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THE POPULATION GENETICS OF SPERM DISPLACEMENT

TIMOTKY PROUT

Department of Genetics, University of California, Davis, California 95626 AND

J0RGEN BUNDGAARD

Institute of Ecology and Geneiics, University of Aarhus, Alarhus, Denmark

Manuscript received January 12, 1976 Revised copy received July 30, 1976

ABSTRACT

This article reports the results of some sperm displacement experiments, as well as the results of a theoretical study of selection arising from genetic differences in displacing ability. The experimental work involved the use of three genetic marker stocks in double and triple matings. The speed of displacement following the matings was determined by scoring the progeny of each female daily. There were clear differences between strains in their displacing ability. It is shown how new information concerning the displacement process results when three markers are used; however, no new light is shed by these experiments on the mechanism of displacement.

The theoretical study of selection resulting from displacement uses a one- locus, two-allele model in which three diploid male genotypes confer different displacing abilities. The results indicate stable equilibria if (I) there is heterosis, and (2) there are certain nontransitive relationships in displacing ability among the different kinds of double matings.

Some evolutionary consequences are discussed in which sperm displacement is regarded as a form of sexual selection.

I N a number of insect species in which the female engages in multiple matings, it has been o’bserved that the eggs laid subsequent to a double mating are f e d -

tilized by a preponderance of sperm from the second male. This phenomenon has been called “sperm displacement” o r “sperm precedence.” The process has been studied intensely in Drosophila melanoguster starting with NONIDEZ (1920), and perhaps most thoroughly by LEFEVRE and JOHNSON (1962). The latter authors and FOWLER (1973) provide references for other other work with D. melanogaster. In addition, BECKENBACH (1975) has shown displacement in D. pseudoobscura and JOHN BRITTNACHER in our laboratory has demonstrated it in D. virilis. But the process is by means confined to Drosophila. A review by G. A. PARKER (1970), as well as other reports, reveals that of 20 insect species tested (including Drosophila), 16 exhibit sperm displacement. In addition to three Dip- terans other than Drosophila, three other orders of insects are represented. For example, sperm displacement is shown by the Coleopteran, Tribolium castaneum ( SCHLAGER 1960), the Orthopteran, Schistccerca gregaria (HUNTER-JONES 1960) and the Lepidopteran, Euphydryas editha (LABINE 1966). Genetxs 85: 95-124 January, 1977

96 'r. PROUT AND J. EUNDGAARD

These cases include rather diverse morphology of the female reproductive system, as well as some species that deliver sperm by means of a spermatophore and some that do not. This and the taxonomic diversity of its occurrence suggest that sperm displacement may be a widespread phenomenon among the insects at least.

The aspect of sperm displacement with which this article is concerned is the possibility that it can provide an opportunity for the operation of natural selection. This is suggested by the fact that in the many experiments designed to detect sperm displacement genetic markers are used, and a study of these data almost invariably reveals that some markers are better than others in the displacement process. Thus, in natural population of species which engage in multiple insemination, genetic variation of this sort should be subject to natural selection.

The following treatment is composed of two parts: experimental and theoretical. Part I reports a set of displacement experiments of our own, done for the purposes of demonstrating once again the effect of male genotype on displacement ability, and also to collect displacement data in a way relevant to its popu- lational consequences.

Part I1 constitutes a theoretical analysis of a simple population model of selection arising from differential sperm displacement.

PART I. EXPERIMENTAL MATERIALS A N D METHODS

We chose for our experiments three unrelated stocks of D. melanogaster in the hope of obtaining maximum differences in displacing ability. The stocks differed in convenient recessive markers: the double-recessive brown, scarlet (bw;st), the recessive brown ( b w ) , and Oregon-R wild type (+). All three stocks %\ere originally obtained from the California Institute of Tech- nology and have been maintained independently in our laboratory. The female parent in all crosses was bw; st so that the paternity of the progeny resulting from two or three different males could be distinguished.

Two sets of experiments were performed: one in which females were exposed to two different males in sequence, and one in which three different males were used.

In order to initiate both sets of experiments, 5-day-old virgin males and females were used to make the following three crosses:

bw; st P 0 x bw; st 8 8 bw; st P P x bw 8 8 b w ; s t P o x + 88

Each of these three matings was made en masse by putting ca. 100 0 0 and 100 6 6 into a 500 ml Erlenmeyer flask for 2 hrs. Previous experience indicated that this would yield a high rate of singly mated females (BUNDG~ARD and CHRISTIANSEN 1972). The flies were then etherized and each female was placed in a 2.5 cm x 10 cm shell vial containing 5 ml of an agar-yeast medium. Whether or not a successful mating occurred was determined by examining vials for the presence of larvae 36 hrs later. Those females which proved to have mated were then allowed to mate again by transferring them without etherization to a new vial into which three 6-day-old males carrying the second type of marker had been introduced. These were then given 24 hrs in which to engage in the second mating. Whether or not they did so was determined from progeny. After this 24 hr period, all females were transferred again, some into new vials without males (for the double mating experiment) and some into new vials containing three 7-day-old marker males

SPERM DISPLACEMENT 97

TABLE 1

Number of double and triple matings made, classified according to the sequence of male genotypes

1 F t d + bw; st + bw bw bw; st

Double matings 2nd d"

bw; s t + bw + bw; st bw

No. females 18 18 18 18 IS 18

108

Triple matings 1st d 2nd d bw + bw bw; st

bw; st + bw; st bw + bw + bw; st

3rd d bw; st 16 + 16 bw 14 + 11 bw; si 9 bw 9

75 ~

for the third mating. All females were transferred daily thereafter for I2 days, after which the transfers were made at two-day intervals. Those females which were presented with a third type of male were given 48 hrs in which to mate. That is, the males were transferred once again with females before being discarded.

Table 1 shows the number of females who, having produced progeny from the first mating, were given the opportunity to engage in a second mating (doubles), or both a second and a third mating (triples).

In the case of the double matings, all possible pairs of males were mated in the two reciprocal orders. In the case of triple matings, the three different males were used in the 3! = 6 different orders in the sequence. Hereafter, a particular mating sequence will be identified by A-+ B, where A is the first male and B the second. Triples will be similarly disignated A + B 3 C.

The offspring were incubated at 25", and all the progeny that were produced by each female in each vial into which she was transferred were classified and counted.

RESULTS

A. The number of double and triple matings: Table 2 shows the sums of the various types of matings which actually occurred. The total number of crosses that provided information is less than the number of matings (Table 1) because of death o r nonproduction of some females. In the double mating experiment, 79/106 females (74.5%) mated twice. In the triple mating experiment, the five females designated A + B * ? mated twice but stopped producing before it was possible to determine whether or not they had mated a third time. Also, 13 females skipped the second male but mated with the third, and these are designated A + C. The first two matings in the triple mating experiment

98 T. PROUT A N D J. BUNDGAARD

TABLE 2

Summary of various types of matings which took place+

A only A-+ B

Total

A only A + B A + B + ? A - + B + C A-C

Total

Doubles 27 79

106

Triples 1

10 5

42 13 71

* A 4 B or A B + C represent first and second males or first, secmd and third males, respectively. A + B + ? indicates cases where the female stopped producing or died before it was possible to be certain whether or not she mated a third time. The totals are less than the number of matings set up (Table 1 ) because of death or nonproduction of some females.

provided additional data on double matings precisely comparable to those of the double mating experiment. Classifying the A-, C mating with singles, there were 57/71 (80.3%) doubles and combining both experiments there were 136/ 177 (76.8%) double matings in all.

Not counting the A + B + ? group, of those females who engaged in two matings, 42/52 (80.8% ) engaged in the third mating.

Another statistic is of interest: olf the 71 females exposed for three days to males in the triple mating experiment, only one failed to mate two or three times. This suggests that a female D. melanogaster may mate many times during her lifetime.

The summary data of Table 2 were broken down by genotype as listed in Table 1 and were found to be statistically homogeneous for both double and triple matings. However, larger numbers would be required to detect specific mating preferences, a subject with which this study is not primarily concerned.

B. Preliminary aspects: All twelve kinds of matings are shown in Figure 1 and Figure 2. These show the proportion of offspring in the daily counts from the second and third male. Each line on a given graph represents the progeny of a single female. Before attempting to characterize and compare the results of all these experiments, we will first discuss some basic aspects of displacement experiments that cannot be deduced from Figures 1 and 2. For this purpose we have chosen just one of the crosses, namely bw + +, to analyze in some detail.

Of the 18 females who mated to bw and were then exposed to +males, 13 actually mated with them, and 5 did not. Most of these 13 females produced progeny yielding information through day 12. This is a typical pattem and is one of the reasons Figures 1 and 2 show the results only through day 12.


1.00

f .50

0 1 2 6 7 8 9 IO 1 1 12

d a y U I 2 1 3 1 4 5 6 7 0 9 10 I 1 12

n dor I +--. bw bw++

1 2 1 3 1 4 5 6 7 8 9 10 1 1 12

U do*

bw -D bwrt

100

f .SO

0

I 2 1 3 1 4 5 8 7 8 9 1 0 1 1 12

U d.v

1 2 1 3 1 , 5 6 1 0 9 10 1 1 12

U d - v

FIGURE 1.-Each line on each graph is the fraction of offspring, f , due to the second male produced by a single female. A single point is based on about 25 offspring in the early part of the experiment and about 10 at day 12. The symbol ‘‘I 1 ” on the abscissa indicates the

24 hr period when the female was confined with the second male. The symbols “A’ 9 “B” mean “A” was the first male and “B” the second. The numbers in the upper left part of the graph are the number of females scored. The number of lines on the graph are less than the number of females, because of frequent coincidence of line segments.

I1

Most experiments on sperm displacement simply report the net lifetime production after the second mating, which can be misleading. For example, in the present case, the 13 doubly mated females produced 2,505 offspring of which 1,553 were 4- and 952 were bw, or 62% + and 38% bw. This result could be interpreted as being due to considerable sperm mixing, with only slight displacement in favor of the second male. However, this conclusion could be disputed on the grounds that there was a 50:50 mix of sperm (no displacement) , but that bw was less viable (for example, see the data and discussion of KAUFMANN and DEMEREC 1942). In the case at hand, we have independent evidence that viability effects in our data are slight. In any case, it is more revealing to analyze this 62: 38 lifetime result by examining the progeny, day by day.


bw- +- b w r t bw- bwrr - + ,.U 0

f . 5 0

0

++ bw-bwst +- bwrt + bw

FIGURE 2.-As in Figure 1, each line shows the progeny of a single female and ‘&A”+ “B” + “C” means “A’ is the first male, “B” the second and “C” the third. The dashed line shows fraction, f, produced by the 2nd male and the solid line that by the 3rd male. The symbol ‘‘l-..I’’ is as in Figure 2 and indicates the period during which the female was

confined with the third male. I1 I11

The bw + 4- cross in Figure 1 shows a great deal of variation among individual females, mainly in the initiation of displacement (although some females showed a gradual displacement), and there were some cases of recovery of the first sperm. These results are summarized in Figure 3, which shows the composition of offspring (frequency of +) averaged over females for each day. The point is that neither bw -+ f Figure 1 nor bw f in Figure 3 can be character- ized as a simple 62:38 mixing of sperm. Quite clearly the sperm from the two males are partitioned in some way and differentially utilized through time; i.e., a process of sperm displacing is occurring. It is easy to imagine delayed or slow displacement resulting in a 50: 50 lifetime result, which would erroneously suggest no displacement process at all. Figure 3 is misleading also in that it obscures the heterogeneity among individual females. However, at the population level it is completely sufficient, since it compares the average effects of the two male genotypes, f and bw, when placed in the order bw + -I-, on the composition ot subsequent progeny. In this sense, this composite effect over females results in “slow”


. 60

f .40

2 4 6 8 10 12 days

FIGURE 3.-Frequency, f , of + offspring in bw + + cross: These are unweighted means over the 13 females. Means weighted by progeny numbers produce only minor alterations in the above. These data were compiled from individual females shown in the bw + f graph in Figure 1.

displacement of bw by + progeny as compared to “rapid” displacement of -I- by bw in the f + bw cross (see Figure 1). In the section on population theory we shall use the words “ s ~ o w ’ ~ and “rapid” to describe the average effects of male genotypes on progeny composition, as illustrated in hypothetical graphs corresponding to Figure 3.

A similar phenomenon showing changes in sperm utilization called the “brooding effect” has been observed by a number of workers when females are mated singly to males heterozygous for various kinds of chromosomal anomalies (CHILDRESS and HARTL 1972). In this case the segregant sperm are used differentially in successive broods. The similarity between this brooding effect and the displacement process may be more than coincidental. However, we wish to emphasize the difference, namely, that in our experiments the two types of sperm were delivered at different times by different males who were homozygous for marker loci and were chromosomally normal.

Further analysis of the bw 3 + case illustrates another important aspect of sperm displacement first established by LEFEVRE and JOHNSON (1962), namely, that in doubly inseminated females, the second sperm do not simply add to the total supply of sperm available, but actively displace the sperm from the first mating. The five females in the bw f case who did not mate with the second male allow a comparison to be made between singly and doubly inseminated females with regard to lifetime productivity. We know that 10 of the 13 doubly inseminated females stopped producing before they died (of the remaining 3 we cannot be sure). All 5 singly mated females stopped producing before they died. If they stopped producing because of depletion of sperm, then the lifetime production of each type should reflect any differences in available sperm. Table 3 shows the mean number of offspring (lifetime) per females for the two types. The two means do not differ from each other statistically ( t I3 = .46, P = .35).

The number of females involved here is small. We merely wish to demonstrate that our data are consistent with the phenomenon firmly established by LEFEVRE and JOHNSON (1962) , who performed extensive experiments specifically designed to answer this question. They concluded that the storage organs of the

102 T. PROUT AND J. BUNDGAARD

TABLE 3

Lifetime production of offspring per female in the bw + + mating for those females who did not mate with + males (bw only) and for those who mated with bw and then + males (bw +)

Type of female No. females Offsprindfemale S.E.'

Mated to bw only 5 182.2 14.2

Mated: bw + + 10 201.2 27.7

* S.E. = standard error.

female have a fixed capacity and that each male delivers more than enough sperm to fill them. Sperm from the second male somehow physically displace the sperm already deposited there by the first male.

To these observations oln lifetime production, we add some observations on the pattern of absolute daily production. In Figure 4, absolute production of the doubly mated females (bw -+ +) is given by the polygon outlined by the solid line. The dotted line partitions the polygon into its bw and -I- components. The dashed line shows the daily production of the singly mated females (to bw). This latter line roughly follows the total production of doubles rather than that part of the doubles composed of bw only. Thus, it appears that the sperm from the second male are not simply making up far a dwindling supply from the first mating, at least as this is expressed in the number of fertilized progeny.

2 4 6 8 10 12 days

FIGURE 4.-Absolute progeny numbers per day per female of the females who mated to bw only, dashed line, and those who mated doubly bw + f , polygon. The polygon is partitioned to show the bw us. + composition of the progeny resulting from double matings.


This preliminary discussion, using the bw 4 + mating as an example, shows that (1) it is necessary to follow the displacement process through time, but not necessarily through lifetime and (2) it is sufficient to present only the relaitve composition of the progeny as we did in Figures 1 and 2, rather than absolute numbers of progeny.

C. Main results: Figure 1 shows the results of the double matings, each line on the graph being the proportion of offspring from one female due to the second male as a function of time. On the abscissa. the 24 hr period during which the second mating took place is indicated by “11.” For the triple matings in Figure 2. the dashed line gives the proportion due to the second male. and the solid line that due to the third m?le. The periods when the second and third matings occurred are indicated on the abscissa by “11” and “111.” The composition of the progeny ( f ) is recorded over the midpoint of these time intervals. It can be seen that in some instances the second mating and sperm displacement occurred so rapidly that the vial in which this took place produced offspring only from the second mating.

We first call attention to the remarkable individual variation among females already mentioned in connection with the detailed analysis of the bw -, + case. Within many types of crosscs, both double and triple- almost any pattern can be found: immediately or delayed initiation of displacement; “rapid” or “slow” displacement, rccovery of first sperm or not; and occasionally the rapid achievement of what appears to be an equilibrium mixture of sperm.

In spite of this heterogeneity, we fec>l that two generalizations are warranted: (1) There is a clear tendency in ever!- cross for the sperm from the last mating

to displace the sperm from the previous mating. This pattern carries over from the double matings to the triple matings in which the second sperm commence to displace, but are suppressed by sperm from the third mating.

(2) There are clear average differences between male genotypes with regard to the intensity with which the displacement process is expressed.

Examining just the double matings, it is quite clear that h i ; s t is much more effective in displacing + than the latter is in displacing bw; s t . Not much less dramatic is the superiority of bw in displacing + as coml’ared to the ability of f to displace bw. The strains hw; s t and bw are nearly equal: but there is a suggestion that hw is somewhat weaker than hw; s t . In fact, this simple inspection of the graphs suggests a transitive relationship hetween the three strains; thus,

bw; st 4-

Inspection of the triples shows that this relationship carries over fairly well. The 4- strain is definitely weak, whether it is second or third in the sequence. Both hw; st and hw are much stronger in either position, but in the triple matings it is not clear which of the two is stronger. In what follows, the emphasis will be on double matings. The data on triple matings are used here merely to justify our concentration on the properties of double matings. The results of the triple

1 04 T. PROUT A N D J. BUNDGAARD

matings suggest that in progressing from doubles to multiples there will be no rad- ical change in basic rules of the system, but merely complications in the details.

A crude measure of the intensity of the displacement process is the time, T, required to accomplish 50 % displacement. This can be estimated graphically for each individual female. Although this may not have much meaning in terms of the displacement mechanism, it is objective and provides a variance for the purpose of statistical testing. In Table 4 the means and standard errors of T for each of the double matings are given. The crosses are grouped in pairs which are reciprocals (A -+ B, B 3 A), and the column headed P is the probability associated with a “t” test comparing the means of the two. In all cases, the reciprocal matings showed statistically significant differences.

The next column records K = 1 / ~ , so that the intensity of displacement can be conceptualized in terms of some sort of rate. The last column gives the ratio of K values for the two reciprocal matings (these are computed directly from the T values). Thus, for the bw; st us. 4- pair, bw; st is 14.36 times “faster” in displacing 4- than + is in displacing bw; st.

By this method of quantification, the system of crosses shows reasonable internal consistency: if bw;st is 14.36 times better than + and bw is 7.47 times better than +, then one might predict that bw; st should be 14.36/7.47 = 1.92 times better than bw. In fact, bw; st is 1.88 better than bw. These calculations provide objective support for the judgment made earlier from graphical inspection that the displacement characteristics of the three strains appear to exhibit transitivity.

Table 5 shows similar calculations for the triple matings. Here the rate, K, was calculated for the displacement due to the last male us. the first two males. These results indicate that both bw; st and bw are about equally superior to f, but the superiority of bw; st over bw has disappeared at least when looked at this

TABLE 4

Quantification of displacement intensity for double matings by graphical determination of the time r i for the achievement of 50% displacement

hlating N’ TT S.E.S PI K = 1/r Kl] ratios

.I6\ P < .01 2.38\ 14.36 + + bw; s t 17 .42

bw; st 4 + 9 6.03 1.27) .I71

++bw

bu,+ + bw + bw; st

bw; s t + bw P = .03 i’7g} 1.88

.65 .95

14 .56

I2 I .05

* N = the number of females on which determinations of ri were made for each mating. + r = mean over females. SE. = standard error. P = the probability associated with a 7” test comparing T between the two reciprocal

matings. 1 1 “Ratios” = larger K + smaller K (computed directly from T’s).

S P E R M DISPLACEMENT 105

TABLE 5

Quantification of displacement intensity for triple matings

Mating N 7 S.E. P K i / ~ K ratios

b w + f + b w ; s t

bw+ b w ; s t + f .75} 3.73 .20

8 1.34

10 5.00 .58J

P < .01 ’63} 3.60 .29 .’”) .18

’”\ P = . 6 0

b w ; s t + + + b w 7 1.59

b w ; s t + bw+ f 6 5.73

.“1 .90 ++ bw+ bw;s t 7 2.41

++ bw;s t+ bw 4 2.18 .18) .&J

The calculations are similar to those of Table 4 except that T was determined for the third male us. the first two males. The triple matings are paired so that the last two matings are the reciprocal matings reported in Table 4. The “Ratios” are the same as in Table 4 except that the last one is the smaller K -+ larger K .

way. It can be argued that the bw; s t superiority to f might be dampened by the presence of bw in the first mating. However, we feel that it would not be fruitful to attempt a serious analysis of the triple matings a t this primitive stage of under- standing of the processes involved.

DISCUSSION O F E X P E R I M E N T A L RESULTS

Experiments such as ours, in principle, could conceivably shed some light on sperm displacement mechanism. However, there are too many uncertainties in cur data, arising from the way in which we collected them. The heterogeneity among individual females must arise in part from the fact that during the 24 hr period allowed for the second mating, and the 48 hr period for the third, there must have been variation in the time when the mating took place; furthermore, some females may have mated more than once with the same male during these intervals.

Factors such as these must affect, to some extent, the subsequent pattern of displacement. In fact, it is possible that there could be some biasing of average genotypic differences in apparent displacement due to females more readily mating with one genotype than another in the second mating. However, we feel that such a bias could not explain all of our results for two reasons. First, we consider average differences among male genotypes with regard to general mating success. If a second (or third) mating occurred, mating ability could influence the time of that mating and also the number of additional but undetected matings during confinement with the female. The only information we have on mating success is the number of females each male genotype was able to induce to re- mate. Although there are differences among genotypes, they were not pro- nounced enough to be detected statistically, and so we judge they were not pro- nounced enough to have a strong influence on apparent sperm displacement ability. Secondly, we can compare two matings where the second male was of

106 T. P R O U T A N D J. BUNDGAARD

the same type, but the first males were different. Here the average time of the second mating and the number of matings with the second male should be the same. When this is done, there are still differences depending on the first male. For instance, in the double matings, displacement was slower in bw; st + f as compared to bw --f +. Also, bw; st + bw was slower in displacement than + --f bw. This observation carries over to the effect of the second mating in the triple matings at least for the case where + is second. We therefore feel confident that in spite of the uncertainties mentioned, as well as some genotype biases, our data do demonstrate differences among the three male genotypes in the actual sperm displacement process.

It is worth noting that more refined double mating experiments involving the six possible crossings using three genotypes might shed some light on the mechanism of displacement, because these crosses force the recognition of two separate properties of a given genotype: ( 1 ) its ability to displace previous sperm, and (2) its ability to resist being displaced by the next type of sperm. The first property is revealed when the first sperm are the same and the second sperm different, as for instance bw; st + -+ and bw; st -+ bw. Here differences are due to differences in displacement ability, i.e., bw is better than +. The second property is brought out by the type of crosses mentioned earlier, where the second sperm are the same and the first sperm different. Thus, the comparison between bw; st -+ + and bw -+ + shows bw; st better able to resist + than is bw.

To be sure, these two properties could be manifestations of the same physi- cal characteristic of a type of spermquant i ty or quality-but this would have to be determined by experiment. Also, using three types of sperm raises the question of transitivity, already alluded to. Such a question is not defined with just two types. Finally, these experiments suggest the question of how much displacement would occur if the second sperm were of the same type as the first. This is not a trivial question. It has already come up as an important unknown in our own experiments. But, more important, when a worker simply wishes to ascer- tain whether or not a species exhibits displacement and if so, the intensity of it, then essentially he is asking what is the normal situation, i.e., what happens when a normal mating by a “wild-type” male is followed by another one? This would not be a problem were it not for the fact that the probe, whether sterilized sperm or genetic marker, has its own peculiar displacement characteristics.

We wish to emphasize that although we, by necessity, have referred to the three strains by the markers they carry, we do not attribute the difference to the markers themselves. The formal genetics of displacement differences is another interesting question which one of us (JB) is now pursuing.

For the present purposes, we deliberately chose stocks with independent his- tories, hoping to obtain maximum differences in displacing ability. Our primary purpose was to display such differences hoping to provide some biological base for our theoretical treatment. We were also interested in the relationship among three types in order to gain some information relevant to a three-genotype model. Finally, for population purposes, it was important to have some information concerning how displacement manifested itself during the course of a female’s egg-


laying period. We think the data serve these purposes, and, together with several reports of frequent multiple matings in natural populations (ANDERSON 1974; COBBS 1975; MILKMAN and ZEITHER 1974), provide biological justification for the investigation of a simple population model. This is the subject of the next section.

PART 11. POPULATION THEORY

In this section we will construct and analyze a population model in which the genetic differences controlling displacement are of the simplest kind. There will be just two alleles, A and a, and the three genotypes which they produce, AA, Aa, and aa. There are two basic biological assumptions to be made. First, we assume the displacing characteristics of a given male type are conferred by his diploid genotype, rather than by the haploid genotype of the sperm. Our data and those of most others, performed with males from homozygous marker stocks, provide no direct evidence on this question (although obvious experiments could be done). We base this assumption on the usual behavior of single locus hetero- zygotes, which simultaneously place two genetic kinds of sperm in competition, but in a single mating. In such experiments one usually finds Mendelian ratios when cultures are raised under optimal conditions; and, in crowded cultures, departures from Mendelism can usually be attributed to viability effects rather than to some form of meiotic drive. In short, meiotic drive is a rare phenomenon. Biologically, we think of differences in displacement ability as being due to the amount of sperm produced, to the amount or quality of seminal fluid, or even to differences in copulation performance, all of which could arise from the diploid male genotype as well as from environmental effects.

Secondly, we assume that females in natural populations die before they use up their supply of sperm, and that there exist different male genotypes that vary in their displacing ability but that, on a single insemination, can fill the female storage organs to capacity and thus supply sufficient sperm for her lifetime under population conditions. Thus, any variation in absolute progeny number in single or double matings is solely a female attribute. Of course, there exist male phenotypes which are sterile, or deliver drastically reduced amounts of sperm, and which would suffer low fitness in single matings because of this, but we are not concerned with this kind of sterility in this treatment.

Thus, we are studying a component of fitness which can come into being only if there are multiple inseminations in which different sperm types compete with one another only when confined together in the same female. We restrict the treatment to double matings. A. The General Model: In order to introduce the basic structure of the model, we first considered just two male phenotypes, A and B. These are mated to an arbitrary population of females, including double matings, and then the relative contribution of the two types to the offspring will be evaluated. Several interpre- tations of the resulting formulation are possible; we commence with the most simplistic one.


TABLE 6

Probabilities of uarious kinds of matings

A B A + A A + B B + A B + B

(1 - M ) z ( 1 - M ) y M x ~ M x y M x y MY2

Consider a cohort of females which emerges into a population of males composed of Type A with probability 5 and Type B with probability y; (x 4- y = 1). The females mate either once with probability ( 1 - M ) or twice in rapid suc- cession with probability M . We assume the females mate at random with the two types of males. Table 6 displays the types of matings and their probabilities. After these matings the females proceed to lay eggs at a constant rate until they die. It is now necessary to construct a displacement parameter which evaluates, in the mixed double matings (A + B, B +. A), the relative contribution of the A us. B sperm to the total eggs laid after these matings.

Figure 5 depicts the displacement process in the two reciprocal matings (A + B, B + A). These two graphs each represent the averages over all individual matings of a particular type, analogous to the data shown in Figure 3 of

t ime, t 1

1.0

f .5

0

t i m e , t 1

FIGURE 5.-Hypothetical displacement curves where A is favored over B over the two reciprocal matings. The solid line is the frequency of A sperm which is from the first male in the A + B cross, and from the second male in the B + A cross. The dashed line is the frequency of B sperm.


the experimental section. The solid line in each case shows the relative composition of eggs fertilized by A sperm. In the A + B matings this will be a decreas- ing function, say, f ( t ) , while in the B -+ A mating it will be an increasing function, say, g ( t ) . Eggs fertilized by B sperm will be the complement of those fertilized by A sperm in each case. Consider the composition of the accumulated fertilized eggs left behind when the female dies at time t = T. The probability that one of them is fertilized by an A male in the A + B cross is the mean of

m f ( t > or

Similarly, for the B -+ A mating this probability is

Since under random mating, the two types of mixed double matings will be equally frequent; namely, z y each (see Table 6) a sufficient parameter for the probability that an egg is fertilized by A sperm can be constructed by weighting the two kinds of matings one-half each. We denote this probability KA (AB) and define it

1 1 T 1 K A (AB) = - [- J f ( t ) d t + - JTg(t)dt] .

T O T O Thus, no matter how “fast” or “slow” displacement occurs, if it is reciprocal, defined g ( t ) = 1 - f ( t ) , then KA (AB) = 1/2. If this is not the case, then displacement is differential, and one type or the other will be favored over the two matings. The graphs in Figure 5 suggest that A is favored; so that KA (AB) > l/e.

We now construct the model in its simplest form. Denoting the total probability that an egg is fertilized by a sperm from an A male by F ( A ) , then from Table 6 we have

F ( A ) = ( ~ - M ) x + M x ’ + M ~ x ~ K A ( A B ) . (1) The several restricting assumptions made in arriving at this equation can be

relaxed in several ways without changing its form. Egg-laying rate does not have to be uniform so long as the male type does not influence egg laying. If more eggs were laid earlier than later, then the value of K , (AB) would be influenced by the early part of the displacement process. The parameter M can be interpreted in several ways. If all females mate twice, then (1 - M ) is the fraction of eggs laid before the second mating. The value M may have a distribution over females, and if so may be regarded as a mean in ( 1 ) as long as K A (AB) is not affected. Equation ( 1 ) also applies when females mate many times, with the restriction that the interval between matings is long enough to permit displacement to be completed before the next mating. In this case all matings are, in effect, doubles. This situation is depicted in Figure 6 in which A and B males are made to alternate merely to illustrate displacement.

110 T. P R O U T AND J. BUNDGAARD

B - D A A 4 6 B+A A+B B*A

l i m e

FIGURE 6.-Repeated matings where all matings are effective doubles because displacement is completed before the next mating. The solid line is the frequency of A sperm in eggs laid and the dashed line the frequency of B sperm. A sperm are favored over B.

We now generalize (1) to accommodate any number of male types and also restructure the displacement parameter. Let there be n male types A,, A,, . . . Ai, . . . A,, with probabilities xl, xz, . . . x i , . . . 5,. Still restricting females to single or double matings and denoting by F ( i ) the probability that an egg is fertilized by a male of type Ai, we have

(2) F ( i ) = (1 - M ) x , + M [ X I + 2 x ~ ~ j K i ( i j ) ] , j = l i f i

where k, (ij) = conditional probability that an egg is fertilized by a male of type A,, given that double matings (both kinds) with males Ai and Aj occurred.

By rearrangement, ( 2 ) naturally yields a more useful displacement parameter.

i # i where dii = 2Ki ( i j ) - 1.

As already indicated, the order of i, j in the parenthetical part of Ki (ij) is arbitrary, since this is the outcome over both reciprocal matings between Ai and Aj. The complement of K , ( i j ) is K j ( i j ) , which is the probability that an egg is fertilized by type Ai. The order of i, i in d,j is not arbitrary. We have defined dij above, but d,, is defined:

Furthermore, by the substitution K j (ii) =I - K i ( i j ) in the above, we have the following useful complementary relationship:

dji = 2 K j ( i j ) - 1.

d.. - - d . . 3 % - 21'

This new parameter, d,j, expresses displacement competition in the following

way: if neither type is favored, K i ( , j ) = -and dij = 0. If i is favored over i, then:

1 2

and

whereas if i is favored over i, then O < d ; j < l .

and

1 2

0 < K i ( i j ) I - ,

- I < d i j I O .


These considerations will now be used to assess the effects in a population where displacement is under genetic control. B. The genetic model: Consider two alleles A , a and three genotypes AA, Aa, and aa, with probabilities of the male genotypes xl, x2 and x3, respectively. We assume the amount of displacement in double matings is independent of female genotype. We first consider the relationships of the three male genotypes when mated to any female type. F ( A A ) , F (Aa) and F (aa) will be the probabilities that an egg is fertilized by sperm from AA, Aa or aa males, respectively.

Substituting into (3) and writing in expanded form, we have F(AA) = ~ [ l 4- M(52d12 +&d13)] (44 9

F(Aa) = x2[1 -t M(sldzl -k d 2 3 ) 1 (4B) 7

F(aa) = x3[1 -t M(x2d32 4- xld3d1 (4C) * There are only three independent dSj since d,, = - d,,, etc. The three dij result from the three possible comparisons between three different genotypes, taken two at a time.

The problem now can be formulated as a special case of sex-dependent selection where selection is in one sex only, namely males. The system does not yield conventional fitnesses (of males); the factors in brackets in (4) we denote by ai and are related to conventional fitnesses as follows

Wid

We call them “fitness factors” and, to be precise, define them as follows:

F(AA)

F(Aa)

al = 9

a2 = 9

5 2

5 1

( 5 )

5 3

Finally, it is possible to show that, as usual, random pairing of male with females genotypes is equivalent to the randomm union of their gametes, so that the variables xl, x2 and x3 can be transformed into male and female gene frequencies.

TABLE 7

Scheme showing system a9 a cuse of sex limited selection ~~ ~ ~~~

Gene pool A a Generation n P r Genotypes before selection AA Aa aa

21 x2 23

Females Males

Fitness factors 1 1 1 a1 a2 a3

Genotypes after selection 21 z2 x3 - v l x2a2 23a3 Gene pool Generation n + 1 P‘ r’

1 1 2 T. P R O U T A N D J . B U N D G A A R D

These will be denoted by r for male gametes containing allele A. and p for female gametes containing A . Table 7 exhibits the system when formulated this way. There appears to be no way to reduce the three d, , parameters to two by parameter normalization.

When the system is looked at this way, the problem becomes an analj-sis of two simultaneous recurrence equations in p and r which are

1 2

p’ = 51 + - Z? , 1 2

r’ = alzl + - azz2 , where,

p‘.r’ = gene frequencies in the subsequent generation.

Substitution for al, a , gives

where .xl = pr? zL’ = r + p - 2pr: 5.: = 1 - p - r + p i .

It is useful to write (7A) (but not 7B) in terms of p , r , thus

1 1 I 3 2

p’ = - p + - r

C. Analysis of the Model: With selection in one sex only, a t equilibrium the male and female gene frequencies are equal to each other, and because of random mating the unselected genotvpes are in Hardy-Weinberg frequencies. This is easily shown by examining equation (7A)’. At equilibrium pf = p = I;.

Therefore, 1 1 2 2

6 = -6 + - r;

A SO, r = i;,

and substitution into x i will give Hardy-Weinberg frequencies for the gmotypes. Substituting I; for r, p and r‘ in equation (7B) gives

Mi;i[I;+i,2 + &?d,, + i;&i,,] = 0 , (8) where, G = l - i ; ,

A necessary condition for a nontrivial equilibrium is that M > 0. which is to’be expected. It is of some interest that M plays no role in the equilibrium value, I;.

Removing the two trivial equilibria I; = 0, I; = 1, leaves the following /?all’ + i%L:{ + /?&i!l:3 = 0 . (9)


This is quadratic in Z;. whose coefficients are functions of dij. If fi is transformed to gene ratio. U

then the following quadratic results with di j themselves as coefficients,

- C*&I + Cd,, -I- d,, = 0 , (10) with solution,

(11) 2 = - d13 \ / d l S ' 4d2,d,:3

-2d,,

Thus. there are a t most two equilibria. The existence of an equilibrium requires that

2 > 0 .

Examination of (11) [or (IO) when d,, = 01 yields the five types of cases shown in Table 8 that will yield equilibria. Except for Case 5 , it is sufficient to know only the signs of d ; j , which are given in the first three columns of the table. Each row specifies a case. The next column of triangular diagrams in the table shows thc relationships between the three genotypes specified by the signs of the

TABLE 8

Conditions for equilibria and their siabiliiy characterislies

R ~ I a 1 i i m 4 i i p Niniilier of Care d,, 4, d,, lietween genotypes equilibria Stability

I + + one stable A A aa

Aa

AA< aa 2 0 + - 0 . q one stable

Aa

A A aa one unstable 4 I\ 3 - -

Aa 4 0 - + one unstable

A A > Qa

Aa

A A < aa 5 - + - 4 4 two* one stahle, one unstable

* Additional condition: dPlj + d,,d,, > 0.


dij. The next column gives the number of equilibria and the last column their stability. The determination of stability will be discussed presently. Case 1 is simple heterosis, and there are no restricts on d,, in order for one equilibrium to exist. Case 2 is a kind of mixture of dominance and heterosis. This yields a single equilibrium as does the symmetrical case, not shown, where Aa = aa, A A < Aa, and A A > aa. Case 3 is negative heterosis and Case 4 is analogous to Case 2, being a mixture of dominance and negative heterosis. Case 5 seems to us unlikely, but it is a formal possibility. There is a symmetrical case not shown. This case yields two equilibria providing the discriminant of ( lo) , dlS2 + 4.d,,d2,, is greater than zero. (This condition is automatically satisfied in Cases 1-4). Note that this case requires d2, and d,, to be of opposite sign. As 14d2,d,, I increases, the two equilibria come closer together until there is just one equilibrium when the discriminant just equals zero. With further increase in 14d2,d2,I the roots of (IO) are imaginary, and there is no equilibrium, so that the allele A is always fixed in Case 5. In the symmetrical case, not shown, the allele a is fixed. One other

0 .2 .4 .6 .8 1.0

P FIGURE ‘/.-This and Figures 8 and 9 show the behavior 01 the population point in the r, p

plane, where r female gene frequency. Here are shown trajectories for maximal directional selection where M = d,, = d,, = dI3 = 1. Genotypes AA, Aa, aa are indexed by 1, 2, 3. The trajectories originate on the boundaries of the state space. In one generation, movement is to the next arrowhead until the “equilibrium path” is achieved. Rate of movement along this path is not indicated by arrowheads but is much slower and can be judged from curve A in Figure I O .

male gene frequency and p

SPERM DISPL.ACE1LIENT 115

0, d,, < 0, d,:$ > 0 case not shown always giving complex roots of (10) i s d,:$ (or the symmetrical case d,, > 0, d,.: < 0) . Thus,

.IA: A A =aa

This case results in fixation of the A allele. In the symmetrical case the a allele is fixed.

Although the set of equilibria p = r = 1 falls in a one dimensional space. naniely the p = r line. examination of the local stability of an equilibrium point must provide for small perturbations about the point into the two dimensional p. I' plane. This state space, 0 5 p 5 1, and 0 5 r 5 1: is shown in Figures 7, 8 and 9.

The Jacobian of equations (7) has the following form

1 1 2 - 1 r

1.0

.a

6

r

.4

.2

0

\ \

\ \

\ \

\ \ y \

.2 .4 .6 .E 1 .o P

0.71

FIGURE 8.--This is a numerical example of the intransitive Case 5. where M = 1, d,, = d2:? = ---0.1 and d , . ; = 0.5. The solid line is the "equilibrium path" where the population point spends niost of the time. U G position of the unstable equilihriurn; S position of stable equilibrium. The dotted line . is the r = p line for comparison. The dashed lines are isoclines for male niean fitnrcs, w 8 as defined in the text.


r

1.0

.8

.6

.4

0

0 .2 .4 .6 .8 1 .o P

FIGURE 9.-This is the case of balanced heterosis where M = 1, d,, = d,, = 1 and d,, = 0. Stable equilibrium is at p = r = 0.5. The “equilibrium path” is shown. The arrowheads indicate direction of movement only. Their spacing is arbitrary and is not meant to indicate rate of movement, The isoclines are lines of equal homozygote fitness which is the same for the two homozygotes, AA and a ~ .

where F is the last term in brackets in (7B) that is

F = M ( 1/2XiXzdiz + 1 / 2 ~ 2 ~ & 3 + xix3di3)

Because of the symmetry in F

aF aF aP P - r

We let f denote their common value, evaluated at r = p = 6, and

-I - =TIp=;

f = i/edzi ( e - 34) + 1 / 2 & 3 4 ’ (4 - 36) + d13 $Q(B - $> . (12) J yields eigenvalues

X , = O , X = l + M f .

It is of interest to note that although the equilibrium value (1 1 ) is not a function of M , its stability is. Substitution of Z;/(l + Z;) for 3 into f, in principal, would allow evaluation of in terms of dij, but we found the resulting expression to be intractable, except for special cases. However, evaluating at the two trivial equilibria, p^ = 0 and p^ = 1 permits one to infer the nature of the stability of the internal points. The eigenvalues of $ = 0 and 6 = 1 are

hp o = 1 + M Gd23 (13A) (13B) = 1 + M % d z l .


= o, Xp = ,) > (1, I), are sufficient conditions for the polymorphism to be protected. These conditions are that ( d 2 3 , d z l ) > (0, 0) which are precisely the existence conditions for Case 1, or heterosis. We infer, therefore, that the single internal equilibrium produced by heterosis is stable. The conditions for both trivial equilibria to be stable, (ha = ,,, X,=l ) < (1,l) are that (d23, &) < (0,O) which are the existence conditions for negative heterosis, Case 3. We infer this point to be unstable. The two “dominance” cases, 2 and 4, have A, =, = 1 because d,, = 0. This merely reflects the fact that when q is small, the homozygotes, aa, are essentially absent and the two remaining dominant phenotypes, AA and Aa, are neutral with respect to each other. Fortunately, the question of the nature of the stability of the internal poht can be determined, because when d,, = 0, f and ia can be evaluated at the internal equilibrium point. When d,, = 0

The conditions for boith trivial equilibria to be unstable (

and M d23d3I2

(d23 + d31) A , = 1 -

For Case 2, where d,, > 0, 0 < A, < 1, so that the point is stable. For Case 4, where d23 < 0, A, > 1, so that the point is unstable.

Two of the existence conditions for Case 5 are that dzl < 0 and dZ3 > 0. This means that of the two trivial equilibria, Z; = 0 is unstable and p^ = 1 is stable. For the case where there are two real roots, this means that the internal equilibrium point nearer p^ = 0 is stable while the one nearer p^ = 1 is unstable. When there are no real roots, there are no equilibria, and movement is simply away from $ = 0 and toward p^ = 1.

Figures 7, 8 and 9 show trajectories for some numerical cases. Figure 7 shows maximal directional selection ( M = d,, = d 2 3 = d13 = 1 ) . The figure reveals a very well defined equilibrium path which the population essentially achieves

I

10 20 30 40

generations

FIGURE 10.-Mean gene frequency between the two sexes; i.e., f i = i/z ( p + I) as a function of generation number. Curve A is for maximal directional selection, shown in Figure 7. Curve B uses values derived from experimental data where d,, = 0.16, d,, = 0.58, d,, = 0.79.

118 T. P R O U T A N D J. BUNDGAARD

within, at most, two ger-erations, even when initiated on the margins of the state space. Such a path appeared in all of the cases we studied numerically and is always located near the p = r line. Figure 8 provides a numerical example symmetrical to Case 5 with two equilibria, and Figure 9 gives a case of balanced heterosis. The isoclines on Figures 8 and 9 will be discussed presently. Figure 10 shows the average of male and female gene frequencies as a function of time, in generations, for two cases of directional selection. One is for the case of maximal directional selection shown in Figure 7. The other is a case where the dij were obtained from our experimental data. We obtained Ki (ij) by empirically inte- grating the results for each female for days 3, 4 and 5, averaging for each cross type, and then identifying bw; s t with AA, bw with Au, and + with uu. This produced the following values:

Aa ( bw)

d,, = .16 d,, = .58 AA ua

d,, = .79

These values were calculated simply to show how raw displacement data such as shown in Figure 1 translate into a force for moving gene frequencies. Thus, the displacement relationships of Figure 1 result in the requirement of about 40 generations to move an allele from 0.1 to near fixation.

Finally, we have one general result for multiple matings, which, although not very useful, we nevertheless set forth. We have the colnditions for a trivial equilibrium to be unstable where the multiplicity of the matings, m, ranges from m = 2,3 . . . to n. As p + 0, a mixed mating of multiplicity m will be of only one kind, namely one heterozygote Aa, index 2, and the remaining m-1 males of type ua, index 3. Denote the probability of sperm from the heterozygote male in such a mating by

If the heterozygote sperm is not favored or disfavored, then l/m of the sperm should come from him. The condition that Au is favored and that therefore p + 0 is unstable is

(bw; s t ) (+I

Kz (293-1) .

,n=2 i ~ m [ m ~ z ( 2 , 3m-1) - 11 > 0,

where, M , probability of mating of multiplicity m. In an analogous way, the condition that p + 1 is unstable is

z Mm[mKz(2, 1-1) - 11 > 0 . m=2

These two conditions together are therefore the conditions under which the polymorphism is protected when there are multiple matings. D. Remarks: The following remarks are not strictly mathematical, but pertain to some of the more formal consequences of the foregoing results.


The component of selection resulting from displacement would be difficult to detect in a natural population at equilibrium. Broadly speaking, this is a “fertility” component operating in just one sex. Even with powerful selection operating, at equilibrium, as we have indicated above, the two sexes will have equal gene frequencies and the frequencies of the three genotypes will be Hardy- Weinberg. Thus, no selection would be detected by a Hardy-Weinberg test. From progenies of individual females captured in the wild, one can calculate the efjective male gene frequencies and compare these with gene frequencies among captured male genotypes (CHRISTIANSFN and FRYDENBERG 1973), but at equilibrium these two gene frequencies would be the same even though strong sex- dependent selection in males is occurring. Displacement could be detected by noting changes in genotype frequencies through time in the laboratory in progenies from individual captured females, but to show that displacement is selec- tive is more difficult.

Selection due to sperm displacement gives rise to frequency-dependent selection. This is essentially because the process depends on mixed double matings. We assume that what is meant by “frequency-dependent selection” is that the fitnesses, as conventionally defined, vary with genotype frequency and that the latter is the cause of such variation. In this case the conventional relative fitness of a male genotype i can be obtained from the fitness factors, a,, equations ( 5 ) , as follows,

a, W . =- aN ’

where W , = fitness olf the ith genotype relative to some other genotype, i aN

N . fitness factor of genotype N , to which the i genotype is normalized.

The W , contain genotype frequencies. This consideration is important for experimental studies in which the objective is to measure relative fitnesses and fitness components. Specifically, the adult male component of fitness would show frequency dependence in experiments such as those of ANDERSON and WATANABE (1974), BUNDGAARD and CHRISTIANSEN (1972), PROUT (1971) and YAMAZAKI (1 971 ) . We illustrate this frequency dependence with two numerical cases, one involving directional selection and the other balanced heterosis. Table 9 shotws the maximal directional selection case where M = d,, = d,, = d,, = 1. The gene frequencies in the two sexes are made equal to each other and N = AA, the most favored genotype. It can be seen that the fitnesses vary directly with frequency; that is, the lower the frequency of a, the more intensely the genotype is selected against. The case of balanced heterosis with parameters M = 1, d,, = d,, = 1 and d,,=O is more interesting. It can be seen that by substituting into a, [Equation ( 5 ) which is the second factor in (4)] the fitness factors of the two homozygotes are always equal to each other, a, = a3 = 1 - x,; so, normalizing to a,. W , = W , = W , say. In Figure 9, the isoclines for equal W are shown on the p , r plane. The W surface is a saddle whose ridge is on the p = r line with saddle point and equilibrium point at p = r = s.

120 rr. PROUT AND J. BUNDGAARD

TABLE 9

Frequency dependence of fitnesses for the case of maximal directional selection, M = d,, = d,, = d,, = 1

Gene frequency Fitnerses A 0 A A Aa aa P' 4. 1 w2 w,

0 1.00 1 1 .50 .10 3 0 1 .90 .41 .25 .75 1 .77 .29 .50 50 1 .57 .14 .75 .25 1 .34 .04. .90 .10 1 .I7 .01

1 .oo 0 1 0 0

* p , 9 were arbitrarily chosen to be on the p = r line.

Since the population moves quickly to the equilibrium path as in Figure 7, and since this path in Figure 9 is near the p = r line on the ridge of the surface, most of the time is spent in lowering W to the mini-max where W = 1/3.

For the general case of balanced heterosis where d,, = d,, = d, say, ( d > 0) and d,, = 0 the equation for W can be arranged in the foillowing form

For a given W , the set of points satisfying (14) constitute the isocline for W , and equation (14) is a rectangular hyperbola with asymptotes parallel to the p, r coordinates and center at %, %. Varying W yields a surface with a saddle (hyperboloid) with saddle point i / z , i/z as shown in Figure 7 for d = 1. It can be easily shown that the ridge of the saddle is on the p = r line.

These considerations naturally raise questions concerning mean fitness, W , on which we now coment briefly. The male mean fitness w, , is defined,

(WO = 1, always). For the case of balanced heterosis wd. can be written

-

- W,? = WlZ, + 2, + W3Z3

- Wd (1 + d ) - 1

P r - % P - % r + 2dwd = o .

This too yields a saddle with saddle point (and equilibrium) at x, i / , but here the ridge is on the r = 1 - p line with low points at ( p , r ) = (0,O) and (1 , l ) . This same analytical procedure can be used to investigate wd for all cases. Generally, we find that for positive and negative heterosis there is a saddle point inside the state space, but only for the case of balanced heterosis (or negative heterosis) is the equilibrium point and saddle point coincident (at 1/2, x) ; other- wise, the equilibrium is on a ridge or in a valley. All other cases have the saddle point outside the state space so that the surface inside the space slopes uniformly. In Figure 8 we show such a wcf surface in the presence of two equilibria.


Although it has not been claimed that in the case of sex-dependent selection W?, wd are useful (BODMER 1965), mean fitness in other modes of selection receives considerable attention, even when it is not a potential function. In the case of selection resulting from sperm displacement, mean fitness appears to be just a simple curiosity. Furthermore, from a more biological polint of view, displacement selection must be totally soft (WALLACE 1968), so that we would not expect mean fitness to have any biological significance either.

-

DISCUSSION OF POPULATION THEORY

There remain many unanswered questions concerning sperm displacement. At the organism level, there is the major question of the mechanism underlying displacement, and certainly the experimental designs presently in the literature are not exhaustive. In this article we have attempted to provoke questions at the population level. At this level, too, there remain questions of a theoretical nature (e.g. , a theory of triple matings might be useful), but the most important question concerns whether or not there is genetic polymorphism in natural populations for displacing ability.

Thus, at both the organism and population levels, there are several lines of inquiry which we hope our treatment will encourage.

A direct application of the theory can be made to what has become known as genetic control of pest species (DAVIDSON 1974). First, it is conceivable that powerful displacers might be found and used for the transport of desirable genotypes into pest populations. This notiofi is at least as probable as the proposal to use meiotic drive for this purpose (DAVIDSON 1974). A more practical application of the theory is to the nongenetic technique of simply releasing sterilized males for the purpose of suppressing the pest. The suggestion here is that the males chosen for sterilization should have high displacing ability if the pest species engages in multiple insemination, and if sterility is of the dominant lethal type. Here, we are simply identifying a component of what has been called male “com- petitiveness,” long recognized as an important requirement for success (BERRY- MAN 1967). It is a simple matter to derive from our theory the fraction of eggs killed by dominant lethals as a function of the frequency of sterile males released, thus

(16) K = p ( 1 f Mds,) -- p2Md.y,,

where, K = fraction of eggs killed, p = frequency of sterile males among all males after release,

M 3 the probability of double mating, d.y, the displacement parameter where “S’ denotes released sterilized

male, and N fertile native male. The behavior of this function is shown in Figure 11, where it can be seen that if there are no doubles ( M = 0) , or doubles ( M > 0) but no displacement selection, ds, = 0, then there is a one-to-one relation between p and K ; but if there are doubles, M > 0, and displacement advantage, dSN > 0, then more eggs are killed


1.0

.8

.6

k

.4

.2

0

-1

0 .2 .4 .6 .8 1.0

P FIGURE 11.-The effect of sterile male release on egg hatch. K I fraction of eggs killed due

to dominant lethals, p E frequency of sterile males among all males after release. The straight line is the result if there are no double matings or no differential displacement. d,, displace- inent parameter of released males, s, relative to fertile native males, N. The Md,, =I line represents the maximum possible kill where M = 1, d,, = 1. The Md,, = -1 line is the minimum kill when M = 1 and dSN = -1.

for the same p . The figure also shows that if there are doubles and a poor displacer is used, dSN <0, because of genotype or even because of the sterilizing treatment, then less eggs are killed for a given p .

Finally, we wish to discuss the evolutionary aspects of sperm displacement. First, we evaluate our contribution to the theoretician’s ongoing quest for new modes of balancing selection. If displacing ability exhibits transitivity, then there is only one case which yields a protected polymorphism and a stable equilibrium and that is the case of heterosis (Case 1 in Table 8). We can hardly claim heterosis as a newly discovered mode of balancing selection, and our experimental data, even by graphical inspection, not only suggest a transitive relationship among the three markers. but also quantitative consistency, and perhaps this is demanded by the mechanism of the process.

There are 14 different nontransitive relationships and only four of these yield internal equilibria; there are two as in Case 2, and two as in Case 5 , and in both of these the polymorphism is not secure since one of the trivial equilibria in each case is either stable (Case 5 ) or neutral (Case 2). I t would appear, therefore, that selection due to sperm displacement by itself contributes little to the problem of explaining the fact of polymorphism in nature. However, there is a possibility that if, like meiotic drive, directional selection due to sperm displacement is


countered by selection in some other component of fitness, a stable equilibrium could result. Theoretically this is only a conjecture, and whether or not such joint systems exist in natural populations can only be determined by experiment.

Finally, we turn to the subject of evolution proper; i.e., the consequences of gene substitution resulting from differential displacement. Since sperm displacement is apparently widespread among insects of diverse taxonomy and diverse reproductive biology (see Introduction), it constitutes a phenomenon requiring some sort of evolutionary explanation. Tm70 aspects of this study bear on the origin and subsequent evolution of the displacement process. First, our experimental work explicitly demonstrates that genetic variation that manifests itself in males can influence displacement. This is not to say that genetic variance among females could not influence it too, but for reasons given presently we wish to emphasize the role of males. Secondly, the theoretical study shows the efficacy of differential displacement in its ability to accomplish a gene substitution when selection is directional, as illustrated in Figure 10.

It is possible, therefore, that the displacement characteristics of a given species have been shaped by a form of competition among males, and, as PARKER (1970) has pointed out, can be classified together with courting strategies and many aspects of male mating biology as being the result of sexual selection (MAYR 1971, 1972), more properly termed “intrasexual selection” among males (BATE- MAN 1948, PARKER 1970).

If we postulate sperm mixing as the primitive state, then a genotype, operating through males, which conferred any displacement at all would spread to fixation, thus bringing displacement into existence, to be subsequently intensified by further intrasexual selection. Of course, this does not have to be the case; the existence of displacement could be the natural consequence of some fundamental aspect of insect reproduction which has yet to be elucidated, in which case sperm displacement per se was not selected for.

However, if it is the product of intrasexual selection then, as with many other such characters, it need not have any “adaptive significance,” in the sense of its contributing to the well being of the species as a whole. We find it entirely plausible that this is a case of completely soft selection (WALLACE 1968); the females should lay just as many eggs, and the species should flourish just as well in every way whether sperm displacement aroje and evolved or not.

Mayr has suggested that in some cases intrasexual selection could even result in detrimental effects to the species (MAYR 1971). We cannot imagine any detrimental effects of sperm displacement, but we propose that there may be no advan- tages to the species either.

It could be that much research directed toward discerning the adaptive significance of a particular structure, function or strategy would be well served by considering the mode of selection which might have brought it into being.

We wish to acknowledge the invaluable technical assistance of DOTH JENSEN. Also, TIMOTHY PROUT wishes to thank the Genetics Institute, and especially the late OVE FRYDENBERG, for providing facilities as well as personal support during the course of the work. This project was supported in part by Grant No. GM-23863, awarded by the Public Health Service.

124 T . P R O U T A N D J. BUNDGAARD

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Corresponding editor: R. W. ALLARD

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