RESEARCH PAPER |
Modelling malic acid accumulation in fruits: relationships with organic acids, potassium, and temperature
1Ctifl, Centre de Balandran, 30127 Bellegarde, France
2INRA, Domaine St Paul, Agroparc, 84914 Avignon Cedex 9, France
* To whom correspondence should be addressed. E-mail: philippelobit{at}yahoo.fr
Received 5 December 2005; Accepted 18 January 2006
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Abstract |
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Malic acid production, degradation, and storage during fruit development have been modelled. The model assumes that malic acid content is determined essentially by the conditions of its storage in the mesocarp cells, and provides a simplified representation of the mechanisms involved in the accumulation of malate in the vacuole and their regulation by thermodynamic constraints. Solving the corresponding system of equations made it possible to predict the malic acid content of the fruit as a function of organic acids, potassium concentration, and temperature. The model was applied to peach fruit, and parameters were estimated from the data of fruit development monitored over 2 years. The predictions were in good agreement with experimental data. Simulations were performed to analyse the behaviour of the model in response to variations in composition and temperature.
Key words: Fruit, malate, vacuole, modelling
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Introduction |
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TopAbstractIntroductionModel developmentMaterials and methodsResultsDiscussionConclusionReferences |
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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, citric acid masks the perception of sucrose (Schifferstein and Fritjers, 1990; Bonnans and Noble, 1993) and fructose (Pangborn, 1963), while malic acid seems to enhance sucrose perception (Fabian and Blum, 1943). Two organic acids, citric and malic, are dominant in most fruit species, and understanding the elaboration of acidity requires studying the mechanisms involved in the accumulation of both. This paper, which focuses on malic acid, is part of a larger study about modelling the processes involved in the elaboration of fruit acidity (Lobit et al., 2002, 2003).
Data about the evolution of organic acid concentrations during fruit development have been published by Ishida et al. (1971), Chapman et al. (1991), Chapman and Horvat (1990), Souty et al. (1999), and Lobit (1999), among others. By contrast to quinic and citric acids, which follow similar patterns among all cultivars studied, the evolution of malic acid concentrations appears to be different in each cultivar and, for a given cultivar, to follow apparently inconsistent patterns, with rapid changes of concentrations during development. These observations suggest a complex determinism, probably under the influence of external factors like temperature. Various agronomic studies have shown that a wide range of factors affect malate concentration. In peach, Génard et al. (1991) and Génard and Bruchou (1993) have shown a positive correlation between fruit growth and sucrose content and malate content. Analysing data published by Genevois and Peynaud (1947a, b) and Souty et al. (1967), who compared the chemical composition of various peach cultivars, one finds a positive relationship between ash alkalinity (closely related to potassium content) and malate content (Lobit, 1999). Also in peach, Cummings and Reeves (1971) find that titratable acidity increases with potassium fertilization, which suggests an increased accumulation of organic acids, probably malic acid, in the fruit.
The purpose of the present work was to propose a model for malate accumulation that integrates the known physiological mechanisms involved, and to account for the observed responses to external factors such as temperature and mineral nutrition. The modelling approach followed was aimed at representing the physiology of the fruit mesocarp cells in a mechanistic manner. The hypothesis put forward was that malate accumulation in the fruit was mostly determined by the thermodynamic conditions of its transport from the cytosol to the vacuole of the fruit mesocarp cells. This led to the development of a model describing the interactions between acid–base reactions in the vacuole, proton transport across the tonoplast, and malate accumulation, with parameters describing essentially the functioning of the tonoplastic proton pumps.
Once the model was established, its parameters were estimated and a sensitivity analysis was performed to identify the most influential ones. The model was then used to simulate responses to changes in temperature, potassium, and organic acid accumulation on malate levels.
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Model development |
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The system studied and its representation
Like most of the organic acids in fruit, malic acid is accumulated essentially in the vacuoles of mesocarp cells that contain >85–90% of the total malic acid content (Yamaki, 1984
). Malic acid can be synthesized in the fruit itself or supplied to the fruit by the phloem and xylem saps. However, the pH of the saps is above 5–6 in the xylem and above 7 in the phloem (Peterlunger et al., 1990: Loewenstein and Pallardy, 1998), which is higher than the pK of the weakest acidity of malic acid (Stecher et al., 1960: pKa=5.2), so that most of it is transported in the form of its conjugated base together with cations like K+ (Peterlunger et al., 1990). Therefore, the acid form has to be synthesized within the fruit itself. Malate metabolism involves several pathways, distributed between the cytosol and the mitochondria of the cells. Synthesis takes place in the cytosol, through the carboxylation of the phospho-enol-pyruvate provided by glycolysis by PEP-carboxylase and further reduction into malate. Degradation takes place both in the cytosol, by oxidation into pyruvate via malic enzyme, and in the mitochondria, where malate is a substrate for the citrate cycle (Anon., 2003). It seems unlikely, however, that malate accumulation is determined by the activity of these pathways. Both the PEP-carboxylase and malic enzyme are regulated by pH in a way that contributes to stabilize pH and malate concentration (Davies, 1973; Garcia, 1975; Lakso and Kliewer, 1975; Drouet and Hartmann, 1977; Possner et al., 1981). Furthermore, Moing et al. (1998) did not find any relationship between malate content and the activities of PEP-carboxylase or malic enzyme (Moing et al., 1998) among peach cultivars of different acidity. For these reasons, it was assumed that malate accumulation in the fruit would be determined not by changes in its metabolism, but by the conditions of its transport into the vacuoles of the mesocarp cells.
Malate concentration in the cytosol is not known with much accuracy. Measurements indicated concentrations around 5–6 mM in maize root tips (Chang and Roberts, 1989) and between 1 mM and 2.5 mM in leaves (Gerhardt and Heldt, 1984). Also, malate concentration is expected to be somewhere between the KM of the malic enzyme—around 0.45–0.5 mM according to Lakso and Kliewer (1975) and Franke and Adams (1992)—and the inhibition constant of the PEP-carboxylase—reported to be around 35 mM by Lakso and Kliewer (1975), but as low as 0.1 mM according to Moing et al. (2000). In this wide range, a possible value for malate concentration in the cytosol seems to be anywhere between 0.1 mM and 6 mM. By contrast, vacuolar concentrations of malate measured in fruit mesocarp can be up to 50 mM. In these conditions, malate transport into the vacuole implies an expense of energy.
Mitchell's chemio-osmotic theory (Mitchell, 1967) describes the mechanisms of energy translocation between chemical reactions and the creation or dissipation of ion gradients across membranes. In thermodynamic terms, any transport is characterized by the variation of free energy of the system considered (ions transported for secondary transports, or ions plus components of the chemical reaction for primary transports). The transport can occur only if the variation in free energy is negative.
The transport of malic acid into the vacuole is passive, and occurs by the diffusion of the di-anion form through a specific ion channel (Lüttge and Ball, 1979; Rentsch and Martinioa, 1991; Ratajczac et al., 1994; Barkla and Pantoja, 1996; Emmerlich et al., 2003; Hafke et al., 2003). It follows the electrochemical potential gradient of the di-anion across the tonoplast, defined as follows:
 | (1) |
and 
are the activities of the malate di-anion in the cytosol and in the vacuole, respectively, T the temperature (°K), R is the gas constant (J mol–1 K–1), and F is Faraday's constant (C mol–1).
The activity of the di-anion is proportional to its activity coefficient 
Mal2– and its concentration [Mal2–]: Mal2–=Mal2– [Mal2–]. Since the malic acid is a weak acid, the di-anion concentration in a solution (vacuole or cytosol) is related to the total malic acid concentration [Mal] by the dissociation equation:
 | (2) |
and 
are the apparent acidity constants of the malic acid.
The activity coefficient of the di-anion, Mal2–, as well as those of other ions, depends on the ionic composition of the solution, including the concentration of the ionized forms of the acids. Therefore, solving the acid–base equilibrium between all acids and bases in solution is required to calculate malate activity and pH simultaneously. Details about this procedure are given in another article (Lobit et al., 2002). However, within the conditions expected in fruit tissues, the activity coefficients and acidity constants vary within a range of only a few per cent (data not shown); the main variable that determines malate dissociation is pH.
Both the pH gradient (due to the accumulation of hydrogen ions) and the electric potential gradient (positive inside the vacuole) are generated by the active transport of protons catalysed by tonoplastic proton pumps (Davies, 1997). Two types of pumps, ATPase and PPiase, exist on the tonoplast of plant cells (Rea and Sanders, 1987; Maeshima, 2000). They catalyse the coupled reactions:
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), pear (Shiratake et al., 1997), and grape (Terrier, 1997). However, in order to simplify the calculations and for theoretical reasons discussed below, only ATPase was taken into account in the present model.
In thermodynamic terms, proton transport can occur as long as the variation of free energy associated with the coupled ATP hydrolysis/proton transport reaction, 
GATPase is positive. GATPase is defined as:
 | (3) |
GATP is the reference free energy of ATP hydrolysis, and n the stoichiometry of the reaction.
The stoichiometry of the ATPase varies with pH; Terrier (1997) found 3H+ transported per ATP hydrolysed in grape in the absence of an pH gradient, but most authors find a stoichiometry of about 2 (Bennet and Spanswick, 1984; Guern et al., 1989; Schmidt and Briskin, 1993) in the presence of a pH gradient. By studying the reversal potential of the ATPase, Davies et al. (1994) showed that the number of H+ transported per ATP hydrolysed varied between 1.75 and 3.28 depending on both cytosolic and vacuolar pH. It was found that an accurate representation of their data (correlation coefficient between stoichiometry measured and predicted: R2=0.9994) was achieved by the following equation:
 | (4a) |
=0.29, and ß=–0.12 are fitted parameters.
Interestingly, Kettner et al. (2003), by applying the same electrophysiology technique on yeast vacuoles, found almost the same relationship between cytosolic pH, vacuolar pH, and coupling ratio of the ATPase. They suggested the following equation to describe both their data and those of Davies et al. (1994):
 | (4b) |
However, it was found that the equation (equation 4a) fitted on the data by Davies et al. (1994) describes Kettner's data (except for one point at pHCyt=8.5) even better than their own (data not shown). Therefore, equation 4a will be the one used to calculate the stoichiometry of the ATPase in the present model.
The approach adopted in the present work is to model the evolution of vacuolar composition as a succession of stationary states during which malate concentration, pH, and electric potential can be considered constant. The choice of representing a succession of stationary states rather than fluxes is based on the hypothesis that malate di-anion and H+ transport operate in conditions close to their respective thermodynamic equilibria. Following these hypotheses, computing malic acid concentration in the vacuole can be reduced to solving a set of three equations representing the thermodynamic equilibrium: (i) of the malate di-anion across the tonoplast; (ii) of the acid–base reactions; (iii) of the ATPase. It will be shown that this system can be summarized in a system of two equations with two unknowns that can be solved to calculate pH and malate concentration simultaneously.
Model equations
The hypothesis adopted in the representation of malic acid transport is to consider that the malate di-anion is permanently at a state of thermodynamic equilibrium across the tonoplast. In this situation, the electrochemical potential gradient of the malate di-anion is equal to zero. With 
rewriting and combining equations 1 and 2 gives:
 | (5) |
This equation can be interpreted as a malate partitioning sub-model. Its input variables are T, 
, and pH. 
and are intermediary variables that depend on both pH and 
and have to be estimated by the acid/base equilibrium procedure (however, within a physiological range of pH and concentrations they are approximately constant and affect malate concentration little). The only parameter to be estimated is 
Like for the representation of malate transport, the hypothesis that proton transport operates in a situation of thermodynamic equilibrium is adopted. With GATPase=0, rewriting and combining equations 3 and 4 gives:
 | (6) |
GATP, n0, , ß, and pHCyt. Its inputs are T and pHVac (which is an output of the acid–base reactions model).
The acid–base composition of the vacuole determines 
and pHVac, all required inputs for the sub-models of malate transport and proton pump functioning. Their calculation was detailed and validated in a previous paper (Lobit et al., 2002). It was shown that the calculation could be simplified by taking into account only the concentrations of the three main organic acids: malate, citrate ([Cit]), and quinate ([Qui]), and potassium concentration [K+] as representative of the cations.
These relationships can be summarized by the function:
 | (7) |
 | (8) |
as unknowns: equation 5 describes the behaviour of the malate transport system, equation 6 that of the proton pumps, and equations 7 and 8 describe the acid–base relationships.
The input variables required to solve this system are T, [K+], [Qui], and [Cit]. The parameters are pHCyt, n0, , ß, 
and 
The numerical solution can be obtained by iteration, as follows. An initial value is given for 
then equations 7 and 8 are applied to determine pH, 
and 
From these values, is calculated from equation 6, and a new value for 
is calculated from equation 5. This value is then used as a starting point for a new iteration, until stabilizes (<1% difference between two iterations).
Validity of model hypotheses
The approach adopted in the present work is to model the evolution of vacuolar composition as a succession of stationary states during which malate concentration, pH, and electric potential can be considered constant. Furthermore, malate di-anion and H+ transport are assumed to operate in conditions close to their respective thermodynamic equilibria. An alternative approach would have been to represent all fluxes across the tonoplast, so as to calculate electric potential, pH, and malic acid content from their balance. However, this would have led to an unrealistic number of unknowns and parameters, since all transport systems would need to be taken into account.
It is obviously impossible to validate the hypothesis chosen without direct measurements of the biophysical variables involved. However, one can evaluate if they constitute reasonable hypotheses by verifying that a number of conditions are met. One condition is that various thermodynamic variables calculated under the assumption of the model fall within the range expected from data in the literature. Another condition is that the capacity of tonoplast transport systems, in particular the malate channel and the ATPase, is not saturated (otherwise these transports would be limited by kinetics considerations). In other words, the observed rate of malate accumulation must be lower than the potential rate of malate transport through the di-anion channel, and the observed rate of accumulation of acidity (i.e. of protons) must be lower than the potential rate of H+ transport through the ATPase.
Thermodynamic conditions of transport
The thermodynamic and kinetics conditions of malate transport have been investigated in electrophysiology studies. Ion transport generates an electric current: i=C(E–), where , E, and C are the electric potential gradient, electromotive force, and conductance. While C reflects the activity of the ion transport system, E is the electric potential gradient at which the thermodynamic equilibrium is reached (that is when i=0 and 
). It can be calculated as E= that solves equation 1 for 
The range of E values to be expected for the malate di-anion channel can be estimated from the values of pH and malate concentration measured in the fruit or published. As the vacuole represents nearly all the fruit volume, malic acid concentration [MalVac], pH, and ionic strength I are close to those in the whole fruit flesh (Lobit et al., 2002). Assuming 20 mM 
[MalVac] 60 mM, 3 pH 5, and 50 mM I 100 mM, the di-anion activity can be estimated to be 0.5 mM 
3 mM. In the cytosol, a reasonable range of malate di-anion activity can be estimated to be 0.2 mM 
1.5 mM corresponding to 0.5 mM 
3 mM (equal to malic acid concentration at neutral or slightly alkaline pH in the cytosol), and an activity coefficient of 0.4 
0.5 or I
150 mM. Based on these assumptions, one can calculate –75 mV E –30 mV at T=300 °K (27 °C). The higher range of these values is comparable with the expected tonoplastic potential gradient, which most authors estimate as –30 mV 0 mV. Therefore, the electric conditions are compatible with the partitioning of the malate di-anion across the tonoplast in a state of thermodynamic equilibrium.
The thermodynamic properties of proton transport can be studied in the same way. The electromotive force E of a pump is defined as the electric potential gradient at which the current through the pump equals zero. For ATPase, it can be calculated by rewriting equation 3 and solving it for GATPase=0. E depends only on the stoichiometry of the pump and on the free energy of hydrolysis of the substrate. The free energy of hydrolysis of ATP can be calculated as a function of its standard energy of hydrolysis and the activities of the products and substrates of the reactions. Assuming the standard energy of ATP hydrolysis as –36.8 kJ mol–1, an ATP/ADP ratio in the cytosol between 1 and 5, and the phosphate concentration between 1 and 5 mM, the computed GATP varies at most between –51 and –58 kJ mol–1 (calculation not shown). This range overlaps most published values of GATP (Rea and Sanders, 1987; Briskin and Reynolds-Niesman, 1991; Davies et al., 1993). In maize root tips, Roberts et al. (1985) measured the nucleotide phosphate concentrations by MNR, and computed that GATP varied between –64 kJ mol–1 when ATP utilization was inhibited, and –45 kJ mol–1 when its production was inhibited, with an average value of –56.3 kJ mol–1 in normal conditions. The stoichiometry of ATPase can be estimated either by the equation proposed here or by that proposed by Kettner et al. (2003) (equations 4a and 4b, respectively).
The calculation of the electromotive force is, in theory, similar for the PPiase. The free energy of PPi hydrolysis, estimated from the PPi concentrations measured by Roberts et al. (1985), is about –26 kJ mol–1 (calculation not shown), while Davies et al. (1993) estimate it as –27.3 kJ mol–1. Concerning the stoichiometry, however, things are complicated by the controversy about whether the PPiase transports K+ together with H+ and, if so, in which direction (Maeshima, 2000). Most authors agree that PPiase actively transports only H+ into the vacuole, with a constant stoichiometry of 1 H+ per PPi hydrolysed (Ros et al., 1995; Terrier, 1997; Maeshima, 2000). However, using electrophysiology techniques on whole vacuoles, Davies et al. (1992) found that the variation in reversal potential with cytosolic and vacuolar pH and K+ concentrations was consistent with the PPiase transporting 1.3 H+ and 1.7 K+ into the vacuole per PPi hydrolysed. On the contrary, Obermeyer et al. (1996) found that PPiase only transports K+ passively (from the vacuole to the cytosol) in the presence of PPi. Though in contradiction with the conclusions of Davies et al. (1992), these findings are not incompatible with their measurements of the reversal potential. The presence of a passive ‘leak’ of K+ (probably increasing with the gradient of K+ electrochemical potential) would generate a current opposite to the one generated by H+ transport, thereby decreasing the reversal potential of the PPiase. For these reasons, it is assumed that the electromotive force of PPiase varies as described by Davies et al. (1992).
Based on these considerations, theoretical calculations of the electromotive force of the proton pumps can be made under different assumptions concerning their stoichiometry for H+ and, in the case of the PPiase, for K+. For the simulations of the PPiase electromotive force, the K+ activities are assumed to be 60 mM on both sides of the tonoplast, consistent with the present measurements of K+ concentrations around 80 mM in the fruit pulp, and estimations of activity coefficient for mono-cations (data not shown), and with data in the literature for cytosolic activity (Cuin et al., 2003).
Two main conclusions appeared from these simulations, as shown in Fig. 1. Concerning ATPase, its electromotive force was highly dependent on the model chosen to represent its stoichiometry (even when both models described equally well the same dataset). In one case the electric potential gradient generated declined when vacuolar pH diminished; in the other case it stabilized or even increased in acidic vacuoles. Concerning the PPiase, its electromotive force depended on whether K+ transport (active or by diffusion) was taken into account or not. When no K+ transport was taken into account the electromotive force seemed excessively high. When it was taken into account, the electromotive force was very similar (or smaller at pH >4.5) to that of ATPase. Since it is believed that the measurements by Davies et al. (1992) do provide valid estimations of the reversal potential of the PPiase, it will be assumed that the PPiase does not generate a stronger electric potential than the ATPase and it will be ignored in the present model.
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Kinetic properties of transport systems
The kinetic properties of the malate di-anion channel have been studied extensively in laboratory experiments using a variety of techniques (Martinoia and Ratajczac, 1997
). It presents a Michaelian behaviour approximately relative to cytosolic concentration. The maximum rate of transport (Vmax) has been measured only in isolated vacuoles or tonoplastic vesicles, most of the time not in fruits, and differs widely between experiments. Martinoia and Ratajczac (1997) report measured activities (expressed on a protein mass basis), ranging from 0.5 nmol mg protein–1 min–1 in Catharanthus roseus to 1300 nmol mg protein–1 min–1 in barley; though most measurements are between 20 and 250 nmol mg protein–1 min–1 (Buser-Suter et al., 1982; Nishida and Tominaga, 1987; White and Smith, 1989; Ratajczac et al., 1994). Hafke et al. (2003) measured rates between 29 and 51 nmol m–2 s–1 on whole vacuoles. These values can be approximately converted into rates per tonoplast area, assuming the protein content of the tonoplast to be between 0.25 and 0.5 mg protein m–2 (Martinoia and Ratajczac, 1997), and per kilogram of tissue, assuming a tonoplast area between 50 and 100 m2 kg tissue–1 (recalculated from histological data published by Reeve, 1958; Yamaki, 1984; Scorza et al., 1991). This indicates a maximum rate of transport in the range 0.36–25 mmol malate d–1 kg fruit–1, with a median value around 3.5 mmol malate d–1 kg fruit–1. The KM relative to cytosolic malate depends on the technique used to measure it. Using electrophysiological techniques, Hafke et al. (2003) found a KM of 2.5 mM. Values for KM between 1 and 5 mM are found when influx is measured with isotopically marked malate (Buser-Suter et al., 1982; Martinoia et al., 1985; Nishida and Tominaga, 1987; Marigo et al., 1988; Ratajczac et al., 1994). However, when malate net flux is measured indirectly through associated H+ transport, values for KM between 14 and 19 mM are found (White and Smith, 1989; Ratajczac et al., 1994). As previously mentioned, little is known about cytosolic malate concentration. However, assuming it to be somewhere between 0.1 and 6 mM, it is likely (but not certain) that it is in the same range as the KM of the di-anion channel. The rate of malate accumulation measured during the development of the peaches studied was between 0 the 1.6 mmol malate d–1 kg fruit–1, depending on the stage of development and on the year. This is in the same range, or lower, than the rate of malate transport allowed with the median values of Vm, KM, and malate concentrations estimated above. Therefore, the assumption that the maximum activity of the malate transport system does not limit its storage seems reasonable, except maybe in the periods of faster malate accumulation, or if cytosolic concentration falls below its normal value.
Little is known of the maximum rates of proton transport allowed by the proton pumps. These rates have been measured in vitro on a variety of plant materials; the problems in interpreting these measurements are the same as for malate transport. However, a range between 0.2 and 50 mmol H+ kg–1 d–1 for the ATPase alone, between 1 and 150 mmol H+ kg–1 d–1 if PPiase is also active, can be estimated from the data of Rea et al. (1992), Milner et al. (1995), and Shiratake et al. (1997). On the other hand, the rate of titratable acidity accumulation in the vacuole is, by definition, the net rate of proton accumulation (i.e. pumping by ATPase or PPiase minus ‘leaks’ in the form of protons and protonated acids). The maximum rate of titratable acidity accumulation observed during the present experiments was 6 mmol H+ kg–1 d–1 (data not shown). Therefore, proton-pumping activities in the middle range of those mentioned above are 4–8 times higher than is required to explain the observed acidity accumulation; saturation of the pumps seems unlikely.
Also, substrate availability may cause the pumps to operate below their maximum rates. The KM of the ATPase for ATP is about 0.8 mM (Oleski et al., 1987), while ATP concentration in the cytosol can be estimated to be between 0.25 and 0.4 mM from NMR measurements (Roberts et al., 1985). The affinity constant of the PPiase for PPi is about 15–17 µM (Maeshima et al., 1996; Obermeyer et al., 1996), while Roberts (1990) estimates the PPi concentration to be about 10 µM. Therefore, substrate availability may limit proton pumping activity to about 50% below its maximum. The remaining activity would still be sufficient to explain acidity accumulation without the pumps reaching saturation.
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Materials and methods |
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Field experiment
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 orchard, planted in 1992 at Balandran (Costières de Nîmes, southern France), was conducted according to common commercial practices. The cultivar studied, ‘Fidelia’, is usually harvested around 15 July and produces fruit with a titratable acidity of about 40 meq kg–1.
The 1995 and 1996 seasons were characterized by different climatic conditions (Fig. 2); 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.
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Fruit analyses
Fruit development was monitored by harvesting and analysing fruit samples weekly between the date of stone hardening and maturity. The fruit were harvested on 48 trees in 1995 and 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. 12 samples of eight fruits each), and two trees in 1996 (i.e. 12 samples of four fruits each).
At each harvest date, 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 nitrogen and pulverized. Organic acids were extracted by diluting 10 g of pulp in 50 ml of distilled water, homogenizing the mixture (Polytron PTA 10–15), and centrifuging it at 3500 g for 20 min. After filtration, the supernatant was frozen and stored at –20 °C pending analysis. Analyses of malic and citric acids were performed with enzymatic kits (Boehringer, nos 139068 and 139076) and automated (BM Hitachi 704). Quinate was measured by HPLC (Waters, l 300 mm, 
6.5 mm, pump VARIAN 9010, detector VARIAN 9050 at 
=210 nm), but because these analyses were too time consuming, only a few samples were measured in 1995. As measurements were not available for 1996, a simple empirical model was used in which quinic acid content was modelled as a linear function of time, and concentration was estimated by dividing the amount per fruit by the fresh weight of the flesh. The potassium content was measured as follows: 5 g fresh pulp samples were calcinated at 500 °C for 24 h, diluted with 50 ml 0.3 N nitric acid, then measured by colorimetry using the nitro-vanado-molybdic reagent (auto-analyseur TDF, France). Concentrations in the fruit pulp were used as approximations of those in the vacuoles.
Model parameterization and sensitivity analysis
The model solving and parameterization was performed using the R software (Ihaka and Gentleman, 1996). None of the classical parameterization procedures could be applied to optimize parameter values, both because the model functions were not derivable and because too many iterations required too much computation time. Therefore, approximated parameter values were obtained as follows. For each of the five parameters to fit, 10 values were chosen in a range between a reasonable minimum and maximum expected from the literature. The quality of fit of the model was calculated for each possible combination (i.e. 100 000 cases), and the best set of parameters was retained. Then, the procedure was done again after narrowing the range to within ±20% around the values obtained in the previous step, and the best parameters were retained.
The sensitivity of the responses to changes in parameter values was quantified by the normalized sensitivity coefficients, defined as the ratio of the relative variations of malate concentration by the relative variation of the parameter. Sensitivity coefficients were calculated throughout the fruit development period in 1995 and 1996. The model was then used to simulate the effects of temperature and citrate, quinate, and K+ concentrations on malate accumulation.
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Results |
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Parameterization and sensitivity analysis
Parameter estimation: Though the model has seven parameters, they are not independent so only five of them need to be adjusted. It was chosen to set
as calculated by the Debye–Huckel relationship for I=100 mM, which is approximately the ionic strength expected in the cytosol. Furthermore, the Debye–Huckel relationship shows that the activity coefficient of a di-anion varies little around this ionic strength. Also, since cytosolic pH is assumed constant in the model, and dependence on cytosolic pH is not a factor included in the investigation, ß=–0.12 was kept, as determined by Davies et al. (1994) (Table 1).
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The estimated value for pHCyt is 7.2, which is consistent with the common notion of a neutral or slightly alkaline cytosol. Concerning the parameters describing the thermodynamics of the ATPase, the estimated value
GATP = –56.75 kJ mol–1 is very close to that found by Roberts et al. (1985), which was GATP = –56.3 kJ mol–1, and also falls within the range published by various authors (Briskin and Reynolds-Niesman, 1991; Davies et al., 1993; Rea and Sanders, 1987). The coefficients n0=4 and =0.3, that define the stoichiometry of the pump and its dependence on vacuolar pH, are also surprisingly close to those (4.06 and 0.29, respectively) found by Davies et al. (1994). By contrast, the value for 
mM may seem to be in the lower range of expected cytosolic concentrations. Simulated malic acid concentrations matched the experimental results fairly well, and the differences between the patterns of malate accumulation observed in 1995 and 1996 were correctly described (Fig. 3).
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Sensitivity analysis:
The responses of the model to variations of parameters Malcyt,
GATP, n0, ß, , and pHCyt were computed both in 1995 and in 1996 (Fig. 4). In spite of the differences in fruit composition, the sensitivity coefficients were almost identical during the two seasons studied. The sensitivity to MalCyt is positive (the more malate in the cytosol, the more in the vacuole), but declines during ripening (from around 10 to between 3 and 5). The parameters GATP and n0 determine the electric potential gradient that can be created by ATPase, when pH is neutral on both sides of the tonoplast (from equation 6, =GATP/n0F, when pHCyt=pHVac=7). With malate accumulation strongly dependent on , the sensitivity coefficients of malate concentration to both parameters are expected to have opposite signs and be approximately equal in absolute value, and this is what the model predicts. Also, their absolute values decrease from 20 and 30 at the beginning of the season, to between 10 and 20 at the end of the season. However, these sensitivities appear extremely high, since it means that a change of only 5% in GATP or n0 would cause a variation of up to 100–200% in malate concentration. This makes it even more noteworthy that the values optimized for GATP and n0 and are extremely close to those of Roberts et al. (1985) and Davies et al. (1994). The parameters and ß control the response of ATPase to pHVac and pHCyt. The sensitivity coefficient to is positive, and declines from around 10 to between 3 and 5 during ripening. This is as expected, since an increase in alleviates the inhibition of the ATPase by vacuolar acidity. The sensitivity to ß, by contrast, remains around 1 throughout the season (this is a justification a posteriori for choosing ß to be a fixed parameter during the optimization procedure). The sensitivity coefficient to pHCyt varies from around –12 to between –7 and –5 during ripening, which means that malate accumulation decreases when cytosolic pH increases.
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Simulations
Effects of pH and temperature:
The theoretical relationships between pH, temperature, and predicted malic acid concentration are determined by equations 5, 6, 7, and 8. In order to investigate the influence of these variables, these equations were solved for a range of pH values and temperatures three times during fruit development in 1995 (on 29 May, 19 June, and 10 July), corresponding to different fruit compositions (Fig. 5). Surprisingly, the relationship between pH and malate concentration appeared not to be monotonous: the minimum malate concentration was obtained around pH 3.8, and increased when the pH shifted from this value. This behaviour reflected the conflicting effects of pH on the proton pumps and on the dissociation of malate. On the one hand, decreasing pH increases the dissociation of malic acid, increases the di-anion concentration gradient, and stimulates accumulation; on the other hand, it inhibits the proton pumps, reduces the electric potential gradient, and reduces malate accumulation. By contrast, increasing temperature consistently reduced malate concentration (by about 50% when temperature increased from 15 to 25 °C), in particular at the beginning of fruit ripening (i.e. at high malate concentration).
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Effects of quinate, citrate, and K+ content on malate accumulation:
The impacts of changes in quinate, citrate, and potassium concentration were estimated by simulating variations of ±5% K+, citric acid, and quinic acid concentrations throughout the season (Fig. 6) and calculating the corresponding sensitivity coefficients.
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At all stages, increasing the concentration of quinic or citric acid reduced pH, whereas increasing K+ concentration increased it (data not shown). However, the resulting effect on malate depended on the stages of development. At early stages, increasing the quinic or citric acid concentration stimulated malate accumulation, while increasing the potassimum concentration reduced malate accumulation. By contrast, at maturity the effect of quinic or citric acid on malate accumulation was neutral, or slightly depressed, while there was a strong positive effect of potassium on malate accumulation.
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Discussion |
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TopAbstractIntroductionModel developmentMaterials and methodsResultsDiscussionConclusionReferences |
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Few models concerning the elaboration of fruit quality exist. Most of those that do deal with fruit growth (Génard, 1996
; Génard and Huguet, 1996; Génard et al., 1996; Ben Mimoun et al., 1999), or eventually combine a representation of carbon and water fluxes to estimate dry matter content (Fishman and Génard, 1998). As far as is known, the model of sugar metabolism developed by Génard and Souty (1996) was the first one to provide a mechanistic representation of the elaboration of biochemical composition at the whole fruit level.
Concerning acidity, the task is made more difficult by the fact that several acids have to be taken into account, and by the complexity of the mechanisms involved. Recently, a model of citrate metabolism in fruit was proposed, focusing on citrate production and degradation in the mitochondria (Lobit et al., 2003). Concerning malate, the lack of relationships between malate accumulation and the activities of the enzymes involved in its metabolism like PEP-carboxylase or malic enzyme (Moing et al., 1998) suggested that malate accumulation was not controlled by metabolism. As it is possible that malate accumulation is regulated by vacuolar storage alone, this was explored, resulting in the present approach.
Model simplifications
The most controversial point in the representation of the tonoplastic functioning is probably to consider it as a succession of thermodynamic equilibrium states. In practice, fluxes across the tonoplast happen continuously. Malic acid concentration is actually determined by the balance between the net influx of the di-anion and the efflux of the protonated form, while the electric potential gradient depends not only on the electromotive force of the proton pump, but also on the electric currents through all transport systems and their resistance. In this situation, the thermodynamic equilibrium is not reached and the electric potential gradient and malate concentration are lower than their theoretical maxima attained at thermodynamic equilibrium. As this was not taken into account, the difference is probably accounted for by biases in parameter estimation. With values for GATP and the stoichiometry of ATPase matching those found in literature remarkably well, most of the bias is likely to be in the low value of cytosolic malate concentration found.
In the present model, except for vacuolar composition, all factors likely to interfere with vacuolar functioning, cytosolic pH and malate concentration, and the free energy of ATP hydrolysis were assumed constant during fruit development. This is unlikely in practice; cytosolic malate concentration and/or pH fluctuate when acids or bases are supplied by the sap, synthesized in the cytosol, or stored in the vacuole, and the free energy of ATP hydrolysis is likely to vary during fruit development with respiratory demand or temperature. More important, the permeability of tonoplastic membranes to the protonated form of malic acid and various ions, which causes them to ‘leak’ out of the vacuole and reduces their accumulation, is known to increase during ripening (Terrier, 1999). However, the fact that the model matches correctly the pattern of evolution of malate concentration suggests that these variations are small enough to be neglected.
Model behaviour
An interesting outcome of the present model highlights the role of temperature in determining malate accumulation; a temperature increase from 15 °C to 25 °C alone causes a 50% decrease in malate accumulation. This is the range of temperature variation observed in the field during development between spring and summer. This effect alone explains much of the seasonal pattern of evolution of malate content, even in the absence of any change in physiological parameters of the fruit. This immediate sensitivity to temperature may also explain why malate concentration in the fruit may vary rapidly during fruit development, and in an apparently inconsistent way (Lobit, 1999), while other acids like citrate or quinate show more stable patterns. However, this effect is exaggerated in the model, since malate concentration fluctuates immediately with temperature, while in reality there are kinetic limitations to influx, efflux (by tonoplastic transport systems), and metabolism (by enzymes), that make it depend also on the history of the fruit.
The sensitivity analyses of the model are in agreement with some relationships commonly found between quality criteria in fruit. Potassium fertilization is known to increase the titratable acidity and malate content of fruits (Cummings and Reeves, 1971; Du Preez, 1985). Malate content at fruit maturity is usually correlated positively with ash alkalinity, which is closely related with potassium content (Genevois and Peynaud, 1947a, b; Souty et al., 1967). By contrast, the model predicts little or no negative correlation between malate and citrate content, as found by Génard et al. (1991, 1999). However, this correlation may be due to the fact that malate and citrate concentrations evolve in opposite directions during ripening, without implying a functional interaction between citrate and malate accumulation.
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Conclusion |
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TopAbstractIntroductionModel developmentMaterials and methodsResultsDiscussionConclusionReferences |
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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. The present approach is based on very restrictive simplifications concerning the regulation of physico-chemical variables in the cytosol and the conditions of functioning of the tonoplast transport systems. This approach is not incompatible with more mechanistic representations of vacuolar functioning. Malate partitioning and electric potential sub-models may easily be replaced by dynamic models of malate transport (taking into account both the di-anion transport and leaks of the protonated form) and electric functioning (taking into account all transport systems). Additional models could be developed to represent possible fluctuations of cytosolic malate, pH, and energy substrate concentrations. However, modelling a dynamic system would require too many parameters; in the current state of knowledge, the present model was the only one for which parameter estimation was possible.
The modelling approach made it possible to highlight the importance of regulating factors that are often neglected in physiology studies. This is the case, in particular, of the free energy of ATP hydrolysis and of the ATPase stoichiometry, which appear to play a big part in regulating malate accumulation. The model also made it possible to quantify the role of temperature, highlighting the important role it plays in determining the evolution of malate concentration during one season as well as inter-annual differences.
In spite of its degree of simplification, the model predicts correctly the contrasting patterns of malate accumulation in 1995 and 1996, and predicted malate concentrations are in good agreement with observed ones. The model responses to acid–base composition and temperature are in general agreement with those observed in agronomic experiments. Therefore, it seems adequate enough for prediction purposes, though further validation with different cultivars and growing conditions would be needed. The mechanisms of malate accumulation described in peaches are common to other fruit species, and the present model should be applicable to a wide range of conditions with only minor modifications.
In further developments, combining the malate accumulation model with those already existing for citrate (Lobit et al., 2003), acid–base relationships and titratable acidity provision (Lobit et al., 2002) will make it possible to predict all the acidity variables from fruit growth, temperature, and mineral composition alone. Further steps will be to combine these models for acidity with the existing ones for sugar content, to provide a basis for a more comprehensive approach to fruit quality.
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Acknowledgements |
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This work was part of a PhD thesis funded by the Ctifl (Centre Technique Interprofessionnel des Fruits et Légumes). The authors thank, in particular, Charles Romieu for the initial idea of the thermodynamics modelling and his valuable advice throughout this work, Maryse Reich and Monique Bonafous for fruit analyses, and Michel Souty and Laurent Gomez for providing laboratory facilities and valuable advice.
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Footnotes |
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Present address: Instituto de Investigaciones Agropecuarias y Forestales, Universidad Michoacana de San Nicolás de Hidalgo, Km 9.5 Carr. Morelia-Zinapécuaro, CP 58880, Tarímbaro, Michoacán, Mexico. 
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