Genetic Diversity at a Single Locus Under Viability Selection and Facultative Apomixis: Equilibrium Structure and Deviations from Hardy-Weinberg Frequencies

Copyright1998 by the Genetics Society of America

Genetic Diversity at a Single Locus Under Viability Selection

and Facultative Apomixis: Equilibrium Structure and Deviations

from Hardy-Weinberg Frequencies

R. Deborah Overath and Marjorie A. Asmussen

Department of Genetics, University of Georgia, Athens, Georgia 30602

Manuscript received May 20, 1997 Accepted for publication January 12, 1998

ABSTRACT

We extensively analyze the maintenance of genetic variation and deviations from Hardy-Weinberg frequencies at a diallelic locus under mixed mating with apomixis and constant viability selection. Analytical proofs show that: (1) at most one polymorphic equilibrium exists, (2) polymorphism requires overdominant or underdominant selection, and (3) a simple, modified overdominance condition is sufficient to maintain genetic variation. In numerical analyses, only overdominant polymorphic equilibria are stable, and these are stable whenever they exist, which happens forz78% of random fitness and mating parameters. The potential for maintaining both alleles increases with increasing apomixis or outcrossing and decreasing selfing. Simulations also indicate that equilibrium levels of heterozygosity will often be statistically indistin-guishable from Hardy-Weinberg frequencies and that adults, not seeds, should usually be censused to maximize detecting deviations. Furthermore, although both censuses more often have an excess rather than a deficit of heterozygotes, analytical sign analyses of the fixation indices prove that, overall, adults are more likely to have an excess and seeds a deficit at equilibrium.

M

re-population diversity and structure. For example, main. For example, studies of the classic selection models

HamrickandGodt(1989) found that predominantly with mixed mating, such as the one including apomixis

selfing species contain less genetic variation than mixed byMarshallandWeir(1979), do not address the full

mating or outcrossing species. In addition, populations equilibrium structure in terms of the precise conditions

of selfing species tend to show greater differentiation, under which a stable polymorphic equilibrium exists

presumably because gene flow is reduced. In contrast, and how often one is maintained. In addition, most

species capable of both sexual and vegetative asexual of these investigations were restricted to overdominant

reproduction have comparable, or even somewhat greater fitnesses, assuming, without proof, that overdominance

variation, within populations and comparable differenti- was the only form of selection that could maintain

ge-ation between populge-ations, relative to sexually repro- netic variation. Finally, little is known about the

com-ducing species (HamrickandGodt1989;Hamricket _{bined effects of mating system and selection on the}

al. 1992). _{generation of detectable deviations from}

Hardy-Wein-These empirical findings confirm key results of a num- _{berg expectations. Here we address these issues by}

de-ber of theoretical investigations on the effects of mating _{limiting the complete equilibrium structure of}

Mar-system. For example, in mixed mating models with con- shall and Weir’s (1979) generalized mixed mating

stant viability selection, selfing reduces the amount of _{model under overdominant, underdominant, and}

di-heterozygosity possible at equilibrium and the range of _{rectional selection. We then quantify the effect of each}

fitness values over which maintenance of allelic variation _{mating parameter on the potential for maintaining}

per-is possible (Hayman1953;WorkmanandJain1966;

Mar-manent genetic variation and generating statistically

sig-shallandWeir1979). Asexual reproduction, such as

nificant deviations from Hardy-Weinberg expectations. apomixis (production of seeds without meiosis), can at

least partially counteract the effect of selfing by main-taining diversity over a wider range of fitness values (

Mar-THE BASIC MODEL

shallandWeir1979).

Although the existing selection models for nonrandom _{We explore the effect of apomixis as well as mixed}

mating populations provide useful insight into the mainte- _{selfing and outcrossing on the maintenance and form}

of genetic variation under the diallelic version of the one-locus constant viability selection models introduced

Corresponding author: R. Deborah Overath, Department of Genetics,

byMarshallandWeir(1979). These models of faculta-University of Georgia, Athens, GA 30602-7223.

E-mail address: [email protected] tive apomixis offer the flexibility to study (1) the

alized mixed mating model with all three forms of repro- equation (10) can be simplified to duction; (2) the classic mixed mating model, which

2(2w122w112w22)w ^ 2

combines outcrossing with selfing; and (3–4) the two alternative mixed mating models, which combine

apo-1{2(11a)(w11w222w212)2s[w12(w111w22)22w11w22]}w ^

mixis with either selfing or outcrossing.Marshalland

Weir’s (1979) formulations are based on the standard

1(2a1s)w12[w12(w111w22)22w11w22]50. (6)

Hardy-Weinberg assumptions, with the exceptions that

A more informative method is to express the internal selection is allowed between the zygote (or seed) stage

(i.e., polymorphic) equilibria in terms of the equilib-and the adult stage equilib-and that in each generation,

individ-rium fixation index fˆ512vˆ/2pˆqˆ, where vˆ is the

equilib-uals outcross with probability t, self with probability s,

rium frequency of heterozygotes in seeds. These formu-and reproduce apomictically with probability a, where

lae can be derived directly from the genotypic and allelic

t1s1a51.MarshallandWeir(1979) considered

recursions or via an extension of a technique developed two cases that differ in the time of censusing. Model I

by Kimura andOhta (1971) for the standard mixed

assumes an adult census followed by mating and then

mating model without apomixis. In either case, the equi-selection before the next generation of adults is

cen-librium fixation index satisfies sused. Model II, on the other hand, assumes a seed

census followed by selection and then mating. We have

g( fˆ )52ifˆ2_{2(222a2s22it)fˆ1s22i(a1s)50}

reparameterized these models in terms of the allele

(7)

frequencies and the fixation index, which measures the

deviation from Hardy-Weinberg frequencies (Wright which determines the equilibrium allele frequencies via

1951). This parameterization greatly simplifies the

anal-yses, as it also does for the standard mixed mating model pˆ 5 w122w222fˆ(w122w11)

(2w122 w112 w22)(12fˆ )

(8) (WorkmanandJain1966). Here we primarily focus on

the seed census model, the easier case to analyze. The _where

adult census model and key equilibrium relationships between the two censuses are presented inappendix a.

i5(w122w11)(w122w22) w12(2w122w112w22)

. (9)

The seed census is represented by lowercase letters,

with p the frequency of allele A1,q512p the frequency _{The equilibrium genotypic frequencies can then be}

cal-of allele A2, and f5 1 2 freq(A1A2)/2pq, the fixation _{culated from the standard relationships,}

index in seeds. The constant viability of genotype AiAj

uˆ 5 pˆ2_{1 fˆpˆ(12pˆ) (10)}

is denoted by wij(i,j51,2). The f and p recursions for

the seed census are then _{vˆ5 2pˆ(12 pˆ)(1 2fˆ ) (11)}

and

f9 5a1 s1(2a1s)(f21)w12w

2w1w2 (1)

wˆ 5(12 pˆ)21_fˆpˆ(1 2_{pˆ ) (12)}

p9 5pw1

w (2) which require (for admissibility) that

and

2min

3

12pˆ

pˆ

4

where

Note that the above formulae also apply to the special

w15p(12 f )(w112w12)1 f(w112w12)1 w12 (3)

case of no outcrossing (t50), for which fˆ5(s/2i)21

or fˆ 5 1; whereas, as noted by Marshall and Weir

w25p(12 f )(w122w22)1w22 (4)

(1979), the approach based on wˆ leads only to the single

are the marginal fitnesses of alleles A1and A2, respec- _{root fˆ}5_{1. [The latter can determine valid polymorphic}

tively, and _{equilibria only when the two homozygotes have equal}

fitness (i.e., w115 w22).]

w5pw11 (12p)w2 (5)

In analyzing the full equilibrium structure of this

sys-is the mean fitness of the population. The prime symbol _{tem, the first issue is the exact number of polymorphic}

(9) denotes a value in the next generation. _{equilibria and when they exist. Our approach uses the}

Intermediate Value Theorem from calculus to bracket the roots of the fˆ quadratic in (7) based on its sign at

ANALYTICAL METHODS AND RESULTS

five critical points, and the basic facts that a quadratic equation has at most two real roots and all frequencies

Analysis of equilibrium structure:MarshallandWeir

(1979) obtained polymorphic equilibria equation as a must be in the interval [0,1]. Details of this analysis

are given in appendix b. The results provide a formal

function of the equilibrium mean fitness wˆ . After

equilib-TABLE 1

rium can exist for a given set of fitnesses and mating

system parameters and that such equilibria exist only _{Sign of the equilibrium fixation indices}

for overdominant and underdominant selection. [The

case of no outcrossing (t5 0) is a slight exception in Criterion Sign of Fˆ Sign of fˆ

that when w115w22it has an infinite number of

(neu-trally stable) equilibria.] 0,i, s

21s 1 1

More important, this approach reveals exactly when

i5 s

21s 0 1

a polymorphic equilibrium will occur. For overdomi-nant or underdomioverdomi-nant fitnesses, (7) always has exactly

one root ( fˆ ) in the maximal admissible interval [21,1]. s

21s,i, s

2(a1s) 2 1

This root determines a valid internal equilibrium if and

only if the corresponding allele frequency in (8) is in i5 s

2(a1s) 2 0

the interval (0,1) and the bounds in (13) are satisfied.

s

2(a1s),i,

1

2 2 2

Consequently, equilibrium frequencies will be valid and a polymorphic equilibrium will exist, if and only if

Fˆ is the equilibrium fixation index for the adult census fˆ,min

3

w122w22,

w122w22

w122 w11

4

Graphing the quadratic in (7) shows that inequality (14)

holds if and only if g( f *) , 0. Thus, a polymorphic

equilibrium will exist for overdominant selection (w12.

gotes (Fˆ,0), whereas seeds will necessarily be in

Hardy-wii. wjj) if and only if

Weinberg equilibrium ( fˆ50).

The overdominant results also reveal an important

sw12(wii2wjj)2 2(w122 wii)[wii2(1 2t)wjj], 0

distinction between the generalized mixed mating model (15)

and the model with only mixed selfing and outcrossing.

and for underdominant selection (wii . wjj . w12) if When all three forms of reproduction are present, both

and only if the adult and seed census can have either negative or

positive fixation indices. For the standard mixed mating

sw12(wjj2 wii)22(w122wjj)[wjj2(1 2t)wii]. 0. _{model (a}5_{0), this is true only when adults are censused}

(16) _[as Workman and Jain (1966) illustrated for specific

values via phase diagrams] because i,1/2 ensures the

Sign of the equilibrium fixation index:The

equilib-seed census always has a positive fixation index. In equilib-seeds, rium analysis based on fˆ shows exactly when the

fre-apomixis is clearly necessary to produce an excess of quency of heterozygotes at an internal equilibrium will

heterozygotes at equilibrium ( fˆ , 0) when selfing is

be above (fˆ , 0), below (fˆ . 0), or equal to ( fˆ 5

present; however, apomixis itself is not sufficient be-0) the Hardy-Weinberg frequency 2pˆqˆ. In particular,

cause in the absence of outcrossing (t50) other condi-although with underdominant selection fˆ is always

posi-tions (s,2i) must hold. tive, with overdominant selection the sign of fˆ is given

Local stability of the equilibria: In the absence of by the sign of the quantity s/2(a 1 s)2 i, where i is

mutation and gene flow, polymorphism maintained in the function of fitness defined in (9). The adult census

adults must be maintained in seeds and vice versa. Con-gives the same results, except that for overdominance

sequently, although genotypic frequencies and fixation the sign of Fˆ, the equilibrium fixation index in adults,

indices for internal equilibria differ between the two is given by the sign of the quantity s/(21 s)2 i.

Com-census times, the criteria for existence of such equilibria, paring these two sign criteria reveals that s/2 (a 1 s)

and the stability properties of all equilibria, apply to

$ s/(2 1 s) (with equality only if s 5 0) and hence

both models. The analytical conditions under which suggests that as long as some selfing occurs, the adult

an internal equilibrium will be locally stable are very census is the more likely to have a negative fixation

complex. The local stability conditions for the boundary index, whereas the seed census is the more likely to

equilibria, that is, the fixation of one allele, however,

have a positive value (Table 1). If s 5 0 and a . 0,

are much more tractable, especially when derived from however, fˆ and Fˆ are both negative at overdominant

the genotypic recursions (appendix c). These show that

polymorphisms. Note also that when t50 an excess of

fixation of allele Ai will be locally stable if

heterozygotes requires s,1 for seeds but not for adults (appendix b). Finally, although the sign criteria are

w12, 2wii

22s (17)

invalid for completely random mating populations (a 5 s5 0, t 5 1), inspection of this case shows that

stw12wjj2 [(12t)wjj2wii][(2 2s)w1222wii], 0 only sufficient and not necessary to prevent the loss of

either allele, it does suggest that simple overdominance (18)

will not always maintain variation when any selfing

oc-for j≠ i5 1,2. Note that these two inequalities imply _{curs. Moreover, ignoring the complications of the}

un-that wii.(1 2t)wjjwhen fixation for Aiis stable. _{likely case of equality in (17) or (18), the modified}

Two general observations are evident from the above

overdominance criterion (19) is, in fact, the exact condi-criteria. First, the stability of each fixation state depends

tion for a protected polymorphism under the special on all three genotypic fitnesses, as long as the population

conditions of no selfing (s 5 0), complete selfing (s5 is not mating completely at random (t,1). [If t51, we

1), no outcrossing (t50), and complete random mating have the classical selection model for which a5s 50

(t5 1).

and (17) and (18) reduce to w12 , wii.] Second, the _{For comparison to the numerical results below, we}

criterion (18) corresponds to the fixation boundary

also calculated analytically how often the sufficient PP

curves inMarshallandWeir’s (1979) phase diagrams

condition (19) holds by evaluating the appropriate mul-[their equation (15)]. However, because this criterion

tiple integrals under a uniform distribution on the rele-does not necessarily imply the first [in fact, it implies

vant parameter space. The results reveal that, on average,

the reverse of (17) when wii, (1 2 t)wjj], the phase _{the modified overdominance condition (19) holds for}

diagram curves do not appear to completely describe

17/72 (23.6%) of all sets of fitnesses and mating parame-the local stability conditions of parame-the fixation states for

ters and for 17/24 (70.8%) of the random overdomi-this system.

nant parameter sets. These two fractions are somewhat

Conditions for protected polymorphism: These

for-lower, 7/36 (19.4%) and 7/12 (58.3%), for both the mal stability conditions determine the exact conditions

standard mixed mating model with selfing and outcross-under which genetic variation is preserved through a

ing (s1t51) and the alternate mixed mating model

protected polymorphism (PP), which prevents the loss

with selfing and apomixis (a1s51). Similar analysis of either allele. In particular, a PP exists for the

general-shows that a given degree of selfing, (19) holds for (2 ized mixed mating model with selection if for both

2 s)2_{/12 of random fitnesses and for (2} 2 _s)2_{/4 of}

boundary equilibria (fixation of A1or A2) either (17)

random overdominant fitnesses. As s increases from 0 or (18) is reversed, because both fixation states are then

to 1, the fraction of fitnesses meeting the sufficient PP unstable. Closer examination of these local stability

con-condition (19) decreases from 33.3 to 8.3% of random ditions, in conjunction with the conditions for a

poly-fitnesses and from 100 to 25% of random overdominant morphic equilibrium in (15) and (16), reveals that for

fitnesses, a fourfold drop in each case. overdominance a PP exists if and only if a valid internal

equilibrium also exists (seeappendix d). Similar

analy-sis indicates that for underdominance both of the

NUMERICAL ANALYSIS

boundary equilibria will always be stable whenever a

polymorphic equilibrium exists, and therefore a PP _{Computer simulations provided further insight into}

never exists under these conditions. Because two adja- _{the number, stability pattern, and nature of the}

equilib-cent equilibria are unlikely to be both stable or both _{ria. The program randomly generated fitnesses}

(over-unstable, these results also suggest that (in the absence _{dominant, underdominant, or directional) and}

associ-of cycling) overdominant polymorphic equilibria will _{ated sets of mating system parameters using a random}

be stable whenever they exist, whereas underdominant _{number generator with a uniform distribution on the}

polymorphic equilibria will always be unstable. _{interval [0,1]. Random fitness sets of a specific form}

Interpretation of the biological implications of the _{were generated by choosing w}

11, w12, and w22at random

precise PP conditions is difficult; however, sufficiently _{and then interchanging their values as necessary to}

strong overdominance, _{achieve the desired pattern (e.g., w}

12 5 max {wij} for

overdominance). When all three modes of

reproduc-w12. 2w11

22s,

2w22

22s (19) tion were present, a uniform distribution of the mating

parameters was ensured by first choosing two random numbers to divide the interval [0,1] into three pieces will clearly [see (17) and (18)] always be sufficient to

and then using the lengths of these pieces as the values maintain both alleles in the population through a PP

of the three mating parameters (Karlin1969, pp. 241–

under any combination of selfing, random outcrossing,

242). Adapting the approach ofAsmussenand

Basna-and apomixis. Analysis of the coefficient 2/(2 2s) as

yake(1990), we determined how many (if any)

polymor-a function of s shows thpolymor-at polymor-as the selfing rpolymor-ate increpolymor-ases

phic equilibria exist [from (7)–(13)], the stability of all from 0 to 1, this fitness condition increases from simple

equilibria (using the analytically derived local stability overdominance (w12 . w11,w22) to “double

overdomi-conditions), and the value of the fixation index at poly-nance” (w12.2w11,2w22), in which the fitness of

hetero-morphic equilibria, for each combined set of parame-zygotes exceeds twice that of both homoparame-zygotes.

parameter sets, and reported values are means of 100 such runs.

Analysis of equilibrium structure: The simulations confirmed the analytical predictions that fitnesses must be overdominant to produce a stable polymorphic equi-librium and that such overdominant equilibria are stable whenever they exist. Moreover, a stable polymorphic equi-librium exists if and only if both fixation equilibria are unstable and selection is overdominant. Consequently, only four equilibrium patterns are possible in these sys-tems: SU, US, SUS, and USU [where the end entries indicate the stability of fixation for A1and A2(S5locally

stable, U5unstable) and the intermediate entry refers to a polymorphic equilibrium (when present)]. The

equilibrium structure, in terms of the number and stabil- Figure1.—Mean probability of a stable polymorphism for

random overdominant selection with fixed rates of apomixis

ity patterns of the equilibria, is thus the same as for the

(solid line), selfing (short-dashed line) or outcrossing

(long-classical selection model for random mating

popula-dashed line). Each curve was generated by setting one of

tions. _{the mating parameters to a constant value (from 0 to 1 in}

Because only overdominance will maintain genetic _{increments of 0.1), while 10,000 sets of the other two and the}

variation, further analysis of the equilibrium structure fitnesses were chosen at random, and then checking for a

stable polymorphic equilibrium using (7)–(13) and the

analyt-was restricted to this case. The results reveal that, on

ical local stability criteria. Points are means of 10,000 random

average, when all three mating parameters are chosen

parameter sets obtained using a random number generator

at random, 78% of random overdominant parameter

with a uniform distribution on the interval [0,1].

sets retain both alleles at equilibrium. Of the three sub-sumed models with only two forms of reproduction,

mixed apomixis and outcrossing (s 5 0, and a and t from 100% to onlyz25%. (Note that the 25% for

com-plete selfing agrees with the analytically derived fraction chosen at random) preserves genetic variation 100% of

the time if selection is overdominant, as predicted by of overdominant fitnesses with a PP, which occurs in

the case of s5 1 if and only if w12.2w11,2w22.)

the PP condition (19). This percentage decreases to

70% for the standard mixed mating model (a50, with Significant deviations from Hardy-Weinberg

expecta-tions:The nature of genetic diversity in this system was

s and t chosen at random) and to 58% for mixed selfing

and apomixis (t 50, with a and s chosen at random), characterized by computing the proportion of

polymor-phic equilibria meeting Hardy-Weinberg expectations indicating the extent to which simple overdominance

is insufficient to maintain genetic diversity when selfing (HWE) as well as the sign distribution of the equilibrium fixation indices at both census times. To facilitate com-occurs. These results also show, in conjunction with the

analytical calculations above, that condition (19) is a parisons with studies of natural populations, these calcu-lations were based on values that are statistically distin-fairly accurate predictor of when genetic polymorphism

will be retained. This criterion is actually a perfect pre- guishable from HWE using Li and Horvitz’s (1953)

test statisticx2 5 _f2_N(k2 _{1) with k(k}2 _{1)/2 degrees}

dictor in the absence of outcrossing (t 50) or selfing

(s 5 0), when it is the exact condition for a PP. On of freedom, where N is the sample size, k is the number

of alleles, and f is the equilibrium fixation index ( fˆ for average, the modified overdominance condition (19)

underestimates the maintenance of genetic variation by seeds, Fˆ for adults). Values are reported for a sample

size of N5100 and k52 alleles, which requires|f|.

17% when apomixis is absent (a 50; i.e., the standard

mixed mating model) and by ,10% when all three 0.2 for significance at the 0.05 level. A lower N of course

would mean that fewer fixation indices would be distin-forms of reproduction are present.

Next, we explored the average effect of each of the guishable from HWE (and that the curves in Figure 2

would be higher and those in Figure 3 would be lower). three forms of reproduction on the maintenance of

genetic diversity by setting one of the mating parameters On average, with overdominant fitnesses and all mating

parameters chosen at random, 32% of equilibria for to a constant value (from 0 to 1 in increments of 0.1)

and generating 10,000 sets of the other two at random, adults and 60% for seeds are statistically

indistinguish-able from HWE. Qualitatively similar results were ob-as described above. The results in Figure 1 show that,

on average, the likelihood of obtaining a stable polymor- tained for the three subsumed models (a5 0, s 5 0,

or t50, with the other two mating parameters chosen

phic equilibrium steadily increases to 100% as apomixis

or outcrossing increases to 1 and is always at least 58% at random). In addition, the mixed selfing and apomixis

models (t50) have the lowest expected fraction

indis-along these two curves (0 # a # 1; 0 # t # 1). As

selfing increases from 0 to 1, however, the potential tinguishable from HWE (15% for adults and 27% for

has the highest (47% for adults and 65% for seeds). The average values for mixed apomixis and outcrossing

(s50) are intermediate, with 22% for adults and 51%

for seeds.

For both censuses of the complete model, the average proportion of equilibria meeting HWE ultimately de-creases to 0 as apomixis inde-creases (Figure 2A). This makes sense intuitively, because with more apomixis the favored heterozygotes are producing more heterozy-gous offspring. Seeds are much more likely to meet HWE than adults until the rate of apomixis approaches 0.7, after which the difference between the two census times rapidly disappears. As s increases to z0.3, the proportion meeting HWE increases under both cen-suses (to 35% for adults and 70% for seeds) and then decreases sharply in seeds (to z20%) but only slightly in adults (Figure 2B). As a result, at high levels of selfing (s.0.8), fewer equilibria meet HWE when seeds, rather than adults, are censused, contrary to the results for all other conditions studied. This is consistent, however, with the greater deviation from HWE expected with high selfing rates because of the loss of heterozygotes in seeds. As t increases, the proportion of equilibria indistinguishable from HWE steadily increases toz40% in adults and to 100% in seeds, reflecting the greater effect of viability selection upon adult frequencies at high levels of outcrossing (Figure 2C).

Sign of the equilibrium fixation index: The results from the numerical sign analysis of the equilibrium fixa-tion indices are in accord with analytical predicfixa-tions: adults have a lower frequency of significantly positive fixation indices and a higher frequency of negative ones than seeds because of increased heterozygosity follow-ing overdominant viability selection. More specifically, in the general model the average proportion of signifi-cantly positive fixation indices is 15% for seeds vs. 4% for adults, whereas for negative values the average pro-portions are 25% for seeds and 64% for adults. Under

the standard mixed mating model (a 5 0), the seed

census has no negative fixation indices and a substantial proportion (35%) of statistically positive ones, as ex-pected. The adult census, however, has a substantial proportion of negative values (46%) and a small propor-tion of positive ones (8%). Both of the two alternate mixed mating models have a fairly high proportion of

negative fixation indices in seeds (49% for s 5 0 and

56% for t50) and adults (78% for s50 and 81% for

t 5 0). However, although the mixed apomixis and

outcrossing models (s 5 0) never have a significant

deficit of heterozygotes, the mixed apomixis and selfing models (t50) do have a low average frequency of such

Figure2.—Mean frequency of overdominant polymorphic equilibria (4% in adults and 17% in seeds).

equilibria that are statistically indistinguishable from Hardy- _{The effect of each mating system parameter on the} Weinberg expectations (HWE) for fixed rates of apomixis _{sign distribution of the fixation indices under the} gener-(A), selfing (B), or outcrossing (C). Fixation indices were

alized model is shown in Figure 3 as the average frequency

considered insignificant if|f| , 0.2, where f is fˆ for seeds

of significantly positive (Figure 3, A–C) or negative (Figure

(dashed line) and Fˆ for adults (solid line). Curves were

Figure 3.—Mean frequency of statistically significant positive (A, B, C) and negative (D, E, F) equilibrium fixation indices

(Figure 3, A and D) naturally increases the proportion of Marshall and Weir (1979) based on individual

phase diagrams. For instance, we are able to compare of negative values at both census times because the

favored heterozygotes produce exact copies of them- the full model with the three subsumed models and

gauge the relative effects of the different mating param-selves when they reproduce apomictically. For selfing

(Figure 3, B and E), the frequency of significantly posi- eters on the overall likelihood of maintaining genetic

diversity. Only mixed apomixis and outcrossing will al-tive fixation indices is essentially zero at low selfing rates

and only increases as selfing increases above 0.2. Adults, ways maintain genetic variation with overdominant

se-lection. The full model does so for 78% of random however, show a decrease in this frequency at very high

levels of selfing (s . 0.9), consistent with Hayman’s overdominant parameter sets, followed by the standard

mixed mating model (70%) and, finally, mixed apo-(1953) individual phase diagrams for the standard

mixed mating model. In the case of increased outcross- mixis and selfing (58%). Clearly, the presence of selfing

in the mating system reduces the overall potential for ing, rather than becoming more positive or more

nega-tive, the average value of the equilibrium fixation index maintaining variation.

In interpreting the numerical results here, however, appears to be approaching 0 in seeds and some small

negative number in adults (Figure 3, C and F, and data it must be realized that they are based on random

fit-nesses and mating parameters drawn from a uniform not shown).

distribution. How often genetic variation or significant deviation from Hardy-Weinberg frequencies is actually

DISCUSSION

maintained in natural populations depends on the un-known, true distribution of fitnesses and mating system The comprehensive analysis here provides the exact

conditions that maintain genetic variation under a gener- parameters in nature. By quantifying the fraction of the parameter space with the specified evolutionary outcome, alized mixed mating model with apomixis, selfing,

out-crossing and constant viability selection at a single, diallelic our results instead provide a baseline for comparing differ-ent biological scenarios and developing hypotheses. For locus. A combination of analytical and numerical results

shows that the full system, as well as the three subsumed example, the numerical investigation of the average effect of each mode of reproduction indicates that an increase cases with only two forms of reproduction, has the same

general equilibrium structure as the classical selection in the selfing rate dramatically reduces the possibility

of maintaining variation. In contrast, increasing apo-model with random mating: (1) at most one

polymor-phic equilibrium exists; (2) a polymorpolymor-phic equilibrium mixis or outcrossing has a positive, but smaller, effect

on the maintenance of genetic diversity. Therefore, con-exists only with overdominance or underdominance;

(3) a stable polymorphic equilibrium exists only when trary to the traditional view that apomictic species

should be genetically depauperate (reviewed in Asker

selection is overdominant; and (4) a protected

polymor-phism, with both fixation states unstable, exists when- and Jerling 1992), we would expect that apomictic

species capable of some sexual reproduction should ever a stable internal equilibrium exists, and vice versa.

The one critical difference is that when any self-fertiliza- have levels of genetic diversity comparable to their sex-ual relatives. Recent allozyme studies suggest that this tion occurs, simple overdominant selection may not be

sufficient to maintain both alleles in the population. may indeed be the case (Bayer1989; Hamricket al.

1992;OverathandHamrick1998).

In addition to delimiting the full equilibrium

struc-ture of the models, we have obtained a simple, sufficient In addition to motivating hypotheses, our numerical

results lead to several practical guidelines for those condition for the existence of a protected

polymor-phism. This shows that genetic variation will be main- studying natural populations. For instance, our

discov-ery of little deviation from Hardy-Weinberg expecta-tained whenever w12.2w11/(22s), 2w22/(22s).

Fur-thermore, numerical analyses demonstrate that this tions (HWE) quantitatively reinforces previous cautions

that the fixation index can be quite unreliable in de-modified overdominance condition is an excellent

pre-dictor of the maintenance of genetic polymorphism, tecting the presence of evolutionary forces (Wallace

1958;LewontinandCockerham1959;Li1959;

Work-because for the full model it holds for .90% of the

parameter sets that produce a stable polymorphic equi- man1969;Schaap 1980;Li 1988;Lessios1992). This

equilibrium analysis also revealed an important differ-librium. In fact, under the alternate mixed mating

mod-els in which apomixis is mixed with either just outcross- ence between the complete and classical mixed mating

models: when apomixis is present, excesses of heterozy-ing (s5 0) or just selfing (t5 0), the over-dominant

criterion is a perfect predictor. Under the standard gotes can occur at both seed and adult censuses.

With-out apomixis, this can occur only in adults. Perhaps mixed mating model, without apomixis, the predictive

power is somewhat less than that for the full model finding a significant excess of heterozygotes in studies

using a seed census could be used as an indication that (83%), but it is still substantial.

Together, our analytical and numerical investigations apomixis is occurring; however, apomixis would need to

be fairly high under most conditions for the equilibrium of the equilibrium structure of this system formally

Lynch, M.,1987 The consequences of fluctuating selection for

iso-HWE (Figure 2A). Finally, our results also indicate that

zyme polymorphisms in Daphnia. Genetics 115: 657–669.

adults should usually be censused, except in highly Marshall, D. R.,and B. S.Weir, 1979 Maintenance of genetic

variation in apomictic plant populations. I. Single locus models.

selfing species, in order to maximize the possibility of

Heredity 42: 159–172.

detecting any significant deviation from HWE.

Overath, R. D., _andJ. L. Hamrick, _{1988 Allozyme diversity in}

One aspect of the maintenance of genetic variation _{Amelanchier arborea and A. Laevis (Rosaceae). Rhodora (in press).}

Schaap, T.,_{1980 The applicability of the Hardy-Weinberg principle}

not considered here is the potential for long-lived

tran-in the study of populations. Ann. Hum. Genet. 44: 211–215.

sient polymorphism. If the time to fixation is rather

Wallace, B.,_{1958 The comparison of observed and calculated}

zy-long, polymorphism may be effectively maintained, even gotic distributions. Evolution 12: 113–115.

Workman, P. L., _{1969 The analysis of simple genetic}

polymor-if fixation is the expected outcome. Given that apomixis

phisms. Hum. Biol. 41: 96–114.

is known to slow the approach to equilibrium in the

Workman, P. L.,_andS. K. Jain,_{1966 Zygotic selection under mixed}

absence of selection (Marshall and Weir 1979), it _{random mating and self-fertilization: theory and problems of}

estimation. Genetics 54: 159–171.

would not be surprising to find that the duration of

Wright, S.,1951 The genetical structure of populations. Ann.

Eu-polymorphism can be quite long in cases with high levels

genics 15: 323–354.

of apomixis. Variation may be maintained in such cases

Communicating editor:A. H. D. Brown

because natural populations are not at or near equilib-rium and before an allele can be lost, selection changes (see Lynch1987) or new alleles enter the population

through migration or mutation. Consequently, the pos- _{APPENDIX A}

sibility of long-lived transient polymorphism merits

care-Recursions for the adult census model:Using upper-ful study to upper-fully understand the dynamics of the models

case letters to designate adults, the adult census re-presented here and to better interpret data from natural

cursions are populations.

We thank C. Basten_{for supplying random number generators}

F9 511[(2a1 s)F1s22]w12W

2W1W2 (A1)

andJ. T. Overathfor suggesting ways to make the programs more efficient. We also thankM. L. Arnold, J. C. Avise, A. H. D. Brown, J. L. Hamrick, K. E. Holsinger,and anonymous reviewers for their

P9 5PW1

W (A2)

comments on an earlier draft. This work was supported by National Institutes of Health Training Grant GM-07103 to R.D.O. and by

Na-and

tional Science Foundation Grant DEB-9210895 to M.A.A.

where

W1 51

2P(w122w11)[(2a1s)F1s22]

LITERATURE CITED

Asker, S. E., _andL. Jerling,_{1992 Apomixis in Plants. CRC Press,}

11

2(w112 w12)[(2a1s)F1s]1w12 (A3)

Boca Raton, FL.

Asmussen, M. A., _andE. Basnayake,_{1990 Frequency-dependent} selection: the high potential for permanent genetic variation in

the diallelic, pairwise interaction model. Genetics 125: 215–230. _W2 51

2P(w222w12)[(2a1s)F1 s2 2]1w22 (A4)

Bayer, R. L.1989 Patterns of isozyme variation in Antennaria rosea (Asteraceae: Inuleae) polyploid complex. Syst. Bot. 14: 389–397.

W5PW11(1 2P)W2. (A5)

Hamrick, J. L.,andM. J. W. Godt,1989 Allozyme diversity in plant species, pp. 43–63, in Plant Population Genetics, Breeding, and Genetic

At equilibrium, the two censuses are related by Pˆ 5 pˆ,

Resources, edited byA. H. D. Brown, M. T. Clegg, A. L. Kahler

Wˆ 5wˆ , and Fˆ5 (2fˆ2s)/2(a 1s) [Equation A.11 in andB. S. Weir._{Sinauer, Sunderland, MA.}

Hamrick, J. L., M. J. W. Godt andS. L. Sherman-Broyles, 1992 _{MarshallandWeir(1979)].}

Factors influencing levels of genetic diversity in woody plant species. New For. 6: 95–124.

Hayman, B. I.,1953 Mixed selfing and random mating when

homo-APPENDIX B zygotes are at a disadvantage. Heredity 7: 185–192.

Karlin, S.,1969 _{A First Course in Stochastic Processes. Academic Press,}

Existence of polymorphic equilibria and the sign of New York.

Kimura, M.,andT. Ohta,1971 _{Theoretical Aspects of Population Genet- their fixation indices:Most of the results follow by using}

ics. Princeton University Press, Princeton, NJ.

the Intermediate Value Theorem to bracket the roots

Lessios, H. A.,1992 Testing electrophoretic data for agreement

of the fˆ quadratic (7) in the maximal admissible

inter-with Hardy-Weinberg expectations. Mar. Biol. 112: 517–523.

Lewontin, R. C.,andC. C. Cockerham,1959 The goodness-of-fit _{val [}2_{1,1], based on the sign of g ( fˆ ) at five critical} test for detecting natural selection in random mating

popula-points: g (21)5 2(12a), g (0)5 s22i(a 1s), g(1) tions. Evolution 13: 561–564.

5 2t(2i 2 1), and g (6∞) 5 (sign of i)∞, where the

Li, C. C.,1959 Notes on the relative fitness of genotypes that forms

a geometric progression. Evolution 13: 564–567. latter is shorthand for

Li, C. C.,1988 Pseudo-random mating populations. In celebration

of the 80th anniversary of the Hardy-Weinberg Law. Genetics _lim fˆ→6∞

g ( fˆ ). 119:731–737.

Li, C. C.,andD. G. Horvitz,1953 Some methods of estimating

gen-TABLE B1 The sign patterns for the Fˆ quadratic under the adult census,

The sign of g ( fˆ ) at five critical points

under three selection schemes _(2a1s)iFˆ2_{1[2(12a)(i21)1s]Fˆ1s2(21s)i50}

g (2∞) g (21) g (0) g (1) g (1∞) differ from those for fˆ with overdominance in two ways. First, under the complete model and the two

sub-Overdominance (w12.w11,w22)

sumed models a 50 and t50, the sign of Fˆ is that of

s

2(a1s),i 1 1 2 2 1 s 2 i(21 s). Second, the exact solutions for t50 are

Fˆ5 1 and Fˆ5 [s2 i(21 s)]/i(22 s), which implies s

2(a1s)51 1 1 0 2 1 that for adults, obtaining a negative Fˆ for t 5 0 does

not require s , 1; in particular, if s5 1 an excess of

s

2(a1s).i 1 1 1 2 1 heterozygotes will occur whenever i.1/3.

Underdominance (w12,w11,w22)

2 1 1 2 2

APPENDIX C Directional selection (w11,w12,w22)

Genotypic recursions and local stability analysis for

w12.

w111w22

2 2 1 1 2 2 the seed census: Letting u 5 freq(A1A1) and v 5

freq(A1A2) in zygotes (seeds), the genotypic recursions

for the seed census model are

eral seed census model with all three forms of

reproduc-u9 5(a1s)w11u

w 1

sw12v

4w 1t(p9)

2

tion present are summarized in Table B1.

When selection is overdominant, 0 , i , 1/2 and

and the sign changes of g (fˆ ) across (2∞,∞) show that one

root of the quadratic is in (21,1) and the other is in

v9 5(2a1s)w12v

2w 12tp9(1 2p9)

(1,∞). Exactly one solution for fˆ therefore exists in the maximal admissible interval for any set of

overdomi-nant fitnesses, and it has the sign of s/2(a 1 s) 2 i. where

With underdominant selection, i, 0; again, only one

root lies in (21,1), but in this case it is always positive. _{p9 5}w11u

w 1

w12v

2w

The overdominant and underdominant roots in [21,1]

determine valid polymorphic equilibria if and only if _and

(14) holds. Directional selection with w11,w12,w22and

w5w221(w112w22)u1(w122 w22)v.

w12 . (w11 1 w22)/2 yields the same sign pattern as

underdominance, but evaluation of (8) shows that the

An equilibrium (uˆ,vˆ) is locally stable if both

eigenval-root in (0,1) is inadmissible because pˆ,0. The

Interme-ues of its local stability matrix have magnitude,1. These

diate Value Theorem is uninformative when w11, w12 _{eigenvalues are given by the roots of the characteristic}

,w22and w12,(w111 w22)/2; however, (8) and (12) _equationl22_Bl 1_C5_{0 where}

show that in this case either pˆ. 1 or wˆ , 0 for any

fˆP(21,1). Therefore, no polymorphic equilibria exist

B5 ]u9 ]u 1

]v9

]v (C1)

under directional selection.

A few differences arise for the three subsumed systems

and with only two of the three forms of reproduction (i.e.,

a50, s50, or t5 0). For t50, the underdominant

C5

1

21

]v9

]v

2

1

]v

21

]u

2

sign pattern differs from Table B1 in that g (1) 5 0,

indicating that fˆ 5 1 is the only admissible root. In

addition, the case t50 has two roots when selection is with the partials evaluated at (uˆ,vˆ). At the fixation

equi-overdominant: fˆ5 1 and fˆ5(s/2i)21. Note that the libria, B.0, and the two roots are real numbers (because latter implies that when t50, an excess of heterozygotes B2 2 _4C . _{0); therefore, fixation for A}

i will be locally

can occur in seeds only if s, 1 because 2i , 1. The stable if Bi, 2 and Bi2Ci,1, where for i≠j51,2,

root fˆ51 determines a valid equilibrium for

underdom-inance and overdomunderdom-inance only if w115w22(in which Bi5(1 2t)wjj

wii 1

1

2

2

w12 wii

case every state in which u1w51 is an equilibrium).

The distinctions of a50 and s50 are that with over- _and

dominance, fˆ . 0 for a 5 0, because then g (0) 5

s(122i). 0, whereas fˆ,0 for s 50, because g(0) 5 _C

i5

1

22t

2

w12wjj w2

ii

are (C1) and (C2), respectively, evaluated at the fixation sw12(w222w11).2(w122w22)[w222(12t)w11] (D4)

of Ai. Rewriting the two stability criteria as

for the fixation of A2(pˆ5 0).

Consider the case of overdominance with w12 . w11

2[(12t)wjj 2wii]1(22 s)w1222wii, 0 (C3)

.w22, which implies, by (15), that a polymorphic

equi-and _{librium exists if and only if}

stw12wjj2[(12t)wjj2wii][(22s)w1222wii],0 (C4) sw12(w112 w22),2(w122w11)[w112(12t)w22]. (D5)

Now suppose that an internal equilibrium exists; thus, it follows that fixation of Aiis locally stable if (C4) holds

(D5) holds. This condition is the reverse of inequality together with

(D2) and therefore implies that fixation of A1is unstable

whenever a polymorphic equilibrium exists. Fixation of

w12, 2wii

22 s (C5) _{A2will also be unstable and is so whenever w12.w11.}

w22, because then either w22$(12t)w11, in which case

and

(D4) fails, or w22,(1 2t)w11, in which case (D3) and

(D4) cannot both hold. By symmetry, pˆ5 1 and pˆ50

wii.(12 t)wjj. (C6)

are also both unstable if a polymorphic equilibrium

This reduces to the requirement that (C4) holds to- _{exists when w}

12.w22.w11. Thus, a PP exists whenever

gether with (C5), because this ensures (C6) is satisfied. _{a valid internal equilibrium exists for overdominant} fit-nesses. Now suppose a PP exists when w12.w11.w22.

The instability of pˆ 5 1 implies that either inequality

APPENDIX D _{(D1) or inequality (D2) is reversed (where we are}

ignor-ing the complications posed by the unlikely event of

Co-occurrence (or not) of a valid internal equilibrium

strict equality). For this fitness order, however,

inequal-and a protected polymorphism:In the case of the seed

ity (D2) is reversed whenever inequality (D1) is reversed, census, recall that a polymorphic equilibrium exists for

and hence (D5) always holds when pˆ 5 1 is unstable.

w12. wii.wjjif and only if inequality (15) holds and

Therefore a PP implies a valid polymorphic equilibrium for wii. wjj. w12if and only if inequality (16) holds

exists. By symmetry, the same is true for the case w12.

(i ≠j5 1,2). Showing that a protected polymorphism

w22.w11. This completes the proof that a polymorphic (PP) and a valid internal equilibrium always

simultane-equilibrium exists whenever a PP exists and vice versa ously exist or not is facilitated by rewriting the local

when selection is overdominant. stability conditions in (17) and (18) as

Consider next the case of underdominance with w11

sw12. 2(w122 w11) (D1) .w22.w12. Note first that this fitness order implies that

(D1), (D2), and (D3) hold. Also, by (16), a polymorphic and

equilibrium exists if and only if (D4) holds. Therefore, in this case, both fixation states are stable, and there

sw12(w112w22).2(w122w11)[w112(12t)w22] (D2)

is no PP whenever an internal equilibrium exists. By

for the fixation of A1(pˆ51) and as _{symmetry, the same is true for w22}._w11._{w12. Thus, in}

the case of underdominance, a polymorphic equilibrium

sw12. 2(w122 w22) (D3)

implies that both pˆ5 1 and pˆ 50 are stable, which in