Classic Weinstein: Tetrad Analysis, Genetic Variation and Achiasmate Segregation in Drosophila and Humans

Corresponding author: Michael E. Zwick, Department of Genetics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-4955., mez4{at}po.cwru.edu (E-mail)

Communicating editor: R. S. HAWLEY

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

A maximum-likelihood method for the estimation of tetrad frequencies from single-spore data is presented. The multilocus exchange with interference and viability (MEIV) model incorporates a clearly defined model of exchange, interference, and viability whose parameters define a multinomial distribution for single-spore data. Maximum-likelihood analysis of the MEIV model (MEIVLA) allows point estimation of tetrad frequencies and determination of confidence intervals. We employ MEIVLA to determine tetrad frequencies among 15 X chromosomes sampled at random from Drosophila melanogaster natural populations in Africa and North America. Significant variation in the frequency of nonexchange, or E₀ tetrads, is observed within both natural populations. Because most nondisjunction arises from E₀ tetrads, this observation is quite unexpected given both the prevalence and the deleterious consequences of nondisjunction in D. melanogaster. Use of MEIVLA is also demonstrated by reanalyzing a recently published human chromosome 21 dataset. Analysis of simulated datasets demonstrates that MEIVLA is superior to previous methods of tetrad frequency estimation and is particularly well suited to analyze samples where the E₀ tetrad frequency is low and sample sizes are small, conditions likely to be met in most samples from human populations. We discuss the implications of our analysis for determining whether an achiasmate system exists in humans to ensure the proper segregation of E₀ tetrads.

A classic, as we have all heard, is a work that is often referred to and never read.

ALEXANDER WEINSTEIN (1955)

IN 1936, Alexander Weinstein presented a mathematical method for inferring the frequency of tetrads with different numbers of exchanges in organisms where only one of the four products of meiosis (referred to as single-spore data) is recovered. His classic article contained the first theoretical model of crossing-over constructed on a four-strand basis and allowed him to infer two main conclusions (WEINSTEIN 1936 ; see WEINSTEIN 1958 for a review). First, he concluded that the vast majority of exchange at any single crossover event occurs between homologous chromatids, and few if any exchanges occur between sister chromatids. Second, he concluded that the choice of homologous chromatids that undergo exchange at different crossovers is random. Weinstein's results also suggest an important third conclusion, which he did not emphasize: a significant number of X chromosome tetrads in Drosophila melanogaster fail to undergo exchange during female meiosis. These tetrads are referred to as E₀ tetrads (see Figure 1).

View larger version (19K): In this window In a new window Download PPT slide   Figure 1. Relationship between observed chromatids and the estimated tetrad frequencies in Drosophila and humans. Exchange tetrads of rank n produce chromatids with up to n exchanges in proportions first derived in WEINSTEIN 1936 . This derivation reflects the fundamental observation that n exchange tetrads produce chromatids that have fewer than n exchanges.

The existence of E₀ tetrads presents something of a paradox, because the commonly accepted model suggests that at least a single exchange, or crossover, is necessary for proper segregation. Chiasmata, the cytological structures formed at the site of crossing-over, have long been thought necessary to ensure proper segregation (DARLINGTON 1932 ; NICKLAS 1977 ; reviewed in HAWLEY 1988 ). Two largely distinct research progams solved this paradox in D. melanogaster. The first involved chromosome mechanics and the direct demonstration that chromosomes that failed to cross over are still able to segregate with high efficiency (BRIDGES 1916 ; STURTEVANT and BEADLE 1936 ; COOPER 1945 ). Subsequent research led to the identification of a "backup" system, referred to as distributive pairing, that acts to ensure the segregation of nonexchange bivalents (GRELL 1962 , GRELL 1976 ). The fundamental inference leading to the identification of a backup system arose from the observation that the frequency of nondisjunction was significantly lower than that expected from a null model assuming random segregation of homologs in meiosis. The second strategy, involving screens for meiotic mutants, led to the identification of genetic loci, some of whose mutant phenotypes were defective in the segregation of nonexchange chromosomes (SANDLER et al. 1968 ; BAKER and CARPENTER 1972 ; CARPENTER 1973 ). Subsequent research along the first path led to the falsification of significant aspects of the distributive pairing model, and instead demonstrated that the "backup" system is composed of two genetically distinct pathways, now referred to as the homologous and heterologous achiasmate systems (HAWLEY et al. 1992 ; HAWLEY and THEURKAUF 1993 ). Progress along the second line has led to the cloning, sequencing, and characterization of genetic loci defective in nonexchange chromosome segregation (ZHANG and HAWLEY 1990 ; ZHANG et al. 1990 ; RASOOLY et al. 1991 , RASOOLY et al. 1994 ; WHYTE et al. 1993 ; AFSHAR et al. 1995A , AFSHAR et al. 1995B ; see HAWLEY et al. 1993 for review).

Accurate estimation of E₀ tetrad frequency is of interest for a number of reasons. Failure of the homologous achiasmate system in D. melanogaster is the most common cause of nondisjunction, with nearly 76% of X chromosome spontaneous nondisjunction events arising from E₀ tetrads (KOEHLER et al. 1996 ). Within D. melanogaster, E₀ tetrad frequency estimates for different chromosomes from several laboratory stocks demonstrate that acrocentric X chromosomes have a higher E₀ tetrad frequency than do metacentric autosomes (WEINSTEIN 1936 ; MERRIAM and FROST 1964 ; CARPENTER 1973 ; CHARLESWORTH et al. 1985 ; RUTHERFORD and CARPENTER 1988 ; HAWLEY et al. 1992 ). No previous study has examined the patterns of variation in the frequency of E₀ tetrads among chromosomes sampled from natural populations. In a separate study, ZWICK et al. 1999 (this issue) report that X chromosomes from natural populations harbor high levels of genetic variation in rates of nondisjunction. They further identify two widespread intermediate frequency alleles at the nod locus, a chromokinesin required for proper segregation of achiasmate chromosomes, associated with increased nondisjunction. These observations make it of interest to determine if natural populations also contain high levels of variation in X chromosome E₀ tetrad frequency.

A number of recent studies report great similarity in the genetic events leading to spontaneous nondisjunction in Drosophila and humans (KOEHLER et al. 1996 ; LAMB et al. 1996 , LAMB et al. 1997A ; BUGGE et al. 1998 ; ROBINSON et al. 1998 ). This suggests the hypothesis that failure of an achiasmate system in humans might lead to nondisjunction, an event that occurs at an extraordinary frequency (HASSOLD et al. 1996 ). The existence of achiasmate systems in organisms other than D. melanogaster and the yeast Saccharomyces cerevisiae (DAWSON et al. 1986 ; GUACCI and KABACK 1991 ; LOIDL et al. 1994 ) remains largely unknown. The observation of substantial E₀ tetrad frequencies in the human genome might suggest a requirement for an achiasmate system to ensure the proper segregation of nonexchange chromosomes. Much as was done in Drosophila, identifying an achiasmate system and the genetic loci that function in this pathway could be expected to make important contributions to understanding the causes of nondisjunction in humans.

Accurate estimation of tetrad frequencies with Weinstein's method requires a dense genetic map and large sample sizes. Until recently, such maps were rare. As a consequence, most quantitative analyses of meiotic crossing over have been focused on ordering markers into maps, accommodating missing data, and modeling interference. For example, mapping functions have been used to infer the genetic distance from the observed exchange fraction of widely spaced markers (HALDANE 1919 ; BAILEY 1961 ; ZHAO and SPEED 1996 ). Chiasma interference, first observed in D. melanogaster, has been modeled using a variety of approaches (STURTEVANT 1915 ; MULLER 1916 ; MORTON and MACLEAN 1984 ; MCPEEK and SPEED 1995 ; ZHAO et al. 1995B ). Statistical methods for analyzing ordered and unordered tetrads have also been published (SNOW 1979 ; ZHAO and SPEED 1998A , ZHAO and SPEED 1998B ). However, the primary concerns of these studies have been the efficient construction of genetic maps, including accurate locus ordering and estimation of genetic distance. While tetrad estimation has long been possible in D. melanogaster, recent progress in the genomics of other model organisms and humans has resulted in both the dense genetic maps and the data from the progeny of a large number of meioses required for E₀ tetrad estimation.

Weinstein's method of tetrad frequency estimation has been successfully applied (WEINSTEIN 1936 ; MERRIAM and FROST 1964 ; KOEHLER et al. 1996 ; LAMB et al. 1997B ), but it suffers from a number of serious limitations. First, when sample sizes are limited, Weinstein's method frequently returns biologically meaningless negative tetrad frequency estimates, especially for the E₀ tetrad class. Application of a method that constrains tetrad frequency estimates within a biologically realistic range (i.e., 0.0 to 1.0) may lead to a significantly better estimator of tetrad frequencies. A recently proposed alternative estimation procedure for Weinstein's model using the EM algorithm eliminates the possibility of negative tetrad frequency estimates (BUGGE et al. 1998 ), but like Weinstein's method, suffers from other limitations. Weinstein's method ignores the sex of the progeny and the reciprocal marker arrangements of the individual chromatids recovered in an experimental cross. His method therefore cannot account for viability effects associated with the genetic markers or the sex of the progeny that might alter tetrad frequency estimates. Finally, we can find no published literature that has examined the effectiveness of Weinstein's estimator of tetrad frequencies with simulated datasets.

To overcome these limitations and improve on Weinstein's method of tetrad estimation, we first derive the multilocus exchange with interference and viability (MEIV) model. This model assumes that chromatid frequencies for single-spore data are multinominally distributed. The parameters of this distribution are derived from a plausible model of exchange, interference, and viability. We employ maximum-likelihood analysis of the MEIV model [MEIV likelihood analysis (MEIVLA)] to estimate tetrad frequencies for single-spore data. Second, we employ MEIVLA to estimate tetrad frequencies of a set of X chromosomes randomly sampled from D. melanogaster natural populations in North America and Africa. This is the first analysis of tetrad frequencies for chromosomes randomly sampled from natural populations. We observed surprising levels of variation in the frequency of E₀ tetrads among X chromosomes from both natural populations. In the majority of cases, a model incorporating the viability effects of phenotypic markers fit significantly better than one lacking such effects. Third, we reanalyze a recently published human dataset (LAMB et al. 1997B ) and show that MEIVLA is superior to WEINSTEIN's (1936) method of tetrad frequency estimation. Finally, using parameters estimated from the Drosophila and human datasets, we employ simulations to investigate the efficiency of point estimation and the power of MEIVLA as compared to the WEINSTEIN 1936 model. We particularly focus on the estimation of E₀ tetrad frequencies and the evidence for an achiasmate system in humans.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Drosophila lines:
D. melanogaster isogenic X chromosome lines were sampled at random from natural populations in North America and Africa. North American lines were collected from Raleigh, North Carolina as described in MIYASHITA et al. 1993 . African lines were collected from Zimbabwe as described in BEGUN and AQUADRO 1993 . All balancers and marker stocks are as described in LINDSLEY and ZIMM 1992 . To minimize the effects of the autosomes on X chromosome exchange, an autosomal isogenic background was constructed by employing a stock whose genotype was T(2;3)CyOTM6/+; mwh ry⁵⁰⁶ e¹; spa^pol (hereafter, spa^pol will be referred to as pol). This allowed the simultaneous isolation of a single second (marked with b) and third chromosome (marked with ri) that were subsequently backcrossed and made homozygous. The resulting genotype of the common isogenic background was b; ri; pol. Each experimental X chromosome, the balancer FM7a, and an X chromosome containing the markers y cv v f car, were substituted into this common genetic background.

D. melanogaster experimental cross:
Experimental females were constructed by crossing FM7/y cv v f car; b; ri; pol virgin females to X_i/B^SY; b; ri; pol males in bottles. Virgin females whose genotype was X_i/y cv v f car; b; ri; pol were collected and aged for 2 days. An experimental cross consisted of crossing 30 males whose genotype was y cv v f car/Y to an equal number of experimental females in bottles containing fresh glucose media. Each experimental cross was brooded, with the original parents transferred to new bottles on days 4 and 8. For any experimental cross, the first bottle was brood 1, the day 4 bottle was brood 2, and the day 8 bottle was brood 3. All experimental crosses were maintained in an incubator at 24° with a 12-hr dark/light cycle. For all broods within each experimental cross, all progeny were scored for their phenotypic markers on days 11 through 18, after which the bottles were discarded. The raw count data for each chromosome line, separated by broods, is contained in APPENDIX A. The North American bottles were uniformly more productive than the African bottles. It has previously been observed that female D. melanogaster from Zimbabwe, Africa exhibit premating isolation with males from other populations or laboratory stocks (WU et al. 1995 ; HOLLOCHER et al. 1997 ). In recent work on sperm displacement, it has been observed that in more than half of the matings of Zimbabwe females with Zimbabwe or non-Zimbabwe males, no sperm is transferred and these females often require multiple copulations to achieve insemination (J. COYNE, personal conversation). Similar processes affecting our extracted Zimbabwe X chromosomes stocks could explain the lower productivity of the African bottles.

Tetrad frequencies were calculated by MEIVLA as described in RESULTS. Two arrangements of the dataset were analyzed. First, tetrad frequencies were estimated from the total number of progeny for each X chromosome line in the study. Second, to examine the effect of brooding the parents in bottles, E₀ tetrad frequencies were estimated for the data from all three broods of a given X chromosome line. All tetrad frequency point estimates and confidence intervals were taken from one of three nested viability models. The model with the most parameters was the full viability model. This model assumes that each phenotypic marker has a different sex-specific viability effect. The single viability model assumes that each phenotypic marker has a specific viability effect that is identical in both males and females. The wild-type viability model assumes that the phenotypic markers have no effect on viability of the progeny in either sex. To determine the best-fitting model, we calculated the likelihood test statistic for each model and performed a likelihood-ratio test. We chose the full viability model as the best-fitting model if the P values from the likelihood-ratio test showed that the full model fit significantly better than the single and wild-type viability models. We chose the single viability model as the best-fitting model when it fit significantly better than the wild-type viability model and the full model did not provide a significantly better fit. We chose the wild-type viability model as the best-fitting model when neither of the two alternative models fit significantly better. We employed a P value of 0.05 as our significance threshold. Appendix 1 contains the pertinent likelihood-ratio test results, the viability parameter point estimates, and their confidence intervals. All other statistical analyses were carried out with JMP 3.2.1 (SAS Institute).

Human data:
Human data were obtained from Table 1 in LAMB et al. 1997B . Tetrad frequencies were calculated as described in RESULTS, using only the wild-type viability model.

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Tetrad analysis model:
The central problem of tetrad analysis is to employ the observed numbers or frequencies of chromatids to infer the unobservable frequencies of meiotic tetrads (Figure 1). We assume a known genetic map with K + 1 diallelic loci that divide an acrocentric chromosome into K regions. Alleles at each locus are labeled either + or -. For the D. melanogaster datasets, the - allele is assumed to be a visible mutant. Assume the regions between markers are small enough that there is at most a single exchange event within each region. Starting with a parent who is heterozygous at all loci, with one chromosome containing all + alleles and the other containing all - alleles, the basic experimental data consist of counting N individual chromatids, which are the products of N meioses. Chromatids are recovered in male or female progeny, contain either of two reciprocal marker arrangements, and can exhibit, or not exhibit, an exchange in any of the K regions. For a dataset with K regions, there are 4(2^K) = 2^K+2 distinct observable exchange classes of chromatids.

Each distinct observable type of chromatid is designated by N^l_i with (1 i 2^K) and (1 l 4). The i's partition the observable chromatids into 2^K exchange classes. We say two chromatids are in the same exchange classes if they exhibit an identical pattern of exchange. Thus, for example, two chromatids are in the same exchange class if they both show exchanges in regions 1 and 3 but no others. The algorithm that relates a specific number i to a specific exchange class is unimportant. We require only that each unique exchange class be assigned to a unique i. The l's divide each of the i exchange classes into four subclasses that account for the sex of the progeny and the reciprocal marker arrangement of the chromatid. Let N^l_i be the observed number of chromatids of exchange class i, recovered in males with the - allele for marker 1, N²_i be the observed number of chromatids of exchange class i, recovered in females with the - allele for marker 1, N³_i be the observed number of chromatids of exchange class i, recovered in males with the + allele for marker 1, N⁴_i be the observed number of chromatids of exchange class i, recovered in females with the + allele for marker 1, where

(1)

reflects all observed chromatids. By assumption, the N_i^l's are multinomially distributed.

The following set notation for each exchange class conveniently indicates the exchange location(s). Let I_i,j be an indicator variable that records whether exchange class i has an exchange in region j, where 1 j K. Thus

To maintain a complete list of the specific regions that have undergone exchange, for each exchange class i, create a set of integers S_i, such that S_i = {j | I_i,j = 1}. In other words, S_i is a list of those regions that are observed to have exchanges in class i. Let

(2)

be the number of elements in set S_i. Thus |S_i| reflects the number of exchanges in exchange class i. To derive the multinomial-likelihood expression, an explicit model incorporating exchange, crossover interference, and viability is required. Chromatid interference occurs when the chromatids that exchange at one site influence the choice of chromatids that undergo exchange at an adjoining site. Detection of significant chromatid interference is fairly rare in a variety of organisms (WEINSTEIN 1936 ; ZHAO et al. 1995A ). We therefore assume no chromatid interference in our model.

Incorporating exchange:
Assuming that all regions are small enough such that there is never more than one exchange per region, then in the absence of interference, the probability of an exchange event in region j is R_j. Let Z_i be the probability that any meiosis has exchanges only in the regions of S_i. In the absence of interference, Z_i would equal U_i, where U_i is given by

(3)

Incorporating crossover interference:
To model crossover interference, first note that if there is more than one exchange in class i (|S_i| > 1), then interference will reduce Z_i from U_i and increase the frequency of classes with fewer exchanges. Let 1 - P_i be the proportion by which the ith exchange class is decreased due to interference, with 0 P_i 1. For P_i = 1, no interference is acting, while P_i = 0 indicates complete suppression of the ith exchange class. For a model with K regions, there are 2^K - K - 1 P_i terms that may differ from one. For a specific example, suppose class i consists of a triple crossover event in regions one, two, and three. Given that interference occurs (i.e., P_i < 1), by assumption, the proportion of triple exchanges will decrease while the number of double exchanges increases. There are three different possible classes of double exchanges that increase in frequency for a given triple exchange. By assumption, the probabilities of these events are

In general, multiple crossovers in other regions can be resolved in a similar fashion. To do this, let

(4)

(5)

Note that Y_i is given in terms of Y_j, where S_i S_j and |S_j| - |S_i| = 1. Therefore, one must first calculate Y for the class with exchanges in all regions, then for all classes with exchange in all but one region, and so forth, down to the class with no exchanges. Z_i, the probability that a meiosis has exchanges only in regions S_i, is given by

(6)

Incorporating viability effects of markers:
The viability effects of markers in progeny of both sexes will act to decrease the recovery of certain chromatids relative to others. To incorporate viability, we assume that the + allele for each marker has no effect on fitness (i.e., has fitness = 1). Assume that the - allele at marker i has fitness effect ^m_i in males and ^f_i in females and that these fitness effects are not influenced by culture conditions. We assume a multiplicative model of epistasis. Let

be the indicator that the allele at the j + 1 locus differs from the allele at the first locus of exchange class i. (W_i,j is 0 if locus 1 and j + 1 are both + or are both -, and equal to 1 otherwise.) Let

(7)

(8)

(9)

(10)

Thus G^l_i is proportional to the frequency of chromatids after accounting for marker genotype and sex of the progeny. Letting

(11)

be the sum of these proportions, we normalize by

(12)

to obtain the final estimate of chromatid frequencies.

Final tetrad frequency estimation:
In organisms where all four products of a single meiosis can be recovered, the N^l_i's are multinomially distributed with means NE^l_i. But with single-spore data, only one of the four products of a single meiosis is recovered. Chromatids derived from any particular exchange are recovered with probability 1/2 because exchange is assumed not to occur between sister chromatids (WEINSTEIN 1936 ). Therefore, our N^l_i are multinomially distributed with means NF^l_i, where

(13)

The overall likelihood, L, of the observations is

(14)

Our approach to solving this likelihood expression is to numerically find the values of our parameters (K exchange frequencies, R's; 2^K - K - 1 P's; and 2K + 2 v's, for a total of 2^K + 2K + 1) that maximize (14). This was accomplished by minimizing -log(L) using the "Powell" algorithm (PRESS et al. 1992 ). Once the parameters associated with the maximum have been found, tetrad frequency E_i can be found by noting

(15)

Confidence intervals for all the tetrad frequency point estimates were determined by using the Powell algorithm to search the surrounding likelihood surface. For a calculated maximum likelihood, L_m, the range of parameter values that gave an L such that log(L_m) - log(L) < 2 are considered within the 95% confidence interval. The minimum and maximum tetrad frequencies implied by parameters in the 95% confidence interval are considered the confidence limits. This procedure does not guarantee that all tetrad frequencies between the minimum and maximum will be within the 95% confidence interval. However, for a sufficiently smooth likelihood surface, all intermediate values will be contained.

Tetrad frequency estimation in Drosophila:
Tetrad frequency estimates calculated from the best-supported viability model for six African X chromosomes and nine North American X chromosomes are contained in Table 1 and Table 2, respectively (see Appendix 1 for raw data, viability parameter estimates, and their confidence intervals). For three of six African X chromosomes and nine of nine North Carolina X chromosomes, a model incorporating viability was the best-fitting model. Despite the observed viability effects of the morphological mutants employed in this study, their effect on estimates of E₀ tetrad frequency appears quite small.

The mean estimated E₀ tetrad frequencies for the African (0.118) and the North American (0.105) populations are not significantly different (P = 0.23). The X chromosome samples, however, exhibit significant variation within both natural populations. This is most clearly seen by the nonoverlapping E₀ tetrad frequency point estimates and their confidence intervals in Figure 2. Because X chromosomes from both populations were substituted into a common isogenic background, the source of this variation should reside on the individual X chromosomes.

View larger version (11K): In this window In a new window Download PPT slide   Figure 2. E0 tetrad frequency point estimates and their confidence intervals as calculated for X chromosomes from North American (circle) and African (square) natural populations. Note that most of the variation is found among chromosomes within populations—the population means (vertical lines with either a circle or a square) were not significantly different (P = 0.23).

The E₀ tetrad frequency estimates for each brood were also calculated from the best-supported viability model for the X chromosomes from both natural populations. The mean E₀ tetrad frequency estimates for the three broods are significantly different (Figure 3; P = 0.02). Comparing each pair of means, corrected for multiple comparisons by the Tukey-Kramer HSD, shows that first and third broods are the only two that are significantly different (P < 0.05).

View larger version (17K): In this window In a new window Download PPT slide   Figure 3. E0 tetrad frequency point estimates and their confidence intervals as calculated for each of three broods for all X chromosomes from North America and Africa. Brood 1 (circle), brood 2 (triangle), and brood 3 (square) are presented along with the mean E0 tetrad frequency for each brood (vertical lines marked with either a circle, triangle, or square). The means of broods 1 and 3 were found to be statistically significantly different (P < 0.05) after correcting for multiple tests.

Tetrad frequency estimation in humans:
We determined the raw count data from the human dataset in LAMB et al. 1997B and reanalyzed their data (Table 3). We conclude that the maximum-likelihood estimate for the frequency of E₀ tetrads in human females is 1.5%. Our E₀ tetrad frequency estimate is nearly identical to that in LAMB et al. 1997B for this single dataset. The E₁ and E₂ tetrad frequency point estimates differ. For all tetrad frequency estimates, however, MEIVLA returns confidence intervals that are significantly smaller than those in LAMB et al. 1997B . Furthermore, the confidence intervals for the E₀ tetrad frequency point estimate do not include biologically meaningless negative tetrad frequency estimates.

Simulation results:
To investigate the efficiency of the MEIVLA point estimation procedures and the accuracy of confidence intervals for various sample sizes, we simulate datasets using parameters estimated from the D. melanogaster NC14X line and the human dataset. For the Drosophila NC14X parameter set, we generated 500 simulated datasets for each of six different sample sizes. For the human parameter set, we generated 500 simulated datasets for each of nine different sample sizes.

Using parameters determined from the D. melanogaster NC14X line, the majority of E₀ tetrad frequency estimates calculated by MEIVLA are closer to the true value (0.104, Table 2) than those of WEINSTEIN 1936 for all sample sizes (Table 4). Employing the human dataset parameters, an even greater majority of E₀ tetrad frequency estimates calculated by MEIVLA are closer to the true value (0.0145, Table 5) than those of WEINSTEIN 1936 . This advantage is particularly evident for small sample sizes. Both methods converge to similar point estimates as sample size increases, but even for large samples sizes, MEIVLA consistently outperforms the model of WEINSTEIN 1936 .

We draw two conclusions from analysis of the human data in LAMB et al. 1997B and the results of our simulation studies. First, the maximum-likelihood estimate for the frequency of chromosome 21 E₀ tetrads in human female meiosis is ~1.5%. Second, on the basis of our simulation results, we conclude that with sample sizes of 276, it is not possible to exclude either that there are no chromosome 21 E₀ tetrads or the alternative, that E₀ tetrad frequencies are similar to those observed for the X chromosome in Drosophila. Our simulation analyses employing estimated human parameters (Table 5) suggest that if chromosome 21 E₀ tetrad frequency were as low as 1.5%, with moderate sample sizes (1000–5000 meioses), it should be possible to determine an upper bound that would exclude the E₀ tetrad frequencies seen for the Drosophila X chromosome. To obtain a chromosome 21 E₀ estimate that excludes zero at the lower bound, significantly larger sample sizes of 20,000–30,000 meioses are required. Our simulations with the Drosophila parameters also point out the inadequacy of current human sample sizes, because with a Drosophila sample size of 276, we cannot exclude the lower bound of zero in an organism where E₀ tetrad frequency is ~10% (see sample size 276 in Table 4).

	DISCUSSION

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

We present MEIVLA, a method of tetrad frequency estimation that significantly improves upon those originally derived by Weinstein 1936 and extended in Lamb et al. 1997b :
First, the MEIV model incorporates a clearly defined model of exchange, interference, and viability whose parameters define a multinomial distribution for single-spore data. The derivation of the MEIV model ensures that biologically meaningless results such as negative tetrad frequency estimates are not produced by MEIVLA of the MEIV model. Second, the MEIV model allows the determination of the magnitude of marker viability effects, permitting their incorporation into MEIVLA. Previous methods of tetrad frequency estimation have not incorporated viability in their estimation procedures. Third, simple methods that explore the likelihood surface surrounding its maximum allow the direct determination of confidence intervals. Finally, MEIVLA point estimates and confidence intervals are consistently superior to previous methods. This advantage is most evident in situations where the E₀ tetrad frequency is low and sample sizes are small. Because both of these conditions are met in most samples from human populations, MEIVLA is ideally suited for the analysis of human datasets.

One potential criticism of our exchange model is that we only allow zero or one exchange between markers. However, we do not believe that this is a significant problem for two reasons. First, our method is conditioned upon known, dense genetic maps—such as those found in model organisms and that are increasingly available in nonmodel organisms. Given a sufficiently dense map, it is possible to choose markers so that this assumption is met. Second, the examination of genetic maps from model organisms supports the view that meiosis is regulated in such a manner as to favor a single exchange per chromosome arm. This is evidenced by the excess of single-crossover (E₁) tetrads (Table 1 and Table 2; LAMB et al. 1997B ; see review in HAWLEY 1988 ). Furthermore, when double crossovers do occur during meiosis, interference acts to space the chiasmata, making it very unlikely that a meiotic double-exchange event would occur even between moderately spaced markers. Analysis of our Drosophila data rejects a MEIV model lacking interference (data not shown). Exchange events of a nonmeiotic origin might occur, leading to apparent closely spaced double crossovers (SUZUKI et al. 1966 ), but these events are expected to be rare. Our model is specifically designed to analyze data from acrocentric chromosomes and assumes that the fitness effects of morphological markers are not influenced by culture conditions. A more general model that relaxes these assumptions, allowing the analysis of metacentric chromosomes while incorporating fitness variation between cultures, is in development.

D. melanogaster natural populations harbor a significant level of variation among X chromosomes in their E₀ tetrad frequency:
This study is the first to examine the E₀ tetrad frequency of X chromosomes sampled from natural populations. Previous studies, concerning a small number of laboratory stocks, provided estimates of E₀ tetrad frequency that largely agree with those presented in this study (WEINSTEIN 1936 ; MERRIAM and FROST 1964 ; CARPENTER 1973 ; CHARLESWORTH et al. 1985 ; RUTHERFORD and CARPENTER 1988 ; HAWLEY et al. 1992 ; KOEHLER et al. 1996 ). However, none of these studies were able to address the patterns of variation in natural populations. It is quite striking that both the African and North American samples contained X chromosomes that significantly differed in their E₀ tetrad frequency. Given the observation that the majority of spontaneous nondisjunction arises from nonexchange (E₀) tetrads (KOEHLER et al. 1996 ) and the uniformly deleterious consequences of nondisjunction, it is quite surprising that natural populations harbor this variation.

One might expect that natural selection would act to decrease the frequency of E₀ tetrads because of the deleterious consequences of aneuploidy arising from nondisjunction. In a separate companion study of the patterns of genetic variation underlying nondisjunction in female meiosis, ZWICK et al. 1999 (this issue) demonstrate that X chromosomes from the same two natural populations harbor high levels of genetic variation in rates of nondisjunction in a sensitized assay of E₀ tetrads. They furthermore identify two widespread intermediate frequency alleles at the nod locus, a chromokinesin required for the proper segregation of achiasmate chromosomes, which are significantly associated with an increased frequency of nondisjunction. To account for the high levels of genetic variation observed in female meiosis, ZWICK et al. 1999 (this issue) present an evolutionary model, referred to as the oötid competition model, that can account for high rates of genetic variation in the efficiency of chromosome segregation during female meiosis.

Although the effects on E₀ tetrad frequency estimation were not large, incorporation of a multiplicative viability model significantly improved the fit of the MEIV model in a majority of cases. Incorporation of a specific viability model can improve tetrad frequency estimation when the viability effects are large. Our data further show a significant decrease in mean E₀ tetrad frequency for the third brood as compared to the first brood. This pattern largely agrees with those in other studies that have shown increased exchange as female Drosophila age. The pattern of variation in exchange in relation to maternal age, however, is not straightforward and exhibits substantial variation in different experiments (see ASHBURNER 1989 for a review). One important point to consider in the observation of elevated exchange in older female Drosophila is the possible nonmeiotic origin of exchange. Mitotic crossing over, perhaps associated with transposable element activity, in meiotic stem cells can result in an apparent increase in observed meiotic exchange. Such nonmeiotic exchange is expected to increase with maternal age. Simple "single-spore" tetrad estimation cannot distinguish between these sources of exchange and normal meiotic crossing-over.

We disagree with the conclusions in Lamb et al. 1997b that analysis of the human dataset supports the concept of obligate exchange and fails to provide evidence for a secondary segregation system in human females:
First, the concept of obligate exchange either requires that all tetrads undergo exchange or that homologous chromosomes in tetrads that fail to exchange segregate at random. Our maximum-likelihood estimate for the frequency of chromosome 21 E₀ tetrads in human females is 1.5%, nearly identical to that observed in LAMB et al. 1997B . Because of the limited sample size upon which this estimate is based, the confidence intervals determined from the MEIV model range from 0.0 to 8.3%. Nevertheless, the maximum-likelihood estimate is not 0, and it seems clear that the proper interpretation of both analyses is that they are consistent with a low frequency of chromosome 21 E₀ tetrads. Therefore, because this analysis and LAMB et al. 1997B suggest that the E₀ frequency of chromosome 21 is not 0, to conclude that the model of obligate exchange is correct for humans, one must demonstrate that chromatids in E₀ tetrads segregate at random. This has not been done, nor can it be done with the data contained in LAMB et al. 1997B . The analysis of BUGGE et al. 1998 suffers from a similar set of problems. Thus, the conclusion that the concept of obligate exchange is supported is not warranted.

For quite different reasons, the analysis of the LAMB et al. 1997B dataset cannot speak to the alternative hypothesis of the existence of a secondary segregation system in humans. In Drosophila, the vast majority of nondisjunction arises from the X chromosome. Thus, to a good approximation, the genome-wide E₀ tetrad frequency, which we refer to as ^GE₀, is simply equal to the X chromosome E₀ tetrad frequency. In humans, ^GE₀ must reflect the probability that one or more chromosome pairs may fail to undergo exchange during female meiosis. Because the human karyotype has 23 pairs of chromosomes, tetrad frequency analysis of a single chromosome cannot be used to estimate ^GE₀.

One possible hypothesis is that the Drosophila and human ^GE₀ tetrad frequencies might both be ~10%. If this were the case, the E₀ tetrad frequency for any individual chromosome in humans would be expected to be lower than that observed for the Drosophila X chromosome. To determine the ^GE₀ tetrad frequency for the human genome, the best experimental design would require the simultaneous analysis of many, if not all, human chromosomes. Simultaneous analysis is required because exchange patterns of different chromosomes may not be independent. One example of such interactions is the interchromosomal effect (STURTEVANT 1919 ; reviewed in LUCCHESI 1975 ), the observation that inversions suppressing exchange on one chromosome act to increase exchange on other chromosomes. Studies of single human chromosomes cannot account for this possible source of variation. The analysis in BUGGE et al. 1998 , which employs a hypothesis-testing approach in calculating simultaneous tetrad frequencies for multiple chromosomes, cannot detect interactions between chromosomes. Furthermore, our simulation studies (Table 4 and Table 5) make it quite clear that sample sizes of ~80 meioses analyzed by BUGGE et al. 1998 are far too small to allow any conclusion. However, given the appropriate data, an achiasmate system could be detected in humans much as was done in Drosophila if the observed level of female-specific nondisjunction arising from nonexchange tetrads was significantly lower than that predicted by a null model assuming random segregation of all chromosomes in ^GE₀ tetrads.

To describe how such an analysis would be performed, we assume a very simple model with n independent chromosomes, each with a probability of forming an E₀ tetrad (^GE₀ n), then

(16)

where D is the expected frequency of normal disjunction and nondisjunction arising from tetrads with at least one exchange in females. Therefore, 1 - D is the expected frequency of female-specific nondisjunction arising from E₀ tetrads. For a single E₀ tetrad, is the expected frequency of normal segregants. This equation reduces to

(17)

Because of the great variation among chromosomes in their frequency of nondisjunction, to simplify this analysis, we chose three different numbers of chromosomes. The sample size of five was chosen to reflect those chromosomes (15, 16, 18, 21, and 22) whose frequencies of nondisjunction are the best characterized (LAMB et al. 1996 , LAMB et al. 1997A ; BUGGE et al. 1998 ; ROBINSON et al. 1998 ; reviewed in HASSOLD et al. 1996 ). A sample size of 10 was chosen to encompass nondisjunction arising from the 10 smallest human chromosomes (13–22). A sample size of 23 was chosen to reflect the assumption that each pair of human chromosomes is equally likely to undergo nondisjunction. For ease of analysis, and are assumed to be identical for each set (5, 10, and 23) of chromosomes. Violation of this assumption, as evidenced by variation among chromosomes in their frequency of nondisjunction (reviewed in HASSOLD et al. 1996 ) or the nonindependence of recombination among different chromosomes, could be incorporated into future, more complicated models.

If we first assume that the actual genomic rate of female-specific nondisjunction is 0.2 (see review in HASSOLD et al. 1996 ) and that only 20% of this total arises from nonexchange tetrads, then 4% of all meioses would be expected to generate female-specific nondisjunction arising from E₀ tetrads. Table 6 contains the expected frequency of female-specific nondisjunction (1 - D) for a small set of representative parameters with = 0.5. In Drosophila, it has long been recognized that nullo exceptions are more frequent than diplo exceptions and that this can cause the rate of nondisjunction to be >0.5 in the absence of an achiasmate system. To reflect these observations, Table 7 contains the expected frequency of female-specific nondisjunction (1 - D) for the same small set of representative parameters with = 0.25. The values in italic type in Table 6 and Table 7 represent expected frequencies of nondisjunction originating from nonexchange tetrads in the absence of an achiasmate system that are greater than our assumed frequency of female-specific nondisjunction arising from E₀ tetrads. Thus the parameter sets that lead to levels of nondisjunction >4% represent values necessary to reject the null model of random segregation and thereby would cause one to conclude that an achiasmate system exists in humans.

View this table: In this window In a new window   Table 6. Expected frequency of nondisjunction in the absence of an achiasmate system ( = 0.5, see text)

View this table: In this window In a new window   Table 7. Expected frequency of nondisjunction in the absence of an achiasmate system ( = 0.25, see text)

Thus, much as in the case of Drosophila, there are two alternative strategies one could employ to detect achiasmate systems. The first method requires estimation of ^GE₀ and an estimate of the female-specific rate of nondisjunction. Large datasets would be required, but in principle, if an achiasmate system exists in a specific organism, it should be possible to eliminate a null model assuming no achiasmate system. Datasets are rapidly becoming available in a number of other model and nonmodel organisms, which should allow detection of putative achiasmate systems. A second strategy, which is more direct but more difficult to carry out, would aim to characterize genetic loci whose null phenotypes specifically affect the segregation of achiasmate chromosomes. The identification and characterization of mammalian homologs of Drosophila genes that function in achiasmate segregation (i.e., TOAKI et al. 1996 ) will likely lead to the direct genetic identification and characterization of an achiasmate system, if such a system exists.

	ACKNOWLEDGMENTS

The authors thank Jennifer Salstrom for her aid in scoring the fly crosses. We thank Michael Turelli, John Gillespie, Mark Grote, and two anonymous reviewers for their discussion and assistance in improving the manuscript. M.E.Z. also thanks R. Scott Hawley for providing valuable insight into the mechanisms of chromosome segregation in Drosophila. Fellowship and research support was provided to M.E.Z. by a National Science Foundation Pre-Doctoral Fellowship, a National Science Foundation Dissertation Improvement Grant DEB 96-23970, the Center for Population Biology at UC Davis, and the Daphne and Ted Pengelley Research Award. This research was also partially funded by a National Science Foundation Grant DEB 95-09548 to C.H.L.

Manuscript received December 4, 1998; Accepted for publication April 26, 1999.

	APPENDIX A

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Drosophila melanogaster RAW DATA

	APPENDIX B

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Viability parameter values and P values for best-supported viability model

	LITERATURE CITED

TOP ABSTRACT MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

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