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AOBPreview originally published online on January 17, 2005
Annals of Botany 2005 95(4):673-683; doi:10.1093/aob/mci067
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Annals of Botany 95/4 © Annals of Botany Company 2005; all rights reserved

Modelling the Effect of Fruit Growth on Surface Conductance to Water Vapour Diffusion

CAROLINE GIBERT1,*, FRANÇOISE LESCOURRET1, MICHEL GÉNARD1, GILLES VERCAMBRE1 and ALEJANDRO PÉREZ PASTOR2

1 INRA, Domaine Saint-Paul, Site Agroparc, Unité Plantes et Systèmes de culture Horticoles, 84914 AVIGNON Cedex 9, France and 2 Area de Producción Vegetal, Dpto de Producción vegetal, Escuela Técnica superior de Ingenería Agronómica, Universidad Politécnica de Cartagena, Paseo Alfonso XIII, 52, 30 203 Cartagena, Spain

* For correspondence. E-mail gibert{at}avignon.inra.fr

Received: 22 August 2004    Returned for revision: 12 October 2004    Accepted: 24 November 2004    Published electronically: 17 January 2005


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 

• Background and Aims A model of fruit surface conductance to water vapour diffusion driven by fruit growth is proposed. It computes the total fruit conductance by integrating each of its components: stomata, cuticle and cracks.

• Methods The stomatal conductance is computed from the stomatal density per fruit and the specific stomatal conductance. The cuticular component is equal to the proportion of cuticle per fruit multiplied by its specific conductance. Cracks are assumed to be generated when pulp expansion rate exceeds cuticle expansion rate. A constant percentage of cracks is assumed to heal each day. The proportion of cracks to total fruit surface area multiplied by the specific crack conductance accounts for the crack component. The model was applied to peach fruit (Prunus persica) and its parameters were estimated from field experiments with various crop load and irrigation regimes.

• Key Results The predictions were in good agreement with the experimental measurements and for the different conditions (irrigation and crop load). Total fruit surface conductance decreased during early growth as stomatal density, and hence the contribution of the stomatal conductance, decreased from 80 to 20 % with fruit expansion. Cracks were generated for fruits exhibiting high growth rates during late growth and the crack component could account for up to 60 % of the total conductance during the rapid fruit growth. The cuticular contribution was slightly variable (around 20 %). Sensitivity analysis revealed that simulated conductance was highly affected by stomatal parameters during the early period of growth and by both crack and stomatal parameters during the late period. Large fruit growth rate leads to earlier and greater increase of conductance due to higher crack occurrence. Conversely, low fruit growth rate accounts for a delayed and lower increase of conductance.

• Conclusions By predicting crack occurrence during fruit growth, this model could be helpful in managing cropping practices for integrated plant protection.

Key words: Fruit surface conductance, stomata, cuticle, crack component, fruit growth


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 
The fruit surface conductance to water vapour diffusion is a variable of major importance both for fruit growth and quality since it controls transpirational water loss (Leonardi et al., 2000aGo; Lescourret et al., 2001Go). Intense fruit transpiration rates are positively related with fruit soluble solids content, a major criterion of quality (Li et al., 2001Go; Wu et al., 2003Go).

Different structures make up the skin of a fruit, such as stomata and the cuticle. These structures are involved in the transpiration pathway and contribute to the surface conductance. The number of stomata is determined at anthesis and remains constant during fruit development (Hieke et al., 2002Go). With expansion of the fruit, stomata are diluted on the fruit surface area, leading to a decrease of the stomatal density (Blanke, 1992Go; Knoche et al., 2001Go). Thus, in the early growth stage, as stomatal density is high, the stomatal component contributes predominantly to the total conductance. During the rapid fruit growth, the stomatal component swiftly declines, leading to the cuticular component becoming the major contributor, as shown in the case of sweet cherry (Prunus avium; Knoche et al., 2001Go).

In addition, some physiological disorders (Kertesz and Nebel, 1935Go; Christensen, 1972Go; Sekse, 1995Go) such as splitting and cuticular cracking, as observed on sweet cherry (Prunus avium; Verner and Blodgett, 1931Go), tomato (Lycopersicon esculentum; Frazier, 1934Go), apple (Malus domestica; Goode et al., 1975Go) and nectarine (Prunus persica ‘nucipersica’; Nguyen-The et al., 1989Go), can lead to large variations in the fruit surface conductance. According to Knoche et al. (2002)Go, the contribution of crack conductance can reach 15 % of the fruit total conductance for sweet cherry.

Cracks in sweet cherry fruits are shallow or deep oblong wounds in the fruit flesh (Sekse, 1998Go; Børve and Sekse, 2000Go). Cracks are assumed to occur when the elastic limit of the cuticle is exceeded as a consequence of high internal pressure, especially during periods of rapid fruit expansion (Christensen, 1973Go; Ohta et al., 1997Go). Cracks result from important mechanical stresses concentrated on preferential areas, such as the pedicel cavity or stylar region (Sawada, 1934Go; Considine and Brown, 1981Go; Yamanoto et al., 1991Go; Sekse, 1995Go). Alternation of water-stress and irrigation, which induces fruit shrinking and swelling, or low crop loading by increasing fruit growth, could favour the occurrence of cracks through intensification of those stresses (Milad and Shackel, 1992Go; McFadyen et al., 1996Go; Børve and Sekse, 2000Go). Crack conductance could decline either as a result of the closure of cracks due to a reduction of turgor pressure in the over-ripened fruit (Verner and Blodgett, 1931Go; Christensen, 1973Go) and/or because of wound healing activity around cracks, as observed on tomato (Lycopersicon esculentum; Moctezuma et al., 2003Go), cucumber (Cucumis sativus; Walter et al., 1990Go) and nectarine fruits (Prunus persica ‘nucipersica’; Nguyen-The et al., 1989Go).

This study uses a modelling approach to analyse how fruit growth can influence fruit surface conductance. A model of fruit surface conductance driven by fruit growth was developed, which distinguished each conductance component (cuticle, stomata and crack). This model could be useful to simulate and quantify, for example, crack occurrence according to different fruit growing situations, opening the way to study their contribution to transpirational water loss and hence to fruit quality. Experimentations were conducted in 2002 and 2003 on peach trees (Prunus persica), with various irrigation regimes and crop loads in order to obtain a wide range of fruit growth, some of which promoted crack occurrence. The model was parameterized and tested using these experimental data sets. In order to identify the most influential parameters and inputs of the model, a sensitivity analysis was performed. The capacity of the model to represent the observed variability of individual fruit conductance was examined.


   MODEL DEVELOPMENT
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 
The model simulates the fruit surface conductance at a daily time step. Surface conductance, g(t) (cm h–1) at time t (days after full bloom, DAFB) is the sum of stomatal conductance, gsto(t), cuticular conductance, gcut(t), and crack conductance, gck(t):

(1)
By extension, the term ‘cuticle’ employed in the text represents the set of exocarp cells plus their covering cuticle layers.

Stomata on the fruit surface area are set at anthesis (Hieke et al., 2002Go) and differentiated during the rapid initial fruit growth (Ishida et al., 1990Go). Thus, the number of stomata on a fruit is determined early, at a given DAFB denoted to, presumed to be independent of the fruit development. The density of stomata per fruit surface area at to, dsto(to) is also assumed to be a constant. In the model, stomatal conductance at t is considered as the product of the stomatal density at t and the stomatal specific conductance, g'sto (cm3 h–1):

(2)
where Sf(t) is the fruit surface area (cm2) at t.

Cracks occur when the expansion rate of the pulp surface area is higher than the expansion rate of the cuticle surface area. Therefore, the new crack surface area generated per day, dSnewck/dt (cm2 d–1) is assumed to be equal to the difference between the expansion rates of the pulp and cuticle surface areas (respectively, dSpulp/dt and dScut/dt) when this difference is positive:

(3)

(3a)

The expansion rate of the pulp surface area is approximated by the expansion rate of the fruit surface area, which is a model input.

The cuticle surface area expansion rate is equal to the product of cuticle surface area, Scut(t) (cm2), and the cuticle surface area relative expansion rate, RER(t) (d–1):

(4)
where Scut is the difference between the fruit surface area and the calculated crack surface area, Srck(t) (cm2; see below for calculations, eqn 6).

On the one hand, the patterns of variation of enzyme activities in the exocarp suggest that the cuticle surface area relative expansion rate follows a pattern of seasonal decrease. Xyloglucan endotransglycosylase (XET), a wall-loosening enzyme abundant in the exocarp, shows a decreasing activity during tomato growth, and activity of peroxidase, a cell wall stiffening enzyme, is only detected in the cell walls of mature tomato fruit exocarp cells (Thompson et al., 1998Go; Andrews et al., 2002Go). On the other hand, Yamaguchi et al. (2003)Go have shown by studying cultivars varying in fruit weight that the smallest and lightest fruits display the latest cessation of cell division of the exocarp cells. This suggests that the cuticle surface area relative expansion rate of smaller fruits is higher than that of bigger fruits. Both hypotheses can be accounted for assuming that the cuticle surface area relative expansion rate varies inversely with the fruit fresh mass, M(t) (model input), assumed to be an indicator of fruit development:

(5)
where {lambda} is a parameter (g day–1).

Thus, the new crack surface area generated per day, dSnewck/dt, is calculated by combining eqns (3), (4) and (5).

According to Nguyen-The et al. (1989)Go, Børve and Sekse (2000)Go and Moctezuma et al. (2003)Go, cuticular fractures and cracks may heal. The wound healing process consists of lignification and suberin formation after wounding (Walter et al., 1990Go). In the model, we assume that each day a ratio ({delta}) of generated cracks heals and that this ratio is constant whatever the crack quantity and the fruit development stage. The surface area of remaining cracks at a given time, Srck(t) (cm2), is therefore calculated as:

(6)

Finally, the crack conductance is equal to the proportion of cracks relative to the fruit surface area (cm2 crack cm–2 fruit) multiplied by the specific crack conductance, g'ck (cm h–1).

(7)

Similarly, the cuticular conductance is equal to the proportion of the cuticular surface area relative to the fruit surface area multiplied by the specific cuticular conductance, g'cut (cm h–1). g'cut was assumed to be constant with fruit age as a first approximation. The cuticular surface is the difference between the total fruit area and the stomata and crack surface areas.

(8)
where Ssto is the surface area of the stomata, and Srck(t) the surface area of the remaining cracks (cm2). As the surface area per stoma is very small (8·12 x 10–6 cm2, according to Ishida et al., 1990Go), the surface area of stomata was considered negligible compared with total fruit surface area (Sf) and so was ignored in the following.

Model inputs
The observed fruit fresh mass, M(t) (g), was fitted using a logistic function:

(9)
where Mmax is the maximal mass (g), Mo is the initial mass (g) and a is the initial relative growth rate (d–1).

Fruit surface area, Sf(t) (cm2), was calculated from fruit fresh mass by means of an allometric relationship:

(10)
where {gamma} and {eta} are parameters depending on the fruit geometry (Fishman and Génard, 1998Go).


   MATERIAL AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 
Plant material and experimental design
In order to obtain a wide range of fruit growth, different experiments were performed in 2002 and 2003 at the INRA Avignon Centre (South East of France). Fruit growing conditions varied through different irrigation regimes and crop load, yielding a total of six experimental data sets.

Trees were goblet-trained and received routine horticultural care including winter pruning, weekly irrigation by microjet sprinklers (in an orchard) or by drip irrigation (in containers), and pest control.

Experiment I was performed in 2002, from 44 DAFB to 101 DAFB (full bloom on 13 March), on early maturing peach trees of the cultivar ‘Alexandra’ grafted on GF 677 rootstock planted in 1999 and cultivated outdoors in 110 L containers (n = 20). The trees were thinned at 50 fruits per tree and two drip-irrigation regimes were applied. Ten trees were well-irrigated, twice a day for 30 min (16 L d–1, treatment I1), and ten trees had irrigation withheld five times (between 67 and 96 DAFB) for 2–3 consecutive days before being re-watered (treatment S1).

Experiment II was performed in 2003, from 35 to 85 DAFB (full bloom on 24 March), on the same peach trees used in 2002 (n = 20). Fruit thinning (40 fruits per tree) was performed at 35 DAFB. Two drip-irrigation regimes were applied. Ten trees were well irrigated, twice a day for 30 min (16 L d–1 treatment I2), whereas the remaining ten trees were submitted to four periods (between 63 and 81 DAFB) of withholding watering (varying from 1–3 d; treatment S2).

The successive water deficits were controlled by fruit diameter changes measured using Linear Variable Differential Transducers (Solartron, 10–1 µm) that were installed on the peach fruits and tree trunks.

Experiment III was performed in 2003, from 50 to 87 DAFB (full bloom on 24 March), on 20 ‘Alexandra’ peach trees, grafted on GF 677 rootstock, planted in 1994 in an orchard (5 x 4 m spacing).

A high (HC) and a low crop load (LC) were applied with five and 50 leaves per fruit respectively at 50 DAFB. Shoots with five leaves per fruit were thinned to two fruits, and shoots with 50 leaves per fruit to one fruit. Treatments were applied to fruit-bearing shoots isolated from the tree by girdling (Ben-Mimoun et al., 1996Go). This technique consists of removing bark and phloem tissues over 1 cm width at the shoot base. By this method, the fruit-bearing shoot is isolated from the rest of the tree in terms of its carbon assimilation via the phloem, but not for water supply via the xylem. In order to maintain a constant ratio of leaves per fruit over the period of the experiment, vegetative growth was stopped by removing the terminal and secondary apices on the selected fruit-bearing shoots.

Calculation of fruit surface conductance
Once a week from the beginning of each experiment, 20 fruits per treatment were harvested. Diameters (cheek, suture and height) were measured to calculate fruit surface area. Assuming an ellipsoid shape, fruit surface area was computed with MAPLE Software (Waterloo Maple Inc.). After sealing the pedicel (Spatex-GEB; Roissy CDG, France), fruits were weighed (Mettler PM 4600, 10–2 g) and placed in a ventilated chamber (wind velocity 1 m s–1) with controlled temperature.

Temperature and relative humidity of the chamber were continuously measured (Sefram Log 1520; St Etienne, France). Each fruit was weighed hourly for about 8 h.

Hourly surface conductance, g(h) (cm h–1) of each fruit was calculated as:

(11)
where h is time since sampling (h), Tf is the transpiration per unit time (g h–1), which is equal to the weight loss, Sf is the fruit surface area at h = 0 (cm2), Mw is the molecular mass of water (18 g mol–1), R is the gas constant (83 cm3 bar mol–1 K–1), Temp the temperature (K), P* is the saturation vapour pressure (bar) [depending on temperature following the equation of Fishman and Génard, 1998Go: P* = 0·008048 x exp(0·0547 x (Temp – 273·15)], Hf is the relative humidity within the fruit (assumed to be equal to 100 %), and Ha is the relative humidity of the atmosphere.

During the measuring time, the hourly surface conductance per fruit could either be relatively constant or decrease linearly with relative water loss, RWLh, defined as:

(12)
where Mo is the initial fruit fresh mass and Mh is the fruit fresh mass at time h. When the linear relationship was significant (P < 0·05), fruit surface conductance was taken as equal to the y-axis intercept of the linear relationship (RWLh = 0), otherwise it was taken as equal to the mean of the hourly conductances.

Dependent upon the measured wind speed in the ventilated and temperature-controlled room, the thickness of the boundary layer, {delta}x (m) was calculated according to Nobel (1975)Go as:

(13)
where D is the fruit diameter (m) and v is the wind velocity (m s–1). The boundary layer thickness was equal to 1 mm for a fruit diameter of approx. 70 mm. The corresponding resistance value, obtained by dividing {delta}x by the water vapour diffusion coefficient, Dwv, is approx. 38·5 s m–1 (with Dwv = 2·6 x 10–5 m2 s–1 at 25 °C; Cussler, 1984Go). This value is at least one order of magnitude less than the values of total resistance measured for ‘Alexandra’ by Lescourret et al. (2001)Go. Thus, conductance measurements performed on detached fruit correspond to the fruit surface and not to the fruit boundary layer.

Fruit growth
Cheek diameter was measured twice a week with a digital vernier calliper (Mitutoyo 500–181 U, 10–5 m) on 40 fruits per treatment in 2002 and on 100 in 2003. An allometric relationship was established between the cheek diameter (mm) and the fresh mass of fruits that were used for conductance measurements, in order to obtain in situ fruit growth curves expressed in fresh mass, M(t):

The parameters for ‘Alexandra’ were estimated by a non-linear least-squares procedure on the 2002 and 2003 data sets.

Model solving and parameterization
The model dynamically simulates fruit surface conductance with a daily time step. The model program-solving and parameterization were performed using Splus 3.4 (MathSoft Inc., 1995). The differential equations were integrated numerically using the first order Euler method with a 1-d integration step.

The parameters of the conductance model were estimated by minimizing the sum of squares of errors between simulated and observed data using the non-linear least-squares regression function. Three weeks after full bloom, the pericarp meristematic activity is declining (Masia et al., 1992Go) and the stomata development phase is assumed to be completely achieved, according to the observations of Ishida et al. (1990)Go. We thus defined to as equal to 21 DAFB. As measurements started earlier for experiments in 2003 (S2 and I2) than for other data sets, the parameters g'cut and dsto(to) were estimated by non-linear regression using those data sets from 35 to 64 DAFB. The period from 35 to 64 DAFB corresponds to the intermediate phase of slow growth, i.e. the period when it is assumed that no cracks have yet been generated (Christensen, 1973Go). The other parameters of the model ({lambda}, {delta} and g'ck) were estimated from the four data sets of 2003 by the non-linear least-squares regression function.

The parameters of model inputs (fresh mass and fruit surface area) were estimated for each experimental data set by a non-linear least-squares regression function. As measurements started earlier in 2003 (S2 and I2) than for other data sets, the parameter Mo of the logistic function for I1, S1, HC and LC was fixed from the mean of the initial mass estimated from values of I2 and S2.

Goodness-of-fit and test criteria
Goodness-of-fit and test criteria were computed using the root-mean-squared error (RMSE), a common criterion to quantify the mean difference between simulation and measurement (Kobayashi and Us Salam, 2000Go), here defined as:

(14)
where N is the number of dates, ni is the number of data at date ti, is the simulation result at date ti and is the averaged observed data at date ti.

The smaller the RMSE is in comparison with the measurements, the better the goodness-of-fit is. This can be represented through the relative RMSE:

(15)
where is the mean of all observed values.

The capacity of the model to predict the observed fruit-to-fruit variability of the conductance was examined on I2 and S2, the data sets for which the time sequence was the longest.

Each individual fruit growth curve was fitted to a logistic function. The fruit growth parameters were used as model inputs to compute the time-course of total conductance corresponding to each fruit. Then the standard deviations of simulated and observed conductances were calculated and compared.

Sensitivity analysis
A sensitivity analysis was performed to identify the parameters having the greatest influence on the conductance. The conditions of the irrigated 2003 data set (I2) were used for this purpose. The sensitivity of the mean response (mean total conductance) to changes in parameter values was calculated over two development periods: from 35 to 64 DAFB, corresponding to the slow growth phase, and from 64 to 86 DAFB, corresponding to the rapid growth period. The range of variation of each parameter of the model was ±20 %.

A sensitivity analysis concerning model inputs was also performed. The parameters of fruit growth functions were varied by ±20 % and the consequences on fruit growth curves and on simulated total conductance were examined.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 
Effects of fruit growth on fruit conductance
The treatments resulted in contrasting patterns of fruit growth (Fig. 1). At harvest, three groups could be characterized: (1) low crop load (LC), with mean fruit mass around 185 g (±35 g); (2) the two irrigation treatments of 2003 (I2 and S2) with mean fruit mass around 145 g (±30 g); and (3) high crop load (HC) and the two irrigation treatments of 2002 (I1 and S1), with mean fruit mass lower than 80 g (±25 g). Within a year, there were few differences in fresh fruit mass between the irrigated and stressed treatments. However, fruit enlargement was twice as high in 2003 than in 2002 for those treatments (Fig. 1). The time-course of fruit fresh mass was fitted for each treatment with a logistic function. The parameters corresponding to each data set are reported in Appendix I and were used as model inputs.



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  		FIG. 1. Mean values of peach ‘Alexandra’ fruit fresh mass for six experimental data sets (S1, stressed 2002; I1, irrigated 2002; S2, stressed 2003; I2, irrigated 2003; HC, high crop 2003; LC, low crop 2003). The curves join the mean values at each date.

 
Measured fruit surface conductance varied with fruit growth conditions (Fig. 2). At the early stage of fruit development, surface conductance per fruit could reach 950 cm h–1 for the I2 and S2 data sets. Mean final values of conductance were calculated using data covering 2 weeks close to maturity (i.e. from 93 to 101 DAFB for I1 and S1, from 78 to 85 DAFB for I2 and S2, from 72 to 79 DAFB for LC, and from 79 to 86 DAFB for HC). Mean values were higher for treatments I2, S2 and LC (324, 331 and 392 cm h–1, respectively) than for treatments S1, I1 and HC (120, 152 and 214 cm h–1, respectively). Mean conductance values were twice as low in 2002 than in 2003 for stressed and irrigated treatments.



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  		FIG. 2. Values of measured total conductance for stressed (S1) and irrigated (I1) treatments in 2002, and stressed (S2), irrigated (I2), high crop (HC) and low crop (LC) treatments in 2003. Each point represents the mean value of the measurements at each date and bars represent the 95 % confidence intervals. The line represents the simulated total conductance. The RRMSE value is indicated for each treatment.

 
At a given date, measured fruit surface conductance was highly variable. The standard deviation was between 50 and 150 cm h–1 depending on the data sets.

Measured fruit surface conductance varied during fruit growth. The pattern of variation differed according to the data set. For the I2 and S2 data sets, mean fruit surface conductance greatly decreased during the early stages of development (until 60–64 DAFB), then significantly increased between 60 and 80 DAFB (as also observed with the LC data set) corresponding with the period of intense growth, and finally decreased at the end of growth. In comparison, fruit surface conductance decreased during the whole period of fruit growth for treatments I1, S1 and HC.

Parameterization, adjustment quality and test
The values of the parameters of the logistic function corresponding to each data set are presented in Appendix I. The allometric relationships relating the fruit fresh mass to the cheek diameter, and the fruit surface area to the fruit mass, respectively, fitted well (RRMSE = 0·091, n = 685, and RRMSE = 0·012, n = 696, respectively). The values of the parameters are presented in Appendix II.

The specific stomatal conductance, g'sto, was estimated from Knoche et al.'s (2001)Go observations on sweet cherry fruit to be equal to 0·38 cm3 h–1.

The estimated values of the model parameters are presented in Table 1. The parameters {lambda} and dsto were estimated well, with a very low standard error. The parameters g'cut, {delta} and g'ck presented higher standard errors, but remained well estimated. By multiplying the density of stomata, dsto, at 21 DAFB with the fruit surface area, Sf, at that time, a mean value of 22 409 stomata (±9107) was obtained for the peach fruit.


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  		TABLE 1. Values, standard error and significance of model parameters

 
The pattern of variation of fruit surface conductance during fruit growth was well simulated whatever the data set (Fig. 2). For the four data sets of 2003, the simulated conductance was within the 95 % confidence interval of the observed data for almost all the dates of measurement. Correspondingly, the adjustment quality was very good for these data sets (RRMSE < 0·17), which had been used for parameterization. The tests realised on data sets from 2002 were acceptable, although simulated conductance tended to be over-estimated in the I1 and S1 data sets (RRMSE = 0·22 and 0·50, respectively).

The between-fruit variability of simulated conductance was in good agreement with the observed variability (Fig. 3).



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  		FIG. 3. Comparison between observed (circles) and simulated (triangles) variability of total conductance in irrigated (I2) and stressed (S2) treatments in 2003. Each symbol represents the mean value at each date and bars represent the standard deviation.

 
Contribution of each conductance component to total fruit surface conductance, and simulated crack surface area
For fruit with the highest growth rates (I2, S2 and LC), the crack component accounted for 0 % initially, then from 40 % up to 60 % of total conductance during the rapid fruit growth (around 70 DAFB; Fig. 4). As a consequence, the contribution of the stomatal component greatly decreased from 80 % at 35 DAFB down to 30–20 % at 70 DAFB, and the contribution of cuticular conductance varied slightly between 20–30 % (Fig. 4). In contrast, for fruits with the lowest growth rates (S1, I1 and HC), as crack contribution was lower than 10 %, the stomatal component accounted for 80 % to 50 % of total conductance and the cuticlar contribution increased from 20–40 % during fruit growth.



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  		FIG. 4. Contribution of stomata (solid line), crack (dashed line) and cuticle (dotted line) to the total simulated conductance (expressed as percentage of total conductance) for stressed (S1) and irrigated (I1) treatments in 2002, and stressed (S2), irrigated (I2), high crop (HC) and low crop (LC) treatments in 2003.

 
The higher the fruit growth rate, the larger was the simulated crack surface area (Fig. 5). The maximal proportion of crack surface area was 20 % for LC data set at 65 DAFB, whereas it was between 15–17·5 % for the S2 and I2 data sets and about 2·5 % for the HC and I1 data sets. For the S1 data set, no crack seems to have been generated according to the model.



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  		FIG. 5. Time-course of the percentage of crack surface area relative to total fruit surface area for stressed (S1) and irrigated (I1) treatments in 2002, and stressed (S2), irrigated (I2), high crop (HC) and low crop (LC) treatments in 2003.

 
Sensitivity analysis
Simulated conductance was highly sensitive (±14 %) to variations of specific stomatal conductance (g'sto) and of stomatal density (dsto) at 21 DAFB during the slow growth phase (Table 2). During rapid growth, specific crack conductance (g'ck) and the parameter {lambda} were the most influential. According to the model, when specific crack conductance increased, total fruit surface conductance was raised. Higher {lambda} meant a higher skin expansion rate and therefore fewer cracks. Simulated conductance was more than two times less sensitive to stomatal parameters than during the slow growth phase (±5·5 %). Specific cuticular conductance (g'cut) influenced the modelled conductance (±4–5 %) independently of the stage of development.


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  		TABLE 2. Effect of a ±20 % variation of the model parameters on the total fruit surface conductance for two stages of development

 
With regard to the model inputs describing fruit growth, simulated conductance was very sensitive to the initial relative growth rate, a (Fig. 6). Indeed, this parameter largely influenced fruit growth by shortening the slow growth phase and by increasing the growth rate during the rapid phase. An increase of the initial relative growth rate leads to an earlier and larger increase of conductance. This increase of conductance was caused by a more significant crack surface area with higher growth rates. An increase of Mo and Mmax influenced fruit growth less, and hence had less effect on fruit surface conductance.



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  		FIG. 6. Effect of ±20 % variation of model inputs on fruit growth (left) and total fruit surface conductance (right). Model inputs are the initial relative fruit growth rate (a), the initial fresh mass (Mo) and the maximal fresh mass (Mmax). The solid line represents the adjusted fruit growth and simulated fruit surface conductance obtained for irrigated treatment in 2003 (I2), the dotted and dashed lines correspond to +20 % and –20 % variation of the parameter, respectively.

 


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 
Measurements of conductance on the peach cultivar ‘Alexandra’ varied between 100–950 cm h–1 (overall mean = 289 cm h–1). These values are similar to and even higher than those obtained on the same cultivar by Lescourret et al. (2001Go; 170–585 cm h–1), than those observed on apple (Malus pumila cv. ‘Bramley’ and ‘Cox’; 14–900 cm h–1 and 22–540 cm h–1, respectively) by Jones and Higgs (1982)Go, and on the sweet cherry cv. ‘Sam’ (36–54 cm h–1) observed by Knoche et al. (2001)Go.

Fruit surface conductance was shown to vary with fruit development as has been observed on apples (Jones and Higgs, 1982Go) and on sweet cherry (Knoche et al., 2001Go). It tended to decrease during fruit growth, except for fruits with large growth rates where an increase of total conductance occurred during the rapid growth phase. The model well described those patterns according to fruit growing conditions. The agreement between the model and the actual data was relatively accurate in 2003, but lower in 2002.

For a given date, the observed fruit surface conductance variability was important whatever the data set considered. As the model described most of the existing variability throughout fruit growth, this could be the result of differences in stomatal dilution and/or in crack occurrence in relation to different fruit growth rate.

Stomatal density at 21 DAFB, dsto(to), was estimated at approx. 2300 cm–2, which is equivalent to that observed in apple cv. ‘Golden Delicious’ (1500–2500 cm–2) but twice to three times lower than the stomatal density observed on avocado cv. ‘Fuerte’ (5000–7500 cm–2) (Blanke, 1992Go). The estimated value of the specific cuticular conductance, g'cut (86·2 cm h–1), was of the same order of magnitude as that of intact cuticles of ‘Valencia’ oranges (79–108 cm h–1; Moreshet and Green, 1980Go), tomato fruit (Lycopersicon esculentum; 50 cm h–1), sweet cherry fruit (Prunus avium; 35 cm h–1), capsicum fruit (Capsicum annuum; 32 cm h–1; Becker et al., 1986Go) and apple fruit (Malus domestica; 10 cm h–1; Maguire et al., 1999Go). Specific crack conductance, g'ck, estimated in the model was 12-fold higher than specific cuticular conductance, g'cut. Maguire et al. (1999)Go observed similar results by comparing crack conductance with intact cuticle conductance on apple cv. ‘Braeburn’. Moreover, the surface area of cracks simulated by our model could vary from 0–20 % of the fruit surface area. On apple fruit, the proportion of crack surface area to total fruit surface area was shown to vary from 0–15 % (Maguire et al., 1999Go), and on nectarines, dense networks of cracks around the stylar scar were observed (Nguyen-The et al., 1989Go).

Conductance values were over-estimated by the model in 2002 compared to the observed values. Firstly, this observation questions our assumption regarding the specific conductance of cuticle, which was considered as constant. According to several authors (Christensen, 1972Go; Leonardi et al., 2000aGo, bGo; Knoche et al., 2001Go), cuticular conductance could be affected by cuticle thickness, which is able to change according to year-to-year microclimatic variations and fruit growing conditions (Norris and Bukovac, 1972Go; Riederer and Schneider, 1990Go; Crisosto et al., 1994Go). It may even be wondered whether taking into account a seasonal variation of the specific cuticular conductance caused by the thinning of the cuticle with fruit age (which should be verified by direct measurements) would question the consideration of the crack component in the total conductance. Attempts at modelling were performed in this way, modifying the model by suppressing the crack component and varying the specific cuticular conductance, g'cut, with either time or fresh mass. However, the obtained time-courses did not agree with the observed data (RRMSE values were 2-fold higher for 2002 data sets than those obtained with the current model). Thus, while we cannot exclude that a variation of the specific cuticular conductance could improve the simulation of variation in fruit surface conductance, the crack component cannot be neglected in the model.

Secondly, the number of stomata per fruit was assumed to be constant with fruit development (Hieke et al., 2002Go) and stomata were considered to have the same specific stomatal conductance, g'sto, during all fruit development. However, observations of stomata on sweet cherry and apple fruit skin revealed that the pores could be partially or completely occluded during fruit development (Blanke and Lenz, 1985Go; Bukovac et al., 1999Go) or stomata could be transformed into lenticels, thereby decreasing the permeability of the skin (Blanke and Lenz, 1989Go). In the future, observations of stomata during fruit growth could be useful to determine whether the model has to be improved by introducing an occlusion coefficient in the stomatal conductance component, which is particularly influential on the model outputs. Moreover, we did not consider a possible change of the specific stomatal conductance, g'sto caused by stomatal closure due to water stress. However, such a change was shown to be insufficient to significantly improve the model prediction for the year 2002: with a specific estimation of g'sto for the 2002 data sets of fruits grown under water stress (the other parameter values being unchanged), the model error remained high (RRMSE = 0·395; data not shown).

Regarding healing properties ({delta}), 8 % of cracks healed each day, which corresponds to a total healing at the end of 110 days. The wound healing process lasts 3–5 weeks on sweet cherry fruit and sugar beet (Stesser, 1984Go; Ibrahim, 2001Go), although in this case the healing coefficient is about 20 %, i.e. 2·5 times higher than our observations. As this wound response differs with regard to plant species and organ (El Hadidi, 1969Go; Rittinger et al., 1987Go), measurements of the healing process on wounded fruits could be helpful to better estimate this parameter.

In conclusion, this study has developed a model linking fruit surface conductance to fruit growth. In particular, we have proposed an original way to integrate crack occurrence as related to growth rate. This model allows an estimation of the specific contribution of several components to the total conductance during the fruit growth. Although improvements have to be done, this kind of model may be very helpful as a structuring framework to study the different conductance components throughout growth, avoiding confounding effects and opening the way to some mechanistic representations. Moreover, it could be very useful in managing cropping practices to limit crack occurrence, and hence to prevent damage caused by fungal pathogens from the genus Monilinia. Indeed, cracks constitute preferential entry sites for the main agents of brown rot in stone and pome fruits, both in the orchard and after harvest, throughout Europe (Byrde and Willetts, 1977Go). By means of our model, simulations with different growth curves varying by cropping practice could be performed to determine the optimal curve that would minimize the crack surface area while preserving fruit size.


   APPENDIX I
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 


View this table:
[in this window]
[in a new window]
  		Values (±s.e.) of growth parameters for each data set

 


   APPENDIX II
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
ACKNOWLEDGEMENTS
 LITERATURE CITED
 


View this table:
[in this window]
[in a new window]
  		Values (±s.e.) of parameters in the allometric relationships

 


   ACKNOWLEDGEMENTS
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
LITERATURE CITED
 
We gratefully acknowledge P. Rouet for his assistance in the field experiments. The research was supported by grants from INRA national program #67 (‘Production Fruitière Intégrée’) and from a program of the ‘Ministère de l'Ecologie et du Développement Durable’, ‘Evaluation et réduction des risques liés à l'utilisation des pesticides’, France.


   LITERATURE CITED
TOP
 ABSTRACT
 INTRODUCTION
 MODEL DEVELOPMENT
 MATERIAL AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX I
 APPENDIX II
 ACKNOWLEDGEMENTS
 LITERATURE CITED
 

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