Results
- Simulating genotypic variation of fruit quality in an advanced peachxPrunus davidiana cross

Sensitivity analysis
The relative variation in the model outputs caused by the variation in the values of each parameter were analysed (see Table SP1 in the supplementary data at JXB online). Thirteen of the 39 parameters for fruit load and an additional 12 parameters in the case of heavy fruit load were considered as important for further study. The 13 parameters were related to fruit growth demand, stone build-up, flesh mass increase, and total sugar accumulation. Among them, two were not estimated: Y, the threshold value of hydrostatic pressure needed for growth and ${pi}$ _oc, the fruit osmotic pressure due to compounds other than soluble sugars. The 12 additional parameters (associated with heavy fruit load) were related to light interception, C assimilation, and fruit growth demand. Nine of them, p and k involved in the regulation of leaf photosynthesis by reserves, the ratio of leaf reserve mass to leafy shoot reserve mass, the growth respiration coefficient of fruit, and the parameters associated with leaf assimilation, radiation intensity in shade, and reserve mobilization (r₂, GRC_fruit, p₄, p₇, r₃, r₄, r₆,) could not easily be measured for a large number of genotypes. Thus, 14 parameters of the 25 parameters selected based on the sensitivity analysis were measured for many or all genotypes studied. The values for the 11 remaining parameters that were not measured were fixed to values taken from the literature or from related experiments (Quilot et al., 2004a).

Variation in parameter values between genotypes and parameterization
A total of 23 parameters of the model were not measured and their values were taken from the literature or from related experiments (Quilot et al., 2004a). The origin of the parameter values used in the simulations is given in Table 1. As regards the 14 parameters selected based on the sensitivity analysis, the variation in their values between genotypes was analysed (Table 3).

For two of them, r₁ and p₁, the variation between genotypes was not significant. Accordingly, for the leaf structural mass to leafy shoot structural mass ratio (r₁), the general average (0.672) was used. The light-saturated leaf photosynthesis (p₁) was set to the single value (19.47 µmol m^–2 s^–1) estimated for all genotypes. For the four parameters involved in allometric relationships (see Equations 10 and 14 of the Appendix in the supplementary data at JXB online), statistical models were compared. and s₁ were found to be genotype-dependent and k_stone and s₂ were constants. The specific leaf area (SLA) was measured for 49 genotypes and significant differences were found among these genotypes. Since SLA appeared to have no effect on the model outputs in the case of light fruit load (see Table SP1 of the Appendix in the supplementary data at JXB online), the averaged value (0.0169 m² g^–1) was used. For heavy fruit load simulations, the specific genotypic values were used (StP00 and GotBC₂02).

For the other six parameters, k_sugar, r_su, ${rho}$ , and aL, large differences were observed between genotypes and the range of the observed values was higher than the range considered in the sensitivity analysis. For the coefficient of the transfer function between sugars and other compounds, k_sugar, the values estimated by Quilot et al. (2004b) were used on the same StPBC₂02 dataset. The estimation of the values of ${rho}$ , the permeation coefficient of fruit surface to water vapour, was very time-consuming and required many fruits. This parameter was estimated for 41 genotypes. A mean value was used for the others, although the 41 genotypes displayed high variation in ${rho}$ values.

The model parameterized as described above was then used to estimate values of aL, the hydraulic conductance per unit of fruit surface, by calibration. Great variations in the values of aL, higher than those tested through the sensitivity analysis, were observed in the population. Since ${rho}$ , as well as aL, are involved in the computation of water accumulation in the fruit, the aL estimation probably contains the genotypic variation that was not included in the ${rho}$ values. Therefore, for the 41 genotypes with individual estimates of ${rho}$ , the effect on aL values of setting a constant value of ${rho}$ , instead of the specific ${rho}$ value, was computed. This accounted for 10% of variation in aL. This is not negligible, but it is small in comparison with genotypic variation in aL.

Finally, four out of the 14 parameters measured or estimated were considered constant, whereas the other ten (s₁, k_sugar, r_su, ${rho}$ , and aL for both fruit loads, and SLA for heavy fruit loads only) were considered as genotypic key parameters.

Goodness-of-fit of the model on the basis of data used for parameterization
The observed genotypic variation in dry and fresh fruit masses was well reproduced by the model (Fig. 1). The equation of fruit demand for growth appeared very robust as it reproduced the large range of growth patterns displayed by the population. The goodness-of-fit criteria (RRMSE) ranged from 0.030 to 0.376 for fruit dry matter growth and from 0.041 to 0.311 for fruit fresh growth depending on genotype (see Table SP2 of the Appendix in the supplementary data at JXB online). The observed data at maturity were also reproduced well for all the output variables (Fig. 2, line 1). For dry and fresh masses of fruit and stone, and flesh dry matter content, individual RRMSE ranged from 0 to 0.4 and mean RRMSE over the population from 0.06 to 0.1 (Table 4). The mean RRMSE of the two variables relating to total sugar were nearly equal to 0.1 and were satisfactory. However, individual RRMSE were highly variable between genotypes since they ranged from 0.002 to 0.6. Hence, for some genotypes total sugar amount and concentration were not well reproduced by the model.

The model performed very well in ranking genotypes for dry and fresh fruit masses, fresh stone mass, and total sugar amount in the flesh, with the Spearman correlation coefficient ranging from 0.91 to 0.98 (Fig. 2). For flesh dry matter content and total flesh sugar concentration, the Spearman correlation coefficient reached 0.8 and 0.73, respectively, still a good performance.

Predictive quality of the model for independent data
The model was used to predict fruit observations from independent data under various fruit load conditions. Observations from 87 BC₂ genotypes grown under light fruit load (StPBC₂01 dataset), from seven BC₂ genotypes grown under heavy fruit load (GotBC₂02 dataset) and from S and P1908 for two contrasted fruit loads (StP00 dataset) were used to test the ability of the model to predict fruit quality.

The model correctly predicted growth kinetics of dry and fresh (see Figures SP1 and SP2 of the Appendix in the supplementary data at JXB online) fruit masses for different environmental and growing conditions (year, site, light and heavy fruit loads). Mean RRMSE for growth of dry and fresh fruit varied between 0.17 and 0.3 depending on the experiment (see Table SP2 of the Appendix in the supplementary data at JXB online). The ability of the model to rank the genotypes at maturity appeared accurate for both dry and fresh fruit masses, with the Spearman correlation coefficients from 0.62 to 0.94 (Fig. 2, lines 2, 3, and 4). However, it was less accurate when leafy shoot growth was to be considered (Fig. 2, line 3). In addition to its ability to reproduce variation between genotypes, the model was able to reproduce variations between fruits of the same tree (see Fig. SP1 of the Appendix in the supplementary data at JXB online).

The model predictions appeared especially good for dry and fresh stone masses. Mean RRMSE at maturity ranged from 0.063 to 0.147 for these two variables, and individual RRMSE values never exceeded 0.19 (Table 4). Similarly, the Spearman correlation coefficient varied between 0.75 and 0.98 depending on the experiment (Fig. 2, lines 2, 3, and 4).

Predictions of flesh dry matter content were good as indicated by the RRMSE criteria, but according to the Spearman correlation coefficient were moderately accurate. Mean RRMSE did not exceed 0.17 (Table 4), whereas the Spearman correlation coefficient varied between 0.49 and 0.83 (Fig. 2, lines 2, 3, and 4).

Considering total sugar variables, predictions were globally less accurate, but the accuracy depended on the experiment and the genotype. These variables appeared to be predicted better in the case of light fruit load. Mean RRMSE varied between 0.23 and 0.51 (Table 4) and the Spearman correlation coefficient ranged from 0.22 to 0.78 (Fig. 2, lines 2 and 3).

Determination of the relative importance of the ten genotypic key parameters
The relative importance of the 10 genotypic key parameters was compared with respect to the three criteria: the sensitivity of the model to the parameter, the variation in the parameter value observed in the population, and the mean error of estimation of the parameter (Table 5).

Thus, rules were established taking into account these aspects in order to define the score of each parameter and to rank them. A qualitative score was defined for each criterion attributing different numbers of stars to different levels of the criteria (Table 5). A global score was then defined as the total number of stars. The parameters SLA and had only four stars, and were considered the least important among the ten. aL and were considered the most important genotypic parameters of the model.