Model description and flow equations
The pedicel-fruit system was conceptually described as shown in Fig. 2.1. We ideally considered the fruit as a big cell made of a symplast surrounded by an apoplast and a vascular system connected to the plant by the pedicel. The pedicel was divided into pedicel xylem and phloem (px and pp, respectively). Water is transported from the pedicel to the fruit vascular system, composed of fruit xylem and phloem (fx and fp, respectively); sugars are transported from the pedicel phloem to the fruit phloem by mass flow. We assumed that the phloem and xylem water potentials are the same and that local water exchanges could maintain this equilibrium (Thompson and Holbrook, 2003b; Hall and Minchin, 2013). We assumed that sugars are the only osmotically Model-assisted analysis of the pedicel-fruit system suggests a fruit sugar uptake active solutes. We supposed that solutes concentrations were negligible both in pedicel and fruit xylem so that the xylem pressure potential equals the xylem water potential. The region where the fruit xylem and phloem terminate was represented with a system formed by the fruit cell apoplast (fa) connected to the vascular system and surrounding the fruit cell symplast (fs) (as in Hall et al., (2017) model). A membrane separates the fruit phloem from the fruit cell apoplast, while we assumed that there was no barrier for solutes between the fruit xylem and the fruit cell apoplast. Water is transported in the fruit cell symplast across a membrane and lost through transpiration by the fruit cell apoplast.
Model calibration and analysis of the model variables response to changing inputs
We estimated the model parameters in order to reproduce the average fruit fresh weight variation during a single day. We assumed that the fruits biophysical parameters did not vary along the measurement period. The system of Eq. 18-21 was reduced for describing the girdled pedicel condition, imposing the absence of phloem flows (Uppfp = 0 and Ufpfa = 0). A consequence of the system reduction was that sugar flows were equal to zero. This hypothesis agreed with the measurements made by Génard et al. (2003), which showed that in girdled fruits, sugar accumulation immediately ceased.
The simulated fruit fresh weight (Wh, g) at a given hour h was calculated as the water and sugars accumulation in the fruit between a reference time h0 .and h. 𝑊ℎ=𝑊𝑟𝑒𝑓+Σ (𝑈fxfa,𝑖+𝑈fpfa,𝑖−𝑇fa,𝑖+𝑆fa→fs,𝑖)ℎ 𝑖=ℎ0 (22).
Ufxfa,i, Ufpfa,i, Tfa,i, Sfa→fs,i are the water flows between fruit xylem and fruit cell apoplast, the water flow between fruit phloem and fruit cell apoplast, the fruit transpiration, and the net fruit symplast sugar inflow at hour i, respectively.
The objective function we minimized is the root mean squared error made in the hourly fruits fresh weight prediction (RMSE (g)). 𝑅𝑀𝑆𝐸=√1𝑁×Σ(𝑊ℎ−𝑊𝑜𝑏𝑠,ℎ)2𝑁ℎ=1 (23).
N is the number of observed points during the measurement period, and 𝑊ℎand 𝑊𝑜𝑏𝑠,ℎ are, respectively, the simulated and the observed values of fruit fresh weight at hour h.
We aimed at minimizing the RMSE index for both the control (C) and the girdled (G) condition simulations, for one and two treatments in 1994 and 1995, respectively. For solving this problem, we found the dominant solutions of a multi-objective optimization problem, minimizing both RMSEC and RMSEG. RMSEC (resp RMSEG) is the mean RMSEs for fresh weight prediction in control (resp girdled fruit) conditions for the different leaf-to-fruit ratios. We solved the calibration problem with the multi-objective genetic algorithm NSGA II (Deb et al., 2002). Among the dominant solutions, we chose the one corresponding to the minimum mean value between RMSEC and RMSEG. The parameters we estimated are compiled in Table 2.3.
In order to further verify the goodness of our model predictions, we confronted our predicted xylem and phloem inflows contributions to the total water inflow to those experimentally observed by Morandi et al., (2007) for peach fruits at stage III of growth. Moreover, we confronted our predicted dry mass accumulation to the mean diurnal dry mass accumulation measured by Fishman and Génard (1998) on the same cultivar (cv ‘Suncrest’) in heavy and light crop load treatments.
In order to assess our results dependence on the model inputs, we analyzed the main model variables outputs response to different levels of pedicel water potential and sugar concentration given as input. We present this analysis results in supplementary material (S2.3).
Model calibration and diurnal contribution of xylem and phloem flow to the total water inflow
Table 2.3 reports the parameters values estimated through the model calibration. Table S2.2.1 shows their variability among the best solutions obtained in calibration. Figure 2.2 depicts the comparison between the predicted and the average observed diurnal fruit fresh weight variations and the 5th and 95th percentile of the diurnal fruit fresh weight variation. The percentiles were computed on the measurement repetitions resumed in Table 2.1. Globally, our simulations well reproduced the fruit mass diurnal variation observed in girdled conditions and the mass increase observed in control conditions in all treatments. Moreover, they well reproduced the differences between fruits fresh mass behavior among crop load treatments in control conditions. However, the simulated girdled fruits fresh mass in the 5 leaf-to-fruit ratio treatment had a smaller variation than the observed one. In both 1994 and 1995 30 leaf-to-fruit ratio treatments, the simulated diurnal phloem and xylem inflows contributions to the total water inflow were 20% and 80% for phloem and xylem inflow, respectively, while values in the 5 leaf-to-fruit ratio treatment in 1995 were 29% and 71% for phloem and xylem inflow, respectively. These agree with measurements made by Morandi et al. (2007) for stage III of peach fruit growth, i.e. 30% and 70% for the phloem and xylem inflows contributions to the total water inflow, respectively. The simulated cumulative diurnal dry mass accumulation of the control fruits was 0.29 g day-1 in the 30 leaf-to-fruit ratio treatments for both 1994 and 1995, and 0.12 g day-1 in the 5 leaf-to-fruit ratio treatment in 1995. These were similar to the mean diurnal dry mass accumulations measured by Fishman and Génard (1998) on the same cultivar, i.e. about 0.37 g day-1 for the light crop load treatment and 0.09 g day-1 for the heavy crop load treatment.
The simulated fruit symplast sugar uptake was buffered compared to the variations of the pedicel phloem sugar concentration
We analyzed the simulated water and sugar flows diurnal behavior in the 5 leaf-to-fruit treatment, which we considered the most interesting pattern of water and sugars transports. The input variables of this simulation are presented in the Material and methods section and their values during the day are compiled in Table 2.2. We confronted the diurnal maximum relative variation of the symplast water inflow/outflow and of the sugar uptake with the diurnal maximum relative variation of the input variables related to water and sugar transports, i.e. pedicel phloem water potential and sugar concentration. The diurnal symplast water inflow maximum and minimum values (Fig. 2.3A) were 0.16 g h-1, and -0.13 g h-1, with a diurnal maximum relative variation of 1.8 (computed as |max𝑣−min𝑣max𝑣|). This value was higher than the maximum diurnal pedicel water potential input relative variation of 1.6 (computed with the same formula) (Fig. 2.3B). Therefore, water potential input variations generated high variations of symplast water inflows and outflows. In order to better highlight this variation, we computed that the simulated symplast water inflow averagely decreased by 0.06 g h-1 at every 0.2 MPa decrease of the pedicel water potential input. The minimum sugar uptake value was 0.0035 g h-1 and the maximum was 0.0082 g h-1 (Fig. 2.3C). The relative variation was 1.4 (computed as |max𝑣−min𝑣min𝑣| ), lower than the variation of the pedicel phloem sugar concentration input, whose value was 2.0 (computed with the same formula). We obtained the same relative variation of 1.4 for both the fruit phloem and the fruit cell apoplast sugar concentrations (Fig. 2.3D). These results suggested that the fruit phloem and the fruit cell apoplast sugar concentrations, and the fruit symplast sugar uptake were buffered in response to the large pedicel phloem sugar concentration variation.
Integrated model description
In the next sections, we will first describe how we connected growth, sugar and acid sub-models in an integrated model by highlighting the connections between sub-models. Secondly, we will describe the version of the fruit growth sub-model used in this work. After, we will describe the sub-models of sugar repartition, citric acid accumulation, and malic acid transport and pH, highlighting their connections to the integrated fruit growth model. Finally, we will describe how we computed the fruit osmotic potential, which is the main linking variable between the sub-models.
The integrated model
In this work, we will present an integrated virtual fruit model (Fig. 3.1), which is built linking a biophysical fruit growth sub-model and metabolic sub-models. To describe fruit growth model we used the model of Fishman and Génard (1998) in a version which includes the descriptions of the pedicel compartment and the fruit cell wall elasticity (Hall et al., 2013; Lechaudel et al., 2007), as we will more specifically describe in the next section. The fruit growth sub-model predicts the hourly dynamic of fruit dry and fresh weight along the growth season.
Table of contents :
Understanding the fruit growth biophysical processes can help fruit producers to improve fruit yield and quality
Process-based models can help to analyze genetic variability of fruit growth-related traits and fruit growth processes responses under diffrent environmental conditions
Objectives and structure of the thesis
Published or submitted chapters
1 Model-assisted estimation of the genetic variability in physiological parameters related to tomato fruit growth under contrasted water conditions
2 Model-assisted analysis of the pedicel-fruit system suggests a fruit sugar uptake regulation and a water saving strategy
Material and methods
Conclusions and perspectives
3 A mechanistic virtual fruit model describing fruit growth and the main fruit pulp solutes metabolisms well predicts fruit growth and highlights possible osmotic potential regulation mechanisms during fruit development
Material and methods
Conclusions and perspectives
Findings and conclusion
ANNEXES – SUPPLEMENTARY MATERIAL i
S2.1 – Linear system analytical solution
S2.2 – Parameters variability among best solutions
S2.3 – Sensitivity to inputs
S3.1 – Supplementary figures for the 2.4 density treatment