Statistical Analysis of Half-Tetrads

Corresponding author: Hongyu Zhao, Department of Epidemiology and Public Health, Yale University School of Medicine, 60 College Street, New Haven, CT 06520., hongyu.zhao{at}yale.edu (E-mail).

Communicating editor: D. BOTSTEIN

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 LITERATURE CITED

Half-tetrads, 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, half-tetrads yield information on chromatid interference, chiasma interference, and centromere positions. In this article, for different half-tetrad types and different marker configurations, we derive the relations between multilocus half-tetrad probabilities and multilocus ordered tetrad probabilities. These relations are used to obtain equality and inequality constraints among multilocus half-tetrad 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 half-tetrad data.

HALF-TETRADS, where two meiotic products from a single meiosis are recovered together, arise in different forms in a variety of organisms. The first well-studied half-tetrad data were attached-X chromosomes in Drosophila (BEADLE and EMERSON 1935 ; WELSHONS 1955 ). Half-tetrads 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 (THORGAARD et al. 1983 ; ALLENDORF et al. 1986 ), salmonid fish (K. R. JOHNSON et al. 1987 ), catfish (LIU et al. 1992 ), and zebrafish (S. L. JOHNSON et al. 1995 ). In mammals, half-tetrads can be studied in the form of autosomal trisomies (MORTON et al. 1990 ; SHERMAN et al. 1991 ) and ovarian teratomas (OTT et al. 1976 ; EPPIG and EICHER 1983 ; CHAKRAVARTI and SLAUGENHAUPT 1987 ; CHAKRAVARTI et al. 1989 ; DEKA et 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 , SHERMAN et al. 1994 ; LAMB et 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 (CUI et al. 1992 ) and cows (JARRELL et al. 1995 ).

Half-tetrads may arise from different mechanisms. In Figure 1, we illustrate how meiosis I (MI) nondisjunctions lead to half-tetrads. It is easy to see that when there is no crossover between centromere and a heterozygous marker, MI nondisjunction results in half-tetrads being heterozygous at the marker. When there is one crossover between centromere and the marker, there is equal chance of producing homozygous and heterozygous half-tetrads. 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 half-tetrads, whereas a single crossover always results in heterozygous half-tetrads. For half-tetrads from MI nondisjunctions, the two strands were attached to different centromeres during meiosis, whereas the two strands in half-tetrads from MII nondisjunctions were attached to the same centromere during meiosis. MI and MII nondisjunction are not the only mechanisms that are responsible for half-tetrads. For example, attached-X chromosomes in Drosophila are the result of a different mechanism (BEADLE and EMERSON 1935 ). In this article, we broadly classify half-tetrads into two types: type I half-tetrads, in which no crossover between centromere and marker always results in heterozygous half-tetrads, and type II half-tetrads, in which no crossover always results in homozygous half-tetrads. On the basis of this classification, half-tetrads from MI nondisjunctions are type I half-tetrads, and those from MII nondisjunctions are type II half-tetrads. Attached-X chromosomes are type I half-tetrad. Half-tetrads from fish are mostly type II half-tetrads. Autosomal trisomies and ovarian teratomas can be of either type. In addition, ovarian teratomas can result from mechanisms other than MI or MII nondisjunctions (SURTI et al. 1990 ). Throughout this article, we also make the assumptions that the parental origin of the half-tetrads is known and that phases are known in parents. These assumptions are usually true for experimental organisms, although human half-tetrad data are more complex and may not satisfy these assumptions. For either type I or type II half-tetrads, a further distinction may be made when two or more markers are studied: haplotype information can be either available (attached-X chromosomes in Drosophila) or unavailable. The two types of half-tetrads are called type Ia and type IIa half-tetrads when haplotype information is available and type Ib and type IIb half-tetrads when such information is not available.

View larger version (19K): In this window In a new window Download PPT slide   Figure 1. Diagram illustrating nondisjunction during the first meiotic division. No crossover between the centromere and the marker always results in heterozygous half-tetrads. One crossover between the centromere and the marker has equal chance of resulting in homozygous half-tetrads and heterozygous half-tetrads.

View larger version (20K): In this window In a new window Download PPT slide   Figure 2. Diagram illustrating nondisjunction during the second meiotic division. No crossover between the centromere and the marker always results in homozygous half-tetrads. One crossover between the centromere and the marker always results in heterozygous half-tetrads.

As with ordered and unordered tetrads, half-tetrads are very valuable in studying crossovers during meiosis because (1) chromatid interference and chiasma interference can be distinguished with half-tetrads (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 half-tetrads (WELSHONS 1955 ; COTE and EDWARDS 1975 ; OTT et 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 half-tetrad 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 four-strand 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 half-tetrad, each of the two strands attached to the same centromere has equal chance of being in the half-tetrad, and when strands attached to the same centromere during meiosis form a half-tetrad, the two pairs have equal chance of being in the half-tetrad. Under this assumption, four nonsister chromatid pairs have the same chance of being observed in a half-tetrad 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 half-tetrad probabilities as functions of multilocus ordered tetrad probabilities. These relations are then used to derive linear equality and inequality constraints among multilocus half-tetrad 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 one-marker and two-marker cases in detail before presenting the general results for multiple markers. The four half-tetrad types are discussed in the order of type IIa, IIb, Ia, and Ib half-tetrads, respectively.

Since only two of the four strands are recovered in a half-tetrad, the original ordering of the four strands is lost. It is impossible in general to detect violation of random spindle to centromere attachment assumption (GRIFFITHS et al. 1996 ; ZHAO and SPEED 1998 ) using half-tetrad data.

We use the following notations in this article. Markers are denoted by script letters. For example, we use and to denote markers. Alleles are denoted by italic letters. For example, A and a denote two alleles of marker . We use [X, Y, Z, W] to denote the observed marker configuration for an ordered tetrad, where X and Y are attached to one centromere and Z and W are attached to the other centromere. For example, [AB, Ab, aB, ab] represents an ordered tetrad with two strands carrying AB and Ab attached to one centromere and two strands carrying aB and ab attached to the other centromere. For type Ia and IIa half-tetrads, two strands are separated by a / . For half-tetrads from MII nondisjunctions, these two strands were attached to the same centromere during meiosis. For half-tetrads from MI nondisjunctions, these two strands were attached to different centromeres. For type Ib and IIb half-tetrads, genotypes at each marker are combined and separated by ; in parentheses. For example, aB/Ab represents a half-tetrad with one strand bearing aB and the other strand bearing Ab, whereas (Aa; Bb) represents a half-tetrad with genotype Aa at and genotype Bb at , without knowing whether A and B are on the same strand or A and b are on the same strand.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 LITERATURE CITED

No chromatid interference (one marker):
With one marker, haplotype information is irrelevant. We need consider only two types: type I and type II half-tetrads.

Type II half-tetrads: For a heterozygous marker with alleles A and a, there are three observed patterns: AA, Aa, and aa. For MII nondisjunction, patterns AA and aa are derived from ordered tetrads having first division segregation (FDS) pattern [A, A, a, a]. Pattern Aa is derived from ordered tetrads having second division segregation (SDS) pattern [A, a, A, a]. Under RRA, tetrads having FDS pattern should give rise to AA and aa with the same probability. Tetrads having SDS pattern can only give rise to Aa. Therefore, P(AA) = P(aa) = and P(Aa) = P(SDS).

Type I half-tetrads: 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) = and P(Aa) = + P(FDS). This leads to the inequality constraint for type I half-tetrads: P(Aa) P(AA) + P(aa) = 2 P(AA). This inequality constraint is imposed by RRA.

No chromatid interference (two markers, type IIa half-tetrads):
For two markers and , with the parent undergoing nondisjunction carrying AB on one chromosome and ab on the homologous chromosome, there are 10 distinguishable half-tetrad types: AB/AB, AB/Ab, Ab/Ab, AB/aB, AB/ab, Ab/aB, Ab/ab, aB/aB, aB/ab, and ab/ab. Under RRA, each of the following four pairs should have the same probability: AB/AB and ab/ab, AB/Ab and aB/ab, Ab/Ab and aB/aB, and AB/aB and Ab/ab. These four pairs plus Ab/aB and AB/ab lead to at most six distinct probabilities for half-tetrads with two markers. Two markers and may be (1) on different chromosomes; (2) on the same chromosome but on different sides of the centromere; or (3) on the same chromosome and on the same side of the centromere. We consider these three cases separately.

Two markers on different chromosomes: Let p and q denote the probability of SDS at and , respectively. It is easy to show that P() = P() =P() = P() = ; P() = P() = q; P() = P() = p; and P() = P() = . These four distinct probabilities are determined by two parameters, p and q.

Two markers on different sides of the centromere (–CEN–): It was shown in ZHAO and SPEED 1998 that there are at most five distinct probabilities for ordered tetrads when two markers are on different sides of the centromere. The corresponding five classes can be described as: (1) FDS at both and and parental ditype between and ; (2) FDS at both and but nonparental ditype between and ; (3) FDS at and SDS at ; (4) SDS at and FDS at ; and (5) SDS at both and . Denote the probabilities of these five classes by , ß, , , and ; the constraints imposed by NCI are the proportionality constraint P([AB, ab, AB, ab]):P([AB, ab, Ab, aB]):P([Ab, aB, Ab, aB]) = 1:2:1, and the inequality constraints ß and + 2ß. Under RRA, P() = P() = , P() = P() = , P() = P() = , P() = P() = , and P() = P() = . Therefore, there are five distinct probabilities for type IIa half-tetrads when two markers are on different sides of the centromere. The equality constraint imposed by NCI is P() = P(). The constraints ß and + 2ß for ordered tetrads imply similar constraints among half-tetrad probabilities.

Two markers on the same side of the centromere (CEN––, the case of CEN–– can be discussed similarly): There are six distinct probabilities for ordered tetrads in this case. These six types can be distinguished by whether shows FDS or SDS pattern and whether and show parental ditype (P), tetratype (T), or nonparental ditype (N). Denote these types by (i^t₁i^t₂), where i^t₁ = 0 or 1 corresponds to FDS or SDS at , and i^t₂ = 0, 1, or 2 corresponds to P, T, or N between and . The probability of (i^t₁i^t₂) is denoted by p_i^t1i^t2. For half-tetrads, there are six distinct probabilities as well. Each of these six types can be denoted by (i₁i₂), where i₁ = 0 or 1 corresponding to being homozygous or heterozygous, and i₂ = 0, 1, or 2 corresponding 0, 1, or 2 strands showing recombination between and . These probabilities are denoted by h_i1i2 . The relations between the h_i1i2 and the p_i^t1i^t2 are h₀ = I x p₀ and h₁ = H x p₁, where h_i1 = (h_i10, h_i11, h_i12)', p_i^t1 = (p_i^t10, p_i^t11, p_i^t12)',

Therefore, there is a one-to-one correspondence between the p_i^t1i^t2 and the h_i1i2: p₀ = I x h₀ and p₁ = H^-1 x h₁ . Because the p_i^t1i^t2 are nonnegative, RRA imposes the constraint that H^-1 x h₁ 0.

It was shown in ZHAO and SPEED 1998 that the inequality constraints imposed by NCI for ordered tetrad probabilities are

where

Using these constraints for the p_i^t1i^t2, it can be shown that, under NCI, the h_i1i2 satisfy the following inequality constraints: h₀₀ h₀₂ , h₀₁ 2h₀₂ , h₁₀ h₁₂ , 3h₁₁ 2h₁₂ , and 2h₁₂ h₁₁ . No equality constraints are imposed by NCI among these six half-tetrad probabilities.

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 half-tetrads):
For type IIb half-tetrads—because haplotype information is unavailable—two patterns, AB/ab and Ab/aB, which are distinguishable in type IIa half-tetrads, 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 half-tetrads.

Two markers on different sides of the centromere (–CEN–): As above, AB/ab and Ab/aB have the same probability, and there is no loss of information under NCI compared to type IIa data. We arrive at the same probabilities and constraints as those for type IIa half-tetrads. The only exception is that there is no longer the equality constraint P() = P() because these two types are not distinguishable for type IIb half-tetrads.

Two markers on the same side of the centromere (CEN––, the case of CEN–– can be discussed similarly): Unlike the above two cases, AB/ab and Ab/aB have different probabilities for type IIa half-tetrads when and are on the same side of the centromere. Thus, there is some loss of information because AB/ab and Ab/aB cannot be distinguished. There are five, instead of six, distinct probabilities. These five classes can be represented as (0, i₂), which corresponds to being homozygous and i₂ = 0, 1, or 2 strands showing recombination between and ; and (1, i₂), which corresponds to being heterozygous and being homozygous (i₂ = 0) or heterozygous (i₂ = 1). Let u_i1i2 denote the probability of type (i₁i₂); we have u₀ = I x p₀ and u₁ = U x p₁, where u₀ = (u₀₀, u₀₁, u₀₂)', u₁ = (u₁₀, u₁₁)',

and p_i^t1i^t2 was defined above as the probability of ordered tetrad pattern (i^t₁i^t₂) .

Therefore, we can obtain only p₁₀ + p₁₂ from the u_i1i2 but not the individual values of p₁₀ and p₁₂. The inequality constraints on the p_i^t1i^t2, imposed by NCI are p₀₀ p₀₂, p₀₁ 2p₀₂, p₁₀ p₁₂, and p₁₁ 2p₁₂ (ZHAO and SPEED 1998 ). As long as p₁₀ + p₁₂ 0 (i.e., u₁₁ - u₁₀ 0) the inequalities involving the p_1i^t2 are always satisfied by setting p₁₂ to 0. Because the relations between the u_0i2 and the p_0i2 are the same as the relations between the h_0i2 and the p_0i2 , the constraints under NCI are u₀₀ u₀₂, u₀₁ 2u₀₂, and u₁₁ u₁₀.

No chromatid interference (two markers, type Ia half-tetrads):
For type Ia half-tetrads, the strands in the same half-tetrad were not attached to the same centromere at the four-strand stage during meiosis. Under RRA, each of the four nonsister chromatid pairs has the same chance of being recovered in a half-tetrad. As for type IIa half-tetrads, 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 and . It can be shown that P() = P() = P() = P() = , P() = P() = p + , P() = P() = + , and P() = P() = + p + q + . These four distinct probabilities are determined by two parameters, p and q.

Two markers on different sides of the centromere (–CEN–): There are five classes, each with a distinct probability. Using the same notations as in the discussion of type IIa half-tetrads to define the probabilities of these five classes, we have P() = P() = P() = P() = ; P() = P() = + ; P() = P() = + ; P() = + + + ; and P() = ß + + + . The equality constraints imposed by NCI are that the probabilities of AB/AB and ab/ab are equal to the probabilities of Ab/Ab and aB/aB. Denote the probabilities of these five classes by p₁, p₂, p₃, p₄, and p₅. The probabilities , ß, , , and for ordered tetrads can be obtained from the p_i, i = 1, · · · , 5:

RRA imposes the constraints that the expressions on the right-hand side of the above equations be nonnegative. Because the inequality constraints among ordered tetrad probabilities imposed by NCI are ß and + 2ß (ZHAO and SPEED 1998 ), the inequality constraints among the p_i imposed by NCI are p₅ p₄ and 3p₂ + 3p₃ 5p₁ + 2p₄.

Two markers on the same side of the centromere (CEN––, the case of CEN–– can be discussed similarly): There are six distinct probabilities. Each of these six types is denoted by (i₁i₂), where i₁ and i₂ were defined in the discussion of type IIa data. Denote the probability of half-tetrad pattern (i₁i₂) by h^I_i1i2 . The relations between the h^I_i1i2 and the p_i^t1i^t2 are h^I₀ = I x p₁ and h^I₁ = H x p₀ + I x p₁, where h^I_i1, p_i^t1 , I, and H are similarly defined as for type IIa half-tetrads. RRA imposes the constraints that the p_i^t1i^t2 inferred from the h^I_i1i2 are nonnegative. The inequality constraints among the h^I_i1i2 imposed by NCI can be derived from the constraints among the p_i^t1i^t2. It can be shown that these constraints are

No chromatid interference (two markers, type Ib half-tetrads):
As type IIb half-tetrads, 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 half-tetrads.

Two markers on different chromosomes: Because AB/ab and Ab/aB have the same probability, there is no loss of information under NCI compared to type Ia data. The probabilities are the same as those of type Ia half-tetrads.

Two markers on different sides of the centromere (–CEN–): Since P(AB/ab) and P(Ab/aB), in general, are different for type Ia half-tetrads, there is information loss in type Ib half-tetrads compared to type Ia half-tetrads. There are four distinct probabilities. Because P() + P() = + ß + + + , ordered tetrad probabilities and ß cannot be uniquely determined from type Ib half-tetrad probabilities. Both linear inequality constraints, ß and + 2ß, can be satisfied by setting ß to be 0. The only constraints imposed by NCI are the equality constraints that the probabilities of AB/AB and ab/ab are equal to the probabilities of Ab/Ab and aB/aB.

Two markers on the same side of the centromere (CEN––, the case of CEN–– can be discussed similarly): There are five different classes with distinct probabilities. These five classes are denoted by (i₁i₂), where i₁ and i₂ were defined in the discussion of type IIb data. Let u^I_i1i2 denote the probability of pattern (i₁i₂). It can be shown that u^I₀ = I x p₁ and u^I₁ = U x p₀ + U x p₁, where u^I₀, u^I₁, U and the p_i^t1i^t2 are similarly defined as in the discussion of type IIb half-tetrads.

From the u^I_i1i2 , we cannot uniquely determine p₀₀ and p₀₂. Only the sum of p₀₀ and p₀₂ can be inferred, and we denote it by p^*₀₀ . It is easy to show the following:

The constraints imposed by RRA are that the p_i^t1i^t2 inferred from the u_i1i2 being nonnegative. We can also derive the inequality constraints imposed by NCI from the above relations.

No chromatid interference (multiple markers on the same side of the centromere):
For n markers in the order of CEN–₁–₂– · · · –_n, there are 2^n-1(2ⁿ + 1) distinguishable patterns (Appendix 1 Proposition 1) for both type Ia and type IIa half-tetrads. For type Ib and type IIb half-tetrads, there are three possible patterns at each marker: A_rA_r, A_ra_r, and a_ra_r. Therefore, there are 3ⁿ distinguishable patterns for type Ib and type IIb half-tetrads. Under RRA, the number of distinct probabilities is at most 4^n-1 + 2^n-1 (Appendix 1 Proposition 2) for type Ia and type IIa half-tetrads. Among the 3ⁿ distinguishable patterns for type Ib and type IIb half-tetrads, there are at most (3ⁿ + 1)/2 distinct probabilities (Appendix 1 Proposition 3).

Type IIa half-tetrads: To simplify the derivation of the general results for n markers, we proceed differently from the discussion of the one- and two-marker cases. We first assume that the two strands have already been labeled and are thus distinguishable. Then there are four possible patterns, (A_rA_r), (A_ra_r), (a_rA_r), and (a_ra_r), denoted by 0, 1, 2, and 3, respectively, at each marker _r. The pattern (i₁i₂ ... i_n) of each type IIa half-tetrad is thus defined with i_r = 0, 1, 2, and 3, respectively. Because two strands in a half-tetrad are not labeled, we can evenly divide the cases by two different labelings of the two strands. The results in the following discussion hold under this even division.

There are 2 x 3^n-1 distinct probabilities for ordered tetrad data under NCI. These different ordered tetrad classes are denoted by (i^t₁i^t₂ ... i^t_n), where i^t₁ = 0 or 1 corresponding to FDS or SDS at ₁, and i^t_r = 0, 1, or 2, r = 2, ... , n, corresponding to parental ditype, tetratype, and nonparental ditype between _r-1 and _r. Let p_{i^t1i^t2...i^tn} be the probability of type (i^t₁i^t₂ ... i^t_n) . In the following, we derive the relations between type IIa half-tetrad probabilities, the h_i1i2...in, and ordered tetrad probabilities, the p_{i^t1i^t2...i^tn}. For one marker,

For two markers ₁ and ₂, we consider four patterns at ₁ separately. If the pattern at ₁ is A₁/A₁, parental ditype between ₁ and ₂ will result in A₂/A₂ at ₂, tetratype between ₁ and ₂ will result in A₂/a₂ or a₂/A₂ at ₂ with the same probability, and nonparental ditype between ₁ and ₂ will result in a₂/a₂ at ₂. If the pattern at ₁ is a₁/a₁, parental ditype between ₁ and ₂ will result in a₂/a₂ at ₂, tetratype between ₁ and ₂ will result in A₂/a₂ or a₂/A₂ at ₂ with the same probability, and nonparental ditype between ₁ and ₂ will result in A₂/A₂ at ₂. If the pattern at ₁ is A₁/a₁, under NCI, there is equal chance that the four-strand bundle during meiosis has configuration [A₁,a₁; A₁,a₁], [A₁,a₁; a₁,A₁], [a₁,A₁; A₁,a₁], or [a₁,A₁; a₁,A₁]. Therefore, parental ditype between ₁ and ₂ will result in A₂/a₂ at ₂, tetratype between ₁ and ₂ will have the same probability resulting in A₂/A₂, A₂/a₂, a₂/A₂, or a₂/a₂ at ₂, and nonparental ditype between ₁ and ₂ will result in a₂/A₂ at ₂. The case that the type at ₁ being a₁/A₁ can be considered similarly. Therefore,

where

For an arbitrary n, the probability h_i1i2...in of the type IIa half-tetrad pattern (i₁i₂ ... i_n) can be expressed in terms of the p_{i^t1i^t2...i^tn} as

Write the ci1i2...init1it2...itn into a matrix Cn such that the columns are labeled by it1it2 ... itn and the rows are labeled by i1i2 ... in, each in lexicographical order. It can be shown that (Appendix 1 Theorem 1) the 4r+1 x (2 x 3r) matrix Cr+1 = (ci1i2...irir+1it1it2...itritr+1) can be obtained recursively by replacing each ci1i2...irit1it2...itr in Cr = (ci1i2...irit1it2...itr) by the 4 x 3 matrix ci1i2...irit1it2...itr Xir .

Because the probabilities h_i1...in can be expressed in terms of the p_i^t1...i^tn through the matrix C_n, for two identical rows in C_n, the corresponding half-tetrad patterns should have the same probability. Note that some of these equalities are the result of RRA and they can be readily identified. Equality constraints under NCI can be established by removing these equality constraints resulting from RRA.

The inequality constraints can be established as follows. Define

The matrix Dr+1 = (cit1it2...itritr+1i1i2...irir+1) is defined recursively from Dr = (dit1it2...itri1i2...ir) by replacing each dit1it2...itri1i2...ir in Dr by the 3 x 4 matrix dit1it2...itri1i2...ir Yir . From the facts that

and Y_iX_i = I_3x3 for i = 0, 1, 2, and 3, it is easy to show that

So the p_0i^t2...i^tn and p_1i^t2...i^tn can be recovered from the h_i1i2...in. RRA imposes the constraint that D_nh 0. It was shown in ZHAO and SPEED 1998 that the inequality constraints for ordered tetrads are

where T₁ and T^-1₁ were defined in (1) and T^-1_n-1 = T^-1(n-1)₁ . The operator is the standard tensor product (see, e.g., BELLMAN 1970 ). Therefore, the inequality constraints among the h_i1i2...in can be established. A likelihood ratio test can be used to test these constraints; see ZHAO et al. 1995A .

Type IIb half-tetrads: For type IIb half-tetrads, 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 _r, r = 1, · · · , n. The probability for each type (i₁i₂ ... i_n) can be expressed in terms of the ordered tetrad probabilities p_{i^t1i^t2...i^tn} as

Write the ai1i2...init1it2...itn into a matrix such that the columns are labeled by it1it2 ... itn and the rows are labeled by i1i2 ... in, each in lexicographical order. It is easy to see

Define

then the matrix Ar+1 = (ai1i2...irir+1it1it2...itritr+1) can be obtained by replacing each ai1i2...irit1it2...itr in Ar by the 3 x 3 matrix ai1i2...irit1it2...itrEir .

In the discussion of two-marker data, it was noted that p₀₀ and p₀₂ cannot be determined from the u_i1i2 . Similarly in the n marker case, not all the p_{i^t1i^t2...i^tn} can be recovered from the u_i1i2...in . Equality constraints can be established as in the discussion of type IIa half-tetrads. To establish inequality constraints, define

The matrix Br+1 = (bit1it2...itritr+1i1i2...irir+1) is defined by replacing each bit1it2...itri1i2...ir in Br = (bit1it2...itri1i2...ir) by the 3 x 3 matrix bit1it2...itri1i2...irFir. It can be shown that RRA imposes the constraints that p = Fnu 0, and NCI imposes the constraints that T-1np = T-1nFnu 0.

Type Ia half-tetrads: For type Ia half-tetrads, let h^I_i1i2...in and p_{i^t1i^t2...i^tn} denote the half-tetrad and ordered tetrad probabilities. The relations between the h^I_i1i2...in and the p_{i^t1i^t2...i^tn} can be expressed as

Write the i1i2...init1it2...itn into a matrix n such that the columns are labeled by it1it2 ... itn and the rows are labeled by i1i2 ... in, each in lexicographical order. Using arguments similar to those used in the proof of type IIa half-tetrads in the Appendix 1 it can be shown that

and that the matrix r+1 = (i1i2...irir+1it1it2...itritr+1) is obtained by replacing each i1i2...irit1it2...itr in r by the 4 x 3 matrix i1i2...irit1it2...itr Xir , where X1, X2, X3, and X4 were defined in the discussion of type IIa half-tetrads. Linear equality and inequality constraints imposed by RRA and NCI can be similarly established.

Type Ib half-tetrads: For type Ib half-tetrad data, the relations between the u^I_i1i2...in and the p_{i^t1i^t2...i^tn} can be expressed as

Write the i1i2...init1it2...itn into a matrix such that the columns are labeled by it1it2 ... itn and the rows are labeled by i1i2 ... in, each in lexicographical order. It can be shown that

and that the matrix r+1 = (i1i2...irir+1it1it2...itritr+1) is obtained by replacing each i1i2...irit1it2...itr in r by the 3 x 3 matrix i1i2...irit1it2...itrEir where E0, E1, and E2 were defined in the discussion of type IIb half-tetrads. The linear equality and inequality constraints under RRA and NCI can be established similarly to those for type IIb half-tetrads.

No chromatid interference (multiple markers on different sides of the centromere):
Consider markers on different sides of the centromere in the order of _m– · · · –₁–CEN–₁– · · · –_n. Here we show only the relations between type IIa and type IIb half-tetrad probabilities and ordered tetrad probabilities. Constraints imposed by RRA and NCI can be derived using these relations following the approach described above. Derivations for type Ia and type Ib half-tetrads are similar, and we omit the details here.

Type IIa half-tetrads: We assume that the two strands have been labeled. Any type IIa half-tetrad pattern can be represented by ij = (i₁i₂ ... i_n; j₁j₂ ... 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 half-tetrad pattern is denoted by h_{(i1i2...in;j1j2...jm)}. If the centromere were observable, ordered tetrad pattern could be represented by i^tj^t = (i^t₁i^t₂ ... i^t_n;j^t₁j^t₂ ... j^t_m), where each i^t_k (j^t_l) is 0, 1, or 2, corresponding to parental ditype, tetratype, or nonparental ditype between _k-1 and _k (_l-1 and _l). Both ₀ and ₀ correspond to CEN. The h_ij can be expressed in terms of the p_i^tj^t as

It is shown in the Appendix 1 (Theorem 2) that

where vi1j1it1jt1 is the element in the (i1j1)th row and the (it1jt1)th column of the following matrix:

Write the coefficients qiit = qi1i2...init1it2...itn into a matrix Qn such that the columns are labeled by it1it2...itn and the rows are labeled by i1i2...in, each in lexicographical order. As in the derivation of Theorem 1 (Appendix 1), it can be shown that

and that the matrix Qr+1 = (qi1i2...irir+1it1it2...itritr+1) is obtained by replacing each qi1i2...irit1it2...itr by the the 4 x 3 matrix qi1i2...irit1it2...itrXir , where X0, X1, X2, and X3 were defined in the discussion of type IIa half-tetrads for markers on the same side of the centromere. The coefficients qjjt = qj1j2...jmjt1jt2...jtm are defined the same as qiit = qi1i2...init1it2...itn.

Type IIb half-tetrads: As for type IIa half-tetrads, the probability u_ij for type IIb half-tetrad pattern ij = (i₁i₂ ... i_n;j₁j₂ ... j_m), where each i_k or j_l is 0, 1, or 2, can be expressed in terms of the p_i^tj^t as

It can be shown that

where wi1j1it1jt1 is the element in the (i1j1)th row and the (it1jt1 )th column of the following matrix:

Write the si1i2...init1it2...itn into a matrix Sn, then

and Sr+1 is obtained by replacing each si1i2...irit1it2...itr in Sr by the 3 x 3 matrix si1i2...irit1it2...itrEir, where E0, E1, E2, and E3 were defined in the discussion of type IIb half-tetrads for markers on the same side of the centromere. The coefficients sjjt = sj1j2...jmjt1jt2...jtm are defined the same as siit = si1i2...init1it2...itn.

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 half-tetrads, being heterozygous at a marker corresponds to SDS for the four strands. Therefore, the relation between the proportion of heterozygous half-tetrads and the map distance is the same as that plotted in Figure 1 and Figure 2 of ZHAO and SPEED 1998 for different map functions.

For type I half-tetrads, 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 chi-square model was found to provide better fit to data from different organisms (ZHAO et al. 1995B ).

The chi-square 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 four-strand 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 chi-square model has recently been generalized to a more general class, the Poisson-skip model (LANGE et al. 1997 ). Both the chi-square model and the Poisson-skip model lead to closed-form 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 chi-square model.

For an arbitrary number of markers on the same side or different sides of the centromere, ZHAO and SPEED 1998 derived general closed-form expressions for ordered tetrad probabilities under the Cx(Co)^m model. Using these results and the relations we derived between half-tetrad probabilities and ordered tetrad probabilities, we can evaluate any half-tetrad 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

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX 1 LITERATURE CITED

In the previous section, we derived general relationships between multilocus half-tetrad probabilities and multilocus ordered tetrad probabilities for different half-tetrad types and different marker configurations. Linear constraints among multilocus half-tetrad 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 (TAVOLETTI et al. 1996 ) and rainbow trout (THORGAARD et 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 (ZHAO et 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 chi-square model for the chiasma process.

Alfalfa:
By assuming complete chiasma interference, TAVOLETTI et al. 1996 introduced a maximum likelihood approach to analyzing half-tetrads from alfalfa. We analyze a subset of three markers in their study. These three markers are in the order of CEN–UWg119–MTSc9–UWg65 and were genotyped in 152 progeny. TAVOLETTI et al. 1996 found that a very small percentage, approximately 6%, of half-tetrads 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 chi-square model to the data set, and the results are presented in Table 1. In our analysis, all half-tetrads were assumed to be type IIb half-tetrads (i.e., they all resulted from meiosis II nondisjunctions). The CxCo model gave the best fit among the chi-square 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 chi-square 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, d_CEN–, cannot be simultaneously identified. The Poisson model is the simplest such model, under which d_CEN– varies according to p_MI. In general, the estimate of d_CEN– increases as the estimate of p_MI decreases. This is because as p_MI decreases, more crossover events are needed between CEN and to explain the observed heterozygous half-tetrads at , thus increasing the estimated map distance between them. As an example, for the alfalfa data, the estimated map distance between CEN and UWg119 is 5 cM when no meiosis I nondisjunctions are assumed, whereas the estimated map distance is 3 cM when the meiosis I nondisjunction rate is estimated at around 6%.

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 half-tetrad 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

TOP ABSTRACT METHODS