Plant material and crosses:

During the growing seasons of 2003–2005, root and shoot fragments of tetraploid Rorippa amphibia (denoted AAAA) and R. sylvestris (denoted SSSS) were collected from several locations throughout Europe and grown in a greenhouse environment (Table 1). In the summer of 2004, we made reciprocal crosses between two independent wild-collected pairs of AAAA and SSSS (Table 1) to create first-generation (F₁) hybrids. From each of the four resulting F₁ hybrid seed families, we germinated ∼50 seeds on filter paper moistened with 2 ml of a 3-μm gibberellic acid solution. Seedlings were transferred to soil and further grown in a common greenhouse environment. In the summer of 2005, one individual of each of the four F₁ hybrid seedling families was backcrossed with an unrelated, wild-collected plant (Table 1) to create first-generation backcrosses (BC₁). From each of the four resulting BC₁ seed families, we again germinated ∼50 seeds on 2 ml of a 3-μm gibberellic acid solution. Seedlings were transferred to soil and further grown in a common greenhouse environment.

DNA extraction and analysis of microsatellite loci:

DNA was extracted from fresh leaves using a modified CTAB protocol (Doyle and Doyle 1987). We genotyped the wild-collected plants and the backcrossed F₁ hybrids for 12 microsatellite loci (Stift et al. 2006). On the basis of this initial screening, we selected the most informative loci for each cross. Thus, ideally, each parent possessed four different alleles (i.e., fully heterozygous) and shared no alleles with its crossing partner. We genotyped each of the four F₁ and the four BC₁ offspring families for the selected loci. Offspring with genotypes that could be explained only by mutation or contamination (i.e., alleles observed that were not present in the parents) or nondisjunctions (i.e., more than two alleles observed from one of the parents) were excluded from the analyses. Such anomalous genotypes were found for two loci—RS44 (five times) and RS101 (eight times)—and never constituted >4% of the offspring within one family.

Testing for reciprocal differences:

From the genotypes observed in the offspring of the experimental crosses, we reconstructed the parental gamete frequencies. In cases where the two parents had alleles in common, we worked with the observed genotype frequencies in the offspring. We used the likelihood G-test for contingency tables (Sokal and Rohlf 1995) to test whether the observed parental gamete frequencies differed between the reciprocal crosses.

Gamete formation model:

Consider a tetraploid where each chromosome is marked by a different allele (e.g., ABCD). Under complete tetrasomic inheritance, assuming no double reduction, gametes carrying the allelic combinations AB, AC, AD, CD, BD, and BC will occur in equal proportions () (Muller 1914). The maximum frequency of double reduction (α) is , which can be reached if quadrivalents are always formed at meiosis, one effective crossover occurs between the locus and its centromere, and the recombined chromosomes migrate to the same pole at anaphase I (adjacent orientation, Mather 1935). Under this scenario, the allelic combinations AB, AC, AD, CD, BD, and BC will still occur in equal proportions (), and there will be double-reduction gametes (AA, BB, CC, and DD) each at an expected frequency of .

Preferential (bivalent) pairing in meiosis leads to expected gamete frequencies characteristic of disomic inheritance. If alleles A and B mark homologous chromosomes that pair exclusively with each other, alleles A and B will never end up in the same gamete, and likewise for C and D marked chromosomes. This AB/CD pattern of pairing thus produces gametes with the allelic combinations AC, AD, BC, and BD in equal proportions (). Double reduction is not possible with bivalent formation. The other possible pairings, namely of AC/BD and AD/BC, result in gametes AB, AD, BC, CD and AB, AC, BD, CD, respectively. The expected proportions (probabilities) of all possible gametes produced by an individual ABCD is calculated by the formulasor in matrix notationThese equations define a set of nonlinear equations with four unknown parameters (by definition δ1 + δ2 + δ3 = 1). The “tetrasomic” parameter (τ) indicates the proportion of gametes formed by random meiotic chromosome associations (i.e., random bivalent or quadrivalent pairing) and can take values from zero (full disomic) to 1 (full tetrasomic). In the latter case, the entire last (disomic) part of the equation cancels out. If τ < 1, the expected gamete proportions depend on the setting of three “disomic” parameters (δ1, δ2, and δ3) that indicate the respective degree of preferential pairing of AB/CD, AC/BD, and AD/BC marked chromosomes (respectively) in the nonrandom meiotic chromosome associations. Each can take values from 0 (no pairing) to 1 (obligate pairing) with the constraints that δ1 + δ2 + δ3 = 1 and δ1 × δ2 × δ3 = 0 (i.e., one of the disomic parameters must be zero). The latter constraint guarantees that random bivalent or quadrivalent meiotic configurations are exclusively expressed in the parameter τ. Without such a constraint, there would be an alternative solution for each parameter setting with τ > 0 (e.g., τ = 1 would be equivalent to τ = 0 with ). Finally, the “double-reduction” parameter β represents the frequency of double reductions relative to the total frequency of random (quadrivalent or random bivalent) meiotic associations, from which the frequency of double reduction as used in the literature (α) can be calculated as α = βτ.

Parameter estimation:

The likelihood of multinomial data can be calculated as the sum of expected frequencies raised to the power corresponding to the observed counts (Kalbfleisch 1985): , in which n is the sample size, x_k is the number of observations of event k, and p_k is the probability of that event k. An event in our case is an observed gamete with two specific parental alleles. The total log-likelihood over all observations can be calculated aswith x_AA the frequency of gamete type AA, and p_AA its probability given the model of inheritance under scrutiny, and so forth for all gamete types. We used the constrained nonlinear regression (CNLR) function as implemented in SPSS to estimate the parameter values that gave the best fit to the data. The model was specified with the COMPUTE PRED_ command as (n*(t*TAU+bt*BETATAU+(1-TAU)*(d1*DI1+d2*DI2+d3*DI3)).

Initial starting values for the iterations were set through the MODEL PROGRAM command. The constrained nonlinear regression (i.e., the actual parameter estimation) was performed with the default iterative procedure (sequential quadratic programming), with the negative log-likelihood as the loss function to be minimized. Within the CNLR command, the values or boundaries for the parameters were specified using the BOUNDS/ subcommand. The supplemental material provides the syntax used (including a user guide) and two examples with data (observed gamete counts); model coefficients; parameter estimates for TAU (τ), BETATAU (βτ), DI1 (δ1), DI2 (δ2), and DI3 (δ3); and the corresponding expected frequencies and likelihoods for the example data.

Application of the gamete formation model:

For plant-locus combinations of the type ABCD or AABC with no alleles shared with the crossing partner, we deduced the parental gamete frequencies from the observed offspring genotypes. Then we used SPSS (constrained nonlinear regression, see specifications above) to obtain the parameter values that gave the largest log-likelihood for the following situations: (1) the full tetrasomic null model (τ = 1), in which only the double-reduction rate (βτ) was estimated; (2) three constrained intermediate models, in which the proportion of random segregations (τ) and the DR rate (βτ) were estimated, while the disomic parameters were fixed at δ1 = 1, δ2 = 1, or δ3 = 1, respectively; and (3) three unconstrained intermediate models, in which τ, βτ, and two of the disomic parameters were estimated, while the third was set to zero.

For partially informative cross-locus combinations (of type ABCD or AABC) with some alleles shared with the crossing partner for that particular locus it is not possible to unambiguously reconstruct the parental gamete frequencies from the observed offspring genotypes. Therefore, we worked the other way around and calculated the expected offspring genotype frequencies from the expected gamete frequencies of the parents under the following parameter settings. We assumed full tetrasomic inheritance for one parent (i.e., τ = 1) and let the τ of the other parent decrease from 1 (full tetrasomic) to 0 (full disomic) in steps of 0.01, at δ1 = 1, δ2 = 1, δ3 = 1, respectively. For each of these parameter settings and their expected offspring genotype frequencies we calculated the log-likelihood of the observed frequencies and identified the parameter settings that gave the largest likelihood (i.e., the best fit). This procedure was executed as a (more tedious) spreadsheet algorithm (Microsoft Excel) scanning the parameter space, rather than as a nonlinear regression problem.

For easier comparison across independent analyses (e.g., different loci, different crosses, different plants) and with published studies using a chi-square-based approach or G goodness-of-fit test, we calculated the likelihood deviance , corrected for use of discrete data (Williams' correction) (Sokal and Rohlf 1995) for the best-fit and null models rather than working with the likelihood scores directly.

We then evaluated whether intermediate models (with τ estimated 0 < τ < 1) provided a significantly better fit than the tetrasomic null model (τ = 1) through a likelihood-ratio test (LRT). The LRT follows a chi-square distribution in which the degrees of freedom (d.f.) correspond to the difference in degrees of freedom between the two models compared (Sokal and Rohlf 1995). However, since the parameter value τ of the null model lies at its upper theoretical boundary (i.e., τ = 1), the LRT has to be adapted to become one-tailed and should be tested against a compound distribution of (Self and Liang 1987). Essentially, this means that the P-value of a conventional should be halved. Statistical comparisons of intermediate models (0 < τ < 1) with disomic null models (τ = 0 and δ1 = 1, δ2 = 1, or δ3 = 1) are not informative, because the log-likelihood of disomic null models becomes infinitely small due to observations in classes where the expectation is zero.

Evaluation of model performance on tetraploid segregation data from the literature:

We selected some specific crosses and loci from the literature to test the general applicability of our gamete formation model (specified in Table 6). We specifically included cases in which tetrasomic inheritance could not be statistically rejected in the original study, but to which we suspected that intermediate inheritance models might apply. Also, we included a case where the observed patterns were clearly disomic (Pairon and Jacquemart 2005).

Reciprocal differences:

There were no significant differences between observed female and male gamete frequencies of the same individual (data not shown). Therefore, in subsequent analyses the reciprocal data were pooled.

Inheritance in natural tetraploids:

For the wild-collected tetraploid R. sylvestris (genotypes SSSS1, SSSS2, and SSSS3), the estimated value of τ of the best-fitting intermediate inheritance models varied from τ = 0.64 to τ = 1 for the different loci (Table 2). In none of these cases was the fit significantly better than the null model of full tetrasomic inheritance τ = 1 (Table 2). The likelihood of intermediate models typically decreased asymptotically upon approaching τ = 0 (reflected in an asymptotically increasing deviance G, Figure 1), flattened out around the minimum, and increased again toward τ = 1 (leading to a decreasing deviance G, Figure 1). For the wild-collected tetraploid R. amphibia (genotypes AAAA1, AAAA2, and AAAA3), the estimated value of τ of the best-fitting intermediate model varied from τ = 0.59 to τ = 0.96 for the different loci (Table 2). For three loci (locus RS44 in SSSS2, locus RS101 in SSSS3, and locus RA12 in AAAA1) an intermediate model provided a significantly better fit than the full tetrasomic null model with estimates of τ of 0.70 (RA12) and 0.78 (RS44, RS101) (Table 2).

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Figure 1.—

Fit (deviance G) of observed segregation of microsatellite loci in a tetraploid R. sylvestris (SSSS2) to inheritance models ranging from complete disomic (τ = 0) to complete tetrasomic inheritance (τ = 1). The “disomic” parameter of the most likely disomic model was constrained to 1. Boxes specify locus names.

Inheritance in artificial F₁ hybrids:

For the artificial F₁ hybrids (genotypes AASS1, AASS2, SSAA1, and SSAA2), the estimated value of τ of the best-fitting (constrained) intermediate models varied from 0.29 to 1 for the different loci (Table 3). Similar to the observations in the parents, the likelihood of intermediate models typically decreased asymptotically upon approaching τ = 0 (leading to an asymptotically increasing deviance G, Figure 2), flattened out around the minimum, and increased again toward τ = 1 (leading to a decreasing deviance G, Figure 2). In 9 of the 12 cases the fit was significantly better than the null model of tetrasomic inheritance τ = 1 (Table 3). In 7 of these 9 cases the preferential pairing involved chromosomes that originated from the same parental species, i.e., preferential pairing of the homologous chromosomes (Table 3). In the two other cases, the preferential pairing involved chromosomes that originated from different parental species, i.e., preferential pairing of homeologous chromosomes (AASS2, locus RS60; SSAA2, locus RS44). An unconstrained intermediate inheritance model never provided a significant improvement in fit over a constrained intermediate model (data not shown).

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Figure 2.—

Fit (deviance G) of observed segregation of microsatellite loci in a tetraploid F₁ hybrid R. sylvestris × R. amphibia (SSAA1) to inheritance models ranging from complete disomic (τ = 0) to complete tetrasomic inheritance (τ = 1). The “disomic” parameter of the most likely disomic model was constrained to 1. Boxes specify locus names.

Double reduction:

For loci RA13, RS10, and RS64 no double-reduction (DR) gametes were detected. For locus RS30 null alleles prevented DR gamete identification. For all other loci we detected DR gametes (Table 4). With only 1 observed DR in >1000 offspring analyzed, the DR rate for locus RS44 was the lowest. With 16 observed DRs in 641 offspring, locus RS101 had the highest DR rate (Table 4). The total number of DRs was higher in female meioses (17 vs. 10). In four cases the parental genotypes had no alleles in common, so that iterative estimation of the DR parameter (βτ) was possible. For the cases involving full heterozygotes (i.e., plants of type ABCD), the observed and estimated DR rates were equal. For the two remaining genotypes (of type AABC), the estimated DR rates were higher than the observed rate (Table 5).

Evaluation of model performance on tetraploid segregation data from the literature:

For the allozyme inheritance data of tetraploid Centaurea jacea (Hardy et al. 2001), the estimated values of τ of the best-fitting intermediate inheritance models varied from τ = 0.71 to τ = 0.98 (Table 6). In one case the fit was significantly better than the null model (and Hardy et al.'s conclusions) of full tetrasomic inheritance τ = 1 (Table 6) and included 29% preferential pairing (i.e., τ = 0.71) of the chromosomes marked by alleles A–C and B–D (i.e., δ2 = 1). For the allozyme inheritance data of tetraploid Tolmiea menziesii (Soltis and Soltis 1988), the estimated value of τ of the best-fitting intermediate model varied from τ = 0.27 to τ = 0.96 (Table 6). In one case the fit was significantly better than the null model (and Soltis and Soltis's conclusions) of full tetrasomic inheritance, suggesting 58% preferential pairing (i.e., τ = 0.42) of the chromosomes marked by alleles A–C (i.e., δ2 = 1). For the microsatellite inheritance data of tetraploid Prunus serotina (Pairon and Jacquemart 2005), in agreement with Pairon and Jacquemart's conclusions, the fit of disomic inheritance was better than any intermediate model and significantly better than the tetrasomic null model (Table 6). The likelihood of the disomic model decreased from τ = 0 to τ = 1 in an almost linear fashion (reflected in a linear increase of the deviance G, Figure 3).

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Figure 3.—

Fit (deviance G) of observed segregation of microsatellite loci in tetraploid Centaurea jacea (Hardy et al. 2001) and Prunus serotina (Pairon and Jacquemart 2005) to inheritance models ranging from complete disomic (τ = 0) to complete tetrasomic inheritance (τ = 1). The “disomic” parameter of the most likely disomic model was constrained to 1. Boxes specify species, individual/cross, and locus name as used in the original publication.