SELECTION AT THE HAPLOID AND DIPLOID PHASES: CYCLICAL VARIATIONlJ

EVELYN P. EWING3

Division of Biological Sciences, Uniuersity of Kansas, Lawrence, Kansas 66044

Manuscript received January 27, 1977 Revised copy received April 27, 1977

ABSTRACT

This paper considers a deterministic model, where selection acts a t both the diploid and haploid phases of development. The stability criteria for the maintenance of a protected polymorphism are derived. The implications of the model are examined through several examples, with emphasis on the com- parison of these conditions to those of models that view selection as acting only at the diploid or haploid phases. It is found that the consideration of both phases broadens the conditions over such models.

I N this paper an idealized, deterministic model for panmictic populations, along with two extensions, will be analyzed for situations in which selection acts

cyclically at both the diploid and haploid stages of development, such as may occur in plants. In animals this is probably not the case, for it is believed that the active enzymes governing sperm and egg behavior are those of the parent rather than those produced by the haploid gametes (HARTL 1977). Somewhat similar models, one deterministic and one involving stochastic selection in conjunction with nonMendelian segregation, have been investigated using different tech- niques by SCUDO (1967) and HARTL (1976), respectively. Specifically, if the stochastic element is removed and Mendelian segregation is assumed, this model becomes that of HARTL and is a somewhat more general summary of the Scum model. Of particular interest is the impact that the consideration of the haploid phase has on the maintenance of a stable, nontrivial polymorphism when com- pared to situations where the diploid phase is examined alone, as was done in the now classical paper of HALDANE and JAYAKAR (1963). As their results will be mentioned a number of times, we will simply refer to their model as the HJM.

With higher plants, there is an alternation of haploid and diploid phases, which can be described by the model and its first extension. HARDING (1975) has given a general discussion of the subject, indicating several ways in which selec- tion may act at both the gametophytic (e.g., pollen grain competition) and sporophytic stages. In his paper, various mating systems are considered as well.

' This work was supported by National Science Foundation Grant BM 573 01305 A01 and by the General Research degree from the

This paper 15 the fourth in the series, "Genetic Vaiiatlon in a Heterogeneoiis Environment " The third p p e r m this

Present addiess Department of Biology, U inerslty of New Mcuico, Albuquerque, New Mexico 87131

Fund of the University of Kansas to PHILIP W HEDRICK, and is partial fulfillment of the P h D Univemiy of Kansas

series IS by H ~ D R I C ~ , P W , hI E GINEVAN and E P EWING, Ann Rev Ecol Syst 7 : 1-32 (197s)

Genetics 87: 195-208 Septembei, 1977

196 E. P. EWING

A gametic selection factor, Ga, that segregates in barley populations has been described by JAIN (1975), who states that, although such factors are known to exist in several species of plants (e.g., maize) their evolutionary significance has not been examined in any depth.

In bryophytes, the dominant, readily observable form is the haploid or game- tophytic phase. This phase is distinguished from that of many lower plants as there are distinct sexual organs, the antheridium and archegonium, that may be on the same or different plants. In some species, these develop at different times so that self-fertilization may be rare. After mitosis, resulting gametes unite in the formation of a diploid zygote or sporophyte which can undergo meiosis to form spores. These shifts from haploid to diploid are a good example of distinct alternation of generations, which can be brought about by environmental changes.

The fungi, which reproduce both sexually and asexually, also provide a per- tinent group of examples. The sexual phase, generally rather complex, may occur annually or with environmental fluctuations depending upon the particular species. But it is the asexual stage that is of greater consequence, since it is generally repeated several times per season and results in the formation of more individuals (ALEXOPOULOS 1952). This situation can be described by the second extension which will be discussed later in the paper. In some higher fungi, a dikaryotic phase is intermediate between the haploid and diploid stages. Depend- ing upon the time of selection, this can be examined by the main model or a slight modification of it. An interesting example is that of wheat rust, Puccinia graminis tritici, which is of special interest with respect to both its biological and economic aspects. Teleutospores are produced in the summer on wheat and over- winter after karyogamy has occurred. In the spring meiosis occurs, resulting in the production of basidiospores which can germinate only on its alternate host, the common barberry. Dikaryotization results in binucleate aeciospores that can only infect the primary wheat host. These develop into the uredospores which germinate and reinfect the wheat throughout the spring and summer, giving a rust-colored appearance to the host.

A great diversity with respect to variation in the diploid-haploid alternation of generations can be found in the algae. Many of the fresh water algae, typically found in shallow ponds, are often very sensitive to environmental fluctuations. For example, Ulothrix, a simple, filamentous form, has a diploid zygote which may remain dormant for long periods as a result of adverse, environmental con- ditions. The remainder of the life cycle is spent in a series of haploid phases that can reproduce asexually by zoospores. In Cephaleuros, which proliferates in the intercellular spaces of trees and shrubs, there is a true but unequal alternation of generations. The definitive plant is haploid and can again reproduce for long periods asexually uia zoospores. In the marine green alga, Ulva, also known as the sea lettuce and found so commonly in estuaries, an isomorphic but seasonal alternation of diploid-haploid phases occurs. In the spring the diploid plants that have overwintered undergo meiosis, giving rise to the haploid plants which pre- dominate during the spring and summer months. Both the diploid and haploid

SELECTION AT HAPLOID AND DIPLOID PHASES 197

plants can reproduce asexually. Another example that is quite similar to Ulva, particularly with respect to this seasonal, isomorphic alternation, is Ectocarpos, a brown alga. Here, too, the alternation of generations is not an obligate one. AS a final example, Laminaria, a giant kelp, exhibits a heteromorphic, balanced alternation of diploid and haploid phases (BOLD 1957). The large, predominant form is a diploid perennial from which the haploid, filamentous gametophytes ;re produced. This balance exhibited by Laminaria is very similar to that of ferns and related vascular plants.

With such examples in mind, we will define the model and derive stability conditions. Two extensions will be examined in order to determine the implica- tions of selection at both haploid and diploid phases. More complicated extensions will be discussed elsewhere.

THE MODEL

The model under consideration consists of a two-phase cycle that repeats itself over time: phase one is diploid; phase two, haploid. I t is assumed that the phases are nonoverlapping and that selection operates at both stages of the cycle. AS only two alleles, A1 and A?, are considered, the genotypes, their frequencies before selection, and their fitness values in the diploid phase are as follows:

Genotype AIA, A,A, A&,

Fitness WO 1 U0

Frequency P: 2P40 q:

After selection, the gene frequencies become

( l a >

7 (Ib)

( I C )

qo(1 - qo + qovo) W D

(1-90) (WO +qo - qowo) W D

q 1 =

and

- PI = --

W,, the average fitness for the diploid phase, is where w, = WO f2qo(l-wo) + y: (WO + U0 - 2 ) .

In the haploid phase, the haploid genotypes, their frequencies before selection, and their fitnesses are given below as follows:

Genotype A, A, Frequency p1 41 Fitness w1 U1

So the gene frequencies after this phase are

and

where W, is the average fitness from the haploid phase and is defined by

198 E. P. EWING

WH = W1-q~ (Wl-vl) (2c) In order to have a protected polymorphism, we require that the change in qo

from its initial value to its value after the haploid phase be zero; i.e., Aqo+H qz-qo

= Aqo+AqH = 0 . (3a)

(3b) Since

where AqDCH = qo(l-qo) Q ( ~ o ) 7

Q(qo) =- (1-w,)+qo(wo+uo-2) (l-qofqovo)

(wo+qo-qowo) (ul-wl)

+ -___- W D W D

7 (3c) ---. -- W D W H

we can exclude the trivial values of zero and one to consider the function Q (qo) . In other words, if Q(0) > 0 and Q(1) < 0, then there exists a protected poly- morphism.

At go = 0, q1 ( 0 )

(4a) Q(o> =-

U 1 > wow1 . (4b)

(5a)

0, W D ( 0 ) = w0, and CVH ( 0 ) = wl, SO that

(Ul-WOwl) > 0, WOW1 f. O, 1

wow1 or

Similarly, at qo = 1, q l ( l ) 1, wD(l) = uo, and m,,(1) = ~ 1 , giving 1

UOUl Q(1) =- {uoul-wl} < 0, V O U l Z 0,

or

Therefore, the conditions for stability are

and

We note that (6a) and (6b) are equivalent to

w1 > uou1.

U 1 wow1

w1 > VOUl .

dAqn+H > 0 dqo o

and

The possibility of further conditions for stability can be examined by con- sidering higher order derivatives when (7a) and/or (7b) became equal to zero, in the manner described by PROUT (1968). In particular, we want to look at

and/or

SELECTION AT HAPLOID AND DIPLOID PHASES 199

(9)

W1(W~U0 + WO - 1) > U1 . (8a>

wouo > 1 (8b)

1 > wouo , (8c)

Ul(WoUofU0- 1) > w1 . @a>

wouo > 1 (9b)

1 > WOUO , (9c)

d 2 A 9 D + H I < 0 , dq2, 1

respectively. Then (8) is equivalent to

When (7a) is equal to zero, i.e., u1 = wow1, condition (8) can be rewritten as

and condition (6b) can be rewritten as

which clearly cannot occur. Similarly, (9) is equivalent to

If ( 7b) is altered so that w1 = uoul, then (9a) becomes

and (6a) becomes

so, again this will not hold. As a result, conditions (6a) and (6b) are both necessary and sufficient for the

existence of a protected polymorphism as implied by PROUT (1 968) and GLIDDON and STROBECK (1974). We note that this type of approach with respect to technique of analysis also yields both necessary and sufficient conditions for haploid populations with fitnesses varying over space under a Levene model (GLIDDON and STROBECK 1974), although it yields only sufficient conditions for diploid populations with fitnesses varying over both space (PROUT 1968) and time (HOEKSTRA 1975).

In addition, we point out that the function Q(q,,), given in (3c), can be rewritten as a function whose numerator is cubic in go. The roots of this cubic can be found explicitly, although tediously, by formula. By examining the deriva- tive of this function on the interval [0, 11, given conditions (6a) and (6b), it can be shown that there exists one and only one qo which is stable and which satisfies this equation on [0, 11. In other words, there is a unique, nontrival solution such that this equilibrium is globally stable (HARTL 1976). So, in fact, the polymorphism is not only protected, but can be shown to be globally stable for this model.

EXTENSIONS O F THE MODEL

The model can be extended in a variety of ways. Two will be examined here. First, suppose that there are k repetitions of two-phase cycles, which are alter- nately diploid and haploid, as might occur in higher plants with cyclical changes in the selective values; i.e., the genotypes, their frequencies, and their fitnesses in the kth cycle are as follows:

2kth Diploid phase 2k+lSt Haploid phase Genotype A A l A A A2A2 AI Az Frequency ptk 2Pzkqzk 4zk pzk+i qzki-1 Fitness wzk 1 Uzk w2k+l U z k + l

200 E. P. EWING

Then, after the diploid phase, but before selection in the haploid phase, the frequency of A, is

where ( l i b ) -

w H 2 k + 2 - wzk+l - q2k+l (W27c+l-U2k+l) . Following k repetitions of the cycle, the change in qo becomes

where k

Q(q0) =jio {QUpi(qo)+QHpj+l(qo)} - (12b) By using Q ( 0 ) > 0 and Q ( 1) < 0, as before, the conditions for a protected poly- morphism become

k n

j = 0 3=o II ~ ~ j + ~ > .II wj, n = 2 k f l ,

and k n

i=0 1=0 II w , ~ + ~ > II uj,n=2k+l .

In other words, the product of the fitnesses of the A, (and A,) genotypes must be greater than the product of the fitness values of the A, and A,A, (and the product of the fitnesses of A , and A2A2) genotypes. That is, the haploid product of selec- tive values is larger than the corresponding haploid-diploid product for each genotype.

As a second extension, assume that a diploid phase is followed by n haploid phases as is often the case in lower plants which spend most of their time in a haploid phase. Even if the diploid phase is only transient but selection operates, polymorphism can be maintained when the conditions given below are satisfied. We define the model as follows:

kth Haploid phase: k : 1,2, . . . , n Diploid phase Genotype A A l A,A, A A , Ai A, Frequency p: 2poqo 42, P k qk Fitness WO 1 U0 w k u k

Again, let

SELECTION AT HAPLOID AND DIPLOID PHASES 20 1 n n

and n n

It should be pointed out here that it is possible for a single diploid phase within a cycle to allow for the maintenance of a protected polymorphism, whereas such an equilibrium could not be maintained when haploid phases alone are con- sidered over time as in the HJM.

DISCUSSION

The implications of the model will be examined by considering several examples. In particular, we will look at the conditions with respect to the geo- metric and arithmetic mean fitnesses, denoted by wG, uG and wA, uA, respectively, in order to compare our conditions to those of the HJM. Unless otherwise stated, the discussion applies to the two-phase cycle repetition model.

Before looking at these examples, however, an interesting relationship (pointed out by J. FELSENSTEIN and C . STROBECK) between this model and the HJM should be noted here. Suppose that the diploid-haploid model is compared to a diploid-diploid model, such that the diploid phase of the HJM corresponding to the haploid phase of this model shows geometric mean intermediacy (i.e., let the fitnesses of the ALA, , AlA2, and A,A2 genotypes be T, I,;, respectively). Then, in this special case, the stability conditions for the two models are com- pletely equivalent.

Neutrality: As a starting point, neutrality in one of the phases will be con- sidered first as the results are quite predictable. When there is neutrality in the haploid phase of the cycle (i.e., w1 = u1 = l ) , then overdominance is required in the diploid phase (i.e., uo < 1 and wo < 1 ) . So the model becomes analogous to the well known case where, when the fitnesses of the homozygotes remain constant over time, the fitnesses of the homozygotes must be less than that of the heterozygote. This extends directly to n two-phase repetitions as geometric mean overdominance is required; i.e., the geometric mean fitness of each of the homozygotes must be less than one. If, however, neutrality occurs in the diploid phase of the cycle, the conditions for polymorphism cannot be met. This is not surprising in view of the virtual impossibility of maintaining a stable, nontrivial equilibrium with the haploid HJM over time.

Complete dominance: Now consider a complete dominance model (also termed absolute dominance by PROUT (1968) and HEDRICK (1974)), such that the genotypes and their fitnesses are as follows:

Diploid phase Haploid phase Genotype A,A, A,A2 A2A2 A , A2

Fitness 1 1 U0 1 U1

Then the conditions for a protected polymorphism are

202 E. P. EWING

FIGURE 1.-Absolute dominance. The areas (2) and (3) refer to the region of stability for the two-phase diploid-haploid model, while the areas marked (1) and (3) refer to the corre- sponding HSM model.

i.e., as u1 increases, the allowable uo values decrease. (See Figure 1). While the condition uoul < 1 is equivalent to stating that the geometric mean

of the fitnesses of the A,A, and A , genotypes be less than one, the arithmetic mean can be either greater than, less than, or equal to one. Suppose, for example, that u1 1.5. If u0 = 0.4, uA = 0.95; if uo = 0.5, uA = 1.0; and, if uo = 0.6, uA = 1.05. In Figure 1, these values correspond to region 2, the boundary between regions 2 and 3, and region 3, respectively. This is, in a sense, a broadening of the situation in the analogous two-phase diploid case of the HJM, which requires that the arithmetic mean be greater than one, while the geometric mean be less than one in order to maintain a nontrivial equilibrium.

Assuming that u1 = 1.5, AqD+H curves for different values of uo are given in Figure 2. The shapes of these curves and the gene frequency equilibrium are also instructive as to the maintenance of the polymorphism in a finite population (HEDRICK 1972, 1974). For example, when uo = 0.4, a polymorphism in a finite population can be maintained for a long period (HEDRICK, unpublished) , whereas if uo = 0.05 or 0.6 the polymorphism is not maintained effectively and disap- pears in approximately the same amount of time as if there were no balancing

SELECTION A T HAPLOID A N D DIPLOID PHASES 203

00 0 2 0.6 0 8 I .o FIGURE 2.-Aq curves for absolute dominance where u1 = 1.5. The rate of convergence to qe

is more rapid with increasing magnitude of the slope near qe. Average fitness, W = dWDW13, is one for all U".

O 4 90 - selection operating at all. In other words, this is another example where some values which give a protected polymorphism in an infinite population offer no increased maintenance of the polymorphism in a finite population as compared to neutrality.

The conditions for an equilibrium are similar when repetitions consist of n two-phase cycles. For example, in a €our-phase repetition cycle with wi = 1, i: 0,1,2,3, the conditions become:

ulu3 > 1 ( 16c) and

1 > n ui, (16d) 3

i=O

where the second condition is again equivalent to the geometric mean fitness of the A,A, and A , genotypes being less than one. Suppose that the selective values for the diploid phases are uo = 0.2 and U , = 1.5, while, for the haploid phases, the fitnesses are u1 = 1.25 and us = 0.889. Then the geometric mean is 0.76, while the arithmetic mean is 0.96; i.e., both means are less than one and there is still a protected polymorphism. It is interesting to look as well at the diploid phase fitness values separately, which are uG = 0.55 and va 0.85. If the haploid phases are ignored or overlooked, these values would indicate the disappearance of the A , allele (HALDANE and JAYAKAR 1963). So, when haploid phases are

204 E. P. EWING

important with respect to the fitness values, their consideration has a dramatic impact upon the maintenance of polymorphism.

Diploid additivity: As a third example, we will examine a model involving additive selection in the diploid phase with genotypes and fitness values as follows:

Diploid phase Haploid phase Genotype AIAl A,A, A,A, Ai A? Fitness 1 -s 1 1 +s l+s U1

1 > U 1 > 1-9 , Now the conditions for stability can be written as

(17a)

where 0 I s < 1. It should be pointed out that if v1 = 1-s, this condition cannot be satisfied. As s increases, so does the range for allowable v1 values. (See Figure 3) . In this situation, the geometric mean of wo and w1 is always less than one. However, the geometric mean of vo and u1 can assume values larger than one. For example, if we let s = u1 = 0.75, then wG = 0.66, while vo = 1.15. This is the situation that occurs in region 2 of Figure 3. So polymorphism will remain even though the average fitness of the A2A, and A, genotypes is greater than that of the fitness of the A,A2 genotype.

If s = 0.7, the A q curves for different values of v1 are given in Figure 4. The

1.00

0.75

s 0.50

0.2 5

0.00 0.00 0.25 0.50 0.75 I .oo

"1

FIGDRE 3.-Additivity in the diploid phase. The shaded areas indicate the regions of stability where wo = I--s and uo = w1 7 l + s . Those areas marked (2) and ( 3 ) refer to regions of stability for the two-phase diploid-haploid model, while the areas marked (1) and (3) refer t 3

the corresponding HJM model.

SELECTION AT HAPLOID AND DIPLOID PHASES 205

I

0.0 qo

FIGURE 4.-Aq curves for additivity in the diploid phase where 5 = 0.7. The rate of conver- gence to q, is more rapid with increasing magnitude of the slope near qe. Arerage fitness,

defined as @= 4pDWH, increases with qe. --

earlier discussion of A q values and their value in determining the maintenance of a polymorphism in a finite population is also relevant here. For both of these examples (the situations described in Figures 2 and 4), the maintenance of the polymorphism in finite populations is generally quite effective as compared to other temporal models (HEDRICK 1974,1976, unpublished).

This model also extends to n repetitions of two-phase cycles. Again, we will examine a four-phase cycle with opposing additive selection in the diploid phases, where wo = 1--s, uo = w1 = Ifs, wz = l+sl, and uz = w3 = 1-s, fo r s, sl:O+l. Then the conditions for a protected polymorphism can be rewritten as

1 > ulu3 > (1-s’) (1-s:) . ( 17b) Suppose that s = 0.75, s1 = 0.25, and let uIu3 = 0.875. There is then stability with W O 0.8, but U G = 1.04. If we examine fitness values in the diploid phases separately where there is opposing additive selection, the geometric mean, zuc = 0.56, is less than one, whereas the geometric mean, U , = 1.15, is greater than one; i.e., there is no average heterozygote advantage. So again, the consideration of the haploid phases changes the condithns for stability considerably.

206 E. P. EWING

The preceding example with diploid additivity can be extended as follows:

Diploid phase Haploid phase Genotype A,Al A,Az AzA, Ai Az Fitness 1-s 1 l+ks 1 +ks U1

1 > U , > (1-s) ( l f k s ) , (18)

SO the conditions for stability are

where k 2 0. The behavior of u1 is as before; i.e., as s increases, the range of u1 increases also. As k increases, the region of stability is narrowed. Here, too, it is not necessary that both wG and uG be less than one. In fact, there are cases where both can be greater than one in n two-phase repetitions. For example, consider a four-phase repetition cycle where uz = wo =wl = 3.0, wz = w3 = 0.5. v o = 0.125, u1 = 0.857, and us = 3.5. Although rather extreme, both geometric means are larger than one (i.e., wG = 1.22 and uG = 1.03) , indicating that selection is acting in a manner similar to a frequency-dependent situation.

Fitness and the adaptiue peak: A brief remark on fitness and the adaptive peak is appropriate at this point. When there is a repetition of a single set of selective values in a diploid population that is repeated over time, the nq function

d W d W is in terms of -, where w is the average fitness, so that nq = 0 implies - 0;

i.e., the equilibrium value for nq maximizes @. This is not the case for our two- phase repetition model. In general, when there is temporal variation in the selective values, the average fitness, m, is taken as the geometric mean fitness. So, in our model, %' = WDW,. The critical point of this function (i.e., the

d W value of q for which __ = 0) does not force nq = 0, although it is the maximum d 4

of the w function. In other words, the adaptive peak of fitness is not achieved at equilibrium. It should be pointed out that this is due in part to the fact that, over time, w is chosen indirectly as some sort of average (e.g., the geometric mean), instead of being taken directly from the equations giving rise to the A q function, such as in the repeating, one-phase diploid case.

In conclusion. we reiterate the major points described above. First, selection at both the haploid and diploid phases of deveIopment can make a considerable difference with respect to the retention of variability in a population. For p o p - lations that remain haploid only over time, the HJM shows that no nontrival polymorphism can be maintained. However, if even a single, diploid phase occurs within a cycle, there is relatively little difficulty in maintaining such an equilib- rium. Further, when populations with both haploid and diploid phases are com- pared to populations with only repeating diploid phases, the conditions for maintaining equilibrium are broadened and/or changed from the HJM. The differences between the two models is with respect to the values that the geometric and arithmetic mean fitnesses can assume, as was previously described in the examples involving complete dominance and diploid additivity.

d q d q

SELECTION A T HAPLOID A N D DIPLOID PHASES 20 7

I want to acknowledge the encouragement and many useful discussions provided by PHILIP W. HEDRICK, who also allowed me to use his unpublished results on finite populations. In addition, I thank each of the following who read and commented on this manuscript: JOSEPH FELSENSTEIN, DANIEL L. HARTL, SUBODH K. JAIN, and CURTIS STROBECK.

LITERATURE CITED

ALEXOPOULOS, C. J., 1952 BOLD, H. C., 1957 GLIDDON, C. and C. STROBECK, 1974 Necessary and sufficient conditions for multiple-niche

polymorphism in haploids. Am. Naturalist 109: 233-235. HALDANE, J. B. S. and S. D. JAYAKAR, 1963 Polymorphism due to selection of varying direction.

J. Genet. 58: 237-242. HARDING, J., 1975 Models for gamete competition and self fertilization as components of natural

selection in populations of higher plants. In: Gamete competition in plants and anb"s. Edited by D. L. MULCAHY. North-Holland Publishing Co., Amsterdam, The Netherlands.

Stochastic selection in both haplophase and diplophase. J. Mathematical Biology (In press). __ , 1977 Applications of meiotic drive in animal breeding and population control. Proc. Intern. Congrecs Quant. Genetics (In press).

Maintenance of genetic variation with a frequency-dependent selection model as compared to the overdominant model. Genetics 72: 771-775. -, 1974 Genetic variation in a heterogeneous environment. 1. Temporal heterogeneity and the absolute dominance model. Genetics 78: 757-770. __ , 1976 Genetic variation in a heterogeneous environment, I1 Temporal heterogeneity and directional selection. Genetics

Iniroductory mycology. Wiley, Inc., New York. Morphology of plants. Harper and Brothers, New York.

HARTL, D. L., 1976

HEDRICK, P. W., 1972

84: 145-157. HOEKSTRA, R. F., 1975 JAIN, S. K., 1975

A deterministic model of cyclical selection. Genet. Res. 24: 1-15. Gametic selection in mixed selfing and random mating plant populations.

In: Gamete competition in plants and animals. Edited by D. L. MULCAHY. North-Holland Publishing Co., Amsterdam, The Netherlands.

Sufficient conditions for multiple niche polymorphism. Am. Naturalist 102: PROUT, T., 1968

SCUDO, F. M., 1967 493-496.

Selection on both haplo and diplophase. Genetics 56: 693-704. Corresponding editor: R. C. LEWONTIN