. In the offspring generation, the three genotypes occur in the following relative frequencies among reproductive-aged adults: AA
Aa
aa
p2U 11
2pqU12
q2U22 •
(1)
In general, the Us are not relative viabilities or Wrightian fitness values. They are weights, as defined by Lewontin (1958), taking on values such that relative genotype frequencies conform to the proportions indicated. The Us could be functions of allele frequencies, genotype frequencies, selection coefficients, and other parameters. Denoting allele frequencies in the offspring generation by pi and q', the allele frequency transformation equations are
,_ q2U22 + pqU I2 q - P2U11 + 2 pq U 12 + q2 U22' and pi = 1 - q'.
(2)
The change in allele frequency over one generation is b.q: !!.q = q' - q =
pq[V12
V" + q(V" - 2V12 + V 22 ) ] p2V" + 2pqV12 + q2V22
(3)
Equilibria of the system defined by (2), denoted q, are q = 0, q = 1, and A
q
=
U 11
U 11 - U 12 2U12 + U22 •
(4)
Now consider the special case of ran-
dom mating, Mendelian segregation, and constant viability selection. Then Uij = wij' where the wij are constant, and (3) reduces to Wright's well-known formulation: b.q = pq dW
(5)
2W dq ' where W =
p2 W I 1
+
+
2pqw I2
q2W22 >
o.
W is the mean fitness of the zygotes of the offspring generation; it is a function of the allele frequencies, but the argument is usually suppressed. Equation (5) defines an "adaptive topography" or "evolutionary landscape" upon which populations move through time. It shows that allele frequencies change in such a way that the mean fitness increases, except at equilibrium, justifying the notion that populations climb out of "adaptive troughs" and scale "adaptive peaks." Equation (5) also implies that stable equilibria lie at maxima of the W function while unstable equilibria lie at minima: W is a Liapunov function, or, in the jargon of evolutionary biology, a maximization function, for the special case of Mendelian segregation, random mating, and constant viability selection. The mean fitness function is not necessarily maximized or nondecreasing in cases that depart from the classical viability selection scheme. Suppose that there exists for those more complex cases some maximization function "T." (T is a function of allele frequencies, but the argument will be suppressed for notational simplicity.) Further suppose that T is defined such that b.q = pq dT 2[; dq' where D = p2U t l + 2pqU12
+
q2 U22 >
(6)
o.
Then, by comparison ofequations (3) and (6), the following expression is obtained for the hypothetical maximization function:
362
T
JAMES W. CURTSINGER
=
2
f
+
+
V 12 - VII
q(V t t - 2Vt2 + V 22) dq.
(7)
Equation (7) is the basic expression for defining new maximization function for complex selection at one locus. The integrand of(7) can be obtained in two ways: by solving explicitly for the weights satisfying (1) (see Lewontin, 1958), or by deriving equation (3). To illustrate the application of the extended method ofweights, consider a discrete generation model incorporating diploid viability selection, meiotic drive, and gametic selection. In a large population there exist haplotypes A and a with relative frequencies p and q, respectively, where p + q = 1. Haplonts fuse at random to form diploid zygotes AA, Aa, and aa with relative viabilities Wl l , w t2, and W 22' respectively. Following viability selection, surviving diploids produce haplonts by generalized segregation, heterozygotes producing (1 - m)/2 A-types and (1 + m)/2 a-types. If m > then meiotic drive favors the a allele, while m = corresponds to Mendelian segregation. Viability selection then discriminates among haplotypes A and a with relative survival probabilities equal to hand 1 respectively. The entire life cycle then repeats. Parameter constraints are wij 2: 0, h 2: 0, and -1 :5 m :5 + 1. Censusing the postselection haplont pool, the allele frequency transformation equations for an entire generation of the life cycle are
°
WHDq'
= q2W22 + pqw 12 (1 +
°
m)
and p' = 1 - q', where W HD = p2Wl l h + pqw t2 .(1 + h + m - hm) + q2W22.
(8)
The change in allele frequency from generation to generation is WHv!lq = pq[wd1
+
m) - WIth
q(W1lh -
.(1
+
+
W22)].
m
+
W l2
h - hm) (9)
There are at most three equilibria: q = 0, q = 1, and •
q=
wllh - w.,(l + rn) wllh - w.2 ( 1 + rn + h - hrn)
+ W2 2
.
(10)
The equilibrium structure is governed by the following inequalities: a. If w l2h(1 - m) > W22 and wdl + m) > WIth then the polymorphic equilibrium exists, is stable, and the boundaries are unstable. If the inequality signs are reversed then the polymorphic equilibrium is unstable and the boundaries are stable. b. If w 12 h(1 - m) < W 22 and wdl + m) > WIth then the polymorphic equilibrium does not exist, q = 1 is stable, and q = is unstable. If the inequality signs are reversed then q = 1 is unstable and q = is stable.
°
°
In all cases allele frequencies converge monotonically to equilibrium (see Appendix). The maximization function is obtained by integrating the term in brackets on the right side of (9):
T
=
q2[wt t h - w12 ·(1 + m + h - hm)
+ +
2q[wdl
+
+
W 22]
m) - Wtth]
(11)
const
where "const" is a constant of integration. Note that when it exists the polymorphic equilibrium (10) corresponds to the critical point of T; in particular, the critical point is a maximum if q is stable and is a minimum if q is unstable. Critical points of W do not coincide with q of (10) if m or h 1. Given that T is a quadratic function of q, that it is locally maximized at stable
*' °
*'
363
EVOLUTIONARY LANDSCAPES
equilibria and locally minimized at unstable equilibria, and that allele frequencies converge monotonically to equilibria, it follows that T is nondecreasing. The change in T from generation to generation is t:J.T
=
pqWHD-ZZZ ·[WHD + pwllh
+
qwzz]'
(12)
where Z = wdl + m) - wl1h + q[wllh - w12 (1 + m + h - hm) + wn ]. t:J.T:::: 0, with equality holding only at q = 0, q = 1, or q = q of (10), the polymorphic equilibrium. Three special cases are of particular interest: a. If m = 0 and h = I then we have the classical viability selection model. In this case T differs from W (the usual diploid mean viability) by a constant of integration. b. If h = I and m *- 0 then we have the autosomal meiotic drive model of Hiraizumi et al. (1960). c. If h *- I and m = 0 then we have the gametic selection model of Scudo (1967). The extended method of weights unites the three cases by providing a single maximization function. Now consider the application of this method to the case of n alleles A I' Az, , An having relative frequencies PI' Pz, , Pn (~Pi = 1). For simplicity only the case h = 1 will be considered, though the conclusion applies to other cases. Individuals of the genotype A;Aj produce Ai-bearing and Arbearing gametes in the proportions (1 + m;)l2 and (1 + mj;)/2, respectively, where -1 -s m =::; + 1, mij = - m ji, and m; = O. The allele frequency transformation equations are Wp;'
=
PiWi
(i
=
1,2, ... ,n)
where Wi = ~ pjwij(1
+ m i)
j
W
= ~ ~ PiPjWij. j
,
(13)
One-generation changes in allele frequencyare Wt:J.Pi = Pi(Wi -
W).
(14)
By analogy with the Mendelian n-allele model, we seek a maximization function T such that t:J.Pi
=
Pi(1 2W
pJ bT bpi·
(15)
Functions satisfying (15) are (16)
For n > 2, equation (16) generates a number of different T, functions that are not generally well behaved. Functions are not constrained to finite values and can be shown to exhibit nonmonotone behavior. There might exist a maximization function for the multiple-allele model, but it is not generated by the method described here. It is conjectured that the extended method of weights fails where allele frequency change is nonmonotonic. DISCUSSION
How can one best describe the way that natural selection changes the genetic composition of a population? The common notion is that natural selection improves "population fitness," but that quantity is not readily defined with either precision or generality. The maximization ofmean viability is an appealing idea, governing the behavior of the classical viability selection model, but it is not robust. In particular, relaxing the assumptions concerning the mode of selection, the segregation pattern, the number ofloci, the constancy of the selection parameters, or the structure of the life cycle can result in nonmaximization behavior. Meiotic drive, gametic selection, epistasis, frequency dependence, and other complex modes of selection are real phenomena; their existence challenges the common notion of adaptation. In this paper a new approach to the
364
JAMES W. CURTSINGER
study of complex modes of selection has been developed. By extending Lewontin's (1958) method of weights, it is possible to define a quantity that is maximized at stable equilibria, minimized at unstable equilibria, and always increasing except at equilibrium. The strength of this approach lies in its generality: the derivation entails no specific assumptions other than a discrete generation treatment of a one-locus, two-allele deterministic model. Because of the generality of equation (7), it is possible to obtain maximization principles for a variety of deterministic models, including frequency dependent selection, nonrandom mating, inbreeding, and other cases. Equations (11) and (6) define evolutionary landscapes for meiotic drive and gametic selection in the same way that Wright's mean viability principle defines an adaptive topography for the classical viability selection model. In fact, the usual mean fitness principle is a special case of equations (11) and (6). While the generality of the extended method of weights has appeal, it also has limitations. The method fails ifthere are more than two alleles, though the advantages of being able to describe maximization principles for many modes of selection may outweigh the limitation associated with the number of alleles. A second limitation is that the method has no obvious application to an important class of complex models: selection at two or more loci. Third, some modes of selection may generate nonintegrable functions of weights. Finally, the maximization principles for meiotic drive and gametic selection are remarkably simple, but defy simple explanation. The quantities succeed in combining the effects of selection in haploid and diploid stages, but contain no simple combination ofintuitively defined haploid and diploid "fitness." Nor are the quantities obviously related to gene "transmissibility." The question of interpreting these maximization principles is an important one that will be taken up in a later paper; the T functions will be shown to be related to Fisher's "average effect."
No work in theoretical population genetics is totally new. There is a scattered literature on the derivation of maximization principles for various models that depart from the assumptions of the classical viability selection model. Li has employed heuristic methods to derive maximization principles for the cases of spatial heterogeneity in selection coefficients, inbreeding, compensation (1955), and selection at a sex-linked locus (1967). Levins and MacArthur (1966) and Cannings (197 1) have furthered the analysis of selection in multiple niches. Wright (1955, 1969) has defined a maximized "fitness function," distinct from the usual mean fitness or mean viability, that always exists in the case of a character determined by a single pair of alleles. A survey of standard population genetics texts suggests that the "fitness function" concept has had minimal impact (but see Arnold and Anderson, 1983). The fitness function seems to be related to the ideas developed in this paper, and deserves further analysis. Another related approach is the extension of Fisher's Fundamental Theorem of Natural Selection to take into account variable fitness and nonrandom mating (Kimura, 1958; Crow and Kimura, 1970), multiple loci (Turner, 1981), meiotic drive (Hartl, 1970), and selection on a sex-linked locus (Hartl, 1972). The outcome of those analyses is an expression for the rate of change in mean fitness, in the form of additive genetic variance plus a correction term, rather than the direct definition ofa maximization function. It can be concluded that, in spite of its limitations, the extended method of weights has much potential as a general description of the effect of natural selection. The application of the method to other complex modes ofselection and the interpretation ofthe maximization function will be taken up in subsequent papers. SUMMARY
Wright's principle of maximized mean viability is known to be violated in a number of biologically significant situa-
EVOLUTIONARY LANDSCAPES
tions involving complex modes of selection, including epistasis, meiotic drive, organismal interaction, fertility selection, selective differences between the sexes, and gametic selection. In this paper, Lewontin's (1958) method of weights is extended to derive a very general maximization principle for general one-locus, two-allele deterministic selection models. New maximization principles are developed for the cases of meiotic drive and gametic selection, thereby defining evolutionary landscapes. The extended method of weights has considerable potential as a general descriptor of complex selective processes. ACKNOWLEDGMENTS
I thank Dr. Richard Lewontin for his helpful comments in the early stages of this work. I also thank Drs. Lisa Brooks, Chris Cannings, Andrew Clark, and Larry Jacobson for their criticisms of the manuscript. Drs. Jonathan Arnold and Richard Lewontin made helpful comments following the presentation of this material at the 1983 Annual Meetings of the Genetics Society of America. Dr. Cannings suggested a concise proof of convergence. Ms. Kae Ebling provided expert secretarial assistance. Research is supported by NSF Grant No. BSR8211667. LITERATURE CITED ARNOLD, J., AND W. W. ANDERSON. 1983. Density-regulated selection in a heterogeneous environment. Amer. Natur. 121:656-668. AYALA, F. J., AND C. A. CAMPBELL. 1974. Frequency-dependent selection. Ann. Rev. Ecol. Syst.5:115-138. BARKER, J. S. F. 1979. Inter-locus interactions: a review of experimental evidence. Theoret. Pop. BioI. 16:323-346. BODMER, W. F. 1965. Differential fertility in population genetic models. Genetics 51:411-424. BRITTNACHER, J. G. 1981. Genetic variation and genetic load due to the male reproductive component of fitness in Drosophila. Genetics 97: 719-730. BUNGAARD, J., AND F. B. CHRISTIANSEN. 1972. Dynamics of polymorphisms. 1. Selection components in an experimental population of Drosophila melanogaster. Genetics 71:439-460. CAMERON, D. R., AND R. MOAV. 1957. Inheritance in Nicotiana tabacum. XXVIII. Pollen killer, an alien genetic locus inducing abortion
365
of microspores not carrying it. Genetics 42:326335. CANNINGS, C. 1971. Natural selection at a multiallelic autosomal locus with multiple niches. J. Genet. 60:255-259. CARLSON, W. R. 1969. Factors affecting preferential fertilization in maize. Genetics 62:543554. CHANTER, D.O., AND D. F. OWEN. 1972. The inheritance and population genetics of sex ratio in the butterfly Acraea encedon. J. Zool. 166: 363-383. CLARK, A. G., AND M. W. FELDMAN. 1981a. Estimation ofepistasis in components offitness in experimental populations of D. melanogaster. II. Assessment of meiotic drive, viability, fecundity, and sexual selection. Heredity 46:347377. - - . 1981b. Density dependent fertility selection in experimental populations of Drosophila melanogaster. Genetics 98:849-869. COCKERHAM, C. c., P. M. BURROWS, S. S. YOUNG, AND T. PROUT. 1972. Frequency dependent selection in randomly mating populations. Amer. Natur. 106:493-515. CROW, J. F., AND M. KIMURA. 1970. An Introduction to Population Genetics Theory. Harper and Row, N.Y. CURTSINGER, J. W. 1980. On the opportunity for polymorphism with sex-linkage or haplodiploidy. Genetics 96:995-1006. - - . 1981. Artificial selection on the sex ratio in Drosophilapseudoobscura. J. Hered. 72:377381. - - . 1983. Evolutionary landscapes for complex selection. Genetics 104:s20. CURTSINGER, J. W., AND M. W. FELDMAN. 1980. Experimental and theoretical analysis ofthe "sexratio" polymorphism in Drosophila pseudoobscura. Genetics 94:445-466. DUNN, L. C. 1956. Analysis ofa complex gene in the house mouse. Cold Spring Harbor Symp. Quant. BioI. 21:187-195. EDWARDS, A. W. F. 1972. Foundations of Mathematical Genetics. Cambridge Univ. Press, Cambridge. EWENS, W. J. 1969. Mean fitness increases when fitnesses are additive. Nature 221:1076. FAULHABER, S. H. 1967. An abnormal sex ratio in Drosophila simulans. Genetics 56: 189-213. FREDGA, K., A. GROPP, H. WINKING, AND F. FRANK. 1976. Fertile XX- and XY-type females in the wood lemming Myopus schisticolor. Nature 261 :225-227. HALDANE, J. B. S. 1962. Conditions for stable polymorphism at an autosomal locus. Nature 193:1108. HANKS, G. D. 1965. Are deviant sex ratios in normal strains of Drosophila caused by aberrant segregation? Genetics 52:259-266. HARTL, D. L. 1970. Population consequences of non-Mendelian segregation among multiple alleles. Evolution 24:415-423. - - . 1972. A fundamental theorem of natural
366
JAMES W. CURTSINGER
selection for sex linkage or arrhenotoky. Amer. Natur. 106:516-524. HARTL, D. L., AND Y. HIRAIZUMI. 1976. Segregation distortion, p. 615-666. In M. Ashburner and E. Novitski (eds.), Genetics and Biology of Drosophila, Vol. lb. Academic Press, N.Y. HICKEY, W. A., AND G. B. CRAIG, JR. 1966. Genetic distortion ofsex ratio in a mosquito, Aedes aegypti. Genetics 53:1177-1196. HIRAIZUMI, Y., AND M. V. GERSTENBERG. 1981. Gametic frequency of second chromosomes of the T-007 type in a natural population of Drosophila melanogaster. Genetics 98:303-316. HIRAIZUMI, Y., L. SANDLER, ANDJ. F. CROW. 1960. Meiotic drive in natural populations of Drosophila melanogaster. III. Population implications of the segregation-distorter locus. Evolution 14:433-444. HIROYOSHI, T. 1964. Sex-limited inheritance and abnormal sex-ratio in strains of the housefly. Genetics 50:373-385. KARLIN, S., AND D. CARMELLI. 1975. Numerical studies on two-loci selection models with general viabilities. Theoret. Pop. BioI. 7:399-421. KEMPTHORNE, 0., AND E. POLLAK. 1970. Concepts of fitness in Mendelian populations. Genetics 64:125-145. KIDWELL, J. F., M. T. CLEGG, F. M. STEWART, AND T. PROUT. 1977. Regions of stable equilibria for models of differential selection in the two sexes under random mating. Genetics 85: I 71183. KIMURA, M. 1958. On the change of population fitness by natural selection. Heredity 12:145167. KINGMAN, J. F. C. 1961. A mathematical problem in population genetics. Proc. Cambridge Philos. Soc. 57:574-582. LANDE, R. 1980. Sexual dimorphism, sexual selection, and adaptation in polygenic characters. Evolution 34:292-305. LEVINS, R., AND R. MACARTHUR. 1966. The maintenance ofgenetic polymorphism in a spatially heterogeneous environment: variations on a theme by Howard Levene. Amer. Natur. 100: 585-589. LEWONTIN, R. C. 1958. A general method for investigating the equilibrium ofgene frequency in a population. Genetics 43:421-433. - - - . 1974. The Genetic Basis of Evolutionary Change. Columbia Univ. Press, N.Y. LI, C. C. 1955. The stability of an equilibrium and the average fitness of a population. Amer. Natur. 89:281-296. - - - . 1963. Equilibrium under differential selection in the sexes. Evolution 17:493-496. - - - . 1967. The maximization of average fitness by natural selection for a sex-linked locus. Proc. Nat. Acad. Sci. USA 57:1260-1261. LOEGERING, W. Q., AND E. R. SEARS. 1963. Distorted inheritance ofstem-rust resistance oftimstein wheat caused by a pollen-killing gene. Can. J. Genet. Cytol. 5:65-72.
LYTTLE, T. W. 1979. Experimenta1populationgenetics of meiotic drive systems. II. Accumulation of genetic modifiers of Segregation Distorter (SD) in laboratory populations. Genetics 91:339-357. MORAN, P. A. P. 1964. On the nonexistence of adaptive topographies. Ann. Hum. Genet. 27: 383-393. MULCAHY, D. L. (ed.). 1975. Gamete Competition in Plants and Animals. North-Holland, Amsterdam. MONTZING, A. 1958. A new category of chromosomes. Proc. X Int. Genet. Congr. 1:453467. POLLAK, E. 1978. With selection for fecundity the mean fitness does not necessarily increase. Genetics 90:383-389. PROUT, T. 1971. The relation between fitness components and population prediction in Drosophila. II. Population prediction. Genetics 68: 151-168. REDEl, G. P. 1965. Non-Mendelian megagametogenesis in Arabidopsis. Genetics 51:857-872. SCUDO, F. M. 1967. Selection on both haplo and diplophase. Genetics 56:693-704. SMITH, D. A. S. 1975. All-female broods in the polymorphic butterfly Danaus chrysippus L. and their ecological significance. Heredity 34:363371. STALKER, H. D. 1961. The genetic systems modifying meiotic drive in Drosophila paramelanica. Genetics 46: 177-202. STURTEVANT, A. H., AND TH. DOBZHANSKY. 1936. Geographical distribution and cytology of sexratio in Drosophila pseudoobscura and related species. Genetics 21:473-490. SVED, J. A. 1971. An estimate ofheterosis in Drosophila melanogaster. Genet. Res. 18:97-105. TURNER, B. c., AND D. D. PERKINS. 1976. Spore killer genes in Neurospora. Genetics 83:suppl. s77. TURNER, J. R. G. 1981. "Fundamental theorem" for two loci. Genetics 99:365-369. WILTON, A., AND J. SVED. 1979. X-chromosomal heterosis in Drosophila melanogaster. Genet. Res. 34:303-315. WRIGHT, S. 1932. The roles of mutation, inbreeding, crossbreeding, and selection in evolution. Proc. VI Int. Congr. Genet. 1:356-366. - - - . 1955. Classification of the factors ofevolution. Cold Spring Harbor Symp. Quant. BioI. 20:16-24. - - - . 1969. Evolution and the Genetics of Populations, Vol. II. Univ. Chicago Press, Chicago. Corresponding Editor: J. Felsenstein
ApPENDIX Proof of monotone convergence to equilibrium by an extension of Edwards' (1972) Theorem 2.2.1 : Define r = p/q. Then
EVOLUTIONARY LANDSCAPES r'
=
r.
p2Wllh + pqW12h(1 - m). q2W22 + pqW 12(l + m)
=
q'
Equilibria are f
=
r = r*
0, f =
and
c.
W12h(l - m) - wn . wdl + m) - wllh
d.
= 00,
Some algebraic manipulation leads to r' - r*
= (r -
r*)Q,
where Q = (rwllh + wn)/(rw 12(l + m) + W22). There are six cases to be analyzed. a. If wllh < w12(l + m) and W22 < w 12h(l - m), then r* > and Q < 1. It follows that r' lies between rand r*, indicating monotonic convergence to the polymorphic equilibrium. b. If wllh > W 12 (1 + m) and W22 > w 12h(l - m),
°
e. f.
367
then r* > 0, Q > I, and rlies between r' and r*, indicating monotonic divergence from the polymorphic equilibrium. If wllh < wdl + m) and W22 2:: w 12h(1 - m), then r* ::'0 0, Q < I, and r decreases monotonically to r = 0. If wllh > wdl + m) and W22 ::'0 w 12h(1 - m), then r* ::'0 0, Q> I, and r increases monotonically to r = 00. If wllh = w12(l + m) and Wn #' w 12h(1 - m), then we redefine r = qlp and proceed as above. If wllh = wdl + m) and W22 = w12h(1 - m), then r' = r.
Thus r is either at equilibrium or monotonically converging to equilibrium. Because zsrand!:>.p must have the same sign, allele frequencies are also monotonically convergent.
