181 metabolic markers as predictor set, the corresponding numbers were 0.27 for C24 crosses and 0.24 for Col-0 testcrosses. These results show that some prediction has been achieved. But the level of predictive power is very low, above all in the case of the 181 metabolic markers. We believe this to be a problem of overfitting, indicated by the fact that the larger predictor set yields the smaller prediction success in cross-validation. Feature selection overcomes the problem of overfitting and leads to an almost equal predictive power of genetic and metabolic predictors In order to rank the available variables according to their contribution to the response, we utilized the variable importance in the projection (VIP), which calculates the contribution of each predictor variable to the response in the respective PLS-model [20,21]. To reduce the number of variables in the respective models we kept only those with a high contribution. We determined the threshold via optimizing the predictive power in LOOV. Subsequently the reduced model that resulted in the highest predictive power was selected. Applying this feature selection approach to our four PLS models (i.e. heterosis according to the Col-0 or C24 parent using either metabolic or genetic markers) led to improvements of the predictive power to values of about 0.40 (cf. Table 1), irrespective of the use of exclusively genetic or exclusively metabolic markers. Due to the different degree of the overfitting problem, improve- ments were more substantial in case of metabolic as compared to genetic markers resulting in a nearly equal predictive power of both marker sets (cf. Figure 2 A–D and Table S1). Combining metabolic and genetic markers leads to substantial improvements of heterosis prediction We wanted to answer the question of whether a combination of genetic and metabolic markers improves the predictability of heterosis as compared to the use of genetic markers alone. Since our variable selection algorithm allowed to overcome the problem of overfitting when dealing with a high number of predictors, we were now in a position to apply this algorithm on the combined set of metabolic and genetic markers (i.e. a total of 291 predictors) and test their suitability for heterosis prediction using PLS regression (cf. Figure 2). As shown in Table 1, the combination of metabolic and genetic markers leads to a strong improvement of the predictive power in LOOV as compared to the genetic or metabolic marker based models. The improvement was highly significant in each case, since the estimated confidence intervals of the respective correlation values did not overlap (cf. Table S1). Table 1. Predictive power (PP) in leave-one-out validation of the respectively optimal selections of predictors for the relative mid-parent heterosis regarding the two different testcross set-ups. Response variable PP of optimal genetic selection PP of optimal metabolic selection PP of optimal combined genetic- metabolic selection doi:10.1371/journal.pone.0005220.t001 C24- heterosis 0.42 0.40 0.50 Col-heterosis 0.41 0.38 0.48 Heterosis Prediction Markers Marker identification via minimum-description-length (MDL) based strategies One advantage of the experimental design used here is the identification of variables that are important for heterosis prediction in both testcross populations as compared to variables that are strongly relevant in the prediction models for only one kind of combination, i.e. either with C24 or with Col-0. However, in order to perform this analysis, the number of variables in the respective models had to be reduced further to a minimal set of variables whose predictive power does not differ significantly from the optimal predictive power. To achieve this goal we developed a MDL based strategy (cf. Methods section). Application of this approach to the single metabolic models, to the genetic models and to the combined models led to a substantial reduction in the number of variables needed, as compared to the number of measured variables, in each case (cf. Table S2 and Figure 2). Thus, 68 out of the 110 available genetic and 98 out of the 181 available metabolic markers were omitted in each of the corresponding four minimal models investigated without any significant loss of predictive power, as compared to the optimally achievable predictive power in the particular predictor set. This means that more than half of the markers measured turned out to be of no significant relevance for heterosis prediction in our analysis, neither in the case of C24 nor in the case of Col-0 testcrosses (cf. Table S3). Genetic markers important as heterosis predictors overlap with heterosis quantitative trait loci (QTL) When comparing the relative importance of the 110 genetic markers as judged by their contribution to heterosis prediction with the genetic analysis of heterotic biomass QTL, a substantial overlap is observed (cf. Figure 3). In particular this is true for the following markers that bear a special meaning. Eight genetic markers turned out to be highly predictive for heterosis in both testcross populations. This was indicated by their presence in the minimal genetic model and in the minimal combined genetic-metabolic model for both, C24- and Col- heterosis prediction. Five of those highly predictive markers are located at the top of chromosome 4, a region containing important QTL for heterosis and biomass per se [22], cf. Meyer et al. (2008 submitted). The same is true for the two highly predictive markers at chromosome 3. The remaining one important marker at bottom of chromosome 1 also coincides with one heterotic QTL (cf. Figure 3). Fifteen genetic markers are specific for predicting heterosis in either the Col-0 or the C24 crosses. They are contained in the minimal genetic model and in the minimal combined genetic- metabolic model for one testcross population, but in none of the corresponding models for the other testcross population. Two of these markers are specific for Col-heterosis prediction. They are both located at the bottom of chromosome 3, while the thirteen markers specific for C24-heterosis prediction are distributed to all five chromosomes. Also in the case of the specific genetic markers there is some overlap with QTL (cf. Figure 3). Identified metabolic markers deviate from normal distribution Applying the MDL based feature selection method to the metabolic markers, we identified the most important ones with respect to heterosis prediction (cf. Tables S4 and S5). Sixteen of them are contributing to the minimal models only in the C24 testcrosses, while ten metabolic markers in our models are of PLoS ONE | www.plosone.org 3 April 2009 | Volume 4 | Issue 4 | e5220