Simulation models can be used to perform virtual profiling in order to analyse eco-physiological processes controlling plant phenotype. To illustrate this, an eco-physiological model has been used to compare and contrast the status of a virtual fruit system under two situations of carbon supply. The model simulates fruit growth, accumulation of sugar, citric acid and water, transpiration, respiration and ethylene emission, and was successfully tested on peach (Prunus persica L. Batsch) for two leaf-to-fruit ratios (6 and 18 leaves per fruit). The development stage and the variation in leaf number had large effects of the fruit model variables dealing with growth, metabolism and fruit quality. A sensitivity analysis showed that changing a single parameter value, which could correspond to a genotypic change induced by a mutation, either strongly affects most of the processes, or affects a specific process or none. Correlation analysis showed that, in a complex system such as fruit, the intensity of many physiological processes and quality traits co-varies. It also showed unexpected co-variations resulting from emergent properties of the system. This virtual profiling approach opens a new route to explore the impact of mutations, or naturally occurring genetic variations, under differing environmental conditions.
In contrast to current approaches in fruit physiology, which mostly focus on one or a few genes at a time, highly parallel profiling technologies probe many genes, transcripts, proteins or metabolites at once. In this way, profiling technologies allow a shift from hypothesis-driven research to the analysis of system-wide responses (Kopka et al., 2004; Hennig, 2007). This approach will undoubtedly help to elucidate how things work at the (sub)cellular level. However, it is not always easy to clearly interpret the observed ‘omic’ changes because gene functions, even if known at the cellular or organelle level, are rarely linked to organ and plant responses. Moreover, small environmental changes or a single mutation of one gene can have dramatic effects on gene expression patterns (Kolotilin et al., 2007). One way to improve our understanding of plant system-wide responses will be to complement ‘omic’ approaches with approaches that integrate the processes interacting at different levels under a continually changing environment.
When the system under study is complex (e.g. the fruit), a simulation model can be used to analyse how the system works (in terms of interacting processes) under the control of environmental, genetic and plant factors. The model offers a theory describing how the components of the system causally interact with one another to produce a given outcome. Thus, simulations can be seen as the creation of a possible world constructed in silico using computer programs to represent relevant aspects of the real system under investigation (Peck, 2004). Such an approach to explore phenotypes under changing conditions may be considered as ‘virtual profiling’. Such eco-physiological models are suitable tools to complement ‘omic’ results because they indicate integrated responses at the organ or plant levels. Our purpose is to show how such a model can be used to perform virtual profiling in the example of fleshy fruit quality. Quality traits are seldom subject to modelling, probably because they result from a poorly understood chain of processes with only partly known complex underlying mechanisms (Struik et al., 2005). Nevertheless, models focusing on fruit size and composition have recently been proposed (Génard et al., 2007) that are able to outline emergent properties, i.e. properties that appear only when a number of simple processes operate in an environment, interacting and forming more complex behaviors as a collective. The first objective of this paper is to test, using a peach fruit dataset, a ‘virtual fruit’ model that integrates the main processes involved in fruit quality development into one complex system. The second objective is to show how such a model can be used to perform virtual profiling to infer the phenotype of plants grown under two contrasting carbon supply conditions. Finally, an analysis of sensitivity of the model to the variations of genotypic parameters was performed in order to analyse the putative effects of genotypic changes on the virtual profile. A correlation network based on the sensitivity analysis was produced in order to determine the eco-physiological processes and quality traits that behave similarly whatever the genotype and the carbon supply. The possible links with ‘omic’ profiling are discussed.
Test of the virtual fruit model
The model represents the development of quality in an ‘average‘ peach fruit on a 1-year-old stem that bears fruit and leafy shoots. The simulated quality traits relate to fruit size, the proportion of the fruit that is stone, and sugar and acid concentrations. Gas emission (CO2 and C2H4) and consumption (O2) were also considered as quality traits that indicate fruit maturity. The model is driven by weather data, plant water potential and fruit load, and integrates seven sub-models that have been published and extensively validated and applied (Figure 1) (see Experimental procedures). It was tested on a 1-year experimental dataset including all quality traits predicted by the model. Fruiting stems with four fruits at two leaf-to-fruit ratios (6 and 18 leaves per fruit) were simulated during the 2 months before fruit maturity. The model accurately simulated the order of magnitude and seasonal variations of several peach fruit traits for the two leaf-to-fruit ratios (Figure 2). It also accurately simulated the observed response of fruit growth in terms of dry and fresh masses. In accordance with experimental data, the predicted percentage of stone in the fruit decreased with time and was lower for the high leaf to fruit ratio. The model predicted the seasonal increase of sucrose concentration and the positive effect of leaf-to-fruit ratio on this trait. Only small variations in glucose and fructose concentrations during growth and between leaf-to-fruit ratios were predicted, consistent with the experimental data. Sorbitol, which is very abundant in peach sieve tubes, was predicted to occur at a very low level in the fruit, in accordance with experimental measurements. The seasonal bell-shaped curve of citric acid content was well simulated by the model, which predicted a higher citric acid concentration at harvest for the low leaf-to-fruit ratio. Gas exchanges (O2, CO2 and C2H4) were also well predicted, with a seasonal decrease in O2 consumption and CO2 emission, and an increase in C2H4 emission, especially for the high leaf-to-fruit ratio.
Virtual profiling during fruit development
The model was used to assess the daily value of 37 functional variables (Table 1) over a 62-day period for each leaf-to-fruit ratio. These variables indicate the variations in the state of the system. Most of functional variables were expressed as relative rates that quantify the activity of the underlying processes at a given time and are thus comparable to ‘omic’ outputs such as gene expression. A common relative rate is the relative growth rate of a sink tissue, which quantifies its activity. In the framework of this study, the rates were relative either to a surface, e.g. photosynthetic rate (g m−2 day−1), or to a dry weight, e.g. relative daily variation of sucrose content (day−1), which quantifies the daily variation of sucrose per unit of dry mass of fruit flesh. Some physical characteristics such as turgor pressure or conductance were also considered as they are proportional to relative growth rate and transpiration rate, respectively, in the model.
Table 1. List of the processes and functional variables considered for the virtual profiling. The label and units are given for each variable
Stem and shoots
Relative rate of carbon storage in the stem
Relative rate of carbon storage in the shoots
Relative maintenance respiration of stem and shoots
Leaf phothosynthetic rate
g m−2 day−1
Light-saturated leaf phothosynthetic rate
μmol CO2 m−2 sec−1
Fruit photosynthetic rate
Relative fruit demand
Relative growth rate of flesh dry mass
Relative growth rate of stone dry mass
Relative flesh respiration rate
mmol CO2 g MS−1 day−1
Relative daily variation of CO2 content
mol g−1 day−1
Relative daily variation of O2 content
mol g−1 day−1
Relative daily variation of ATP content
mol g−1 day−1
Relative water influxes
Flesh turgor pressure
Flesh osmotic potential
Relative growth rate of flesh fresh mass
Relative growth rate of stone fresh mass
Skin area increase rate
Sugar and citric acid metabolisms
Relative daily variation in sucrose content
Relative daily variation in sorbitol content
Relative daily variation in glucose content
Relative daily variation in fructose content
Enzyme activity metabolizing glucose and fructose in other compounds than sugars
Enzyme activity metabolizing fructose for respiration
Enzyme activity metabolizing glucose for respiration
Relative daily variation in citrate content
Relative daily variation in ACC content
mol g−1 day−1
Relative daily variation in MACC content
mol g−1 day−1
Relative daily variation in C2H4 content
mol g−1 day−1
ACC oxidase activity
μmol g−1 h−1
ACC synthase activity
μmol g−1 h−1
μmol g−1 h−1
A heatmap of the evolution of these 37 variables over time, sorted according to their functionality, is presented in Figure 3. The 37 variables showed contrasting temporal patterns and could be divided into two groups. Whatever the leaf-to-fruit ratio, variables in the first group had high-frequency oscillations due to changes in the climatic environment (Table 2). Some of these variables represented temporary pools of biochemical compounds such as sorbitol (SO), glucose (GL), fructose (FR), ATP and gas (O2 and CO2). These temporary pools were positively or negatively correlated with the radiation fluctuations. Leaf photosynthesis (Lph) also showed high-frequency oscillations due to changes in radiation and air humidity, which were themselves negatively correlated (r = −0.66), although the light-saturated leaf photosyntesis rate (Pm), which is controlled by the leaf storage pool size, was much more stable. Respiration of vegetative parts (MRs) was highly correlated with temperature. Fruit flesh growth (Ff) and to a lesser extent fruit transpiration (Tr) were correlated with radiation and humidity. At high humidity, transpiration decreased and flesh growth increased. The opposite was noted at high radiation. The skin area increase (Skin) fluctuated with temperature and humidity, probably because of its strong link with fruit growth (r = +0.63).
Table 2. Correlations between highly oscillating functional variables and climatic factors for leaf-to-fruit ratios of 6 and 18
Photosynthetically active radiation
Air relative humidity
Correlations > 0.4 or < −0.4 are shown in bold. Abbreviations are defined in Table 1.
The second group of variables showed a temporal gradient with two distinct periods. The system was characterized early in the season by large changes in fruit skin characteristics (Sco, Cuco), high water influxes (Flux) and osmotic potential (Os) in the fruit, low turgor pressure (P), high stone growth (Sf, Sd), high fruit demand for carbon (Fde) and high fruit growth (Fd), high fruit photosynthesis and respiration (Fph, Fre), high production of citric acid in the fruit (CI), and high activity of enzymes involved in glucose and fructose metabolism (Eglf) or sucrose accumulation (SU). The second period, which roughly corresponds to the fruit maturation period, started with increased ACC (1-aminocyclopropane-1-carboxylic acid), ethylene and MACC (malonyl-ACC) production. It was characterized by high activities of ACC synthase and oxidase (ACCs, ACCo). Ethylene emission (C2H4) and stem and shoot storage processes (Cst, Csh) were also high.
How does leaf-to-fruit ratio affect the virtual profile?
The leaf-to-fruit ratio strongly affected both the intensities and timing of variables as various as carbohydrate storage in the plant, fruit cuticular cracks, and respiration and ethylene emission (Figure 3). However, the effect of threefold increase in a leaf number was not as great as may have been expected, mainly because regulation of maximal leaf photosynthesis contributed to a decrease in the photosynthesis rate when the leaf-to-fruit ratio was increased to 18. Nevertheless, according to our model, changes in the leaf-to-fruit ratio led to a significant time shift in the progression of many processes.
An analysis of the simulations using principal component analysis (PCA) of the 37 variables and 62 days of simulation helped in tracking the developmental variations (Figure 4). The PCA was carried out for the leaf-to-fruit ratio of 6, and the principal components (PC) scores of the days of simulations were plotted on the first two axes, which explained 65% of the variation. The first factor described the seasonal change of the system. It opposed variables defined previously as having high values either early or late in the season. The second factor, which concerned mainly fluctuating variables, was positively correlated with the leaf photosynthetic rate (Figure 4) and also with the sun global radiation (r =0.79, P <0.001). In order to analyse how an increase in the source:sink ratio acts on the seasonal pattern of the system, data for the leaf-to-fruit ratio of 18 were plotted as inactive data on the first plan, which means that they were passively positioned on the axis defined by the data for the leaf-to-fruit ratio of 6. The clear discrepancies for PC2 indicated that increasing the leaf-to-fruit ratio has the same effect as an increase in sun radiation (Figure 4). For PC1, which described the seasonal changes in the system, the distance between two successive dates indicated the speed at which the system state changed. The speed was similar for both treatments until day 95, and then it was higher for the leaf-to-fruit of 18. For this ratio, the higher values reached on PC1 indicate a higher degree of maturity. Indeed, ethylene emission was initiated 10 days earlier and more intensively than for the leaf-to-fruit ratio of 6 (Figure 3). Although the leaf-to-fruit ratio had strong effects on quality traits throughout fruit development, the functional variables had comparable levels at maturity (Figure 3), and this is why both treatments reached the same position on the first plan at maturity (Figure 4). Comparison of the two virtual profiles indicated that the instantaneous picture of the differences between the two treatments at maturity only is a highly restricted view of the effects of leaf-to-fruit ratio. The state at maturity hides the complexity of the system and the interactions and regulations involved during fruit growth.
How do the genotypic parameters affect the virtual profile?
Model parameters are constant with time and independent of environment. A lot of them are genotype-dependent and account for the genetic variability of fruit quality traits. In the integrated model, 12 parameters (hereafter called genotypic parameters) representative of the main processes were selected to analyse the putative effects of genotypic changes on the virtual profiles (Table 3). For this, a sensitivity analysis of model outputs was performed considering independent variations of these 12 genotypic parameters. The normalized sensitivity coefﬁcients (SC), deﬁned as the ratios between the relative variation of functional variables and the relative ratio of parameters, are presented in Figure 5 for each parameter and variable (the sensitivity to a parameter was small when SC∼O). The pattern of sensitivity was different between the two leaf-to-fruit ratios, especially for parameters related to leaf photosynthesis, fruit respiration and carbon demand. Moreover, these parameters had strong effects on a large set of functional variables. For instance, variation in p1 (maximal leaf photosynthesis) largely affected processes related to leaf photosynthesis, plant carbohydrate reserves, and many features of fruit functioning (growth, cuticular crack conductance, respiration and ATP production, gas exchange, sugar and citric acid accumulation, and ethylene metabolism). In contrast, parameters involved in plant reserve mobilization, fruit water management (cell-wall extensibility, conductivity for water transport, specific cuticular and crack conductances) and metabolism of non-sugar compounds had very small effects. Parameters involved in citric and ethylene metabolism had very specific effects on such metabolism.
Table 3. Genotypic parameters for the virtual fruit used in the sensitivity analysis
p1 (μmol CO2 m−2 sec−1)
Light-saturated maximal leaf photosynthesis
Leafy shoot mobile fraction of reserves
RGRini (degree day−1)
Initial relative fruit growth rate
Fruit respiration sub-model
MRRfruit (g CHO g−1 sec−1)
Maintenance respiration rate of fruit at the reference temperature
Cell wall extensibility coefficient in Lockhart’s equation
L (g cm−2 bar day−1)
Conductivity of the composite membrane for water transport
Skin conductance sub-model
gc (cm h−1)
Specific cuticular conductance
gck (cm h−1)
Specific cuticular crack conductance
Synthesis of other compounds than sugars (cell walls, acids, etc.)
b1 (mmol day−1)
Rate of citrate production at day b3 (day for which the rate of citrate production equals b1)
ACC synthase rate constant
ACC oxidase rate constant
Thus a change in a single parameter value, which could correspond to a genotypic change induced by a mutation, either strongly affects most of the processes, only affects specific processes or affects none of them.
A correlation network linking functional variables and fruit quality traits
A network analysis was performed in order to capture the key functional variables affecting quality traits. Each functional variable and quality trait was characterized by a vector made up of the 24 sensitivity coefficients (the 12 genotypic parameters presented above x two treatments). We hypothesize that two variables or traits are linked if they present similar sensitivity to the 12 genotypic parameters. The coefficients of correlation among vectors were used to quantify the link between the corresponding variables. Figure 6 shows the network constructed from all pairwise correlations greater than 0.5. Of the 1176 [(37 + 12)2 − (37 + 12)]/2 pairwise correlations, 430 were greater than 0.5. Eighty per cent of the quality traits and 60% of the functional variables were highly connected nodes, which outlines the high degree of integration of the fruit system. The network was represented, using the Kamada–Kawai algorithm, as a physical system at a state corresponding to its minimal energy (see Experimental procedures). A clear pattern emerged, with three main groups of variables that were highly correlated. The first and most important group constituted fruit traits such as O2 consumption and CO2 emission, fruit size and sugar content. Logically, the corresponding functional variables were the relative growth rates of fruit and stone, photosynthesis, carbohydrate storage in leaves, fruit respiration and sugar metabolism. A second group linked the mass of the stone and the citric acid concentration with functional variables to which they were not functionally linked, except for the relative variation rate of citric acid (CI). Similarly, correlations between independent functional variables such as ATP and stomatal conductance (Sco) could not be easily explained from the theory on which the model is based. The third group represented ethylene metabolism. It was independent of the two other groups even though ethylene metabolism is functionally connected with respiration and ATP production in the model. Nevertheless, a significant coefficient of correlation of 0.37 was found between C2H4 and the relative flesh respiration rate. Two small groups were satellites of the first group, to which they were correlated. They grouped functional variables involved in fruit fresh mass (water influxes, transpiration, conductance, turgor pressure) and in stem/leaf maintenance respiration and carbon storage, respectively. Lastly, relative fruit demand and relative variation of sorbitol formed a small group that was negatively correlated to the first one and to its second satellite. This network analysis showed that, in a complex system such as fruit, many functional variables and quality traits co-vary. It also showed that co-variations result from emergent properties of the system, as they were not included in the basis hypotheses of the model.
The virtual fruit model allowed prediction of the seasonal variation of several fruit traits as well as the effect of leaf-to-fruit ratio on those traits. The validation of such a complex model is difficult. All intermediate variables are good candidates to test the model, but experimental data are difficult to obtain. However, focusing on the output variables only (quality traits), the goodness of fit of the model has been checked and gave satisfactory results. It is important to note that some quality traits considered in this work are also important physiological variables, e.g. sucrose, sorbitol, glucose and fructose contents or ethylene emission, so the model may be considered to have been partly checked at the physiological level.
The approach described here aims to integrate current knowledge in a theoretical model of fruit functioning and use that model to analyse the system and test hypotheses via simulations. Such an approach has been used for several years in the field of ecology (Peck, 2004). Its utility for phenotyping is exemplified in this work, and a comprehensive analysis of how the fruit system behaves is proposed based on virtual profiling. For instance, the increase in the leaf-to-fruit ratio did not simply lead to higher fruit mass but had strong effects on most of the fruit and leaf processes. Cascading effects and regulation occurred in such a way that the fruit state at maturity was similar at both leaf-to-fruit ratios. Consequently, comparison of the two situations at maturity only would have led to mis-interpretation of the effects of leaf-to-fruit ratio on functioning of the system. This information could be helpful in choosing the date at which sampling is done for ‘omic’ studies. Moreover, the way the system varies over the season and with treatments as analysed through the virtual profiling will be of great assistance in interpreting variations observed in ‘omic’ studies.
The sensitivity analysis of the model to variation of its genotypic parameters provided new information on the possible effect of genes on functioning of the whole system. Some appeared to have a large influence on the system as a whole, while others had very specific effects. Such information is of real interest as it provides a physiological explanation of the phenotypic effect resulting from a genotypic variation. More global information was provided by the results of the network analysis, which could be compared to networks of genes or metabolites.
The present paper illustrates how virtual profiling can help understanding of complex systems such as fruit. A challenge for the future is to connect virtual and ‘omic’ profiling. Two main approaches, empirical and mechanistic, are possible. A means to facilitate virtual and ‘omic’ profiling connection is to perform ‘omic’ and virtual profiling on the same subjects, and to use data-mining technologies such as the 02PLS method recently proposed by Bylesjöet al. (2007) to look for links between them. A simpler approach may be based on correlation networks that include changes in gene transcription level and virtual profiles.
To go further, mechanistic integration of information generated by ‘omics’ technologies into models would provide a global view of how plants operate and how they interact with the environment (European Plant Science Organization, 2005). Some attempts at integration from gene to cell have been successfully undertaken (Tomita et al., 1999), but scaling-up to the organ or plant level has not yet been performed. A similar situation existed in the past decade in the field of heart research, with independent researchers providing bottom-up models of cell physiology and top-down models of heart 3D geometry. By developing so-called ‘middle-out’ modelling, scientists successfully coupled both approaches (Noble, 2002). The virtual fruit model belongs to a new generation of models that should provide an important link between molecular research and whole-plant physiology (Struik et al., 2005). An interesting avenue would be to embed mechanistic metabolic models into the virtual fruit model, for instance by replacing the present simplified sugar model by the sugar metabolism model developed by Uys et al. (2007) for sugarcane. The virtual fruit model would produce input for the metabolic model, which could simulate the production of metabolites along the described metabolic pathway over the season and under various growing conditions. Comparison with real metabolic profiles obtained through metabolomic studies would enable testing of the reliability of insertion of the metabolic model within the virtual fruit model.
Integration of genetic control into the virtual fruit model is another important step that has partly been performed by replacing the genotypic parameters of the model by the effects of the corresponding quantitative trait loci (QTL) (Quilot et al., 2005). In the future, our objective is to include the control of processes at the gene level into the virtual fruit model, which requires a better scientific knowledge of the genetic control. This will open the way for the simulation of genomic outputs.
The present approach illustrates the need for a new generation of eco-physiological models at the intersection between metabolic models and crop models. They will form an important cornerstone for the development of plant system science in the future.
The virtual peach fruit model represents the quality of an ‘average’ fruit on a ‘average’ fruit-bearing stem, which is the basic production unit for peach growers. This unit is a 1-year-old stem that bears fruit and leafy shoots. The model integrates seven sub-models describing fruit growth and quality (Figure 1).
A first step of integration was proposed by Lescourret and Génard (2005), who combined three existing process-based models describing dry mass, sugar and water accumulation in the flesh of fruit into a single model. This model was successfully validated using the results of experiments from three different years. The relative root mean squared error of prediction was calculated using a cross-validation approach (Wallach et al., 2001), and was smaller than 20% in most cases (Lescourret and Génard, 2005). Moreover, a simplified version was tested on 87 peach genotypes, and the error of prediction was also smaller than 20% (Quilot et al., 2005).
In the present work, additional sub-models describing skin conductance and microcracking (Gibert et al., 2005), respiration and citric acid accumulation (Lobit et al., 2003; Wu et al., 2007) and ethylene emission (Génard and Gouble, 2005) were incorporated. These sub-models were successfully validated by a cross-validation approach using the results of experiments from different years. A short description of each sub-model is presented below.
The virtual fruit model is driven by its parameters, which are constant over time and independent of environment (for details, see Table S1). As their values are often genotype-dependent, they can be considered as a fingerprint of the genotypes. The outputs of the virtual fruit model are quality traits and functional variables. The quality traits are variables related to fruit quality, such as fruit size, the proportion of the fruit that is stone, sugar and acid concentrations. Gas emission (CO2 and C2H4) and consumption (O2) were also considered as quality traits indicating the level of fruit maturity. The functional variables describe variations in the state of the system. Most functional variables were expressed as relative rates that quantify the activity of the underlying processes.
This sub-model has been described by Lescourret et al. (1998). It has been used to analyse the within-tree variability of fruit growth (Walcroft et al., 2004), and the genotypic and management effect on fruit growth (Génard et al., 1998; Quilot et al., 2002). The daily pool of carbon assimilates available for growth comes from leaf assimilation plus reserve mobilization. The leaf photosynthesis rate may be affected by feedback inhibition through the leaf storage reserves. Fruit carbon assimilation and respiration are also considered. If required, reserves are mobilized first from the leafy shoot, then from the 1-year-old stem. Carbon is allocated according to organ demands and priority rules. The equation for daily carbon demand (D) for fruit growth accounts for the fruit ‘history’ through the fruit mass (Mdry), both in terms of sink size and sink activity. It also takes into account the role of time and temperature as cumulated degree days (dd):
where RGRini is the initial relative growth rate and refers to the limiting final potential dry mass (Mdry). The assimilates not used for maintenance and growth accumulate in the reserve pools.
Fruit respiration sub-model
The growth-maintenance paradigm (Cannell and Thornley, 2000) was used to calculate the respiration in terms of CO2 production. The growth respiration is proportional to fruit growth rate, and the maintenance respiration is proportional to dry mass and temperature (Thornley and Johnson, 1990). The fruit respiration is calculated as the sum of both growth respiration and maintenance respiration:
where Mdry is the fruit dry mass, qg is the growth respiration coefficient, qm is the maintenance respiration coefficient at 20°C, Q10 is the temperature ratio of maintenance respiration, and T is temperature. This sub-model strongly interacts with the carbon sub-model because fruit growth in terms of carbon accumulation depends on fruit respiration.
This sub-model was adapted from the biophysical model of fruit growth described by Fishman and Génard (1998). In the original model, water enters the fruit by peduncular flow (U) and leaves via transpiration (Tra), such that the fruit volume changes as:
The total pedoncular flow is:
where Ψf = Pf − πf, A is the vascular network area, L is the hydraulic conductivity coefficient of vascular network membranes, Ψstem and Ψf are the stem and fruit water potential, and Pf and πf are the turgor and osmotic fruit pressure. The fruit osmotic pressure induced by sugars is calculated from the sugar sub-model outputs.
The hydrostatic pressure of the fruit is calculated by solving Lockhart’s equation describing the fruit volume V increase (Lockhart, 1965):
where φ is the extensibility of the cell walls and Y is the yield threshold value that the hydrostatic pressure of the fruit has to exceed before irreversible expansion occurs. The change in fruit volume can also be calculated as:
Under the condition of steady irreversible growth, both equations must be equal. By setting them as equal, the resulting equations for Pf can be solved and dV/dt can be calculated.
Skin conductance sub-model
This sub-model was proposed by Gibert et al. (2005). Fruit skin conductance to water is calculated as the sum of stomata, cuticle and crack conductances. Using the kinetics of fruit size and initial stomata number as input, the stomatal density and cuticle area may be calculated at any time. Cracks are assumed to be generated when the pulp expansion rate exceeds the cuticle expansion rate. A percentage of cracks are assumed to heal each day. This sub-model strongly interacts with the water sub-model because fruit transpiration is driven by the skin conductance, which also depends on the fruit volume increase.
This sub-model (Génard et al., 2003) simulates carbon partitioning in peach flesh into sucrose, sorbitol, glucose and fructose. The rates of change in the four sugars are described through a set of differential equations, formulated as:
where Cj is the carbon amount in sugar j, Sj and Rj, which can be equal to zero, depending on the compartment, are the incoming carbon supply and the carbon loss by respiration, respectively, and kij is a function of parameters (θ) and variables (x) describing the relative rate of sugar transformation of sugar i into sugar j.
Citric acid sub-model
This sub-model, previously reported by Wu et al. (2007), is based on hypotheses proposed by Lobit et al. (2003). During fruit development, various mechanisms allow the rates of the citrate cycle reactions to match the respiratory demand. For the purpose of modelling, the citrate cycle was simplified and a corresponding system of equations was proposed. After solving the system of equations, the rate of citrate synthesis or degradation (dCI/dt) was expressed as a simple function of days (t) and respiration:
where bi are genotypic parameters and Resp is respiration.
A theoretical model of fruit climacteric ethylene emission was developed by Génard and Gouble (2005), according to which the biosynthetic pathway of ethylene is supplied by ATP and is regulated by ACC synthase and ACC oxidase. The conjugation of ACC with malonate to form MACC was taken into account as a means to decrease the availability of ACC. The main equation of the sub-model is:
where [C2H4 ref] is a reference concentration equal to 1 mol m−3, are Michaelis constants, ko is a parameter, is the apparent skin permeability to ethylene, A is the skin area and V is the fruit volume. The equation is composed of three terms. The first term calculates the ethylene biosynthesis, the second term its diffusion to the atmosphere, and the third term the dilution effect due to fruit volume increase.
Experiments and measurements used to test the model
The model was tested using an experiment performed in 1993 (see Souty et al., 1999) on peach trees of the late-maturing cultivar Suncrest planted in 1982 at the INRA Centre in Avignon (South East France). Trees were goblet-trained and received routine horticultural care, including non-limiting irrigation. In early June, two leaf-to-fruit ratios (6 and 18 leaves per fruit) were applied to 160 fruit-bearing stems from 45 trees isolated from the rest of the tree by removing a strip of bark including phloem tissue. Vegetative growth was prevented by tipping, in order to focus on fruits. Five replicates (made up of fruits from one to two similar shoots) were harvested weekly until fruit maturation. The fresh fruit mass, the fresh and dry mass of the stone, the dry matter content of flesh (evaluated after drying at 70°C for 72 h), and the concentrations of the four sugars and citric acid found in the peach fruit (percentage of fresh flesh mass) were measured. Sugar and acid contents were measured by HPLC (Génard and Souty, 1996; Wu et al., 2002).
At each harvest, O2 consumption and CO2 and ethylene production were measured at a constant room temperature (23°C) by confining intact fruit in a gas-tight 400 ml jar. The internal atmosphere of the jar was analysed by gas chromatography. CO2 and O2 were separated on a Porapak Q column (Waters, http://www.waters.com) at 110°C, followed by a 13X molecular sieve column (Varian, http://www.varianinc.com) at room temperature (Chambroy et al., 1984), and quantified using a catharometric detector. The amount of ethylene released by the fruit was determined using a ﬂame ionization detector gas chromatograph with an activated alumina column at 120°C.
Model implementation, inputs, parameterization, test and sensitivity
The model has been implemented in S language, using R software (R Development Core Team, 2008). The code is available on request. The inputs of this model are the photosynthetically active radiation, mean daily temperature, air relative humidity and stem water potential. Values for photosynthetic active radiation, temperature and relative humidity data were obtained from INRA weather stations close to the experimental fields (for details, see Figure S1). Because the experimental trees were well-watered, a constant daily value of – 0.45 MPa was taken for stem water potential (Lescourret and Génard, 2005). The state of the fruit-bearing stem (number of leafy shoots and fruits on the stem, and initial values of the state variables) was also described.
As virtual fruit parameters can vary greatly according to genotype (between 3 and 90% according to Quilot et al., 2005), a sensitivity analysis of the model to variations in a set of important parameters was investigated (Table 3). This was performed using the environmental conditions for the growing season in 1993, for leaf-to-fruit ratios of 6 and 18. To simplify the analysis, only the mean values (over time) of these variables were considered. The sensitivity of each variable to each parameter was quantiﬁed using the normalized sensitivity coefﬁcient, which is deﬁned as the ratio between the relative variation of the predicted variable and the relative variation of the parameter (±10%).
Principal component analysis and correlation network analysis
A principal component analysis (PCA) was performed to analyse how the leaf-to-fruit ratio affects the fruit-bearing system behaviour. The goal of the PCA is to summarize a multi-variate dataset (the functional variables and quality traits of the virtual fruit model) as accurately as possible using a reduced number of uncorrelated components (Jolliffe, 2002). A correlation network analysis was performed in order to capture, among the 37 key functional variables, those affecting the 12 quality traits. Each functional variable and quality trait was characterized by a vector made up of 24 sensitivity coefficients. The coefficient of linear correlation was chosen to characterize the link between the vectors. The network consisted of a set of nodes (functional variables and quality traits) connected by a system of lines that represented the correlations (positive or negative) between variables and traits when the absolute values were higher than 0.5 (significant at α < 0.001). The Kamada–Kawai algorithm was used for automatic layout generation (Kamada and Kawai, 1989). The network is represented like a physical system, and the idea is to minimize the energy of the system by moving the nodes and changing the forces between them. The networks were constructed using Pajek graph-drawing software (Batagelj and Mrvar, 2003; http://vlado.fmf.uni-lj.si/pub/networks/pajek/).
We gratefully acknowledge R. Stevens (UR1052, Génétique et Amélioration des Fruits et Légumes, INRA, Avignon, France) for revising the manuscript. This research was financially supported by the 6th Framework Program EU Project ISAFRUIT. The ISAFRUIT Project is funded by the European Commission under Thematic Priority 5 – Food Quality and Safety of the 6th Framework Program of Research and Technological Development (contract number FP6-FOOD-CT-2006-016279).