THEORY OF LINKAGE MAP CONSTRUCTION

Model and notation: The theoretical analysis considers a full-sib family derived from crossing two autotetraploid parental lines. Let M_i (i = 1... m) be m marker loci (with dominant or codominant inheritance). Let G₁ and G₂ be the genotypes at the marker loci for two parental individuals, respectively. G_i (i = 1, 2) can be expressed as a m × 4 matrix. Because two tetraploid individuals have at most eight distinct alleles, we represent each element of G_i as a letter A-H or O, where O represents the null allele due to mutation within primer sequences (see, for example, Callenet al. 1993). It is important to note that allele A at marker locus 1 is different from allele A at marker locus 2.

When we are considering linked loci, it is often necessary to specify how the alleles at different loci are grouped into homologous chromosomes, i.e., the linkage phases of the alleles. Alleles linked on the same homologous chromosome will appear in the same column of the matrix G_i. For a two-locus genotype with four different alleles at each locus, one possible genotype is (ABCDABCD). This indicates that allele A of locus 1 is on the same chromosome as allele A of locus 2, allele B of locus 1 is on the same chromosome as allele B of locus 2, etc. Alternatively we could have a genotype matrix, for example, (ABCDBACD). In this case allele A of locus 1 is on the same chromosome as allele B of locus 2, etc. If the phase is uncertain, alleles will be enclosed in parentheses. For brevity in the text of this article, two-locus genotypes of known phase are also written using a slash to separate the chromosomes, so that the above genotypes would be written as AA/BB/CC/DD and AB/BA/CC/DD, respectively.

We define P₁ and P₂ to be the phenotypes of the two parents, i.e., their gel band patterns at the marker loci. P_i (i = 1, 2) can be denoted by a m × 8 matrix, each of whose elements may take a value of 1 indicating presence of a band at the corresponding gel position or 0 indicating absence of a band. These matrices carry no information about phase. The jth rows of G_i and P_i correspond to locus M_j. Let O_Mi be the n × 8 matrix of phenotypes of the n offspring at the marker locus M_i.

In general, there is no simple one-to-one relationship between the phenotype and the genotype of markers scored in tetraploid individuals. There are two reasons for this. First, a multiple dosage of an allele cannot be distinguished from a single dosage on the basis of the gel band pattern. Second, some alleles may not be revealed as the presence of a corresponding gel band, i.e., the null alleles. Table 1 summarizes the relationship between genotype and phenotype at a marker locus in which all possible cases of null alleles and multiple dosages of identical alleles are taken into account. It can be seen from Table 1 that there may be four, six, four, or one corresponding genotype(s) if the parental phenotype shows one, two, three, or four bands. An individual genotype can be uniquely inferred from its phenotype if and only if the individual carries four different alleles and these alleles are also observed as four distinct bands.

Luo et al. (2000) recently developed a method for predicting the probability distribution of genotypes of a pair of parents at a codominant (for example, RFLPs, microsatellites) or dominant (for example, AFLPs, RAPDs) marker locus on the basis of their and their progeny’s phenotypes scored at that locus. This approach infers the number of possible configurations of the parental genotypes with the corresponding probabilities, conditional on the parental and offspring phenotypes. For each of the predicted parental genotypic configurations, the expected number of offspring phenotypes and their frequencies can be calculated and compared to the observed frequencies. Results from a simulation study and analysis of experimental data showed that in many circumstances both the parental genotypes can be correctly identified with a probability of nearly 1. A tetrasomic linkage analysis can then be carried out using the most probable parental genotype, or using each of a set of possible parental genotypes in turn if more than one genotype is consistent with all the phenotypic data. This is illustrated in the following analyses of data from simulation and experimental studies.

The steps of the linkage analysis are (i) the prediction of the parental genotype(s) that is consistent with the parental and offspring phenotype data using the method described in Luo et al. (2000); (ii) the detection of linkage between pairs of marker loci and their partition into linkage groups; (iii) the estimation of linkage phase, recombination frequency, and LOD score for pairs of markers within each linkage group; and (iv) the ordering of markers within each linkage group. The power to detect linkage and the variance of the estimates of the recombination frequency are shown to vary considerably with parental configuration and phase, and this will be examined.

Test for independent segregation of loci: The first step of the linkage analysis is to test whether pairs of loci are segregating independently. We propose that this may be investigated for each pair of markers by representing their joint segregation in a two-way contingency table and testing for independent segregation, as discussed by various authors (e.g., Maliepaardet al. 1997) for diploid crosses. Let n_ij be the observed number of progeny with the ith (i = 1, 2,..., I) marker phenotype at the first locus and the jth(j = 1, 2,..., J) marker phenotype at the second locus. The expected number under independent segregation is e_ij = n_i·n_·j/n, where ni⋅=∑j=1Jnij and n⋅j=∑i=1Inij . The observed and expected numbers may be compared by Pearson’s chisquare statistic, χ2=∑i=1I∑j=1J(nij−eij)2eij. (1) Other possible test statistics are the likelihood-ratio test G2=2∑i=1I∑j=1Jnijlognijeij (2) or the Cressie-Read family of power divergence statistics C(λ)=2λ(λ+1)∑i=1I∑j=1Jnij[(nijeij)λ−1]. (3) These statistics have an asymptotic chi-square distribution with d.f. = (I - 1)(J - 1). However, in this application the contingency tables may be sparse, as the number of cells can be as large as 36² = 1296, and so the asymptotic distribution cannot be assumed without investigation. Table 2 compares the percentage points for the distribution of χ², G², and C(λ) with λ = ⅔ [as recommended by Cressie and Read (1984) for sparse tables] for 500 simulations of the configuration AA/BB/CC/DD × EE/FF/GG/HH, with the two loci segregating independently and 200 offspring. The percentage points for Pearson’s chi-square statistic are closest to the true values, but the other two have lower percentage points. The three distributions were compared for several other configurations, but Pearson’s chi-square statistic always had percentage points closest to the true distribution.

The power of Pearson’s chi-square test to detect linkage was examined for 100 simulations of each of a range of configurations, linkage phases, and true recombination frequencies. For true recombination frequencies r ≤ 0.2, the power was generally 100% (i.e., the hypothesis of independent segregation was always rejected) for a significance level α = 0.01 and >90% for r ≤ 0.3. The exceptions to this were configurations with alleles restricted to simplex repulsion or duplex mixed configurations; e.g., for cross AB/AA/BA/BB × CC/CD/DD/DC, with all alleles in duplex mixed configurations, and a true recombination frequency of 0.2, independent segregation was rejected for 3/100 simulations. When the markers were genuinely unlinked, the rejection rate for a significance level α = 0.05 was found to be close to 5% for all configurations examined.

Partition of loci into linkage groups: Cluster analysis is a suitable technique to partition the marker loci into linkage groups, so that a marker segregates independently of markers in different linkage groups and shows a significant association with at least some of the other markers within its linkage group. The above test statistics depend on the number of marker phenotypes at each locus, but the significance level of the test for independent segregation is comparable for all pairs and could be regarded as a distance between loci. Although it ranges from 0 for the most tightly linked loci to 1, the range (0, 0.05) is of most interest for indicating pairs of loci that are likely to be linked. We therefore prefer to transform the significance level, say s, to a measure of distance that gives more discrimination between the distances of most interest. The transformation d = 1 - 10^-2s, which maps the range of the significance level (0, 0.05) to the range of the distance measure (0, 0.21), was used here, although many alternative transformations are possible. Different clustering methods will give slightly different dendrograms: the nearest-neighbor cluster analysis adds a marker to a cluster according to its distance to the closest marker in the cluster, but can combine large groups on the strength of one marker from each subgroup. We prefer to compare the dendrogram from nearest-neighbor cluster analysis with that from average linkage cluster analysis to avoid such “chaining.” Inspection of the clustering at distances corresponding to different levels of significance will indicate how the marker loci should be partitioned into linkage groups. In practice, the criterion for partitioning the dendrogram into different linkage groups can be determined as the distance measure by which significant linkage is inferred. However, the Bonferroni correction for the overall significance level may be necessary to take the multiple linkage tests into account. The calculation of recombination frequencies and LOD scores then proceeds for each linkage group in turn.

Calculation of segregation probabilities: One of the major difficulties in linkage analysis with tetraploid species is to calculate the conditional distribution of the offspring genotypes, and hence phenotypes, at two linked loci for any given pair of parental genotypes. This involves consideration of a large number of segregation and recombination events. In this section, a general computer-based algorithm is described to compute the probability distribution.

For simplicity but without loss of generality, we use A and B for two loci in this section and subscripts to represent the alleles. Consider a parental genotype A_iB_i/A_jB_j/A_kB_k/A_lB_l. During gametogenesis of the individual, three equally likely pairs of bivalents can be generated, i.e., A_iB_i/A_jB_j//A_kB_k/A_lB_l, A_iB_i/A_kB_k//A_jB_j/A_lB_l, and A_iB_i/A_lB_l//A_jB_j/A_kB_k, where is used to distinguish paired homologous chromosomes. The gametes created from each of these pairs of bivalents can be sorted into three classes: (i) nonrecombinants, A_ξB_ξA_ηB_η(ξ ≠ η; ξ and η may be i, j, k, or l), four gametic genotypes, each of which has a frequency of (1 - r)²/4; (ii) single recombinants A_ξB_ηA_γB_γ(ξ ≠ η ≠ γ; ξ, η, or γ may be i, j, k, or l), eight gametic genotypes, each with a frequency of r(1 - r)/4; (iii) double recombinants A_ξB_ηA_γB_∼ (ξ ≠ η ≠ γ ≠ ∼; ξ, η, γ, or ∼ may be i, j, k or l), four gametic genotypes, each with a frequency of r²/4. Thus, when the three possible pairs of bivalents are considered, a general form for frequency of the gametic genotype i can be written as gi=xi012(1−r)2+xi112r(1−r)+xi212r2, (4) where x_i0, x_i1, and x_i2 are numbers of the nonrecombinants, single recombinants, and double recombinants, respectively, within the ith gametic genotype class. With random union between all possible gametes generated from two parents and sorting the zygotes according to their genotype, a general formula for the frequency of zygote genotype i may be expressed as hi=1144[yi0(1−r)4+yi1r(1−r)3+yi2r2(1−r)2+yi3r3(1−r)+yi4r4]=1144∑j=04yijrj(1−r)4−j, where y_ij is the number of zygotes with j recombinations within the ith zygote genotype.

To evaluate the coefficients y_ij manually is obviously very tedious. A computer algorithm was developed to calculate the offspring’s genotypic distribution for any given pair of tetraploid parental genotypes. The computer subroutine outputs the number of all possible distinct offspring genotypes k and {y_ij} (i = 1, 2,..., k) from the two parental genotypes. For example, if two parental genotypes are AA/BB/BB/OB and CA/DAEC/EO, there are a total of 225 possible genotypes in their offspring. Many of these offspring genotypes correspond to the same phenotype. Thus, the phenotypic distribution of the offspring can be readily derived by combining the probabilities of those genotypes that result in the same phenotype, so that the general formula for the probability of zygote phenotype i is fi=∑g∈ihg=1144∑g∈i∑j=04ygjrj(1−r)4−j. (5) In the above equation, R_g∊ih_g indicates the sum over the frequencies of all those genotypes g that correspond to the same phenotype i. For instance, the 225 offspring genotypes in the above example are classified into 36 distinct phenotypes when the marker genes are assumed to be codominant, and these are illustrated in Table 3. It can be seen that the frequency of the first phenotype is f₁ = [8(1 - r)⁴ + 32r(1 - r)³ + 40r²(1 - r)² + 24r³(1 - r) + 8r⁴]/144 = (1 - r² + r³)/18.

Maximum-likelihood estimate of r: If the parental genotypes and their linkage phase are known, the joint expected phenotypic distribution of their offspring can be derived using the method suggested above. The corresponding observed offspring phenotypes at the marker loci can be recognized as a random sample from a multinomial distribution with probabilities f_i (i = 1, 2,..., k) and sample size n = R^k_i=1n_i, where k is the number of possible phenotypes and n_i is the observed number of offspring in the ith phenotype class. Thus, the log-likelihood of the recombination frequency, r, given the observed data at loci M_i and M_j, is given by L{r∣OMi,OMj,G1,G2}=ln{(nn1,n2,…,nk)f1n1f2n2⋯fknk}=C+∑i=1kniln(fi), (6) where f_i (i = 1, 2,..., k) are functions of r and given by Equation 5.

The maximum-likelihood estimate (MLE) of the recombination frequency r may be obtained by solving ddrL{r∣OMi,OMj,G1,G2}=∑i=1knifiddr(fi)=0. (7) Only in a limited number of cases can the likelihood equation be solved analytically because the equation is usually a polynomial with a power ≥5. An iterative solution may be obtained, however, using the expectation-maximization (EM) algorithm (Dempsteret al. 1977). Maliepaard et al. (1997) applied the EM algorithm to give a general formulation for all possible genetic configurations in a cross between two outbreeding diploid plant species, and here we modify their approach for the autotetraploid case.

In Equation 5, define z_ij = Σ_g∊iy_gjr^j (1 - r)^{4 - j}/144, so that the probability of phenotype i is f_i = R⁴_j=0z_ij. Substituting this into Equation 7, we obtain ddrL{r∣OMi,OMj,G1,G2}=∑i=1knifi∑j=04ddr(zij)=∑i=1kni∑j=04zijfiddr(ln(zij))=∑i=1kni∑j=04zijfij−4rr(1−r). (8) So the derivative of the likelihood is equal to zero when r=14n∑i=1kni∑j=04jzijfi. (9) From an initial estimate of r, we can calculate z_ij/f_i, the expected proportion of individuals of phenotype i with j recombinants (the expectation step of the EM algorithm), and substitute this into Equation 9 to give an updated estimate of r (the maximization step). The algorithm is iterated until the sequence of estimates of r converges.

Estimation of parental pairwise linkage phases: In the above analyses, it was assumed that the parental genotypes and their linkage phases were known. In practice, only the parental and offspring phenotypes are observable. As pointed out in the Model and notation section, Luo et al. (2000) calculate the genotypic distribution for any pair of tetraploid parents at a single dominant or codominant marker locus using data on the marker phenotypes scored on the parents and their offspring. However, the method does not provide information about the linkage phases of the alleles the parents carry at different loci. Knowledge about the linkage phase of the parental genotypes is not only required in the linkage analysis, but it is also important in using the map information in locating QTL (e.g., Lander and Botstein 1989) or optimizing schemes of marker-assisted selection for quantitative traits (Luoet al. 1997).

The number of possible different linkage phases depends on the number of distinct alleles at each locus and increases exponentially with the number of loci under consideration. We therefore consider here the phase for each pair of linked loci and use these as building blocks to estimate the multilocus linkage phase. In a two-locus system of tetrasomic inheritance, an individual genotype may have a maximum of 4 × 3 × 2 = 24 distinct linkage phases, and for a pair of individuals there may be a maximum of 24 × 24 = 576 distinct linkage phase configurations. A Fortran-90 computer subroutine was developed to work out all possible linkage phase configurations for any given pair of parental genotypes G₁ and G₂ at any two loci i and j. Let S₁ and S₂ be possible two-locus linkage phases for parents 1 and 2, respectively. The likelihood of r, S₁, and S₂, given the observed phenotypic data O_Mi and O_Mj at the loci, may be written as lp[r,S1,S2∣OMi,OMj,G1,G2]=Pr{OMi,OMj∣r,S1,S2}=(nn1,n2,…,nk)f1n1f2n2⋯fknk, (10) where the expected frequency of the ith offspring phenotype, f_i (i = 1, 2,..., k), is calculated for given r, S₁, and S₂ as demonstrated in Equation 5. As discussed above, the EM algorithm can be used to maximize the likelihood function of Equation 10 for every possible configuration of S₁ and S₂. The configurationŜ_1,Ŝ₂ for which this maximum is the highest is taken as the maximum-likelihood estimate of the parental genotypic linkage phase and the corresponding value of rˆ is the maximum-likelihood estimate of r. The LOD score for each pair of marker loci is calculated as LOD=log10lp[r^,S^1,S^2∣OMi,OMj,G1,G2]lp[0.5,S^1,S^2∣,OMi,OMj,G1,G2]. (11)

Ordering the markers: The above analyses give the maximum-likelihood estimate of the recombination frequency and the linkage phase for each pair of markers in a linkage group. This information can be used to order the markers in linkage groups and to calculate map distances between them. One possible approach, the least-squares method for estimation of multilocus map distances as implemented in the JoinMap linkage software (Stam and Van Ooijen 1995), was examined by Hackett et al. (1998) in a simulation study of dominant markers in a tetraploid population. They concluded that the reconstructed marker order and map distance using the JoinMap analysis of simulation data was in good agreement with the simulated ones. The same method was used here.

Estimation of parental multilocus linkage phases: Once the markers have been ordered, we need to reconstruct the phase of the complete linkage group. Prediction of the multilocus linkage phase in tetrasomic linkage analysis is not feasible: there are a huge number of configurations of possible phases and no appropriate theory of multilocus linkage analysis for tetraploid species. Here we propose an intuitive algorithm to predict the multilocus parental linkage phase in the tetrasomic linkage analysis, on the basis of the range of likelihood values of the alternative linkage phases obtained in the above two-locus analysis. Let d_ij be the difference in the log-likelihood value between the most likely and the second most likely linkage phases predicted for the marker loci i and j on a linkage group. The phase of the marker pair with the largest log-likelihood difference d_ij is reconstructed first, and further markers are then placed relative to this pair, placing markers with large d_ij before those with smaller d_ij. There may be a contradiction between the phase of two markers estimated directly and the phase estimated when each of the pair is referred to a third marker; we reject an overall configuration with such contradictions for a pair with large d_ij, but accept the overall configuration if d_ij is close to zero.

INFORMATION AND POWER OF THE MAXIMUM-LIKELIHOOD ESTIMATION

The information of the maximum-likelihood estimate of the recombination frequency r is given by I(r)=−E[d2dr2Lp[r,S1,S2∣OMi,OMj,G1,G2]]=−E[∑i=1kni[1fid2dr2(fi)−1fi2(ddr(fi))2]], (12) where E denotes expectation. It can be shown that the expectation of the first term on the right-hand side of this expression is equal to zero. Substituting for the second term, we obtain I(r)=E[∑i=1knifi2(ddr(fi))2]=E[∑i=1knifi2(∑j=04ddr(zij))2]=nr2(1−r)2[∑i=1k1fi(∑j=04jzij)2−16r2]. (13) The details of the derivation of this equation are given in the appendix.

Hackett et al. (1998) demonstrated that the simplex coupling linkage phase was the most informative for estimating recombination frequency among dominant marker configurations. For this the information is n/r(1 - r) and the information content of other configurations is examined relative to this by means of the relative information RI(r)=I(r)n∕r(1−r)=1r(1−r)[∑i=1k1fi(∑j=04jzij)2−16r2]. (14) We can also examine the power of the likelihood-ratio test. The likelihood-ratio test statistic is given by G2=2{ln[lp[r^,S^1,S^2∣OMi,OMj,G1,G2]]−ln[lp[r=0.5,S^1,S^2∣OMi,OMj,G1,G2]]}. (15)

It has been shown by Agresti (1990, pp. 98, 241) that G²has an approximate large-sample noncentral chisquare distribution with 1 d.f. and the noncentral parameter in the present context is λ=2n∑i=1kfi(r)ln[fi(r)fi(0.5)]. (16) Thus, the statistical power for the linkage test at a given significance level α is given by the probability βL=Pr{χ1,λ2>χ12(α)}, (17) where χ1,λ2 represents a random variable with a noncentral chi-square distribution with 1 d.f. and the noncentrality parameter λ, and χ12(α) is the 1 - α percentile of a central chi-square distribution, also with 1 d.f.

For two parents, there are 128 configurations at a single locus where the parents share one or more alleles, which are informative about recombination in both parents. This count does not include permutations of the parents; i.e., AAOO × AOOO and AOOO × AAOO are considered as the same configuration. To consider all pairs of such loci, and to allow for the different phases, would give a very large number of configurations. We therefore examined the information and power of the likelihood-ratio test for each configuration when linked to a locus with eight alleles, ABCD × EFGH. The most informative configurations are those with seven or eight alleles: AA/BB/CC/DD × EE/FF/GG/HH and AA/BB/CC/DD × EA/FE/GF/HG, which are four times as informative as the simplex coupling configuration for all values of the recombination frequency. For many configurations, the relative information varies with the recombination frequency. At a recombination frequency of 0.2, 20 of the configurations examined were less informative than simplex coupling: these configurations were characterized by a small number of alleles occurring as simplex or duplex in each parent. The least informative configuration was AA/BA/CO/DO × EA/FA/GO/HO, with a relative information of 0.14. There was a strong linear relationship between the information and the noncentrality of the likelihood-ratio test, for example, a correlation of 0.996 using a recombination frequency of 0.2. For a recombination frequency of 0.2 and a population of 200 offspring, the power of the likelihood-ratio test was >0.9 for all configurations except the least informative AA/BA/CO/DO × EA/FA/GO/HO, although the power decreases with decreasing population size or increasing marker separation.

When the two parents do not share any alleles, the information can be calculated for each parent separately and then summed. The most informative configuration for a single parent is AA/BB/CC/DD, which is twice as informative as the simplex coupling configuration for all values of the recombination frequency. The relationship between the information and the noncentrality is the same as for two parents with shared alleles. The least informative configurations are some of those with a single informative allele: duplex-duplex mixed (AA/AO/OA/AA, relative information = 0.04), simplex repulsion (AO/OA/OO/OO, relative information = 0.07), and duplex-duplex repulsion (AO/AO/OA/OA, relative information = 0.11). Some configurations with two informative alleles also have very low information, for example, AO/AB/BA/OA, where the two duplex alleles at each locus are in repulsion and so are the two simplex alleles. The relationship between the relative information and the recombination frequency is illustrated for a range of configurations in Figure 1.

As the information depends on the configurations of both loci and on their phase, it is difficult to exclude any single-locus configurations as uninformative. The more alleles at a locus, the more informative it is likely to be, especially if these loci are present in a simplex configuration. A locus with an allele that occurs in both parents is likely to have a low information content, unless we are considering a configuration such as AA/OO/OO/OO × AA/OO/OO/OO, with single-dose alleles in coupling in both parents. The configurations with low power for detecting nonindependent segregation by Pearson’s chi-square test also had low information and low power in the likelihood-ratio test.

SIMULATION STUDY

To validate the theoretical analyses represented above and to investigate their statistical properties, we conducted a simulation study using the method developed above.

Simulation model: Computer programs were developed to simulate meiosis in a tetraploid individual with any genotype at the simulated marker loci, random pairing of four homologous chromosomes to give two bivalents (i.e., no double reduction), random sampling of gametes from meiosis, random union of gametes randomly sampled from the gamete pool, and generation of the phenotype from any given individual genotype. In a single meiosis, the “random walk” procedure suggested by Crosby (1973) was extended to simulate genetic recombination between linked loci. Chiasmata interference, sexual differentiation in recombination frequency, and segregation distortion were assumed to be absent in the simulation model.

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Figure 1.

—Relative information about the recombination frequency for different parental genotype configurations.

A full-sib family was simulated by crossing two tetraploid parental lines. Twenty-two codominant marker loci were generated, 10 linked on the first chromosome, 5 on each of the second and third chromosomes, and 2 isolated loci that were independent of the rest. The simulated parental genotypes at each of the marker loci were determined by sampling independently from six possible alleles whose population frequencies were assumed to be 0.3 (allele A), 0.2 (allele B), 0.2 (allele C), 0.1 (allele D), 0.1 (allele E), and 0.1 (null allele O), respectively. Loci with more than six alleles were not simulated, as these appear to be rare in practice (R. C. Meyer, personal communication). The main purpose for choosing parental genotypes in such a way is to test the theory and method on a general basis. The parental genotypes at these marker loci and the recombination frequencies between the adjacent loci are shown in Table 4. It should be noted that the alleles listed in the same column for loci on the same chromosome have the same linkage phase. The phenotypes of the two parents and 200 offspring (a realistic number for actual experiments) were scored at all 22 marker loci. To elucidate statistical properties, some pairs of these loci were studied in 100 repeated simulation trials.

Analysis of the simulated data: The genotypes of the two parents were predicted for each of the loci using the method proposed by Luo et al. (2000), on the basis of the phenotypes of the parents and their offspring. The predicted parental genotypes are tabulated in Table 4 together with the corresponding probabilities. It can be seen that the parental genotypes at 18 of the 22 marker loci were diagnosed correctly with a prediction probability of nearly 1.0. However, there were two almost equally likely parental genotypes predicted for the marker loci L₂, L₅, L₂₀, and L₂₂. For locus L₂ the parental phenotypes are the same (1110000), but the most likely parental genotypes are different (AABC and ABCC) and it is not possible at this stage to tell which parent has which genotype. Both genotypes at this locus were used in the linkage analysis. For the other three loci, allele A is present for all offspring, and this is consistent with more than one configuration with multiple dosages of A. The dosages of the informative alleles are the same for the two possible configurations for L₅, L₂₀, and L₂₂, and so estimates of recombination frequencies are the same for the two configurations.

Pearson’s chi-square tests of independence were performed for all possible pairs of these marker loci using the test statistic given in Equation 1. Figure 2 displays the significance probabilities, transformed to distances as described previously, as dendrograms calculated using nearest-neighbor cluster analysis and average linkage cluster analysis. The nearest-neighbor analysis shows the three clusters (loci L₁-L₁₀, L₁₃-L₁₇, and L₁₈-L₂₂) have each grouped at a distance of zero. Loci L₁₁ and L₁₂ remained isolated until the distance exceeded 0.13. However, the three linkage groups also merge at a very small distance. Inspection of the significance levels shows that this is due to a single (spurious) significant association between L₁ and L₁₈. Inspection of the average linkage cluster analysis shows that the same initial groupings form more slowly, but that locus L₁₈ is clearly associated with L₁₉ and L₂₀, and the average distance between locus L₁₈ and loci L₁-L₁₀ is large. The distantly linked group L₁₈-L₂₂ finally merges at a large distance using average linkage cluster analysis, but inspection of the significance levels shows highly significant associations between 6 of the 10 pairs of this group, and we proceed assuming that they form a linkage group.

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Figure 2.

—Cluster analysis of the 22 simulated loci, using (a) nearest-neighbor cluster analysis and (b) average linkage cluster analysis.

Linkage analysis was performed on all pairs of loci within each linkage group. For brevity only the results from the largest linkage group (loci L₁-L₁₀) are presented. Table 5 shows the significance of this test (the first rows of the lower diagonal), the maximum-likelihood estimates of recombination frequencies (the upper diagonal) for the most likely phase, and the corresponding LOD scores (the second rows of the lower diagonal) among the pairs of loci. It can be seen that the true parental genotype at L₂ has consistently higher LOD scores in its pairings with L₁, L₃, L₄, L₆, and L₇ (i.e., all the highly significant linkages) than the other predicted parental genotype (L_2′) with the parental genotypes reversed. The estimate of the recombination frequency and LOD score were unaffected by the choice between the alternative genotypes for locus L₅. Most cases where the independence test was significant (P < 0.05) corresponded to LOD scores >3, although a small number of pairs with a significant independence test (e.g., L₁, L₈) had large recombination frequencies and lower LODs.

The maximum-likelihood estimates of the pairwise recombination frequencies and the LOD scores in Table 5 were used to construct a linkage map of these genetic marker loci using the JoinMap linkage software (Stam and Van Ooijen 1995), as summarized in Figure 3. The best-fitted map predicted from JoinMap indicates that loci L₁-L₁₀ were joined into a correct order except that the relative simulated positions of the marker loci L₇ and L₈ were reversed. The map distances of the linkage group agreed well with the actual ones.

The linkage phases of the parental genotypes were reconstructed using the procedure described in the above analysis. Table 6 illustrates the parental linkage phases at every pair of loci with a difference d_ij > 3 in the log-likelihood between the most likely and second most likely phase. Locus L₅ does not appear in Table 6, as there was only one phase with a recombination frequency <0.5 in each case. The reconstructed linkage phases of the parental genotypes are shown in Figure 3. This reconstruction uses the most likely phase for all pairs except for four [(L₁, L₈), (L₁, L₉), (L₂, L₇), (L₆, L₉)]. For these four pairs, the largest difference in the log-likelihood between the most likely phase and the reconstructed phase was 0.56, and the difference in the estimates of the recombination frequency was always <0.01. The reconstructed phase is identical to that simulated.

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Figure 3.

—The best-fitted map, the estimated map distance (in centimorgans), and parental linkage phases reconstructed from the codominant marker loci L₁-L₁₀ from the simulation study.

Linkage maps of loci L₁₃-L₁₇ and L₁₈-L₂₂ were estimated using the same approach. In each case the order and phase were reconstructed correctly.

To investigate the reliability of the pairwise linkage phase estimation, separate simulation trials were performed. The simulated recombination frequencies were 0.05, 0.1, and 0.3 and the sample size was 200. Figure 4 illustrates the maximum-likelihood estimate of r between L₉ and L₁₀ (Figure 4a), and between L₇ and L₁₀ (Figure 4b), and the corresponding LOD scores calculated at all possible parental linkage phase configurations. It can be seen that the correct parental linkage phases were the most likely when the marker loci were closely linked (i.e., r ≤ 0.1), although the difference in the likelihood value between the most likely and the second most likely linkage phases reduced as the value of r increased. When the loci were loosely linked (i.e., r = 0.3), the most likely parental linkage phase could differ from the simulated phase, but when this occurred the MLE of r at the most likely phase was always very close to that calculated at the simulated phase.

Further simulations were carried out to examine the power to detect linkage and the bias in estimates of the recombination frequency. Table 7 shows the means and standard deviations of the maximum-likelihood estimates of recombination frequencies for 100 replicate simulations of some pairs of marker loci considered above. Linkage was detected as significant (P < 0.05) by both the independence test and the likelihood-ratio test with a frequency ≥90% when the recombination frequency r ≤ 0.3, except for the least informative pair (L₂, L₅). For r = 0.5, the frequency of significant tests was close to 5%. The means of the MLEs of r were close to the corresponding simulated values for r ≤ 0.3. For r = 0.5, the marker estimates were biased downward, due to the selection of the most likely phase. The parental linkage phases at the marker loci were correctly predicted for at least 89% of simulations with cases with r ≤ 0.3.

LINKAGE ANALYSIS OF EXPERIMENTAL DATA IN AUTOTETRAPLOID POTATO

Some preliminary data from the Scottish Crop Research Institute were used to test this approach, using five SSR marker loci (STM0017, STM1017, STM1051, STM1052, and STM1102) and six AFLP marker loci (e35m61-18, e35m61-21, e37m39-14, e39m61-7, e46m37-12, and p46m37-12) scored on 77 offspring from a cross between two parental lines: the advanced potato breeding line 12601abl and the cultivar Stirling (Bradshawet al. 1998). Details of scoring the DNA molecular markers are described in Meyer et al. (1998) and Milbourne et al. (1998). Preliminary analysis of the AFLP markers (Meyeret al. 1998), and of the SSR markers in diploid and tetraploid populations (Milbourneet al. 1998) suggested that these markers are all on the same linkage group.

Table 8 summarizes the parental phenotypes and the phenotype distribution of the offspring at the marker loci. Of a total of 77 offspring scored at these marker loci, there were 73, 73, 72, and 70 progeny whose phenotypes at the marker loci STM1017, STM1051, STM1052, and STM1102, respectively, were unambiguously observed. The phenotypic data were used to predict the parental genotypes using the method of Luo et al. (2000). The predicted parental genotypes at the marker loci are also shown in Table 8 together with the corresponding prediction probabilities and the χ² values of the goodness-of-fit test. It was found from the analysis that the number of possible genotype configurations with probability ≥0.1 varies from 1 (at STM0017, STM1051, and the AFLP marker loci) up to 8 (STM1017). For this locus, all that can be deduced is that allele 1 occurs in a simplex condition in parent 1.

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Figure 4.

—The maximum-likelihood estimates of recombination frequencies and the corresponding LOD scores for all possible linkage phases for (a) loci L₉ and L₁₀ and (b) loci L₇ and L₁₀. Arrows indicate the true linkage phases.

The independence tests were performed for all possible pairs of the marker loci and the significance probabilities of the tests are listed as the first rows of the lower diagonal in Table 9, along with the results of a pairwise linkage analysis. The maximum-likelihood estimates of recombination frequency between the pairs of marker loci are listed on the upper diagonal and the LOD scores are given in the second rows of the lower diagonal of Table 9. Both possible genotypes for locus STM1052 are shown, as these gave slightly different estimates of the recombination frequencies and LOD scores. The likelihood for genotype AABO × AACO was always larger than for the other genotype. The use of the alternative parental genotypes at loci STM1017 and STM1102 did not affect the estimates of recombination frequencies and LOD scores.

The MLEs of the pairwise recombination frequencies and the LOD scores were used to map the marker loci using JoinMap. The 11 markers were mapped as a linkage group with a length of 48.9 cM (using genotype AABO × AACO for STM1052). The order was the same using the alternative genotype, and the calculated length in this case was 48.7 cM. The allelic linkage phases of the parental genotypes at the marker loci were reconstructed as described. The linkage map and the reconstructed phases are illustrated in Figure 5. For loci STM1017 and STM1102, the dosage of the alleles that are present for all offspring is uncertain, but the phase of the segregating alleles can be reconstructed. For STM0017, there is uncertainty about the phase for parent 2 as this marker is well separated from the other SSR markers that are informative about parent 2, although allele A of this marker is unlikely to be linked in coupling to the simplex alleles of the other SSR markers (A for STM1102, C for STM1052, and D for STM1051). The only difference between the inferred phase in Figure 5 and the most likely phase is for the pair STM0017 and STM1102, for which the inferred phase has a log-likelihood 1.16 less than the most likely, although both phases correspond to loose linkages (recombination frequency ≈ 0.4, LOD 0.90). The conclusion that the SSR markers form a single linkage group agrees with the analysis of these markers in a diploid cross (Milbourneet al. 1998).

APPENDIX: DERIVATION OF THE INFORMATION MEASURE

By definition, I(r)=−Eddr[∑i=1knifiddr(fi)]=E∑i=1kni[1fid2fidr2−1fi2(dfidr)2]. The expectation of the first item on the right-hand side of the above equation is zero, and then E∑i=1kni1fi2(dfidr)2=E∑i=1knifi2(∑j=04dzijdr)2=E∑i=1knifi2(∑j=04zijd(logzij)dr)2=E∑i=1knifi2(∑j=04zij(j−4r)r(1−r))2=E[1r2(1−r)2∑i=1knifi2(∑j=04jzij−4r∑j=04zij)2]=E[1r2(1−r)2∑i=1knifi2(∑j=04jzij−4rfi)2]. Substituting E(n_i) = nf_i, we derive the information measure as I(r)=1r2(1−r)2∑i=1knfi(∑j=04jzij−4rfi)2=nr2(1−r)2∑i=1k1fi[(∑j=04jzij)2−8rfi∑j=04jzij+16r2fi2]=nr2(1−r)2[∑i=1k1fi(∑j=04jzij)2−16r2], noting that ∑i=1k∑j=04jzij=∑j=04j(4j)rj(1−r)4−j=4r and ∑i=1kfi=1 .