natural populations of Drosophila melanogaster (reviewed by LANGLEY. 1977),. IPaper No. 5710 of the ... D. pseudoobscura (e.g., PRAKASH and LEWONTIN.
ALLOZYMIC VARIATION AND LINKAGE DISEQUILIBRIUM IN SOME LABORATORY POPULATIONS O F DROSOPHILA MELANOGASTERI C. C. LAURIE-AHLBERG
AND
B. S . WEIR2
Department of Genetics, North Carolina State University, Raleigh, North Carolina 27650 Manuscript received July 20, 1978 Revised copy received December 8, 1978 ABSTRACT
Nine laboratory populations of D . melanogaster were surveyed by starch gel electrophoresis for variation at 17 enzyme loci. A single-fly extract could be assayed for all 17 enzymes, so that the data consist of 17-locus genotypes.Pairwise linkage disequilibria were estimated from the multilocus genotypic frequencies, using both BURROWS’ and HILL’S methods. Large amounts of linkage disequilibrium were found, i n contrast to the results reported for natural populations.-Knowledge of the approximate sizes of these populations was used to compare the observed heterozygosities and linkage disequilibria with predictions of the neutral allele hypothesis. The relatively large amount of linkage disequilibrium is consistent with the small sizes of the populations. However, the levels of heterozygosity in at least some populations suggest that some mechanism has been operating to retard the rate of decay by random drift. Several examples of significant deviation from Hardy-Weinberg frequencies and the large amount of linkage disequilibnim present i n these populations indicate that a likely mechanism is selective effects associated with neutral alleles because of linkage disequilibrium with selected loci (e.g., “associative overdominance”). The results are therefore consistent with both neutralist, and selectionist hypotheses, but suggest the importance of considering linkage disequilibrium between neutral and selected loci when attempting to explain the dynamics of enzyme polymorphisms.
intense interest in the adaptive significance of allozymic variability in eT net:: years has generated many descriptions of natural populations in terms of single-locus variability measures, such as heterozygosities and electromorph frequency distributions. However, relatively few descriptions of the associations of allozymes at different loci exist in spite of the fact that information about the frequencies of multilocus genotypes has long been available through the method of slicing and multiply staining starch gels. There have been several studies designed to detect linkage disequilibrium in natural populations of Drosophila melanogaster (reviewed by LANGLEY 1977), IPaper No. 5710 of the Journal Series of the North Carolina Agricultural Experiment Station, Raleigh, North Carolina. This investigation was supported by Public Health Service Research Grant No. GM 11546 from the National Institute of General Medical Sciences. * Department of Statistics, North Carolina State University, Raleigh, North Carolina 27650. Genetics 92: 1295-1314 August, 1979.
1296
C . C . LAURIE-AHLBERG A N D B. S. WEIR
D.pseudoobscura (e.g., PRAKASH and LEWONTIN 1968 and 1971; PRAKASH and MERRITT 1972), D.subobscura (e.g., ZOUROS and KRIMBAS1973; ZOUROS et al. 1974; LOUKAS and KRIMBAS1975) and a few other Drosophila species (PRAKASH and LEVITAN1973; BAKER 1975). Although many examples of linkage disequilibrium between allozymes and inversions have been found in Drosophila, there is very little evidence for stable linkage disequilibria among allozymes in the absence of inversions. Linkage disequilibria among allozymes have been detected in a few other organisms-the blue mussel, Mytilus edulis (MITTON and KOEHN 1973), a marine fish, Fundulus heteroclitus (MITTON and KOEHN 19751, a salamander, PZethodon cinereus (WEBSTER 1973) and the Yanomama Indians (SMOUSEand NEEL1977). I n no case has the mechanism causing the disequilibrium been elucidated. It has been suggested that information about linkage disequilibria among allozymes might be useful for determining the adaptive significance of allozyme polymorphisms (e.g.,LEWONTIN 1974). There are, however, a number of possible causes of virtually any pattern of linkage disequilibrium-those involving selection may result in the same type of geographic distribution of disequilibria as those that do not (see THOMSON 1977 for a discussion), Thus, it seems unlikely that it will be any easier to interpret patterns of linkage disequilibria among allozymes than it has been to interpret patterns of electromorph distributions or heterozygosities. However, a knowledge o i the nature and extent of linkage disequilibria in nature and laboratory populations may influence the design of more critical experiments. Here we report the results of an electrophoretic survey of nine laboratory populations of Drosophila melanogaster. Heterozygosities and linkage disequilibria are estimated from multilocus genotypic frequencies and compared with values for natural populations reported in the literature. Since a range of approximate sizes and the (closed) structure of these laboratory populations is known, some comparisons with the predictions of the neutral allele theory can be made that are not possible with data from natural populations. MATERIALS A N D METHODS
The populations: Two sets of populations were used in this study; one set has been selected for resistance to DDT (2,2-bis (p-chlorophenyl) -1,l,i-trichloroethane) and the other consists of the corresponding control populations. These populations were originally established and have been maintained for many years by D. J. MERRELL, University of Minnesota. They are designated as follows: Control ORC 731C 91c J#1
Selected (Resistant) ORR 731R 91R J#2 and J#3
ORC and ORR were derived in 1952 froin the well-known stock, Oregon-R, which has been kept i n the laboratory since 1925 (BRIDGESand BREHME1944). The pairs 731C, 731R and 91C, 91R were established in 1952 from collections of several hundred wild flies made in a residential area
1297
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
of St. Paul, Minnesota about one month apart. The J populaticns were established from another large collection made in the same locality in 1959. The populations have been maintained on standard cornmeal medium in population units described by REEDand REED(1948). These units are relatively small and cannot support very (1956) made large populations. During the early years of selection, MERRBLL and UNDERHILL counts of the adults in the populations. They found that each population unit contains a maximum of 600 to 700 adults and, at times, as few as ten or 12 adult flies per population were observed. T h e selected populations sometimes fell to such low numbers that exposure to D D T was temporarily discontinued to avoid possible extinction. Since generations are overlapping in these populations, generation length is not known precisely. The "effective generation length," which relates discrete generation models to populations with overlapping generations, has been estimated as 15 to 18 days for D. mezanogsrster a t 25" (CROWand CHUNC1967) and as 24 days at 21" (MERRELL1953). W e use conservative values of 25 days per generation at 21" and 20 days at 24". The age of the 731, 91 and OR populations is then at least 390 generations (18 years at 21" and seven years a t 24"). The J populations are approximately 288 generations old. Selection for resistance to DDT is accomplished by placing a piece of one inch by three inch filter paper impregnated with DDT on the surface of the food in the population units. All of the selected populations have become highly resistant as compared to the controls (DAPRUS1975 1977). and DAPKUSand MERRELL The populations were sampled by attaching a half pint bottle containing fresh medium to the population units and allowing adults to lay eggs in it for 48 hours. Adults emerging from the sample bottles were used for electrophoresis. Electrophoresis: Electrophoresis was performed in horizontal starch gels. The 17 enzyme loci surveyed are listed i n Table 1, along with their map positions and the buffer systems used. The procedures for electrophoresis and staining are described by LANGLEY, TOBARI and KOJIMA ITOand VOELKER(1977). Each single-fly extract was divided into three (1974) and by LANGLEY, TABLE 1 Enzymes surveyed, their map position and the bu8er system used for electrophoresis Enzyme
Esterase-C (EST-C) Esterase-6 (EST-6) Alcohol dehydrogenase (ADH) a-glycerophosphate dehydrogenase ((Y-GPDH) Acid phosphatase (ACPH) Octanol dchydrogemse (ODH) Phosphoglucomutase (PGM) 6-phosphogluconate dehydrogenase (6-PGD) Aldehyde oxidase (AOX) Glucose-6-phosphate dehydrogenase (G-6PD) Malate dehydrogenase-I (MDH-1) Isocitrate dehydrogenase (IDH) Hexokinase-C (HEX-C) Hexokinase-B (HEX-B) Glutamateoxaloacetate transaminase (GOT) Malic enzyme (ME) Dipeptidase-A (DIP-A)
Map position
buffer
Gel
Bridge buffer
3-51.7 3-36.0 2-50.1 2-20.5 3-101.1 3-49.2 3-43.4 I- 0.6 3-56.6 1-63 2-35.3 3-27.1 2-74.5 1-29.2 2- 3.0 3-53.1 2-52.2
1 1 2a 2a 1 1 2a 2a 3a 2a 2a 2a 3a 3a 1 2a 1
1 1 2b 2b 1
2b 2b 3b 2b 2b 2b 3b 3b 1 2b 1
Citrate p H 7.0; (2b) 0 . 0 0 5 ~Histidine pH 7.0; (3a) Buffers: (1) POULIK(1957); (2a) 0 . 5 ~ 0 . 1 Tris-Citrate-EDTA ~ pH 8.5; (3b) 0 . 0 2 ~ Tris-Citrate-EDTA pH 8.5.
1298
C . C . LAURIE-AHLBERG A N D
B. S. WEIR
parts, run on each of the three buffer systems in Table 1 and assayed for all 17 enzymes. The data therefore consist of 17-locus genotypes. The map positions are from O’BRIENand MACINTYRE (1976) except for HEX-B, HEX-C and DIP-A, which came from R. VOELKER(personal communication). Es/iniafion of linkage disequilibrium: In the following, we write the frequency of a n individual formed by the union of gametes AiBi and A,B, as Pi: (=Pkz).Dots denote summation II cver all pocsihlr values of the index replaced by the dot. For alleles A ; , B i at loci A and B , respectively, linkage disequilibrium is usually defined as the deviation of the product of the allele frequencies p i , q i from the frequency of the AiBi gamete. P i j : Dii = P i ] - p i q j . W e find it helpful to partition this “usual” linkage disequilibrium. D i j into two components. Following COCKERHAM and WEIR (1977), we write
where P i . p i q j are the frequencies of the pair of nongametic alleles A i , Bi within and between I individuals, respectively. The two components are known as the within-individuals linkage disequilibria. and they sum to the “uwal” disequilibrium, D i i . The within-individuals component is equal to one half the difference in frequency of the two type5 of double heterozygotes, ~
The between-individuals component is a measure of the nonrandom union of gametes. The ideal situation, when both gametic and genotypic frequencies are available, is to estimate DYj and Db . separately. In this study, coupling and repulsion heterozygotes could not be distinI J guished; and we therefore work with composite measures. A single composite disequilibrium was suggested by P. M. BURROWS (COCKERHAM and WEIR 1977): Ajj
DYi+ 2 DbZi
+ Pi.. -2
,
which has an unbiased estimate in samples of N individuals:
N N-1
k
A I.1. =-
(Pi’ .‘
PiQj)
Dyj
-
+ 2D!j ,
where tildes denote sample values. Note that estimation of A i j does not require distinguishing between the two types of double heterozggotes and does not require knowledge of the mating system. A ji: We report correlation coefficients hssed on BURROWS’
RB ,=--__ 21
Aii
___
This measure incorporates the departures from Hardy-Weinberg equilibrium for the sample frequencies at each locus. These one-locus disequilibrium coefficients are:
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
1299
The measure R?. is discussed by WEIR(1979). 23 The usual approach t o investigating associations between loci from genotypic data is to estimate D i f as
which for large N is approximately
- - -
Dij=DC+D!i
.
HILL(1574) provides a maximum likelihood method for estimating this quantity that requires an assumption of random union of gametes (which is equivalent to assuming that DP. = 0). We 23 also report correlation coefficients based on the estimates of Dii obtained by HILL’Smethod
As expected in cases where each locus can have an allele with low frequencies, the present data set contains many samples with missing genotypic classes. These cases necessitate some modifications of HILL’S procedure (WEIRand COCKERHAM, 1975). Only one locus, AOX was polymorphic for more than two electromorphs. In these cases, the least common alleles were pooled so that there were just two allelic classes for the estimation of linkage disequilibrium. For pairs of loci involving a sex-linked locus, only female genotypic frequencies were used. The Rii values were tested for significance by using the z-transformation of FISHER(19332). Although the z-transformation assumes normally distributed variables, WEIR(1979) has shown that it may be used with confidence in the present case of discrete allele frequency data. RESULTS
The loci MDH-I, IDH, HEX-G, HEX-B, GOT, M E and DIP-A are monomorphic in all nine populations ( 2 N = 180-240). The loci EST-C, EST-&,A D H , a-GPDH, ACPH ODH, PGM, 6-PGD and G-6PD are segregating for two allozymes in one or more populations; AOX is segregating for either two or three allozymes in eight of the populations. Table 2 shows the frequencies of the allozymes at the ten polymorphic loci. If the populations are ordered from highest to lowest frequency of a given allozyme, the sequences in Table 3 result. Only the sequence for the AOX’ allozyme (CCCRCRRRR) strongly suggests some relationship to DDT resistance. It was previously shown that the AOX allozymes are in strong linkage disequilibrium with a chromosome 3 inversion polymorphism in these populations, which can account for the apparent selection with respect to DDT resistance (LAURIE1979). AHLBERG and MERRELL Table 4 shows the observed and expected heterozygosities for each locus and population, as well as the results of the x2 tests for goodness-of-fit to HardyWeinberg proportions. Appropriate sample size corrections were made in these calculations (COCKERHAM 1973). There are ten cases of significant excess of heterozygotes involving EST C , (U-GPDH,ODH, PGM and AOX and one case of
1300
C. C .
LAURIE-AHLBERG A N D B. S. WEIR TABLE 2
The allozymic frequencies at ten loci--N is the number of genomes sampled Allozyme
EST-C6 EST-@
N EST-66 EST-64
N ADH6 ADH4
N ~u-GPDH~ a-GPDH4
N ACPHG ACPH8
N ODH~ ODH6
N PGM4 PGM2
N AOXl AOX2 ~0x3
AOX4
N 6-PGD2 6-PGD4 N G-6PD2 G-6PD4
N
ORC
0.884 0.116 99 0.625 0.375 100 0.729 0.271 96 0
1.ooo 99 0.995 0.005 100 0.580 0.420 100 1.Ooo 0 99 1.ooo 0 0 0
97 1.000 0 97 I .000 0 79
ORR
731C
0.005 0.900 0.995 0.100 110 120 0.682 1,000 0 0.318 120 100 0 0.832 0.168 1.ooo 110 120 1.000 1.Ooo 0 0 120 110 1.000 1.ooo 0 0 110 120 0.814 1.WO 0 0.186 110 119 1.a00 1.000 0 0 100 12.0 0.509 0.668 0.401 0.332 0 0 0 0 110 119 0.995 1.om 0 0.005 110 120 1.om 1.WO 0 0 90 120
731R
0.153 0.847 98 0.247 0.753 99 0.460
0.540 100 0.910 0.090 100 1.000 0 100 0.685 0.315 100 1.000 0 loo 0.610 0 0 0.390 loo 0.910 0.090 100 1.000 0 120
Population
91C
0.542 0.458 95 0.895 0.105 95 0.815 0.185 100 1.000 0 loo 1.000 0 100 0.681 0.319 99 1.000 0 1023 0.965 0.035 0 0 100 1.000 0 100 1.156 0.844 93
91R
J#l 0.335 0.630 0.370 0.665 115 loo 0.985 0.717 0.015 0.283 115 loo 0.495 0.374 0.505 0.626 95 99 0.571 0.695 0.429 0.305 115 99 1.OOO 0.970 0.030 0 115 100 0.925 0.504 0.075 0.496 115 100 1.oQo 0.996 0.004 0 100 115 0.705 0.868 0 0.127 0.035 0 0.260 0.004 loo 115 0.850 0.665 0.150 0.335 100 115 1.000 1.ooo 0 0 100 115
J#Z
0.543 0.467 128 0.603 0.397 131 0.647 0.353 99 0 1.ooo 112 1.om 0 137 1.ooo 0 138 0.754 0.246 138 0.575 0 0 0.425 134 0 1.ooo 120 1.om 0
120
J#3
0
1.om 100 0.445 0.555 100 0.725 0.275 100 1.om 0 99 1.ooo 0 100 0.595, 0.4Q5 100 1.ooo 0 100 0.335 0.415 0
0.250 100 1.m 0 100 0.861 0.139 97
a heterozygote deficiency (EST-6 in 91C). The average of the expected heterofrom 0.037 ( 7 3 1 0 to 0.176 (91C) and the grand zygosities over loci ranges average over loci is = 0.12. The percent of polymorphic loci averaged over populations is 34%. Each of the 17 loci used in this study has also been surveyed in one o r more ,of the following natural populations of D. melanogaster: Katsunuma, Japan 1970; LANGLEY, TOBARI and KOJIMA1974); (KOJIMA, GILLESPIEand TOBARI 1970; MUKAI, Raleigh, North Carolina (KOJIMA, GILLESPIEand TOBARI WATANABE and YAMAGUCHI 1974) ; Carpenter, North Carolina (LANGLEY, ITO and VOELKER1977) ; Brownsville, Texas (LANGLEY, TOBARI and KOJIMA1974) and Amherst, Massachusetts (BAND 1975). The expected heterozygosities reported in the studies cited above were averaged over populations and loci to get
a
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
1301
TABLE 3 The sequences that result from ordering the five DDT-resistant (“R’)and four unselected control ((V)populations from highest to lowest frequency of the indicated allozyme Allozvme
Order
EST-C6
CCRRCCRR (R) (C) RCCRCRRR RCCRRRRC (C) (RCCR) RCR (RC) (RCRCRRR) CC (CR) RRRCRCC (CRCRCRR)CR (CCCR) RRRC (R) (C) CCRCRRRR (CRCRRCR)RC
EST-@ ADHG (u-GPDHG ACPH6 ODH6 PGM4 6-PGD2 AOXl G-6PDZ
Letters inside of parentheses represent monomorphic populations.
-
an average expected heterozygosity, = 0.1784. This expected heterozygosity for natural populations is substantially larger than the values for most of these laboratory populations, but only slightly greater than the value for the J # l population (G = 0.1 762). The values of RYj and R t for all pairs of polymorphic loci are given in Table 5. Overall, the two estimates are very similar, but HILL’S(1974) method seems to give larger absolute values. The 137 situations for which R i j could be estimated may be categorized as follows: 75 had IRyj I > IRZ I, 55 had IRZ 1 < IRfjI and in seven cases R f j = RE.. Discrepancies between R: and R f jindicate that D:j # 0, which may be caused by nonrandom union of gametes due to the mating system, by factors such as selection, migration, etc., or by sampling effects that make D!j # 0 even when DZ = 0 (see COCKERHAM and WEIR1977). BURROWS’ estimates may tend to be do not necessarily smaller in absolute value than HILL’Sbecause Dyj and have the same sign and because they have been corrected for inbreeding. If D t j # 0, we cannot say with certainty what the maximum likelihood method of HILL(1974) is estimating (or what his test statistic is testing), but clearly the RYj’s would reflect the effects of both Dz and Dt (see WEIR1979). Fortunately, D\j can be estimated directly from the genotypic frequencies in those-special cases where there are no double heterozygotes in the sample (and thus Dyj = 0). There were a total of 13 such-cases in five of the nine populations and involving all of the loci. In no case was significantly different from zero, which of course lends support to the assumption of random union of gametes that underlies HILL’S(1974) method. Moreover it should be noted that perhaps the most common cause of nonrandom mating, spatial subdivision of the population, is not a plausible mechanism in laboratory population cages. The Rij values in Table 5 show that there is extensive linkage disequilibrium in these laboratory populations. For HILL’S(1974) method, 44.8% (26/58) of
-
-
Rj
1302
C . C. LAURIE-AHLBERG A N D B. S. WEIR
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
** I* * *
* ** I
!$zI I q 20 ++ c o h
*
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con
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na.-
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*
1303
1304
C. C. LAURIE-AHLBERG AND B. S . WEIR
** **
&%
2" %0 I 1
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22
(Do
$5
2s
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I I
0 0
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9 9
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ma2 *a 0 0
II++++++l
I I
31 X
X
2
2
2
Q 1 X
X
x2
0 0
I
++
2
2
Q 1 X \D
X
2
2
X
X
2
2
X
0 X
3
T
X
X
1305
ALLOZYMIC VARIATION A N D L I N K A G E D I S E Q U I L I B R I U M
8 3
0
*+ x*
x
x
x
x
x
x
x
x
x
x
x
x
* + 3-
1306
C. C.
LAURIE-AHLBERG A N D
B. S. WEIR
the pairs of linked loci show significant disequilibria, as do 10.1% (8/79) of the pairs of unlinked loci. For BURROWS’ method. the percent significant pairs is 34.5% (20/58) for linked loci and 8.9% (7/79) for unlinked loci. These results contrast strongly with those from natural populations of D. melanogaster in which the proportion of pairs of allozyme loci that show linkage disequilibrium is much smaller. A commonly used method for detecting linkage disequilibria in D. melanogaster populations is the extra’ction of whole chromosomes by, for example, the C y / P m method (MUKAI,METTLERand CHIGUSA1971; CHARLESWORTH and CHARLESWORTH 1973; LANGLEY, TOBARI and KOJIMA1974; MUKAI,WATANABE and YAMAGUCHI 1974; MUKAI and VOELKER1977; LANGLEY, TOBARI and KOJIMA1977). A survey of these studies revealed that only 9.2% (3.5/38) of the pairs of linked allozyme loci showed significant disequilibria (only those enzyme loci used in this study are included here and the number of significant pairs was first averaged over different oamples of the same population and then over populations). The study of LANGLEY, SMITH and JOHNSON (1978) involves the use of genotypic data to calculate BURROWS’ Rij and is thus more strictly comparable to the data reported here. They report linkage disequilibria for a number of East Coast natural populations and for two laboratory population cages. The percent of significant linkage disequilibria is, for natural populations, 5.1 % for linked loci and 6.7% for unlinked loci. For the two population cages, they report that 37.5% linked pairs and 10.3% unlinked pairs are significant, a result very similar to ours. It therefore appears that linkage disequilibria among allozymes are much more common in laboratory than in natural populations of D.melanogaster. I n this study, the R,j values for a given pair of loci are clearly heterogeneous among populations, except in a few cases. For example, the locus pair EST-C X EST-6 has significant disequilibria in both positive and negative directions. In only one case where R,j’s could be computed for more than two populations were the signs of the R,j’s all the same-ODH X AOX. The homogeneity of the six Rij values for ODH x AOX was tested by the method of FISHER (1932). Both BURROWS’ and HILL’S( 1974) Rij values are significantly heterogeneous. For the unlinked pair of loci 6-PGD x EST-C in ORR, the correlation coefficients are essentially one. This results Irom the fact that all individuals sampled were homozygous at both loci, except for a single doubly heterozygous individual. DISCUSSION
Heterozygosities: We have shown that the expected heterozygosity averaged over several samples 01 natural populations of D.melanogaster is greater than the values observed for the laboratory populations considered here. This qualitative result is, of course, predicted whether allozymes are selectively neutral or whether they are maintained by balancing selection. The reduction in population size that results from collecting wild flies and maintaining them in small
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
1307
containers in the laboratory is expected to lead to a loss of variability for both neutral and selected alleles. The amount of reduction in heterozygosity expected for neutral alleles is ( 1931 ) , given by WRIGHT
where H t is the heterozygosity of a population after t generations in the laboratory, H , is the heterozygosity of the founding population sample and Ne is the effective size of the laboratory population. Of course, Ho for the laboratory populations studied here is not known since they were established in the 1950’s, before gel electrophoresis came to be used for measuring genetic variability. The effective sizes of the populations are not known either, although we do have some information about population size: as discussed earlier, census data indicate that the maximum size of the population is 600 to 700. At least two factors probably operate to make the effective size considerably smaller than the census valuedifferential productivity (as, for example, shown by CROWand MORTON 1955) and fluctuations in population size. Both control and DDT-selected populations have experienced great fluctuations in size, sometimes falling to very low numbers. W e may, therefore, conclude that the effective sizes of these populations are well below 600 and that the selected (resistant) populations have had smaller sizes than the control populations because of the great intensity of selection. The age of the laboratory populations in generations was discussed earlier. Although we cannot use equation (1) to predict H t values for lack of precise knowledge of H , and Ne, we can use the observed H t values to predict (a) the relative sizes of the R and C populations derived from the same founding population sample and therefore having the same Ho and t values, and (b) values of N e using an approximate value for derived from the literature on natural populations (i.e., N 0.18, as discussed earlier). These relative and absolute values of N e can then be compared with the conclusions we independently drew about population sizes: i.e.,that R populations should be smaller than their controls and that effective sizes should be less than 600. The method of estimating population sizes from the observed heterozygosities is described in detail in APPENDIX A and summarized here. We have either two or three populations that originated from a single founding population and that have been maintained isolated from each other in the laboratory for t generations. Because they have been subjected to different environments that influence population size (Le., DDT exposure), we allow them to have different effective population sizes, Nj ( j = 1,2,3), and we define
a,
eo
nj=(1--)
I t 2Nj
.
C. C. LAURIE-AHLBERG A N D B. S. WEIR
1308 The values of is given by
nj
are chosen so that the expected heterozygosity for the ith locus
where H,,i is the initial heterozygosity at the ith locus. The method of least squares is employed to minimize the difference between observed and expected heterozygosities. Table 6 summarizes the population size estimates based on observed heterozygosities. Relative values of ni(n7) can be computed for a set of lines that descended from a single founding population without any knowledge of the initial heterozygosity values. The relative values in Table 6 are such that the sum of the ( n ; ) 2over the two or three lines from a single founding population is one. Estimation of the absolute values of rzi requires knowledge of H,,the initial heterozygosity averaged over loci. The values of n; for resistant populations are not consistently smaller than the values for their controls as predicted. The nt values for the resistant J populations and for ORR are less than their controls, but greater for the 731 and 91 population pairs. The nj and Nj values in Table 6 were calculated on the assumption that = 0.1784 for all the populations, which is the average heterozygosity derived from the literature on natural populations. We predicted that the Xi values would all be substantially less than 600, but we found this for only four out of nine values. These results suggest that some mechanism is operating in at least some of the populations to retard the loss of heterozygosity by random drift. Balancing selection operating on the allozymes themselves, is, of course, one possibility. Another possibility is the allozymes are themselves neutral, but are in linkage disequilibrium with selected loci. The latter possibility seems very likely for the following reasons: (a) we observed extensive linkage disequilibrium even among loosely linked loci in these laboratory populations. It therefore seems likely that each of the allozyme loci studied here may be in linkage disequilibrium with a large number of loci, some of which will be selected. (b) Previous studies have shown that all but one of the populations (ORC) are polymorphic f o r one or two
a,
TABLE 6 Estimation of effective population sizes from heterozygosities Population
ORC ORR 731C 731R 91c 91R J#lC J#2R J#3R
i
nl*
0.7250' 0.6888 0.3269 0.946 1 0.5987 0.8010 0.6086 0.5435 0.5781
"J
0.5261 0.4998 0.2603 0.7527 0.5800 0.7759 0.8808 0.7866 0.8366
1
Nj
390.
304 28 1 144i 609 359 769 1135 601 807
390 390 288
1309
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
chromosome 3 inversions, In(3L)P and Zn(3R)P, which seem to be maintained by heterokaryotype advantage ( LAURIE-AHLBERG and MERRELL1979). The AOX locus is in strong linkage disequilibrium with these inversions, which accounts for the high degree of heterozygosity of this locus and also for the significant deviation from Hardy-Weinberg frequencies observed in several of the populations. Many of the other chromosome 3 lcci are in linkage disequilibrium with AOX, indicating that they toe are associated with the inversion(s). The inversions occur at higher frequencies in resistant than in control populalions (causing more inversion heterozygosity) , which may explain why the allozymic heterozygosity in resistant populations is not consistently lower than controls as we predicted. (c) The significant deviations from Hardy-Weinberg in the direction of heterozygote excess that we observed for five loci clearly indicates the existence of “associative overdominance,” which can lead to a retardation of the loss of heterozygosity in small populations (OHTAand KIMURA 1971). Linkage disequilibrium: The laboratory populations studied here and the two laboratory populations studied by LANGLEY, SMITH and JOHNSON(1978) show extensive linkage disequilibria among allozymes, in contrast to the results of studies of natural populations. Of course, the small size of laboratory populations seems to be a likely cause of this result. Under an hypothesis of neutrality, linkage disequilibrium values, like heterozygosities, can provide information about population size. Even though Dij is expected to become zero over time within populations for neutral loci, the sample squared correlation coefficient R:, has a nonzero expectation. We cannot predict the value of this expectation exactly, but we can work with the very similar ratio of expected values:
If Burrows’ measure is used, U
u2 Ad
=E(A:j) / E [,&(I-pi)+@]
[qj(I-qj)+@i]
.
An exact method for determining these quantities from the descent measures appropriate for the particular mating system has been found (WEIRand HILL, in preparation), and, as part of this treatment, good approximate formulae were derived that seem preferable to earlier approximations (HILLand ROBERTSON 1968; OHTAand KIMURA 1969; SVED1971). If nij individuals are sampled to provide the estimate Aii for loci i and i for which the recombination fraction is
-
Ci j 7
Here N e is the inbreeding effective population size for either a monoecious popu-
1310
C . C. LAURIE-AHLBERG A N D B. S. WEIR
lation mating at random (excluding selfing) or a dioecious population mating at random. If monogamy is imposed in the dioecious case, the numerator of the first term in these expressions is changed to 1+ci3+2c223. These approximations are best when loci i and j are unlinked, c , = ~ 0.5, and this seems to be the only case of relevance here, as we now argue. Some information about the recombination fraction, ct3, f o r the pairs of loci is available from their published map positions. However, the occurrence of polymorphic inversions in most populations of D. melanogaster can greatly modify recombination fractions obtained from mapping experiments. LANGLEY, TOBARI and KOJIMA (1974), LANGLEY,SMITH and JOHNSON (1978) and LANGLEY ( 1977) have devised a parameter called “effective recombination” ( r e ) ,which is meant to take into account the effects of inversion heterozygosity on recombination among allozyme loci. For a pair of loci on the same chromosome arm as a polymorphic inversion, re = % ( 1-H) r, where H is the frequency of inversion heterozygotes and r is the map distance. This modification of map distances is an accurate reflection of the effects of inversion heterozygosity on recombination only when the loci involved are not in linkage disequilibrium with the inversion. When one o r both loci are nonrandomly associated with the inversion, then the recombination fraction between them is a function of the linkage disequilibrium between them. Since many allozymes are associated with TOBARI and KOJIMA1974), inversions in Drosophila populations (e.g.,LANGLEY, use of the “effective recombination” parameter appears to be invalid. Because we cannot determine the recombination fractions for linked loci when inversions are present, we have used only the R;, values for unlinked loci to estimate population size, which also bypasses the problem of no male recombination. Although it makes little difference, we prefer to use BURROWS’ disequilibrium measure, and choose N e to make observed values ( R f J ) 2as close as possible to 2 = 1/3N, 4- l / n , 3 .If a population has m pairs of loci for which R f l is measAt] ured, the least squares estimator of N e is just
The summation in the denominator is over m pairs of loci. Details of this estimation procedure are given in Table 7, from which it is clear that the procedure is not completely satisfactory. In four populations-ORC, 731R, J#2R, J#3Rthe denominator in the above expression is negative and the only possible estimate of N , is infinity. As to be discussed by WEIRand HILL(in preparation), one trouble with this approach to the estimation of N e is the restriction to unlinked loci. I n this case, for the values of N e and n i j we have, the sampling effect l / n i i swamps the population structure effect 1/3N,. I n other words, the observed values of R:i are due almost entirely to sampling and contain practically no information about N e . Therefore, these population size estimates based on linkage disequilibrium data provide no evidence either for or against a model of neutrality.
1311
ALLOZYMIC VARIATION A N D LINKAGE DISEQUILIBRIUM
TABLE 7 Estimation of population size from observed values of RB.
11
Population
ORC ORR 731C 731R 91c 91R J#lC J#2R J#3R
m
4 9 0 14 9 14 18 4 7
22-
'
1
J
0.0417 0.1364 0.2006 0.1568 O.U)08 0.2316 0.0436 0.1161
0.0178 1.3203 0.1615 0.3254 0.4205 0.4647 0.0298 0.0599
CO
3* CO
18 21 27 CO CO
* This value becomes 13 if the value of 6-PGD x EST-C is omitted.
The general lack of consistency in the magnitude and direction of linkage disequilibrium among the populations is also consistent with both neutralist and selectionist hypotheses. In only one case, ODH x AOX, were the R,j values of the same sign in a number of populations. This observation could be accounted for by epistatic selection, but it may also have an historical basis. Linkage disequilibria between In(3R)P and both AOX and ODH have been observed in natural populations (LANGLEY, TOBARI and KOJIMA1974). Since our laboratory populations are polymorphic for Zn(3R)P and since AOX and ODH are closely linked to it, the present levels of linkage disequilibrium between AOX and ODH may be greatly influenced by the initial conditions. I n conclusion, we have shown that laboratory populations of Drosophila may have extensive linkage disequilibrium, even among loosely linked loci. These results suggest the great importance of considering linkage disequilibrium between neutral and selected loci as a possible explanation for the behavior of allozyme polymorphisms in small populations. We thank D. Z. BEATTIEand E. W. RAINES for their expert technical assistance and D. J. MERWL for kindly providing the laboratory populations. W. G. HILLfirst pointed out to us the problems in estimating population size from lingage disequilibrium estimates. LITERATURE CITED
BAKER,W. K., 1975 Linkage disequlibrium over space and time in natural populations of Drosophila montana. Proc. Natl. Acad. Sci. U.S. 72: 4095-4099. BAND, H. T., 1975 A survey of isozyme polymorphism in a Drosophila melanogaster natural population. Genetics 80: 761-771. BRIDGES, C. B. and K. F. BREHME,1944 The Mutants of Drosophila melanogaster. Carnegie Inst. Wash. Publ. No. 552. CHARLESWORTH, B. and D. CHARLESWORTH, 1973 A study of linkage disequilibrium in populations of Drosophila melanogaster. Genetics 7 3 : 351-359. C O C K E R H ~ R I ,c. c., 1973 Analyses of gene frequencies. Genetics 74: 679-700.
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C. C. LAURIE-AHLBERG A N D B. S. W E I R
COCKERHAM, C. C. and B. S. WEIR, 1977 Digenic descent measures for finite populations. Genet. Res. 30: 121-147. CROW,J. F. and Y. J. CHUNG,1967 Measurement of effective generation length in Drosophila population cages. Genetics 57: 951-955. 1955 Measurement of gene frequency drift in small populations. CROW,J. F. and N. E. MORTON, Evolution 9: 202-214. DAPKUS,D. C., 1975 Genetic studies on DDT-resistant Drosophila melanogaster. Ph.D. Thesis. University of Minnesota. DAPKUS,D. C. and D. J. MERRELL,1977 Chroniosomal analysis of DDT-resistance in a longterm selected population of Drosophila melanogaster. Genetics 87 : 685-697.
FISHER, R. A., 1932 Statistical Methods for Research Workers. Oliver and Boyd, London. HILL, W. G., 1974 Estimation of linkage disequilibrium in randomly mating populations. Heredity 33: 229-239.
HILL, W. G. and A. ROBERTSON, 1968 Linkage disequilibrium in finite populations. Theoret. Appl. Genet. 38: 226-231. KOJIMA,K. I., J. GILLESPIEand Y. M. TOBARI,1970 A profile of Drosophila species’ enzymes assayed by electrophoresis. I. Number of alleles, heterozygosities and linkage disequilibrium i n glucose-metabolizing systems and some other enzymes. Biochem. Genet. 4: 627437. LANGLEY, C. H., 1977 Nonrandom associations between allozymes in natural populations of Drosophila melanogaster. In: Lecture Notes in Biomathematics, 19: Measuring Selection in Natural Populations. Edited by F. B. CHRISTIANSEN and T. M. FENCHEL. Springer-Verlag, New York. LANGLEY,C. H., K. ITOand R. A. VOELKER,1977 Linkage disequilibrium in natural populations of Drosophila melanogaster. Seasonal Variation. Genetics 86: 447-454. LANGLEY,C. H., D. B. SMITH and F. M. JOHNSON, 1978 Analysis of linkage disequilibria between allozyme loci in natural populations of Drosophila melanogaster. Genet. Res. 32: 215-230. LANGLEY, C. H., Y. N. TOBARI and K. KOJIMA,1974 Linkage disequilibrium i n natural populations of Drosophila melanogaster. Genetics 78: 921-936. LAURIE-AHLBERG, C. C. and D. J. MERRELL,1979 Aldehyde oxidase allozymes, inversions and DDT resistance in some laboratory populations of Drosophila melanogaster. Evolution 33 : 342-349. LEWONTIN,R. C., 1974 The Genetic Basis of Euolutionary Change. Columbia University Press, New York. LOUKAS, M. and C. B. KRIMBAS, 1975 The genetics of Drosophila subobscura populations. V. A study of linkage disequilibrium in natural populations between genes and inversions of the E chromosome. Genetics 80: 331-347. MERRELL,D. J., 1953 Selective mating as a cause of gene frequency changes in laboratory populations of Drosophila melanogaster. Evolution 7 : 287-296. MERRELL, D. J. and J. C . UNDERHILL, 1956 Selection for DDT resistance i n inbred, laboratory and wild stocks of Drosophila melanogaster. J. Econ. Entomol. 49 :300-306. MITTON,J. B. and R. K. KOEHN,1973 Population genetics of marine Pelecypods. 111.. Epistasis between functionally related isoenzymes of Mytilus edulis. Genetics 73 : 487-496. -, 1975 Genetic organization and adaptive response of allozymes to ecological variables in Fundulus heteroclitus. Genetics 79:97-1 11. MUKAI,T., L. E. METTLERand S. I. CHIGUSA,1971 Linkage disequilibrium in a local population of Drosophila melanogaster. Proc. Natl. Acad. Sci. U.S. 68: 1065-1069.
ALLOZYMIC VARIATION A N D L I N K A G E DISEQUILIBRIUM
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MUKAI, T. and R. A. VOELKER, 1977 The genetic structure of natural populations of Drosophila mdanogaster. XIII. Further studies on linkage disequilibrium. Genetics 86: 175-185. and 0. YAMAGUCHI, 1974 The genetic structure of natural popuMUKAI,T., W. K. WATANABE lations of Drosophila melanogaster. XII. Linkage disequilibrium in a large local population. Genetics 77:771-793. OBRIEN,S. J. and R. J. MACINTYRE, 1976 Biochemical loci of Drosophila melanogaster. ISOzyme Bull. 9: 15-17. OHTA,T. and M. KIMURA,1969 Linkage disequilibrium due to random genetic drift. Genet. Res. 13: 47-55. , 1971 Behavior of neutral mutants influenced by associated overdominant loci in finite populations. Genetics 69: 247-960. PRAKASH, S. and M. LEVITAN,1973 Associations of alleles of the esterase-1 locus with gene arrangements of the left arm of the second chromosome i n Drosophila robusta. Genetics 75: 371-379.
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PRAKASH, S. and R. C. LEWONTIN,1968 A molecular approach to the study of genic heterozygosity in natural populations. 111, Direct evidence of coadaptation i n gene arrangements of Drosophila. Proc. Natl. Acad. Sci. US. 59: 398405. -, 1971 A molecular approach to the study of genic heterozygosity in natural populations. V. Further direct evidence of coadaptation in inversions of Drosophila. Genetics 69: 4 0 5 4 8 . S. and R. B. MERRITT,1972 Direct evidence of genic differentiation between sex ratio PRAKASH, and standard gene arrangements of X chromosome in Drosophila pseudoobscura. Genetics 72: 169-175. REED,S. C. and E. W. REED,1948 Natural selection in laboratory populations of Drospohila. Evolution 2: 176-186. SMOUSE,P. E. and J. V. NEEL, 1977 Multivariate analysis of gametic disequilibrium in the Yanomama. Genetics 85: 733-752. SVED,J. A., 1971 Linkage disequilibrium and homozygosity of chromosome segments in finite populations. Theoret. Pop. Biol2: 125-141. THOMSON, G., 1977 The effect of a selected locus on linked neutral loci. Genetics 85: 753-788. WEBSTCR, T. P., 1973 Adaptive linkage disequilibrinm between two esterase loci of a salamander. Proc. Natl. Acad. Sci. US. 70: 1156-1160. WEIR,B. S., 1979 Inferences about linkage disequilibrium. Biometrics 35: 235-254. WEIR,B. S. and C. C. COCKERHAM, 1974 Behavior of pairs of loci in finite monoecious populations Theoret. Pop. Biol. 6: 323-354. __ , 1979 Estimation of linkage disequilibrium in randomly mating populations. Heredity 42 : 105-1 11. WRIGHT,S., 1931 Evolution in Mendelian populations. Genetics 16 : 97-159. ZOUROS, E. and C. B. KRIMBAS,1973 Evidence for linkage disequilibrium maintained by selection in two natural populations of Drosophila subobscura. Genetics 73: 659-674. ZOUROS, E., C. B. KRIMBAS, S. TSAKAS and M. LOUKAS, 1974 Genic us. chromosomal variation in natural populations of Drosophila subobscura. Genetics 78: 1223-1244. Corresponding editor: W. W. ANDERSON
APPENDIX A
Estimation of population sizes from heterozygosity Suppose 1 lines all originated from the same original population and have all been maintained for t generations. Because they have been subjected to different regimes, we allow them to have different effective population sizes Nj,j = 1,2, . . ,1 and we define
.
1314
C. C. LAURIE-AHLBERG AND B. S . WEIR
In this study j = 1,2 o r i = 1,2,3. The effective population sizes are chosen so that the expected heterozygosity, for the ith locus, i = 1,2, . . . ,m,is given by E H j i = njHPi where Hoi is the initial heterozygosity a t the ith locus. We wish to estimate the ni values from the observed heterozygosities Hii and employ least squares. The appropriate sum of squares is 1
Q
3=1 .Z
m *=1 ,Z
(Uii - njHoi >'
and
Substituting (2) into (1) provides
where 7n
sky
& HkiHkri .
Equation (3) shows that the least squares estimates of n j form an eigenvector of the matrix S who,se kth row, k'th column element is S,,,. In other words, the least squares procedure can provide the relative magnitude of the nits but not their actual values, since if n = {n,,n,, . . . ,nl)' as an eigenvector of S, then so is any constant multiple of n. Additional information is needed, and appears to be most simply obtained from summing equation ( 2 ) over i to obtain
where I?, and Rj are the average heterozygosities in the initial population and the jth line, respectively. For r?, we use the average of all published heterozygosities, averaged over all the loci studied, which eliminates the need for any assumptions about the individual locus values Hoi. 2
For convenience we choose an eigenvector n* of S with elements obeying . E ( n f ) ? = 1. I=1 J To find the required values nj we take n = Kn* and find K from
i.e.,
No correction €or the sample size used to estimate Hi,is required here since the samples from different populations are very similar in size and thus do not affect the relative effective size estimates.
