QTL mapping with partially informative markers

QTL mapping with partially informative markers

This material must be read as Supplementary Information of Gazaffi et al. (2014) paper and provide a number of details about how to deal with different degree of information provided by markers in outcrossing species.

The model presented by Gazaffi et al. (2014) was developed considering a given locus that segregates in a \(1:1:1:1\) fashion. This would result in four different conditional probabilities for QTL genotypes (\(P^1Q^1\), \(P^1Q^2\), \(P^2Q^1\), \(P^2Q^2\)). However, for situations in which the genetic map has regions with partially informative markers, may result in limitations on the level of information and the statistical power for mapping. Also, some hypothesis could not be tested. This is quite common in the absence in fully informative markers.

The statistical-genetic model of Gazaffi et al. (2014) was developed considering a locus with the four genotypic classes. With such classes, it is possible to model up to three genetic effects using three contrasts: additive effects of the QTL alleles of parent P, additive effects of the QTL alleles of parent Q, and the intra-locus interaction (dominance) between both additive effects. However, depending upon the markers presenting in the study and despite the use of multipoint probabilities, some regions of the linkage map could no have enough information to estimate all the genetic effects. This may occur in cases where a linkage group has only one or two marker types (e.g., \(C\) and \(D_1\), or, \(C\) and \(D_2\)).

In this situation, it is necessary to avoid problems with collinearity between QTL conditional probabilities. To do so, we used a criteria described in Belsley et al. (1980), based on diagnosis of singular value decomposition using condition index. Belsley et al. (1980) suggests that collinearity problems occur at condition index over 30. However, in our studies, we found that for QTL mapping with our model, good results can be achieved using a threshold of 3.5.

In cases where the problems of colinearity are not present, contrasts 1, 2, and 3 can be used. Segregation and linkage phase are inferred using the proceeding presented by Gazaffi et al. (2014). Although, when it is identified a dependence relationship between QTL genotypes probabilities, it is suggested the removal or adaptation of the contrasts to define the genetic effects (contrasts 4 to 13). Table 1 estabilishes all the contrasts that the algorithm can verify. Contrasts 1 and 2 are similar to a backcross model. Contrasts 3 and 9 are similar to an \(F_2\) model.

When contrasts 1, 2 and 3 can be used, the segregation pattern is inferred based on the following complementary hypotheses. First, it is verified the significance of the effects \(\alpha_p^{\ast}\), \(\alpha_q^{\ast}\) e \(\delta_{pq}^{\ast}\) using the hypotheses: \(H_{01}: \alpha_p^{\ast}=0\), \(H_{02}: \alpha_q^{\ast}=0\), and \(H_{03}: \delta_{pq}^{\ast}=0\), respectively. If it is observed more than one significant effect, we need to check the following additional hypotheses to test if estimates are equal: \(H_{04}:\) \(|\alpha_p^{\ast}|=|\alpha_q^{\ast}|\), \(H_{05}:\) \(|\alpha_p^{\ast}|\) = \(|\delta_{pq}^{\ast}|\), and \(H_{06}:\) \(|\alpha_q^{\ast}|=|\delta_{pq}^{\ast}|\). If just one genetic effect is significant, it is assumed the QTL has segregation 1:1. If two genetic effects are significant, a complementary test is performed to infer if the segregation pattern is \(1:2:1\) (non-rejected complementary hypothesis) or \(1:1:1:1\) (rejected complementary hypothesis). If there are three significant genetic effects, it is considered QTL segregation is: i) \(3:1\) if the three complementary hypotheses were not rejected, ii) \(1:2:1\) if just one between the three hypotheses was not rejected, and, iii) \(1:1:1:1\) for other cases.

The linkage phase between QTL and markers is inferred through the signal interpretation of \(\alpha_p^{\ast}\) and \(\alpha_q^{\ast}\). The default model assumes the configuration where both parents are on coupling phase.

If this is the case, the estimates of \(\alpha^{\ast}_p\) and \(\alpha^{\ast}_q\) will be positive. In the other configurations, the estimates will show inverted signals.

In genomic regions where it is not possible to fit a model with all the parameters (three contrasts), the characterization will be constrained. From Table 2 (based upon the dependence relationship between QTLs genotypes), it is possible to infer which QTL effects are estimable, as well as their expectations and characterizations.

For cases \(i\) and \(ii\), it only possible to estimate one parent effect, which allows to detect a QTL with 1:1 pattern (similar to a backcross population). Some additional considerations:

• For a QTL presented in a region with such colinearity, it is only possible to test the QTL presence segregating in one parent. In this case, even if the QTL would present more significant effects, they cannot be estimated. This is due to low marker information which indicates that the observed QTL segregation would only be partially consistent with the real QTL segregation.

• Even though some regions permit QTL detection with a genetic effect similar to a backcross, which can be compared to the fullsib progeny of a pseudo-testcross approach (Grattapaglia and Sederoff, 1994). The present model has the advantage of allowing the usage of cofactors from regions with different segregation pattern, increasing the statistical power.

In case \(iii\), it is possible to estimate the effect \(\alpha_p^{\ast}\). However, due to the conditional probabilities of genotypes \(P^1Q^1\) and \(P^1Q^2\) being the same, it is not possible to isolate the effects \(\alpha_q^\ast\) e \(\delta_{pq}^\ast\). In such case, the strategy of the algorithm is to test the existence of a genetic effect \(\alpha_q^\ast\) within the informative genotypes (\(P^2Q^1\) and \(P^2Q^2\)) - contrast 4 of Table 1. However, this contrast tests a possible combination between \(\alpha_q^\ast\) and \(\delta_{pq}^\ast\) effects which would difficult the identification of the effect origin and its correct segregation pattern. Moreover, if only the effect \(\alpha_p^{\ast}\) is significant, the QTL would segregate in the progeny in a \(1:1\) fashion. If only the effect \(\alpha_q^\ast\) is significant, the segregation pattern in the progeny would be \(1:2:1\). In cases where both effects are significants, the observed segregation will be \(1:2:1\) or \(3:1\). The \(3:1\) pattern rises when the absolute value of \(\alpha_q^\ast\) is the double of the absolute value of \(\alpha_p^\ast\). In this type of colinearity the identification of the QTL segregation pattern - as well as the linkage phase between the markers - is partially informative because there are markers which are not possible to distinctly estimate their genetic effects. Therefore, the future input of markers with different segregation patterns could provide results more precises for the QTL mapping, even changing the segregation pattern previously inferred. This same situation can be analysed for cases \(iv\), \(v\), \(vi\), \(vii\), and \(viii\).

Case \(viii\) is similar to an \(F_2\) population, therefore, \(\alpha_p^\ast\) and \(\alpha_q^\ast\) would be equal providing a unbiased estimate for the segregation and linkage phase between QTLs and markers \((P^1P^2 \times Q^1Q^2\) (if \(\alpha_p^\ast\) positive) or \(P^2P^1 \times Q^2Q^1\) (if \(\alpha_p\) negative). To obtain the dominance estimate, one needs to consider half of the effect of \(\delta_{pq}^\ast\) (similar to what was proposed by Cockerham, 1954).

For cases \(ix\) to \(xii\), the colinearity is originated due to the similarity between three genotypic classes, which only allows to estimate a genetic effect represented by a linear combination between \(\alpha_p^{\ast}\), \(\alpha_q^{\ast}\) and \(\delta_{pq}^{\ast}\). In such cases, it is not possible the correct understanding of the segregation and linkage phases between QTLs and markers. Moreover, it is emphasized that to limit those contrasts, it was used the coefficient \(1/3\) for similar genotypes and \(-1\) for the different genotype. Such contrasts are not orthogonals in relation to each other and can lead to some problems on the mapping process.