J. theor. Biol. (1978) 73, 347-357
The Population Genetics of Anisogamy BRIAN CHARLESWORTH School of Biological Sciences, University
of Sussex, Brighton, England
(Received 30 August 1977, and in revised form 13 January 1978) This paper analyses the population genetics of anisogamy controlled by a single locus, in both the haploid and diploid cases. The conclusions of Parker et al. (1972), based on computer calculations, are confirmed analytically. The effects of the existence of two mating types on the evolution of anisogamy are examined. Close linkage between a mating type locus and the gamete size locus may produce non-random associations of alleles, leading to disassortative fusion with respect to gamete size. With loose linkage, there is random association of alleles, but selection favours closer linkage.
1. Introduction Parker, Baker dc Smith (1972) proposed a model for the evolution of anisogamy (the production of gametes of two different sizes by a sexual species). They showed, by a mixture of graphical argument and computer simulation,
how selection can result in the stable maintenance of two alleles at a locus controlling gamete size, owing to a conflict between selection for the production of large numbers of gametes and selection for large zygote size. Random fusion of gametes with respect to gamete size was assumed. This model has been extended by Parker (1978), who investigated the process of selection for non-random fusion of gametesin an anisogamouspopulation, and by Maynard Smith (1978) and Bell (1978). The purpose of this paper is to point out some properties of the population genetics of a locus affecting gamete size, and to consider the effects of the existence of mating types on
selection for anisogamy. 2. The Haploid Model Consider a haploid, sexual organism. The life cycle is such that, every generation, haploid vegetative individuals produce gametes by asexual
divisions. In a species lacking mating types, these gametes fuse at random to produce diploid zygotes which persist for some time and then divide to 341
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produce a new generation of vegetative individuals. (This is a reasonably accurate picture of the life cycle of many green algae, for example.) There is a locus A with alleles A,, Al, . . . A,. It is assumed that the number of gametes produced by a vegetative individual is a function of its (haploid) genotype, so that an Ai individual produces on average a characteristic number, n, of gametes. The size, mi, of these gametes is inversely proportional to n,, and the size, mij, of a zygote produced by fusion of Ai and A, gametes is equal to mi+mi. The survival rate of such a zygote, Vii, is some increasing function of its size. All these assumptions are similar to those made by Parker et al. (1972, p. 539) for their Model 1. Let Xi be the frequency of A,, among the vegetative individuals of a given generation. Since Ai individuals produce ni gametes, the frequency of A, among the pool of gametes is: Xl
=
Xtni
i I
Xjnj.
j=l
The frequency of Ai among the vegetative individuals is thus: Xl =
f Xf X:Vj,).
Xf~~Vij/(
(1) of the next generation (2)
If we write wjk = njnkvjk
Pa)
and w = p,XkWjk jk
equations (2) reduce to: OX:
=
Xi ~
XjWij.
j=l
(3b)
(4)
Equation (4) is identical with the standard equation for gene frequency change in a single-locus, multiple-allele system in a random-mating diploid (Crow 8c Kimura, 1970). We can therefore use the well-known properties of this system to investigate equation (4). If there are just two alleles, Al and AZ, the condition for the stable maintenance of variation in gamete size is thus the “heterozygote advantage” condition : WlZ > WllY wzz9 (5) and the equilibrium frequencies of the two alleles among the vegetative individuals are given by:
21
w12-w22
g2 = w12-wll’
(6)
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It can be shown as follows the conditions (5) require that the relation of zygote viability to size be steeper than linear, and that the difference in size between the small and large gametes should exceed some critical level, dependent on the departure from linearity. Following Parker et al. (1972), a simple model is to write: = W?lfj, vij (7) where c( is a constant of proportionality, and k determines the steepness of slope of the dependence of zygote viability on size. Writing 0 = n2/n1, and assuming that A, specifies the smaller gametes so that 8 < 1, we obtain the following necessary and sufficient conditions for anisogamy from conditions (5) : ‘V12
-
= 2-k e’-k(l+e)k
> 1,
(W
= 2-k e-‘(l+B)k
> 1
t7b)
Wll w12
~w22
We have 2-k(l + 0)” < 1 for k > 0, so that when k < 1 condition (7a) cannot hold. If k > 1, then or-k > Zk(l +0)-k for sufficiently small 8, and (7a) can be satisfied. Condition (7b) can be satisfied for arbitrary k > 0 provided that 0 is sufficiently small. The critical values of k given by conditions (7) for different values of 0 are shown in Fig. 1. Condition (7a) enables one to find the minimum value of k for a given 8 which permits A, to invade an A, population (i.e. the condition for macrogametes to establish themselves in a microgamete population). Condition (7b) gives the maximum value of k for a given 8 which permits A, to invade an A, population, i.e. microgametes to establish themselves in a macrogamete population. For all values of 8 < 1 these two conditions are compatible; the smaller 0, i.e. the larger the degree of anisogamy, the easier they are to satisfy. It can be seen from Fig. 1 that when k is close to 1, only genes which induce a high degree of anisogamy can be maintained polymorphic. We can also examine the relation of the equilibrium frequencies of Al and A, to 8 and k. Combining equations (6) and (7), we obtain 1-2k e(i+eyk 21 (8) z2= 1 -2k ek-l(i +ep It can be seen from this equation that if k < 2, vegetative forms producing microgametes are more frequent than those producing macrogametes; the reverse is true when k > 2. Furthermore, as 0 tends to zero, the frequencies of the two forms approach equality. Hence, with high anisogamy, a 1 :l sex ratio is approached.
350
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CHARLESWORTH
z
4 6 - log20
8
IO
FIG. 1. The lower curve shows the critical value of k from equation (7a) as a function ef 0; the upper curve shows the value from equation (7b). Anisogamy is possible for k values which lie between the two curves; increased gamete size is favoured above the upper curve and decreased gamete size below the lower curve.
3. The Diploid Model
Parker et al. (1972) also considered a model in which the number and size of gametes are determined by the constitution of the diploid zygote from which they are ultimately derived. [This model seems less plausible than the haploid one for the green algae, in which anisogamy presumably first evolved (Lewin, 1976).] The model described above can easily be modified for this situation. For simplicity, only the case with two alleles will be analyzed. Allele A, is dominant over A,, so that A,A, zygotes give rise to vegetative individuals producing n, gametes of size m, (type 1 gametes), and A,/- zygotes produce n2 gametes of size mZ (type 2 gametes). As before, the size of a zygote is the sum of the size of the two gametes from which it was formed, and its viability is an increasing function of its size. Let x, 2y and z be the frequencies of A,A,, A,A, and A,A2 zygotes after selection in a given generation. Table 1 shows how the corresponding frequencies in the
GENETICS
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OF ANISOGAMY TABLE
1
Frequenciesof zygotes (after selection)produced by fusions of gametesderived from all possible zygotes of the preceding generation, and their genotypic distributions. The diploid model with two alleles and dominance is assumed Genotypes of parental zygotes
Frequency of zygotes (unnormalized)
Distribution of zygotic genotypes AlAl Adz AA, : 0 t 0 0
next generation can be found, by between gametes from all possible quencies of the resulting zygotes. We Wx’ = X2Wll
04 t 4 0
0 0 t 1
calculating the probabilities of fusions pairs of parents, and the genotype freobtain the following recurrence relations. +2xyw,,+yZw22 (94
WY’ = (Y+zmwl2+Yw22)
(9b) Wz’ = (y+z)2w,, PC> where Wij = ninjvij (as in the haploid model), and W is the sum of the unnormalized frequencies in the second column of Table 1. The stability of this system can be studied by examining its behaviour at the endpoints when A, or AZ are fixed. In the former case, equation (9b) can be linearized in the neighbourhood of x = 1 to give: WllY’ = YW129 so that A, increases when rare if wr2 > w1r. Similarly, equations (9a) and (9b) give:
when A 1 is rare,
x’+y’ z x;z+y, Hence, A, increases when rare if wrZ > wZ2. These two conditions are identical with the “heterozygote advantage” condition (5) for the haploid model, so that the conditions for maintenance of anisogamy are the same for both cases. It can be shown as follows that a similar statement is true for the equilibrium frequencies of macrogamete and microgamete producers. Adding equations (9b) and (SC), and ‘equating corresponding genotype frequencies in successive generations, gives : w = Rw,2+(29+2)w22, (loa)
352
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Adding equations (9a) and (9b), and substituting from equation (lOa), gives : w = Rw,,+(29+)Pw,,, (lob) so that R = w12-w22 ___ _--~~--(11)
cv+Q
w12-Wll’
Equation (11) is identical in form with equation (6) for the haploid model. 4. Mating Types and Anisogamy
Experiments on a large number of isogamous green algae have demonstrated the existence of genetically determined mating types, such that only gametes of opposite mating type can fuse to form zygotes (Lewin, 1976). The usual mode of inheritance involves a locus with two alleles, such that the mating type of a gamete is determined by the nature of the allele which it carries. Since many isogamous relatives of anisogamous species are known to possess mating type systems, it is of interest to consider the dynamics of selection for anisogamy in a population which is segregating for mating type alleles. Only the haploid model of anisogamy, which seems to be the normal mode of genetic control of gamete size in the Volvocales (Wiese, 1976), will be considered here. With two alleles, A, and A2, at the gamete size locus and two mating-type alleles, M, and M,, there are four haploid genotypes: AiMi, AIM,, A2M, and A,M,. Let the frequencies of these among the vegetative forms be x1, x2, x3 and x4 respectively. Assume, as before, that A, individuals produce n, gametes and A2 produce n2 gametes. Let the recombination fraction for the two loci be R, and let the coefficient of linkage disequilibrium be D = x1x4-x2x3. Then, taking into account the fact that only gametes of different mating type can fuse, we obtain the following recurrence relations, using the same definition of fitness wij as in equation (3a):
wx; = ~X1(X2W11+XqW,2)-3RDW12 icx; = +x2hwll +%Wl2)+M~Wl2 wx;
=
3x3(x2wlz+XqW22)+~RDWl2
(12)
iiix: = ~x~(x~w~~+x~w~~)--~RDw~~ where w = XlXZWll +(x,x,+x,x,)w,,+x,x,w,2. It is immediately obvious from these equations that xi + xi = xi + xi, so that after one generation M, and M2 must be equally frequent. It can be shown as follows that, if linkage is sufficiently tight, linkage disequilibrium tends to build up, so that the gamete size alleles become
GENETICS
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353
associated with different mating types. Assume first that R = 0, and that A2 is introduced into a population fixed for A1 but at equilibrium for the mating type locus. The mutation to AZ occurs in an MZ individual, so that x3 is always zero. From equations (12) we thus have: 4
= Xk -~ -4
x4w121w119 =
x4w12 xzw11
It follows from these equations that, if wr2 > wlr so that there is selection for anisogamy, A,M, tends to displace A1M2. We would therefore expect that the final equilibrium should consist of AIMI and A,M,. It can be verified that, if the “heterozygote advantage” condition (5) is satisfied, such an equilibrium is indeed stable to the introduction of AI&f2 and A,M,, and computer calculations of population trajectories confirm that it is globally stable. By continuity, for small but non-zero R a similar stable equilibrium but with a low frequency of AIM2 and AZ&f, individuals generated by recombination will exist, under the same conditions. With close linkage, therefore a population results in which there is disassortative fusion with respect to gamete size, as a result of the associations between the mating type and gamete size alleles. With loose linkage, however, it can be shown as follows that an anisogamous population results with the mating types distributed equally among the two gamete size classes. Consider the above situation when A, is introduced into an h4, gamete, but with arbitrary R. Ignoring second-order terms in x3 and x4, we obtain the equations: WIIX;
z
~11~:
zz x~w,~-Rw,~(x~-x~).
~3~12+R~12(~4-~3),
Adding these equations, we have: x;+xl$ = (x3 +X4)W12/W11, (W and subtracting them we get: xi-xi z (x4- x3(1 -2R)w,th1. WI Equations (13a) shows that linkage has no effect on the initial rate of increase in frequency of the gene for gamete size. Furthermore, if we note that D w +(x4-x3) when A2 is rare, then from equation (13b) is follows that D increases only when : w,,(l-2R) > wI1. (14) This condition clearly can never be satisfied for a pair of unlinked loci; if linkage is loose, then extremely strong selection, i.e. an extremely high degree
354
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of anisogamy, is required for linkage disequilibrium to be built up. Computer calculations of population trajectories show that condition (14) must indeed be satisfied if a population with an association between the two loci is to be established; such an association can be strong only if R is considerably smaller than the critical value given by (14). We may next consider the properties of populations where linkage is so loose that no linkage disequilibrium is possible. Setting D = 0 in equations (12), and solving for the equilibrium frequencies, gives: $1 = 22 = (WI2 - %*PmQ
-
Wll
-
w22)9 (1%
2,
=
24
=
(w1z-w11)/2(2w,2-wl1-w22).
This equilibrium can be shown to be stable if condition (5) is satisfied. Equations (15) shows that the frequencies of the vegetative forms producing microgametes and macrogametes are the same as in the case with no mating types; within each size class, there is an equal frequency of each mating type. According to Wiese (1976), there is no evidence in anisogamous Volvocales for the existence of different mating types within the sexes. This raises the question of whether there can be selection for reduction of the recombination fraction so that linkage disequilibrium can be established. From conventional multi-locus studies of random-mating diploid populations, we know that selection for a rare modifier of the recombination rate between selected loci is slow or non-existent unless the loci are held in linkage disequilibrium by selection (Feldman, 1972; Charlesworth & Charlesworth, 1973). Since equations (12) are similar in form to those for the diploid case, the same conclusion is expected to hold in the present case. This question was investigated by computer calculations of the fate of an “inversion” introduced into AIMI gametes, which suppresses recombination between inverted and normal gametes (cf. Charlesworth & Charlesworth, 1973). The inversion thus remains permanently associated with AIM,. (Any genetic element which suppresses recombination when heterozygous will behave in the same way.) When inverted AIMI individuals were introduced at a low initial frequency into a population at the equilibrium of equations (15), it was found that the inversion increased initially at a rate of the order of the square of its frequency, as is predicted analytically. Eventually, when its frequency has increased sufficiently, it becomes strongly selected, with the result that the final population consists of (inverted) A 1M, and (non-inverted) A&f2 gametes, between which no recombination is possible. An example is shown in Fig. 2. These results show that a very rare element reducing recombination between the gamete size and the mating type loci is almost neutral. If it reaches a
GENETICS
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355
Generation
FIG. 2. This shows the course of change in frequency of inverted AIMI and non-inverted A,M, chromosomes when the inversion is introduced at a frequency of 1% into a population segregating at the A and M loci. R = 03 and w 11 = was = 1, wla = 2. There is initially complete linkage equilibrium.
frequency of 1% or so as a result of genetic drift or other selective forces, it may come to experience significant selection, and eventually create a situation in which there is a strong association between alleles at the two loci. As Fig. 2 shows, even unlinked loci are subject to this selection pressure, so that it is possible for reciprocal translocations or centric fusions, which link the two loci when originally on separate chromosomes, to be selected by this mechanism. We can conclude, therefore, that despite the lack of linkage disequilibrium when the mating type and gamete size loci are loosely linked, there may indeed be selection favouring the establishment of close linkage. 5. Discussion
The analytic results presented in sections 2 and 3 of this paper confirm the graphical and simulation arguments of Parker et al. (1972), as corrected by Parker (1978). It is particularly interesting to note that, with the power model of equation (7) for the relation between zygote size and viability, anisogamy can always be selected when k > 1, provided that there is a
356
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sufficiently large effect on gamete size and number of the mutation at the A locus. Furthermore, when zygote viability is relatively insensitive to zygote size, so that k is only slightly greater than unity, only a mutant which has a very large effect on gamete size can be selected. There is no difficulty in principle in seeing how a very striking gamete size dimorphism could be set up by a single mutational change affecting the timing and frequency of the cell divisions involved in gamete production. We have also seen that the pre-existence of mating types does not affect the conditions for the evolution of anisogamy, and how close linkage between the mating type locus and the gamete size locus can bring about a situation in which there is disassortative fusion with respect to gamete size. There is selection for closer linkage between the two loci even when linkage is initially so loose that the mating type alleles are equally frequent in microgametes and macrogametes. Parker (1978) has presented some models of the evolution of disassortative fusion; he assumes the existence of genes which can cause small gametes which carry them to fuse selectively with large gametes, and vice versa. Such genes are perhaps of somewhat dubious plausibility, whereas mating type loci are frequently found in isogamous green algae (Lewin, 1976). Close linkage between the mating-type locus and the locus controlling gamete size is thus an attractive alternative explanation of disassortative fusion, although there are some difficulties in understanding how genes on separate chromosomes could be brought together, as may be necessary on this view. The genetic dissection of sexuality in species of algae with anisogamy and disassortative fusion may provide a way of discriminating between these hypotheses. On the mating-type hypothesis, the microgametes and macrogametes should be found to differ at more than one locus, although it may be difficult to separate the loci by recombination, just as the male- and femaledetermining chromosomes of dioecious flowering plants can be shown to differ at several loci (Westergaard, 1958). It would also be expected that the two sorts of gamete should differ with respect to sexual agglutinins in the same way as do the mating types of isogamous species (Wiese, 1976). The models presented above have assumed that the differences in gamete size in anisogamous species are of direct genetic causation. It is known, however, that in many Volvocales, for example, sex may be environmentally determined, so that members of the same haploid alone can differentiate into macrogametes or microgametes; in multicellular forms, macrogametes and microgametes may be produced by the same individual (Wiese, 1976). It is easy to construct a model for the evolution of this sort of situation on the following lines. Suppose we have an initial isogamous A, population. A mutation to A2 occurs, such that vegetative A, individuals differentiate a
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OF ANISOGAMY
proportion p of the time into gametes of the same size as A,(nz,) and q of the time into gametes of a different size (mz). There are corresponding differences in the numbers of gametes produced, n, and n,. Depending on the values of p and q and on the viabihties of the zygotes formed by fusion of gametes of different sizes, it is possible to show that A, may spread; under suitable conditions it may even become fixed. Under conditions where it cannot be fixed, one would find at equilibrium two classes of vegetative individuals; one capable of producing gametes of both sizes and the other capable of producing only one. Disassortative fusion in a species where gamete size is determined non-genetically obviously cannot evolve by associations between mating type alleles and gamete size alleles, but must arise by a mechanism similar to that proposed by Parker (1978). REFERENCES BELL, G. (1978). J. theor. Biol. 73, 247. CHARLESWORTH, B. & CHARLFSWORTH, D. (1973). Genet. Res., Camb. CROW, J. F. & KIMKJRA, M. (1970). Introduction to Population Genetics Harper & Row. FELDMAN, M. (1972). Theor. pop. Biol. 3, 324. LEWIN, R. A. (1976). (ed.) The Genetics of Algae. Oxford: Blackwell. MAYNARD SMITH, J. (1978). The Euolution of Sex and Recombination.
21, 167. Theory. New York:
Cambridge: bridge University Press. PARKER, G. A. (1978). J. theor. Biol. (in press). PARKER, G. A,, BAKER, R. R. &SMITH, V. G. F. (1972). J. theor. Biol. 36, 529. WESTERGAARD, M. (1958). Adv. Genet. 9,217. WIESE, L. (1976). In The Genetics of AIgae (R. A. Lewin, ed.). Oxford: Blackwell.
Cam-
The Population Genetics of Anisogamy BRIAN CHARLESWORTH School of Biological Sciences, University
of Sussex, Brighton, England
(Received 30 August 1977, and in revised form 13 January 1978) This paper analyses the population genetics of anisogamy controlled by a single locus, in both the haploid and diploid cases. The conclusions of Parker et al. (1972), based on computer calculations, are confirmed analytically. The effects of the existence of two mating types on the evolution of anisogamy are examined. Close linkage between a mating type locus and the gamete size locus may produce non-random associations of alleles, leading to disassortative fusion with respect to gamete size. With loose linkage, there is random association of alleles, but selection favours closer linkage.
1. Introduction Parker, Baker dc Smith (1972) proposed a model for the evolution of anisogamy (the production of gametes of two different sizes by a sexual species). They showed, by a mixture of graphical argument and computer simulation,
how selection can result in the stable maintenance of two alleles at a locus controlling gamete size, owing to a conflict between selection for the production of large numbers of gametes and selection for large zygote size. Random fusion of gametes with respect to gamete size was assumed. This model has been extended by Parker (1978), who investigated the process of selection for non-random fusion of gametesin an anisogamouspopulation, and by Maynard Smith (1978) and Bell (1978). The purpose of this paper is to point out some properties of the population genetics of a locus affecting gamete size, and to consider the effects of the existence of mating types on
selection for anisogamy. 2. The Haploid Model Consider a haploid, sexual organism. The life cycle is such that, every generation, haploid vegetative individuals produce gametes by asexual
divisions. In a species lacking mating types, these gametes fuse at random to produce diploid zygotes which persist for some time and then divide to 341
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CHARLESWORTH
produce a new generation of vegetative individuals. (This is a reasonably accurate picture of the life cycle of many green algae, for example.) There is a locus A with alleles A,, Al, . . . A,. It is assumed that the number of gametes produced by a vegetative individual is a function of its (haploid) genotype, so that an Ai individual produces on average a characteristic number, n, of gametes. The size, mi, of these gametes is inversely proportional to n,, and the size, mij, of a zygote produced by fusion of Ai and A, gametes is equal to mi+mi. The survival rate of such a zygote, Vii, is some increasing function of its size. All these assumptions are similar to those made by Parker et al. (1972, p. 539) for their Model 1. Let Xi be the frequency of A,, among the vegetative individuals of a given generation. Since Ai individuals produce ni gametes, the frequency of A, among the pool of gametes is: Xl
=
Xtni
i I
Xjnj.
j=l
The frequency of Ai among the vegetative individuals is thus: Xl =
f Xf X:Vj,).
Xf~~Vij/(
(1) of the next generation (2)
If we write wjk = njnkvjk
Pa)
and w = p,XkWjk jk
equations (2) reduce to: OX:
=
Xi ~
XjWij.
j=l
(3b)
(4)
Equation (4) is identical with the standard equation for gene frequency change in a single-locus, multiple-allele system in a random-mating diploid (Crow 8c Kimura, 1970). We can therefore use the well-known properties of this system to investigate equation (4). If there are just two alleles, Al and AZ, the condition for the stable maintenance of variation in gamete size is thus the “heterozygote advantage” condition : WlZ > WllY wzz9 (5) and the equilibrium frequencies of the two alleles among the vegetative individuals are given by:
21
w12-w22
g2 = w12-wll’
(6)
GENETICS
OF
349
ANISOGAMY
It can be shown as follows the conditions (5) require that the relation of zygote viability to size be steeper than linear, and that the difference in size between the small and large gametes should exceed some critical level, dependent on the departure from linearity. Following Parker et al. (1972), a simple model is to write: = W?lfj, vij (7) where c( is a constant of proportionality, and k determines the steepness of slope of the dependence of zygote viability on size. Writing 0 = n2/n1, and assuming that A, specifies the smaller gametes so that 8 < 1, we obtain the following necessary and sufficient conditions for anisogamy from conditions (5) : ‘V12
-
= 2-k e’-k(l+e)k
> 1,
(W
= 2-k e-‘(l+B)k
> 1
t7b)
Wll w12
~w22
We have 2-k(l + 0)” < 1 for k > 0, so that when k < 1 condition (7a) cannot hold. If k > 1, then or-k > Zk(l +0)-k for sufficiently small 8, and (7a) can be satisfied. Condition (7b) can be satisfied for arbitrary k > 0 provided that 0 is sufficiently small. The critical values of k given by conditions (7) for different values of 0 are shown in Fig. 1. Condition (7a) enables one to find the minimum value of k for a given 8 which permits A, to invade an A, population (i.e. the condition for macrogametes to establish themselves in a microgamete population). Condition (7b) gives the maximum value of k for a given 8 which permits A, to invade an A, population, i.e. microgametes to establish themselves in a macrogamete population. For all values of 8 < 1 these two conditions are compatible; the smaller 0, i.e. the larger the degree of anisogamy, the easier they are to satisfy. It can be seen from Fig. 1 that when k is close to 1, only genes which induce a high degree of anisogamy can be maintained polymorphic. We can also examine the relation of the equilibrium frequencies of Al and A, to 8 and k. Combining equations (6) and (7), we obtain 1-2k e(i+eyk 21 (8) z2= 1 -2k ek-l(i +ep It can be seen from this equation that if k < 2, vegetative forms producing microgametes are more frequent than those producing macrogametes; the reverse is true when k > 2. Furthermore, as 0 tends to zero, the frequencies of the two forms approach equality. Hence, with high anisogamy, a 1 :l sex ratio is approached.
350
B.
0
CHARLESWORTH
z
4 6 - log20
8
IO
FIG. 1. The lower curve shows the critical value of k from equation (7a) as a function ef 0; the upper curve shows the value from equation (7b). Anisogamy is possible for k values which lie between the two curves; increased gamete size is favoured above the upper curve and decreased gamete size below the lower curve.
3. The Diploid Model
Parker et al. (1972) also considered a model in which the number and size of gametes are determined by the constitution of the diploid zygote from which they are ultimately derived. [This model seems less plausible than the haploid one for the green algae, in which anisogamy presumably first evolved (Lewin, 1976).] The model described above can easily be modified for this situation. For simplicity, only the case with two alleles will be analyzed. Allele A, is dominant over A,, so that A,A, zygotes give rise to vegetative individuals producing n, gametes of size m, (type 1 gametes), and A,/- zygotes produce n2 gametes of size mZ (type 2 gametes). As before, the size of a zygote is the sum of the size of the two gametes from which it was formed, and its viability is an increasing function of its size. Let x, 2y and z be the frequencies of A,A,, A,A, and A,A2 zygotes after selection in a given generation. Table 1 shows how the corresponding frequencies in the
GENETICS
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OF ANISOGAMY TABLE
1
Frequenciesof zygotes (after selection)produced by fusions of gametesderived from all possible zygotes of the preceding generation, and their genotypic distributions. The diploid model with two alleles and dominance is assumed Genotypes of parental zygotes
Frequency of zygotes (unnormalized)
Distribution of zygotic genotypes AlAl Adz AA, : 0 t 0 0
next generation can be found, by between gametes from all possible quencies of the resulting zygotes. We Wx’ = X2Wll
04 t 4 0
0 0 t 1
calculating the probabilities of fusions pairs of parents, and the genotype freobtain the following recurrence relations. +2xyw,,+yZw22 (94
WY’ = (Y+zmwl2+Yw22)
(9b) Wz’ = (y+z)2w,, PC> where Wij = ninjvij (as in the haploid model), and W is the sum of the unnormalized frequencies in the second column of Table 1. The stability of this system can be studied by examining its behaviour at the endpoints when A, or AZ are fixed. In the former case, equation (9b) can be linearized in the neighbourhood of x = 1 to give: WllY’ = YW129 so that A, increases when rare if wr2 > w1r. Similarly, equations (9a) and (9b) give:
when A 1 is rare,
x’+y’ z x;z+y, Hence, A, increases when rare if wrZ > wZ2. These two conditions are identical with the “heterozygote advantage” condition (5) for the haploid model, so that the conditions for maintenance of anisogamy are the same for both cases. It can be shown as follows that a similar statement is true for the equilibrium frequencies of macrogamete and microgamete producers. Adding equations (9b) and (SC), and ‘equating corresponding genotype frequencies in successive generations, gives : w = Rw,2+(29+2)w22, (loa)
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Adding equations (9a) and (9b), and substituting from equation (lOa), gives : w = Rw,,+(29+)Pw,,, (lob) so that R = w12-w22 ___ _--~~--(11)
cv+Q
w12-Wll’
Equation (11) is identical in form with equation (6) for the haploid model. 4. Mating Types and Anisogamy
Experiments on a large number of isogamous green algae have demonstrated the existence of genetically determined mating types, such that only gametes of opposite mating type can fuse to form zygotes (Lewin, 1976). The usual mode of inheritance involves a locus with two alleles, such that the mating type of a gamete is determined by the nature of the allele which it carries. Since many isogamous relatives of anisogamous species are known to possess mating type systems, it is of interest to consider the dynamics of selection for anisogamy in a population which is segregating for mating type alleles. Only the haploid model of anisogamy, which seems to be the normal mode of genetic control of gamete size in the Volvocales (Wiese, 1976), will be considered here. With two alleles, A, and A2, at the gamete size locus and two mating-type alleles, M, and M,, there are four haploid genotypes: AiMi, AIM,, A2M, and A,M,. Let the frequencies of these among the vegetative forms be x1, x2, x3 and x4 respectively. Assume, as before, that A, individuals produce n, gametes and A2 produce n2 gametes. Let the recombination fraction for the two loci be R, and let the coefficient of linkage disequilibrium be D = x1x4-x2x3. Then, taking into account the fact that only gametes of different mating type can fuse, we obtain the following recurrence relations, using the same definition of fitness wij as in equation (3a):
wx; = ~X1(X2W11+XqW,2)-3RDW12 icx; = +x2hwll +%Wl2)+M~Wl2 wx;
=
3x3(x2wlz+XqW22)+~RDWl2
(12)
iiix: = ~x~(x~w~~+x~w~~)--~RDw~~ where w = XlXZWll +(x,x,+x,x,)w,,+x,x,w,2. It is immediately obvious from these equations that xi + xi = xi + xi, so that after one generation M, and M2 must be equally frequent. It can be shown as follows that, if linkage is sufficiently tight, linkage disequilibrium tends to build up, so that the gamete size alleles become
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associated with different mating types. Assume first that R = 0, and that A2 is introduced into a population fixed for A1 but at equilibrium for the mating type locus. The mutation to AZ occurs in an MZ individual, so that x3 is always zero. From equations (12) we thus have: 4
= Xk -~ -4
x4w121w119 =
x4w12 xzw11
It follows from these equations that, if wr2 > wlr so that there is selection for anisogamy, A,M, tends to displace A1M2. We would therefore expect that the final equilibrium should consist of AIMI and A,M,. It can be verified that, if the “heterozygote advantage” condition (5) is satisfied, such an equilibrium is indeed stable to the introduction of AI&f2 and A,M,, and computer calculations of population trajectories confirm that it is globally stable. By continuity, for small but non-zero R a similar stable equilibrium but with a low frequency of AIM2 and AZ&f, individuals generated by recombination will exist, under the same conditions. With close linkage, therefore a population results in which there is disassortative fusion with respect to gamete size, as a result of the associations between the mating type and gamete size alleles. With loose linkage, however, it can be shown as follows that an anisogamous population results with the mating types distributed equally among the two gamete size classes. Consider the above situation when A, is introduced into an h4, gamete, but with arbitrary R. Ignoring second-order terms in x3 and x4, we obtain the equations: WIIX;
z
~11~:
zz x~w,~-Rw,~(x~-x~).
~3~12+R~12(~4-~3),
Adding these equations, we have: x;+xl$ = (x3 +X4)W12/W11, (W and subtracting them we get: xi-xi z (x4- x3(1 -2R)w,th1. WI Equations (13a) shows that linkage has no effect on the initial rate of increase in frequency of the gene for gamete size. Furthermore, if we note that D w +(x4-x3) when A2 is rare, then from equation (13b) is follows that D increases only when : w,,(l-2R) > wI1. (14) This condition clearly can never be satisfied for a pair of unlinked loci; if linkage is loose, then extremely strong selection, i.e. an extremely high degree
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of anisogamy, is required for linkage disequilibrium to be built up. Computer calculations of population trajectories show that condition (14) must indeed be satisfied if a population with an association between the two loci is to be established; such an association can be strong only if R is considerably smaller than the critical value given by (14). We may next consider the properties of populations where linkage is so loose that no linkage disequilibrium is possible. Setting D = 0 in equations (12), and solving for the equilibrium frequencies, gives: $1 = 22 = (WI2 - %*PmQ
-
Wll
-
w22)9 (1%
2,
=
24
=
(w1z-w11)/2(2w,2-wl1-w22).
This equilibrium can be shown to be stable if condition (5) is satisfied. Equations (15) shows that the frequencies of the vegetative forms producing microgametes and macrogametes are the same as in the case with no mating types; within each size class, there is an equal frequency of each mating type. According to Wiese (1976), there is no evidence in anisogamous Volvocales for the existence of different mating types within the sexes. This raises the question of whether there can be selection for reduction of the recombination fraction so that linkage disequilibrium can be established. From conventional multi-locus studies of random-mating diploid populations, we know that selection for a rare modifier of the recombination rate between selected loci is slow or non-existent unless the loci are held in linkage disequilibrium by selection (Feldman, 1972; Charlesworth & Charlesworth, 1973). Since equations (12) are similar in form to those for the diploid case, the same conclusion is expected to hold in the present case. This question was investigated by computer calculations of the fate of an “inversion” introduced into AIMI gametes, which suppresses recombination between inverted and normal gametes (cf. Charlesworth & Charlesworth, 1973). The inversion thus remains permanently associated with AIM,. (Any genetic element which suppresses recombination when heterozygous will behave in the same way.) When inverted AIMI individuals were introduced at a low initial frequency into a population at the equilibrium of equations (15), it was found that the inversion increased initially at a rate of the order of the square of its frequency, as is predicted analytically. Eventually, when its frequency has increased sufficiently, it becomes strongly selected, with the result that the final population consists of (inverted) A 1M, and (non-inverted) A&f2 gametes, between which no recombination is possible. An example is shown in Fig. 2. These results show that a very rare element reducing recombination between the gamete size and the mating type loci is almost neutral. If it reaches a
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Generation
FIG. 2. This shows the course of change in frequency of inverted AIMI and non-inverted A,M, chromosomes when the inversion is introduced at a frequency of 1% into a population segregating at the A and M loci. R = 03 and w 11 = was = 1, wla = 2. There is initially complete linkage equilibrium.
frequency of 1% or so as a result of genetic drift or other selective forces, it may come to experience significant selection, and eventually create a situation in which there is a strong association between alleles at the two loci. As Fig. 2 shows, even unlinked loci are subject to this selection pressure, so that it is possible for reciprocal translocations or centric fusions, which link the two loci when originally on separate chromosomes, to be selected by this mechanism. We can conclude, therefore, that despite the lack of linkage disequilibrium when the mating type and gamete size loci are loosely linked, there may indeed be selection favouring the establishment of close linkage. 5. Discussion
The analytic results presented in sections 2 and 3 of this paper confirm the graphical and simulation arguments of Parker et al. (1972), as corrected by Parker (1978). It is particularly interesting to note that, with the power model of equation (7) for the relation between zygote size and viability, anisogamy can always be selected when k > 1, provided that there is a
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sufficiently large effect on gamete size and number of the mutation at the A locus. Furthermore, when zygote viability is relatively insensitive to zygote size, so that k is only slightly greater than unity, only a mutant which has a very large effect on gamete size can be selected. There is no difficulty in principle in seeing how a very striking gamete size dimorphism could be set up by a single mutational change affecting the timing and frequency of the cell divisions involved in gamete production. We have also seen that the pre-existence of mating types does not affect the conditions for the evolution of anisogamy, and how close linkage between the mating type locus and the gamete size locus can bring about a situation in which there is disassortative fusion with respect to gamete size. There is selection for closer linkage between the two loci even when linkage is initially so loose that the mating type alleles are equally frequent in microgametes and macrogametes. Parker (1978) has presented some models of the evolution of disassortative fusion; he assumes the existence of genes which can cause small gametes which carry them to fuse selectively with large gametes, and vice versa. Such genes are perhaps of somewhat dubious plausibility, whereas mating type loci are frequently found in isogamous green algae (Lewin, 1976). Close linkage between the mating-type locus and the locus controlling gamete size is thus an attractive alternative explanation of disassortative fusion, although there are some difficulties in understanding how genes on separate chromosomes could be brought together, as may be necessary on this view. The genetic dissection of sexuality in species of algae with anisogamy and disassortative fusion may provide a way of discriminating between these hypotheses. On the mating-type hypothesis, the microgametes and macrogametes should be found to differ at more than one locus, although it may be difficult to separate the loci by recombination, just as the male- and femaledetermining chromosomes of dioecious flowering plants can be shown to differ at several loci (Westergaard, 1958). It would also be expected that the two sorts of gamete should differ with respect to sexual agglutinins in the same way as do the mating types of isogamous species (Wiese, 1976). The models presented above have assumed that the differences in gamete size in anisogamous species are of direct genetic causation. It is known, however, that in many Volvocales, for example, sex may be environmentally determined, so that members of the same haploid alone can differentiate into macrogametes or microgametes; in multicellular forms, macrogametes and microgametes may be produced by the same individual (Wiese, 1976). It is easy to construct a model for the evolution of this sort of situation on the following lines. Suppose we have an initial isogamous A, population. A mutation to A2 occurs, such that vegetative A, individuals differentiate a
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proportion p of the time into gametes of the same size as A,(nz,) and q of the time into gametes of a different size (mz). There are corresponding differences in the numbers of gametes produced, n, and n,. Depending on the values of p and q and on the viabihties of the zygotes formed by fusion of gametes of different sizes, it is possible to show that A, may spread; under suitable conditions it may even become fixed. Under conditions where it cannot be fixed, one would find at equilibrium two classes of vegetative individuals; one capable of producing gametes of both sizes and the other capable of producing only one. Disassortative fusion in a species where gamete size is determined non-genetically obviously cannot evolve by associations between mating type alleles and gamete size alleles, but must arise by a mechanism similar to that proposed by Parker (1978). REFERENCES BELL, G. (1978). J. theor. Biol. 73, 247. CHARLESWORTH, B. & CHARLFSWORTH, D. (1973). Genet. Res., Camb. CROW, J. F. & KIMKJRA, M. (1970). Introduction to Population Genetics Harper & Row. FELDMAN, M. (1972). Theor. pop. Biol. 3, 324. LEWIN, R. A. (1976). (ed.) The Genetics of Algae. Oxford: Blackwell. MAYNARD SMITH, J. (1978). The Euolution of Sex and Recombination.
21, 167. Theory. New York:
Cambridge: bridge University Press. PARKER, G. A. (1978). J. theor. Biol. (in press). PARKER, G. A,, BAKER, R. R. &SMITH, V. G. F. (1972). J. theor. Biol. 36, 529. WESTERGAARD, M. (1958). Adv. Genet. 9,217. WIESE, L. (1976). In The Genetics of AIgae (R. A. Lewin, ed.). Oxford: Blackwell.
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