JXB Advance Access originally published online on September 25, 2003
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Journal of Experimental Botany, Vol. 54, No. 392, pp. 2489-2501,
November 1, 2003
© 2003 Oxford University Press
Modelling citrate metabolism in fruits: responses to growth and temperature
Received 17 March 2003; Accepted 11 July 2003
1 Ctifl, Centre de Balandran, 30127 Bellegarde, France
2 Unité Plantes et Systèmes de culture Horticoles, INRA, Domaine St Paul, Site Agroparc,84914 Avignon Cedex 9, France
3 Institute of Botany, Chinese Academy of Science, 100093 Beijing, PR China
* To whom correspondence should be addressed. Fax: +33 4 32 72 24 32. E-mail: mic{at}avignon.inra.fr
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Abstract |
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Citrate production and degradation during the last stage of fruit development were modelled by representing the fluxes through the enzymes of the citrate cycle and the malic enzyme, the transport of metabolites between the cytosol and the mitochondria, and the stoichiometry equations that relate these reactions. After solving the corresponding system of equations, the rate of citrate synthesis (or degradation) was expressed as a simple function of temperature, mesocarp weight, and respiration. The model was applied to peach fruit, and its parameters were estimated from the data of a 2-year field experiment. The predictions of the model were in agreement with experimental data. Simulations were made to analyse the responses to variations of temperature and fruit growth. Increasing fruit growth before stone hardening stimulated citrate production, while increasing fruit growth after stone hardening reduced it. Delaying the date at which the maximum growth rate was reached enhanced citrate production during most of the period. In the last weeks before harvest, increasing temperature depressed citrate production, while, at the beginning of the period studied, it enhanced it.
Key words: Citrate, fruit growth, model, temperature.
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Introduction |
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TopAbstractIntroductionMaterials and methodsResultsDiscussionConclusionAppendicesReferences |
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Acidity plays an important part in the perception of fruit quality. It affects not only the sour taste of the fruit (Lyon et al., 1993), but also sweetness, by masking the taste of sugars. According to Moreau-Rio et al. (1995), the overall consumer appreciation is related more to the titratable acidity/refractometric index ratio than to the soluble sugars content alone. The proportions of individual acids are also important: for example, citrate masks the perception of sucrose (Schifferstein and Fritjers, 1990; Bonnans and Noble, 1993) and fructose (Pangborn, 1963), while, on the contrary, malate seems to enhance sucrose perception (Fabian and Blum, 1943).
Two organic acids, citrate and malate, are dominant in most fruit species (Tucker, 1993), and understanding the elaboration of acidity requires studying the mechanisms involved in the accumulation of both. In this paper, the focus was on modelling citrate metabolism.
Most of the organic acids of the fruit, including citrate, are stored in the vacuoles of mesocarp cells (Yamaki, 1984) and permanently exchanged between cytosol and vacuole by a variety of mechanisms (Martinoia and Ratajczak, 1997). Citrate synthesized in the cytosol appears to be easily transported into the vacuole as soon as its cytosolic concentration increases (Gout et al., 1993). Citrate efflux probably occurs by diffusion of the protonated form through the tonoplast, as demonstrated for malate (Lüttge and Smith, 1984). However, since the balance between influx and efflux depends on citrate concentration in the cytosol, it is likely that the rate of citrate production or degradation, by driving the changes in cytosolic concentration, determines most of its vacuolar accumulation.
Only a few enzymes are known to catalyse citrate synthesis or degradation. All of them intervene in the citrate cycle and are located in the mitochondria (except citrate lyase, which is in the glyoxysomes). In addition to their role in respiration, isolated mitochondria metabolize di- and tri-carboxylic acids (Ulrich, 1970). Citrate can be either oxidized or produced (Bowman et al., 1976), depending on the experimental conditions.
The purpose of this article is to propose a model representing citrate production and degradation during fruit development. Although it is based on an analysis of citrate metabolism at the cellular level, its purpose is not to develop an exhaustive representation of metabolic pathways. Instead, the analysis of metabolism is used to identify the main factors involved and to clarify their interrelations in order to derive an expression of citrate production and degradation at the whole fruit level. The result is a simple relation between the rate of net citrate production and fruit initial dry weight, temperature, and respiration. This model has been parameterized and tested using the data from a 2-year experiment in a peach orchard.
Model development
Governing equations: the citrate cycle: Mitochondrial functioning has been extensively studied (Douce, 1985). The main pathways in the mitochondria are represented in Fig. 1A. They include the citrate cycle and the oxidation of malate into pyruvate by an NAD-specific malic enzyme. In addition to the citrate cycle enzymes, mitochondria are also equipped with transport systems on the inner membrane that exchange the citrate cycle intermediates between the matrix and the cytosol. During fruit development, different mechanisms allow the rates of the citrate cycle reactions to match the respiratory demand. However, not all reactions are regulated with the same intensity, and relative variations in metabolic fluxes may affect citrate metabolism.
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For the purpose of modelling, this system was simplified as represented in Fig. 1B. The only metabolites considered, pyruvate, malate and citrate, were chosen because they are at branch points between several reactions and because they are exchanged between the cytosol and the mitochondria.
A mathematical formulation of this system is described below. It includes stoichiometry equations, enzymatic reactions and transports. Emphasis is put on the responses of enzymatic reactions and transports to the factors that take part in their regulation.
Stoichiometry equations: The maintenance of the pools of metabolic intermediates imposes, for any given metabolite, that the sum of flux that synthesize it and the sum of those that consume it be equal. Mathematically speaking, these reactions translate into the following set of stoichiometry equations:
for pyruvate:
3+4–1 = 0(1a)
for malate:
2 + 5 – 1 – 3 = 0(1b)
for citrate:
1 – 6 – 2 = 0(1c)
The respiratory flux, approximated as the flux of CO2 produced by the citrate cycle, is:
Resp = 1 + 22 + 3(1d)
The net balance of citrate metabolism in the mitochondria (6) is obviously equal to the difference between the rates of citrate production and degradation (equation 1c), but another property of the system is that any citrate production has to be compensated for by the entry of dicarboxylic acids in the cycle. As a result, the net rate of citrate production is also equal to the difference between the rate of importation of dicarboxylates from the cytosol and the rate of malate oxidization through the malic enzyme (equation 1b + c).
Enzymatic reactions: The reactions of the citrate cycle are regulated by various factors, which can be grouped into three classes (Douce, 1985): (i) redox state (NAD+, NADH, FAD+, FADH availability), (ii) energy availability (ADP, ATP, GDP, GTP, and Pi concentration), (iii) substrate availability (i.e. the concentration of the cycle intermediates, including pyruvate, di-carboxylic acids, CoA, acetyl-CoA, and succinyl-CoA). Oxidation reactions are catalysed by enzymes that use NAD+ as substrate and produce NADH (or FAD+ and FADH for succinate dehydrogenase). Also, specific activation by NAD+ and inhibition by NADH have been demonstrated, in particular, for the pyruvate dehydrogenase complex (Reed et al., 1978), isocitrate dehydrogenase (Cox and Davies, 1970; Duggleby and Dennis, 1970a, b), and 
-ketoglutarate synthase (Poulsen and Wedding, 1970). Among the enzymes stimulated by ADP and inhibited by ATP are pyruvate dehydrogenase complex (Rubin and Randall, 1977; Randall et al., 1981), citrate synthase (Douce, 1985), and the -ketoglutarate synthase complex (Douce, 1985). Special mention has to be made for malate dehydrogenase and malic enzyme, which compete for the oxidation of malate. The role of NADH and ADP in orienting the reactions of malate oxidation in intact mitochondria, demonstrated by Palmer et al. (1982), can be explained by the contrasting responses of malate dehydrogenase and malic enzyme to NADH and NAD+ (Macrae and Moorhouse, 1970; Hatch et al., 1974; Day et al., 1984). Metabolite concentrations also regulate mitochondrial reactions in several ways. Enzymes that operate far from saturation respond to the substrate concentrations: for example, oxaloacetate concentration is below 1 µmol, while the KM of citrate synthase is in the 1–5 µmol range (Douce, 1985). Allosteric effects also account for specific regulations: the pyruvate dehydrogenase complex is stimulated by CoA and inhibited by acetyl-CoA (Reed et al., 1978), -ketoglutarate synthase is inhibited by succinyl-CoA in a similar way, and isocitrate dehydrogenase is activated by citrate (Cox and Davies, 1970; Duggleby and Dennis, 1970a, b). The thermodynamic conditions of the reactions also depend on metabolite concentrations: for example, malate oxydation into oxaloacetate can be reversed when oxaloacetate accumulates (Palmer et al., 1982). Finally, all reaction rates depend on temperature. However, the lack of experimental data about the responses of the enzymes concerned makes it difficult to predict which steps are most affected.
For the purpose of modelling, regulations by energy and redox state were grouped under a generic ‘regulation state’, represented by a single variable Reg. Redox or energy state are indeed linked by the functioning of the respiratory chain, which couples NADH oxidation with ATP synthesis: depending on the conditions, the NADH/NAD+ ratio is more or less tightly related with ADP availability, ADP/ATP ratio, or free energy of ATP hydrolysis (Douce, 1985). Also, when a series of reactions was considered as a single reaction, possible regulations by intermediary metabolites were omitted and only those by substrates, products, ‘regulation state’, and temperature were taken into account: for example, citrate degradation into malate was considered to be regulated only by citrate and malate concentrations, temperature and ‘regulation state’, not by isocitrate, oxoglutarate, succinate, or succinyl-CoA. These properties were formalized in the following set of equations:
1 = a1f1(Pyr, Mal, Cit, Reg,T)(2a)
2 = a2f2(Mal, Cit, Reg,T)(2b)
3 = a3f3(Pyr, Mal, Reg,T)(2c)
where Pyr, Mal and Cit are the mitochondrial concentrations of pyruvate, malate and citrate (mol m–3), Reg is the ‘regulation state’ (dimensionless) and T the temperature (°K). ai 1
i3 are enzyme activities. fi 1i3 are functions, the characteristics of which are not determined for the moment. Properties used for the calculation of citrate metabolism will be detailed in the appendix.
Transports: Concentration gradients between cytosol and mitochondria are the driving force for the transport of the cycle substrates. Pyruvate transport involves a specific system that exchanges its anion against OH– (Papa et al., 1971). Di- and tri-carboxylate transport involve several specific systems that exchange phosphate and OH–, dicarboxylate and phosphate, tricarboxylate and phosphate respectively (Phillips and Williams, 1973; De Santis et al., 1975; Douce, 1985). For citrate, malate and pyruvate, the balance of the exchanges amounts to the transfer of the protonated form of the acid: therefore the transport, which is electroneutral, is likely to depend essentially on the concentration gradients. Finally, temperature, which affects the kinetic properties of the transport systems involved, also has to be taken into account. In order to simplify the modelling of transports, the formalism adopted was derived from diffusion equations. Fluxes were considered proportional to the difference of concentrations between the cytosol and the mitochondrial matrix, while the activity of the transport systems varies with temperature:
4 = a4f4(T) (PyrCyt – Pyr)(3a)
5 = a5f5(T) (MalCyt – Mal)(3b)
6 = a6f6(T)(CitCyt – Cit)(3c)
where ai 4i6 are transport activities, fi 4i6 are functions of temperature. Pyr, Mal, Cit are the concentrations of pyruvate, malate and citrate in the mitochondria, and PyrCyt, MalCyt, CitCyt those in the cytosol assumed to be constant for sake of simplicity.
Activities of enzymes and transport systems: During development, the activities of mitochondrial enzymes vary, as well as the metabolic fluxes in the mitochondrial pathways. As no biochemical assays were available to estimate directly the number of mitochondria and the activities of their enzymes, they had to be estimated indirectly.
With peach cell multiplication taking place in the first month after bloom and mesocarp growth after pit hardening due essentially to cell enlargement (Masia et al., 1992; Zanchin et al., 1994), the metabolic machinery of the fruit is likely to vary little during mesocarp growth: the number of mitochondria and the activities of their enzymes were considered constant during the period modelled. Furthermore, all enzymatic and transport activities were assumed to be proportional to a ‘mitochondrial equipment’, itself proportional to mesocarp dry weight (DW1, in g) at pit hardening:
for 1i6 ai = ki DW1(4)
Solving the system
The approach followed to represent the regulation of fluxes was derived from the theory of metabolic control analysis (MCA) (Fell, 1997). It consisted of computing the variations of fluxes when the conditions of mitochondrial functioning (i.e. ‘regulation state’, temperature, and metabolite concentrations) shift from a ‘reference state’, defined, for an arbitrary temperature (here T0 = 298 °K), by a set of values of metabolite concentrations in the mitochondria Pyr0, Mal0, Cit0, by a ‘regulation state’ Reg0, and a rate of respiration Resp0 = k0DW1 for which the citrate net production is null (i.e. 60 = 0).
The mathematical developments are detailed in the appendix. They yielded an expression, the net rate of citrate production (or degradation), that could be expressed in the following form:
where 6 is the rate of net citrate production in mol d–1, a (mol g–1 d–1) and d (g mol–1 d–1) are coefficients, b and c (°K–1) coefficients of response to temperature. The temporal step considered was 1 d.
A simple empirical interpretation of this expression is the following. The rate of citrate synthesis is defined as the product of two terms: a ‘synthesis potential’, function of ‘mitochondrial equipment’ and temperature, and an ‘efficiency level’, function of temperature and of the ratio respiration/‘mitochondrial equipment’.
Total citrate content of the fruit was obtained by integrating 6 over the monitored period, starting with citrate content observed at the beginning of this period. Citrate concentration in the fruit was obtained by dividing the total amount of citrate by the mesocarp fresh weight:
where [Cit] (in mol kg–1) is the citrate concentration in the mesocarp, FW the mesocarp fresh weight (g), t the date (in days after bloom), t0 the date of the beginning of the experiment, and 6 the rate of net citrate synthesis (mol d–1).
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Field experiment and fruit analysis
The data needed to parameterize and test the model were obtained from a field experiment in which fruit development and evolution of composition were monitored during the 1995 and 1996 growing seasons. The cultivar ‘Fidelia’ was planted in an orchard of the ‘Costières de Nîmes’ region (southern France) in 1992 and received common commercial practices. Temperatures were measured in the shade in a nearby orchard. Average temperature T was computed as the mean of minimum and maximum daily temperatures. Degree-days (dd °D) were accumulated from full-bloom for each year of the study, with upper and lower temperature threshold at 308 and 280 °K. The 1995 and 1996 seasons were generally characterized by contrasted climatic conditions: during the first half of the period studied (mid-May to mid-July), temperatures were higher in 1996 than in 1995, whereas during the second half, temperatures were higher in 1995 (Fig. 2).
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Fruit development was monitored by harvesting and analysing fruit samples weekly between the date of stone hardening and maturity. The fruit were harvested on 96 trees in 1995, 48 trees in 1996. Two fruits per tree were picked at each date, and samples were made by pooling fruits from four trees in 1995 (i.e. 24 samples of eight fruits each), two trees in 1996 (i.e. 24 samples of four fruits each).
At each harvest the fresh weight and dry matter content of fruit flesh and stones were measured after peeling and stoning. Each sample of pulp was frozen in liquid N and pulverized. Citrate was extracted by diluting 10 g pulp in 50 ml distilled water, homogenizing the mixture (Polytron PTA 10–15), and centrifuging it at 5000 rpm for 20 min. After filtration, the supernatant was frozen and stored at –20 °C pending analysis. Analysis of citrate was performed with enzymatic kits (Boehringer, no. 139068) and automated (BM Hitachi 704).
Model inputs
In order to provide the entries of the model, daily estimates of mesocarp fresh weight and respiration were needed. Fruit growth being measured weekly, a model was used to interpolate these variables between the dates of the observations.
Fresh weight growth of the mesocarp after pit hardening was fitted, considering two growth phases (i.e. before and after pit hardening). Time being expressed in °D after bloom, the corresponding equations was as follows:
where FW1 (g) is the fresh mesocarp weight at the time of pit hardening, FW2 (g) the maximum fresh mesocarp weight produced after pit hardening, RGR2 (°D–1) the maximum relative growth rate during mesocarp growth, dd2 (°D after bloom) the date when maximum growth rate is reached, and dd the date in degree-days.
The daily mesocarp dry weights were estimated also from the weekly measurements on the basis of estimated parameters FW1, FW2, RGR2, and dd2 following the equations:
where DM1 (g g–1) is the dry matter content at the date of pit hardening, DM2 (g g–1) the dry matter content when the maximum mesocarp fresh weight is attained. The mesocarp dry weight (DW1) in the citrate accumulation model (equation 9) was taken from FW1xDM1.
The respiration of fruit mesocarp was estimated as a function of mesocarp dry weight, rate of dry weight growth, and temperature. The model used for this purpose was adapted from DeJong and Goudriaan (1989). Fruit respiration (mol CO2 fruit–1 d–1) was the sum of two components, maintenance respiration Respm and growth respiration Respg:
where DW is mesocarp dry weight (g), qm (mol CO2 g–1 d–1) is the maintenance respiration coefficient at 293 °K, qg (mol CO2 g–1) is the growth respiration coefficient, Q10 is the temperature ratio of maintenance respiration (dimensionless), and T (°K) is the temperature. Though the model was developed initially to describe the respiration of the whole fruit, it was applied without modification to estimate the respiration of the mesocarp alone. qm, qg and Q10 values were taken from DeJong et al. (1987) and DeJong and Goudriaan (1989): qm = 5.43 10–5 mol CO2 g–1 d–1, qg = 7.023 10–3 mol CO2 g–1 and Q10 = 1.96.
Model parameterization, goodness of fit and sensitivity analysis
The model program solving and parameterization was performed using S-Plus (MathSoft Inc., 1999). The differential equations were integrated numerically using the first order Runge Kutta method with a 1 d integration step. During each season, fresh and dry matter growth curves were fitted by minimizing the sum of squares of errors using the Nonlinear Least Squares Regression function (MathSoft Inc., 1999). The citrate accumulation model was fitted to the data from the 1995 and 1996 seasons, by minimizing the sum of squares of errors for both simultaneously.
Goodness-of-fit criteria were computed to evaluate the predictive quality of the model. The adopted criterion was the root mean squared error (RMSE), a common criterion to quantify the mean difference between simulation and measurement (Kobayashi and Us Salam, 2000), here defined as:
with ni being data at each date ti, N being dates, xi being the simulation data and yi being the mean of observed data at date ti.
The smaller the RMSE in comparison to measurements, the better the goodness-of-fit. This idea can be represented through the relative RMSE:
A sensitivity analysis was performed to identify the most influential parameters on the model response. The sensitivity of the responses to changes in parameter values was quantified by the normalized sensitivity coefficients, defined as the ratio between the relative variations of citrate concentration and the relative variation of parameters. These coefficients were estimated by computing the responses of citrate concentrations to variations of ±10% on the model parameters.
The model was then used for simulating effects of changes in growth curves, by changing the corresponding parameters. Effects of variations of temperature were also simulated by increasing or decreasing temperature throughout fruit development.
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Results |
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Parameterization
The growth model fitted well both fresh weight and dry weight growth curves (Fig. 3). The standard errors of the estimated parameters were small (Table 1). The RMSE values were confidently acceptable compared with the mean measured mesocarp fresh and dry weights in each year (RRMSE = 0.03 to 0.07).
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Although standard errors of the parameters for the citrate accumulation model were rather high (Table 2), and a high correlation was found between parameters a and c (Table 3), simulated citrate concentrations matched well the experimental results (Fig. 4) with RRMSE = 0.22 in each year.
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Sensitivity analysis
The responses of the model to variations of parameters a, b, c, and d, were computed both in 1995 and 1996. The sensitivity coefficients were different between the two seasons studied, and varied during the season (Fig. 5). Coefficients a and b intervene in the definition of the citrate ‘synthesis potential’, and reflect its dependence on the initial ‘mitochodrial equipment’ and temperature respectively. Sensitivity to a increased and then was close to be constant, which means that for a given ‘mitochondrial equipment’ and temperature, variations in citrate accumulation were nearly proportional to variation of citrate ‘synthesis potential’. Sensitivity to b increased from 0 to 1.7 during the season in 1995 while in 1996 it remained high (between 2.4 and 3.5) throughout the season, which reflected the strong dependence of the ‘synthesis potential’ on temperature. The ‘efficiency level’ of the mitochondria depends on temperature and on the respiration/‘mitochondrial equipment’ ratio, through parameters c and d, respectively. Sensitivity to c was close to unity near maturity and similar between the two seasons studied, which suggests that c was not directly affected by annual temperature changes. Sensitivity to d was low in the beginning of the season and increased near maturity to reach values about 0.5 in 1995, 0.1 in 1996.
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Simulations: effects of growth and temperature
Simulations of different growth conditions were performed by changing the parameters of the growth model (Fig. 6). The sensitivity coefficients to FW1 and DM1 were positive, which means that increasing mesocarp fresh and dry weight favoured citrate accumulation. By contrast, increasing mesocarp growth (FW2 and DM2) decreased citrate accumulation. The evolutions of the absolute values of sensitivity coefficients to FW1, and FW2 were very similar, with a continuous increase from 0 to about 1.5 during the period studied. This observation is not surprising for FW1 and DW1, which play a major role in the definition of the ‘mitochondrial equipment’. The sensitivity to FW2 can be explained by its impact on respiration (increasing FW2 and DW2 increased respiration and reduced the ‘efficiency level’ of the mitochondria). Sensitivity coefficients to RGR2 were first positive and then negative near maturity but remained small. Delaying the period of pulp growth (i.e. increasing dd2) increased citrate concentration through diminishing citrate degradation. The effects of temperature were strong. Increasing temperature by 1 °C during the first part of mesocarp growth increased citrate accumulation by up to 26–32%. During the second part of mesocarp growth, increasing temperature by 1 °C decreased citrate accumulation by up to 45–50%. These apparently conflicting responses reflected opposite effects of temperature, which increased the citrate ‘synthesis potential’ while reducing the ‘efficiency level’ of the mitochondria.
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Discussion |
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Model development and simulations
Few models exist about the elaboration of fruit quality. Most of them deal with fruit growth (Génard and Huguet, 1996; Génard and Souty, 1996; Génard et al., 1996; Lescourret et al., 1998), or eventually combine a representation of carbon and water fluxes to estimate dry matter content (Fishman and Génard, 1998). Although statistical approaches have been applied to predict the kiwifruit quality at harvest (Gonzalez et al., 1995), only the model of sugar metabolism developed by Génard and Souty (1996) and Génard et al. (2003) provides a mechanistic representation of the elaboration of biochemical composition. In the case of acidity, the task is made more difficult not only by the fact that several acids have to be taken into account, but also by the lack of knowledge about the mechanisms involved in their metabolism and storage. It appears difficult to identify single steps controlling acid accumulation. For example, studies focusing on tonoplastic transport system (Canel et al., 1995) or mitochondrial citrate synthase (Canel et al., 1996) failed to explain the differences in citrate accumulation between high- and low- acidity citrus cultivars. Similarly, Moing et al. (1999) found little relation between PEP-carboxylase activity and acid accumulation in peach.
The approach presented here was based on an analysis of the mitochondrial pathways involved in citrate metabolism, represented as a series of enzymatically regulated steps. Its formalism was derived from that used in MCA theory (Fell, 1997): instead of representing individual complex kinetics and regulatory behaviour for the enzymes involved, relative variations of the fluxes around a ‘reference state’ were represented as linear combinations of the regulating factors (temperature, energy state, and concentrations of metabolites). The system was further simplified by assuming that groups of successive reactions could be modelled as a single reaction, following a ‘top-down’ approach (Kell and Mendes, 2000). The mathematical resolution of this system led to a simplified expression, in which citrate metabolism was directly related to input variables (mesocarp growth curve, respiration and temperature) estimated at the whole fruit level.
Simulations of citrate concentrations showed an agreement between predicted and observed data, while the need to estimate the model parameter on the data from two seasons prevented it being validated on an independent data set. Simulations made with the model confirmed the relations between assimilate supply, fruit growth and citrate concentration observed in several studies (Chapman and Horvat, 1990; Liverani and Cangini, 1991; Wu et al., 2002). Citrate concentration is usually correlated positively with fruit weight at the beginning of fruit development and negatively near fruit maturity (Génard et al., 1994, 1999).
Limits of the model
Considerable simplifications and approximations were necessary to build the model: analysing the theoretical behaviour of the complete system makes it possible to identify them better and discuss their relevance.
Only mitochondrial functioning was considered in the model: other pathways that may interfere with the citrate cycle were not taken into account. Among these pathways is the use of metabolic intermediates of the citrate cycle to provide skeletons for various syntheses (Popova and Pinheiro de Carvalho, 1998). By removing metabolic intermediates from the citrate cycle, this use may depress citrate formation during periods of active biosynthesis (e.g. in the first stage of fruit growth). Another omission was the glyoxylic cycle, in which isocitrate is degraded into succinate and glyoxylate, further reduced into malate (Douce, 1985). Though this is an additional pathway for citrate degradation, it may also favour citrate production, by allowing the replenishment of the pool of metabolic intermediates without supply of exogenous dicarboxylates. However, the glyoxylic cycle being involved essentially in the metabolism of fatty acids, its importance in fleshy fruit mesocarp is probably small.
Another limit of the model is that the fluctuations of the concentrations of cytosolic metabolites (in particular pyruvate, malate and citrate) and their effects on mitochondrial functioning were not taken into account. Though pyruvate supply by the glycolysis is highly regulated, large changes in its concentration have been observed when respiration is altered (Ulrich, 1970). Malate metabolism being involved in the regulation of cytosolic pH (Davies, 1973), its concentration may fluctuate when acids or bases are supplied by the saps, synthesized in the cytosol or stored in the vacuole. Finally, little is known about citrate concentration in the cytosol, but its repartition between cytosol and vacuole is likely to vary when tonoplastic transports are affected by changes in temperature and vacuolar pH. These omissions probably resulted in biases in the estimation of citrate metabolism.
The mathematical representation of the pathways was simplified by linearizing the relationships between reaction rates and regulating variables. Strictly speaking, this approximation is valid only in the case of small relative variations. And yet, variations of the input variables from the ‘reference state’ can be high, especially for temperature and respiration. This approximation was adopted in the model because of the mathematical difficulties in solving non-linear systems of equations. Without affecting the qualitative behaviour of the model, it is likely to result in biases when input variables are too far from those in the ‘reference state’, either at the beginning of the season (when temperature and respiration are low) and at the end (when they are high).
A more serious question was that of estimating the ‘mitochondrial equipment’. Little information is available about the activities of mitochondrial enzymes and their evolution during fruit development, and it is less than certain that they remain constant during development. For example, Terrier (1998) finds that succinate dehydrogenase activity increases by two to three times during the second growth stage. An increase in mitochondrial equipment would result in an increase in the citrate ‘synthesis potential’. Relative variations of mitochondrial enzyme activities may also bias the balance between citrate synthesis and degradation in the mitochondria, causing variations in the ‘efficiency level’. In both cases, responses to variations of the activities of mitochondrial enzymes would be attributed to temperature and respiration, and the high sensitivity coefficients of the model with respect to these variables may be a consequence of this bias.
To end with, the model presented here described citrate metabolism only during the second stage of fruit growth. Initial citrate concentrations at the beginning of the modelled period had to be provided as inputs. Further understanding of the events that occur in the early stages of fruit development would be needed. In particular, the multiplication of cells and the establishment of their metabolic machinery are under the control of genetic and environmental factors that remain to be studied and quantified. Also, active biosyntheses occur in the first growth stage, and their interactions with the mitochondrial functioning are probably not negligible.
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Conclusion |
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The model proposed here represents a first step towards integrating the physiological knowledge available at the cellular level to represent the elaboration of acidity at the whole fruit level. In spite of many unknowns and approximations, the approach adopted provides a framework in which future advances may be integrated. Further experiments about the activities of mitochondrial enzymes may provide data to estimate parameters. Feedback on citrate metabolism by vacuolar storage may also be incorporated in the model as new data become available to quantify the activities of tonoplastic transport systems.
The pattern of citrate accumulation described in peaches is common to other fruit species, such as tomato for example (Knee and Finger, 1992). So are the pathways involved in its metabolism. Therefore, the present model is likely to be applicable to a variety of species with only minor modifications. In a more immediate future, however, it still needs to be validated by experiments with different peach cultivars and growing conditions. Also, it has to be combined with models for the other organic acids, so that the acid-base composition of the fruit can be known and acidity predicted (Lobit et al., 2002). Only then will it be possible to compute the main acidity variables. Further steps will be to combine these models for acidity with the existing ones for sugar content, to provide a basis for a more accurate approach to fruit quality.
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Acknowledgements |
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The authors thank M Souty and L Gomez for the laboratory facilities and M Reich and M Bonafous for fruit analyses.
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Appendices |
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The analysis of the pathways involved in citrate metabolism (see governing equations) lead to the definition of a system of interrelated equation, as follows:
Stoichiometry equations:
3+4–1 = 0(1a)
2+5–1–3 = 0(1b)
1–6–2 = 0(1c)
Resp = 1+22+3(1d)
Enzymatic reactions:
1 = a1f1(Pyr, Mal, Cit, Reg,T)(2a)
2 = a2f2(Mal, Cit, Reg,T)(2b)
3 = a3f3(Pyr, Mal, Reg,T)(2c)
Transport reactions:
4 = a4f4(T) (PyrCyt–Pyr)(3a)
5 = a5f5(T) (MalCyt–Mal)(3b)
6 = a6f6(T)(CitCyt–Cit)(3c)
Values of enzyme and transport activities:
for 1i6 ai = ki DW1 (4)
Variations of T (temperature) and Resp (respiration) with climate and fruit growth imply that all variables and fluxes of mitochondrial functioning have to adjust simultaneously, which in turns determines 6 (citrate production or degradation). The technique adopted to calculate these variations is derived from the theory of MCA (Fell, 1997). It consists in computing the variations of fluxes when the conditions of mitochondrial functioning (i.e. ‘regulation state’, temperature, and metabolite concentrations) shift from a ‘reference state’, defined here by a set of conditions where the citrate net production is null (i.e. (60 = 0)). These conditions are: an arbitrary temperature (T0 = 298 °K), a set of values of metabolite concentrations in the mitochondria Pyr0, Mal0, Cit0, a ‘regulation state’ Reg0, and a rate of respiration Resp0 = k0DW1 for which (60 = 0). Furthermore, the relative variation of each flux is approximated by a linear combination of the relative variations of the variables that control it.
For each enzymatic reaction modelled, the regulating variables taken into account are the concentrations of the substrates and products, the temperature T, and the ‘regulation state’ Reg (equations 2a–c). Defining the variations of the variables 
Pyr = Pyr–Pyr0, Mal = Mal–Mal0, Cit = Cit–Cit0, T = T–T0, and Reg = Reg–Reg0, and of the fluxes: 1 = 1–(10), 2 = 2– (20), 2 = 2– (30), the variations of fluxes can be written:
where (40) 1i3 (mol d–1) and 
i 1i3 (dimensionless) are coefficients.
Concerning transports, defined by equations 3a, b, c, the variations of fluxes are calculated when conditions shift from the ‘reference state’, defined as:
40 = a4(PyrCyt – Pyr0)(8a)
50 = a5(MalCyt – Mal0)(8b)
60 = a6(CitCyt – Cit0) = 0(8c)
In the range of temperature considered, the dependence of transports on temperature can be approximated by a linear relation:
where (iT 4i6) (dimensionless) are coefficients.
When combining equations 3a, b, c, 8a, b, c and 9a, b, c, simplifying them and dropping terms of second order (i.e. terms that include products of variations), variations of fluxes can be expressed:
Since by definition at the ‘reference state’ 6 = 6,
The variation of respiration relative to the reference state can be written:
Resp=Resp–K0DW1(11)
where Resp is the respiration of the fruit mesocarp, estimated empirically from the fruit growth curve and temperature as described in ‘Materials and methods’.
Finally, since all enzyme and transport activities are assumed to be proportional to DW1, this is also the case for the fluxes in the reference state:
For 1 i 5 i0 = KiDW1)(12)
Stoichiometry equations 1a, b, c, d being valid in any conditions, they can also be written for variations of fluxes. Combining them with equations 4, 7, 10, 11, and 12 provides a set of four equations with four unknowns (Pyr, Mal, Cit, Reg), 30 constant parameters (1Pyr, 3Pyr, 1Mal, 2Mal, 3Mal, 1Cit, 2Cit, 1Reg, 2Reg, 3Reg, 1T, 2T, 3T, 4T, 5T, 6T, k4, k5, k6, K0, K1, K2, K3, K4, K5, Pyr0, Mal0, Cit0, Reg0, T0) and three input variables (DW1, Resp and T).
This system can be solved to compute Cit and to estimate the rate of citrate synthesis or degradation 6 as a function of the input variables DW1, Resp and T. The algebraic resolution was made using Maple software (Char et al., 1992). The expression for 6 was a polynomial function of Resp, DW1, and T–T0, too complex (one page formula) to be developed here. However, by grouping all constant terms together, it could be rewritten in the shape of equation 4:
where 6 is the rate of net citrate production in mol d–1, a (mol g–1 d–1) and d (g mol–1 d–1) are coefficients, b and c (°K–1) coefficients of response to temperature.
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