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Selection in plant populations of effectively infinite size II. Protectedness of a biallelic polymorphism
J. theor. Biol. (1982) 96,689-705
Selection in Plant Populations of Effectively Infinite II. Protectedness of a Biallelic Polymorphism HANS-R•
Size
LF GREGORIUS
Lehrstuhl fib Forstgenetik und Forstpflanrenziichtung der Universitiit G&tingen, Biisgenweg 2, 3400 Giittingen- Weende, West Germany. (Received 29 June 1981, and in revised form 11 January 1982) The biologically important problem of protectedness of genetic polymorphisms in monoecious plant populations exhibiting genotypically determined variation in rates of self-fertilization and sexually asymmetrical fertilities has hitherto escaped exact, analytical treatment for the reason that appropriate mathematical techniques relying on allelic frequencies do not seem to exist. For the particular case of one locus and two alleles it was possible to develop such a technique which provides conditions of high precision for protectedness of an allele. A comparison of the results with those already known from models that appear to be specializations of the present model showed that some of the earlier conclusions can be generalized, while others have to be handled with great care or should even be rejected. Above all, this concerns the role of self-fertilization, which is frequently considered to counteract the establishment of genetic polymorphisms. However, it turned out that increasing the heterozygote selfing rate also increases protectedness for both alleles in all situations. Moreover, even if the amount of self-fertilization is the same for all genotypes, asymmetry in the production of ovules and pollen, which is more the rule than an exception, may imply protectedness only for comparatively large selfing rates. The probably most outstanding finding is that, depending on the ovule and pollen fertilities, protectedness may be realized only within small ranges of selfing rates, and these ranges may vary from arbitrarily low to arbitrarily high rates. On the other hand, if the ovule fertilities show strong overdominance for the heterozygotemore precisely, if the heterozygote produces more than twice as many ovules as either of the homozygotes-both alleles are protected irrespective of the pollen fertilities and rates of self-fertilization; this generalizes earlier results obtained for more specific models.
1. Introduction In the classical biallelic the allelic polymorphism
viability selection model with is protected if the genotypic
random mating, viabilities show
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0 1982 Academic Press
Inc. (London)
Ltd.
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GREGORIUS
overdominance. However, as soon as sexually asymmetrical fertility differences occur, no such simple ranking relationship between genotypic selection values can be applied to prove protectedness (cf. Mandel, 197 1). This situation requires a fitness concept that combines all selection values into a single value termed “realized fitness”, as was pointed out in the first paper of this series (Gregorius & Ross, 1981). Moreover, sexually asymmetrical fertility selection results in increased heterozygosity as compared to pure viability selection, and thus conditions under which a polymorphism is protected arise more readily. This accords well with our intuitive expectation that a high level of heterozygosity also suggests a high degree of protectedness. In most plant species, however, at least partial selffertilization is very common and is therefore likely to reduce the heterozygosity gained by cross-pollination and asymmetry in the production of ovules and pollen. Colonizing plant species, for example, may depend heavily on their capacity to self-fertilize in order to be able to establish a population (Baker, 1967). On the other hand, such a newly established population would be severely endangered if it were not equipped with the means to conserve the genetic variability needed for adaptation to variable environmental conditions. Allard (1965, 1969) and Jain & Marshall (1967) reported that largely self-fertilizing plant species maintained stores of genetic variability that were quite as large as those found in related outcrossing species. Consequently, there must be effective mechanismsfor protecting genetic polymorphisms even for low degrees of heterozyosity. This raisesthe more general question as to whether the conjecture is always true that increased rates of self-fertilization and homozygosity reduce protectedness of an allelic polymorphism. Probably becauseappropriate mathematical tools were not yet sufficiently developed, most analytical treatments of the subject are confined either to completely self-fertilizing populations (for a review see e.g. Karlin, 1968, p. 12; or Nagylaki, 1977, p. 96) or to very particular models of selection with partial selfing (Moran, 1962, p. 58; Kimura & Ohta, 1971, p. 190; Nagylaki, 1976; Charlesworth, 1980). The present paper is devoted to the biallelic case and is based on the model introduced in the first publication of this series (Gregorius & Ross, 1981). This model allows arbitrary selection with respect to viability and to male and female fertility, as well as arbitrary rates of self-fertilization. It should be pointed out that this model applies to purely monoecious (hermaphrodite) populations as well as to populations composed of a mixture of unisexual (males or females) and bisexual genotypes. As was to be expected, the analytical details are rather
PROTECTEDNESS
OF
691
ALLELES
involved and are therefore presented in the Appendix. However, since it was possible to develop a new kind of analysis of protectedness which is likely to be applicable to a wider range of models, the reader might wish to become familiar with its features. The wealth of opportunities to apply the coefficients of protectedness resulting from this analysis to particular cases requires restriction of the presentations. In this paper emphasis shall therefore be put primarily on critically reviewing a few known results and conjectures.
2. The Model The model, which is elaborated in Gregorius & Ross (1981), refers to plant populations of effectively infinite size reproducing in separated (nonoverlapping) generations. An unordered genotype AiAj is characterized by the average number 4ij of ovules that are produced by A&,-plants, the average proportion gij of self-fertilized ovules and the average number pij of pollen grains not used in selfing (free pollen). Since all these parameters are assumed to be measured from birth (zygotic stage) of the plants they already reflect the effect of viability. The (1 - aii) . dij non-selffertilized ovules of an AiAj-plant are fertilized by pollen randomly drawn from the pool of free pollen (as specified by the p’s) produced in the population. With the help of this notation, bisexual genotypes have 4ij > 0 and /.Lij > 0, while unisexual genotypes represent females if dii > 0, pij = uij = 0 and males if CLij> 0, 4ij = gij = 0. The case that a plant partially or completely self-fertilizes but produces no free pollen, i.e. 4ij > 0, mij >O and Fij = 0, constitutes a biological rarity and is excluded from the present considerations. Denoting by Pjj the relative frequency of AlAi-plants at the zygotic stage, 4 := 1 Pij a4ij, id isj
/.Z:= 1 Pij . pii hi isj
and
Cr:= 1 Pi,vi&ij/& ii isj
are the population averages of the number of ovules, number of free pollen grains and the proportion of self-fertilized ovules, respectively, per individual plant measured from birth. Under the assumption of regular Mendelian segregation, the one-locus-two-allele version of the general transition equation in Gregorius & Ross (1981) has the following representation (primes, e.g. Pii, denote genotypic frequencies in the next
692
H.-R.
GREGORIUS
generation): pli =
gii
* dii
- Pii
+ $ * fll2412P12 i
(1 -vii).
4ii.
Pi<
++I.
(1-~12)412P12
Piipii
+$-412P12
+ B
Uii4ii aPii
z
1 _ Piipii
+ b12p12
(
4 ++iiPii
>
fi
+t412P12
Piipii
6
+b12p12
*
h2412Pl2
+
2
1
6 Piipii
+&12P12
6
xi- (
for homozygotes,
where
i = 1 or i = 2, and
+ a-
(+12)412P12
Pi2 =
h2h2Pl2
(14
)
CL
+ Cl-
(+22M22P22
6
6 xlLllP11+~cL12P12
+:(1-(T12)~12P12+(1-~ll)~llPll B
ii x cL22p22
+ b-4 12p12 cc
-S4l,Pl2
I ~1-mM11P11 4
+(1
CL22p22+tCL12p12 6
*
-u22)422P22
CLllPll
6
*
cz +b12P**
(lb)
CL
for the heterozygote. Since pi := Pii +iP12 describes the allelic frequency letting pi denote this frequency in the next generation, may be combined to give 1 (I+ Pi
ciiJ4iiPii
=2’
+t(l+(712)412P12
of A; (i = 1, 2) and the above equations
.$(l-5)
i x Piipii
+ tCL12P12 CL
(2) .
These formulations of the transition equations provide us with some basic insights into the role played by self-fertilization in the changing of germtypic
PROTECTEDNESS
OF
ALLELES
693
and allelic frequencies from one generation to the next. To simplify interpretation, consider that (PiJ’ii +&r#&/fi is the frequency of Ai in the pool of free pollen produced in the population. From equation (la) it follows that the frequency of a homozygote increases if its rate of self-fertilization is increased. An increase in the heterozygote rate of self-fertilization is accompanied by an increase in frequency of that homozygote whose allele is the less frequent one in the pool of free pollen; the other homozygote declines in frequency.
Equation
(lb) implies that
the frequency of the heterozygote is not affected by changing the heterozygote rate of self-fertilization, while it decreases if either one of the homozygote rates of self-fertilization grows.
These observations confirm some of the intuitive expectations about the _ relationship between heterozygosity and self-fertilization, but not all. A simple reformulation of equation (2) reveals in connection with the above statements about genotypic frequencies that the frequency of Ai increasesas the rate of self-fertilization of the corresponding homozygote AiAi grows; it decreaseswith growing rate of selffertilization of the other homozygote A&(j# i). Whether Ai decreasesor increasesin frequency with a growing heterozygote rate of self-fertilization dependson whether this allele is the more or lessfrequent one, respectively, in the pool of free pollen.
To conclude this list of basic properties of self-fertilization, it might be worthwhile to emphasize that a change in allelic frequencies in the course of the generations can be caused merely by genotypic differences in the rates of self-fertilization. This is easily verified with the help of equation (2) if one sets dii = C$and pii = p for all three genotypes. 3. The Conditions ior Protectedness of an Allele
It is a well-known fact that even comparatively simple population genetic selection models, such as one-locus-two-allele models, can exhibit complex dynamic behavior caused by the existence of multiple attractive and unstable equilibria or cycles (cf. e.g. M&at, 1969; Kidwell et al., 1977; Hadeler & Liberman, 1975; Pollak, 1978). In principle, it is also possible to construct models showing completely undirected, i.e. chaotic dynamics (cf. Li & Yorke, 1975). Presumably because of these observations it has become rather common during the last years to attack the problem of maintenance of genetic variability by selection from a different point of view: instead of analyzing the existence and stability of polymorphic equilibria, the repulsivity of the extinction states for alleles (i.e. where the alleles
694
H.-R.
GREGORIUS
have zero frequencies) is investigated (cf. Gregorius, 1979). In selection theory it has becc-me usual to call an allele protected if its extinction state is repulsive in some sense. More specifically, the term “protectedness” will be used in the sense Karlin (1977) applied to it. An allele is called protected if it cannot be lost where all possible genotypes are present with it is required that the allele cannot stay for an at arbitrarily low frequencies. If all alleles are allelic polymorphism is called protected.
from any state of the population positive frequencies: moreover, indefinite number of generations protected in this sense, then the
This definition of protectedness is sharper than Karlin’s in two respects: since the allelic frequency in the next generation is a function of the genotypic rather than the allelic frequencies in the present generation, the allelically polymorphic initial conditions in Karlin’s definition have to be replaced by genotypically polymorphic initial conditions; situations in which an allele may start at arbitrarily low frequency and remain there without becoming extinct are excluded. The first of these, i.e. that the transition equation (2) for the alellic frequencies relies on the genotypic frequencies, is also responsible for the fact that protectedness cannot directly be analyzed in the classical way by considering spectral radii of derivatives (Jacobi matrices). This is a consequence of the fact that generations generally do not start with HardyWeinberg proportions at the zygotic stage. A type of analysis that takes this into account and relies on allelic frequencies as variables of primary interest is elaborated in the Appendix. The principle of this analysis consists in showing ?. that the multiplication (growth) rate pi/pi of Ai always approaches a value Bi after many generations in the close vicinity of the extinction state for A,. To exclude trivial cases, the selection values are assumed not to represent situations in which an allele or a genotype can be lost in a single generation (:l.e. PII, P12, PI2 > 0 implies Pi,, Pi2, Pi2 >O). Then A, is protected if 0, > 1, and if ii < 1, Ai eventually becomesextinct at least when starting at low frequency. In the Appendix it is proven that (consult equations (Al) and (A2)) 6i = $(a; + di - 6,) + Ji(b; + di -a, )’ + b,(ci -d,) =~(a;+di-b;)+J~(ai+bi-d,) if AjAj(i
2
+bi(c;-a;)
# j) is a bisexual genotype, i.e. if ~ij, ~,j > 0, where Ui := Uii~ii/~jj, Ci
:= $(1 -
bz+ $(+,241,/4jp Ujj)/.Lii//.L,j
+ +(1 + Uii)ki/djj
(3a) (3b)
PROTECTEDNESS
OF
695
ALLELES
and di
:=$(l-Ujj)PlZ/Pjj
+f(l
+al2)412/4jp
Note that Ai is trivially protected if the genotype homozygous with respect to the other allele is unisexual, i.e. if 4ij = 0 or pjj = 0 (j # i). Before analyzing in more detail the significance of the rates of selffertilization for protectedness, some special cases of the present model shall be considered which refer to known results. No self-fertilization,
i.e. Uij = 0 for i, j = 1,2
In this case ai = bi = 0, di = $(p12/pjj + dr2/4jj) and, consequently, si = di* In accordance with the pres:nt results,lMandel (1971) and also Kidwell et al. (1977) pointed out that Bi > 1 and Bi < 1 indicate protectedness (initial increase of a new allele) and non-protectedness of Ai, respectively. Complete self-fertilization,
i.e.
ffij
= 1 for i, j = I,2
The effect of the p001 of free pollen is eliminated. and therefore
ai = ci = 4ii/4jj, 2bi =
di = 412/4jj
6i = $(a; + bi) +ilai - bil= max {ai, bi} = max {4ii, 44,2}/4jp
Hence, the biallelic polymorphism is protected if e^i> 1 and 62 > 1, which is equivalent to max {4i1, d22} < &512. This coincides with the result reported by Karlin (1968, p. 15, bottom of Table 2). Constant rates of self-fertilization and no differences between male and female selection coefficients, i.e. uij = u and 4ij = pij =: uij for i, j = 1, 2
ai = (+vii/vjj, bi = $(+V12/Vjj, ci = Vii/Vjj and di = v12/vjb After rearrangement
one arrives at
ii = U12+~(Uii-~U12)+~(1-~)t):2 2
+(r2(Vii--tV*2)2
* Vjj
Since the numerator is an increasing function of Vii, Uii 5 Vjj implies ii 5 I$. Consequently, the biallelic polymorphism is protected if Vii 5 Ujj and di > 1, which is the same conclusion Kimura & Ohta (1971, p. 194) arrived at. This condition is met if and only if either max {ulr, ~22) < $12, in which case protectedness of the biallelic polymorphism is guaranteed for all ~(01~51), or ViisUjj
Ujj)+
Ujj(DlZ-Vii)’
696
H.-R.
GREGORIUS
The appertaining analysis follows easily from (A3) in the Appendix. Unfortunately, Kimura & Ohta (1971, p. 155) draw a too narrow conclusion from this result when claiming “that overdominance cannot be a major mechanism for polymorphism in predominantly self-fertilizing organisms, For ull =uZ2l for unless overdominance is very strong”. all m, 0 5 u < 1; consequently, even for very small overdominance and a very high rate of selfing, the polymorphism can still be protected. In the frame of this special case it becomes evident that if a protected polymorphism exists for some value of g, then it is automatically maintained for all smaller values of CT,including (+ = 0. This is not true in general, as shall be pointed out in the next section. All genotypes bisexual, no differences among female selection values and no se&g for one homozygote, i.e. 4ii = # for i, j = 1,2 and uii = 0 This model type has been treated by Nagylaki (1976). It will be shown whether the author is correct in claiming that the selfing allele A, “is established in the population if and only if at least one of the selfing genotypes contributes to cross-fertilization” (i.e. kii >O or ~12 >O) and that the cross-pollinating allele Ai is not protected if ~12, @ii I Fij and (1
-(+lZ)U&u ai
>(l =
-@ii)fll2Wl2.
ai =O,
ffji,
Cj = 1(1
-
6, = $u,, = bi,
Ci
=
$Jdii/Fjj
+ it 1 + CT;,1,
di=S/112I~jj+kCl+u12)
uii)Cljjl,lLii,
and di=~(l-crii)~-12/cLii+1(1+~12). Hence, ~i=t(Uji+3(1+CLlZ/CLjj)) +Y(gii ij
=$((l
-40 -~ii)P12/Pii
+4J((1-c+ii)@121Pir
Consider
+/L12//Ljj)~2+IT12~/-Lii/~jj
+ 1 -(Tit)
+ l) +
1)2+~12(1-(Tii)~jjlcLii.
the example cii = ~12 = 0.5 and 0 < piz/pjj,
pi;/pjj < 0.5. Then
~i=3(1+tCL1ZI~jj)+~J~(CL1ZICLjj)2+~(CLiiICLjj+t) <;+&J&T$=l, which, contradicting Nagylaki’s claim, proves that the selfing allele Ai may not be protected although both selfing genotypes contribute (free) pollen to cross-fertilization. NOW, choose uii = (+I2 = 5, /JI 2= 2 and pii = 3 < pjj,
PROTECTEDNESS
OF
ALLELES
697
such that the condition (1 -c~ii)~lzplz < (1 -~&~~i~ii is met. Then 6j > i(( 1 - ~~i)~1z/~ii + 1) = 1, i.e. the cross-pollinating allele Aj is protected, which also contradicts the second above-mentioned claim of Nagylaki. These examples may suffice to demonstrate the insecurity that still exists in judging the significance of self-fertilization for the maintenance of genetic polymorphisms even with respect to comparatively simple models of selection. The last section of the present paper shall therefore be devoted to a few basic comments on the role of self-fertilization. 4. Protectedness
and Self-fertilization
In order to simplify understanding of the following considerations, note that the sums in the brackets appearing in equation (3b) have the representation ai+di-bi=~iAii/9jj+t~~~/~jj+3(1-~ji)~~~/~jj Ui+bi-di=~iidii/~jj-~~12/~jj-~(l-~jjii)CL12/~jj Ci
- Uj = $( 1 -
Uii)&ii/4jj
+ $( 1 -
Ujj)pii/pjp
Consequently, as long as 41~ > 0 and ii - ai > 0, & is a strictly increasing function of the haterozygote selfing rate u12. In other words, if all other selection coefficients remain the same and only the heterozygote selfing rate u12 is increased, then both coefficients of protectedness e^,and &, i.e. the degree of protectedness of the biallelic polymorphism, also increase. This observation points out a phenomenon of selfing which does not become apparent in models where all selfing rates are assumed to be the same, such as in the case mentioned in the last section. There, protectedness for a certain selfing rate implied protectedness for all smaller rates; now, however, protectedness might not be realized for all small heterozygote selfing rates, while it might be guaranteed for all high rates. The probably most trivial example for this may be found in a model of no differences within female- and within male-selection values and identical selfing rates for the tw? hopozygotes, i;e. +ij 3 4, pij c p for i, j = 1,2 and ~11s ~22 = (+. Clearly, & = 13~=: 13,and 13= 1 if cr12= v which refers to the case of no selection in allelic frequencies. Consequently, e^< 1 for ~12 1 for cr12> (+. Moreover, this example shows that selection might operate exclusively through differences between rates of self -fertilization (cf. WGhrmann & Lange, 1970). Performing changes in the homozygote selfing rates only, the direction of change in the 6’s depends on the relationships between the male and
698
H.-R.
GREGORIUS
fernal: selec$on coefficients. It may happen that as uii, say, moves from 0 to 1, Bi or 0, declines for some time and then grows again or vice versa. Nevertheless, it is possible to formulate a principle of general validity. Since bi(ci -a,) 2 0 one obtains the inequality ,. Bi~~(Ui+di-bi)+tlUi-(di-b;)l=max{U;,d,-bi} =max{uiiidii/4jj,
If both $$,,/4,, therefore
t412/4jj
and i+r,/#~,
+hl
-~jjii,cL12/cLjj)~~~l2l~j~
(4)
exceed one, then 6; > 1 for i = 1,2 and
the biallelic polymorphism is protected, irrespective male selection values and the rates of self-fertilization,
of the magnitudes of the if max {QS,,, q&}< &#J,,.
Under the condition that all genotypes are bisexuals, it was shown in the last section that the above inequalities become equalities, i.e. b;,= +4,,/4ij for i = 1,2 if complete self-fertilization is realized for all three genotypesAand dri, & 5 &#~i,. There are two additional possibilities to arrive at 8i = $,2/d, for i = 1,2, both of which naturally require that the two homozygotes are bisexuals: the heterozygote is a female, i.e. p rZ= cl1 = u22&2
PROTECTEDNESS
OF
699
ALLELES
appropriate normalization. It follows that in this case cl = dl = i(l-a)/p+$(l+a)#, az=a/& b,=’ 2u, c2 = i( 1 - a)~ + f( 1 + cr)/d and dz= 1. The respective explanations in the Appendix show that
and, consequently,
A1 is protected if
$(d+i)>l
and
From (A3) in the Appendix
Osa<
1-@-4h
1-b * one concludes that A2 is protected if either
l
i.e.
*
u’&pe
Or -7
The last condition
24
and
is equivalent U>
O/4-1).
to
o-FM--1 l-&h
u>O ’
’
Since ,>h4+1< l-#p
24 2-4
one obtains in summary: A2 is protected if a> o-PM--1 1-h * Combining the conditions for protectedness of A1 and A2 and taking account of the inequality (2 - F)#J - 1 < 1 - (2 - 4)~ one arrives at the result that the biallelic polymorphism is protected for all positive values of u located in the open interval O
and
(2-PM-1
l-4.P
l-4.P
provided (2 - 4)~ < 1. A comparison of the statements inherent in this inequality with those made for sexual symmetry in the preceding section reveals that sexually
700
H.-R.
GREGORiUS
asymmetrical selection may alter the effect of selfing considerably. Under the conditions of the above model, polymorphisms may be protected only for rates of self-fertilization that belong to intervals whose lower and upper bounds are greater than zero and smaller than one, respectively. Such intervals may be arbitrarily small and located anywhere in the unity interval. To see this, denote Y:=
l-4@ 1-d
and notice that y>l
and
4=-
y-1
Y-FL’
Inserting this representation of C#J into both of the above bounds yields (2-PM-1
1-h
=12
Y
which proves the assertion. For example, if p = 9/10 and 4 = 50/51, then y = 6 and the biallelic polymorphism is protected for all selfing rates (T with $< (+< $; if CJ< 3 or cr >&, the polymorphism is not protected. This, obviously, is in sharp contrast with any model of sexually symmetrical (such as pure viability) selection. The probably most impressive demonstration of the significance of selffertilization can be achieved by considering the other extreme where selection takes place only with respect to the selfing rates, i.e. 4ij = 4, pLr,= p for i, j = 1,2. In this case ? ~j=~(l+(Tjr-t~jj)+~/t(l-~~i-3~jj)-+~cT1~(1-~((T~~+(T?~)).
From (A3) in the Appendix it follows that $, > 1 is equivalent to 1 -ff,r (5) 1 - fh + md Applying this formulation, it may be shown by contradiction that for e”r, & > 1 it is necessary (but not sufficient) that vI z> ~11, (~22.However, as was pointed out earlier, if gll = uz2, then crl 1= a22 < u12 implies ill i2 > 1. Since 6; is a continuous function of its parameters, small deviations of the ~ij’s and Pii’S from constancy might therefore still produce a protected polymorphism if condition (5) is met. These small deviations comprise all kinds of ranking relationships between the sexual specific selection coefficients, including, for example, underdominance in both the 4ij’s and the Nij’s, which would lead to fixation in the absence of self-fertilization. ~((T~~+(T~~)<~ and
a12>~,,.
PROTECTEDNESS
OF
ALLELES
701
The findings of this section demonstrate the multiplicity of interactions between sexual specific selection values and selfing rates that can lead to protectedness of a biallelic polymorphism. Some of these offer solutions to the problems mentioned in the introduction and others contradict known statements. However, it is difficult to derive general qualitative rules other than those concerning the action of the heterozygote selfing rate or resulting from the inequality (4) without considering more specific aims. Among the wealth of interesting questions that can at least in part be treated meaningfully with the help of a biallelic model, one is of fundamental importance for the evolution of plants: what are the conditions for the maintenance of mixed systems of sexuality, such as gynodioecy (bisexual and female plants), androdioecy (bisexual and male plants) and trioecy (bisexual, male and female plants)? In such not merely monoecious populations it might happen that the allelic but not the genotypic polymorphism is protected since, for example, the heterozygote is always maintained at a positive frequency while one of the homozygotes eventually disappears. This requires consideration of both types of protectedness, allelic and genotypic, simultaneously and will be the subject of further investigation within the frame of the present series. It is also of general interest to study in more detail some important cases of purely monoecious populations. The author is grateful to M. D. Ross for important suggestions to improve the contents and representations of this paper. The work was supported by a Heisenberg fellowship. REFERENCES R. W. (1965). Genetic systems associated with colonizing ability in predominantly self-pollinated species. In: The Genetics of Colonizing Species (Baker and Stebbins). New York: Academic Press, pp. 49-76. ALLARD, R. W. (1969). Breeding systems and population structures: synthesis (abstr.) Abstracts of the papers presented at the XI International Botanical Congress, Seattle p. 2. BAKER, H. G. (1967). Evolution 21, 853. CHARLESWORTH, B. (1980). J. fheor. Biol. 84,655. GREGORIUS, H-R. (1979). Inr. J. Systems Sci. 10,863. GREGORIUS, H-R. & Ross, M. D. (1981). Math. Biosci. 54,291. HADELER, K. P. & LIBERMAN,~. (1975).J. math. Biol.2,19. JAIN,S.K.&MARSHALL, D.R.(1967). Am. Nat. 101,19. KARLIN, S. (1968). Equilibrium of Population Genetic Models with Non-random Mating. New York, London, Paris: Gordon and Breach. KARLIN,% (1977). Am.Nat. 111,1145. KIDWELL,J. F., CLEGG, M.T.,STEWART,F. M. & PROUT,T. (1977). Genetics 85,171. KIMURA, M. & OHTA, T. (1971). Theoretical Aspects of Population Genetics. Princeton: Princeton University Press. ALLARD,
LI, T. Y. & YORKE,
J. A. (1975).
Am. Math.
Monthly
82,985.
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GREGORIUS
MANDEL, S. P. H. (1971). Heredity 26, 49. MBRAT, P. (1969). Ann. G&z&. Ski. anim. 1,49. MORAN, P. A. P. (1962). The Sfufisfical Process of Evolutionary Theory. Oxford: Clarendon Press. NAGYLAKI,T.(~~~~). J. theor. Biol.58,55. NAGYLAKI, T. (1977). Selecfion in One- and Two-locus Systems. Berlin, Heidelberg, New York: Springer-Verlag. POLLAK, E. (1978). Genetics 90,383. W~HRMANN,K. & LANGE, P.(1970). Theor. appi. Genet. 40,289.
APPENDIX The central quantity to be considered in the present analysis of protectedness of an allele Ai is its multiplication rate Pi/m over many generations at very low frequencies. For this purpose consider the variable 4t := Pii/pl (hence 1 -4i = iPrz/pi) and rewrite the equations (2) and (la) in the form Pi/Pi
1 (l+~ii)~iiqi+(l+(T12)~12(l-qi) = 2 ’ 6
/J-ii
’ 4i +/412(1-4;) ti
For a given value of 4 look at the limit of pi/pi as pi approaches 0. Since the case& > 0, cii > 0 and pii = 0 is excluded from the considerations, this limit equals infinity if either 4jj = 0 or ,uii = 0. Hence, denoting the limit (given 4;) by @i(4i),one arrives at 6, (si) = co if the homozygote Afi?( j f i) produces no pollen (kj; = 0) or no ovules (C$jj= O), and 8~(4j)=4i.Ci+(l-q,).d;
for
/Ajj.+jj>O
:= $(1 -
+ $(1 + Uii)4ii/4jj
where C,
Ujj)/.Lii//Ljj
and di
:=3(1
-ujjii)Pl2/&jj
+3U
+(+12M12/&
As it should be, Ai is protected if the homozygote A/Aj(j # i) represents a unisexual genotype (8i(4i) = 00> 1 for all 4i). Therefore, from now on only the case kjj. 4jj > 0 is of interest.
PROTECTEDNESS
OF
To establish the relationship
703
ALLELES
of Pii/pi to Pli/p:
as pi approaches zero and
Pii/pi = qi remains constant, note that = WpiPI/Pi’
pgp;
Denoting
this limit of P:i/pi by 4: one obtains qf=4i*ai+(1mqi)-bi I ei(4i)
where ai := giiAiJ4jj
and
bi := $alzdlz/+j+
This proves that, as the frequency of Ai approaches zero, Pii/pi = 4i possesses a dynamics that is governed by the transition equation 4!
=f(4,)=
qi(ai-bi)+bi
1 ’
qi(Ci-di)+di’
Again, two trivial situations can be excluded from further analysis: ci = di, since then tii(Si) z ci = di for all possible values of 4s and ~i = bi = 0, since then after the first generation ei(qi) = @i(O)= di for all possible initial values qi. Note that the latter case is met if aii = ~12 = 0. Since 0 5 ai 5 ci and 0 I bi 5 di, f is a function that maps the closed zero-one-interval [O, l] into itself. Consequently, the dynamics of 4; (defined by the iterates of f) are completely determined by the properties f exhibits on [0,1]1. Omitting the indices for convenience, one obtains for the derivatives of f with respect to q: $=f(4)=
a*d-c’b
[dc - 4 + 4
2 and
$=fy4)=
2(c - d)(cb -ad) [q(c-d)+d13
’
Hence, f is a montone and convex or concave function on 10, 11, and can therefore not have more than one fixed point 4 = f(t) with 0 < 4 < 1, since c # d. This point, if it exists and f is increasing, i.e. ad 2 cb, attracts all trajectories starting with 4, 0 < 4 < 1. If f is decreasing, the trajectories perform oscillations around 4. To see that in this case 4 still attracts all trajectories starting in 10, I[, consider that (a-b)[q(a-b)+b]+b[q(c-d)+dl fZ(q)=f(f(q))=(c-d)[q(a-b)+b]+d[q(c-d)+d] =q[(a-b)2+b(c-d)]+b(a+d-b) q(c-d)(a+d-b)+b(c-d)+d*
704
H.-R.
GREGORIWS
Hence
The numerator
can be rewritten in the form
q(u+d-b)(a-b-d)-(a+d-b)[b-q2(c-d)] =(a+d-b)[q(u-b)+b-q(q(c-d)+d)]
=(u+d-b)[f(q)(q(c-d)+d)-q(q(c-d)+d)l =(a+d-b)(q(c-d)+d)(f(q)-q).
Consequently, (u+d-b)(q(c-d)+d) f2(4)-4=(f(4)-4)*q(c-d)(u+&b)+b(c-d)+d2’
where the ratio on the right side is positive for all 4, 014 5 1, since by assumption c # d and a + b > 0. It follows, that for 4 # 4, (f2(4) - 4) X (f(4) -4) > 0, which in turn implies that the oscillations around 4 are damped. This proves that the trajectories produced by the iterates of f always converge to a fixed point off. If O< 4 = f(J) C 1 exists, it is straightforward to show that 4 has the representation 1 ld+b-a q=2 d-et
--dm
ford>c
and * ld+b-a q=T d-c
+Jm
ford
Insertion of both these values for 4 into e(4) = d(c -d) + d results in the same expression for e^:= 8(d), namely e^=e(&=$(u+d-b)+x&d+b-u)2+b(c-d).
(AlI
In fact, the root is always real, since $(d+b-u)2+b(c-d)=$(u+b-d)Z+b(c-u)~0.
Therefore,
e^may also be given the equivalent e^=+(u+d-b)+&u+b-d)2+b(c-a).
representation t-42)
PROTECTEDNESS
OF
ALLELES
705
The cases in which f does not have a fixed point in the open interval 10, l[ are the following: f(0) = 0 and f(1) = 1: according to whether f is convex or concave, 4 = 0 or 4 = 1 are the only attractive fixed points, respectively; f(0) > 0, f( 1) = 1 and f’(1) 5 1: 4 = 1 is the only attractive fixed point; f(0) = 0, f(1) < 1 and f(0) 5 1: 4 = 0 is the only attractive fixed point. If a, b, c and d are chosen such that they reflect the above conditions for f, it is easily verified that 0 evaluated at the respective &values can also be obtained directly with the help of (Al) or (A2). The same is true for the initially excluded cases c = d and a= b = 0. Recall that 8i as given by (Al) or (A2) represents the multiplication rate for Ai after the population has stayed for a sufficiently large numbfr of generations at low freque?cies for ai. Consequently, Ui is protected if 0i > 1 and it is not protected if @< 1.
Selection in Plant Populations of Effectively Infinite II. Protectedness of a Biallelic Polymorphism HANS-R•
Size
LF GREGORIUS
Lehrstuhl fib Forstgenetik und Forstpflanrenziichtung der Universitiit G&tingen, Biisgenweg 2, 3400 Giittingen- Weende, West Germany. (Received 29 June 1981, and in revised form 11 January 1982) The biologically important problem of protectedness of genetic polymorphisms in monoecious plant populations exhibiting genotypically determined variation in rates of self-fertilization and sexually asymmetrical fertilities has hitherto escaped exact, analytical treatment for the reason that appropriate mathematical techniques relying on allelic frequencies do not seem to exist. For the particular case of one locus and two alleles it was possible to develop such a technique which provides conditions of high precision for protectedness of an allele. A comparison of the results with those already known from models that appear to be specializations of the present model showed that some of the earlier conclusions can be generalized, while others have to be handled with great care or should even be rejected. Above all, this concerns the role of self-fertilization, which is frequently considered to counteract the establishment of genetic polymorphisms. However, it turned out that increasing the heterozygote selfing rate also increases protectedness for both alleles in all situations. Moreover, even if the amount of self-fertilization is the same for all genotypes, asymmetry in the production of ovules and pollen, which is more the rule than an exception, may imply protectedness only for comparatively large selfing rates. The probably most outstanding finding is that, depending on the ovule and pollen fertilities, protectedness may be realized only within small ranges of selfing rates, and these ranges may vary from arbitrarily low to arbitrarily high rates. On the other hand, if the ovule fertilities show strong overdominance for the heterozygotemore precisely, if the heterozygote produces more than twice as many ovules as either of the homozygotes-both alleles are protected irrespective of the pollen fertilities and rates of self-fertilization; this generalizes earlier results obtained for more specific models.
1. Introduction In the classical biallelic the allelic polymorphism
viability selection model with is protected if the genotypic
random mating, viabilities show
689
0022-5193/82/120689+17$03.00/0
0 1982 Academic Press
Inc. (London)
Ltd.
690
H.-R.
GREGORIUS
overdominance. However, as soon as sexually asymmetrical fertility differences occur, no such simple ranking relationship between genotypic selection values can be applied to prove protectedness (cf. Mandel, 197 1). This situation requires a fitness concept that combines all selection values into a single value termed “realized fitness”, as was pointed out in the first paper of this series (Gregorius & Ross, 1981). Moreover, sexually asymmetrical fertility selection results in increased heterozygosity as compared to pure viability selection, and thus conditions under which a polymorphism is protected arise more readily. This accords well with our intuitive expectation that a high level of heterozygosity also suggests a high degree of protectedness. In most plant species, however, at least partial selffertilization is very common and is therefore likely to reduce the heterozygosity gained by cross-pollination and asymmetry in the production of ovules and pollen. Colonizing plant species, for example, may depend heavily on their capacity to self-fertilize in order to be able to establish a population (Baker, 1967). On the other hand, such a newly established population would be severely endangered if it were not equipped with the means to conserve the genetic variability needed for adaptation to variable environmental conditions. Allard (1965, 1969) and Jain & Marshall (1967) reported that largely self-fertilizing plant species maintained stores of genetic variability that were quite as large as those found in related outcrossing species. Consequently, there must be effective mechanismsfor protecting genetic polymorphisms even for low degrees of heterozyosity. This raisesthe more general question as to whether the conjecture is always true that increased rates of self-fertilization and homozygosity reduce protectedness of an allelic polymorphism. Probably becauseappropriate mathematical tools were not yet sufficiently developed, most analytical treatments of the subject are confined either to completely self-fertilizing populations (for a review see e.g. Karlin, 1968, p. 12; or Nagylaki, 1977, p. 96) or to very particular models of selection with partial selfing (Moran, 1962, p. 58; Kimura & Ohta, 1971, p. 190; Nagylaki, 1976; Charlesworth, 1980). The present paper is devoted to the biallelic case and is based on the model introduced in the first publication of this series (Gregorius & Ross, 1981). This model allows arbitrary selection with respect to viability and to male and female fertility, as well as arbitrary rates of self-fertilization. It should be pointed out that this model applies to purely monoecious (hermaphrodite) populations as well as to populations composed of a mixture of unisexual (males or females) and bisexual genotypes. As was to be expected, the analytical details are rather
PROTECTEDNESS
OF
691
ALLELES
involved and are therefore presented in the Appendix. However, since it was possible to develop a new kind of analysis of protectedness which is likely to be applicable to a wider range of models, the reader might wish to become familiar with its features. The wealth of opportunities to apply the coefficients of protectedness resulting from this analysis to particular cases requires restriction of the presentations. In this paper emphasis shall therefore be put primarily on critically reviewing a few known results and conjectures.
2. The Model The model, which is elaborated in Gregorius & Ross (1981), refers to plant populations of effectively infinite size reproducing in separated (nonoverlapping) generations. An unordered genotype AiAj is characterized by the average number 4ij of ovules that are produced by A&,-plants, the average proportion gij of self-fertilized ovules and the average number pij of pollen grains not used in selfing (free pollen). Since all these parameters are assumed to be measured from birth (zygotic stage) of the plants they already reflect the effect of viability. The (1 - aii) . dij non-selffertilized ovules of an AiAj-plant are fertilized by pollen randomly drawn from the pool of free pollen (as specified by the p’s) produced in the population. With the help of this notation, bisexual genotypes have 4ij > 0 and /.Lij > 0, while unisexual genotypes represent females if dii > 0, pij = uij = 0 and males if CLij> 0, 4ij = gij = 0. The case that a plant partially or completely self-fertilizes but produces no free pollen, i.e. 4ij > 0, mij >O and Fij = 0, constitutes a biological rarity and is excluded from the present considerations. Denoting by Pjj the relative frequency of AlAi-plants at the zygotic stage, 4 := 1 Pij a4ij, id isj
/.Z:= 1 Pij . pii hi isj
and
Cr:= 1 Pi,vi&ij/& ii isj
are the population averages of the number of ovules, number of free pollen grains and the proportion of self-fertilized ovules, respectively, per individual plant measured from birth. Under the assumption of regular Mendelian segregation, the one-locus-two-allele version of the general transition equation in Gregorius & Ross (1981) has the following representation (primes, e.g. Pii, denote genotypic frequencies in the next
692
H.-R.
GREGORIUS
generation): pli =
gii
* dii
- Pii
+ $ * fll2412P12 i
(1 -vii).
4ii.
Pi<
++I.
(1-~12)412P12
Piipii
+$-412P12
+ B
Uii4ii aPii
z
1 _ Piipii
+ b12p12
(
4 ++iiPii
>
fi
+t412P12
Piipii
6
+b12p12
*
h2412Pl2
+
2
1
6 Piipii
+&12P12
6
xi- (
for homozygotes,
where
i = 1 or i = 2, and
+ a-
(+12)412P12
Pi2 =
h2h2Pl2
(14
)
CL
+ Cl-
(+22M22P22
6
6 xlLllP11+~cL12P12
+:(1-(T12)~12P12+(1-~ll)~llPll B
ii x cL22p22
+ b-4 12p12 cc
-S4l,Pl2
I ~1-mM11P11 4
+(1
CL22p22+tCL12p12 6
*
-u22)422P22
CLllPll
6
*
cz +b12P**
(lb)
CL
for the heterozygote. Since pi := Pii +iP12 describes the allelic frequency letting pi denote this frequency in the next generation, may be combined to give 1 (I+ Pi
ciiJ4iiPii
=2’
+t(l+(712)412P12
of A; (i = 1, 2) and the above equations
.$(l-5)
i x Piipii
+ tCL12P12 CL
(2) .
These formulations of the transition equations provide us with some basic insights into the role played by self-fertilization in the changing of germtypic
PROTECTEDNESS
OF
ALLELES
693
and allelic frequencies from one generation to the next. To simplify interpretation, consider that (PiJ’ii +&r#&/fi is the frequency of Ai in the pool of free pollen produced in the population. From equation (la) it follows that the frequency of a homozygote increases if its rate of self-fertilization is increased. An increase in the heterozygote rate of self-fertilization is accompanied by an increase in frequency of that homozygote whose allele is the less frequent one in the pool of free pollen; the other homozygote declines in frequency.
Equation
(lb) implies that
the frequency of the heterozygote is not affected by changing the heterozygote rate of self-fertilization, while it decreases if either one of the homozygote rates of self-fertilization grows.
These observations confirm some of the intuitive expectations about the _ relationship between heterozygosity and self-fertilization, but not all. A simple reformulation of equation (2) reveals in connection with the above statements about genotypic frequencies that the frequency of Ai increasesas the rate of self-fertilization of the corresponding homozygote AiAi grows; it decreaseswith growing rate of selffertilization of the other homozygote A&(j# i). Whether Ai decreasesor increasesin frequency with a growing heterozygote rate of self-fertilization dependson whether this allele is the more or lessfrequent one, respectively, in the pool of free pollen.
To conclude this list of basic properties of self-fertilization, it might be worthwhile to emphasize that a change in allelic frequencies in the course of the generations can be caused merely by genotypic differences in the rates of self-fertilization. This is easily verified with the help of equation (2) if one sets dii = C$and pii = p for all three genotypes. 3. The Conditions ior Protectedness of an Allele
It is a well-known fact that even comparatively simple population genetic selection models, such as one-locus-two-allele models, can exhibit complex dynamic behavior caused by the existence of multiple attractive and unstable equilibria or cycles (cf. e.g. M&at, 1969; Kidwell et al., 1977; Hadeler & Liberman, 1975; Pollak, 1978). In principle, it is also possible to construct models showing completely undirected, i.e. chaotic dynamics (cf. Li & Yorke, 1975). Presumably because of these observations it has become rather common during the last years to attack the problem of maintenance of genetic variability by selection from a different point of view: instead of analyzing the existence and stability of polymorphic equilibria, the repulsivity of the extinction states for alleles (i.e. where the alleles
694
H.-R.
GREGORIUS
have zero frequencies) is investigated (cf. Gregorius, 1979). In selection theory it has becc-me usual to call an allele protected if its extinction state is repulsive in some sense. More specifically, the term “protectedness” will be used in the sense Karlin (1977) applied to it. An allele is called protected if it cannot be lost where all possible genotypes are present with it is required that the allele cannot stay for an at arbitrarily low frequencies. If all alleles are allelic polymorphism is called protected.
from any state of the population positive frequencies: moreover, indefinite number of generations protected in this sense, then the
This definition of protectedness is sharper than Karlin’s in two respects: since the allelic frequency in the next generation is a function of the genotypic rather than the allelic frequencies in the present generation, the allelically polymorphic initial conditions in Karlin’s definition have to be replaced by genotypically polymorphic initial conditions; situations in which an allele may start at arbitrarily low frequency and remain there without becoming extinct are excluded. The first of these, i.e. that the transition equation (2) for the alellic frequencies relies on the genotypic frequencies, is also responsible for the fact that protectedness cannot directly be analyzed in the classical way by considering spectral radii of derivatives (Jacobi matrices). This is a consequence of the fact that generations generally do not start with HardyWeinberg proportions at the zygotic stage. A type of analysis that takes this into account and relies on allelic frequencies as variables of primary interest is elaborated in the Appendix. The principle of this analysis consists in showing ?. that the multiplication (growth) rate pi/pi of Ai always approaches a value Bi after many generations in the close vicinity of the extinction state for A,. To exclude trivial cases, the selection values are assumed not to represent situations in which an allele or a genotype can be lost in a single generation (:l.e. PII, P12, PI2 > 0 implies Pi,, Pi2, Pi2 >O). Then A, is protected if 0, > 1, and if ii < 1, Ai eventually becomesextinct at least when starting at low frequency. In the Appendix it is proven that (consult equations (Al) and (A2)) 6i = $(a; + di - 6,) + Ji(b; + di -a, )’ + b,(ci -d,) =~(a;+di-b;)+J~(ai+bi-d,) if AjAj(i
2
+bi(c;-a;)
# j) is a bisexual genotype, i.e. if ~ij, ~,j > 0, where Ui := Uii~ii/~jj, Ci
:= $(1 -
bz+ $(+,241,/4jp Ujj)/.Lii//.L,j
+ +(1 + Uii)ki/djj
(3a) (3b)
PROTECTEDNESS
OF
695
ALLELES
and di
:=$(l-Ujj)PlZ/Pjj
+f(l
+al2)412/4jp
Note that Ai is trivially protected if the genotype homozygous with respect to the other allele is unisexual, i.e. if 4ij = 0 or pjj = 0 (j # i). Before analyzing in more detail the significance of the rates of selffertilization for protectedness, some special cases of the present model shall be considered which refer to known results. No self-fertilization,
i.e. Uij = 0 for i, j = 1,2
In this case ai = bi = 0, di = $(p12/pjj + dr2/4jj) and, consequently, si = di* In accordance with the pres:nt results,lMandel (1971) and also Kidwell et al. (1977) pointed out that Bi > 1 and Bi < 1 indicate protectedness (initial increase of a new allele) and non-protectedness of Ai, respectively. Complete self-fertilization,
i.e.
ffij
= 1 for i, j = I,2
The effect of the p001 of free pollen is eliminated. and therefore
ai = ci = 4ii/4jj, 2bi =
di = 412/4jj
6i = $(a; + bi) +ilai - bil= max {ai, bi} = max {4ii, 44,2}/4jp
Hence, the biallelic polymorphism is protected if e^i> 1 and 62 > 1, which is equivalent to max {4i1, d22} < &512. This coincides with the result reported by Karlin (1968, p. 15, bottom of Table 2). Constant rates of self-fertilization and no differences between male and female selection coefficients, i.e. uij = u and 4ij = pij =: uij for i, j = 1, 2
ai = (+vii/vjj, bi = $(+V12/Vjj, ci = Vii/Vjj and di = v12/vjb After rearrangement
one arrives at
ii = U12+~(Uii-~U12)+~(1-~)t):2 2
+(r2(Vii--tV*2)2
* Vjj
Since the numerator is an increasing function of Vii, Uii 5 Vjj implies ii 5 I$. Consequently, the biallelic polymorphism is protected if Vii 5 Ujj and di > 1, which is the same conclusion Kimura & Ohta (1971, p. 194) arrived at. This condition is met if and only if either max {ulr, ~22) < $12, in which case protectedness of the biallelic polymorphism is guaranteed for all ~(01~51), or ViisUjj
Ujj)+
Ujj(DlZ-Vii)’
696
H.-R.
GREGORIUS
The appertaining analysis follows easily from (A3) in the Appendix. Unfortunately, Kimura & Ohta (1971, p. 155) draw a too narrow conclusion from this result when claiming “that overdominance cannot be a major mechanism for polymorphism in predominantly self-fertilizing organisms, For ull =uZ2
-(+lZ)U&u ai
>(l =
-@ii)fll2Wl2.
ai =O,
ffji,
Cj = 1(1
-
6, = $u,, = bi,
Ci
=
$Jdii/Fjj
+ it 1 + CT;,1,
di=S/112I~jj+kCl+u12)
uii)Cljjl,lLii,
and di=~(l-crii)~-12/cLii+1(1+~12). Hence, ~i=t(Uji+3(1+CLlZ/CLjj)) +Y(gii ij
=$((l
-40 -~ii)P12/Pii
+4J((1-c+ii)@121Pir
Consider
+/L12//Ljj)~2+IT12~/-Lii/~jj
+ 1 -(Tit)
+ l) +
1)2+~12(1-(Tii)~jjlcLii.
the example cii = ~12 = 0.5 and 0 < piz/pjj,
pi;/pjj < 0.5. Then
~i=3(1+tCL1ZI~jj)+~J~(CL1ZICLjj)2+~(CLiiICLjj+t) <;+&J&T$=l, which, contradicting Nagylaki’s claim, proves that the selfing allele Ai may not be protected although both selfing genotypes contribute (free) pollen to cross-fertilization. NOW, choose uii = (+I2 = 5, /JI 2= 2 and pii = 3 < pjj,
PROTECTEDNESS
OF
ALLELES
697
such that the condition (1 -c~ii)~lzplz < (1 -~&~~i~ii is met. Then 6j > i(( 1 - ~~i)~1z/~ii + 1) = 1, i.e. the cross-pollinating allele Aj is protected, which also contradicts the second above-mentioned claim of Nagylaki. These examples may suffice to demonstrate the insecurity that still exists in judging the significance of self-fertilization for the maintenance of genetic polymorphisms even with respect to comparatively simple models of selection. The last section of the present paper shall therefore be devoted to a few basic comments on the role of self-fertilization. 4. Protectedness
and Self-fertilization
In order to simplify understanding of the following considerations, note that the sums in the brackets appearing in equation (3b) have the representation ai+di-bi=~iAii/9jj+t~~~/~jj+3(1-~ji)~~~/~jj Ui+bi-di=~iidii/~jj-~~12/~jj-~(l-~jjii)CL12/~jj Ci
- Uj = $( 1 -
Uii)&ii/4jj
+ $( 1 -
Ujj)pii/pjp
Consequently, as long as 41~ > 0 and ii - ai > 0, & is a strictly increasing function of the haterozygote selfing rate u12. In other words, if all other selection coefficients remain the same and only the heterozygote selfing rate u12 is increased, then both coefficients of protectedness e^,and &, i.e. the degree of protectedness of the biallelic polymorphism, also increase. This observation points out a phenomenon of selfing which does not become apparent in models where all selfing rates are assumed to be the same, such as in the case mentioned in the last section. There, protectedness for a certain selfing rate implied protectedness for all smaller rates; now, however, protectedness might not be realized for all small heterozygote selfing rates, while it might be guaranteed for all high rates. The probably most trivial example for this may be found in a model of no differences within female- and within male-selection values and identical selfing rates for the tw? hopozygotes, i;e. +ij 3 4, pij c p for i, j = 1,2 and ~11s ~22 = (+. Clearly, & = 13~=: 13,and 13= 1 if cr12= v which refers to the case of no selection in allelic frequencies. Consequently, e^< 1 for ~12 1 for cr12> (+. Moreover, this example shows that selection might operate exclusively through differences between rates of self -fertilization (cf. WGhrmann & Lange, 1970). Performing changes in the homozygote selfing rates only, the direction of change in the 6’s depends on the relationships between the male and
698
H.-R.
GREGORIUS
fernal: selec$on coefficients. It may happen that as uii, say, moves from 0 to 1, Bi or 0, declines for some time and then grows again or vice versa. Nevertheless, it is possible to formulate a principle of general validity. Since bi(ci -a,) 2 0 one obtains the inequality ,. Bi~~(Ui+di-bi)+tlUi-(di-b;)l=max{U;,d,-bi} =max{uiiidii/4jj,
If both $$,,/4,, therefore
t412/4jj
and i+r,/#~,
+hl
-~jjii,cL12/cLjj)~~~l2l~j~
(4)
exceed one, then 6; > 1 for i = 1,2 and
the biallelic polymorphism is protected, irrespective male selection values and the rates of self-fertilization,
of the magnitudes of the if max {QS,,, q&}< &#J,,.
Under the condition that all genotypes are bisexuals, it was shown in the last section that the above inequalities become equalities, i.e. b;,= +4,,/4ij for i = 1,2 if complete self-fertilization is realized for all three genotypesAand dri, & 5 &#~i,. There are two additional possibilities to arrive at 8i = $,2/d, for i = 1,2, both of which naturally require that the two homozygotes are bisexuals: the heterozygote is a female, i.e. p rZ= cl1 = u22&2
PROTECTEDNESS
OF
699
ALLELES
appropriate normalization. It follows that in this case cl = dl = i(l-a)/p+$(l+a)#, az=a/& b,=’ 2u, c2 = i( 1 - a)~ + f( 1 + cr)/d and dz= 1. The respective explanations in the Appendix show that
and, consequently,
A1 is protected if
$(d+i)>l
and
From (A3) in the Appendix
Osa<
1-@-4h
1-b * one concludes that A2 is protected if either
l
i.e.
*
u’&pe
Or -7
The last condition
24
and
is equivalent U>
O
to
o-FM--1 l-&h
u>O ’
’
Since ,>h4+1< l-#p
24 2-4
one obtains in summary: A2 is protected if a> o-PM--1 1-h * Combining the conditions for protectedness of A1 and A2 and taking account of the inequality (2 - F)#J - 1 < 1 - (2 - 4)~ one arrives at the result that the biallelic polymorphism is protected for all positive values of u located in the open interval O
and
(2-PM-1
l-4.P
l-4.P
provided (2 - 4)~ < 1. A comparison of the statements inherent in this inequality with those made for sexual symmetry in the preceding section reveals that sexually
700
H.-R.
GREGORiUS
asymmetrical selection may alter the effect of selfing considerably. Under the conditions of the above model, polymorphisms may be protected only for rates of self-fertilization that belong to intervals whose lower and upper bounds are greater than zero and smaller than one, respectively. Such intervals may be arbitrarily small and located anywhere in the unity interval. To see this, denote Y:=
l-4@ 1-d
and notice that y>l
and
4=-
y-1
Y-FL’
Inserting this representation of C#J into both of the above bounds yields (2-PM-1
1-h
=12
Y
which proves the assertion. For example, if p = 9/10 and 4 = 50/51, then y = 6 and the biallelic polymorphism is protected for all selfing rates (T with $< (+< $; if CJ< 3 or cr >&, the polymorphism is not protected. This, obviously, is in sharp contrast with any model of sexually symmetrical (such as pure viability) selection. The probably most impressive demonstration of the significance of selffertilization can be achieved by considering the other extreme where selection takes place only with respect to the selfing rates, i.e. 4ij = 4, pLr,= p for i, j = 1,2. In this case ? ~j=~(l+(Tjr-t~jj)+~/t(l-~~i-3~jj)-+~cT1~(1-~((T~~+(T?~)).
From (A3) in the Appendix it follows that $, > 1 is equivalent to 1 -ff,r (5) 1 - fh + md Applying this formulation, it may be shown by contradiction that for e”r, & > 1 it is necessary (but not sufficient) that vI z> ~11, (~22.However, as was pointed out earlier, if gll = uz2, then crl 1= a22 < u12 implies ill i2 > 1. Since 6; is a continuous function of its parameters, small deviations of the ~ij’s and Pii’S from constancy might therefore still produce a protected polymorphism if condition (5) is met. These small deviations comprise all kinds of ranking relationships between the sexual specific selection coefficients, including, for example, underdominance in both the 4ij’s and the Nij’s, which would lead to fixation in the absence of self-fertilization. ~((T~~+(T~~)<~ and
a12>~,,.
PROTECTEDNESS
OF
ALLELES
701
The findings of this section demonstrate the multiplicity of interactions between sexual specific selection values and selfing rates that can lead to protectedness of a biallelic polymorphism. Some of these offer solutions to the problems mentioned in the introduction and others contradict known statements. However, it is difficult to derive general qualitative rules other than those concerning the action of the heterozygote selfing rate or resulting from the inequality (4) without considering more specific aims. Among the wealth of interesting questions that can at least in part be treated meaningfully with the help of a biallelic model, one is of fundamental importance for the evolution of plants: what are the conditions for the maintenance of mixed systems of sexuality, such as gynodioecy (bisexual and female plants), androdioecy (bisexual and male plants) and trioecy (bisexual, male and female plants)? In such not merely monoecious populations it might happen that the allelic but not the genotypic polymorphism is protected since, for example, the heterozygote is always maintained at a positive frequency while one of the homozygotes eventually disappears. This requires consideration of both types of protectedness, allelic and genotypic, simultaneously and will be the subject of further investigation within the frame of the present series. It is also of general interest to study in more detail some important cases of purely monoecious populations. The author is grateful to M. D. Ross for important suggestions to improve the contents and representations of this paper. The work was supported by a Heisenberg fellowship. REFERENCES R. W. (1965). Genetic systems associated with colonizing ability in predominantly self-pollinated species. In: The Genetics of Colonizing Species (Baker and Stebbins). New York: Academic Press, pp. 49-76. ALLARD, R. W. (1969). Breeding systems and population structures: synthesis (abstr.) Abstracts of the papers presented at the XI International Botanical Congress, Seattle p. 2. BAKER, H. G. (1967). Evolution 21, 853. CHARLESWORTH, B. (1980). J. fheor. Biol. 84,655. GREGORIUS, H-R. (1979). Inr. J. Systems Sci. 10,863. GREGORIUS, H-R. & Ross, M. D. (1981). Math. Biosci. 54,291. HADELER, K. P. & LIBERMAN,~. (1975).J. math. Biol.2,19. JAIN,S.K.&MARSHALL, D.R.(1967). Am. Nat. 101,19. KARLIN, S. (1968). Equilibrium of Population Genetic Models with Non-random Mating. New York, London, Paris: Gordon and Breach. KARLIN,% (1977). Am.Nat. 111,1145. KIDWELL,J. F., CLEGG, M.T.,STEWART,F. M. & PROUT,T. (1977). Genetics 85,171. KIMURA, M. & OHTA, T. (1971). Theoretical Aspects of Population Genetics. Princeton: Princeton University Press. ALLARD,
LI, T. Y. & YORKE,
J. A. (1975).
Am. Math.
Monthly
82,985.
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H.-R.
GREGORIUS
MANDEL, S. P. H. (1971). Heredity 26, 49. MBRAT, P. (1969). Ann. G&z&. Ski. anim. 1,49. MORAN, P. A. P. (1962). The Sfufisfical Process of Evolutionary Theory. Oxford: Clarendon Press. NAGYLAKI,T.(~~~~). J. theor. Biol.58,55. NAGYLAKI, T. (1977). Selecfion in One- and Two-locus Systems. Berlin, Heidelberg, New York: Springer-Verlag. POLLAK, E. (1978). Genetics 90,383. W~HRMANN,K. & LANGE, P.(1970). Theor. appi. Genet. 40,289.
APPENDIX The central quantity to be considered in the present analysis of protectedness of an allele Ai is its multiplication rate Pi/m over many generations at very low frequencies. For this purpose consider the variable 4t := Pii/pl (hence 1 -4i = iPrz/pi) and rewrite the equations (2) and (la) in the form Pi/Pi
1 (l+~ii)~iiqi+(l+(T12)~12(l-qi) = 2 ’ 6
/J-ii
’ 4i +/412(1-4;) ti
For a given value of 4 look at the limit of pi/pi as pi approaches 0. Since the case& > 0, cii > 0 and pii = 0 is excluded from the considerations, this limit equals infinity if either 4jj = 0 or ,uii = 0. Hence, denoting the limit (given 4;) by @i(4i),one arrives at 6, (si) = co if the homozygote Afi?( j f i) produces no pollen (kj; = 0) or no ovules (C$jj= O), and 8~(4j)=4i.Ci+(l-q,).d;
for
/Ajj.+jj>O
:= $(1 -
+ $(1 + Uii)4ii/4jj
where C,
Ujj)/.Lii//Ljj
and di
:=3(1
-ujjii)Pl2/&jj
+3U
+(+12M12/&
As it should be, Ai is protected if the homozygote A/Aj(j # i) represents a unisexual genotype (8i(4i) = 00> 1 for all 4i). Therefore, from now on only the case kjj. 4jj > 0 is of interest.
PROTECTEDNESS
OF
To establish the relationship
703
ALLELES
of Pii/pi to Pli/p:
as pi approaches zero and
Pii/pi = qi remains constant, note that = WpiPI/Pi’
pgp;
Denoting
this limit of P:i/pi by 4: one obtains qf=4i*ai+(1mqi)-bi I ei(4i)
where ai := giiAiJ4jj
and
bi := $alzdlz/+j+
This proves that, as the frequency of Ai approaches zero, Pii/pi = 4i possesses a dynamics that is governed by the transition equation 4!
=f(4,)=
qi(ai-bi)+bi
1 ’
qi(Ci-di)+di’
Again, two trivial situations can be excluded from further analysis: ci = di, since then tii(Si) z ci = di for all possible values of 4s and ~i = bi = 0, since then after the first generation ei(qi) = @i(O)= di for all possible initial values qi. Note that the latter case is met if aii = ~12 = 0. Since 0 5 ai 5 ci and 0 I bi 5 di, f is a function that maps the closed zero-one-interval [O, l] into itself. Consequently, the dynamics of 4; (defined by the iterates of f) are completely determined by the properties f exhibits on [0,1]1. Omitting the indices for convenience, one obtains for the derivatives of f with respect to q: $=f(4)=
a*d-c’b
[dc - 4 + 4
2 and
$=fy4)=
2(c - d)(cb -ad) [q(c-d)+d13
’
Hence, f is a montone and convex or concave function on 10, 11, and can therefore not have more than one fixed point 4 = f(t) with 0 < 4 < 1, since c # d. This point, if it exists and f is increasing, i.e. ad 2 cb, attracts all trajectories starting with 4, 0 < 4 < 1. If f is decreasing, the trajectories perform oscillations around 4. To see that in this case 4 still attracts all trajectories starting in 10, I[, consider that (a-b)[q(a-b)+b]+b[q(c-d)+dl fZ(q)=f(f(q))=(c-d)[q(a-b)+b]+d[q(c-d)+d] =q[(a-b)2+b(c-d)]+b(a+d-b) q(c-d)(a+d-b)+b(c-d)+d*
704
H.-R.
GREGORIWS
Hence
The numerator
can be rewritten in the form
q(u+d-b)(a-b-d)-(a+d-b)[b-q2(c-d)] =(a+d-b)[q(u-b)+b-q(q(c-d)+d)]
=(u+d-b)[f(q)(q(c-d)+d)-q(q(c-d)+d)l =(a+d-b)(q(c-d)+d)(f(q)-q).
Consequently, (u+d-b)(q(c-d)+d) f2(4)-4=(f(4)-4)*q(c-d)(u+&b)+b(c-d)+d2’
where the ratio on the right side is positive for all 4, 014 5 1, since by assumption c # d and a + b > 0. It follows, that for 4 # 4, (f2(4) - 4) X (f(4) -4) > 0, which in turn implies that the oscillations around 4 are damped. This proves that the trajectories produced by the iterates of f always converge to a fixed point off. If O< 4 = f(J) C 1 exists, it is straightforward to show that 4 has the representation 1 ld+b-a q=2 d-et
--dm
ford>c
and * ld+b-a q=T d-c
+Jm
ford
Insertion of both these values for 4 into e(4) = d(c -d) + d results in the same expression for e^:= 8(d), namely e^=e(&=$(u+d-b)+x&d+b-u)2+b(c-d).
(AlI
In fact, the root is always real, since $(d+b-u)2+b(c-d)=$(u+b-d)Z+b(c-u)~0.
Therefore,
e^may also be given the equivalent e^=+(u+d-b)+&u+b-d)2+b(c-a).
representation t-42)
PROTECTEDNESS
OF
ALLELES
705
The cases in which f does not have a fixed point in the open interval 10, l[ are the following: f(0) = 0 and f(1) = 1: according to whether f is convex or concave, 4 = 0 or 4 = 1 are the only attractive fixed points, respectively; f(0) > 0, f( 1) = 1 and f’(1) 5 1: 4 = 1 is the only attractive fixed point; f(0) = 0, f(1) < 1 and f(0) 5 1: 4 = 0 is the only attractive fixed point. If a, b, c and d are chosen such that they reflect the above conditions for f, it is easily verified that 0 evaluated at the respective &values can also be obtained directly with the help of (Al) or (A2). The same is true for the initially excluded cases c = d and a= b = 0. Recall that 8i as given by (Al) or (A2) represents the multiplication rate for Ai after the population has stayed for a sufficiently large numbfr of generations at low freque?cies for ai. Consequently, Ui is protected if 0i > 1 and it is not protected if @< 1.