GENETIC LOADS IN HETEROGENEOUS ENVIRONMENTS CHARLES E. TAYLOR1 Department of Genetics, University of Culifornia, Davis, California 95616
Manuscript received August 1, 1974 Revised copy received January 6, 1975 ABSTRACT
A model of population structure in heterogeneous environments is descnbed with attention focused on genetic variation a t a single locus. The existence of equilibria at which there is no genetic load is examined.-The absolute fitness of any genotype is regarded as a function of location in the niche space and the population density at that location. It is assumed that each organism chooses to live in that habitat in which it is most fit (“optimal habitat selection”).Equilibria at which there is no segregational load (“loadless equilibria”) may exist. Necessary and sufficient conditions for the existence of such equilibria are very weak. If there is a sufficient amount of dominance or area in which the alleles are selectively neutral, then there exist equilibria without segregational loads. In the N , p phase plane defined by papulation size, N , and gene frequency, p , these equilibria generally consist of a line segment which is parallel to the p axis. These equilibria are frequently stable.
natural selection acts simultaneously at many loci is enormously compliH:gd. This process is frequently simplified by assuming that genotypic fitnesses are constant and that each locus contributes independently to the total fitness of an organism. But these simplifications sometimes lead to erroneous conclusions. A notable example is calculations of genetic load based on small but constant selection coefficientsat a large number of loci which act independently. These calculations yield estimates of genetic load requiring much more variation in fitness than is actually observed in nature (LEWONTIN and HUBBY 1966; CROW1970). This discrepancy may be due to one or more of the following factors: (1) loci do not independently determine fitness (KING 1967; SVED,REEDand BODMER 1967; MILKMAN 1967; FRANKLIN and LEWONTIN 1970) ; (2) selection coefficients are only trivially different from unity (KINGand JUKES1969; ARNHEIM and TAYLOR 1969; KIMURA and OHTA 1971); and (3) selection coefficients are not constant, but frequency-dependent (MAYR1963; KOJIMAand YARBROUGH 1967). The first two possibilities have received much theoretical investigation but little experimental verification. Frequency-dependent selection, in turn, has received much experimental validation (e.g., LEWONTIN 1955; ALLARD and ADAMS 1969; TAYLOR and SOKAL 1973), but theoretical studies have been comparatively few. Where the ramifications of frequency-dependent selection for Present address: Department of Biology, University of California, Riverside, California 92502 Genetlcs 8 0 : 621-635 July, 1975
C . E. TAYLOR
genetic loads have been examined the results are mixed: in some instances loads are reduced (CLARKE 1972) and in other cases there is no effect (NEI 1971), and they may even be increased (OHTA and KIMURA1971). I n this paper a model for frequency-dependent selection is examined to determine how it may resolve the difference between the theory and observation of genetic loads. The model studied beIow has been elaborated elsewhere (TAYLOR 1973,1974). It assumes: (1) fitness is density-dependent; (2) the various genotypes differ in their fitnesses in the available niches; and ( 3 ) individual behavior is adaptive, SO that habitat preferences are correlated with fitnesses in the chosen habitat. Elsewhere it is shown that conditions f o r the maintenance of polymorphisms are 1974). It will be shown here that quite unrestrictive with this model (TAYLOR conditions for equilibria without segregational loads are similarly lax. The primary feature of this paper is the mathematical description of how a preference for favorable environments might affect population structure. This description is surprisingly difficult, and only perfect choice will be discussed. It should be pointed out that several authors have made verbal arguments pointing to the ease with which loads might be reduced by such a model (e.g., MAYR1963, p. 244; DOBZHANSKY 1970, p. 225; LEWONTIN 1974, p. 259). This paper focuses on polymolrphisms at a single locus; multi-locus models will be discussed elsewhere. I will begin with a brief description of the model and derivation of conditions for load-free equilibria, then discuss the stability of such equilibria, and finally apply these results to the load problem. THE MODEL
The set of all possible environmental states is termed the Hutchinsonian niche space, S. It is convenient, though not necessary, to assume that S can be described by one axis, n, which corresponds to density of the species in question, and a set of m other axes orthogonal to this density axis. The environment is assumed to be heterogeneous in space but homogeneous in time. Thus, each physical location in the environment can be described by a vector x (x = (xr,. . . , z,) together with a density, n. The set of all environmental states in the region inhabited by the population in question is termed the niche of that population. I t is assumed that this niche can be described by a function cp(x). This function describes the frequency of environmental states available to individuals of the population. For simplicity it is assumed that Q, (x)is constant through time. The niche describes the environment in which a n organism might live. The subset of the niche in which an individual actually lives is termed that individual’s habitat. Each habitat is thought to correspond to a single x. For example, the amount of a lake at each temperature, x, might be described by @ (x). This would be the niche of a trout population. That part of the lake in which an individual trout was to be found, its habitat, would be described by a single x. This usage resembles the standard ecological nomenclature proposed by HUTCHINSON (1957) and by WHITTAKER, LEVINand ROOT(1973) but is not identical to it.
The habitat chosen by any particular individual depends on its fitness there.
It is assumed that each individual chooses to dwell where its fitness is greatestcalled “optimal habitat selection”. The assumption of optimal habitat selection is the distinguishing feature of this model. The consequences of this assumption are discussed below. A n individual‘s fitness is the expected number of zygotes produced by that individual for the next generation. This is an “absolute” rather than “relative” fitness and is represented by the function Wij(x,n). Such a fitness depends on the genotype, the type of habitat, and the density of conspecifics there. It will be assumed that Wij (x,n)is continuously differentiable and strictly decreasing with
n, so that - Wij (x,n)< 0 whenever Wcj (x,n)# 0. For a two-dimensional niche an space W i j ( x , n ) is a surface, shown graphically in Figure 1. In general, each genotype would have a different such fitness surface. The life cycle is assumed to consist of discrete generations. At the beginning of each generation the zygotes are released into a niche characterized by cp (x). These organisms may redistribute themselves by habitat selection, each selecting a habitat where its fitness is greatest. Measurements of density are made after this redistribution. The zygotes then mature to adulthood. Differences in fitness correspond to differences in survival to adulthood. When mature, the adults shed gametes which combine at random to produce zygotes for the next generation. All adults then die, and a new generation begins. LOADLESS EQUILIBRIA
The genotypes in the population, their frequencies and mean fitnesses may be represented as follows: AA Aa aa genotype: frequency: P2 2p(l-p) (1--P)2 fitness: c11 ClZ cz 2
FIGURE1.-Fitness surface in a two-dimensional niche space. Fitness ( W ) , is shown to depend upon both the type of niche (x), and density (n).In general the niche space would be represented by a surface d many dimensions. The isocline on this fitness surface where W ( x , n ) = 1 determines the population densities indicated by k ( x ) .
C . E. TAYLOR
The mean fitness of the population,
calculated from the formula
At equilibrium = 1. This paper is concerned largely with relating the function Wi,(x,n)to the constants ci, when there is no genetic load. Genetic load, L, is defined by the equation
Cmad L=--.-.-. CUlax
where cmax is the mean fitness of that genotype with the greatest fitness. Represent the populatioa size by N and generation by t . If there is no change in population structure over one generation, and
then the population is said to be at a loadless equilibrium. To have a loadless equilibrium it is necessary that the maximum fitness equal the mean fitness, that is, cl1= clz = cZ2,while the genotypes are present in HardyWeinberg proportions. In this absence of load no one genotype is more fit than any other, guaranteeing that gene frequencies are not changing. The only additional requirement for an equilibrium is an unchanging population size. 1. Thus, necessary and sufficient conditions for a loadless This implies that equilibrium are CIl = Cl? = c22 -1 (1) while the genotypes are in Hardy-Weinberg proportions. The next task is to find the fitness surfaces and patterns of environmental heterogeneity f o r which p and N exist and satisfy the conditions of loadless equilibria. The procedure is first to find how an environment can be divided up by the population so that each individual is most fit where it is and at the same time has a fitness of 1; and secondly, to determine if this division of the environment corresponds to a possible division by genotypes in Hardy-Weinberg equilibrium for some gene frequency. If it does not, then no loadless equilibrium exist for this combination of environments and fitness surfaces. If, however, such a correspondence does exist, a loadless equilibrium exists; and the gene frequencies and population size allotwing for such a division of the environment correspond to that loadless equilibrium. The set of all such combinations constitutes the set of conditions for the existence of loadless equilibria. The equilibria distribution of organisms depends on the fitness surface of each genotype. Consider first a population comprising one genotype and practicing optimal habitat selection. Because each individual is assumed to dwell where its fitness is greatest, all individuals are equally fit. (Otherwise those who were less fit in one location could improve their fitness by relocating elsewhere. This would be contrary to hypothesis. So all must be equally fit already.) The density of individuals in each habitat, n(x), would be such that it defines a level curve, or isocline, on the fitness surface. For some constant c, n ( x ) is thus defined
implicitly by the equation Wij(x,n(x)) =c. In particular, a population composed entirely of the ijthgenotype when the mean fitness is 1 defines a special function, kif(x): wij ( X , k i j (x)) = 1 . This function, shown in Figure 1, is similar to the carrying capacity of the various niches for that genotype. When the population comprises several genotypes the population density in each habitat is difficult to calculate if fitnesses are arbitrary. But they are easily calculated at loadless equilibria. The mean fitness of each genotype is 1 at such an equilibrium (Equation 1 ), so individuals of one genotype would be found only in those regions where its own k t j ( x )was greater than that of the others. (Otherwise there could be relocation with subsequent improvement of fitnesses, contrary to the hypothesis of optimal selection,) No individual of that genotype would be found where the population density was greater than kij (x). Consequently, the total population density at each x, k(x), (= % kij(x)), would be 'L3
k (XI = max (kll (x),klz(x)$22 (x) ) . The niche space can be partitioned into areas according to which genotype has a superior kij (x) as shown in Figure 2. These regions into which the niche space is divided need not be contiguous nor have equal carrying capacities. But because of the restrictions on where each genotype will live, the genotypic composition of individuals living in environments associated with each area of the niche space may be determined. Let each of the regions in which any one genotype is superior (R,)and the number of individuals in that region (a;) be labeled as follows:
( > k 1 Z (x),kz,(x>
FIGURE&.-Appartianment of the niche space into regions (R;),or set d points, in which one genotype OT another is superior. At a loadless equilibrium the number d individuals found in each region is 51,.
C. E. TAYLOR
( > kll
(x)) k , , ( x ) = k,,(x) (> k22(X) 1 kl, (XI = k,, (4 (> kl, (XI kl,(X) = kzz(x) (> kll(X)) k,,(x) = k , , ( x ) = k,,(x)
a 4 0 5
a 6 a 7
ai =jRs @(x)k(x)dx at loadless equilibria. The R; divide the niche space into disjoint regions. The total number of organisms must equal the sum of individuals in each region. Thus,
@(x)k(x)dx = Ea2, .
Because this value of I\i is unique there is a unique population size at which loadless equilibria can occur. Genotype frequencies at loadless equilibria can be calculated. The total number of individuals of each genotype in each region, summed over all regions, must equal the total number of individuals of that genotype. For example, in R, only the A,A, individuals have a fitness as high as 1 so only A,A, individuals will live there, and the composition of the o1individuals in this region will be entirely A,A,. I n R4, where there are a4 individuals, both A,A, and A,A, individuals might live, for k,, (x)and k,, (x)are equal over this region. The genotypic composition at any point in the region is therefore indeterminate. But when integrated over the entire region the individuals must be composed of A,A, and AIAz individuals in the proportions. say, a and 1-a. The other regions of niche space in which AIAl individuals might live are R5 and R,. The numbers of organisms living in these regions are as, of which a proportion, b, are A,A,, and a7,of which the proportion, f , are A,A,. This exhausts the locations where A,A, individuals might live. Thus, a total of a, an, -I- ba, fa7individuals with the A,A, genotype must be somewhere in the environment at a loadless equilibrium. But if the gene frequency at equilibrium is p”, then the number of A,A, indi-
viduals at equilibrium must be fi2i$. Thus, at equilibrium,
JPN = 0, an4 bo, f0, . (3) The total number of heterozygotes and A,A, homozygotes may be similarly described: 2$( 1 - $)& = 0 2
(1 - a)n,
+ ca6 + f a 7
= Q 3 -k ( 1 - b)n, ( 1 - c)06 (1 - f - f ) a 7 , (5) where ay by cyf., f‘., and f -I-f’ represent proportions which lie between zero and unity. (1 - $)‘l\j
If there does exist a gene frequency p* which satisfies Equations (3)-(5), it represents an equilibrium gene frequency at which there is no segregational load, If no such value of p^ exists then there is no p and N with cI1= c12= cZ2,and therefore no loadless equilibrium exists. Thus, the existence of solutions to equations (3)-( 5 ) represents necessary and sufficient conditions for the existence of loadless equilibria. Moreover, since I? is uniquely determined, if more than one solution to these equations exists the different solutions represent different gene frequencies. Conditions for the existence of loadless equilibria may be viewed another way. The solution to equations (3)-(5) is the intersection of solutions to the set of equations
+ f0,)fi-l H = (0, + (1 + + R = (Q, + (1 - b)Q5+ (1 + (1 - f-f’)Qi)fi-’ D = (Q, 4- aQ, 4- bn,
(subject to the restrictions on a, byc, f, and j’) with the parametric curve
p;(l-pj olpll (7) (l-P)2 of Hardy-Weinberg equilibria. Whether the image of (6) intersects this parametric curve determines if a solution exists to (3)-(5). The set of all p for which the Hardy-Weinberg curve intersects the image of (6) is the set of all loadless equilibria. This alternative view makes possible a graphical description of the conditions for loadless equilibria. Both the image of (6) and the Hardy-Weinberg curve can be represented in a de Finetti diagram (LI 1955), An example of a de Finetti diagram is shown in Figure 3. It is an equilateral triangle corresponding to the plane H = 1 - D - R in the octant of R3 in which all of H , D, R, are 2 0. The scale has been adjusted so that the perpendicular lengths from any interior point to that side labeled D, H , or R represent the D, H , R component of that point. Two additional properties of these diagrams are that: (1) the perpendicular projection from any interior point onto the side labeled H divides this side into two portions, the relative lengths of which are equal to the gene frequency. p , of that point and its complement, 1 - p ; and (2) the locus of points describing Hardy-Weinberg equilibria is a parabola in the de Finetti diagram as shown in Figure 3. For proofs of these assertions and for additional properties see LI (1955). The set of genotypic proportions permitting loadless equilibria can be illustrated in these de Finetti diagrams. Referring to Figure 2, homozygous A A individuals would have a fitness of 1 anywhere in region R,, R,, R5,or R, at a loadless equilibrium. Anywhere else in the habitat the density of individuals would be > k , , ( x ) with the fitness of A A individuals < 1. The number of A A individuals consequently must be > n1 but 5 -I- Q 4 R5 0;. Therefore D
C . E. TAYLOR
H FIGURE 3.-de
Finetti diagram. Each combination of genotype frequencies colmsponds to a point in the triangle. The parabola shows the set of Hardy-Weinberg equilibria. The hexagon illustrates an example of genotype frequencies at loadless equilibrium.
lies between two lines drawn parallel to the D axis in Figure 3. I n a similar fashion, the number of A a individuals at equilibrium must be 2 n2,but 5 Q2 4fi4 Q, n7between lines parallel to the H axis) and the number of aa individuals 2 n3but 5 a3 a4 fi6 o7 (between lines parallel to the R axis). The intersection of these conditions is the hexagon in the middle of the diagram. Such a hexagon exists when all of the 0% are non-zero. If one or more &i are equal to zero the hexagon may degenerate to a parallelogram, line, or point, depending upon which of the 0;'s disappear. This hexagon is the set of genotypic proportions which might constitute a population at loadless equilibrium. If the hexagon intersects the parabola of Hardy-Weinberg equilibria, then all gene frequencies which are in the intersection of the hexagon and the parabola represent equilibria without genetic loads. If there is no intersection then there is no gene frequency p which will simultaneously satisfy equations ( 3 ) - ( 5 ) , and so there is no gene frequency which, at Hardy-Weinberg equilibrium, permits the existence of an equilibrium without incurring a genetic load. If and only if the genotypic proportions circumscribed by the hexagon are also Hardy-Weinberg equilibria will there be points of loadless equilibria. All positions and sizes of hexagons are possible. ?"he sets of equilibrium gene frequencies may be: ( 1 ) a closed interval of gene frequencies; (2) two closed intervals of gene frequencies; or finally, ( 3 ) a single unique gene frequency. Which of these possibilities is eventually realized depends upon the shape and location of the hexagon, which in turn depends upon the Q'S. TWO cases will be discussed in more detail-one is the case of complete dominance and the other is its opposite where there is neither any dominance nor neutrality.
+ + +
In the case of complete dolminance, let it be A , which is dominant olver the A , allele throughout the niche space. Assume that there is no neutrality. Then kl,(x) = k,, (x),and either
ICl2 (x),kz,(x) > kll (x) or kll (x) > klz (4,kzz( 4 except over regions of zero volume. Under such circumstances only Ol and a6 are non-zero. The hexagon in this case degenerates to a straight line segment along D = ~ ~ / f Since i . fi = nl,+a6 (from equation 2),and because the degenerate hexagon intersects the Hardy-Weinberg parabola where D=p*2, there is a loadless equilibrium with
Thus, with complete dominance throughout the niche space, there is always a unique gene frequency equilibrium at which there is na genetic load. If there is neither dominance nor neutrality, however, then only one of kll(x), klz(x), and k,, (x) is superior in the niche space except over regions of zero volume. In such a case only nl,a,,as, are non-zero. From equation (6) it is apparent that the hexagon degenerates to a single point, (D,H,R) ( O l , ~ , , 0 3 ) . Unless this point itself lies on the Hardy-Weinberg parabola there will be no equilibrium without a genetic load. The foregoing examples are special cases. In general there should be areas in the niche space in which one allele is dominant, other areas in which the olther allele is dominant, and perhaps areas of heterozygote inferiority, areas of neutrality, and those in which the fitness of one genotype is uniquely superior. Consequently each ni would generally be non-zero. Even when these numbers are small, the polygon formed is a non-degenerate hexagon and may intersect the Hardy-Weinberg parabola along one or two line segments. Generally then, it would be expected that there is either no gene frequency which represents a loadless equilibrium, or else all gene frequencies along one or two intervals represent such equilibria. The length of the interval(s) is determined by the size of the hexagon. The size of the hexagon is a function of the amount of dominance and neutrality; the more habitats in which there are dominance and neutrality the larger the hexagon, and the larger the equilibrium interval. Throughout this discussion the conditions for a loadless equilibrium have been described geometrically, rather than algebraically, because these terms are intuitively pleasing. Nevertheless, one can describe conditions for a loadless equilibrium in terms of the algebraic relationships among the 0’s. This has been done in the APPENDIX. In summary, the necessary and sufficient conditions for the existence of a loadless equilibrium are the existence oif solutions to equations (3)-(5). Dominance and neutrality are favorable to these conditions. When all ai are non-zero and an equilibrium exists, this equilibrium is generally not unique.
C. E. TAYLOR
Stability of Loadless Equilibria
If a population is not at equilibrium size and gene frequency it will change from generation t to generation t 1. This change is governed by the equation
c = pzcll + 2p(l-p)c12
T h e expression for AN/At follows directly from the definition of fitness as the expected number of offspring in generation t i1 from a n individual i n genera1969). tion t. The expression for A p / A t also follows from this definition (WRIGHT From the behavior of (8) it is possible to tell if a population not already at a loadless equilibrium will eventually arrive at one, and if the population will stay there when mildly perturbed. Finally, if an equilibrium point is stable slight deviations from the model may move it slightly, but will not cause it to disappear. This “robustness” is not necessarily possessed by equilibria which are unstable ( KARLINand MCGREGOR 1972). T h e stability of loadless equilibria for (8) has been examined in detail and reported (TAYLOR 1973). The main features can be easily summarized, but first some definitions are necessary. The N , p plane is called the “phase plane” for (8). One can imagine an arrow being drawn from each point on this plane to the point where it would go from generation t to t 4-1. This provides a pictorial representation of the system. A n example is shown in Figure 4.
FIGURE 4.-Phase space of gene frequency ( p ) and population size ( N ) . Arrows show the direction of population change from the origin of each arrow. In (a) d l the equilibria are stable, while in (b) all interior equilibria are neutrally stable, but the endpoints and isolated equilibrium point are locally unstable.
Loosely speaking, if the arrows drawn on a phase plane around a point are directed toward that point then the system is stable there; otherwise it is not. The area of the phase plane which directs arrows to a particular point determines the global stability of that point. The term local stability refers only to arrows in the immediate vicinity of the point. Determining the global stability of non-linear systems such as (8) is typically quite difficult and will not even be attempted here. As used below, “stability” will refer to local stability exclusively. A point is asymptotically stable if any displacement away from equilibrium will dampen through time and the population will return to that equilibrium. An equilibrium is unstable if a small displacement will become larger so the population will move away from that equilibrium. Between these extremes is neutral stability. A point in the phase plane is neutrally stable if two or more equilibria lie immediately adjacent to one another so a small perturbation will not necessarily return, but neither will it go farther away. Both neutrally stable and asymptotically stable equilibrium points are “stable”. A set of equilibria will be “stable” only if all elements of the set are stable, and will be “unstable” otherwise. No point on the interior of a line segment of equilibria can be asymptotically stable. At best such a point (fi,6) can be neutrally stable because every neighborhood of a point on that line segment includes other equilibrium points at which the solution would remain instead of returning to ( f i , p ) as t increased. Only an isolated equilibrium point, one not lying on a line segment of equilibria, can be asymptotically stable. The possible types of stability are shown in Figure 4. The set of loadless equilibria is depicted as a line segment and a point in these drawings. In Figure 4a the isolated equilibrium point is asymptotically stable and all other equilibrium points are neutrally stable. In the right-hand figure, Figure 4b, the equilibria on the interior of the line segment are neutrally stable, but the endpoints of this line segment and the isolated equilibrium point are all unstable. Such a set of loadless equilibria is unstable. Two types of equilibria need to be distinguished: ( 1 ) loadless equilibria on the interior of a line segment of such equilibria (“internal equilibria”) at which SE points - and = 0; and (2) endpoints of these line segments together aP CP with isolated equilibrium points where the partial derivatives are not necessarily zero. It can be shown that all internal equilibria are neutrally stable. I n terms of the phase diagrams in Figure 4 all arrows point toward the line segment of equilibria, except possibly a t the extreme endpoints of that line segment. The arrows need not be perpendicular to the line segment, but must point toward it. It seems that equilibria which are endpoints or isolated equilibria are sometimes locally stable and sometimes not. If, however, the fitnesses of all genotypes respond “about equally” to density, then it can be shown that such equilibria are always locally stable. That is, if the slopes of the W i j ( x , n ) are “approximately equal” then the set of loadless equilibria is stable.
C. E. TAYLOR
It should be pointed out that local instability does not necessarily imply global instability. Even when loadless equilibria are unstable the arrows from some directions point toward the equilibrium; so it seems possible, and even likely, that the system is unstable only locally, and in a more global sense may really be stable. This, however, is just conjecture; global stability must await further investigation. DISCUSSION
In spite of their name it should be recognized that loadless equilibria are not really without loads. Any finite population at a loadless equilibrium, for which frequency-dependent selection is acting, will drift away from this equilibrium by chance processes. CROW(1970) has pointed out that such a population will incur a more or less permanent substitutional load because it will constantly be returning to the equilibrium from which it has drifted, and from which it will drift again. OHTA and KIMURA(1971) have shown that these loads due to drift may sometimes be greater with frequency-dependent selection than with heterosis. In their model, holwever, the equilibrium was a single point. When the population may drift up and down a line segment of equilibria, as may happen in the model presented here, then the load from drift would be lessened, though the amount is not known. In any real population some individuals would fail to inhabit that niche where they are most fit, resulting in variance in fitness. This too would induce some sort of load, termed the “dysmetric load” by HALDANE (see CROW1970). Although strictly optimal habitat selection is unlikely to be achieved by any organism, there are reasons for thinking that animal behavior should approach this ideal. An organism whose drive mechanisms (sense of comfort, happiness, or whatever motivates behavior) are such that its choice of habitat most nearly maximizes its fitness would be at a selective advantage over those individuals whose behavior is more random. Frequency dependence generally arises when individuals interact with one another. as in mating, mimicry, or larval conditioning. In this model the frequency dependence may be thought to arise from the avoidance of olvercrowded habitats. Environmental heterogeneity is important to this model only when such choices can be made. When dispersion is uniform, without habitat selection, the model with environmental heterogeneity does not produce a radical departure from one in which the environment is uniform (TAYLOR 1973, 1974). It was shown that polymorphisms may be maintained without incurring any segregational loads, but only at a single locus. Genetic loads pose a dilemma only when many loci are considered simultaneously. If the frequency-dependent selection examined here is important f o r natural populations, then it must be possible for loadless equilibria to exist simultaneously at a great many loci. This model can be extended to many loci if it is recognized that there is a different fitness surface for each genotype. With m loci, at each of which there are two alleles segregating, there would be 3” genotypes, each with its own fitness
surface and carrying-capacity function. If there are some environments in which each genotype has a higher carrying capacity than the other genotypes, and if it doesn’t make too much difference which genotype lives in most of the environment, then it can be shown that an equilibrium will exist without a segregational load (TAYLOR, manuscript in preparation). Depending upon the correlations which exist between the ecological requirements at the different loci, linkage disequilibrium might be sustained. If, for example, there were alleles A and a at one locus and alleles B and b at a second locus, linkage disequilibrium would tend to evolve if aabb, AABB, or AaBb generally have higher carrying capacities than the other genotypes. Recombinational loads would, of course, accompany this linkage disequilibrium. This will be discussed in more detail elsewhere. The p i n t to be made here is that no contradictions are necessarily inherent in extending these results to many loci. The single-locus analysis above shows that animals can maintain genetic variation without sustaining segregational loads. It underlines the tremendous adaptive potential which behavior confers to a species. And while the assumption of random distribution implicit in many simpler models may be convenient, these models may also be ignoring some essential features of adaptation. I am indebted to W. J. HELTON, J. S. FARRIS, J. B. MITTON,and MINNATAYLOR for their encouragement and assistance. Some of this work was supported by NIH training grant 5TOlGM0070101 to the Department of Genetics, University o t California, Davis. LITERATURE CITED
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APPENDIX Consider the vertices (D,,H,,R,) and the associated gene frequency pi in the hexagon shown i n Figure 3. The genotype frequencies corresponding to the vertices am: Vertices of hexagon
Vertices in Figure 3
+ + + + + + + kQ1 + + + + h - 4+ + + + WQ1 + + + + fi-% + + + + Q4,Q2,
The least and greatest gene frequencies of points in the hexagon, corresponding to vertices 1 and 4, are p1 and p 4 . Since p i = D, 1/2H,,
3- 1/2(Q2 Q*)l P4 = fi-l[Q1 Q4 as Q,
a t these vertices.
+ + + + 1/2(3, + Q,)l
If a loadless equilibrium exists it can be represented as the intersection of the Hardy-Weinberg parabola and a hexagon. Three sets of requirements for the existence d loadless equilibria may be distinguished. Case I: plrp4 2 .5
If both p1 and p4 are less than or equal to 5,the Hardy-Weinberg parabola is non-decreasing on the interval where the parabola and hexagon might intersect. If in this case they do intersect, then vertex 5 must lie on or below the parabola and vertex 2 must lie on or above it. Conversely, if vertex 5 lies on or below the parabola and vertex 2 lies above it, then the parabola and hexagon must intersect. Algebraically these conditions may be stated
H22 2p2.(l - pZ) H5I 2p5(l - p6) . This may be expressed in terms of the Qz's. When p1 and p4 are both
5 .5, then if
and mly if
+ + 1 / w 2 + ++1/m2 + + + + a711 < 2fi-'[s21 + + + + + +
fi-w,+ Q4 + 0, +a7] > , C'
h-ln 2 -
does there exist a loadless equilibrium.
Case II: pl, p4 2 .5 If both p1 and p4 are 2 .5, then the Hardy-Weinberg parabola is non-increasing o n the interval of possible intersection with the hexagon. In such a case an intersection occurs if and only if vertex 6 is on or below the parabola and vertex 3 is on or above it. That is, if
H , 2 2P,(l and
H , I ~ P , ( I- P,) . Equivaltmtly, there exists a loadless equilibrium if and only if
+ + 0,+ +2 + ++ f+ + + f fi-1c't21 I 2h--2[01+ a4+ 1/m21 + a5+ + + l/wl,I 2fi-2[01
Lc14 6 '
when both p 1 and pg are greater than or equal to .5.
Case III: pl
< .5 > p 4
Finally, the only other alternative is that pl an intersection ixcurs if and only if
< .5 and p4 > .5. Under these circumstances
H , 5 .5 and max([H,-92p2(l
-P~)I,[H~--~P - ~ (d~l }2 0.
In other words, there is a loadless equilibrium if and only if
d-laz5 .5 and either fi-1(Q2
+ + a, +a7)-'k 2 Q4
2 2fiU2[01 1/2(a2,
< .5 > pa.
++ "01 + 1/2w2, + + 0,)l Sz5
f 0G 07)]
+ 1/2(0, +