POLYPLOIDY is an important evolutionary force in For allopolyploids derived from the chromosome

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1 Copyright 2001 by the Genetics Society of America A Multivalent Pairing Model of Linage Analysis in Autotetraploids Samuel S. Wu,*,1 Rongling Wu,*,1 Chang-Xing Ma,*, Zhao-Bang Zeng, Mar C. K. Yang* and George Casella* *Department of Statistics, University of Florida, Gainesville, Florida 32611, Department of Statistics, Nanai University, Tianjin , People s Republic of China and Department of Statistics, North Carolina State University, Raleigh, North Carolina Manuscript received May 2, 2001 Accepted for publication August 29, 2001 ABSTRACT Polyploidy has been recognized as an important step in the evolutionary diversification of flowering plants and may have a significant impact on plant breeding. Statistical analyses for linage mapping in polyploid species can be difficult due to considerable complexities in polysomic inheritance. In this article, we develop a novel statistical method for linage analysis of polymorphic marers in a full-sib family of autotetraploids. This method is established on multivalent pairings of homologous chromosomes at meiosis and can provide a simultaneous maximum-lielihood estimation of the double reduction frequencies of and recombination fraction between two marers. The EM algorithm is implemented to provide a tractable way for estimating relative proportions of different modes of gamete formation that generate identical gamete genotypes due to multivalent pairings. Extensive simulation studies were performed to demonstrate the statistical properties of this method. The implications of the new method for understanding the genome structure and organization of polyploid species are discussed. POLYPLOIDY is an important evolutionary force in For allopolyploids derived from the chromosome flowering plants (Stebbins 1971; Grant 1981; Bever combination of distinct genomes and subsequent chro- and Felber 1992; Jacson and Jacson 1996; Soltis mosome doubling (Soltis and Soltis 2000), statistical and Soltis 2000). It is estimated that as much as 30 80% methods developed for molecular linage mapping by of angiosperms are polyploids or have experienced one estimating recombination fractions between different or more episodes of polyploidization (Stebbins 1971; loci in diploid species (Lander and Green 1987) will Grant 1981; Masterson 1994). Evidence for the cre- also apply. However, these methods cannot be used in ative role of polyploidy in evolution is well synthesized in autopolyploids that are formed due to the chromosome a recent review by Otto and Whitton (2000), although doubling of the same genome by fusion of unreduced they estimated that only 2 4% of speciation events in gametes (Soltis and Soltis 2000). Autopolyploids may flowering plants involve polyploidization. The fremultivalent undergo either bivalent (two chromosomes pair) or quency of polyploidy in domesticated plant taxa is also pairing (more than two chromosomes pair) high (75%); alfalfa, banana, canola, coffee, cotton, poone or both, at meiosis, in which a gene has more than tato, soybean, strawberry, sugarcane, sweet potato, and possible partner (or set of partners). Polysomic wheat represent excellent examples of polyploids of ecotion. inheritance could result from the multivalent formanomic importance (Hilu 1993). To study the evolutionpolyploid Most of the available statistical methods for auto- ary consequences of polyploidy on genome organization linage analysis assume bivalent pairings (Wu and develop superior varieties of polyploid plant speal. et al. 1992; Hacett et al. 1998; Ripol et al. 1999; Luo et cies, a number of genome projects have now been 2000, 2001). Statistical analysis assuming multivalent launched to construct genetic linage maps using mothe pairings has not been explored thoroughly because of lecular marers and identify genes responsible for economically complexity of polysomic inheritance. important traits in polyploid populations Double reduction is a phenomenon that two sister ranging from tetraploid (potato) to octoploid (sugar- chromatids of a chromosome sort into the same gamete cane; Wu et al. 1992; da Silva et al. 1993; Yu and Pauls (Darlington 1929; de Winton and Haldane 1931; 1993; Grivet et al. 1996; Hacett et al. 1998; Meyer Mather 1936; Fisher 1947). It may be generated due et al. 1998; Brouwer and Osborn 1999; Ripol et al. to multivalent pairings in autopolyploids. Figure 1 shows 1999). how different types of gametes are formed. At anaphase I, chromatids located on a chromosome may migrate either to the same pole (reductional separation) or to different poles (equational separation). The type of sep- Corresponding author: Samuel S. Wu, Division of Biostatistics, P. O. Box , University of Florida, Gainesville, FL aration depends on the number and the type of cross- samwu@biostat.ufl.edu overs located between the centromere and the locus 1 These authors contributed equally to this wor. under consideration. We consider the segregation of Genetics 159: (November 2001)

2 1340 S. S. Wu et al. Figure 1. A diagram displaying the segregation patterns of loci A and B during meiosis in an autopolyploid (modified from Mather 1936 and Bever and Felber 1992). Locus A having no crossover with the centromere undergoes path X of reductional separation (no double reduction), whereas locus B displaying a crossover with the centromere undergoes either path Y of equational separation with no double reduction or path Z of equational separation with double reduction. Gametes having undergone double reductions are underscored. In this article, we use Fisher s model to devise a maximum-lielihood method for simultaneously estimating the frequency of double reduction and the recombina- tion fraction between different marers in autopoly- ploids whose gamete formation is predominately due to multivalent pairings. The method relied on an expec- tation-maximization (EM) algorithm (Dempster et al. 1977). Mathematically, we prove that the difference in the frequency of double reduction between two loci is bounded by two times the recombination fraction in tetraploid. Our linage analysis here is based on fully informative codominant marers of eight different alleles at each marer between the two autotetraploid parents. Statistical properties of this autopolyploid meth- od are examined using a simulation study. two loci A and B in autotetraploid demonstrating quadrivalent formation during meiosis. Locus A is so close to the centromere that no crossover happened between them. The first division for this locus is reductional and double reduction never occurs (path X). Locus B has one crossover with the centromere and, thus, undergoes equational separation. If the four homologous chromosomes segregate randomly, they may migrate to the same cell in two different ways. In the first way, chromosomes 1 and 2 and their respective homologues migrate to the same cells, and therefore alleles located on sister chromatids reach different gametes and double reduc- tion never occurs (path Y). In the second way, chromosomes 1 and 2 and chromosomes 3 and 4 migrate to the same cells, which may cause double reduction when chromatids segregate randomly. Fisher (1947) formulated a pioneering theoretical model for analyzing two lined loci in an autotetraploid undergoing quadrivalent pairings during meiosis. Although Fisher elegantly described the modes of gamete formation in terms of the recombination number between the two loci and the frequency of double reduction at each locus, he was not able to provide a tractable computational method for estimating these parameters. AUTOTETRAPLOID MODEL A quadrivalent pairing model for two lined marers: Consider two lined marers M and M l on the same chromosome in an autotetraploid. At marer M, four alleles, each assigned to one of the four homologous chromosomes, are labeled by P 1, P 2, P 3, and P 4 for parent P and by Q1, Q2, Q3, and Q 4 for parent Q. Accordingly,

3 Linage Analysis in Autotetraploids 1341 four different alleles at marer M l are labeled by P l 1, Each gamete has two chromosomes and these will be P l 2, P l 3, and P l 4 for parent P and by Q1, l Q2, l Q3, l and Q4 l of different possible types. For one for parent Q. The recombination fraction between the parent, all these types of gametes can be classified into two marers is denoted by P for parent P and Q for 11 basic modes according to double reduction and the parent Q. For the two autotetraploid parents used for number of recombination events (Mather 1936; Table the cross, there are a total of 576 allelic configurations 1): or linage phase assignments between the two marers, 1 and 2: The first two modes of gametic formation shown one of which is schematically expressed as in Table 1 involve double reduction at both marers P M and M l 1. Of the two modes, the first has no recom- l P2 P 3 l P3 P 4 l P4 Q 1 l Q1 Q 2 l Q2 Q 3 l Q 3 Q 4 l Q4, (1) l bination between the chromosomes and thus entails only four parental types of gamete: where lines indicate the individual homologous chro- P1 P1 P 1 l P1, P 2 l P2 P 2 l P2, P 3 l P3 P 3 l P3, P 4 l P4 P 4 mosomes on which the two marers are located. The l P4. recombination fractions l P and Q are estimated on the basis of the segregation of the two-marer joint geno- The second mode has 12 possibilities as a result of types observed in the progeny of the family. However, recombination between a pair of the four chromothe observations of the joint marer genotypes are consomes: founded by the models of meiotic pairings (bivalent or quadrivalent) and parental linage phases of different P1 P2 P 1 l P2, P 1 l P3 P 1 l P3, P 1 l P4 P 1 l P4 ; P 2 l l P1, P 2 l P3 P 2 l P3, P 2 l P4 P 2 alleles across the two maternally and two paternally derived chromosomes. To mae accurate estimates for P l P4 ; l and Q, therefore, it is essential to select a most liely P3 P1 P 3 l P1, P 3 l P2 P 3 l P2, P 3 l P4 P 3 l P4 ; P 4 l P1 P 4 l P1, P 4 l P2 P 4 l P2, P 4 l P3 P 4 pairing model and linage phase configuration over the l P3. l two parents. In this article, we proposed a model for fully information 3 and 4: The second two modes include double reductive codominant marers, i.e., those of eight different only at marer M. For mode 3, one parental alleles between the two autotetraploid parents at each chromosome is unchanged, but the other is made up marer. We assume that the four homologous chromochromosome by all possible types of recombination between this somes form quadrivalents. Thus, for a particular marer and the remaining three. There are 12 M, we must consider the full chromatid complement possibilities and typical gametes are lie that may be represented as gametes P 1P 1, P 2P 2, P 3P 3, P1 P1 P 1 l P2, P 1 l P1 P 1 l P3, P 1 l P1 P 1 and P l P4 4P. 4 for parent P and gametes Q1Q 1, Q2Q 2, Q3Q 3, and Q4Q l 4 for parent Q. The generation of these gametes is typical of the four-strand model in which both chro- And for mode 4, both chromosomes are derived from matids of a single chromosome may be passed to the recombination between the four parental chromosame gamete, forming the so-called double reduction somes. There are also 12 possibilities such as (Darlington 1929; Mather 1936; Fisher 1947). The frequency of double reduction is a constant for any P1 P2 P 1 l P3, P 1 l P2 P 1 l P4, P 1 l P3 P 1 given locus, depending on its distance from the centromere. l P4. l We denote the frequencies of double reduction l at M by P for parent P and Q for parent Q. Similarly, 5 and 6: The next two modes involve double reduction P and Q are denoted for marer M l. Following the only at marer M l and have classifications similar to classification in Fisher (1947), with two lined loci in the second two modes. tetrasomics there are four different combinations in Other 5: The last five modes (7A, 7B, 8A, 8B, and 9), terms of the existence of double reduction: in which neither marer M nor M l has double reduc- tion, can be sorted into three types. In the first type, 1. Both marers display double reductions; mode 7, two gametic chromosomes are derived from 2. only marer M displays double reductions; two of the parental chromosomes either without recom- 3. only marer M l displays double reductions; and bination (mode 7A) or with recombination (mode 7B). 4. none of the marers display double reductions. There are six possibilities for each group. Typical Since there are four sources for the allele at any given gamete types are locus, a gametic chromosome with two loci can be made up 16 ways: P Pr P (r, s 1, 2, 3, 4). s l 7B P1 P and l P2 P l P2 P l P1. l Because the same genotype is represented, 7A and cannot be distinguished on the basis of the marer

4 1342 S. S. Wu et al. TABLE 1 The model of quadrivalent formation for a diploid gamete in a tetrasomic parent with two lined marers M and M l Double reduction No. of Relative frequency No. of recombination combination Mode Typical gamete types of formation events Both marers 1 Only marer M 3 Only marer M l 5 Neither marer A 7B 8A 8B P 1 P1 P 1 4 f l P1 l 1 0 P 1 P2 P 1 12 f l P2 l 2 2 P 1 P1 P 1 12 f l P2 l 3 1 P 1 P2 P 1 12 f l P3 l 4 2 P 1 12 f l P1 l 5 1 P 2 P1 P 3 12 f l P1 l 6 2 P 1 6 f l P2 l 7A 0 P 1 P2 P 2 6 f l P1 l 7B 2 P 1 24 f l P3 l 8A 1 P 1 P3 P 2 24 f l P1 l 8B Total There are 136 different gametes with the two lined marers P 1 P3 P 12 f l P4 l 9 2 Pr 1 r 2 P l P l s 2 (r 1, r 2, s 1, s 2 1, 2, 3, 4). We use a single f 7 to denote the mixed frequency with which both 7A and 7B occur at meiosis; i.e., f 7 f 7A f 7B. For the same reason, the mixed frequency of 8A and 8B is denoted by f 8. s 1 P phenotypes. The second type (mode 8) of nondouble reduction is that two gametic chromosomes are derived from three of the parental chromosomes with one event of recombination (8A, 24 possibilities) or two events of recombinations (8B, 24 possibilities). Gamete examples for modes 8A and 8B are P 1 and l P3 P 1 l P3 P 2 l P1. l P 1 P3 P 2 l P4. l Mode 9 has 12 possibilities. They are also indistinguishable because they have iden- tical genotypes. The third type (mode 9) of nondou- ble reduction includes recombination between all four different chromosomes such as Because gametes for fully informative marers are unique to the two parents and because the two parents are assumed to behave independently in terms of double reduction and recombination, gamete genotypes can provide adequate information for linage analysis as much as zygote genotypes. Therefore, to simplify our treatments, we base our linage analysis on the segregation of the gamete genotypes in each parent. Thereafter, only parent P is considered because a symmetrical inference can be made for parent Q. We refer to the frequencies of double reduction and recombination fraction between the marers for parent P by,, and without the subscript P, unless otherwise specified. Parameter estimation: For marer M, assume a fixed assignment for the four alleles of parent P in the order P 1, P 2, P 3, and P 4. Given such a fixed assignment for marer M, we randomly assign the four observed alleles of marer M l, P l 1, P l 2, P l 3, and P l 4, with a total of 24

5 Linage Analysis in Autotetraploids different possibilities. One of the possibilities should P1 and l P2 P 1 l P2 P 2 present a correct assignment for the alleles of the two l marers among the four homologous chromosomes. The estimates of the frequencies of double reduction are two reciprocal assignments, but they have the same and the recombination fraction between the two mar- genotype and are mixed in the same cell at row 5 and ers should be based on their best, but unnown, allelic column 5. assignment across the parental chromosomes. For lin- Because formation mode 7 is a mixture of double age analysis in autotetraploid populations, therefore, a recombinants and nonrecombinants, the determinavector of unnown parameters can be denoted by tion of the expected number of recombination events (A,,, ) T, where A is the th allelic assignment under this mode requires information about the relative for marer M l relative to the fixed allelic assignment proportions of these two types of offspring. Given the of marer M. relative proportion of double recombinants in mode 7 Given a particular allelic assignment for parent P as (φ f 7B /f 7, see appendix), the expected number of shown in expression (1), four double reduction gametes recombination events is 2φ. Similarly, for mode 8, which and six nondouble reduction gametes generated by is a mixture of single recombinants and double recombimarer M can be arrayed in the order {P 1P 1,P 2P 2, nants, the expected number of recombination events is P 3P 3,P 4P 4,P 1P 2,P 1P 3,P 1P 4,P 2P 3,P 2P 4,P 3P 4} and {P l 1P l 1,P l 2P l 2, calculated as 1 (1 ) 2 1, where is P l 3P l 3,P l 4 P l 4,P l 1 P l 2,P l 1P l 3,P l 1P l 4,P l 2P l 3,P l 2P l 4,P l 3P l 4} at marer M l. the proportion of double recombinants in mode 8 Thus, we can identify two-marer gamete ( f 8B /f 8 ; see appendix). The expected numbers of genotypes for parent P. Following notation in Fisher recombination events between the two marers can be (1947), we define f, the relative frequencies of the 11 expressed in matrix notation as different modes of gamete formation, which must sum to unity (Table 1). However, because the marer phenotypes of 7A and 7B cannot be distinguished, we use a D {Ds r 1 r 2 1 s 2 } single f 7 to denote the mixed frequency with which both P 1P 1 P 2P 2 P 3P 3 P 4P 4 P 1P 2 P 1P 3 P 1P 4 P 2P 3 P 2P 4 P 3P 4 7A and 7B occur at meiosis. For the same reason, the mixed frequency of 8A and 8B is denoted by f 8.Itisnot difficult to express the joint relative frequencies of twomarer diploid gametes in matrix notation: H {H r 1 r 2 s 1 s 2 } P l 1P l 1 P l 2P l 2 P l 3P l 3 P l 4P l 4 P l P l 1P l 2/P l 2P l φ P l 1P l 3/P l 3P l φ P 1P 1 P 2P 2 P 3P 3 P 4P 4 P 1P 2 P 1P 3 P 1P 4 P 2P 3 P 2P 4 P 3P 4 P P l 1P l 4/P l 4P l 1 1P l φ l 1 4 f f 6 P l 2P l 3/P l 3P l 2 P φ 1 1 2P 2 l f f 6 P l 2P l 4/P l 4P l φ 1 P 3P 3 l f f 5 P l 3P l 4/P l 4P l φ P 4P l 4 l f f 5 The above information allows us to express the recom- P 1P l 2/P l 2P l 1 l f f f 9. bination fraction and the two double reduction parameters, at marer M P 1P l 3/P l 3P l 1 l f f f f f 8 and at marer M l, in terms of P 1P l 4/P l 4P l 1 l f f f f f 8 f 1,...,f 9 and φ,. We have P 2P l 3/P l 3P l 2 l 1 12 f f f f f f 8 P 2P l 4/P l 4P l 2 l 1 12 f f f f f 1 f 1 f 2 f 3 f 4 ; 7 24 f 8 P 3P l 4/P l 4P l 3 l f f f 7 f 1 f 2 f 5 f 6 ; (2) 2 (f 3 f 5 ) 2(f 2 f 4 f 6 f 9 ) 2φf 7 (1 )f 8. However, as illustrated earlier, there are as many as 136 From the above equations, it follows that f 3 gamete formations for any two lined marers. The 36 f 4 f 5 f 6 ) f 3 f 4 f 5 f 6 2. Therefore the extra gamete formations are each due to a reciprocal difference in the frequency of double reduction beallelic assignment of marer M l and are located in the tween two loci is bounded by two times the recombina cells of the above matrix s bottom-right tion fraction in tetraploid. This inequality is consistent corner, in which neither of the two marers displays with the fact that when two marers are close, their double reduction (Table 1). Of these 36 formations, 6 double reduction rates tend to be similar. We believe are under mode 7, 24 are under mode 8, and the reever, similar inequalities exist for other ploidy levels. How- maining 6 are under mode 9. For example, gamete due to complexity of gamete types for those cases, formations we are not able to generalize the result at this moment. 1343

6 1344 S. S. Wu et al. For a fully informative marer, every gamete genotype can be well distinguished. Thus, N offspring in a fullsib family can be sorted into the nine distinguishable gamete formation modes of size N 1, N 2,...,N 9, respectively (see Table 1). It is not difficult to derive the explicit expressions of the maximum-lielihood estimates for the frequencies of these nine formation modes f 1, f 2,...,f 9 in terms of the corresponding sample frequencies N 1, N 2,...,N 9 on the basis of the following lielihood function given the observed marer data (M): ( f M) N N 1...N 9 9 i 1 f N i i. Since this a fourth-order polynomial of, closed-form solutions exist and can be calculated very easily. The characterization of linage phase: We derived statistical procedures for estimating,, and when the allelic assignment as shown in expression (1) is assumed. The estimates of parameters (,, ) for any one of the other 23 assignments can be similarly obtained by changing the positions of the corresponding elements in matrices H and D. One remaining issue is how to determine the best assignment, i.e., one corre- sponding to a most liely parental linage phase of the two marers. The most liely linage phase can be determined using the posterior probability of (A,,, ) T conditional on the marer data M, where A is the th allelic assignment for marer M l relative to the fixed allelic assignment of marer M. From Bayes theorem: From the above matrix H, which indicates where double reduction has occurred for each of the marers, the two double-reduction parameters, and, can be esti- mated in terms of the corresponding frequencies of formation modes; i.e., ˆ (N 1 N 2 N 3 N 4 )/N and ˆ (N 1 N 2 N 5 N 6 )/N. Since these are simply estimates of binomial proportions, the variances of ˆ and ˆ are (1 )/N and (1 )/N, respectively. Suppose we could distinguish the two f 7 modes and the two f 8 modes; the lielihood function given complete data (N 1, N 2,...,N 6, N 7A, N 7B, N 8A, N 8B, N 9 )is ( f M) N N 1...N 9 6 i 1 f N i i f N 7A f 7A N 7B f 7B N 8A f 8A N 8B f 8B N 9 9. (3) On the basis of the observed incomplete data N 1, N 2,..., N 7, N 8, N 9, the EM algorithm is used to estimate the recombination fraction by maximizing the lielihood Equation 3 (Dempster et al. 1977; Lander and Green 1987). The general equations formulating the iteration of the 1)th EM step are given as follows: E step: Calculate the expected number of recombination events between marers M and M l for all offspring with no occurrence of double reduction. This is equivalent to estimating φ for mode 7 and for mode 8, respectively, by [ ( ) ] 2 2[ ( ) ] 2 18 ( ) 9, ˆ ( ) ( ) 1 2N [N 3 N 5 2(N 2 N 4 N 6 N 9 ) N N 8]. (6) P( M) P( )P(M ) R 24 1P( )P(M ). These posterior probabilities for all possible assignments depend on the prior probabilities P( ). In practice, the prior distribution can be assumed to be uniform among all 24 assignments and, in this case, the posterior probabilities are proportional to the lielihoods L( ) P(M ). The final MLEs for the parameters (,, ) are based on the most liely assignment with the highest posterior probability. Sved (1964) demonstrated that, unless they solely form bivalents, autotetraploids have a recombination fraction bounded by 1 1/x, where x is the level of ploidy. Thus, for autotetraploids undergoing quadriva- lent pairings, the maximum value of recombination fraction is The test of whether or not the two given marers are lined is based on the log-lielihoodratio test statistic under the full model (Equation 3), φˆ( ) 3 2 ( ). (4) which corresponds to the parameter estimators derived from the most liely assignments, and the reduced M step: Maximize the expected log-lielihood of. model with the restraint of The lielihood- This gives an updated estimate for the recombination ratio test (LRT) statistic calculated in this way has a fraction and is obtained as 2 -distribution with 1 2 d.f. under the null hypothesis (Self and Liang 1987). Thus, two marers M and M l ˆ( 1) 1 2N [N 3 N 5 2(N 2 N 4 N 6 N 9 ) can be declared to be lined if the LRT is 1/2, 2 for 2φˆ( ) N 7 (1 ˆ ( ) )N 8 ]. (5) an appropriate choice of the type I error rate (for example, 1/2, ). These two steps are repeated until the estimate of converges to a stable value. Such a stable value is the maximum-lielihood estimate (MLE) of. If we plug φ and from Equations 4 into 5, we can see that the stable values of the iterative procedure are solutions of the following polynomial equation in : SIMULATION Analysis of a simulated data set: We illustrate the autotetraploid model through analyzing a simulated example. Since gamete genotypes can provide adequate information for linage analysis as much as zygote types,

7 Linage Analysis in Autotetraploids we consider analysis only on the segregation of the ga- P1 P1 P 3 mete genotypes in parent P, which is assumed to have l frequencies of double reduction (0.05, 0.1) and recombination fraction These parameters correspond to should be in mode 7 instead of mode 8. The counts relative frequencies of the nine different gamete formation for all nine gamete formation modes, under the new modes f ( , , , , assignments, are N 1 7, N 2 7, N 3 2, N 4 1, N , , , , ), which give 5, N 6 5, N 7 53, N 8 123, N 9 0. Consequently, we the joint relative frequencies of two-marer diploid ga- can obtain the MLE ( ˆ, ˆ, ˆ) (0.07, 0.105, ) metes in the matrix H. A random sample of N 200 with log-lielihood ll A Similar to the first gametes was simulated from multinomial distribution assignment, we also have MLE ( ˆ, ˆ, ˆ) (0.118, 0.208, with probabilities given by H. The marer data M, e.g., 0.75) with log-lielihood ll N under the null the counts of all gamete types, can be presented in the hypothesis. following matrix form: This procedure needs to be repeated for all of the other 22 assignments. In Table 2, we present MLEs and P 1P 1 P 2P 2 P 3P 3 P 4P 4 P 1P 2 P 1P 3 P 1P 4 P 2P 3 P 2P 4 P 3P 4 log-lielihood for all 24 different allelic assignments. P l 1P l Figure 2 (top left) plotted the log-lielihood values P l 2P l against the 24 assignments of marer M l with a dictionary order 1234, 1243,..., Estimates of the recom- P l 3P l P l 4P l bination fraction for different assignments are indicated by different insets in the figure. It shows that a true assignment has the largest log-lielihood value. P l 1P l 2/P l 2P l 1 P l 1P l 3/P l 3P l 1 P l 1P l 4/P l 4P l 1 P l 2P l 3/P l 3P l 2 P l 2P l 4/P l 4P l 2 P l 3P l 4/P l 4P l Suppose parent P has alignment P 1 l P2 P 3 l P3 P 4 l P4 ; l. then there are 11 offspring in the first gamete formation mode (N ). Similarly, counts for the other eight modes are N 2 0, N 3 3, N 4 0, N 5 9, N 6 1, N 7 173, N 8 2, and N 9 1. Hence we have MLEs of the relative frequencies of the nine different gamete formation modes fˆ (11/200, 0, 3/200, 0, 9/200, 1/200, 173/200, 2/200, 1/200), which correspond to ˆ (N 1 N 2 N 3 N 4 )/N 0.07, ˆ (N 1 N 2 N 5 N 6 )/N 0.105, and ˆ with log- lielihood (ll) A Furthermore, under the null hypothesis 0.75, the MLEs of mode frequencies are fˆ ( , 0, , 0, , , , 0.01, ), and the parameter estimates are ˆ 0.022, ˆ with ll N For a second assignment P 1 l P2 P 3 l P4 P 4 l P3, l gamete classification is different. For example, gamete P3 P4 P 3 l P4 l LRT was calculated for each simulation to test for the significance of linage. In Figure 2, the log-lielihood values are plotted against the 24 different allelic assignments of marer M l with a dictionary order 1234, 1243,...,4321. For different assignments, different estimates of the recom- has no recombination and should be classified into mode 1 instead of mode 2, and gamete P l Since assignment 1 has the largest log-lielihood, we choose the final MLEs for the parameters on the basis of the first assignment; e.g., ( ˆ, ˆ, ˆ) (0.07, 0.105, 0.453) with log-lielihood ll A However, under the null hypothesis, the final MLE comes from the last assignment with log-lielihood ll N Thus the LRT statistic equals 2 ( ) , which is much larger than the cut point value 1/2, , implying that there is very strong evidence that the two marers are lined. More simulations: Extensive simulation studies were performed to investigate the properties of our statistical method by evaluating the effectiveness of determining a correct allelic assignment, the precision of the parame- ter estimates, and the power to detect linage. A number of genetic scenarios are designed to explore the effects of different parameter values on their estimation from this new method. A segregating full-sib family of size N 80, 200, 400, or 800 is simulated by hypothesizing different recombination fractions ranging from tight linage to free recombination, 0.05, 0.15, 0.25, 0.50, 0.65, and 0.75, and different pairs of double reduction rates with various degrees of difference between two marers, (, ) (0.05, 0.1), (0.15, 0.2), (0.25, 0.3), (0.1, 0.2) and (0.05, 0.3). For 0.05, however, only the first three pairs of (, ) are considered because the other two combinations are impossible (recall 2 ). The simulation is repeated 1000 times for each scenario. For each replication, the maximum-lielihood estimates ( ˆ, ˆ, ˆ) and the log-lielihood value are obtained for all 24 possible assignments. In addition, the

8 1346 S. S. Wu et al. TABLE 2 Maximum-lielihood estimates ˆ of the recombination fraction with all 24 different allelic assignments of marer M l for the simulated example Allelic assignments MLE Log-lielihood (ll A ) Log-lielihood (ll N ) of marer M l ˆ (alternative) (null hypothesis) P1P l l 2P l 3P l P1P l l 2P l 4P l P1P l l 3P l 2P l P1P l l 3P l 4P l P1P l l 4P l 2P l P1P l l 4P l 3P l P2P l l 1P l 3P l P2P l l 1P l 4P l P2P l l 3P l 1P l P2P l l 3P l 4P l P2P l l 4P l 1P l P2P l l 4P l 3P l P3P l l 1P l 2P l P3P l l 1P l 4P l P3P l l 2P l 1P l P3P l l 2P l 4P l P3P l l 4P l 1P l P3P l l 4P l 2P l P4P l l 1P l 2P l P4P l l 1P l 3P l P4P l l 2P l 1P l P4P l l 2P l 3P l P4P l l 3P l 1P l P4P l l 3P l 2P l The last two columns are the log-lielihood (ll A ) under alternative hypothesis and (ll N ) under null hypothesis ( 0.75). bination fraction are obtained, as indicated by different insets in the figure. It is shown that a true assignment usually corresponds to the largest log-lielihood value. There is a distinct difference between the largest and the second-largest log-lielihood values, especially when is small. This implies that our method can well be used to characterize the marer linage phase in parents. In some cases, the second-largest log-lielihood value is associated with the estimate of 0.75, so it is easy to avoid the assignment corresponding to such an esti- mate. We did not report simulation results about double reduction rate estimates ˆ and ˆ because we have closed- form formulas for their variances. To evaluate the preci- sion of the recombination fraction estimates, square- rooted mean square errors (RMSEs) are calculated for all simulation scenarios (Table 3). As expected, the RMSEs decrease with increasing sample sizes. However, sample size effects also decrease with increasing sample sizes. This means that a sample size of is adequate for providing a precise estimate of. It is also worth noting that the estimate wors reasonably well when N 80. In addition, the RMSEs of ˆ values increase with decreasing but decrease at 0.75 because of the boundary effect. It is seen that the precision of ˆ depends on true double reduction rates (, ) with two tendencies (Table 3). First, the RMSEs tend to be larger when there are larger double reduction rates. Second, the RMSEs tend to increase when the difference of double reduction between the two marers increases. For example, the RMSEs of ˆ 0.5 or above are larger for (, ) (0.10, 0.20) than (0.25, 0.30), although the latter combination has larger double reduction rates. The power to detect a significant linage is examined on the basis of 1000 replicates (Figure 3). Obviously, the power of the test increases with increasing sample sizes. However, the effect of sample size depends on the double reduction rates and recombination fraction. For example, the effect is larger for (, ) (0.1, 0.2) than for (, ) (0.15, 0.2) when 0.65, but this is reversed for 0.5. DISCUSSION The main difficulty in performing linage analysis for autopolyploids stems from the complexities of polyso- mic inheritance. With the occurrence of polysomic in- heritance, the recombination fraction alone is no longer sufficient to specify the frequencies of gamete genotypes and their segregation patterns. To simplify linage anal-

9 Linage Analysis in Autotetraploids 1347 Figure 2. Plot of the log-lielihood (ll) vs. 24 assignments for 18 sets of parameters (,, ) from one simulation with sample size N 200. The x-axes are the 24 assignments in dictionary order, i.e., 1234, 1243,..., Different symbols used in the plots indicate the range of MLE ˆ for each assignment: circles for ˆ in (0, 1/8]; crosses for (1/8, 3/8]; stars for (3/8, 5/8]; diamonds for (5/8, 3/4]; and squares for others. The MLE ˆ s corresponding to the most liely assignment are indicated in each plot.

10 1348 S. S. Wu et al. TABLE 3 Square-rooted mean square error (RMSE) for the estimator ˆ of the recombination fraction for all 28 combinations of parameters (,, ) and four sample sizes N (, ) N (0.05, 0.10) (0.10, 0.20) (0.15, 0.20) (0.05, 0.30) (0.25, 0.30) ysis in autopolyploids, many earlier methods assume a bounded by two times their recombination fraction in pure bivalent pairing model between homologous chromosomes tetraploid. during meiosis (Wu et al. 1992; Hacett et With these underpinning mechanisms of quadriva- al. 1998; Ripol et al. 1999; Luo et al. 2001). Although the lent pairings, Fisher (1947) formulated a pioneering statistical merits of these methods were demonstrated by genetic model to count all possible modes of gamete extensive simulations, their underlying assumption may formation in autotetraploids. But, in his time, he could significantly deviate from biological reality. For an auto- not separate and further estimate two different modes polyploid, multivalent pairings during gametogenesis generating the same gamete genotypes (e.g., mode 7A may result in double reduction (Darlington 1929; de vs. 7B or mode 8A vs. 8B; Table 1). Thans to the Winton and Haldane 1931; Mather 1936; Fisher development of the maximum-lielihood method im- 1947), a phenomenon that adds extra complexity in plemented with the EM algorithm (Dempster et al. the establishment of a worable model for polysomic 1977), we are now able to well discriminate and estimate linage analysis. the proportions of these different modes by viewing In this article, we derive a statistical method for simultaneously them as a missing data problem. estimating the linage and linage phase be- The advantage of the EM algorithm is that it resulted tween different marers in a full-sib family of autotetraploids in closed-form solution for the recombination fraction. undergoing quadrivalent pairings at meiosis. This However, if we forego this, it is also possible to perform method based on quadrivalent pairings is not a simple a Bayesian analysis. We may assign a Dirichlet prior for extension of the existing models on bivalent pairing. the frequencies of the nine formation modes f (f 1, f 2, Rather, the method has incorporated the cytological...,f 9 ), which yields a Dirichlet posterior distribution mechanisms underlying gamete formation derived from of f given the sample frequencies N 1, N 2,...,N 9. Thus multivalent pairings, some of which (i.e., double reduction) we can easily sample from the posterior of f and obtain are unique and do not happen with bivalent pair- a posterior sample of (,, ) by letting f 1 f 2 ings. We also showed that the difference in the frequency f 3 f 4, f 1 f 2 f 5 f 6 and solving using Equation of double reduction between two marers is 6 with each N i /N replaced by f i. Moreover if we extend

11 Linage Analysis in Autotetraploids 1349 Figure 3. The size and power of the lielihood-ratio test of linage using 2 1/2, on the basis of 1000 replicates. The power (or size) of the test vs. true for all five sets of double reduction rates was plotted. gote genotypes, because the segregation at the gamete level cannot provide adequate information for linage analysis. Second, our method is based on a single pairing model quadrivalent. Chromosome pairings in autopolyploids indeed are a function of the homology between the genomes involved, with a propensity in pairing between homologous over homeologous chromo- somes, which is defined as the preferential pairing factor (Sybenga 1994). Such a preferential pairing factor de- termines the relative importance of bivalent vs. multivalent pairings in autopolyploids and, therefore, can be used to model the frequency of double reduction and recombination fraction when both bivalent and multiva- lent pairings happen simultaneously during meiosis. Last, our method is developed for autotetraploids, but its extension to autohexaploid, autooctoploid, and autodexaploid species is important because many important plant species have such high ploidy levels (Soltis and Soltis 2000). For an autohexaploid plant, for instance, triploid gametes are generated at meiosis, including three gamete types of pure double reduction, partial double reduction, and no double reduction. The statistical method proposed in this article describes a mapping framewor for studying the genome structure and organization in complex autopolyploid species, providing a sophisticated model for linage analysis in autopolyploids. It provides a necessary plat- form on which researchers can map quantitative trait loci (QTL) underlying economically and biologically this to the 11 basic gamete modes f* (f 1, f 2,...,f 6, f 7A, f 7B, f 8A, f 8B, f 9 ), then a Gibbs sampler could be set up to obtain posterior samples (Robert and Casella 2000). Although we have devised a statistical method for resolving a fundamentally important problem in autopolyploid linage analysis, one that has puzzled geneticists for over one-half century, there is still much room for improvement. First, our model is proposed for fully informative codominant marers, i.e., those of eight different alleles between the two autotetraploid parents at each marer. For these marers, an explicit expression exists for the MLE of the frequency of double reduction, although the estimate of the recombination fraction must rely upon EM iterations. In a practical full-sib mapping population, other types of marers, such as dominant or partially informative, may be common. For autopolyploids, dominant marers derived from randomly amplified polymorphic DNA or amplified fragment length polymorphism technologies typically cannot be distinguished among simplex (single dose), duplex (double dose), and multiplex (multiple dose) types, because they present an identical genotype (Wu et al. 1992; Yu and Pauls 1993; Luo et al. 2000). For these dominant or partially informative marers, gametes formed with double reduction may have the same genotypes as those formed without double reduction. Thus, estimating the frequency of double reduction will have to require the EM algorithm. Also, linage analysis for these marers must be based on the segregation of zy-

12 1350 S. S. Wu et al. important traits in autopolyploids. Although some prepolyploidy in majority of angiosperms. Science 264: Masterson, J., 1994 Stomatal size in fossil plants evidence for liminary studies have been reported for QTL mapping Mather, K., 1936 Segregation and linage in autotetraploids. J. in autopolyploids, assuming pure bivalent pairings Genet. 32: (Doerge and Craig 2000; Xie and Xu 2000), all of these Meyer, R. C., D. Milbourne, C. A. Hacett, J. E. Bradshaw, J. W. McNichol and R. Waugh, 1998 Linage analysis in tetraploid should be viewed as premature until a comprehensive potato and association of marers with quantitative resistance to model is framed to tae both bivalent and multivalent late blight (Phytophthora infestans). Mol. Gen. Genet. 259: pairings into account. Otto, S. P., and J. Whitton, 2000 Polyploid incidence and evolu- tion. Annu. Rev. Genet. 34: We are grateful to Dr. S. Xu and Dr. M. Gallo-Meagher for stimulat- Ripol, M. I., G. A. Churchill, J. A. 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If genome organization in a highly polyploid and aneuploid inter- we form gametes with two chromosomes by selecting, specific hybrid. Genetics 142: with replacement, from the 16 alleles twice, this yields Hacett, C. A., J. E. Bradshaw, R. C. Meyer, J. W. McNicol, D. Milbourne et al., 1998 Linage analysis in tetraploid species: possibilities (16 with no recombination, a simulation study. Genet. Res. 71: with one recombination, and 144 with two). Hilu, K. W., 1993 Polyploidy and the evolution of domesticated Recall that φ and are the proportions of gamete plants. Am. J. Bot. 80: Jacson, R. C., and J. W. Jacson, 1996 Gene segregation in autotetraploids: prediction from meiotic configurations. Am. J. Bot. 83: 7 and 8, respectively. Note that f 7A contains 12 out of types that have two recombination events under modes gametes with no recombination and f 7B contains 12 Lander, E. S., and P. Green, 1987 Construction of multilocus geout of 144 gametes with two recombinations; thus the netic linage maps in human. Proc. Natl. Acad. Sci. USA 84: relative proportions should be 12(1 ) 2 /16:12 2 / Luo, Z. W., C. A. Hacett, J. E. Bradshaw, J. W. McNicol and 144 9(1 ) 2 : 2. Similarly, f 8A contains 48 out of 96 D. Milbourne, 2000 Predicting parental genotypes and gene segregation for tetrasomic inheritance. Theor. Appl. Genet. 100: gametes with one recombination and f 8B contains out of 144 gametes with two recombinations; thus the Luo, Z. W., C. A. Hacett, J. E. Bradshaw, J. W. McNicol and D. relative proportions should be 48 2 (1 )/96:48 Milbourne, 2001 Construction of a genetic linage map in tetraploid species using molecular marers. Genetics 157: /144 3(1 ):. Consequently, we may assume φ /(9(1 ) 2 2 ) and /(3 2 ).