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First published online September 2, 2005; 10.1104/pp.105.063339 Plant Physiology 139:531-545 (2005) © 2005 American Society of Plant Biologists ETHY. A Theory of Fruit Climacteric Ethylene Emission1Unité de Recherche Plantes et Systèmes de culture Horticoles (M.G.), and Unité Mixte de Recherche A408, Sécurité et Qualité des Produits d'Origine Végétale (B.G.), Institut National de la Recherche Agronomique, Domaine Saint-Paul, Site Agroparc, 84914 Avignon cedex 9, France
A theory of fruit climacteric ethylene emission was developed and used as the basis of a simulation model called ETHY. According to the theory, the biosynthetic pathway of ethylene is supplied by ATP and is regulated by 1-aminocyclopropane-1-carboxylic acid (ACC) synthase and ACC oxidase. The conjugation of ACC with malonate to form MACC was taken into account as a way to decrease the availability of ACC. Because of the seasonal increase of fruit volume, the dilution of biochemical compounds used in ETHY was taken into account. Finally, the ethylene diffusion across the skin was considered. The theory took into account the effect of temperature and O2 and CO2 internal concentrations on ethylene. The model was applied to peach (Prunus persica) fruit over 3 years, several leaf:fruit ratios, and irrigation conditions. An adequate ethylene increase was predicted without considering any increase in respiration during the ripening period, which suggests that the respiratory climacteric may not be required for ripening. Another important result of this study is the high sensitivity of ETHY to the parameters involved in the calculation of ACC oxidase and ACC synthase activities, ATP production, and skin surface and permeability. ETHY was also highly sensitive to changes in fruit growth and temperature.
The plant hormone ethylene (C2H4) plays a major role in the ripening process of climacteric fruits. Ripening parameters such as flesh softening (Haji et al., 2003  ; Hiwasa et al., 2003), color change (Flores et al., 2001), and production of aromas depend strongly on C2H4 production (Rupasinghe et al., 2000; Alexander and Grierson, 2002; Flores et al., 2002).
C2H4 production in plant tissues has been studied extensively (Arshad and Frankenberger, 2002
The aim of this work was to develop a theory of C2H4 emission, based on a mathematical representation of the respiration process and C2H4 pathway. This representation was chosen to be as simple as possible because a lot of quantitative information about the regulation of this pathway is missing. The theory relates C2H4 production to fruit growth and environmental conditions. Indeed, fruit C2H4 production is known to vary during the development of the fruit depending on environmental conditions such as temperature; levels of O2 and CO2 in the air (Lelièvre et al., 1997
The theory concerns C2H4 emission during the last period of fruit development (for a peach, this is the last 2–3 months before maturity). It concerns (1) the biosynthetic pathway of C2H4, which includes the Yang cycle and the SAM-C2H4 pathway schematized in Figure 1, and (2) dilution and diffusion of C2H4 through the fruit skin.
It is known that Met and SAM pools are too low in plant tissues to sustain normal rates of ACC production, implying that they must be continuously regenerated (Baur and Yang, 1972 ). This is the role of the Yang cycle. As this cycle is supplied by ATP (Kushad et al., 1983; Theologis, 1992), ATP production through respiration is an important aspect of ETHY. The SAM-C2H4 pathway is regulated by ACCs and ACCo, which are the two key enzymes considered in ETHY. The stress was laid on the autocatalytic C2H4 production, which results in up-regulation by C2H4 of these two enzymes. The conjugation of ACC with malonate to form malonyl-ACC (MACC) was taken into account in ETHY as a way to decrease the availability of ACC for C2H4 production. Because of the seasonal increase of fruit volume, the dilution of biochemical compounds used in ETHY was taken into account. Finally, C2H4 diffusion across the skin was considered. Metabolic and biophysical processes are strongly connected in ETHY. For example, the diffusion of C2H4 in the ambient atmosphere depends on the internal C2H4 concentration, which depends itself on the intensity of the diffusion and on other processes (dilution, metabolism). Yang cycle, respiration, production of MACC, regulation of ACCs and ACCo, dilution, and C2H4 release in ambient atmosphere are the basic processes represented in ETHY. The fruit is described as one compartment separated from the exterior by the skin. The ambient atmosphere plays the role of the exterior compartment. Gas exchange occurs by diffusion through the fruit skin. The temperature of the fruit is assumed to be equal to that of the ambient atmosphere.
Fruit growth is considered to influence C2H4 production because it is related to respiration, dilution, and to the skin area through which C2H4 is released. The temperature is an important external factor controlling respiration in ETHY. The model also takes into account the effect of O2 and CO2 internal concentrations on C2H4 through their action on ACCo, which requires O2 and is inhibited by high concentrations of CO2 (Lelièvre et al., 1997 The main state variables of the system are MACC, ACC, CO2, O2, and C2H4 concentrations in the fruit. The hourly information coming from the external compartment is the temperature and the concentrations of O2 and CO2 in the ambient atmosphere. This information and the fruit growth components (dry and fresh mass, and dry growth rate) represent the ETHY inputs. Using the inputs together with a theoretical analysis quantified by the governing equations, we compute the main reaction rates of the C2H4 biosynthesis pathway and gas transfer processes. The lists of the ETHY variables and parameters are presented in "Materials and Methods."
Dilution versus Metabolism and/or Diffusion
As [X] = X/V, with V (m–3)
Yang Cycle, and the SAM-C2H4 and ACC-MACC Pathways
The variation of ACC results from the balance between its synthesis, its degradation in C2H4 controlled by ACCo (k3; h–1), and its conjugation with malonate to form MACC catalyzed by the ACC N-malonyltransferase (k4; h–1). The rate law equation for ACC is then:
Assuming the Yang cycle at steady-state
The ACC concentration depends on the amount of ACC per fruit and on dilution by water according to Equations 1 and 3. The rate law equation for ACC concentration is then:
The variation of MACC concentration is:
The C2H4 concentration in the fruit depends on the one hand on the balance between C2H4 biosynthesis and diffusion to the external atmosphere (
Enzyme Regulations
 (mol m–3) and  (mol m–3) Kms.
C2H4 Diffusion to the External Atmosphere
 (m h–1) is the apparent skin permeability and A (m2) is the skin area.
A is estimated from fruit mass through an empirical equation:
 and  are parameters.
According to Lescourret et al. (2001)
 (m h–1) and  (m g–1 h–1) are parameters.
ACC and C2H4 Concentrations, and ACCs and ACCo Activities
 and ACC ([ACC0]).
ATP, O2, and CO2 Concentrations
 is the rate of CO2 production caused by respiration.
Spoelstra et al. (2002)
).
The oxygen and carbon dioxide concentrations are calculated assuming that they are under steady-state condition [i.e. respiration rates for O2 (
 (h m–1) and  (h m–1) are resistances to diffusion of CO2 and O2, respectively. The air CO2 and O2 concentrations in mol m–3 are calculated from the percentage of O2 in the air using the ideal-gas equation.
The respired oxygen is calculated from the CO2 produced by the respiration using the respiratory quotient (RQ) concept:
) was used to calculate the respiration in terms of CO2. The growth respiration is considered proportional to fruit growth rate and the maintenance respiration to dry mass and temperature (Penning de Vries and van Laar, 1982; Thornley and Johnson, 1990). The effect of temperature is described with the Q10 concept. The fruit respiration in terms of CO2 production (mol h–1) is then calculated as:
Equations 16 to 20 are combined to calculate ATP, CO2, and O2 concentrations as a function of fruit dry mass, volume, area, and temperature:
Equations 5, 8 to 12, and 21 to 23 form a system of equations for [ACC], [MACC], [C2H4], [ATP], [CO2], [O2], and
The MACC concentration reported in the literature during the early growth of fruits is very variable, from 3 10–5 to 8 10–3 mol m–3 according to the authors (Amoros et al., 1989
The Michaelis-Menten constant for O2 and CO2 inhibition of C2H4 production were deduced from the experiment of de Wild et al. (1999)
The seasonal trends of fruit masses are shown in Figure 2. Fresh and dry masses increased with both fruit age and leaf:fruit ratio. The suppression of tree irrigation in 1997 had only a minor negative effect on fresh mass for the 30 leaf:fruit ratio. There was high variability between fruit masses for a given treatment and date. That is why the mean, maximal, and minimal growth curves (Fig. 3) were used for each treatment and year as inputs of ETHY. The mean temperature was 22.5°C in 1993 and 1996, and 21°C in 1997. The mean daily temperature fluctuated greatly each year between 16°C and 26°C (Fig. 4).
ETHY Goodness-of-Fit and Predictive Quality The model was able to simulate the order of magnitude of fruit respiration, the lack of irrigation effect (data not shown), and the increase of maximal respiration with the leaf:fruit ratio (Fig. 5). Root mean squared errors (RMSE) varied from 0.16 to 0.38 mol h–1 m–3 according to year and leaf:fruit ratio treatment, and the mean value of the error was 0.27 mol h–1 m–3. The predictive quality of the model was quite correct: the root mean squared errors of prediction (RMSEP), which ranged from 0.16 to 0.54 mol h–1 m–3 with a mean equal to 0.3 mol h–1 m–3, were close to the RMSE.
With regard to C2H4, the RMSE varied from 8 to 234 µmol h–1 m–3, according to year and leaf:fruit ratio, and the mean value of the error was 89 µmol h–1 m–3. The RMSEP ranged from 7 to 260 µmol h–1 m–3 with an average of 107 µmol h–1 m–3. The parameters related to ACCs and ACCo were estimated at ks = 0.778 ± 0.042 h–1, ko = 0.268 ± 0.017 h–1. In accordance with the measured data, the model predicted an increase in C2H4 production with the leaf:fruit ratio, no effect of irrigation treatment, and lower C2H4 production in 1996 than in the other two years (Fig. 6). The intratreatment variation of C2H4 production was high (Fig. 6), which was fairly well depicted by the model using as input the maximal and minimal growth curves of each treatment.
Seasonal Variation of ATP, Internal O2 and CO2, ACC, MACC, and C2H4 Concentrations in Response to Year and Fruit Growth The model predicted significant variations of ATP concentration from 0.008 to 0.023 mol m–3 with a general decrease following that of fruit respiration per unit of fruit volume (Fig. 7). The increase of fruit growth from six to 30 leaves per fruit has for consequence a 25% increase of ATP concentration. At a short time step, fairly high fluctuations of ATP concentrations are simulated, which results mainly from the effect of temperature on fruit respiration (Figs. 4 and 7).
The predicted concentrations of O2 in the fruit for the different treatments and years were in the14% to 18.7% range, which were slightly lower than in the air (21%). The predicted CO2 concentrations in the fruit (1.3%–3.9%) were much higher than in the air (0.03%). Under the steady-state assumption, the O2 and CO2 concentrations in the fruit were fairly stable during the season, but short time variations were observed in response to temperature fluctuations (Fig. 8). The increase of fruit growth with leaf:fruit ratio resulted in an [O2] decrease and [CO2] increase in the fruit.
The model predicted a seasonal increase of ACC, MACC, and C2H4 concentrations and a positive effect of leaf number per fruit (Fig. 9). The concentration of C2H4 was 10 times lower than that of MACC and ACC, and MACC concentrations were always lower than those of ACC. The ACC reached its maximal concentration before MACC and C2H4, especially in the case of high leaf:fruit ratio. The activity of ACCs and ACCo was simulated for [ATP0] = 0.015 mol m–3, [ACC] = 0.015 mol m–3, and ambient [O2] and [CO2] for the treatment with six and 30 leaves per fruit in 1993. The activity of the two enzymes increased from 100 to 140 d after bloom (DAB) and this all the more strongly as the leaf:fruit ratio was high (Fig. 10). The C2H4 production started much later and this more especially as the leaf:fruit ratio was low.
Key Internal Variables for C2H4 Production According to ETHY, C2H4 emission depends on three main variables, the concentration of ATP on which ACCs depends, the concentration of ACC on which C2H4 synthesis depends, and the concentration of C2H4 itself on which the intensity of the diffusion in the external atmosphere depends. The link between C2H4 emission and these three variables was analyzed using the simulations of the leaf:fruit ratio done in 1993. As expected, there was, whatever the leaf:fruit ratio, a unique very strong positive linear link between internal C2H4 concentration and C2H4 emission (Fig. 11). A strong positive exponential link was also found with ACC concentration. There was no obvious connection between C2H4 emission and the ATP concentration, which shows that there is no link between C2H4 emission at a given time and ATP concentration at this time. This could be due to the low ATP cost of C2H4 production. The amount of ATP needed for C2H4 production was computed considering that two ATP are needed to produce one ACC molecule (Fig. 1). The computed ATP cost of C2H4 production was always below 0.04% of ATP production by respiration. Nevertheless, ATP concentration is a key variable for C2H4 production. Indeed, keeping ATP concentration as a constant (0.013 mol m–3), whatever the leaf:fruit ratio, and re-estimating the parameters (ks = 0.5 ± 0.08 h–1 and ko = 0.36 ± 0.06 h–1) led to a poor adjustment to the experimental data (RMSE = 131 µmol h–1 m–3). With a constant concentration of ATP, the model was able to simulate correct C2H4 production for low leaf:fruit ratio but predicted too low the C2H4 production for high leaf:fruit ratio (Fig. 6).
Key Parameters of Fruit Metabolism
A sensitivity analysis to the model parameters was performed to look for the key parameters, i.e. the parameters whose variation has a strong effect on the model outputs. The analysis was performed using the inputs of 1993. To see if the model was sensitive to different degrees to small or large parameter variations, each parameter was varied from X = ±1%, 5%, 10%, 20%, 40%, 60%, 80% and 100% around the value reported in "Materials and Methods." The effect of this variation on the average value of (1) O2, CO2, ATP, ACC, MACC, and C2H4 concentrations and (2) fruit respiration and C2H4 emission was assessed. The model was considered sensitive to a parameter when an X percentage of variation of this parameter resulted in at least an X/2 percentage of variation of one or more of the above-mentioned variables. As the model was similarly sensitive to parameters whatever the level of X (data not shown), only the results obtained for X = ±20% are presented (Table I). The sensitivity increased from six to 30 leaves per fruit. However, as the general trends were very similar, only the results obtained for 18 leaf:fruit ratio in 1993 were presented (Table I). The oxygen internal concentration was only slightly sensitive to parameter
It is interesting to note that an increase in skin permeability to C2H4 induced a decrease in ACC, MACC, and C2H4 concentrations as well as in C2H4 emission. Equation 10 allows an increase of skin permeability to C2H4 with fruit mass if  For small values of  the C2H4 emission started earlier and reached a higher level for high than for low leaf:fruit ratio as observed in our experiments. But for high values of the opposite situation was predicted by the model. In the latter case, the increase of C2H4 synthesis with the leaf:fruit ratio was overcompensated by the increase of C2H4 diffusion to the external atmosphere (Fig. 12). For medium values of  no effect of leaf:fruit ratios was predicted.
Effect of O2 and CO2 Air Concentration and Temperature on C2H4 Production Sensitivity to environmental variables was analyzed using, as previously, the results obtained with an 18 leaf:fruit ratio in 1993. The environmental variables had a variation of ±X% throughout the season, and the effect of this variation on the average value of C2H4 emission was assessed. The values of X considered in the analysis were the same as those used when examining the sensitivity to parameters. The model was similarly sensitive whatever the level of X. Considering a ±20% variation, the effect was very small for CO2 (±0.2%), and no effect of the natural variation of CO2 concentration in the air seemed possible. Although larger, the effect of O2 remained negligible (–11% to +7%). On the contrary, sensitivity to temperature was high (–71% to +170%). Indeed, a temperature increase as low as +5% had a large effect on C2H4 production (+31%).
The theoretical framework developed here enables us to provide a description of the relationship between environment, fruit growth, and C2H4 emission. It pieces together elements of present knowledge on autocatalytic C2H4 in a biophysical and metabolic theory focusing on processes of regulation by C2H4 itself, ATP, CO2, O2, and temperature. The simulation model based on this theory was able to predict the effect of fruit growth intensity on C2H4 emission for several years in peach.
ETHY predicted significant variations of ATP concentration from 0.007 to 0.023 mol m–3 with a general decrease following that of fruit respiration per fruit mass. Such a variation of more than 100% of ATP concentration is commonly observed during postharvest fruit storage (Saquet et al., 2003a
The different steps leading to C2H4 production have been considered, with a focus on ACCs and ACCo. The strong sensitivity to parameters related to these enzymes shows the importance of metabolism in the control of C2H4 production. In agreement with our results, studies on gene expression showed the importance of these enzymes in controlling the rate of endogenous C2H4 production (Lelièvre et al., 1997
Our main assumption is that C2H4 production is highly dependent on ATP availability and thus on fruit respiration. This assumption is in agreement with results of Saquet et al. (2003b)
Another important assumption of the model is that the diffusion process through the skin has to be considered as shown by the high sensitivity of the model to the permeability to C2H4 and to the skin area. The importance of such a process has been put forward by postharvest scientists studying the effect of film permeability of packaging on gas concentrations (Fishman et al., 1995 It was also assumed that C2H4 production was regulated by factors such as O2 and CO2 concentrations in the tissues and temperature. Effects of O2, CO2, and temperature have been well established, especially in postharvest studies. Our model is only slightly sensitive to air O2 concentrations and almost insensitive to CO2 concentrations. Only strong variations, such as those encountered during postharvest fruit storage in packages, can be effective. On the contrary, the temperature has a strong effect on C2H4 emission, and small variations can have a significant effect.
One interesting result of our study is that it was possible to simulate the onset of C2H4 production without considering any initial event such as the increase of endogenous concentrations of jasmonates (Fan et al., 1998
An interesting perspective of this work is based on the high ETHY sensitivity to parameters involved in calculating ACCo and ACCs activities, ATP production, and skin surface and permeability. This would enable the use of ETHY to analyze comprehensively the high genetic variability in C2H4 production (Miccolis and Saltveit, 1991
Plant Material and Experimental Design The ETHY model was parameterized, calibrated, and validated for the late-maturing peach (Prunus persica L. Batsch) cv Suncrest/GF 677. The measurements were performed on peach trees planted in 1981 in the orchard of the Institut National de la Recherche Agronomique (INRA) Avignon Centre. Trees were goblet trained and received routine horticultural care suitable for commercial orchards. This care included winter and summer pruning, weekly irrigation from June to harvest, hand thinning in April, and pest control. Fungicides and insecticides were applied every 2 weeks from February to July. Weeds were destroyed by weed-killer. The first experiments in 1993 and 1996 varied assimilate supply to the fruits. Treatments were applied to fruit-bearing shoots located on the southern part of each tree and isolated from the tree by girdling. Three treatments set leaf:fruit ratios at six, 18, and 30 leaves per fruit to obtain minimum, mean, and maximum growth curves that were representative of the Suncrest cultivar. Shoots with six or 18 leaves per fruit were thinned to four fruits, and shoots with 30 leaves per fruit were thinned to two or three fruits. In 1993 and 1996, 240 fruit-bearing shoots were prepared on 64 and 36 trees, respectively, to provide 80 sets of three neighboring shoots with leaf:fruit ratios of six, 18, and 30. Fruits were harvested from five replicates per treatment each week from mid-June to the beginning of fruit maturation. A replicate was made up of fruits from two shoots when fruits were small and from one shoot later in the season. The last harvests were on August 16, 1993, and August 9, 1996. Experiment 2, performed in 1997, varied both assimilate and water supply to the fruits. The treatments were applied to 240 fruit-bearing shoots isolated from 40 trees by girdling and located on the southern part of each tree. Half of the trees were irrigated in June and July, whereas irrigation was withheld from the remaining trees. For each irrigated treatment, leaf:fruit ratio of the selected fruit-bearing shoots was 10 or 30. Sixty pairs of neighboring shoots with leaf:fruit ratios of 10 and 30 were prepared for each irrigation treatment. Fruits from five replicates per treatment were harvested each week from mid-June to fruit maturation (July 24).
At each harvest, O2 consumption and CO2 and C2H4 production were measured individually at a constant room temperature (23°C) by confining intact fruit in a gas-tight 400-mL jar. The internal atmosphere of the jar was analyzed by gas chromatography. CO2 and O2 were separated on a Porapak Q column at 110°C, followed by a 13x molecular sieve column at room temperature (Chambroy et al., 1984
After C2H4 measurement, the stone was removed and the fresh mass of fruit flesh was measured. The dry mass was measured by drying the stone and the flesh at 70°C for 72 h. The hourly dry and fresh fruit masses, which were inputs of the model, were interpolated from our measurements by local regression (Chambers and Hastie, 1992
Simulation of both CO2 and C2H4 emission were based on an hourly scale. Accordingly, the results were displayed hourly on the figures, using DAB as units (e.g. for hour 12 of DAB 110, the x value was 110.5). An hour was also used as the time frame in the numerical integration. The computer program was written using Splus simulation language (Becker et al., 1988
For testing model quality, a goodness-of-fit criterion was calculated separately for C2H4 and CO2 emission in each of the 10 yearxtreatment combinations. The basic criterion was a RMSE, a very common criterion describing the mean distance between simulation and measurement (Kobayashi and Us Salam, 2000
 is the C2H4 (or the CO2) emission at date i calculated by the model, and  the average of the ni measures. The smaller the RMSE in comparison to measurements, the better the goodness-of-fit.
The predictive quality of the model, which is not the same as adjustment quality (Wallach et al., 2001
The ETHY variables are as follows: Mfresh (g), fruit fresh mass; Mdry (g), fruit dry mass; V (m–3), fruit volume; A (m2), skin area; T (°C), temperature;
The ETHY parameters are as follows: qg = 0.0025 (mol g–1), growth respiration coefficient; qm = 12 10–5 (mol g–1 h–1), maintenance respiration coefficient at 20°C; Q10 = 2, temperature ratio of maintenance respiration; RQ = 0.834, respiratory quotient;
We thank J. Hostallery and R. Laurent for their assistance during the field experiments, G. Jacquemin for the gas exchange analysis, and M. Reich for mass measurements. We thank F. Lescourret and M.G. Grotte for helpful comments on this paper, and C. Young (INRA Translation Unit, Unité Centrale de Documentation, Jouy-en-Josas) for revising the manuscript. Received March 24, 2005; returned for revision June 6, 2005; accepted June 13, 2005.
1 This work was supported by Région Provence-Alpes-Côte d'Azur (grant nos. 2002–06290 and 2003–10048).  Article, publication date, and citation information can be found at www.plantphysiol.org/cgi/doi/10.1104/pp.105.063339. * Corresponding author; e-mail michel.genard{at}avignon.inra.fr; fax 33–432–72–22–82.
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ABSTRACT
THEORETICAL ANALYSIS OF THE...
10–6 Mfresh and Mfresh (g) the fruit fresh mass, we get:
; mol h–1), and on the other hand on dilution:
(h–1) and dilution in water. One mol of CO2 produced by respiration was assumed to be coupled with the production of 5 mol of ATP (Cannell and Thornley, 2000
). Assuming that such a property is valid for the fruit, the ATP concentration can be approximated by the following:
) and CO2 are equal to the fluxes of these gases through the skin to the external atmosphere], and the gas diffusion to the external atmosphere can be described by Fick's first law:
which can be solved numerically with given (inputted) functions of time Mdry(t), Mfresh(t) and T(t); O2 and CO2 concentrations in the air (assumed to be constant); and the initial values of respective variables. 
). The apparent peach skin permeability to C2H4 was measured by Cameron and Reid (1982)
Parameters 
The estimation was RQ = 0.834 ± 0.006. The ranges of CO2 and O2 concentrations for fleshy fruits are 0.4% to 8% and 15% to 19%, respectively, according to measurements obtained for cantaloupe (Cucumis melo; Lyons et al., 1962
and 
which are close to the minimal resistance values obtained by Cameron and Reid (1982)
; and parameters 
increase on seasonal variation of C2H4 emission simulated by ETHY at 23°C over six (thin lines) and 30 (thick lines) leaves per fruit.
(mol m–3), atmospheric O2 concentration; 
(mol m–3), atmospheric CO2 concentration; 
C2H4 diffusion to the external atmosphere. 
Michaelis constant involved in effect of O2 on ACCo; 
(mol m–3), Michaelis constant involved in effect of CO2 on ACCo; 
resistances to diffusion of O2; 
(h m–1), resistances to diffusion of CO2; 
(m h–1), skin permeability; and 
