EVELYN P. EWING3. Manuscript received January 27, 1977 Revised copy received April 27, 1977 ABSTRACT

Transcription

1 SELECTION AT THE HAPLOID AND DIPLOID PHASES: CYCLICAL VARIATIONlJ EVELYN P. EWING3 Division of Biological Sciences, Uniuersity of Kansas, Lawrence, Kansas Manuscript received January 27, 1977 Revised copy received April 27, 1977 ABSTRACT This paper considers a deterministic model, where selection acts at both the diploid 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 comparison 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. IN 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 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 egg behavior are those of the parent rather than those produced by the haploid gametes (HARTL 1977). Somewhat similar models, one deterministic one involving stochastic selection in conjunction with nonmendelian segregation, have been investigated using different techniques by SCUDO (1967) HARTL (1976), respectively. Specifically, if the stochastic element is removed Mendelian segregation is assumed, this model becomes that of HARTL 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 compared to situations where the diploid phase is examined alone, as was done in the now classical paper of HALDANE 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 diploid phases, which can be described by the model its first extension. HARDING (1975) has given a general discussion of the subject, indicating several ways in which selection may act at both the gametophytic (e.g., pollen grain competition) sporophytic stages. In his paper, various mating systems are considered as well. ' This work was supported by National Science Foundation Grant BM A01 by the General Research Fund of the University of Kansas to PHILIP W HEDRICK, is partial fulfillment of the Ph D degree from the Univemiy of Kansas This paper 15 the fourth in the series, "Genetic Vaiiatlon in a Heterogeneoiis Environment " The third pper m this series IS by H~DRIC~, P W, hi E GINEVAN E P EWING, Ann Rev Ecol Syst 7: 1-32 (197s) Present addiess Department of Biology, U inerslty of New Mcuico, Albuquerque, New Mexico Genetics 87: Septembei, 1977

2 196 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 gametophytic phase. This phase is distinguished from that of many lower plants as there are distinct sexual organs, the antheridium 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 asexually, also provide a pertinent 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 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 diploid stages. Depending 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 economic aspects. Teleutospores are produced in the summer on wheat overwinter 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 reinfect the wheat throughout the spring 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 conditions. 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 shrubs, there is a true but unequal alternation of generations. The definitive plant is haploid can again reproduce for long periods asexually uia zoospores. In the marine green alga, Ulva, also known as the sea lettuce 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 predominate during the spring summer months. Both the diploid haploid

3 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 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 related vascular plants. With such examples in mind, we will define the model derive stability conditions. Two extensions will be examined in order to determine the implications of selection at both haploid 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. It is assumed that the phases are nonoverlapping that selection operates at both stages of the cycle. AS only two alleles, A1 A?, are considered, the genotypes, their frequencies before selection, their fitness values in the diploid phase are as follows: Genotype AIA, A,A, A&, Frequency P: 2P40 q: Fitness WO 1 U0 After selection, the gene frequencies become qo(1 - qo + qovo) where q1= W D (1-90) (WO +qo - qowo) PI = -- - W,, W D the average fitness for the diploid phase, is w, = WO f2qo(l-wo) + y: (WO + U0-2). (la> 7 (Ib) In the haploid phase, the haploid genotypes, their frequencies before selection, their fitnesses are given below as follows: Genotype A, A, Frequency p1 41 Fitness w1 U1 So the gene frequencies after this phase are (IC) where W, is the average fitness from the haploid phase is defined by

4 198 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) Since AqDCH = qo(l-qo) Q(~o) 7 (3b) where Q(qo) =- (1-w,)+qo(wo+uo-2) (l-qofqovo) WD W D -. (wo+qo-qowo) -- (ul-wl) 7 (3c) W D W H we can exclude the trivial values of zero one to consider the function Q (qo). In other words, if Q(0) > 0 Q(1) < 0, then there exists a protected poly- morphism. At go = 0, q1 (0) 0, W D (0) = w0, CVH (0) = wl, SO that 1 Q(o> =- (Ul-WOwl) > 0, WOW1 f.o, wow1 or U1 > wow1. Similarly, at qo = 1, ql(l) 1, wd(l) = uo, m,,(1) = ~ 1 giving, 1 Q(1) =- {uoul-wl} < 0, VOUlZ 0, UOUl or w1 > uou1. Therefore, the conditions for stability are U1 wow1 w1 > VOUl. We note that (6a) (6b) are equivalent to daqn+h > 0 dqo o (4a) (4b) (5a) The possibility of further conditions for stability can be examined by considering higher order derivatives when (7a) /or (7b) became equal to zero, in the manner described by PROUT (1968). In particular, we want to look at /or

5 d2a9d+h I < 0, SELECTION AT HAPLOID AND DIPLOID PHASES 199 (9) dq2, 1 respectively. Then (8) is equivalent to W1(W~U0 + WO - 1) > U1. (8a> When (7a) is equal to zero, i.e., u1 = wow1, condition (8) can be rewritten as wouo > 1 (8b) condition (6b) can be rewritten as 1 > wouo, (8c) which clearly cannot occur. Similarly, (9) is equivalent to Ul(WoUofU0-1) > If ( 7b) is altered so that w1 = uoul, then (9a) becomes wouo > 1 (9b) (6a) becomes 1 > WOUO, (9c) so, again this will not hold. As a result, conditions (6a) (6b) are both necessary sufficient for the existence of a protected polymorphism as implied by PROUT (1968) GLIDDON STROBECK (1974). We note that this type of approach with respect to technique of analysis also yields both necessary sufficient conditions for haploid populations with fitnesses varying over space under a Levene model (GLIDDON STROBECK 1974), although it yields only sufficient conditions for diploid populations with fitnesses varying over both space (PROUT 1968) 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 derivative of this function on the interval [0, 11, given conditions (6a) (6b), it can be shown that there exists one only one qo which is stable 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 OF 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 alternately diploid haploid, as might occur in higher plants with cyclical changes in the selective values; i.e., the genotypes, their frequencies, their fitnesses in the kth cycle are as follows: 2kth Diploid phase 2k+lSt Haploid phase Genotype AAl A A A2A2 AI Az Frequency ptk 2Pzkqzk 4zk pzk+i qzki-1 Fitness wzk 1 Uzk w2k+l Uzk+l

6 200 E. P. EWING Then, after the diploid phase, but before selection in the haploid phase, the frequency of A, is where wh2k+2 - wzk+l - q2k+l (W27c+l-U2k+l). Following k repetitions of the cycle, the change in qo becomes (lib) where k Q(q0) =jio {QUpi(qo)+QHpj+l(qo)} - (12b) By using Q(0) > 0 Q( 1) < 0, as before, the conditions for a protected polymorphism become k n II ~ ~ j + ~ j=0 3=o k i=0 1=0 >.II wj, n = 2kfl, n II w,~+~> II uj,n=2k+l. In other words, the product of the fitnesses of the A, ( A,) genotypes must be greater than the product of the fitness values of the A, A,A, ( the product of the fitnesses of A, A2A2) genotypes. That is, the haploid product of selective 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: Diploid phase kth Haploid phase: k: 1,2,..., n Genotype AAl A,A, AA, Ai A, Frequency p: 2poqo 42, Pk qk Fitness WO 1 U0 w k uk Again, let

7 SELECTION AT HAPLOID AND DIPLOID PHASES 20 1 n n 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 considered 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 geometric arithmetic mean fitnesses, denoted by wg, ug 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 C. STROBECK) between this model 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, A,A2 genotypes be T, I,;, respectively). Then, in this special case, the stability conditions for the two models are completely equivalent. Neutrality: As a starting point, neutrality in one of the phases will be considered 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 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) HEDRICK (1974)), such that the genotypes 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

8 202 E. P. EWING FIGURE 1.-Absolute dominance. The areas (2) (3) refer to the region of stability for the two-phase diploid-haploid model, while the areas marked (1) (3) refer to the corresponding 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, A, genotypes be less than one, the arithmetic mean can be either greater than, less than, or equal to one. Suppose, for example, that u If u0 = 0.4, ua = 0.95; if uo = 0.5, ua = 1.0;, if uo = 0.6, ua = In Figure 1, these values correspond to region 2, the boundary between regions 2 3, 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 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 disappears in approximately the same amount of time as if there were no balancing

9 SELECTION AT HAPLOID AND DIPLOID PHASES I.o O4 90 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". 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) 3 1> n ui, (16d) i=o where the second condition is again equivalent to the geometric mean fitness of the A,A, A, genotypes being less than one. Suppose that the selective values for the diploid phases are uo = 0.2 U, = 1.5, while, for the haploid phases, the fitnesses are u1 = 1.25 us = Then the geometric mean is 0.76, while the arithmetic mean is 0.96; i.e., both means are less than one 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 va If the haploid phases are ignored or overlooked, these values would indicate the disappearance of the A, allele (HALDANE JAYAKAR 1963). So, when haploid phases are

10 204 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 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 Now the conditions for stability can be written as 1 > U1 > 1-9, (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 w1 is always less than one. However, the geometric mean of vo u1 can assume values larger than one. For example, if we let s = u1 = 0.75, then wg = 0.66, while vo = This is the situation that occurs in region 2 of Figure 3. So polymorphism will remain even though the average fitness of the A2A, A, genotypes is greater than that of the fitness of the A,A2 genotype. If s = 0.7, the Aq curves for different values of v1 are given in Figure 4. The s I.oo FIGDRE 3.-Additivity in the diploid phase. The shaded areas indicate the regions of stability where wo = I--s uo = w1 7 l+s. Those areas marked (2) (3) refer to regions of stability for the two-phase diploid-haploid model, while the areas marked (1) (3) refer t3 the corresponding HJM model. "1

11 SELECTION AT HAPLOID AND DIPLOID PHASES 205 I 0.0 FIGURE 4.-Aq curves for additivity in the diploid phase where 5 = 0.7. The rate of convergence to q, is more rapid with increasing magnitude of the slope near qe. Arerage fitness, -- defined 4pDWH, increases with qe. qo earlier discussion of Aq values 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 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, uz = w3 = 1-s, for 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, let uiu3 = There is then stability with WO 0.8, but UG = 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.

12 206 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 SO the conditions for stability are 1 > U,> (1-s) (lfks), (18) 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 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. vo = 0.125, u1 = 0.857, us = 3.5. Although rather extreme, both geometric means are larger than one (i.e., wg = 1.22 ug = 1.03), indicating that selection is acting in a manner similar to a frequency-dependent situation. Fitness the adaptiue peak: A brief remark on fitness 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; dq dq i.e., the equilibrium value for nq This is not the case for our twophase 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 dw value of q for which = 0) does not force nq = 0, although it is the maximum d4 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 Aq 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 diploid phases of deveiopment can make a considerable difference with respect to the retention of variability in a population. For poplations 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 equilibrium. Further, when populations with both haploid diploid phases are compared to populations with only repeating diploid phases, the conditions for maintaining equilibrium are broadened /or changed from the HJM. The differences between the two models is with respect to the values that the geometric arithmetic mean fitnesses can assume, as was previously described in the examples involving complete dominance diploid additivity.

13 SELECTION AT HAPLOID AND DIPLOID PHASES 207 I want to acknowledge the encouragement 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 commented on this manuscript: JOSEPH FELSENSTEIN, DANIEL L. HARTL, SUBODH K. JAIN, CURTI STROBECK. ALEXOPOULOS, C. J., 1952 BOLD, H. C., 1957 LITERATURE CITED Iniroductory mycology. Wiley, Inc., New York. Morphology of plants. Harper Brothers, New York. GLIDDON, C. C. STROBECK, 1974 Necessary sufficient conditions for multiple-niche polymorphism in haploids. Am. Naturalist 109: HALDANE, J. B. S. S. D. JAYAKAR, 1963 Polymorphism due to selection of varying direction. J. Genet. 58: HARDING, J., 1975 Models for gamete competition self fertilization as components of natural selection in populations of higher plants. In: Gamete competition in plants anb"s. Edited by D. L. MULCAHY. North-Holl Publishing Co., Amsterdam, The Netherls. HARTL, D. L., 1976 Stochastic selection in both haplophase diplophase. J. Mathematical Biology (In press)., 1977 Applications of meiotic drive in animal breeding population control. Proc. Intern. Congrecs Quant. Genetics (In press). HEDRICK, P. W., 1972 Maintenance of genetic variation with a frequency-dependent selection model as compared to the overdominant model. Genetics 72: , 1974 Genetic variation in a heterogeneous environment. 1. Temporal heterogeneity the absolute dominance model. Genetics 78: , 1976 Genetic variation in a heterogeneous environment, I1 Temporal heterogeneity directional selection. Genetics 84: HOEKSTRA, R. F., 1975 A deterministic model of cyclical selection. Genet. Res. 24: JAIN, S. K., 1975 Gametic selection in mixed selfing rom mating plant populations. In: Gamete competition in plants animals. Edited by D. L. MULCAHY. North-Holl Publishing Co., Amsterdam, The Netherls. PROUT, T., 1968 Sufficient conditions for multiple niche polymorphism. Am. Naturalist 102: SCUDO, F. M., 1967 Selection on both haplo diplophase. Genetics 56: Corresponding editor: R. C. LEWONTIN