Download Modelling malic acid accumulation in fruits: relationships

Journal of Experimental Botany, Vol. 57, No. 6, pp. 1471–1483, 2006
doi:10.1093/jxb/erj128
RESEARCH PAPER
Modelling malic acid accumulation in fruits: relationships
with organic acids, potassium, and temperature
Philippe Lobit1,2,*,†, Michel Genard2, Patrick Soing1 and Robert Habib2
1
Ctifl, Centre de Balandran, 30127 Bellegarde, France
2
INRA, Domaine St Paul, Agroparc, 84914 Avignon Cedex 9, France
Received 5 December 2005; Accepted 18 January 2006
Abstract
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.
Introduction
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.
* To whom correspondence should be addressed. E-mail: [email protected]
y
Present address: Instituto de Investigaciones Agropecuarias y Forestales, Universidad Michoacana de San Nicolás de Hidalgo, Km 9.5 Carr. MoreliaZinapécuaro, CP 58880, Tarı́mbaro, Michoacán, Mexico.
ª The Author [2006]. Published by Oxford University Press [on behalf of the Society for Experimental Biology]. All rights reserved.
For Permissions, please e-mail: [email protected]
1472 Lobit et al.
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.
Model development
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-enolpyruvate 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:
ÿ
Mal2
Vac
DGMal2 = 2FDW + RT ln ÿ
ð1Þ
2
MalCyt
ÿ
ÿ
2
where Mal2
Cyt and MalVac are the activities of the
malate di-anion in the cytosol and in the vacuole, respectively, T the temperature (8K), R is the gas constant
(J mol1 K1), and F is Faraday’s constant (C mol1).
The activity of the di-anion is proportional to its activity
coefficient aMal2 and its concentration [Mal2]:
Mal2=aMal2 [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:
½Mal = ½Mal2 ðK91 K92 Þ=ð1 + K91 h + K92 h2 Þ
ð2Þ
where ½Mal is the total concentrations of all forms of malic
acid, h=10pH, and K91 and K92 are the apparent acidity
constants of the malic acid.
Modelling malate accumulation in fruits
2
The activity coefficient of the di-anion, aMal , 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:
+
+
ATP + nHCyt
, ADP + Pi + nHVac
and
+
+
, 2Pi + nHVac
PPi + nHCyt
Both are probably active in most fruits, as found in tomato
(Milner et al., 1995), 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,
DGATPase is positive. DGATPase is defined as:
ð3Þ
DGATPase = DGATP + nFDW
nRT ln 10 pHVac pHCyt
where DGATP 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:
ðpHCyt 7Þ
n = n0 + aðpHVac 7Þ + b10
ð4aÞ
where n is the observed stoichiometry, pHVac and pHCyt
are the vacuolar and cytosolic pH, respectively, and
n=4.06, a=0.29, and b=–0.12 are fitted parameters.
1473
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):
n = 4:9 0:69 pHCyt pHVac
ð4bÞ
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
DGMal2 = 0; rewriting and combining equations 1 and 2
gives:
ÿ
ð2FDWÞ
1
K91 K92
2
RT
½MalVac = ÿ
ð5Þ
2
2 MalCyt e
a MalVac 1 + K91 h + K92 h
This equation can be interpreted as a malate partitioning
Its input variables are T, DW, and pH.
ÿ sub-model.
a Mal2
,
K9
,
variables that
1 and K9
Vac
ÿ2 are intermediary
,
and
have
to
be estimated
depend on both pH and Mal2
Vac
by the acid/base equilibrium procedure (however, within
a physiological range of pH and concentrations they are
approximately constant and affect malate concentration
little).
to be estimated is
ÿ
Theÿ only
parameter
2
2
2
MalCyt = a MalCyt MalCyt :
Like for the representation of malate transport, the
hypothesis that proton transport operates in a situation of
1474 Lobit et al.
thermodynamic equilibrium is adopted. With DGATPase=0,
rewriting and combining equations 3 and 4 gives:
DGATP
DW = n0 + aðpHVac 7Þ + b10ðpHCyt 7Þ F
ð6Þ
ÿ
RT
lnð10Þ pHVac pHCyt
F
The five parameters of this proton transport sub-model are
DGATP, n0, a, b, and pHCyt. Its inputs are T and pHVac
(which is an output of the acid–base reactions model).
composition of the vacuole determines
ÿ The acid–base
a Mal2
,
K9
,
K9
1
2 , and pHVac, all required inputs for the
Vac
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 ½K+ as representative of the cations.
These relationships can be summarized by the function:
+
pHVac = f ð½Mal; ½Cit; ½Qui; ½K + Þ
ÿÿ
a Mal2
; K91 ; K92 Þ = gð½Mal; ½Cit;
Vac
ð7Þ
ð8Þ
+
½Qui; ½K Þ constant
Solving the model means solving a set of four equations
with malate concentration, pH, and DW 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, a, b,
2
aðMal2
Cyt Þ; and ½MalCyt : The numerical solution can be
obtained by iteration, as follows. An initial value is given
for ½Mal2
Vac ; then equations 7 and 8 are applied to
determine pH, aðMal2
Vac Þ, K91 , and K92 : From these values,
DW is calculated from equation 6, and a new value for
½Mal2
Vac is calculated from equation 5. This value is then
used as a starting point for a new iteration, until ½Mal2
Vac 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(EDW), where DW, 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 DGMal2 = 0).
It can be calculated as E=DW that solves equation 1 for
DGMal2 = 0: 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 <ðMal2
Vac Þ <3 mM. In the
cytosol, a reasonable range of malate di-anion activity can
be estimated to be 0.2 mM <ðMal2
Cyt Þ <1.5 mM corresponding to 0.5 mM <½Mal2
<3
mM
(equal to malic acid
Cyt
concentration at neutral or slightly alkaline pH in the
cytosol), and an activity coefficient of 0.4 <aðMal2
Cyt Þ
<0.5 or I’150 mM. Based on these assumptions, one can
calculate 75 mV <E < 30 mV at T=300 8K (27 8C).
The higher range of these values is comparable with
the expected tonoplastic potential gradient, which most
authors estimate as 30 mV <DW <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 DW
at which the current through the pump equals zero. For
Modelling malate accumulation in fruits
ATPase, it can be calculated by rewriting equation 3 and
solving it for DGATPase=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 mol1, an ATP/ADP ratio
in the cytosol between 1 and 5, and the phosphate
concentration between 1 and 5 mM, the computed DGATP
varies at most between 51 and 58 kJ mol1 (calculation
not shown). This range overlaps most published values of
DGATP (Rea and Sanders, 1987; Briskin and ReynoldsNiesman, 1991; Davies et al., 1993). In maize root tips,
Roberts et al. (1985) measured the nucleotide phosphate
concentrations by MNR, and computed that DGATP varied
between 64 kJ mol1 when ATP utilization was inhibited,
and 45 kJ mol1 when its production was inhibited, with
an average value of 56.3 kJ mol1 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 mol1 (calculation not
shown), while Davies et al. (1993) estimate it as 27.3 kJ
mol1. 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+
1475
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.
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
Fig. 1. Calculation of the electromotive forces of the ATPase and PPiase
under different assumptions. In all cases, pHCyt=7.2, DWATP = 56.3 kJ
mol1, DGPPi = 27, kJ mol1, (K+)=60 mM on both sides of the
tonoplast, and T=198 8K. Two models were considered to represent the
variation the stoichiometry of the ATPase: (a) our model (equation 4a);
(b) the model of Kettner et al. (equation 4b). Concerning the PPiase, the
cases considered were (c) the electromotive force varying as described
by Davies et al. (1992) (calculations not shown), (d) a stoichiometry of
1 H+ per PPi hydrolysed.
1476 Lobit et al.
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 protein1
min1 in Catharanthus roseus to 1300 nmol mg protein1
min1 in barley; though most measurements are between
20 and 250 nmol mg protein1 min1 (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 m2 s1 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 m2
(Martinoia and Ratajczac, 1997), and per kilogram of
tissue, assuming a tonoplast area between 50 and 100 m2
kg tissue1 (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 d1 kg fruit1, with a median value around
3.5 mmol malate d1 kg fruit1. 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 d1 kg fruit1, 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+ kg1 d1 for the ATPase alone, between 1 and
150 mmol H+ kg1 d1 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+ kg1 d1 (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 lM (Maeshima et al., 1996; Obermeyer et al., 1996),
while Roberts (1990) estimates the PPi concentration to be
about 10 lM. 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.
Materials and methods
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
Nı̂mes, 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 kg1.
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.
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 8C 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, U 6.5 mm, pump VARIAN
Modelling malate accumulation in fruits
9010, detector VARIAN 9050 at k=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 8C for 24 h, diluted with 50 ml
0.3 N nitric acid, then measured by colorimetry using the nitrovanado-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 620% 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.
Results
Parameterization and sensitivity analysis
Parameter estimation: Though the model has seven
parameters, they are not independent so only five of them
30
Temperature (°C)
25
20
1477
ÿ
need to be adjusted. It was chosen to set a Mal2
Cyt = 0:47;
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,
b=0.12 was kept, as determined by Davies et al. (1994)
(Table 1).
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
DGATP = 56.75 kJ mol1 is very close to that found by
Roberts et al. (1985), which was DGATP = 56.3 kJ mol1,
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 a=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 ½Mal2
Cyt =0.58 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).
Sensitivity analysis: The responses of the model to variations of parameters Malcyt, DGATP, n0, b, a, 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 DGATP 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,
DW=DGATP/n0F, when pHCyt=pHVac=7). With malate
accumulation strongly dependent on DW, 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.
15
Table 1. Fitted parameters of the malate accumulation model
10
Average temperature 95
Average temperature 96
5
15 Mar
15 Apr
16 May
16 Jun
17 Jul
17 Aug
Fig. 2. Average daily temperatures measured in the orchard during the
1995 and 1996 seasons (7 d shifting average).
Fixed parameters
ðMal2
Cyt Þ
b
Fitted parameters
0.47 mM
0.12
½Mal2
Cyt pHCyt
DGATP
n0
a
0.58 mM
7.2
56.750 kJ mol1
4
0.3
1478 Lobit et al.
1995
Malate content (mmol Kg-1)
60
1996
50
40
30
20
10
0
15 May 30 May 14 Jun
29 Jun
14 Jun
29 Jul
15 May 30 May
14 Jun
29 Jun
14 Jul
29 Jul
Fig. 3. Comparisons between malate concentration measured (crosses) and predicted (lines) in 1995 and 1996.
1995
30
1996
Sensitivity Coefficient
20
10
0
-10
-20
-30
15 May
29 May
12 Jun
26 Jun
10 Jul
28 May
17 Jun
07 Jul
27 Jul
Fig. 4. Sensitivity of malate concentration to model parameters. The sensitivity coefficient to one parameter is defined as the ratio between the relative
response and the relative variation in the corresponding parameter.
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
DGATP 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 DGATP and n0 and are
extremely close to those of Roberts et al. (1985) and
Davies et al. (1994). The parameters a and b control the
response of ATPase to pHVac and pHCyt. The sensitivity
coefficient to a is positive, and declines from around 10 to
between 3 and 5 during ripening. This is as expected, since
an increase in a alleviates the inhibition of the ATPase by
vacuolar acidity. The sensitivity to b, by contrast, remains
around 1 throughout the season (this is a justification
a posteriori for choosing b 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.
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 dianion 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
Modelling malate accumulation in fruits
Discussion
reduced malate concentration (by about 50% when temperature increased from 15 to 25 8C), in particular at the beginning of fruit ripening (i.e. at high malate concentration).
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 65% K+, citric acid, and quinic acid
concentrations throughout the season (Fig. 6) and calculating the corresponding sensitivity coefficients.
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.
Modelled malate content (mmol Kg-1)
a
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
29 May 95
19 Jun 95
10 Jul 95
b
29 May 95
19 Jun 95
10 Jul 95
120
1479
100
80
60
40
20
0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
5
pH
10
15
20
25
30
35
40
Temperature (°C)
Fig. 5. Sensitivity of malate concentration to vacuolar pH (a) and temperature (b) at three different dates: 29 May 1995, 19 June 1995, and 10 July 1995.
1995
1996
Sensitivity Coefficient
1
0.5
Qui
Qui
Cit
Cit
K+
K+
0
-0.5
-1
15 May 29 May
12 jun
26 Jun
10 Jul
28 May
18 Jun
09 Jun
30 Jun
Fig. 6. Sensitivity of malate concentration to concentrations of other organic acids and cations during the 1995 and 1996 seasons.
1480 Lobit et al.
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 DGATP 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 8C to 25 8C 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.
Conclusion
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
Modelling malate accumulation in fruits
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.
Acknowledgements
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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