Abstract
Halftetrads, where two meiotic products from a single meiosis are recovered together, arise in different forms in a variety of organisms. Closely related to ordered tetrads, halftetrads yield information on chromatid interference, chiasma interference, and centromere positions. In this article, for different halftetrad types and different marker configurations, we derive the relations between multilocus halftetrad probabilities and multilocus ordered tetrad probabilities. These relations are used to obtain equality and inequality constraints among multilocus halftetrad probabilities that are imposed by the assumption of no chromatid interference. We illustrate how to apply these results to study chiasma interference and to map centromeres using multilocus halftetrad data.
HALFTETRADS, where two meiotic products from a single meiosis are recovered together, arise in different forms in a variety of organisms. The first wellstudied halftetrad data were attachedX chromosomes in Drosophila (Beadle and Emerson 1935; Welshons 1955). Halftetrads were also constructed using autosomes in Drosophila (Baldmin and Chovnick 1967). They have been used in the study of many other organisms, including maize (Rhoades and Dempsey 1966), potatoes (Mendiburu and Peloquin 1979), leopard frog (Volpe 1970), rainbow trout (Thorgaardet al. 1983; Allendorfet al. 1986), salmonid fish (K. R. Johnsonet al. 1987), catfish (Liuet al. 1992), and zebrafish (S. L. Johnsonet al. 1995). In mammals, halftetrads can be studied in the form of autosomal trisomies (Mortonet al. 1990; Shermanet al. 1991) and ovarian teratomas (Ottet al. 1976; Eppig and Eicher 1983; Chakravarti and Slaugenhaupt 1987; Chakravartiet al. 1989; Dekaet al. 1990). However, the material required of trisomies and teratomas is rare, and the recombination pattern in meiosis that generates trisomies and teratomas can differ from that in normal meiosis (Sherman et al. 1991, 1994; Lambet al. 1996). Cui et al. (1992) introduced one technique that uses the polymerase chain reaction to analyze the products of meiosis I in individual secondary oocytes. This method has been since used to map genetic markers in mice (Cuiet al. 1992) and cows (Jarrellet al. 1995).
Halftetrads may arise from different mechanisms. In Figure 1, we illustrate how meiosis I (MI) nondisjunctions lead to halftetrads. It is easy to see that when there is no crossover between centromere and a heterozygous marker, MI nondisjunction results in halftetrads being heterozygous at the marker. When there is one crossover between centromere and the marker, there is equal chance of producing homozygous and heterozygous halftetrads. In Figure 2, the mechanism of meiosis II (MII) nondisjunctions is shown. Given no crossovers between centromere and a heterozygous marker, MII nondisjunction always results in homozygous halftetrads, whereas a single crossover always results in heterozygous halftetrads. For halftetrads from MI nondisjunctions, the two strands were attached to different centromeres during meiosis, whereas the two strands in halftetrads from MII nondisjunctions were attached to the same centromere during meiosis. MI and MII nondisjunction are not the only mechanisms that are responsible for halftetrads. For example, attachedX chromosomes in Drosophila are the result of a different mechanism (Beadle and Emerson 1935). In this article, we broadly classify halftetrads into two types: type I halftetrads, in which no crossover between centromere and marker always results in heterozygous halftetrads, and type II halftetrads, in which no crossover always results in homozygous halftetrads. On the basis of this classification, halftetrads from MI nondisjunctions are type I halftetrads, and those from MII nondisjunctions are type II halftetrads. AttachedX chromosomes are type I halftetrad. Halftetrads from fish are mostly type II halftetrads. Autosomal trisomies and ovarian teratomas can be of either type. In addition, ovarian teratomas can result from mechanisms other than MI or MII nondisjunctions (Surtiet al. 1990). Throughout this article, we also make the assumptions that the parental origin of the halftetrads is known and that phases are known in parents. These assumptions are usually true for experimental organisms, although human halftetrad data are more complex and may not satisfy these assumptions. For either type I or type II halftetrads, a further distinction may be made when two or more markers are studied: haplotype information can be either available (attachedX chromosomes in Drosophila) or unavailable. The two types of halftetrads are called type Ia and type IIa halftetrads when haplotype information is available and type Ib and type IIb halftetrads when such information is not available.
As with ordered and unordered tetrads, halftetrads are very valuable in studying crossovers during meiosis because (1) chromatid interference and chiasma interference can be distinguished with halftetrads (Welshons 1955); (2) when chromatid interference is absent, chiasma interference can be detected with two loci and may be detected with just one locus if the locus is sufficiently far from the centromere; and (3) the position of the centromere can be mapped.
Most studies on halftetrads (Welshons 1955; Cote and Edwards 1975; Ottet al. 1976; Chakravarti and Slaugenhaupt 1987) used only three loci for the detection of chromatid interference and one locus for the mapping of centromeres. In the context of chromatid interference, Zhao et al. (1995a) and Zhao and Speed (1998) derived a set of linear equality and inequality constraints on the multilocus probabilities of unordered and ordered tetrad patterns under the assumption of no chromatid interference (NCI). These constraints can be used to test the assumption of NCI and to order markers. Risch and Lange (1983) and Zhao et al. (1995b) fitted chiasma interference models to multilocus unordered tetrad data. For halftetrad data analysis, Chakravarti et al. (1989) proposed two approaches for multilocus analysis. One was to assume that there are at most three chiasmata across the region under genetic study. The other was to treat the proximal marker as a pseudocentromere relative to the distal marker. Because the first approach does not apply to tetrads with more than three chiasmata and the second approach applies only in the absence of chiasma interference, neither is completely satisfactory. Da et al. (1995) presented an approach to analyzing two markers under the assumptions of NCI and no chiasma interference. Assuming complete chiasma interference, Tavoletti et al. (1996) proposed a maximum likelihood method. Lamb et al. (1997) followed Weinstein's (1936) approach to inferring joint chiasma probabilities at the fourstrand stage. There is no assumption on the chiasma process in this approach except that there are at most two chiasmata in each marker interval.
In this article, we assume that when strands attached to different centromeres during meiosis form a halftetrad, each of the two strands attached to the same centromere has equal chance of being in the halftetrad, and when strands attached to the same centromere during meiosis form a halftetrad, the two pairs have equal chance of being in the halftetrad. Under this assumption, four nonsister chromatid pairs have the same chance of being observed in a halftetrad from MI nondisjunctions. For MII nondisjunctions, two sister chromatid pairs have the same chance of being recovered. This assumption is abbreviated as RRA (random recovering assumption) in the following discussion.
Under RRA and the assumption of NCI, we derive multilocus halftetrad probabilities as functions of multilocus ordered tetrad probabilities. These relations are then used to derive linear equality and inequality constraints among multilocus halftetrad probabilities imposed by NCI. The constraints can be used to test NCI, order markers, and construct genetic maps under a certain chiasma process model. We discuss onemarker and twomarker cases in detail before presenting the general results for multiple markers. The four halftetrad types are discussed in the order of type IIa, IIb, Ia, and Ib halftetrads, respectively.
Since only two of the four strands are recovered in a halftetrad, the original ordering of the four strands is lost. It is impossible in general to detect violation of random spindle to centromere attachment assumption (Griffithset al. 1996; Zhao and Speed 1998) using halftetrad data.
We use the following notations in this article. Markers are denoted by script letters. For example, we use
METHODS
No chromatid interference (one marker): With one marker, haplotype information is irrelevant. We need consider only two types: type I and type II halftetrads.
Type II halftetrads: For a heterozygous marker
Type I halftetrads: For MI nondisjunction, patterns AA and aa can result only from SDS ordered tetrads. Pattern Aa can result from both FDS and SDS tetrads. Under RRA, SDS gives rise to AA, Aa, and aa with probability 1/4, 1/2, and 1/4. Therefore, P(AA) = P(aa) = P(SDS)/4 and P(Aa) = P(SDS)/2 + P(FDS). This leads to the inequality constraint for type I halftetrads: P(Aa) ≥ P(AA) + P(aa) = 2 P(AA). This inequality constraint is imposed by RRA.
No chromatid interference (two markers, type IIa halftetrads): For two markers
Two markers on different chromosomes: Let p and q denote the probability of SDS at
Two markers on different sides of the centromere
Two markers on the same side of the centromere (
Therefore, there is a onetoone correspondence between the
It was shown in Zhao and Speed (1998) that the inequality constraints imposed by NCI for ordered tetrad probabilities are
Using constraints under NCI, we can distinguish different configurations for two markers: on different chromosomes, on the same side, or on different sides of the centromere. Zhao and Speed (1998) discussed how to apply constraints imposed by NCI to order markers using ordered tetrads.
No chromatid interference (two markers, type IIb halftetrads): For type IIb halftetrads—because haplotype information is unavailable—two patterns, AB/ab and Ab/aB, which are distinguishable in type IIa halftetrads, are no longer distinguishable. This leads to 9, instead of 10 distinguishable patterns: (AA; BB), (AA; Bb), (AA; bb), (Aa; BB), (Aa; Bb), (Aa; bb), (aa; BB), (aa; Bb), and (aa; bb). Under RRA, the following four pairs should have the same probability: (AA; BB) and (aa; bb), (AA; Bb) and (aa; Bb), (AA; bb) and (aa; BB), and (Aa; BB) and (Aa; bb).
Two markers on different chromosomes: Because AB/ab and Ab/aB have the same probability when markers are on different chromosomes, there is no loss of information under NCI compared to type IIa data. The probabilities are the same as type IIa halftetrads.
Two markers on different sides of the centromere (
Two markers on the same side of the centromere (
Therefore, we can obtain only p_{10} + p_{12} from the u_{i}_{1}_{i}_{2} but not the individual values of p_{10} and p_{12}. The inequality constraints on the
No chromatid interference (two markers, type Ia halftetrads): For type Ia halftetrads, the strands in the same halftetrad were not attached to the same centromere at the fourstrand stage during meiosis. Under RRA, each of the four nonsister chromatid pairs has the same chance of being recovered in a halftetrad. As for type IIa halftetrads, there are 10 distinguishable types and at most six distinct probabilities.
Two markers on different chromosomes: We also use p and q to denote the probability of SDS at
Two markers on different sides of the centromere (
RRA imposes the constraints that the expressions on the righthand side of the above equations be nonnegative. Because the inequality constraints among ordered tetrad probabilities imposed by NCI are α ≥ β and γ + δ ≥ 2β (Zhao and Speed 1998), the inequality constraints among the p_{i} imposed by NCI are p_{5} ≥ p_{4} and 3p_{2} + 3p_{3} ≥ 5p_{1} + 2p_{4}.
Two markers on the same side of the centromere (
No chromatid interference (two markers, type Ib halftetrads): As type IIb halftetrads, because haplotype information is unavailable, patterns AB/ab and Ab/aB cannot be distinguished. There are nine distinguishable patterns. Under RRA, the following four pairs of patterns have the same probability: (AA; BB) and (aa; bb), (AA; Bb) and (aa; Bb), (AA; bb) and (aa; BB), and (Aa; BB) and (Aa; bb). Therefore, there are at most five distinct probabilities for type Ib halftetrads.
Two markers on different chromosomes: Because AB/ab and Ab/aB have the same probability, there is no loss of information under NCI comparedto type Iadata. The probabilities are the same as those of type Ia halftetrads.
Two markers on different sides of the centromere (
Two markers on the same side of the centromere (
From the
The constraints imposed by RRA are that the
No chromatid interference (multiple markers on the same side of the centromere): For n markers in the
Type IIa halftetrads: To simplify the derivation of the general results for n markers, we proceed differently from the discussion of the one and twomarker cases. We first assume that the two strands have already been labeled and are thus distinguishable. Then there are four possible patterns, (A_{r}A_{r}), (A_{r}a_{r}), (a_{r}A_{r}), and (a_{r}a_{r}), denoted by 0, 1, 2, and 3, respectively, at each marker
There are 2 × 3^{n}^{−1} distinct probabilities for ordered tetrad data under NCI. These different ordered tetrad classes are denoted by (
Because the probabilities h_{i1…in} can be expressed in terms of the
The inequality constraints can be established as follows. Define
Type IIb halftetrads: For type IIb halftetrads, there are three patterns at each marker. These three patterns are denoted by 0, 1, and 2, corresponding to observing 0, 1, and 2 copies of allele A_{r} at
In the discussion of twomarker data, it was noted that p_{00} and p_{02} cannot be determined from the u_{i}_{1}_{i}_{2}. Similarly in the n marker case, not all the
Type Ia halftetrads: For type Ia halftetrads, let
Type Ib halftetrads: For type Ib halftetrad data, the relations between the
No chromatid interference (multiple markers on different sides of the centromere): Consider markers on different sides of the centromere in the order of
Type IIa halftetrads: We assume that the two strands have been labeled. Any type IIa halftetrad pattern can be represented by ij = (i_{1}i_{2} … i_{n}; j_{1}j_{2} … j_{m}), where each i_{k} (k = 1, · · ·, n) or j_{l} (l = 1, · · ·, m) is 0, 1, 2, or 3. The probability of this halftetrad pattern is denoted by h_{(i1i2…in:j1…jm)}. If the centromere were observable, ordered tetrad pattern could be represented by
Type IIb halftetrads: As for type IIa halftetrads, the probability u_{ij} for type IIb halftetrad pattern ij = (i_{1}i_{2} … i_{n};j_{1}j_{2} … j_{m}), where each i_{k} or j_{l} is 0, 1, or 2, can be expressed in terms of the p_{i}^{t}_{j}^{t} as
Multilocus genetic mapping: In the studies of ordered tetrads, Zhao and Speed (1998) compared various map functions that relate the map distance between the centromere and a marker to the observed FDS and SDS proportions at the marker. It was found that most map functions proposed in the literature agree fairly well for SDS proportions up to 2/3.
For type II halftetrads, being heterozygous at a marker corresponds to SDS for the four strands. Therefore, the relation between the proportion of heterozygous halftetrads and the map distance is the same as that plotted in Figures 1 and 2 of Zhao and Speed (1998) for different map functions.
For type I halftetrads, the probability of being heterozygous at a marker is the sum of the FDS proportion and half of the SDS proportion. The probability of being heterozygous at a marker as a function of the map distance between this marker and the centromere can be easily derived.
A crossover process model is needed for multilocus analysis. Different models have been proposed in the literature to model the crossover process (McPeek and Speed 1995). Among them, the chisquare model was found to provide better fit to data from different organisms (Zhaoet al. 1995b).
The chisquare model, which was first introduced by Fisher et al. (1947), was suggested as a plausible biological model by Foss et al. (1993). The model, which is represented as Cx(Co)^{m}, assumes that the crossover intermediates, C events, are randomly distributed along the fourstrand bundle, and every intermediate resolves either as a crossover (Cx) or not (Co). When an intermediate resolves as a Cx, the next m intermediates must resolve as a Co, and after m Co's the next intermediate must resolve as a Cx. The process is made stationary by letting the leftmost crossover intermediate have the same chance of being one of Cx(Co)^{m}. The chisquare model has recently been generalized to a more general class, the Poissonskip model (Langeet al. 1997). Both the chisquare model and the Poissonskip model lead to closedform expression for the probability of any ordered tetrad pattern. This gives a rather flexible and tractable class of models for genetic linkage analysis. Note that the Poisson model is a special case of the chisquare model.
For an arbitrary number of markers on the same side or different sides of the centromere, Zhao and Speed (1998) derived general closedform expressions for ordered tetrad probabilities under the Cx(Co)^{m} model. Using these results and the relations we derived between halftetrad probabilities and ordered tetrad probabilities, we can evaluate any halftetrad probability. Therefore, maximum likelihood estimates of the interference parameter m and the genetic distances among the markers and the centromere are tractable under this class of models.
RESULTS
In the previous section, we derived general relationships between multilocus halftetrad probabilities and multilocus ordered tetrad probabilities for different halftetrad types and different marker configurations. Linear constraints among multilocus halftetrad probabilities were also obtained under the assumption of NCI. If marker order is known, these constraints can be used to test the assumption of NCI. Assuming NCI, if marker order is unknown, procedures similar to those proposed by Zhao and Speed (1998) can be applied to use these constraints to order markers under general chiasma crossover processes. If marker order is known and NCI is assumed, map distances between centromere and genetic markers can be estimated under a specific chiasma process model. In this section, the methods developed above are used to analyze data from alfalfa (Tavolettiet al. 1996) and rainbow trout (Thorgaardet al. 1983) via the method of maximum likelihood. Haplotype information is unavailable in both data sets. Because there is little consistent evidence of chromatid interference (Zhaoet al. 1995a) and both data sets yield little evidence of chromatid interference, NCI is assumed in the following analyses. For both data sets, we assume known marker order and use the chisquare model for the chiasma process.
Alfalfa: By assuming complete chiasma interference, Tavoletti et al. (1996) introduced a maximum likelihood approach to analyzing halftetrads from alfalfa. We analyze a subset of three markers in their study. These three markers are in the order of CEN–UWg119– MTSc9–UWg 65 and were genotyped in 152 progeny. Tavoletti et al. (1996) found that a very small percentage, approximately 6%, of halftetrads in this organism were the results of meiosis I nondisjunctions. To study whether the observed data can be explained by a moderate chiasma interference and no meiosis I nondisjunctions, we fitted the chisquare model to the data set, and the results are presented in Table 1. In our analysis, all halftetrads were assumed to be type IIb halftetrads (i.e., they all resulted from meiosis II nondisjunctions). The CxCo model gave the best fit among the chisquare models. The estimated map distances under the CxCo model were 5, 4, and 11 cM in the three intervals CEN– UWg119, UWg119–MTSc9, and MTSc9–UWg65, respectively. The standard errors were estimated using the parametric bootstrap method by (1) simulating data sets with the same sample size under the CxCo model assuming the estimated parameter values; (2) estimating model parameters for each simulated data set; and (3) approximating the standard errors of the parameter estimates using the standard errors of the estimated parameter values from these simulated data sets. Using this method, the standard errors were estimated to be 1, 1, and 2 cM, respectively. The above estimated genetic distances agree fairly well with the estimates by Tavoletti et al. (1996), which were 3, 4, and 11 cM in the three intervals. The CxCo model, which imposes moderate amount of chiasma interference, gave almost perfect fit to the observed data under the assumption of no meiosis I nondisjunctions. Recall that complete chiasma interference was assumed in deriving the estimate of the meiosis I nondisjunction proportion in alfalfa by Tavoletti et al. (1996). Therefore, it is difficult to distinguish the model studied by Tavoletti et al. (1996) and the chisquare model studied here using this data set. Note that in some cases, the meiosis I nondisjunction proportion, p_{MI}, and the map distance between the centromere and the most proximal marker,
Rainbow trout: Two markers in the order of CEN–Idh2–Est1 in rainbow trout were studied by Thorgaard et al. (1983). A total of 138 progeny were genotyped. The number of progeny for each of the six observed halftetrad types are given in Table 2. When the Cx(Co)^{m} models, where m = 0, · · ·, 6, were fitted to the data, the model with the greatest degree of chiasma interference, the Cx(Co)^{6} model, gave the best fit to this data set. This is consistent with the conclusion of Thorgaard et al. (1983) that there is high interference in rainbow trout. The estimated map distances under the Cx(Co)^{6} model were 36 and 11 cM in the two intervals CEN–Idh2 and Idh2–Est1. The corresponding standard errors were estimated to be 4 and 2 cM. Using a different method, Thorgaard et al. (1983) estimated that the genetic distances in these two intervals are 35 and 9 cM, respectively. Our estimates agree fairly well with their estimates.
In both examples, we have assumed no chromatid interference. If chromatid interference indeed exists, map distances can be either over or underestimated depending on the specific pattern of chromatid interference. In addition, chiasma interference can be incorrectly “detected” even if it is absent. A detailed study of chromatid interference is reported in H. Zhao and T. P. Speed (unpublished results).
DISCUSSION
In this article, four types of halftetrad data, type Ia, Ib, IIa, and IIb, were studied. Halftetrads, just like ordered and unordered tetrads, provide information on both chromatid interference and chiasma interference. In addition, they can be used to map centromeres.
Under the assumptions of no chromatid interference and random recovering of halftetrads, the relations between ordered tetrad probabilities and halftetrad probabilities were established. Using these relations, we derived constraints among halftetrad probabilities imposed by RRA and NCI. These constraints can be used to test for NCI. When the order of the markers and their relations to the centromere are not known, the best order for these markers can be established through examining the constraints imposed by NCI.
The relations between tetrad probabilities and halftetrad probabilities can be used to construct genetic maps and to locate centromeres under a given chiasma process model. This provides an approach to incorporating chiasma interference in genetic analysis. Because of the presence of chiasma interference in most organisms, the approach of Da et al. (1995), which assumes no chiasma interference, is not consistent with biological evidence. Several articles in the literature on halftetrads (Ottet al. 1976; Chakravarti and Slaugenhaupt 1987; Chakravartiet al. 1989; Mortonet al. 1990) aimed to address chiasma interference issue in their halftetrad analysis. But their methods either were only applicable to onemarker data or made different assumptions about chiasma interference in the same analysis for multiple markers. In contrast, our proposed approach in this article applies to any crossover process model that incorporates chiasma interference. Zhao and Speed (1998) noted that most map functions proposed in the context of centromere mapping can be well approximated by the map function under the Cx(Co)^{2} model. Therefore, multilocus genecentromere mapping, as described above using the chisquare model as the crossover process model, may provide a tractable and flexible approach to analyzing multilocus halftetrad data. The chisquare model has recently been extended to allow a more general class of interarrival distributions, yet the tractability of the model is still preserved (Langeet al. 1997). This class of generalized chisquare models can also be used to analyze multilocus halftetrad data.
H. Zhao and T. P. Speed (unpublished results) studied a Markov model for chromatid interference. They showed how this chromatid interference model can be applied with the chisquare model to study both chromatid and chiasma interference. Their approach can be easily adopted here to study both types of genetic interference using halftetrads.
Throughout this article, we have assumed that the type of halftetrads observed is known (type Ia, Ib, IIa, or IIb) and the phases in parents are known. Although these assumptions cover many data sets from experimental organisms, they are often violated in human data. For example, autosomal trisomies could be the result of MI or MII nondisjunction events. If the probability of each halftetrad type is known, the maximum likelihood method can still be applied for multilocus genecentromere mapping. Alternatively, we can introduce parameters to account for the uncertainties in determining halftetrad types (H. Zhao, unpublished results).
Another assumption we have made is that the observed halftetrads constitute a random sample of all halftetrads derived from tetrads. For the attachedX chromosome data of Drosophila in the literature, only halftetrads showing certain phenotypes were further genotyped. In this case, likelihood method should be modified to take into account the specific sampling scheme.
In summary, the multilocus analysis approach discussed in this article makes use of all the information available from the observed halftetrad data. It offers a general and consistent framework for analyzing different types of halftetrad data.
Acknowledgments
This work was supported by National Institutes of Health grant HG 0109301. The authors thank two referees for their constructive comments.
APPENDIX
Proposition 1. For both type Ia and type IIa halftetrads, there are 2^{n}^{−1}(2^{n} + 1) distinguishable patterns.
Proof. When two strands in the halftetrad are labeled, there are four possible types at each marker
(a) When all markers are of type 0 or 3, each pattern is distinguishable from other patterns, this gives 2^{n} distinguishable patterns.
(b) When at least one marker, say
Adding (a) and (b) there are 2^{n}^{−1}(2^{n} + 1) distinguishable patterns.
Proposition 2. For type Ia and type IIa halftetrad data, there are at most 4^{n}^{−1} + 2^{n}^{−1} distinct probabilities under RRA.
Proof. (a) When all markers show type 0 or 3, each pattern has a corresponding one for which the pair forms a tetrad. Under RRA, these two patterns should have the same probability. This gives 2^{n}/2 = 2^{n}^{−1} distinct probabilities.
(b) When all markers show type 1 or 2, the two strands can be labeled according to which one carries
(c) When at least one marker has type 0 or 3 and at least one has type 1 or 2, after the top and bottom strands are defined according to the marker showing type 1 or 2, each type has a corresponding type with the same probability, determined by one of the markers showing type 0 or 3. This gives (4^{n} − 2^{n} − 2^{n})/4 distinct probabilities.
Adding (a) to (c), there are at most 4^{n}^{−1} + 2^{n}^{−1} distinct probabilities under RRA.
Proposition 3. For type Ib and type IIb halftetrad data, there are (3^{n} + 1)/2 distinct probabilities under RRA.
Proof. (a) When all markers are heterozygous, there is one pattern.
(b) When at least one marker is homozygous, for each pattern, there is a corresponding pattern with the opposite configuration at each marker. These two patterns have the same probability under RRA. This gives (3^{n} − 1)/2 probabilities.
Adding (a) and (b), there are (3^{n} + 1)/2 distinct probabilities.
Theorem 1. The matrix C_{n} can be obtained using the procedure illustrated in the text.
Proof. When n = 1, both parental ditype and nonparental ditype have the same chance of yielding type 0 or 3, and tetratype between the centromere and the marker has the same chance of giving rise to type 1 or 2. So we have
Theorem 2. With notations introduced in the text,
Proof. When n = 1 and m = 1, if there is parental ditype CEN and
Footnotes

Communicating editor: D. Botstein
 Received January 20, 1998.
 Accepted May 18, 1998.
 Copyright © 1998 by the Genetics Society of America