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Genotype frequencies associated with incompatibility systems in tristylous plants
THEORETICAL
POPULATION
27, 318-336 (1985)
BIOLOGY
Genotype Frequencies Associated with Incompatibility Systems in Tristylous Plants IVAR
HEUCH
Departmenr
AND ROLV
of Marhemarics. :V-5014 Bergen.
Received February
T. LIE
Unicersiry :Voruq
qf Bergen,
16. 1984
A general procedure has been given previously for calculating frequencies of the morphs Long. Mid, and Short in equilibrium populations of tristylous plants. It is now demonstrated that an equilibrium state actually exists at the genotype level if this procedure produces admissible morph frequencies. This result holds in a diploid model. with or without linkage between the two loci involved. It is shown how the genotype frequencies may bc determined for any set of mating probabilities. It is also explained how these frequencies may be calculated in a tetraploid model incorporating double reduction. The general theory is applied to a particular situation where the Mid morph is at a selective disadvantage as a seed parent. ,C, 1985 Academic Press. Inc
1. INTRODUCTION The incompatibility systems in flowering plants based on heterostyly occupy a special position among mating systems favouring outbreeding. This is partly because the incompatibility types are easily identified by morphological differences. and also because there seems to be a relatively simple common underlying pattern among the genetic mechanisms involved. These systems therefore offer good opportunities for contrasting observed population structure with results predicted by theoretical models. Other cases of negative assortative mating which have been studied mathematically often do not correspond closely to any mechanisms known to operate in nature. This paper is concerned with heterostylous systems involving three different incompatibility types or “morphs.” Tristylous species of this sort occur in the families Lythraceae, Oxalidaceae, and Pontederiaceae. The same basic morphological differentiation can be found in nearly all tristylous plants, with each individual being classified as belonging to one of the morphs Long, Mid, and Short according to the length of its styles. It is also possible to distinguish between morphs on the basis of the lengths of 318 0040-5809j85 83.00 Copyright C. 1985 by Academic Press. Inc. All rights of reproduction in any form reserxd.
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the stamen whorls, and in some casesby considering other characteristics such as pollen size as well. Normally, physiological mechanisms operating in combination with the morphological differentiation ensure that the great majority of matings take place between plants belonging to distinct morphs only. However, when the systems break down in various ways, illegitimate matings may also occur. Since Darwin (1877) first determined morph frequencies in various natural populations of Lythrum salicaria (purple loosestrife), much work of this kind has been carried out in this species. The main results were reviewed by Heuch (1979a). Population frequencies in Lythrum junceum have been studied by Duiberger (1970) and Ornduff (1975). There has also been considerable interest in morph frequencies observed among tristylous species of Oxafis (e.g. Ornduff, 1964; Weller, 1979). More recently, extensive investigations have focused on morph frequencies in species belonging to the third tristylous family, Pontederiaceae; thus, in Eichhorniu crussipes (water hyacinth: Barrett and Forno: 1982) and in Pontederia cordata (pickerelweed; Barrett et al., 1983). In order to interpret such data collected in nature, theoretical models are needed to predict morph frequencies under various hypotheses concerning the mating systems. Fisher (1935) set up the first mathematical model of this kind, but its purpose was mainly to explore the consequences of an earlier incorrect theory about the genetics of tristyly. Parallel with breeding experiments in L. saficuria. Fisher (1941) subsequently carried out a mathematical analysis to determine equilibrium genotype frequencies in three alternative models assuming diploid, tetraploid, and hexaploid inheritance for the important A4 locus (see Section 3). When later experiments indicated that the tetraploid model was correct in this species, the results were extended to take account of double reduction at the M locus (Fisher. 1944). In these papers attention was confined to the particular “isoplethic” equilibria with all morph frequencies equal to li3, but no rigorous mathematical justification was given for this restriction. The possibility of analysing this general type of incompatibility system was also discussed by Finney ( 1952) and Spieth ( 1971). Heuch (1979a) showed that isoplethic equilibrium frequencies were in fact the only possible ones in the symmetric mating models of “pollen elimination” and “zygote elimination” introduced by Finney (1952, 1979). The derivation given by Heuch (1979a) was based on a general proposition which can also be used in asymmetric situations, resulting in unequal equilibrium morph frequencies (Heuch, 1979b, 1982). This method was recently applied by Barrett et al. (1983) in connection with an observed anisoplethic population structure in P. corduta. and by Taylor (1984) in a model for allocation of reproductive energy among ovule and pollen types. However, until now this approach has suffered from the defect that it deals
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with morph frequencies only and does not allow any statements regarding genotype frequencies. The purpose of this paper is to show how the genotype frequencies may be determined in models of this kind with intermorph matings only. Section 2 provides an outline of the general procedure for calculating morph frequencies and applies it to a new practical situation. In Section 3 it is shown that this general procedure used in the diploid model always leads to equilibria at the genotypic level. Section 4 indicates how linkage may be accommodated. In Section 5 we consider problems connected with the considerably more complex tetraploid models, and in Section 6 some practical implications of the mathematical results are discussed.
2.
DETERMINATION
OF MORPH
FREQUENCIES
Consider an infinite population with relative frequencies of the morphs Long, Mid, and Short equal to L. :M and S. respectively. where L + M+ S= 1. We assume that matings occur according to a set of rules depending on the morph frequencies. Let the frequency of matings between the unordered pair of morphs i, j be R,, where the indices 1. m, and s will be used for i, j to indicate Long, Mid, and Short. Thus the quantities R,, will be given as functions of L, M. and S which may incorporate any special features of the breeding system. The symbol R, corresponds to P(S, x S,) in the notation of Heuch (1979a). Since we do not allow for the possibility of intramorph matings, there are only three mating frequencies R,, R,, and R,, of interest, with
R,,,,+ R, + R,,, = 1. We assume that R, = 0 if and only if either or both of the morphs i and j have a population frequency equal to zero. Proposition 1 of Heuch (1979a) now enables us to determine the equilibrium frequencies L, IV, and S in an infinite population. Although this theorem actually applies to a more genera1 situation. in the present context it may be formulated in this way: Select at random an individual from the equilibrium population, and then consider with equal probabilities either the seed or pollen parent of this individual. The probabilities that this randomly chosen parent plant from a previous generation will belong to Long, Mid, or Short can be written as
Qt = t(R,n,+ R,.y1,
Q, = f( R,,, + Rx, ).
Q.,= 84, + Km). (2.1)
NOW, if all morphs are present in the equilibrium
population, each
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probability of this type must be equal to the corresponding morph frequency: Ql= L, Q,,,= AM, QS=S. (2.2) When the frequencies R, are known as functions of L, M, and S, insertion of (2.1) into (2.2) yields a system of three equations which can be solved for the unknowns L, M, and S. The justification of this theorem depends on the genetic mechanism involved, but the assumptions will hold in practically all known casesof tristyly. We shall indicate in Sections 3 and 5 how the theorem alternatively may be derived from the complete set of genotype equations. Taylor (1984) has pointed out that this theorem is a particular case of a more general result derived by Brian Charlesworth (Lloyd, 1977) dealing with overall fitness values depending on morph frequencies. According to this result all morphs must (under certain prescribed conditions) have equal fitness values at equilibrium. In our notation, the quantities QJL, Q,,,;‘:M. and Q.,:S may be regarded in a general sense as morph fitness values. so that this condition is clearly equivalent with (2.2). Each morph considered in this situation may comprise several genotypes, all of which are assumed to behave in the same way with regard to mating and have equal selective values. In contrast, selective differences between morphs with respect to fertility may be built into the frequencies R,,. These quantities may also take account of other forms of selection between morphs, provided that the selection operates subsequent to the particular stage in the life cycle at which L, M, and S represent population frequencies. Selection prior to this stage is not permitted as it may change the marginal allele distributions among genes derived from parent plants belonging to any particular type. The derivations given by Heuch (1979a) and Taylor (1984, Appendix I) both assume that these distributions are identical to those obtained with ordinary Mendelian segregation. However, even with selective differences between morphs at an earlier stage, it is always possible to reformulate the model in terms of morph frequencies prior to selection. The corresponding equilibrium values may be determined utilizing (2.2), and it is then easy to calculate separately the analogous frequencies after selection. This approach was used by Heuch (1979a) in connection with estimation of fitness values in L. salicaria assuming that census followed selection. It should be noted that the expressions (2.1) simply reflect the fact that each individual has two parent plants belonging to different morphs. There are no assumptions inherent in these expressions concerning the distribution of offspring morphs. In the equilibrium models of Sections 3 and 5 it is found that Short plants produce the same number of Short and nonShort offspring, but the offspring distributions from Long or Mid do not
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possesssimilar simple properties. This illustrates a basic problem in these models; although the fractions Q,/L and Q,,,/M may be viewed as overall fitness values, a high fitness of this kind does not necessarily guarantee a large proportion of the same morph in the next generation. Nevertheless, it is possible to conclude that (2.2) holds at equilibrium. with the implication of equal fitness values. but this result can only be justified by referring to the general underlying genetic systems. As an illustrative example we consider more closely a situation in which one morph differs from the other two in the fertility as a seed parent. To fix ideas, suppose that the Long, Mid. and Short morphs produce average numbers of seedsin the proportions 1 : u’ : 1, irrespective of the morphs of the pollen parents as long as legitimate matings are involved. The intrinsic fertilities as pollen parents are assumed to be identical for all morphs. This model gives an idealized description of the situation studied by Mulcahy (1964) in Oxalis priceae ssp. colorea, where Mid appeared to have fewer surviving offspring produced by seedsthan Long and Short. perhaps with 1~~0.5. If we assume in a cross-pollinated population that sufficient pollen is deposited on the stigmas to fertilize all ovules, this description leads to the following mating frequencies: R/,,, = [ LM;( 1 - L) + wML;‘( 1 - M)]/f R,., = [LS:( 1 -L)
+ SL,‘( 1 - S)];k
(2.3)
R,,,, = [ vMSI( 1 - M) + SM.:‘( 1 - S)]: ~7.
Here C = L + ICM+ S. For )I‘ = 1 this reduces to the pollen elimination model. After substitution in (2.2) we obtain the equations L[l+rrM/(l-M)+S/(l-S)]/(2C)=L M[w + L:‘( 1 - L) + S/( 1 - S)]/(2)?) = M.
(2.4a) (2.4b)
The third equation is obtained from (2.4a) by interchanging L and S. For every )t’ > 0 this system has a unique set of solutions with L, M, S > 0:
(For ~+=l we have, of course, L=M=S=1/3.) When 0<)1;<1 this yields 0 < M < l/3, while fertilities )C> 1 lead to l/3 < MC l/2, with an increasing frequency of Mid for increasing values of IV. In connection with 0. priceae, Mulcahy (1964, p. 1048) argued that the reduced number of surviving offspring from Mid as a pistillate parent “will
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affect reproduction in long and short in no smaller degree than it will affect mid’. He concluded that “mid would suffer no disadvantage in a crosspollinating population.” Although this intuitive argument may seem quite reasonable, it fails to take into account the different contributions an individual can make to the next generation as a seed and pollen parent. At least in the mathematical model defined by (2.3), selection operating against Mid as a seed parent will be reflected in a reduced relative frequency. If, for example, IV= 0.5, the model predicts M= 0.219 and L = s = 0.390.
A similar model with a reduced fertility of one morph both as a seed and pollen parent was studied by Heuch (1979a). This led to a more restrictive condition on the relative fitness values for an equilibrium to be maintained. If one morph differs in fertility as a pollen parent only, the simplest corresponding model is that considered by Heuch (1979a) for Ox&s alpina. 3. THE DIPLOID
MODEL
The basic genetic model for determination of incompatibility types in tristylous plants is that originally proposed by von Ubisch (1921) for Osalis aaldiuiensis. According to this model each individual is diploid, and there are two underlying loci, with alleles S, s and M, m, respectively. Any plant with an S allele belongs to the Short morph, whereas a plant with an ss genotype and at least one M allele is of the Mid form. The Long morph is represented by the single genotype ssmm. When matings Short x Short are not allowed, no SS genotypes will be produced. In accordance with the notation introduced by Fisher (1941), we use the symbols a’ for the frequency of the Long genotype ssmm and b’ and c’ for the Mid genotypes ssMm and s&M. The frequencies of the Short genotypes Ssmm, SsMm, and SsMM are denoted by a, 6, and c. The region of interest for these frequencies is given by a’+b’+c’+a+b+c=l,
a’, b’, c’, a, 6, c > 0,
with the additional restriction that at least two genotypes belonging to distinct morphs should have positive frequencies. The same genetic model has been considered for Lythrum (Fisher, 1941), for various other species of Oxalis (Weller, 1976), and for species in Pontederiaceae (Barrett et al., 1983). We shall primarily specify the composition of a population using genotype frequencies, but we shall also be concerned with the morph frequencies L = a’,
M=b’+c’,
S=a+b+c,
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HEUCH
AND
LIE
and frequencies of the various gametic types that are produced. Following Fisher (1941), let p’ and q’ be the frequencies (among all gametes) of Mid gametes carrying m and A4 alleles, respectively, and let p and q be the corresponding frequencies of Short gametes. Then p’ = b’j2,
q’ = b’j2 + c’,
p = a + b/2,
q = b/2 + c,
with p’ + q’ = M and p + q = S. The conditional frequencies of the gametic types sm and sh4 from Mid are thus p’/M and q’/M, while the conditional frequencies of the gametic types sm, Sm, sM, and SM from Short are p/(2S), p/(2S), q/(2S), and q/(2S) if the two loci are assumed to be unlinked. We now have the following equations relating the genotype frequencies a’, b’, c’, a, b, c to the frequencies ii’, 6’, 7, 5, a, c’ in the next generation: 6’ =
R, p’,‘M + R, p/( 2s) + R,, p’p/( 2MS)
6’ =
R, q’/M + R,q/(2S)
E’ =
+ R,,( p’q + q’p))/( 2MS) Rmq’dW’W
ii=
R,piW)+
g=
R,dW)
z=
R,, + R,,(
(3.1)
P’PIWW p’q + q’p)lWfS)
Kmq’dWS).
The quantities p’, q’, p, q, M, S, R,,,,, R,,T, and R,, appearing on the righthand side should here be considered as functions of the basic genotype frequencies. Assume from now on that the population under study is in equilibrium with 2’ = a’, ii = a, etc. The relations above have been set up under the assumption of nonoverlapping generations, but the equilibrium equations will also be correct in a more realistic model with a random sample of individuals in each generation surviving to the next. It is readily seen that the only equilibria with exactly two morphs remaining are given by a’ = b’ = j (with 1oss of the Short morph), a’ = a = f (with loss of Mid), and c’ = c = 4 (with loss of Long). For the equilibrium with all three morphs present we wish to replace the set of equations (3.1) with a simpler equivalent set. Let the first equation be formed by considering rl + 6+ F. This equilibrium relation reduces to a + b + c = f( R, + R,,),
or S=Qs.
(3.2)
FREQUENCIES
A similar, $+2’+$+(1:
slightly
IN TRISTYLOUS
more complicated
325
PLAN-S
equation
is found
q’ + q = (q’lM) Qn,+ (q/S) Qy
considering (3.3)
The third equation, formed by considering ci’ + &’ + 2’ + ci + F+ c’, is simply a’+h’+c’+a+h+c=
1.
(3.4)
Eqs. (3.2) (3.3): (3.4). considered conjointly, are easily shown to be equivalent to the conditions (2.2) used for finding the morph frequencies. Further simplifying equations are set up considering i;’ - C, which yields (‘, = (‘*
(3.5)
and p.‘2 + S’ - fi,‘2 - 2: which gives
q’ - y = R,,,,q’,‘wf), or: since 2M = R ,,,,+ R,,,, q’ = ZMq:R,,,,.
(3.6)
The final equation is obtained considering G’, and taking advantage of (3.6) and the relations p’ = 52 - q’. p = S - q, we write this as q’,:S - (4M,!R,,, )q + M = 0.
(3.7)
Now (3.2)-(3.7) constitute a system of equations in the genotype frequencies equivalent to the original equilibrium equations given by (3.1). The natural first step in solving these equations is to start solving (2.2) for the morph frequencies L: M. and S. In most biologically realistic situations (2.2) has a unique solution, but in general these equations may also admit no solution or several solutions. We proceed to show that any solution of (2.2) with admissible values L, M, S corresponds to a unique admissible set of equilibrium genotype frequencies. Let L, M, and S now represent the particular values obtained from (2.2). with corresponding values R,,,,: R,,: and R,,,. Equation (3.7) yields q = ZMS:‘R,,, - (4M”S’;R;,,
- MS)’ ‘.
(3.8)
By virtue of (2.1) and (2.2): 4MS=
f&t,, + R,,,, NJ43+ R,,,) 2 R;,,,
and thus q given by (3.8) is always a real number. In a similar way we see that 2MS:R,, > S. and hence only (3.8) can actually produce an admissible
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HEUCHANDLIE
solution of (3.7). In the special isoplethic situation expression (3.8) reduces to the value q= (2-$)/3 found by Fisher (1941). Now let q stand for the particular value given by (3.8). On the basis of (3.2b(3.7) we obtain another equivalent set of equations with explicit expressions for the genotype frequencies: b’ = 2M( 1 - 2q/R,,),
a’ = L,
c’ = c = M(4qjR,,
- 1),
(3.9) l)q,
a=S-M+2(2M/R,,-
b=2(M-(4M/R,,-
1)q).
Here 6’ > b for all q, and similarly a 3 (2M - R,,)c;(2M). Hence, in order to get an admissible solution it is sufficient to check that b, c 3 0. However. if the left-hand side of (3.7) is denoted by 11/(q), it is easily seen that +(R,,/4)>0.
IC/(MR,,i(4M-
R,,)) Q 0.
- R,,), and that all genotype This guarantees that R,,/4 < q < MRJ(4M frequencies given by (3.9) are in fact nonnegative. (In particular, this result implies that q d S for q given by (3.8).) 4. THE EFFECTOF LINKAGE For 0. oaldiaiensis Fyfe (1950) showed that the M and S loci are rather tightly linked. We shall outline how the model in Section 3 may be adapted to this situation. The same notation can be used for genotype frequencies as before except in connection with the double heterozygote. Let 6, and b, be the frequencies of the genetic types SMjsm and SmisM, corresponding to b = 6,. + b, in our earlier notation. Moreover, let pO, q,,, p, , q, be the frequencies of the gametic types sm, sM, Sm, and SM from Short. With a recombination fraction E.> 0 we obtain p,=f(a+i.b,+(l-A)b,), q,=$((l-i)b,+).b,+c),
with similar expressions for p0 and qo. We still have p,, + q. = p, + q, = S/2, but it is no longer obvious that p,, = p, and q. = ql. The terms p/(2S) and qi(2S) in the equations (3.1) must be replaced by p,,/S or p,/S and q,JS or ql/S, and it is easy to set up the seven equations involved. We write down two of these only:
6, = &q,iS+
R,,p’q,l(MS)
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FREQGENCIESIN TRISTYLOUS PLANTS
In the system corresponding to the equilibrium, the equation based on 6, may for instance be replaced by that found from
which is
or h, -h, = 0. Hence, there will be linkage equilibrium between the genes in the M and S loci, and the further analysis in Section 3 applies with linkage as well.
5. THE TETRAPLOID MODEL
Fisher and Martin (1947) found that the genetic system in L. salicaria involved tetraploid inheritance at both the M and S loci, but that genotypes could be assigned to phenotypes according to the same general pattern as in the basic diploid model. This system includes the possibility of double reduction, permitting both genes at a given locus in a diploid gamete to be copies of the same gene in the tetraploid parent. Various tristylous speciesof Oxalis are also polyploid (Weller, 1976). In this situation the Long morph will be represented by the genotype ssssmmmm or s,m,; Mid will consist of the four genotypes s,Mm,, s&,m,, s,Mjm, s,M,; and Short will comprise ten possible genotypes Ss,m,,.... Ss,M,, SZsZm4,..., S2szM,. No genotype will be formed with more than two S alleles when matings Short x Short are excluded. With a relative frequency a, of the genotype S,s,- ,M,m, pj (i = 0, 1, 2; j = 0, 1, 2, 3,4) the morph frequencies are L=a,,
M= i ,= I
aoi.
S= i
i
ak.
i= I /=o
Denote the probabilities of double reduction at the M and S loci by a and /I, respectively. According to the standard interpretation of double reduction we have 0 Q r, fl d l/7. We assume that the two loci are unlinked (as indeed found by Fisher and Martin (1947) in L. sahcaria). Suppose that plants with the genotype M,m, Pk at the M locus produce gametes Mjmzpj (j= 0, 1, 2) with probabilities Irki( It follows immediately from elementary probability arguments that the nonzero terms h,,(a) are
328
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h,(r) = h,,(r) = 1,
AND
LIE
h,,(a) = h,,(l)
h,,(r)=h,,(x)=(l
-r),!2.
h2lJx)=k2(x)=(l
+2x),%,
= (2 +x)/4.
lz,2(1)=h&x)=x.:‘4, h,,(x) = 2( I - r),‘3.
Analogous expressions h,,(p) apply to segregation at the S locus. The frequency of the gametic type s,M,ml Pj among Mid gametes (standardized relative to the entire gametic population) can now be written as p;=
i
r=l
lI,;(X) %I,.
j=o.
1. 2.
and the frequency of the Short gametic type .S,s,~ ,M,m,pi is pii= i i: ki(P) h,.,(x) a,,,., u= I r=O
i.j=o,
1, 2.
(5.1)
This leads to the following equations for the genotype frequencies in the next generation: 6, = doiR/m,PA/M + R, P,o:‘S+ R,,, pb P,o/(MS) di, = h,,&., P’,IM+
R,.sp,,:S+
R,,.,( P;P;, + P; pioMK3
Ei2 = 6,iRtrn P;lM+
R/x PiziS + Rms(Pb Pi2 + Pi Pi1 + pi pio)!!(MS)
d;, =
Rmt( Pi Pi2 + Pi PiI ):‘(MS)
2, =
L
(5.2)
P; P,Y’(MS)
(i=O, 1,2). Here 6,= 1, d,,=6,~=0. At equilibrium, all ii, = a,. The conditions Q, = S and Q,,,= M pertaining to the equilibrium with L, M, S> 0 may be derived by considering C,.jiCii and Z,,, jcS,. We now assume that the equations (2.2) have been solved for L, M, S. On the basis of the expressions for c7, with i = 1, 2 we see that the Short genotype frequencies may be written as 2
aur = c ~rI\ Pm : II =o
u = 1, 2. I’ = 0: 1. 2. 3, 4,
(5.3)
where c,.,. depends on morph and mating frequencies and pb, p’,, pi, but not on U. Hence it follows from (5.1) that the frequencies of Short gametic types with one or two S alleles can be expressed as Pii=
i
II= I
‘C(B)
i
i
hcj(X)Crbl.Pl,,,;
w=o c=O
i= 1, 2, j=O, 1, 2.
(5.4)
Thus with dj,.=xC /I,~(z) c,.,., Eq. (5.4) shows that (p,,,, pzo, pII, P?~.
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PLANTS
p,?, pII)’ is an eigenvector of the Kronecker product of the two matrices (dj,,,)j,,. and (h,i(~))j,, (corresponding to an eigenvalue equal to unity). Because of the structure of the eigenvectors of such matrices (e.g., Pease (1965, p. 328)), it follows that the ratio pyjp,, has the same value, say y, for j = 0, 1, 2. Equation (5.3) implies that we also have a,/acj = y for j = 0, 1, 2. With pI. = cjpii and aj. = xialj, we obtain from (5.1) that PI- =III1(B)al.
+h2,(B)a2..
(5.5)
On the other hand. the equilibrium equations for the genotype frequencies yield i= 1, 2. a,. = R,p,./S+ R,,pi.iS=2p,., (5.6) Inserting (5.6) into (5.5) we obtain J’=p?
:b,. =3/Q(4(l-/I))
(5.7)
as the ratio between frequencies of Short genotypes with two and only one S allele. It also follows from (5.6) that po. =S-(p,.
+pz. )=S-(a,.
+az.)/2=S/2.
Thus we have the important result that exactly one half of the Short gametesat equilibrium will carry no S allele, irrespective of the probability /I of double reduction. This means that the Short morph will always produce the same number of Short and non-Short offspring. Equation (5.1) now implies that poi = plj + pr, for j = 0, 1, 2. This simplifies comparison of the equations (5.2) for Mid and Short genotypes, and it is easily found that
which gives p; + 2~; = (2M’R,,,)( p., + 2p. 2)
4(1 -x)p;-3~p’,=(6M:‘(6M-(l-r)
(5.8)
R,))(4(l-~)p.~-33rp.~).
(5.9)
Thus Mid gametic frequencies may be expressed in terms of Short gametic frequencies. Write for the sake of brevity x = p , + 2p. 2. By considering the value of 4( 1 - a) p. 2- 3zp., in the next generation and utilizing (5.8) and (5.9), we obtain an expression involving the same quantity and x in the present generation. This procedure yields after some algebra 2(1 -a)(2+a)(6M-(l--u) 4(1-r)p~2-3rp~1=(24M-2R,m+(12M+R,m)cc+R,a2)SX2~
RI,)
(5.10)
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HEUCH AND LIE
We denote the fraction on the right-hand side of (5.10) by C. The equilibrium relation for Long (found from (5.2)) may now be written as
(p.o+yy(pb+$y=~(L+y).
(5.11)
By means of (5.8), (5.9), (5.10), and the conditions 1, pi= A4, cj p.,= S, the quantities p.,, and pb may both be expressed in terms of x. Inserting this in (5.11) we find the following quartic equation in x: S(l+2(M-S)) R m., [ 2MS --x [ Rm
(4-r)x-Cx’ 2(2+a) (4-r)M.u 1 2+ x R,,,
1 3MCx2
-6&(1-r)
R,
(5.12)
+JL+F).
This is the generalized version of the quartic equations in x derived by Fisher (1941, 1944). When x has been found for any given t(, the gametic frequencies may be determined using p , + 2p. 2= x, (5. lo), (5.8), and (5.9 ), and the genotype frequencies are given by (5.2). In the tetraploid situation the equilibrium with only Long and Short remaining is given by a,=;,
a ,0=2(1-/Q/(4-/?).
a,, = 3/3/(2(4-p)).
(5.13)
There is a corresponding equilibrium with the Mid and Short morphs only, involving similar expressions, but now for the genotype frequencies ao4, a14, and a24. The third two-morph equilibrium, for Long and Mid with positive frequencies a,. a,,, . and aOz, is obtained by replacing /I in the expressions in (5.13) by x
6. DISCUSSION
Our analyses of the diploid and tetraploid models differ somewhat in scope. In the diploid situation we may, for example, conclude that the restriction w >O on the fertility of Mid is both a necessary and sufficient condition for a trimorphic equilibrium to exist in the selection model in Section 2. Our results for diploidy seem to represent an extension of Fisher’s (1941) analysis, even in the isoplethic case with L = A4 = S = l/3. Fisher’s mathematical treatment has frequently been referred to as implying
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the existence of such an equilibrium, but this seemsdoubtful, as he did not consider the complete set of genotype equations. In fact Fisher’s verbal argument suggeststhat he had in mind a general proposition analogous to that given by Charlesworth (Lloyd, 1977) on the equality of fitness values at equilibrium. The relevant passage from Fisher (1941, p. 33) is the following: .... in the absence of selection among genotypes, the only selective agency at work lies in the number of plants exposed to legitimate pollination by pollen from different sources, and such opportunities for pollination will evidently be equalized if the three style lengths are present in equal numbers. Hence in all cases u’ = i, b’ + c’ f = f. a - b + c = +.
In practice. however, this conclusion is dependent on the structure of the underlying genetic system. As shown by Finney (1983), there are a great many theoretical trimorphic incompatibility systems based on two diallelic loci which only possessanisoplethic equilibria under pollen elimination. By contrast, because of the complexity of the algebra involved, we have not attempted to give any discussion of the solution in the tetraploid situation. Thus our presentation here represents a direct generalization of Fisher’s (1944) analysis to cases with anisoplethic equilibria, at the same time covering the more comprehensive model with double reduction at the S locus. (Because of similar segregation distributions in the heterozygotes Ss and Sass,Fisher’s results for a tetraploid M locus and a diploid S locus may formally be obtained from those for both loci tetraploid by setting fl =O.) In this connection a remark made by Fisher (1944, p. 169) is of some interest: “The important possibility that the different style lengths, or certain genotypes within these, should have unequal fertility or viability has not been introduced. partly because this would alter the form of the analysis very considerably...“. In fact the treatment given in Section 5, covering situations with fitness differencesbetween style lengths, follows the same general approach as that used by Fisher, although, of course, it relies on a new procedure for finding morph frequencies. Fisher published no mathematical analysis of the mode1 with a tetraploid S locus, but he was evidently aware of the relatively minor changes that would be needed in the genotype frequencies at equilibrium. Numerical values of such frequencies were given by Fisher and Martin ( 1947), and the proportion 38/(4 - fi) of Short plants carrying two S alleles was quoted by Fisher (1949). The analysis given here is in a certain sense also incomplete in that no investigation has been made of the stability of the equilibria. In standard models such as those based on pollen or zygote elimination, or in the selection model in Section 2, numerical iterations always indicate stability. However, no obvious procedure for proving this presents itself, and any general approach considering local stability using a linearization technique
332
HEUCH AND LIE
seemsto involve at !east four variables related by rather complicated transformations. Moreover, the naive assumption that the procedure described here will always lead to stable equilibria is false. Consider as an example a rather unrealistic set of mating frequencies defined by RI,,,= 1 - 2.X
R,= 1-2M,
R,,=1-2L
for combinations of population frequencies with L, iW. S< 4, and arbitrarily defined otherwise. In this situation it follows that any set of morph frequencies in the region L, M. S < f defines a neutral equilibrium. However, as witnessed by the recent overview of Finney (1983), attempts to determine genotype or even morph frequencies, let alone the equilibrium stability, in other related and more general incompatibility systems very often run into serious problems. Thus it may not be so surprising that it should be difficult to attain a complete analysis of the tristylous equilibria. In order to compare genotype frequencies in the selection model of Section 2 with those in the ordinary isoplethic equilibria, numerical values were found utilizing (3.8) and (5.12). All such calculations for tetraploidy, based on a set of admissible morph frequencies, invariably produced a unique set of genotype frequencies. Tables I and II present the results for w = 0.5, 1.0, and 2.0, and in the tetraploid situation with no double reduction and with LY= l/7, ,4= 0. Analogous genotype frequencies for any j3> 0 can be written down immediately using (5.7). The increased frequency of the Short morph when w = 0.5 is seen to be entirely due to an increase of the Short genotype without any A4 allele. A reduced pistillate Mid fertility also leads to a marked increase in the relative proportion among Mid plants of individuals with a single M allele. In contrast, when w = 2.0 and Mid is superior, the genotypes carrying a single M allele become the most frequent ones among the Short combinations. Double reduction in tetraploids mostly affects the uncommon genotypes with several M alleles, but may in such cases produce a considerable relative increase in population frequencies. All equilibrium freTABLE Equilibrium
Genotype Frequencies in a Diploid
0.5 1.0 2.0
Tristylous
SSMM
sshfm
0.390 0.333 0.293
0.211 0.309 0.369
Species Short
Mid
Long Relative fertility of Mid, H
I
SSMM
0.008 0.024 0.045
Ssmm
SsMm
0.285 0.179 0.108
0.097 0.131 0.140
SSb4.M
0.008 0.024 0.045
FREQUENCIES IN TRISTYLOUS TABLE Equilibrium
II
Genotype Frequencies in a Tetraploid
Tristylous
Relative fertility of Mid, II
Probability of double reduction at M-locus, r
sG”4
SJMm,
s,M,m,
0.5 0.5 1.0 I.0 2.0 7.0
0 17 0 17 0 17
0.390 0.390 0.333 0.333 0.293 0.293
0.206 0.182 0.296 0.260 0.345 0.302
0.013 0.035 0.035 0.066 0.062 0.097
(’ All calculations
0 I ‘7 0 1 ‘7 0 1.7 with a probability
0.285 0.285 0.180 0.180 0.1 IO 0.110
Species”
Mid
Long
0.5 0.5 1.0 1.0 2.0 2.0
333
PLANTS
0.094 0.083 0.123 0.108 0.128 0.111
0.010 0.020 0.028 0.038 0.048 0.056
s,M,m 5x 1om4 0.002 0.002 0.007 0.007 0.014
s4 M4 6x 1O-6 2x 10-a 6x lO-5 7x 10-4 3x10-4 0.002
Ss,M,m
ss,M,
4x 10-4 0.002 0.002 0.007 0.007 0.014
6x 1O-6 1x10~~ 6x 1O-5 7x 10-4 3x10-4 0.002
of double reduction at the S-locus. /I, equal to zero.
quencies were checked by the more tedious procedure based on recursive calculations relating genotype frequencies in one generation with those in the next. Such calculations have sometimes been used in more general situations to determine equilibria in complex tristylous mating systems (Charlesworth, 1979; Barrett et al., 1983). In the more restricted models studied in this paper, the direct method outlined here seemsto offer a simpler approach, although it still involves some numerical work solving a quartic equation in the tetraploid case. It should in principle be possible to extend this approach to models including intramorph matings, such as those considered by Charlesworth (1979), Heuch (1979b), and Barrett et al. (1983), but this seemsto lead to very complex equations. Fisher (1941) used the numerical values in the isoplethic equilibria to derive the morph distributions among offspring from a given morph of the seed parent. Since it is difficult to determine the genotypes of Mid and Short individuals selected for breeding experiments, such calculations may be relevant to special experiments set up in “open-pollinated plots.” Thus, for Long seed parents the morph frequencies among daughter plants are found to be 0.415 Long, 0.335 Mid, and 0.250 Short when M’= 1.0 (as deter-
334
HEUCH
AND
LIE
mined by Fisher (1941)), and 0.447 Long, 0.233 Mid, and 0.320 Short when M’= 0.5. These results apply in the diploid model, but it is remarkable that the corresponding figures for tetraploidy with or without double reduction at the M locus differ only by less than about 10-j. The values given in this connection by Fisher (1944) for x = 0.08 apparently involve some numerical errors and seem the exaggerate the effect of double reduction. Identical numbers (except for a misprint) were quoted by Fisher (1949), but the modified frequencies given by Fisher (1965) for z = 0.10 are correct. Double reduction at the S locus does not affect these results. We also indicate another use of information on genotype frequencies. In order to explain observed anisoplethic population structure in tristylous plants, various authors have considered the consequences of “founder effects” (e.g. Ornduff, 1964; Barrett. 1977). A new population may be created in different ways involving seeds and pollen, but one possibility is that two seedsare drawn at random from a large trimorphic neighbouring population. If the plants produced by these seeds represent identical morphs, no further sexual reproduction will occur (unless illegitimate matings are successful). With plants belonging to two distinct morphs, the gene pool created may or may not be capable of establishing a trimorphic population. We assume that the new population is expanding rapidly so that all potential forms will actually occur, and that generations overlap. Consider for simplicity the diploid model. The combination of genotypes SSMM, SsMM will lead to a population consisting of Mid and Short only. The combination ssmm, Ssmm will produce a population of Long and Short plants, while the combinations ssrrr111z. ssMm and ssmm. ssMM will give rise to Long and Mid only. All other combinations of distinct morphs will produce trimorphic populations, at least after two generations. Using the equilibrium genotype frequencies determined above we may calculate the conditional probabilities of the various founder combinations, given that the new population is not monomorphic. The results in the diploid situation are displayed in Table III. We note that the probability of a dimorphic population exceeds the probability of all morphs being represented both for by= 1 and 1t:=0.5. What is really striking, however, is that the recently founded population will practically never consist of Mid and Short only. With tetraploidy, the probabilities are essentially the same except for the Mid/Short combination, which is now even less likely to occur (this probability is 1.2x 10p-8for kt’= 1.0 and r = 0, regardless of the value of a). In a situation with no fertility differences approximately two thirds of the dimorphic populations will consist of Long and Mid. Of course, with later gene exchange with other populations, a great many populations must be expected to become trimorphic anyway. But if such events involving the foundation of a new population are relatively frequent, these calculations may suggest why overall relative morph frequencies deter-
FREQUENCIES IN TRISTYLOUS TABLE
335
PLANTS
III
Probabilities of Morph Combinations in a Population Founded by Two Seeds Selected at Random, Conditional on the Population Being at least Dimorphic” Morphs represented in the population Relative fertility of Mid. II 0.5 1.0 2.0
Long, Mid
Long, Short
Mid, Short
All three morphs
0.264 0.333 0.369
0.344 0.179 0.097
2.1 x 1o-4 0.002 0.006
0.392 0.486 0.528
il Values presented for a diploid tristylous species.
mined over large areas may give higher values for Mid than Short, with even higher frequencies for Long. This hypothesis may be relevant to L. salicaria. which has overall morph frequencies in this order. It is related to the theory studied by Heuch (1980) involving random extinction in finite populations, but the stochastic element now enters at a different stage in the evolution of a population.
REFERENCES S. C. H. 1977. Tristyly in Eichhorniu crassipes (Mart.) Solms (water hyacinth), Biotropica 9. 23@238. BARRETT. S. C. H., AND FORNO. I. W. 1982. Style morph distribution in New World populations of Eichhorniu crussipes (Mart.) Solms-Laubach (water hyacinth). Aquat. Bot. 13, ‘99-306. BARRETT. S. C. H.. PRICE. S. D., AND SHORE. J. S. 1983. Male fertility and anisoplethic population structure in tristylous Ponkderiu corduru (Pontederiaceae), ELlolufion 37, 745-759. CHARLESWORTH. D. 1979. The evolution and breakdown of tristyly, Er:olufion 33, 48-98. DARWIS. C. 1877. “The Different Forms of Flowers on Plants of the Same Species,” John Murray. London. DCLBERGER. R. 1970. Tristyly in LJthrum junceum. Yew Phytol. 69, 751-759. FINSEY. D. J. 1952. The equilibrium of a self-incompatible polymorphic species, Geneticu 26, 33-64. FI~NEY. D. J. 1979. The equilibrium of a self-incompatible polymorphic species-A correction, Geneticu 51. 77. FINSEY. D. J. 1983. Genetic equilibria of species with self-incompatible polymorphism, Biomeirics 39. 573-585. FISHER. R. A. 1935. On the selective consequences of East’s (1927) theory of heterostylism in Lxthrum. J. Genet. 30. 369-382. FISHER. R. A. 1941. The theoretical consequences of polyploid inheritance for the Mid style form of Lyrhrum salicaria, Ann. Eugenics 11, 31-38. FISHER. R. A. 1944. Allowance for double reduction in the calculation of genotype frequencies with polysomic inheritance. Ann. Eugenics 12, 169-171. FISHER. R. A. 1949. “The Theory of Inbreeding,” 1st ed., Oliver & Boyd, Edinburgh. BARRETT.
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FISHER, R. A. 1965. “The Theory of Inbreeding,” 2nd ed.. Oliver & Boyd, Edinburgh. FISHER, R. A.. AND MARTIN, V. C. 1947. Spontaneous occurrence in Lyrhrum salicaria of plants duplex for the short-style gene, Nature (London) 160, 541. FY~: V. C. 1950. The genetics of tristyly in Uxalis aaldioiensis, Herediry 4, 365-371. HELXH. I. 1979a. Equilibrium populations of heterostylous plants. Theor. Pop. Biol. 15, 43-57. HEUCH, 1. 1979b. The effect of partial self-fertilization on type frequencies in heterostylous plants. Ann. Bor. 44, 611-616. HEUCH, I. 1980. Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria, Herediras 92, 53-51. HEUCH, I. 1982. A mathematical analysis of some incompatibility systems in higher plants, in “Nordic Symposium in Applied Statistics and Data Processing 1982” (A. Hoskuldsson, J. Stene, and K. Esbensen, Eds.). pp. 379-392, NEIJCC, Lyngby. LLOYD, D. G. 1977. Genotypic and phenotypic models of natural selection, J. Theor. Biol. 69, 543-560. MULCAHY, D. L. 1964. The reproductive biology of Oxalis priceae. Amer. J. Bat. 51, 1045-1050. ORNDUFF, R. 1964. The breeding system of Uxalis suksdorfii, Amer. J. Bof. 51, 307-314. ORNDUFF. R. 1975. Pollen flow in Lythrum junceum. a tristylous species, Xew Phytol. 75, 161-166. PEASE,M. C. 1965. “Methods of Matrix Algebra,” Academic Press. New York. SPIETH, P. T. 1971. A necessary condition for equilibrium in systems exhibiting self-incompatible mating, Theor. Pop. Biol. 2, 404418. TAYLOR, P. D. 1984. Evolutionary stable reproductive allocations in heterostylous plants, Eaolurion 38, 408416. VON UatSCH. G. 1921. Zur Genetik der trimorphen Heterostylie sowie einige Bemerkungen zur dimorphen Heterostylie, Biol. Zenrralbl. 41, 88-96. WELLER, S. G. 1976. The genetic control of tristyly in Oxalic section fonoxalis. Heredity 37. 387-393. WELLER. S. G. 1979. Variation in heterostylous reproductive systems among populations of Ox-al& alpina in Southeastern Arizona. Syst. Bat. 4, 57-71.
POPULATION
27, 318-336 (1985)
BIOLOGY
Genotype Frequencies Associated with Incompatibility Systems in Tristylous Plants IVAR
HEUCH
Departmenr
AND ROLV
of Marhemarics. :V-5014 Bergen.
Received February
T. LIE
Unicersiry :Voruq
qf Bergen,
16. 1984
A general procedure has been given previously for calculating frequencies of the morphs Long. Mid, and Short in equilibrium populations of tristylous plants. It is now demonstrated that an equilibrium state actually exists at the genotype level if this procedure produces admissible morph frequencies. This result holds in a diploid model. with or without linkage between the two loci involved. It is shown how the genotype frequencies may bc determined for any set of mating probabilities. It is also explained how these frequencies may be calculated in a tetraploid model incorporating double reduction. The general theory is applied to a particular situation where the Mid morph is at a selective disadvantage as a seed parent. ,C, 1985 Academic Press. Inc
1. INTRODUCTION The incompatibility systems in flowering plants based on heterostyly occupy a special position among mating systems favouring outbreeding. This is partly because the incompatibility types are easily identified by morphological differences. and also because there seems to be a relatively simple common underlying pattern among the genetic mechanisms involved. These systems therefore offer good opportunities for contrasting observed population structure with results predicted by theoretical models. Other cases of negative assortative mating which have been studied mathematically often do not correspond closely to any mechanisms known to operate in nature. This paper is concerned with heterostylous systems involving three different incompatibility types or “morphs.” Tristylous species of this sort occur in the families Lythraceae, Oxalidaceae, and Pontederiaceae. The same basic morphological differentiation can be found in nearly all tristylous plants, with each individual being classified as belonging to one of the morphs Long, Mid, and Short according to the length of its styles. It is also possible to distinguish between morphs on the basis of the lengths of 318 0040-5809j85 83.00 Copyright C. 1985 by Academic Press. Inc. All rights of reproduction in any form reserxd.
FREQUENCIES IN TRISTYLOUS
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the stamen whorls, and in some casesby considering other characteristics such as pollen size as well. Normally, physiological mechanisms operating in combination with the morphological differentiation ensure that the great majority of matings take place between plants belonging to distinct morphs only. However, when the systems break down in various ways, illegitimate matings may also occur. Since Darwin (1877) first determined morph frequencies in various natural populations of Lythrum salicaria (purple loosestrife), much work of this kind has been carried out in this species. The main results were reviewed by Heuch (1979a). Population frequencies in Lythrum junceum have been studied by Duiberger (1970) and Ornduff (1975). There has also been considerable interest in morph frequencies observed among tristylous species of Oxafis (e.g. Ornduff, 1964; Weller, 1979). More recently, extensive investigations have focused on morph frequencies in species belonging to the third tristylous family, Pontederiaceae; thus, in Eichhorniu crussipes (water hyacinth: Barrett and Forno: 1982) and in Pontederia cordata (pickerelweed; Barrett et al., 1983). In order to interpret such data collected in nature, theoretical models are needed to predict morph frequencies under various hypotheses concerning the mating systems. Fisher (1935) set up the first mathematical model of this kind, but its purpose was mainly to explore the consequences of an earlier incorrect theory about the genetics of tristyly. Parallel with breeding experiments in L. saficuria. Fisher (1941) subsequently carried out a mathematical analysis to determine equilibrium genotype frequencies in three alternative models assuming diploid, tetraploid, and hexaploid inheritance for the important A4 locus (see Section 3). When later experiments indicated that the tetraploid model was correct in this species, the results were extended to take account of double reduction at the M locus (Fisher. 1944). In these papers attention was confined to the particular “isoplethic” equilibria with all morph frequencies equal to li3, but no rigorous mathematical justification was given for this restriction. The possibility of analysing this general type of incompatibility system was also discussed by Finney ( 1952) and Spieth ( 1971). Heuch (1979a) showed that isoplethic equilibrium frequencies were in fact the only possible ones in the symmetric mating models of “pollen elimination” and “zygote elimination” introduced by Finney (1952, 1979). The derivation given by Heuch (1979a) was based on a general proposition which can also be used in asymmetric situations, resulting in unequal equilibrium morph frequencies (Heuch, 1979b, 1982). This method was recently applied by Barrett et al. (1983) in connection with an observed anisoplethic population structure in P. corduta. and by Taylor (1984) in a model for allocation of reproductive energy among ovule and pollen types. However, until now this approach has suffered from the defect that it deals
320
HEUCH AND LIE
with morph frequencies only and does not allow any statements regarding genotype frequencies. The purpose of this paper is to show how the genotype frequencies may be determined in models of this kind with intermorph matings only. Section 2 provides an outline of the general procedure for calculating morph frequencies and applies it to a new practical situation. In Section 3 it is shown that this general procedure used in the diploid model always leads to equilibria at the genotypic level. Section 4 indicates how linkage may be accommodated. In Section 5 we consider problems connected with the considerably more complex tetraploid models, and in Section 6 some practical implications of the mathematical results are discussed.
2.
DETERMINATION
OF MORPH
FREQUENCIES
Consider an infinite population with relative frequencies of the morphs Long, Mid, and Short equal to L. :M and S. respectively. where L + M+ S= 1. We assume that matings occur according to a set of rules depending on the morph frequencies. Let the frequency of matings between the unordered pair of morphs i, j be R,, where the indices 1. m, and s will be used for i, j to indicate Long, Mid, and Short. Thus the quantities R,, will be given as functions of L, M. and S which may incorporate any special features of the breeding system. The symbol R, corresponds to P(S, x S,) in the notation of Heuch (1979a). Since we do not allow for the possibility of intramorph matings, there are only three mating frequencies R,, R,, and R,, of interest, with
R,,,,+ R, + R,,, = 1. We assume that R, = 0 if and only if either or both of the morphs i and j have a population frequency equal to zero. Proposition 1 of Heuch (1979a) now enables us to determine the equilibrium frequencies L, IV, and S in an infinite population. Although this theorem actually applies to a more genera1 situation. in the present context it may be formulated in this way: Select at random an individual from the equilibrium population, and then consider with equal probabilities either the seed or pollen parent of this individual. The probabilities that this randomly chosen parent plant from a previous generation will belong to Long, Mid, or Short can be written as
Qt = t(R,n,+ R,.y1,
Q, = f( R,,, + Rx, ).
Q.,= 84, + Km). (2.1)
NOW, if all morphs are present in the equilibrium
population, each
FREQUENCIES
IN TRISTYLOUS
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321
probability of this type must be equal to the corresponding morph frequency: Ql= L, Q,,,= AM, QS=S. (2.2) When the frequencies R, are known as functions of L, M, and S, insertion of (2.1) into (2.2) yields a system of three equations which can be solved for the unknowns L, M, and S. The justification of this theorem depends on the genetic mechanism involved, but the assumptions will hold in practically all known casesof tristyly. We shall indicate in Sections 3 and 5 how the theorem alternatively may be derived from the complete set of genotype equations. Taylor (1984) has pointed out that this theorem is a particular case of a more general result derived by Brian Charlesworth (Lloyd, 1977) dealing with overall fitness values depending on morph frequencies. According to this result all morphs must (under certain prescribed conditions) have equal fitness values at equilibrium. In our notation, the quantities QJL, Q,,,;‘:M. and Q.,:S may be regarded in a general sense as morph fitness values. so that this condition is clearly equivalent with (2.2). Each morph considered in this situation may comprise several genotypes, all of which are assumed to behave in the same way with regard to mating and have equal selective values. In contrast, selective differences between morphs with respect to fertility may be built into the frequencies R,,. These quantities may also take account of other forms of selection between morphs, provided that the selection operates subsequent to the particular stage in the life cycle at which L, M, and S represent population frequencies. Selection prior to this stage is not permitted as it may change the marginal allele distributions among genes derived from parent plants belonging to any particular type. The derivations given by Heuch (1979a) and Taylor (1984, Appendix I) both assume that these distributions are identical to those obtained with ordinary Mendelian segregation. However, even with selective differences between morphs at an earlier stage, it is always possible to reformulate the model in terms of morph frequencies prior to selection. The corresponding equilibrium values may be determined utilizing (2.2), and it is then easy to calculate separately the analogous frequencies after selection. This approach was used by Heuch (1979a) in connection with estimation of fitness values in L. salicaria assuming that census followed selection. It should be noted that the expressions (2.1) simply reflect the fact that each individual has two parent plants belonging to different morphs. There are no assumptions inherent in these expressions concerning the distribution of offspring morphs. In the equilibrium models of Sections 3 and 5 it is found that Short plants produce the same number of Short and nonShort offspring, but the offspring distributions from Long or Mid do not
322
HEUCH AKD LIE
possesssimilar simple properties. This illustrates a basic problem in these models; although the fractions Q,/L and Q,,,/M may be viewed as overall fitness values, a high fitness of this kind does not necessarily guarantee a large proportion of the same morph in the next generation. Nevertheless, it is possible to conclude that (2.2) holds at equilibrium. with the implication of equal fitness values. but this result can only be justified by referring to the general underlying genetic systems. As an illustrative example we consider more closely a situation in which one morph differs from the other two in the fertility as a seed parent. To fix ideas, suppose that the Long, Mid. and Short morphs produce average numbers of seedsin the proportions 1 : u’ : 1, irrespective of the morphs of the pollen parents as long as legitimate matings are involved. The intrinsic fertilities as pollen parents are assumed to be identical for all morphs. This model gives an idealized description of the situation studied by Mulcahy (1964) in Oxalis priceae ssp. colorea, where Mid appeared to have fewer surviving offspring produced by seedsthan Long and Short. perhaps with 1~~0.5. If we assume in a cross-pollinated population that sufficient pollen is deposited on the stigmas to fertilize all ovules, this description leads to the following mating frequencies: R/,,, = [ LM;( 1 - L) + wML;‘( 1 - M)]/f R,., = [LS:( 1 -L)
+ SL,‘( 1 - S)];k
(2.3)
R,,,, = [ vMSI( 1 - M) + SM.:‘( 1 - S)]: ~7.
Here C = L + ICM+ S. For )I‘ = 1 this reduces to the pollen elimination model. After substitution in (2.2) we obtain the equations L[l+rrM/(l-M)+S/(l-S)]/(2C)=L M[w + L:‘( 1 - L) + S/( 1 - S)]/(2)?) = M.
(2.4a) (2.4b)
The third equation is obtained from (2.4a) by interchanging L and S. For every )t’ > 0 this system has a unique set of solutions with L, M, S > 0:
(For ~+=l we have, of course, L=M=S=1/3.) When 0<)1;<1 this yields 0 < M < l/3, while fertilities )C> 1 lead to l/3 < MC l/2, with an increasing frequency of Mid for increasing values of IV. In connection with 0. priceae, Mulcahy (1964, p. 1048) argued that the reduced number of surviving offspring from Mid as a pistillate parent “will
FREQUENCIES
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323
affect reproduction in long and short in no smaller degree than it will affect mid’. He concluded that “mid would suffer no disadvantage in a crosspollinating population.” Although this intuitive argument may seem quite reasonable, it fails to take into account the different contributions an individual can make to the next generation as a seed and pollen parent. At least in the mathematical model defined by (2.3), selection operating against Mid as a seed parent will be reflected in a reduced relative frequency. If, for example, IV= 0.5, the model predicts M= 0.219 and L = s = 0.390.
A similar model with a reduced fertility of one morph both as a seed and pollen parent was studied by Heuch (1979a). This led to a more restrictive condition on the relative fitness values for an equilibrium to be maintained. If one morph differs in fertility as a pollen parent only, the simplest corresponding model is that considered by Heuch (1979a) for Ox&s alpina. 3. THE DIPLOID
MODEL
The basic genetic model for determination of incompatibility types in tristylous plants is that originally proposed by von Ubisch (1921) for Osalis aaldiuiensis. According to this model each individual is diploid, and there are two underlying loci, with alleles S, s and M, m, respectively. Any plant with an S allele belongs to the Short morph, whereas a plant with an ss genotype and at least one M allele is of the Mid form. The Long morph is represented by the single genotype ssmm. When matings Short x Short are not allowed, no SS genotypes will be produced. In accordance with the notation introduced by Fisher (1941), we use the symbols a’ for the frequency of the Long genotype ssmm and b’ and c’ for the Mid genotypes ssMm and s&M. The frequencies of the Short genotypes Ssmm, SsMm, and SsMM are denoted by a, 6, and c. The region of interest for these frequencies is given by a’+b’+c’+a+b+c=l,
a’, b’, c’, a, 6, c > 0,
with the additional restriction that at least two genotypes belonging to distinct morphs should have positive frequencies. The same genetic model has been considered for Lythrum (Fisher, 1941), for various other species of Oxalis (Weller, 1976), and for species in Pontederiaceae (Barrett et al., 1983). We shall primarily specify the composition of a population using genotype frequencies, but we shall also be concerned with the morph frequencies L = a’,
M=b’+c’,
S=a+b+c,
324
HEUCH
AND
LIE
and frequencies of the various gametic types that are produced. Following Fisher (1941), let p’ and q’ be the frequencies (among all gametes) of Mid gametes carrying m and A4 alleles, respectively, and let p and q be the corresponding frequencies of Short gametes. Then p’ = b’j2,
q’ = b’j2 + c’,
p = a + b/2,
q = b/2 + c,
with p’ + q’ = M and p + q = S. The conditional frequencies of the gametic types sm and sh4 from Mid are thus p’/M and q’/M, while the conditional frequencies of the gametic types sm, Sm, sM, and SM from Short are p/(2S), p/(2S), q/(2S), and q/(2S) if the two loci are assumed to be unlinked. We now have the following equations relating the genotype frequencies a’, b’, c’, a, b, c to the frequencies ii’, 6’, 7, 5, a, c’ in the next generation: 6’ =
R, p’,‘M + R, p/( 2s) + R,, p’p/( 2MS)
6’ =
R, q’/M + R,q/(2S)
E’ =
+ R,,( p’q + q’p))/( 2MS) Rmq’dW’W
ii=
R,piW)+
g=
R,dW)
z=
R,, + R,,(
(3.1)
P’PIWW p’q + q’p)lWfS)
Kmq’dWS).
The quantities p’, q’, p, q, M, S, R,,,,, R,,T, and R,, appearing on the righthand side should here be considered as functions of the basic genotype frequencies. Assume from now on that the population under study is in equilibrium with 2’ = a’, ii = a, etc. The relations above have been set up under the assumption of nonoverlapping generations, but the equilibrium equations will also be correct in a more realistic model with a random sample of individuals in each generation surviving to the next. It is readily seen that the only equilibria with exactly two morphs remaining are given by a’ = b’ = j (with 1oss of the Short morph), a’ = a = f (with loss of Mid), and c’ = c = 4 (with loss of Long). For the equilibrium with all three morphs present we wish to replace the set of equations (3.1) with a simpler equivalent set. Let the first equation be formed by considering rl + 6+ F. This equilibrium relation reduces to a + b + c = f( R, + R,,),
or S=Qs.
(3.2)
FREQUENCIES
A similar, $+2’+$+(1:
slightly
IN TRISTYLOUS
more complicated
325
PLAN-S
equation
is found
q’ + q = (q’lM) Qn,+ (q/S) Qy
considering (3.3)
The third equation, formed by considering ci’ + &’ + 2’ + ci + F+ c’, is simply a’+h’+c’+a+h+c=
1.
(3.4)
Eqs. (3.2) (3.3): (3.4). considered conjointly, are easily shown to be equivalent to the conditions (2.2) used for finding the morph frequencies. Further simplifying equations are set up considering i;’ - C, which yields (‘, = (‘*
(3.5)
and p.‘2 + S’ - fi,‘2 - 2: which gives
q’ - y = R,,,,q’,‘wf), or: since 2M = R ,,,,+ R,,,, q’ = ZMq:R,,,,.
(3.6)
The final equation is obtained considering G’, and taking advantage of (3.6) and the relations p’ = 52 - q’. p = S - q, we write this as q’,:S - (4M,!R,,, )q + M = 0.
(3.7)
Now (3.2)-(3.7) constitute a system of equations in the genotype frequencies equivalent to the original equilibrium equations given by (3.1). The natural first step in solving these equations is to start solving (2.2) for the morph frequencies L: M. and S. In most biologically realistic situations (2.2) has a unique solution, but in general these equations may also admit no solution or several solutions. We proceed to show that any solution of (2.2) with admissible values L, M, S corresponds to a unique admissible set of equilibrium genotype frequencies. Let L, M, and S now represent the particular values obtained from (2.2). with corresponding values R,,,,: R,,: and R,,,. Equation (3.7) yields q = ZMS:‘R,,, - (4M”S’;R;,,
- MS)’ ‘.
(3.8)
By virtue of (2.1) and (2.2): 4MS=
f&t,, + R,,,, NJ43+ R,,,) 2 R;,,,
and thus q given by (3.8) is always a real number. In a similar way we see that 2MS:R,, > S. and hence only (3.8) can actually produce an admissible
326
HEUCHANDLIE
solution of (3.7). In the special isoplethic situation expression (3.8) reduces to the value q= (2-$)/3 found by Fisher (1941). Now let q stand for the particular value given by (3.8). On the basis of (3.2b(3.7) we obtain another equivalent set of equations with explicit expressions for the genotype frequencies: b’ = 2M( 1 - 2q/R,,),
a’ = L,
c’ = c = M(4qjR,,
- 1),
(3.9) l)q,
a=S-M+2(2M/R,,-
b=2(M-(4M/R,,-
1)q).
Here 6’ > b for all q, and similarly a 3 (2M - R,,)c;(2M). Hence, in order to get an admissible solution it is sufficient to check that b, c 3 0. However. if the left-hand side of (3.7) is denoted by 11/(q), it is easily seen that +(R,,/4)>0.
IC/(MR,,i(4M-
R,,)) Q 0.
- R,,), and that all genotype This guarantees that R,,/4 < q < MRJ(4M frequencies given by (3.9) are in fact nonnegative. (In particular, this result implies that q d S for q given by (3.8).) 4. THE EFFECTOF LINKAGE For 0. oaldiaiensis Fyfe (1950) showed that the M and S loci are rather tightly linked. We shall outline how the model in Section 3 may be adapted to this situation. The same notation can be used for genotype frequencies as before except in connection with the double heterozygote. Let 6, and b, be the frequencies of the genetic types SMjsm and SmisM, corresponding to b = 6,. + b, in our earlier notation. Moreover, let pO, q,,, p, , q, be the frequencies of the gametic types sm, sM, Sm, and SM from Short. With a recombination fraction E.> 0 we obtain p,=f(a+i.b,+(l-A)b,), q,=$((l-i)b,+).b,+c),
with similar expressions for p0 and qo. We still have p,, + q. = p, + q, = S/2, but it is no longer obvious that p,, = p, and q. = ql. The terms p/(2S) and qi(2S) in the equations (3.1) must be replaced by p,,/S or p,/S and q,JS or ql/S, and it is easy to set up the seven equations involved. We write down two of these only:
6, = &q,iS+
R,,p’q,l(MS)
327
FREQGENCIESIN TRISTYLOUS PLANTS
In the system corresponding to the equilibrium, the equation based on 6, may for instance be replaced by that found from
which is
or h, -h, = 0. Hence, there will be linkage equilibrium between the genes in the M and S loci, and the further analysis in Section 3 applies with linkage as well.
5. THE TETRAPLOID MODEL
Fisher and Martin (1947) found that the genetic system in L. salicaria involved tetraploid inheritance at both the M and S loci, but that genotypes could be assigned to phenotypes according to the same general pattern as in the basic diploid model. This system includes the possibility of double reduction, permitting both genes at a given locus in a diploid gamete to be copies of the same gene in the tetraploid parent. Various tristylous speciesof Oxalis are also polyploid (Weller, 1976). In this situation the Long morph will be represented by the genotype ssssmmmm or s,m,; Mid will consist of the four genotypes s,Mm,, s&,m,, s,Mjm, s,M,; and Short will comprise ten possible genotypes Ss,m,,.... Ss,M,, SZsZm4,..., S2szM,. No genotype will be formed with more than two S alleles when matings Short x Short are excluded. With a relative frequency a, of the genotype S,s,- ,M,m, pj (i = 0, 1, 2; j = 0, 1, 2, 3,4) the morph frequencies are L=a,,
M= i ,= I
aoi.
S= i
i
ak.
i= I /=o
Denote the probabilities of double reduction at the M and S loci by a and /I, respectively. According to the standard interpretation of double reduction we have 0 Q r, fl d l/7. We assume that the two loci are unlinked (as indeed found by Fisher and Martin (1947) in L. sahcaria). Suppose that plants with the genotype M,m, Pk at the M locus produce gametes Mjmzpj (j= 0, 1, 2) with probabilities Irki( It follows immediately from elementary probability arguments that the nonzero terms h,,(a) are
328
HEUCH
h,(r) = h,,(r) = 1,
AND
LIE
h,,(a) = h,,(l)
h,,(r)=h,,(x)=(l
-r),!2.
h2lJx)=k2(x)=(l
+2x),%,
= (2 +x)/4.
lz,2(1)=h&x)=x.:‘4, h,,(x) = 2( I - r),‘3.
Analogous expressions h,,(p) apply to segregation at the S locus. The frequency of the gametic type s,M,ml Pj among Mid gametes (standardized relative to the entire gametic population) can now be written as p;=
i
r=l
lI,;(X) %I,.
j=o.
1. 2.
and the frequency of the Short gametic type .S,s,~ ,M,m,pi is pii= i i: ki(P) h,.,(x) a,,,., u= I r=O
i.j=o,
1, 2.
(5.1)
This leads to the following equations for the genotype frequencies in the next generation: 6, = doiR/m,PA/M + R, P,o:‘S+ R,,, pb P,o/(MS) di, = h,,&., P’,IM+
R,.sp,,:S+
R,,.,( P;P;, + P; pioMK3
Ei2 = 6,iRtrn P;lM+
R/x PiziS + Rms(Pb Pi2 + Pi Pi1 + pi pio)!!(MS)
d;, =
Rmt( Pi Pi2 + Pi PiI ):‘(MS)
2, =
L
(5.2)
P; P,Y’(MS)
(i=O, 1,2). Here 6,= 1, d,,=6,~=0. At equilibrium, all ii, = a,. The conditions Q, = S and Q,,,= M pertaining to the equilibrium with L, M, S> 0 may be derived by considering C,.jiCii and Z,,, jcS,. We now assume that the equations (2.2) have been solved for L, M, S. On the basis of the expressions for c7, with i = 1, 2 we see that the Short genotype frequencies may be written as 2
aur = c ~rI\ Pm : II =o
u = 1, 2. I’ = 0: 1. 2. 3, 4,
(5.3)
where c,.,. depends on morph and mating frequencies and pb, p’,, pi, but not on U. Hence it follows from (5.1) that the frequencies of Short gametic types with one or two S alleles can be expressed as Pii=
i
II= I
‘C(B)
i
i
hcj(X)Crbl.Pl,,,;
w=o c=O
i= 1, 2, j=O, 1, 2.
(5.4)
Thus with dj,.=xC /I,~(z) c,.,., Eq. (5.4) shows that (p,,,, pzo, pII, P?~.
FREQUENCIES IN TRISTYLOUS
329
PLANTS
p,?, pII)’ is an eigenvector of the Kronecker product of the two matrices (dj,,,)j,,. and (h,i(~))j,, (corresponding to an eigenvalue equal to unity). Because of the structure of the eigenvectors of such matrices (e.g., Pease (1965, p. 328)), it follows that the ratio pyjp,, has the same value, say y, for j = 0, 1, 2. Equation (5.3) implies that we also have a,/acj = y for j = 0, 1, 2. With pI. = cjpii and aj. = xialj, we obtain from (5.1) that PI- =III1(B)al.
+h2,(B)a2..
(5.5)
On the other hand. the equilibrium equations for the genotype frequencies yield i= 1, 2. a,. = R,p,./S+ R,,pi.iS=2p,., (5.6) Inserting (5.6) into (5.5) we obtain J’=p?
:b,. =3/Q(4(l-/I))
(5.7)
as the ratio between frequencies of Short genotypes with two and only one S allele. It also follows from (5.6) that po. =S-(p,.
+pz. )=S-(a,.
+az.)/2=S/2.
Thus we have the important result that exactly one half of the Short gametesat equilibrium will carry no S allele, irrespective of the probability /I of double reduction. This means that the Short morph will always produce the same number of Short and non-Short offspring. Equation (5.1) now implies that poi = plj + pr, for j = 0, 1, 2. This simplifies comparison of the equations (5.2) for Mid and Short genotypes, and it is easily found that
which gives p; + 2~; = (2M’R,,,)( p., + 2p. 2)
4(1 -x)p;-3~p’,=(6M:‘(6M-(l-r)
(5.8)
R,))(4(l-~)p.~-33rp.~).
(5.9)
Thus Mid gametic frequencies may be expressed in terms of Short gametic frequencies. Write for the sake of brevity x = p , + 2p. 2. By considering the value of 4( 1 - a) p. 2- 3zp., in the next generation and utilizing (5.8) and (5.9), we obtain an expression involving the same quantity and x in the present generation. This procedure yields after some algebra 2(1 -a)(2+a)(6M-(l--u) 4(1-r)p~2-3rp~1=(24M-2R,m+(12M+R,m)cc+R,a2)SX2~
RI,)
(5.10)
330
HEUCH AND LIE
We denote the fraction on the right-hand side of (5.10) by C. The equilibrium relation for Long (found from (5.2)) may now be written as
(p.o+yy(pb+$y=~(L+y).
(5.11)
By means of (5.8), (5.9), (5.10), and the conditions 1, pi= A4, cj p.,= S, the quantities p.,, and pb may both be expressed in terms of x. Inserting this in (5.11) we find the following quartic equation in x: S(l+2(M-S)) R m., [ 2MS --x [ Rm
(4-r)x-Cx’ 2(2+a) (4-r)M.u 1 2+ x R,,,
1 3MCx2
-6&(1-r)
R,
(5.12)
+JL+F).
This is the generalized version of the quartic equations in x derived by Fisher (1941, 1944). When x has been found for any given t(, the gametic frequencies may be determined using p , + 2p. 2= x, (5. lo), (5.8), and (5.9 ), and the genotype frequencies are given by (5.2). In the tetraploid situation the equilibrium with only Long and Short remaining is given by a,=;,
a ,0=2(1-/Q/(4-/?).
a,, = 3/3/(2(4-p)).
(5.13)
There is a corresponding equilibrium with the Mid and Short morphs only, involving similar expressions, but now for the genotype frequencies ao4, a14, and a24. The third two-morph equilibrium, for Long and Mid with positive frequencies a,. a,,, . and aOz, is obtained by replacing /I in the expressions in (5.13) by x
6. DISCUSSION
Our analyses of the diploid and tetraploid models differ somewhat in scope. In the diploid situation we may, for example, conclude that the restriction w >O on the fertility of Mid is both a necessary and sufficient condition for a trimorphic equilibrium to exist in the selection model in Section 2. Our results for diploidy seem to represent an extension of Fisher’s (1941) analysis, even in the isoplethic case with L = A4 = S = l/3. Fisher’s mathematical treatment has frequently been referred to as implying
FREQUENCIES IN TRISTYLOUS
PLAKTS
331
the existence of such an equilibrium, but this seemsdoubtful, as he did not consider the complete set of genotype equations. In fact Fisher’s verbal argument suggeststhat he had in mind a general proposition analogous to that given by Charlesworth (Lloyd, 1977) on the equality of fitness values at equilibrium. The relevant passage from Fisher (1941, p. 33) is the following: .... in the absence of selection among genotypes, the only selective agency at work lies in the number of plants exposed to legitimate pollination by pollen from different sources, and such opportunities for pollination will evidently be equalized if the three style lengths are present in equal numbers. Hence in all cases u’ = i, b’ + c’ f = f. a - b + c = +.
In practice. however, this conclusion is dependent on the structure of the underlying genetic system. As shown by Finney (1983), there are a great many theoretical trimorphic incompatibility systems based on two diallelic loci which only possessanisoplethic equilibria under pollen elimination. By contrast, because of the complexity of the algebra involved, we have not attempted to give any discussion of the solution in the tetraploid situation. Thus our presentation here represents a direct generalization of Fisher’s (1944) analysis to cases with anisoplethic equilibria, at the same time covering the more comprehensive model with double reduction at the S locus. (Because of similar segregation distributions in the heterozygotes Ss and Sass,Fisher’s results for a tetraploid M locus and a diploid S locus may formally be obtained from those for both loci tetraploid by setting fl =O.) In this connection a remark made by Fisher (1944, p. 169) is of some interest: “The important possibility that the different style lengths, or certain genotypes within these, should have unequal fertility or viability has not been introduced. partly because this would alter the form of the analysis very considerably...“. In fact the treatment given in Section 5, covering situations with fitness differencesbetween style lengths, follows the same general approach as that used by Fisher, although, of course, it relies on a new procedure for finding morph frequencies. Fisher published no mathematical analysis of the mode1 with a tetraploid S locus, but he was evidently aware of the relatively minor changes that would be needed in the genotype frequencies at equilibrium. Numerical values of such frequencies were given by Fisher and Martin ( 1947), and the proportion 38/(4 - fi) of Short plants carrying two S alleles was quoted by Fisher (1949). The analysis given here is in a certain sense also incomplete in that no investigation has been made of the stability of the equilibria. In standard models such as those based on pollen or zygote elimination, or in the selection model in Section 2, numerical iterations always indicate stability. However, no obvious procedure for proving this presents itself, and any general approach considering local stability using a linearization technique
332
HEUCH AND LIE
seemsto involve at !east four variables related by rather complicated transformations. Moreover, the naive assumption that the procedure described here will always lead to stable equilibria is false. Consider as an example a rather unrealistic set of mating frequencies defined by RI,,,= 1 - 2.X
R,= 1-2M,
R,,=1-2L
for combinations of population frequencies with L, iW. S< 4, and arbitrarily defined otherwise. In this situation it follows that any set of morph frequencies in the region L, M. S < f defines a neutral equilibrium. However, as witnessed by the recent overview of Finney (1983), attempts to determine genotype or even morph frequencies, let alone the equilibrium stability, in other related and more general incompatibility systems very often run into serious problems. Thus it may not be so surprising that it should be difficult to attain a complete analysis of the tristylous equilibria. In order to compare genotype frequencies in the selection model of Section 2 with those in the ordinary isoplethic equilibria, numerical values were found utilizing (3.8) and (5.12). All such calculations for tetraploidy, based on a set of admissible morph frequencies, invariably produced a unique set of genotype frequencies. Tables I and II present the results for w = 0.5, 1.0, and 2.0, and in the tetraploid situation with no double reduction and with LY= l/7, ,4= 0. Analogous genotype frequencies for any j3> 0 can be written down immediately using (5.7). The increased frequency of the Short morph when w = 0.5 is seen to be entirely due to an increase of the Short genotype without any A4 allele. A reduced pistillate Mid fertility also leads to a marked increase in the relative proportion among Mid plants of individuals with a single M allele. In contrast, when w = 2.0 and Mid is superior, the genotypes carrying a single M allele become the most frequent ones among the Short combinations. Double reduction in tetraploids mostly affects the uncommon genotypes with several M alleles, but may in such cases produce a considerable relative increase in population frequencies. All equilibrium freTABLE Equilibrium
Genotype Frequencies in a Diploid
0.5 1.0 2.0
Tristylous
SSMM
sshfm
0.390 0.333 0.293
0.211 0.309 0.369
Species Short
Mid
Long Relative fertility of Mid, H
I
SSMM
0.008 0.024 0.045
Ssmm
SsMm
0.285 0.179 0.108
0.097 0.131 0.140
SSb4.M
0.008 0.024 0.045
FREQUENCIES IN TRISTYLOUS TABLE Equilibrium
II
Genotype Frequencies in a Tetraploid
Tristylous
Relative fertility of Mid, II
Probability of double reduction at M-locus, r
sG”4
SJMm,
s,M,m,
0.5 0.5 1.0 I.0 2.0 7.0
0 17 0 17 0 17
0.390 0.390 0.333 0.333 0.293 0.293
0.206 0.182 0.296 0.260 0.345 0.302
0.013 0.035 0.035 0.066 0.062 0.097
(’ All calculations
0 I ‘7 0 1 ‘7 0 1.7 with a probability
0.285 0.285 0.180 0.180 0.1 IO 0.110
Species”
Mid
Long
0.5 0.5 1.0 1.0 2.0 2.0
333
PLANTS
0.094 0.083 0.123 0.108 0.128 0.111
0.010 0.020 0.028 0.038 0.048 0.056
s,M,m 5x 1om4 0.002 0.002 0.007 0.007 0.014
s4 M4 6x 1O-6 2x 10-a 6x lO-5 7x 10-4 3x10-4 0.002
Ss,M,m
ss,M,
4x 10-4 0.002 0.002 0.007 0.007 0.014
6x 1O-6 1x10~~ 6x 1O-5 7x 10-4 3x10-4 0.002
of double reduction at the S-locus. /I, equal to zero.
quencies were checked by the more tedious procedure based on recursive calculations relating genotype frequencies in one generation with those in the next. Such calculations have sometimes been used in more general situations to determine equilibria in complex tristylous mating systems (Charlesworth, 1979; Barrett et al., 1983). In the more restricted models studied in this paper, the direct method outlined here seemsto offer a simpler approach, although it still involves some numerical work solving a quartic equation in the tetraploid case. It should in principle be possible to extend this approach to models including intramorph matings, such as those considered by Charlesworth (1979), Heuch (1979b), and Barrett et al. (1983), but this seemsto lead to very complex equations. Fisher (1941) used the numerical values in the isoplethic equilibria to derive the morph distributions among offspring from a given morph of the seed parent. Since it is difficult to determine the genotypes of Mid and Short individuals selected for breeding experiments, such calculations may be relevant to special experiments set up in “open-pollinated plots.” Thus, for Long seed parents the morph frequencies among daughter plants are found to be 0.415 Long, 0.335 Mid, and 0.250 Short when M’= 1.0 (as deter-
334
HEUCH
AND
LIE
mined by Fisher (1941)), and 0.447 Long, 0.233 Mid, and 0.320 Short when M’= 0.5. These results apply in the diploid model, but it is remarkable that the corresponding figures for tetraploidy with or without double reduction at the M locus differ only by less than about 10-j. The values given in this connection by Fisher (1944) for x = 0.08 apparently involve some numerical errors and seem the exaggerate the effect of double reduction. Identical numbers (except for a misprint) were quoted by Fisher (1949), but the modified frequencies given by Fisher (1965) for z = 0.10 are correct. Double reduction at the S locus does not affect these results. We also indicate another use of information on genotype frequencies. In order to explain observed anisoplethic population structure in tristylous plants, various authors have considered the consequences of “founder effects” (e.g. Ornduff, 1964; Barrett. 1977). A new population may be created in different ways involving seeds and pollen, but one possibility is that two seedsare drawn at random from a large trimorphic neighbouring population. If the plants produced by these seeds represent identical morphs, no further sexual reproduction will occur (unless illegitimate matings are successful). With plants belonging to two distinct morphs, the gene pool created may or may not be capable of establishing a trimorphic population. We assume that the new population is expanding rapidly so that all potential forms will actually occur, and that generations overlap. Consider for simplicity the diploid model. The combination of genotypes SSMM, SsMM will lead to a population consisting of Mid and Short only. The combination ssmm, Ssmm will produce a population of Long and Short plants, while the combinations ssrrr111z. ssMm and ssmm. ssMM will give rise to Long and Mid only. All other combinations of distinct morphs will produce trimorphic populations, at least after two generations. Using the equilibrium genotype frequencies determined above we may calculate the conditional probabilities of the various founder combinations, given that the new population is not monomorphic. The results in the diploid situation are displayed in Table III. We note that the probability of a dimorphic population exceeds the probability of all morphs being represented both for by= 1 and 1t:=0.5. What is really striking, however, is that the recently founded population will practically never consist of Mid and Short only. With tetraploidy, the probabilities are essentially the same except for the Mid/Short combination, which is now even less likely to occur (this probability is 1.2x 10p-8for kt’= 1.0 and r = 0, regardless of the value of a). In a situation with no fertility differences approximately two thirds of the dimorphic populations will consist of Long and Mid. Of course, with later gene exchange with other populations, a great many populations must be expected to become trimorphic anyway. But if such events involving the foundation of a new population are relatively frequent, these calculations may suggest why overall relative morph frequencies deter-
FREQUENCIES IN TRISTYLOUS TABLE
335
PLANTS
III
Probabilities of Morph Combinations in a Population Founded by Two Seeds Selected at Random, Conditional on the Population Being at least Dimorphic” Morphs represented in the population Relative fertility of Mid. II 0.5 1.0 2.0
Long, Mid
Long, Short
Mid, Short
All three morphs
0.264 0.333 0.369
0.344 0.179 0.097
2.1 x 1o-4 0.002 0.006
0.392 0.486 0.528
il Values presented for a diploid tristylous species.
mined over large areas may give higher values for Mid than Short, with even higher frequencies for Long. This hypothesis may be relevant to L. salicaria. which has overall morph frequencies in this order. It is related to the theory studied by Heuch (1980) involving random extinction in finite populations, but the stochastic element now enters at a different stage in the evolution of a population.
REFERENCES S. C. H. 1977. Tristyly in Eichhorniu crassipes (Mart.) Solms (water hyacinth), Biotropica 9. 23@238. BARRETT. S. C. H., AND FORNO. I. W. 1982. Style morph distribution in New World populations of Eichhorniu crussipes (Mart.) Solms-Laubach (water hyacinth). Aquat. Bot. 13, ‘99-306. BARRETT. S. C. H.. PRICE. S. D., AND SHORE. J. S. 1983. Male fertility and anisoplethic population structure in tristylous Ponkderiu corduru (Pontederiaceae), ELlolufion 37, 745-759. CHARLESWORTH. D. 1979. The evolution and breakdown of tristyly, Er:olufion 33, 48-98. DARWIS. C. 1877. “The Different Forms of Flowers on Plants of the Same Species,” John Murray. London. DCLBERGER. R. 1970. Tristyly in LJthrum junceum. Yew Phytol. 69, 751-759. FINSEY. D. J. 1952. The equilibrium of a self-incompatible polymorphic species, Geneticu 26, 33-64. FI~NEY. D. J. 1979. The equilibrium of a self-incompatible polymorphic species-A correction, Geneticu 51. 77. FINSEY. D. J. 1983. Genetic equilibria of species with self-incompatible polymorphism, Biomeirics 39. 573-585. FISHER. R. A. 1935. On the selective consequences of East’s (1927) theory of heterostylism in Lxthrum. J. Genet. 30. 369-382. FISHER. R. A. 1941. The theoretical consequences of polyploid inheritance for the Mid style form of Lyrhrum salicaria, Ann. Eugenics 11, 31-38. FISHER. R. A. 1944. Allowance for double reduction in the calculation of genotype frequencies with polysomic inheritance. Ann. Eugenics 12, 169-171. FISHER. R. A. 1949. “The Theory of Inbreeding,” 1st ed., Oliver & Boyd, Edinburgh. BARRETT.
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FISHER, R. A. 1965. “The Theory of Inbreeding,” 2nd ed.. Oliver & Boyd, Edinburgh. FISHER, R. A.. AND MARTIN, V. C. 1947. Spontaneous occurrence in Lyrhrum salicaria of plants duplex for the short-style gene, Nature (London) 160, 541. FY~: V. C. 1950. The genetics of tristyly in Uxalis aaldioiensis, Herediry 4, 365-371. HELXH. I. 1979a. Equilibrium populations of heterostylous plants. Theor. Pop. Biol. 15, 43-57. HEUCH, 1. 1979b. The effect of partial self-fertilization on type frequencies in heterostylous plants. Ann. Bor. 44, 611-616. HEUCH, I. 1980. Loss of incompatibility types in finite populations of the heterostylous plant Lythrum salicaria, Herediras 92, 53-51. HEUCH, I. 1982. A mathematical analysis of some incompatibility systems in higher plants, in “Nordic Symposium in Applied Statistics and Data Processing 1982” (A. Hoskuldsson, J. Stene, and K. Esbensen, Eds.). pp. 379-392, NEIJCC, Lyngby. LLOYD, D. G. 1977. Genotypic and phenotypic models of natural selection, J. Theor. Biol. 69, 543-560. MULCAHY, D. L. 1964. The reproductive biology of Oxalis priceae. Amer. J. Bat. 51, 1045-1050. ORNDUFF, R. 1964. The breeding system of Uxalis suksdorfii, Amer. J. Bof. 51, 307-314. ORNDUFF. R. 1975. Pollen flow in Lythrum junceum. a tristylous species, Xew Phytol. 75, 161-166. PEASE,M. C. 1965. “Methods of Matrix Algebra,” Academic Press. New York. SPIETH, P. T. 1971. A necessary condition for equilibrium in systems exhibiting self-incompatible mating, Theor. Pop. Biol. 2, 404418. TAYLOR, P. D. 1984. Evolutionary stable reproductive allocations in heterostylous plants, Eaolurion 38, 408416. VON UatSCH. G. 1921. Zur Genetik der trimorphen Heterostylie sowie einige Bemerkungen zur dimorphen Heterostylie, Biol. Zenrralbl. 41, 88-96. WELLER, S. G. 1976. The genetic control of tristyly in Oxalic section fonoxalis. Heredity 37. 387-393. WELLER. S. G. 1979. Variation in heterostylous reproductive systems among populations of Ox-al& alpina in Southeastern Arizona. Syst. Bat. 4, 57-71.