Summary Changes in elastic and plastic components ofmango (Mangifera indica L. cv ‘Cogshall’) fruit growth wereanalyzed with a model of fruit growth over time and in responseto various assimilate supplies. The model is based on water re-lations (water potential and osmotic and turgor pressures) at thefruit level. Variation in elastic fruit growth was modeled as afunction of the elastic modulus and variation in turgor pressure.Variation in plastic fruit growth was modeled using the Lock-hart (1965) equation. In this model, plastic growth parameters(yield threshold pressure and cell wall extensibility) variedduring fruit growth. Outputs of the model were diurnal and sea-sonal fruit growth, and fruit turgor pressure. These variableswere simulated with good accuracy by the model, particularlythe observed increase in fruit size with increasing availabilityof assimilate supply. Shrinkage was sensitive to the surfaceconductance of fruit peel, the elasticity modulus and the hy-draulic conductivity of fruit, whereas fruit growth rate washighly sensitive to parameters linked to changes in wall exten-sibility and yield threshold pressure, regardless of the assimi-late supply. According to the model, plastic growth wasgenerally zero during the day and shrinkage and swelling werelinked to the elastic behavior of the fruit. During the night, plas-tic and elastic growths were positive, resulting in fruit expan-sion.

Keywords: expansion, Mangifera indica, shrinkage, turgor

pressure, water relations.

Introduction

After a short period of cell division, fruit growth consists in theenlargement of fruit cells. For fleshy fruit, this period of cellenlargement is mainly characterized by a large accumulationof water that results from the balance between incoming andoutgoing fluxes (Ho et al. 1987). Water is supplied by thephloem as well as the xylem, and is lost through transpiration.Fruits are subjected to large variations in volume resultingfrom the changing balance between these various fluxes,

which have elastic and plastic components.Diurnal variation, described mainly by daily shrinkage, re-

sults from elastic variation in tissue volume. When water lossoccurs during the day, the fruit shrinks and its turgor pressureremains positive because of elastic adjustment (Jones et al.1985). According to Ortega (1985), a variation in turgor pres-sure leads to a proportional elastic variation in volume. Thismodel of elastic deformation relies on one parameter, cell wallelasticity, described by the elastic modulus. The elastic re-sponse was described by Génard et al. (2001) to simulatepeach stem and plum root diameter variations. In this model,the elastic modulus varies with turgor and cell size, as shownby pressure-volume relations in plant cells (Steudle et al. 1977,Tyree and Jarvis 1982).

Plastic fruit variation, which is an irreversible process, is afunction of cell wall extensibility and the turgor pressure atwhich wall yielding occurs (Lockhart 1965). Plastic growthrelies on two parameters: the yield threshold pressure beyondwhich growth could occur, Y, and cell wall extensibility, φ(Dale and Sutcliffe 1986), representing the irreversible andmetabolism-linked deformability of the wall (Cosgrove 1985).Rapid growth responses to changes in plant water status andlinear relationships between growth rate and turgor pressuresuggest that changes in growth are caused by changes in turgorpressure, and that φ and Y are constant. This approximationwas adopted by Nonami and Boyer (1990) to describe stemgrowth and by Fishman and Génard (1998) to simulate growthof peach fruit. However, some studies have demonstrated asubstantial lack of correlation between turgor pressure andgrowth rate (Schackel et al. 1987). Variation in Y and φ couldbe involved in the complex process of cell expansion. Proseuset al. (2000) suggested that factors involving cell wall metabo-lism controlled φ and the long-term growth of the cell. Thecessation of cell growth is controlled by alteration of wallstructure (McQueen-Mason 1995). Okamoto-Nakazato et al.(2000) isolated wall-bound proteins affecting Y.

Fishman and Génard (1998) presented a growth model forpeach fruit assuming only irreversible plastic growth. This

Tree Physiology 27, 219–230© 2007 Heron Publishing—Victoria, Canada

An analysis of elastic and plastic fruit growth of mango in response to

various assimilate supplies

MATHIEU LECHAUDEL,1,2 GILLES VERCAMBRE,3 FRANÇOISE LESCOURRET,3

FREDERIC NORMAND1 and MICHEL GÉNARD3

1Centre de Coopération Internationale en Recherches Agronomique pour le Développement, Département des Productions Fruitières et Horticoles,

Station Bassin Plat, BP 180, 97455 Saint Pierre Cedex, Ile de la Réunion, France

2Corresponding author (lechaudel@cirad.fr)

3INRA-PSH, Domaine St Paul, 84914 Avignon Cedex 9, France

Received February 27, 2006; accepted April 3, 2006; published online November 1, 2006

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

model predicts null turgor pressure during fruit shrinkage,which conflicts with the positive turgor pressure usually mea-sured in fruit tissues (Mills et al. 1997). This simulated nullturgor pressure probably results from the lack of a reversibledeformation process in the model, i.e., an elastic growthcomponent.

We present a model of mango fruit growth, inspired by thepeach fruit growth model of Fishman and Génard (1998), thattakes account of both the reversible elastic and the irreversibleplastic components of growth. We modeled diurnal and sea-sonal fruit growth and effects of various assimilate supply re-gimes. The predicted diurnal and seasonal dynamics of fruitfresh mass and fruit turgor pressure were tested against fieldmeasurements collected over different years, including fromleaf-to-fruit ratio treatments. Sensitivity of daily fruit shrink-age and fruit growth rate to model parameters was analyzed fortwo leaf-to-fruit ratios. The contributions of the elastic andplastic components in fruit growth are discussed for two stagesof fruit growth.

Materials and methods

Plant material

The experimental study was conducted on 11-year-old (in2000) mango (Mangifera indica L.) trees cv. ‘Cogshall’grafted on ‘Maison Rouge’ on Reunion Island (20°52′48″ S,55°31′48″ E) during the 2000, 2001 and 2002 growing sea-sons. The 2000 experimental plot comprised 10 rows, 7 mapart, of nine trees spaced 5 m apart. The trees observed in2001 and 2002 were in an adjacent plot, spaced at 5 m by 6 m.During the experiment, trees were irrigated every two days at100% replacement of evapotranspiration.

Each year, six weeks after flowering, 10–15 branches wereselected per tree, representing less than 10% of the totalbranches of the tree. The branches were chosen from the top ofeach tree to reduce the variability in light received by leaves ofthe selected branches which could significantly affect carbonassimilation and fruit growth. To obtain various assimilatesupplies, branches were girdled by removing a 10–15 mmwide band of bark, and were defruited or defoliated or both toadjust leaf-to-fruit ratios to 10, 25, 50, 100 or 150 leaves perfruit (with 50 leaves for 5 fruits, 100 leaves for 4, 2 and 1 fruitand 150 leaves for 1 fruit, respectively) during the 2000 grow-ing season. Girdled branches were prepared by the samemethod to provide two leaf-to-fruit ratios (10 and 100) duringthe 2001 growing season, and a single ratio (100) during the2002 growing season. To keep leaf-to-fruit ratios constant dur-ing fruit growth, new leaves emerging on girdled brancheswere removed. The girdling treatment was set up after thephysiological fruit drop, when fruit length was about 5 cm.

Model presentation

The model is based on a biophysical representation of peachfruit growth proposed by Fishman and Génard (1998). It simu-lates hourly diurnal and seasonal fruit growth after the periodof cell division. The fruit has two components, the flesh and

the stone. The flesh is described as a single homogeneouscompartment, separated from the exterior by a compositemembrane (Fishman and Génard 1998). Growth in fresh massof the stone is deduced from growth in fresh mass of the fleshby an empirical relationship. The main variable of the systemis the mass of water in fruit flesh (w). Unlike previous models(Fishman and Génard 1998, Lescourret and Génard 2005), ourmodel explicitly takes into account the reversible elastic en-largement of the fruit by including the parameter elasticmodulus, as proposed in the cell growth model of Ortega(1990). Moreover, we have considered that parameters of irre-versible plastic enlargement (cell wall extensibility, φ (MPa –1

h –1) and yield threshold pressure, Y (MPa)) varied during fruitgrowth. Hourly climatic data (global radiation, temperatureand relative humidity of the ambient atmosphere) and drymass accumulated in the fruit flesh are model inputs. Modeloutputs are changes in fruit fresh mass with time, and interme-diate variables linked to fruit water relations (osmotic andturgor pressures) and to plastic and elastic components of fruitgrowth. Although some governing equations were establishedat the cell level, we considered the accuracy of the values of themodel parameters mainly at the fruit level because fruit fleshwas represented as a single homogeneous compartment.

Water fluxes The rate of change in flesh water mass (dw/dt) isthe difference between water inflow from xylem and phloem(U) and water outflow through transpiration (Tf).

dw

dtU – T= f (1)

Fruit transpiration is calculated as described by Fishmanand Génard (1998):

T A H Hf f f a= αρ( – ) (2)

where Af is fruit area (cm2), α is the saturation concentration ofwater vapor (dimensionless), α = (MwP*) / (RT), with Mw = 18g mol –1 the molecular mass of water, R = 8.3 cm3 MPa mol –1

K –1 is the gas constant, T is temperature in Kelvin, P* (MPa) isthe saturation vapor pressure derived according to Nobel(1974), ρ is surface conductance (cm h –1) of the fruit peel, Hf

is relative humidity in the air-filled space within the fruit(equal to 0.996, cf. Fishman and Génard (1998)) and Ha is rela-tive humidity of the ambient atmosphere.

Fruit area Af, is related to fruit fresh mass (FMf, g) as:

Af fFM= γ η( ) (3)

where γ (cm2 g – η) and η (dimensionless) are parameters.A composite membrane separates the xylem vessels and

phloem sieve tubes of the stem from the fruit cell. The reflec-tion coefficient of this membrane for sugars is assumed to beequal to 1, as proposed by Nobel (1974). This implies that sol-utes and sugars do not enter the fruit by mass flow, but only viaactive uptake. Water flow (U) between the stem and the fruit istherefore:

220 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

( ) ( )U AL A a L= =f s f f f s fΨ Ψ Ψ Ψ– – (4)

where A is external area of the vascular network, assumed tobe proportional to fruit area Af, a (dimensionless) is the ratio ofthe area of the vascular network to the fruit area, L f (g cm – 2

MPa –1 h –1) is hydraulic conductivity between the stem and thefruit, and is the sum of the hydraulic conductivities of xylemand of phloem, Ψs (MPa) is xylem water potential of the stemand Ψf (MPa) is fruit water potential. Parameters a and L f ap-peared in a multiplicative form and could not be estimated sep-arately. Their product, aL f, was thus considered a uniqueparameter.

Fruit water potential is:

Ψf f f= P – π (5)

where Pf (MPa) is turgor pressure and πf (MPa) is osmoticpressure in the fruit. Turgor pressure is calculated from equa-tions related to changes in flesh volume and osmotic pressureis calculated from flesh composition in osmotically activecompounds.

Osmotic pressure According to the definition of van’t Hoff,the osmotic pressure πf is:

π fs=

RT n

w(6)

where n s is the number of moles of osmotically active solutesin the fruit flesh and w is the total volume of water in the flesh(cm3).

The various osmotically active solutes that contribute to theosmotic pressure (i.e., sugars, organic acids, minerals andamino acids) were individually expressed in the model, exceptfor amino acids, which were considered as a global pool:

n n njjs aa= +∑ (7)

where nj is number of moles of osmotically active solute j in‘Cogshall’ mango pulp, and naa is number of moles of totalamino acids.

The concentration of total nitrogen in mango flesh was rela-tively constant during fruit growth at about 0.05 to 0.12 g 100gFM

–1 (data not shown). In peach, about two-thirds of total nitro-gen is in soluble form, mainly amino acids, in fruit flesh (Lobitet al. 2002). In ‘Cogshall’ mango flesh, the concentration ofsoluble nitrogen was approximated as two-thirds of the totalnitrogen. The contribution of amino acids, naa, to the osmoticpressure is calculated based on the mean molecular mass ofamino acids found in mango (Hall et al. 1980) and the esti-mated total soluble nitrogen concentration.

The number of moles of the other osmotically active soluteswas expressed for each solute as:

n j

j

j

=( )prop DM

MM(8)

where propj (dimensionless) is the mass proportion of the os-motically active solute j in the dry mass of the flesh (DM, g)and MMj (g mol –1) is the molecular mass of the osmoticallyactive solute j. The proportion of osmotically active solute j

(propj) in the flesh dry mass was estimated from results of bio-chemical analyses by linear regression with the following ex-planatory variables: (1) the sum of degree days after full bloom(dd); (2) the dry mass of flesh (DM); and (3) their interaction:

prop dd DM DM ddj j j j j= + + +δ δ δ δ1 2 3 4( ) (9)

with parameters, depending on compound j.

Turgor pressure The change in flesh volume (V) can be writ-ten as the sum of water and dry matter changes:

dV

dt D

dw

dt D

d

dt= +

1 1

w s

DM(10)

where dw/dt and dDM/dt are the rates of change in water anddry mass of flesh, respectively, and Dw and Ds are the densityof water and carbohydrates, respectively. The density of themain compounds of dry matter such as citric acid (1.67), glu-cose (1.56), fructose (1.60) and sucrose (1.58) is around 1.6(Weast et al. 1984). Therefore, we used 1.6 for the density ofdry matter (Ds) and 1.0 for that of water (Dw). CombiningEquations 1, 4 and 5, Equation 10 can be developed as:

( )( )dV

dt DA aL P T

D

d

dt= + +

1 1

wf f s f f f

s

DMΨ – –π (11)

The change in flesh volume can also be written as the resultof both plastic (dVplas) and elastic (dVelas) volume variations:

dV

dt

dV

dt

dV

dt= +plas elas (12)

Because plastic growth follows the classical equation ofLockhart (1965), and the model of elastic deformation of tis-sues proposed by Ortega (1985), Equation 12 can be expandedas:

dV

dtV P Y V

dP

dtP Y

dV

dtV

dP

dtP

= + >

= ≤

φε

ε

( – )1

if

1if

ff

f

ff Y

(13)

where Pf is turgor pressure (MPa), and ε is the elastic modulus(MPa). The elastic modulus was considered to be constant in afirst approximation.

Combining Equations 11 and 13, fruit turgor pressure is cal-culated as:

TREE PHYSIOLOGY ONLINE at http://heronpublishing.com

ELASTIC AND PLASTIC FRUIT GROWTH IN MANGO 221

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

dP

dt V

A aL P T

D

D

d

dtV P Y

f

f f s f f f

w

sf

( – ) –

1 (DM)– ( – )

=

++

ε

π

φ

Ψ

≥

=

++

if ( )

( – ) –

1

f

f

f f s f f f

w

P Y

dP

dt V

A aL P T

D

D

ε

πΨ

s

f(DM)if ( )

d

dt

P Y

<

(14)

Fruit turgor pressure is expressed as a differential equationthat can be solved numerically providing initial values ofturgor pressure, Pf, volume, V, and fruit dry mass, DM.

Cell wall properties The threshold pressure (Y) and φ reflectproperties of the cell wall and may vary during fruit growth.Two processes called “strain-hardening” and “wall-loosening”affect Y according to the following equation (Green et al.1971):

dY

dth

dV

dt–= s (15)

where h (MPa cm – 3) is a parameter accounting for the harden-ing of the cell wall in response to stretching, and parameter s

denotes a process that softens the cell wall. The rate of wallloosening (s, MPa h –1) is probably a complex function of envi-ronmental conditions, metabolism and plant hormones ac-tively involved in fruit maturation. We assumed that s wasequal to zero because we considered the period of fruit growthbefore fruit maturation.

Equation 15 is then integrated over a given period of time,and at time t after full bloom becomes:

Y t Y h V t V( ) ( ( ) – )o o= + (16)

where Vo (cm3) is the flesh volume at to, the time at which fruitgrowth begins, corresponding to full bloom. We assumed thatVo and Yo are equal to zero at to. Therefore:

Y t h V(t)( ) = (17)

A correlation between the decrease in wall extensibility, φ,and the cessation of growth has been reported (Bütenmeyer etal. 1998). As cells mature, φ decreases and may reach zero(Proseus et al. 1999). We propose an equation where φ is con-stant and maximal (φmax, MPa –1 h –1) until a given time (ddini,°C day) and then decreases according to a rate parameter(τ, °C –1 day –1):

φ φ

φ φ τ

= ≤

= >

max ini

max(dd – dd )

ini

if dd dd

if dd ddini(18)

Fruit transpiration, osmotic and turgor pressures were cal-culated by Equations 2, 6 and 14, allowing estimation of waterfluxes. Integrating Equation 1 over time enabled us to calcu-

late the water mass of the fruit flesh mass. The dry mass offruit flesh was added to the water mass to calculate the fruitfresh mass of the flesh, which was added to the fresh mass ofthe stone to obtain the fruit fresh mass.

This general fruit growth model is valid for any stone fruit.Mango fruit contains latex located in laticifers. Plants withlaticifers are generally drought tolerant, and it is believed thatlatex is involved in the modulation of plant water status(Whiley and Schaffer 1997). However, because our experi-mental trees were well-watered, we considered mango fruit la-tex as an indistinct component of flesh, without influence onwater fluxes. Moreover, because laticifers are located in a thinlayer of the exocarp and are absent from the mesocarp, theirvariation in diameter, related to changes in latex pressure, isassumed to be negligible compared with variation in fruitgrowth caused by changes in water fluxes.

Model inputs and initial conditions

Climatic data were collected by the local meteorological sta-tion located close to the orchard (CIRAD, Saint Pierre). Maxi-mum and minimum daily temperatures were used to assess thesum of growing degree days after full bloom.

Hourly xylem water potential of the stem, Ψs, was calcu-lated from temperature, relative humidity and global radiationby an empirical relationship derived from concomitant mea-surements of xylem water potential of the stem and climaticdata (data not shown).

The rate of dry matter accumulation in the fruit flesh was de-rived from data on fruit diameter changes. The daily rate of drymass accumulation was converted to an hourly rate, assuminga constant rate throughout the day, which is consistent with ob-servations of constant sugar concentrations in phloem during aday (Sovonick-Dunford 1986) and simulations of diurnalchange in peach dry mass (Fishman and Génard 1998).

Estimation of ε, aLf and h required an initial value for fruitturgor pressure. We used the mean value of fruit turgor pres-sures measured at predawn of the day of estimation. When themodel was run throughout the season, it required an initialvalue for turgor pressure. The mean value of all measurementsassessed at predawn in 2001 and 2002 was used. Initial valuesfor fresh and dry masses of fruit were obtained from measure-ments assessed the day when simulation began.

Determination of model parameters and allometric

relationships

The experimental setup used to estimate model parameters isdescribed in Table 1. Some model parameters (ρ, ε, φmax, Y, andδ δ δ δj j j j

1 2 3 4, , , ) were estimated by independent measurementsdescribed in the Model parameterization section, whereas oth-ers (aLf, h, τ, and ddini) were estimated by model calibrationdescribed in the corresponding paragraph. A list of model pa-rameter values is presented in Table A1 of the Appendix.

Model parameterization By combining Equations 1 and 2,surface conductance of the fruit peel, ρ, was estimated from cli-matic data, fruit surface area, and the rate of change in flesh wa-ter mass of fruit for which water flow between the stem and the

222 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

fruit was null (U = 0 in Equation 1). Fruit transpiration wasmeasured on four detached fruits from the 10 and 100 leaf-to-fruit ratio treatments, during six days in November and De-cember 2001. Fruits were detached by cutting the panicle axis,and not the fruit peduncle, to avoid latex exudation. The panicleaxis was covered with adhesive, and fruits were suspended inthe tree canopy in their previous position. Transpiration was as-sumed to be equal to the rate of change in flesh water mass(dw/dt).

The elastic modulus, ε, of mango fruit was estimated duringthe last stage of fruit growth when net accumulation of freshmatter was close to zero, i.e., when plastic growth became neg-ligible. During this period, measured variation in fruit volumewas assumed to result solely from elastic variation. Diurnalvariation in flesh volume and turgor pressure were measuredduring one day from 0500 to 2000 h on fruits from the 10 and100 leaf-to-fruit ratio treatments in 2001 (December 4, 2001)and 2002 (December 17, 2002), respectively. The elasticmodulus was estimated with Equation 13 using measured val-ues of flesh volume variation and turgor pressure.

To estimate φmax, growth was measured during rapid fruitgrowth, when dd < ddini and φ = φmax (from Equation 18). Fromthe classical equation of Lockhart (1965), a linear relationshipbetween the estimated relative rate of plastic volume variationand turgor pressure is expected, with a slope equal to wall ex-tensibility φ, and a y-intercept equal to the opposite of theproduct of wall extensibility and yield threshold pressure(–φY). The relative rate of plastic volume variation dVplas/Vdt

was determined after subtracting elastic extension from totalrelative rate of volume variation (from Equation 13). Relativerate of elastic volume variation dVelas/Vdt was calculated fromthe variation in turgor pressure and the previously estimated

elastic modulus. Turgor pressure and diurnal variation in fleshvolume were measured between 0500 and 2200 h, on fruitsfrom the 100-leaf-to-fruit ratio treatment during the period ofrapid fruit growth (November 8, 25 and 29, 2002). We thentested if φ and Y varied between these dates during rapid fruitgrowth, and estimated φmax as the mean of the values.

To estimate parameters of Equation 9, we analyzed fruitsfrom the 10- and 100-leaf-to-fruit ratio treatments harvestedduring the 2001 growing season (every 15 days between Octo-ber 19, 2001 and January 21, 2002). Fruit pulp was thawed andfinely homogenized by a Polytron (PT1600E, Kinematica AG,Switzerland) in a vessel held in ice. Concentrations of calcium,magnesium and potassium were determined in the dilutedjuice by capillary ion analysis (CIA, Waters, MA), as pre-sented by Léchaudel et al. (2005). Concentrations of organicacids, sucrose, glucose and fructose were analyzed byhigh-performance liquid chromatography (HPLC, DIONEXCo., Sunnyvale, CA), as described by Léchaudel et al. (2005).

Model calibration Parameters aLf, h, τ, and ddini were esti-mated through model calibration by fitting output values of di-urnal water mass of fruit flesh and seasonal fruit fresh mass tothe observations by nonlinear least squares regression. The cal-ibration procedure for aLf and h used averaged data of volumevariations monitored on fruits from the 100-leaf-to-fruit ratiotreatment during three periods of three to four days in Novem-ber 2002. Because fruit growth was rapid during this period,the value of φmax was used as the wall extensibility value in themodel. Parameters τ and ddini (Equation 18) were estimatedfrom averaged data of the seasonal variation in fruit growthfrom the 100-leaf-to-fruit ratio treatment in 2000 and 2001.

Allometric relationships The allometric relationship be-

TREE PHYSIOLOGY ONLINE at http://heronpublishing.com

ELASTIC AND PLASTIC FRUIT GROWTH IN MANGO 223

Table 1. The experimental setup to estimate model parameters and validate the model. Abbreviations: LGR = low growth rate; and HGR = highgrowth rate.

Parameters Year Leaf-to-fruit ratio Measurement Date

Model parameterization

δ δ δ δj j j j1 2 3 4, , , 2001 10 Fruit composition October 19, 2001 – January 21, 2002

100 Fruit compositionε 10 Diurnal fruit growth (LGR) December 4, 2001

10 Fruit water relations (LGR)2002 100 Diurnal fruit growth (LGR) December 17, 2002

100 Fruit water relations (LGR)φmax 2002 100 Diurnal fruit growth (HGR) November 8, 25 and 29, 2002

100 Fruit water relations (HGR)

Model calibration

h, aLf 2002 100 Diurnal fruit growth (HGR) Periods beginning on November 8, 25 and 29, 2002τ, ddini 2000 100 Seasonal fruit growth November 15, 2000 – January 4, 2001

2001 100 Seasonal fruit growth October 19, 2001 – January 21, 2002

Model validation

2000 10 Seasonal fruit growth November 15, 2000 – January 4, 200125 Seasonal fruit growth50 Seasonal fruit growth

150 Seasonal fruit growth2001 10 Seasonal fruit growth October 19, 2001 – January 21, 2002

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

tween fruit fresh mass and fruit skin area was studied on 61fruits picked during the 2001 growing season. Each fruit waspeeled and its peel area was measured. Parameters relating fruitarea (cm2) to fruit mass (g), as described in Equation 3, were es-timated as (mean ± SE): γ = 3.65 ± 0.21 cm2 g – h and η = 0.73 ±0.10 (n = 61).

Dry mass (DM) and water mass (w) of fruit flesh were re-lated to fruit diameter (D) by empirical relationships obtainedfrom measurements of fruits harvested in 2000 from fiveleaf-to-fruit ratios, 10, 25, 50, 100 and 150 leaves per fruit:DM = 0.8191e(0.051D), R2 = 0.90, and w = 14.168e(0.035D), R2 =0.94, n = 253.

Fresh mass of the stone (FMstone) was calculated empiricallyfrom fresh mass of the flesh: FMstone = 0.1167FMflesh, R2 = 0.72and n = 253.

Measurements of stem and fruit water relations and fruit

growth

Water relations in the fruit flesh and in the stem Diurnalvariations in fruit water potential and fruit osmotic pressurewere measured during five days between November and De-cember in 2001 and in 2002. Two dates of measurement (De-cember 4, 2001 and December 17, 2002) corresponded to theperiod of low growth rate at the end of fruit growth (mean dailywater mass accumulation: 0.9 ± 0.6 g of water). The three otherdates (November 8, 25 and 29, 2002) corresponded to periodsof high fruit growth rate (mean daily water mass accumulation:4.4 ± 0.9 g of water). Measurements were performed at about3-h intervals between 0500 and 2200 h. Fruit water potentialwas measured with a WP4 psychrometer (Decagon Devices,Pullman, WA). For each measurement, two fruits from the10-leaf-to-fruit ratio treatment were used in 2001, and threefruits from the 100-leaf-to-fruit ratio treatment were used in2002. The water potential was determined on a disc of fruitflesh of about 2.5 cm in diameter. Four replicates per fruit wereused. As soon as the water potential was measured, each discwas frozen. The osmotic pressure of the juice extracted fromthe thawed fruit was later determined with a vapor pressureosmometer (Wescor, Logan, UT). Turgor pressure was calcu-lated by subtracting osmotic pressure from fruit water poten-tial.

Xylem water potential of the stem was measured with apressure chamber on a sample of 8–10 leaves, just before mea-suring fruit water potential. Four hours before the measure-ment, and the previous day for the first measurement in themorning, leaves were enclosed in a bag wrapped in aluminumfoil to inhibit leaf transpiration so that leaf water potential wasin equilibrium with that of the stem xylem at the point of leafinsertion.

Seasonal variation in fruit growth During the 2000 growingseason, six fruits from five leaf-to-fruit ratio treatments (10, 25,50, 100 and 150 leaves per fruit) were harvested each week be-tween November 15, 2000 and January 4, 2001. In the secondyear of the experiment, six fruits from two leaf-to-fruit ratiotreatments (10 and 100 leaves per fruit) were harvested every15 days between October 19, 2001 and January 21, 2002. For

each fruit, total fresh mass and masses of the different fruitcomponents (peel, pulp and stone) were measured. Two sam-ples of each component were taken from each fruit. One wasweighed and dried at 75 °C for 48 h to determine dry mattercontent. The other sample was frozen at –20 °C for futurebiochemical analysis.

Diurnal variation in fruit growth Diurnal variations in fruitdiameter were measured on two fruits from the 10-leaf-to-fruitratio treatment during one day on December 2001, and on 4–10fruits from the 100-leaf-to-fruit ratio treatment during four toseven successive days in November and December 2002.Changes in fruit diameter were recorded hourly with a linearvariable differential transformer (LVDT) mounted on an IN-VAR frame (Li et al. 1989) and connected to a data logger(Model 21 X, Campbell Scientific Ltd).

Model evaluation and simulations

Root mean squared error (RMSE), which describes the meandistance between simulations and measurements (Kobayashiand Us Salam 2000), was used for (1) internal validation ondata used for calibration and (2) external validation on inde-pendent data. This criterion was expressed as:

( )RMSE

–i ic

i

2

i 1

N

ii 1

N= =

=

∑

∑

n V V

n

where Vic is the studied variable (fruit turgor pressure or fresh

mass) simulated by the model on date i (i ∈ [1,N]) and Vi is themean value of the studied variable (fruit turgor pressure orfresh mass) measured on date i on ni fruits. A relative RMSE,RRMSE, was calculated as the ratio between RMSE and themean of all measurements.

The model was used to analyze elastic and plastic growthrate on a diurnal basis. Periods of high and low fruit growthrate, corresponding to 91–93 and 129–131 days after fullbloom, respectively, were simulated for fruit from the 100-leaf-to-fruit ratio treatment. Simulated elastic and plastic com-ponents of fruit volume variations were assessed.

Results

Model parameters

The fitted parameter values of the empirical model of fleshcomposition in osmotically-active compounds are summa-rized in Table A1 in the Appendix. The relationship betweensimulated and observed number of moles of total solutes wasclose to 1:1, with R2 = 0.7 (data not shown).

Model parameter values are reported in Table A2 in the Ap-pendix. Estimated fruit surface conductance (± SE) was ρ =231.0 ± 5.9 cm h –1 (n = 12). This value is higher than the esti-mate of 123.3 cm h –1 for mature green mangoes cv. Keitt(Fishman et al. 1996). However, Lescourret et al. (2001)showed that, among peach varieties, fruit surface conductance

224 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

varies between 62 to 901 cm h –1. Our value for unripeCogshall mango fruit is within this range.

We obtained an elastic modulus of about 15.32 ± 2.14 MPafor fruits from the 10- and 100-leaf-to-fruit ratio treatmentsover two growing seasons. This estimate is within the range ofvalues obtained for apple fruit cells (2–16 MPa), stem tissue(2–50 MPa), and for cells of higher plant tissues (0–30 MPa)(Tyree and Jarvis 1982, Steudle and Wieneke 1985, Dale andSutcliffe 1986, Génard et al. 2001).

Estimated φmax was 0.01725 ± 0.00131 MPa –1 h –1, which islower than the value obtained for leaves (Serpe and Matthews1992) and generally lower than published values of between0.03 and 0.2 MPa –1 h –1 for cells. Estimated φmax was alsolower than the value of 0.1 MPa –1 h –1 used in the peach fruitmodel of Fishman and Génard (1998). The y-intercept of thelinear relationship between plastic relative growth rate andturgor pressure, divided by estimated φmax, varied from 0.22 to0.60 MPa (data not shown), which is consistent with our hy-pothesis of a variable yield threshold pressure, Y. The coeffi-cient of cell wall hardening, h, was estimated to be 2.027 10 – 3

± 3.6 10 – 5 MPa cm – 3. The rate of decrease in cell wall extensi-bility, τ, was constant at 0.966 ± 0.071, whereas ddini variedwith the year when the experiment was performed: ddini =1005 ± 85 °Cd in 2000, and ddini = 686 ± 108 °Cd in 2001.These dates corresponded to the beginning of the growthslowdown for each growing season.

Parameter aL f had a value of 1.555 10 – 2 ± 8.3 10 – 4 g cm – 2

MPa –1 h –1. Based on a composite membrane area to fruit arearatio of 0.0273 as found by Fishman and Génard (1998) forpeach, hydraulic conductivity of the mango fruit would be0.570 g cm – 2 MPa –1 h –1. This value is of the same order ofmagnitude as the hydraulic conductivity values found formaize roots (0.097 g cm – 2 MPa –1 h –1; Steudle et al. 1993) andplant membranes (0.266 g cm – 2 MPa –1 h –1; Nobel 1974).

Internal and external validation of the model

Turgor pressure was satisfactorily simulated by the model forall leaf-to-fruit ratios and for both years of measurement (Fig-ure 1). The root mean squared error of the model (RMSE) wasabout 0.1 MPa (range 0.2–0.7 MPa) for observed values, andthe relative error (RRMSE) was 24% (Figure 1).

The model accurately simulated diurnal shrinkage and ex-pansion of the fruit during periods of rapid (Figures 2A–2C)and slow (Figure 2D) growth in November and December2002 (RRMSE < 1%). To study the behavior of the model withcontrasting assimilate supplies, fruit fresh mass was simulatedwith leaf-to-fruit ratio varying from 10 to 150 and during twosuccessive seasons (Figure 3). Regardless of the growing sea-son, the model was able to reproduce the pattern of fruitgrowth as well as the marked and positive effect of leaf-to-fruitratio on final fruit mass, even on data not used for model cali-bration (Figures 3C–3G). This can be considered a successfultest of the model. The RRMSE values for internal validation(Figures 3A and 3B) and external validation (Figures 3C–3G)of the model may be considered acceptable because they were

lower than 16 and 18%, respectively. A slight underestimationof fruit fresh mass in treatments with a high leaf-to-fruit ratiowas observed in 2000 (Figures 3A and 3F).

Analysis of model sensitivity to parameters

An analysis of the sensitivity of the model to parameters wasperformed with climatic data of the 2001 growing season andseasonal fruit dry mass from the 10- and 100-leaf-to-fruit ratiotreatments as model inputs. Sensitivities of the seasonal meanof daily shrinkage and of fruit growth rate to model parameterswere investigated (Table 2). The model was considered sensi-tive to a parameter for a given output when a 20% change inthis parameter value led to a change in the output value of morethan 5%.

Mean daily shrinkage was sensitive to the parameters re-lated to water fluxes and elastic extension (fruit surface con-ductance, ρ, the parameter aLf, and the elasticity modulus, ε).Sensitivity to parameters related to plastic extension was vari-able. Mean daily shrinkage was sensitive to a 20% decrease ofddini, but not to a 20% increase, or to the parameters φmax, τand h.

Among the parameters related to plastic extension, fruitgrowth rate was highly sensitive to ddini, τ and to a lesser extentto h, but not to ρ, aL f, ε or φmax. Fruit growth rate was twice assensitive to variations in h when wall extensibility φ was maxi-mal (i.e. with dd < ddini) than when φ decreased (data notshown). Leaf-to-fruit ratio did not affect mean daily shrinkageor fruit growth rate sensitivity to the parameters.

TREE PHYSIOLOGY ONLINE at http://heronpublishing.com

ELASTIC AND PLASTIC FRUIT GROWTH IN MANGO 225

Figure 1. Simulated values of fruit turgor pressure plotted against cor-responding observed values assessed in 2001 from the 10 leaf-to-fruitratio treatment (�) and 2002 from the 100 leaf-to-fruit ratio treatment(�). The root mean squared error (RMSE) and relative mean squarederror (RRMSE) are indicated on the graph.

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

226 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

Figure 2. Hourly variations in flesh wa-ter mass in fruits from the100-leaf-to-fruit ratio treatment usedfor model calibration during three mea-surement periods of rapid fruit growthbeginning on November 8 (A), 25 (B)and 29 (C), 2002, and for modelparameterization during one measure-ment period of slow fruit growth begin-ning on December 17, 2002 (D). Solidlines and dashed lines are observed andsimulated masses, respectively. Theroot mean squared error (RMSE) andrelative mean squared error (RRMSE)are indicated on each graph.

Figure 3. Observed (�) and simulated(lines) values of growth in fruit freshmass, for five leaf-to-fruit ratio treat-ments (10, 25, 50, 100, and 150). Eachvalue is the mean of six fruits and ver-tical bars represent standard deviationof measurements. Internal validation ofthe model: 100-leaf-to-fruit ratio treat-ment in 2000 (A) and 2001 (B). Exter-nal validation of the model: 10- (C),25- (D), 50- (E) and 150- (F) leaf-to-fruit ratio treatments in 2000 and 10-(G) leaf-to-fruit ratio treatment in2001. Year of experiment, leaf-to-fruitratio treatment, root mean squared er-ror (RMSE) and relative mean squarederror (RRMSE) are indicated on eachgraph.

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

Elastic and plastic growth rates

Simulated elastic and plastic components of fruit volume vari-ation are presented for periods of high (Figure 4A) and low(Figure 4B) growth rate, corresponding to a mean increase inflesh water mass of 4 and 2 g day –1, respectively. Elasticgrowth rate was negative in the morning and became positivein the afternoon. Plastic extension rate was positive during thenight and became zero a few hours after sunrise. Because plas-tic fruit growth was zero during the day, the contraction of fruitvolume observed during sunny hours depended only on theelastic behavior of the fruit. During the period of low growthrate (Figure 4B), the plastic component of fruit growth waslower and the elastic component of fruit growth was twice ashigh as during the period of high growth rate (Figure 4A). Thedaily sum of the simulated rate of elastic deformation variedbetween –1 and 1 g, regardless of the period of fruit growth.The sum of elastic deformation simulated over the growingseason was less than 2% of the final fruit mass.

Discussion

The mango fruit growth model is based on a theoretical ap-proach to water fluxes and cell growth in fruit. Calibration andvalidation of the model with field data demonstrated that it sat-isfactorily simulates diurnal and seasonal mango fruit growth.It needs however to be validated on larger data sets. Particular

features of the model that might be improved through furtherstudies are included in the following discussion.

The model accurately simulated the diurnal variation in fruitwater mass during the periods of rapid and slow fruit growth in2002. The concomitant diurnal variations in fruit transpiration(maximum at midday) and in xylem water potential of thestem (minimum at midday) induced a negative fruit water bal-ance between morning and mid-afternoon. Consequently, fruitturgor pressure decreased. Because the model considered ex-plicitly elastic growth, it predicted positive turgor pressureeven during high shrinkage; in contrast, the model of Fishmanand Génard (1998), which ignores elastic deformation, pre-dicted a null turgor pressure in such a case. Our model predic-tion was in accordance with our measurements and with mea-surements made on peaches (McFadyen et al. 1996) andapples (Mills et al. 1997). Plastic growth occurred mainly dur-ing the night. During daytime, plastic growth, as described bythe model, was generally zero and shrinkage and expansionwere linked to the elastic behavior of the fruit. The elasticmodulus was a parameter of fruit elasticity that affectedshrinkage, whereas parameters related to plastic growth didnot, as already observed for stem diameter variations (Génardet al. 2001). Contrary to variations in stem diameter (Génard et

TREE PHYSIOLOGY ONLINE at http://heronpublishing.com

ELASTIC AND PLASTIC FRUIT GROWTH IN MANGO 227

Table 2. Analysis of model sensitivity to parameters: variations, ex-pressed as a percentage of the reference value, of the mean dailyshrinkage over the growing season and of fruit growth rate after± 20% variation in each model parameter. Simulations used for thecalculation of mean daily shrinkage over the growing season and fruitgrowth rate were performed on fruits from the 10- and 100-leaf-to-fruit ratio treatments during the 2001 growing season. Abbrevia-tion: LFR = leaf-to-fruit ratio.

Parameter Parameter Shrinkage Fruit growth ratevariation (%)

10 LFR 100 LFR 10 LFR 100 LFR

Water inflow and transpiration

aLf +20 +6 +6 +2 +2–20 –8 –8 –3 –2

ρ +20 +8 +8 –1 –1–20 –8 –8 +1 +1

Elastic extension

ε +20 –10 –10 0 0–20 +13 +12 0 0

Plastic extension

φ max +20 +2 +2 +1 +1–20 –2 –2 –2 –1

ddini +20 +1 +1 +21 +18–20 –8 –7 –21 –19

τ +20 –4 –5 +46 +38–20 –2 –1 –7 –7

h +20 –4 –5 –5 –5–20 +4 +5 +6 +6 Figure 4. Simulated water accumulation in fruit flesh (solid line), and

elastic (dotted line) and plastic (dashed line) components of the fleshvolume variation for periods of high (A) and low (B) growth rate oc-curring in 2000 for fruit from the 100-leaf-to-fruit ratio treatment. Pe-riod A: 91–93 days after full bloom; and Period B: 129–131 days afterfull bloom. Shading indicates nighttime.

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

al. 2001), fruit shrinkage was sensitive to a parameter of waterinflow, aL f.

Study of diurnal volume variations have shown that fruitshrinkage is higher in the later stage of fruit growth for peach(Huguet and Génard 1995). We found a similar result formango. The model was able to account for this process be-cause the rate of simulated elastic deformation was directlyproportional to fruit size (Equation 13) and thus increased dur-ing fruit growth. Yield threshold pressure also increased withincreasing fruit size (Equation 17). Consequently, the differ-ence between fruit turgor pressure and Y (Equation 13) de-creased, leading to the observed decrease in the plasticcomponent of fruit growth during the season.

The climatic effect and the positive effect of leaf-to-fruit ra-tio on fruit fresh mass, observed experimentally, were accu-rately simulated (Figure 3). However, fruit mass from treat-ments with the highest leaf-to-fruit ratio in 2000 was slightlyunderestimated. Urban et al. (2002) showed that increasing theleaf-to-fruit ratio from 25 to 100 leaves per fruit on mango de-creased the leaf diffusive conductance to water vapor by about45%. Leaves from trees with a light crop load have a higherwater potential according to Berman and DeJong (1996). Aconstant xylem water potential of the stem was used for thesimulations, regardless of the leaf-to-fruit ratio. Increasing thexylem water potential of the stem when leaf-to-fruit ratio ishigher may lead to an increase in fruit water uptake and fruitmass. The model might be improved in this way.

Fruit growth rate was unaffected by aLf and ε, as observedfor stem diameter variations (Génard et al. 2001). Fruit growthrate was highly sensitive to parameters related to yield thresh-old pressure Y (i.e., h), and to variation in cell wall extensibil-ity φ (i.e., ddini and τ). Growth rates of stem diameter (Génardet al. 2001) and peach fruit (Fishman and Génard 1998) aresensitive to Y. The relationship between turgor pressure andfruit volume variations indicated that Y was likely variable. Pa-rameter Y may change rapidly, as demonstrated by Green et al.(1971) in Nitella. In the absence of wall softening, as assumedhere, Y changed with fruit growth rate. In vitro studies havedemonstrated an acid-induced decrease in the yield thresholdtension in the cell wall of hypocotyl segments (Okamoto andOkamoto 1994). Hormonal and pH control of yield thresholdpressure Y need to be considered in detail to develop the modelfurther.

Fruit growth rate was highly sensitive to variations in wallextensibility. Several enzymes are involved in the regulation ofcell wall extensibility. McQueen-Mason (1995) demonstratedthat, as cells mature, expansin activity is reduced and the cellwall is modified as a result of disruption of hydrogen bondsconnecting the cellulose microfibrils and the hemicelluloses.Based on an analysis of the correlation between cell wall enzy-matic activities and tomato fruit growth, Thompson et al.(1998) proposed that fruit growth rate is determined by epider-mal xyloglucan endotransglycosidase activity, and the termi-nation of growth by the rise in peroxidase activity. In ourmodel, cessation of fruit growth was partly regulated bychanges in cell wall extensibility. The model might be im-proved by increasing the representation of enzymatic activity

changes in the cell wall, and including the sources ofvariations of these enzymes.

Acknowledgments

The authors gratefully acknowledge C. Soria (CIRAD-Réunion, PôleFruits et Maraîchage) for his technical assistance and G. Wagman andL. Urban for revising the manuscript and improving the English.

References

Berman, M.E. and T.M. DeJong. 1996. Water stress and crop load ef-fects on fruit fresh and dry weights in peach (Prunus persica). TreePhysiol. 16:859–864.

Bütenmeyer, K., H. Lüthen and M. Böttger. 1998. Auxin-inducedchanges in cell wall extensibility of maize roots. Planta 204:515–519.

Cosgrove, D.J. 1985. Cell wall yield properties of growing tissue.Plant Physiol. 78:347–356.

Dale, J.E. and J.F. Sutcliffe. 1986. Water relations of plant cells. In

Plant Physiology. Ed. F.C. Steward. Academic Press, Orlando, FL,pp 1–48.

Fishman, S. and M. Génard. 1998. A biophysical model of fruitgrowth: simulation of seasonal and diurnal dynamics of mass. PlantCell Environ. 21:739–752.

Fishman, S., V. Rodov and S. Ben-Yehoshua. 1996. Mathematicalmodel for perforation effect on oxygen and water vapor dynamicsin modified-atmosphere packages. J. Food Sci. 61:956-961.

Génard, M., S. Fishman, G. Vercambre, J.-G. Huguet, C. Bussi,J. Besset and R. Habib. 2001. A biophysical analysis of stem androot diameter variations in woody plants. Plant Physiol.126:180–202.

Green, P.B., R.O. Erickson and J. Buggy. 1971. Metabolic and physi-cal control of cell elongation rate. Plant Physiol. 47:423–430.

Hall, N.T., J.M. Smoot, R.J. Knight and S. Nagy. 1980. Protein andamino acid compositions of ten tropical fruits by gas-liquid chro-matography. J. Agric. Food Chem. 28:1217–1221.

Ho, L.C., R.I. Grange and A.J. Picken. 1987. An analysis of the accu-mulation of water and dry matter in tomato fruit. Plant Cell Envi-ron. 10:157–162.

Huguet, J.G. and M. Génard. 1995. Effets d’une contrainte hydriquesur le flux pédonculaire massique et la croissance de la pêche.Agronomie 15:97–107.

Jones, H.G., A.N. Lakso and J.P. Syvertsen. 1985. Physiological con-trol of water status in temperate and subtropical fruit trees. Hortic.Rev. 7:301–344.

Kobayashi, K. and M. Us Salam. 2000. Comparing simulated andmeasured values using squared deviation and its component.Agron. J. 92:345–352.

Léchaudel, M., J. Joas, Y. Caro, M. Génard and M. Jannoyer. 2005.Leaf:fruit ratio and irrigation supply affect seasonal changes inminerals, organic acids and sugars of mango fruit. J. Sci. FoodAgric. 85:251–260.

Lescourret, F. and M. Génard. 2005. A virtual peach fruit model simu-lating changes in fruit quality during the final stage of fruit growth.Tree Physiol. 25:1303–1315.

Lescourret, F., M. Génard and S. Fishman. 2001. Variation in surfaceconductance to water vapor diffusion in peach fruit and its effectson fruit growth assessed by a simulation model. Tree Physiol.21:735–741.

Li, S.H., J.G. Huguet and C. Bussi. 1989. Irrigation scheduling in amature peach orchard using tensiometer and dendrometers. Irriga-tion Drainage System 3:1–12.

228 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

Lobit, P., P. Soing, M. Génard and R. Habib. 2002. Theoretical analy-sis of relationships between composition, pH, and titratable acidityof peach fruit. J. Plant Nutr. 25:2775–2792.

Lockhart, J.A. 1965. An analysis of irreversible plant cell elongation.J. Theor. Biol. 8:264–275.

McFadyen, M.L., R.J. Hutton and E.W.R. Barlow. 1996. Effects ofcrop load on fruit water relations and fruit growth in peach.J. Hortic. Sci. 71:469–480.

McQueen-Mason, S.J. 1995. Expansins and cell wall expansion.J. Exp. Bot. 46:1639–1650.

Mills, T.M., M.H. Behboudian and B.E. Clothier. 1997. The diurnaland seasonal water relations, and composition, of ‘Braeburn’ applefruit under reduced plant water status. Plant Sci. 126:145–154.

Nobel, P.S. 1974. Introduction to biophysical plant physiology.W.H. Freeman and Company, San Fransisco, 488 p.

Nonami, H. and J.S. Boyer. 1990. Primary events regulating stemgrowth at low water potentials. Plant Physiol. 94:1601–1609.

Okamoto, H. and A. Okamoto. 1994. The pH-dependent yield thresh-old of the cell wall in a glycerinated hollow cylinder (in vitro sys-tem) of cowpea hypocotyl. Plant Cell Environ. 17:979–983.

Okamoto-Nakazato, A., T. Nakamura and H. Okamoto. 2000. Theisolation of wall-bound proteins regulating yield threshold tensionin glycerinated hollow cylinders of cowpea hypocotyl. Plant CellEnviron. 23:145–154.

Ortega, J.K.E. 1985. Augmented growth equation for cell wall expan-sion. Plant Physiol. 79:318–320.

Ortega, J.K.E. 1990. Governing equations for plant cell growth.Physiol. Plant. 79:116–121.

Proseus, T.E., J.K.E. Ortega and J.S. Boyer. 1999. Separating growthfrom elastic deformation during cell enlargement. Plant Physiol.119:775–784.

Proseus, T.E., G.L. Zhu and J.S. Boyer. 2000. Turgor, temperature andthe growth of plant cells: using Chara corallina as a model system.J. Exp. Bot. 51:1481–1494.

Schackel, K.A., M.A. Matthews and J.C. Morrison. 1987. Dynamicrelation between expansion and cellular turgor in growing grape(Vitis vinifera L.) leaves. Plant Physiol. 84:1166–1171.

Serpe, M.D. and M.A. Matthews. 1992. Rapid changes in cell wallyielding of elongating Begonia argenteo-guttata L. leaves in re-sponse to changes in plant water status. Plant Physiol. 100:1852–1857.

Sovonick-Dunford, S. 1986. Water relations parameters of white ashsieve tubes. In Phloem Transport. Eds. J. Cronshaw, W.J. Lucas andR.T. Giaquinta. Colloquium 08–85, Plant Biology IV, Asimolar,CA, pp 187–191.

Steudle, E. and J. Wieneke. 1985. Changes in water relations and elas-tic properties of apple fruit cells during growth and development.J. Am. Soc. Hortic. Sci. 110:824–829.

Steudle, E., U. Zimmermann and U. Luttge. 1977. Effect of turgorpressure and cell size on the wall elasticity of plant cells. PlantPhysiol. 59:285–289.

Steudle, E., M. Murrmann and C.A. Peterson. 1993. Transport of wa-ter and solutes across maize roots modified by puncturing theendodermis. Further evidence for the composite transport model ofroot. Plant Physiol. 103:335–349.

Thompson, D.S., W.J. Davies and L.C. Ho. 1998. Regulation of to-mato fruit growth by epidermal cell wall enzymes. Plant Cell Envi-ron. 21:589–599.

Tyree, M.T. and P.G. Jarvis. 1982. Water in tissue and cells. In Physio-logical Plant Ecology II. Water Relations and Carbon AssimilationEd. H. Ziegler. Springer-Verlag, Berlin, pp 35–77.

Urban, L., F. Bertheuil and M. Léchaudel. 2002. A coupled photosyn-thesis and stomatal conductance model for mango leaves. ActaHortic. 584:81–88.

Weast, R.C., M.J. Astle and W.H.E. Beyer. 1984–1985. Handbook ofchemistry and physics. 65th Edn. CRC Press, Boca Raton, FL.

Whiley, A.M. and B. Schaffer. 1997. Stress physiology. In TheMango. Botany, Production and Uses. Ed. R.E. Litz. CAB Interna-tional, Wallingford, U.K., pp 147–173.

TREE PHYSIOLOGY ONLINE at http://heronpublishing.com

ELASTIC AND PLASTIC FRUIT GROWTH IN MANGO 229

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from

Appendix

230 LECHAUDEL, VERCAMBRE, LESCOURRET, NORMAND AND GÉNARD

TREE PHYSIOLOGY VOLUME 27, 2007

Table A2. Parameters and their estimated values for the mango fruit growth model.

Parameter Equation Value Significance

ρ (cm h–1) 2 231.0 ± 5.9 Fruit surface conductanceγ (cm– 2 g– η) 3 3.65 ± 0.21 Empirical parameters relating fruit areaη (dimensionless) 3 0.73 ± 0.10 (cm2) to fruit mass (g)ε (MPa) 13 15.32 ± 2.14 Volumetric elastic modulusaLf (g cm– 2 MPa–1 h–1) 4 1.555 10– 2 ± 8.3 10– 4 Product of the ratio of the composite membrane area to the fruit area

by the hydraulic conductivity of the composite membrane of thefruit for water transport

h (MPa cm– 3) 17 2.027 10– 3 ± 3.6 10– 5 Coefficient of “cell wall hardening”φmax (MPa–1 h–1) 18 1.725 10– 2 ± 1.31 10– 3 Maximal cell wall extensibilityτ (°C–1d–1) 18 0.966 ± 0.071 Rate of decrease in cell wall extensibilityddini (°C day) in 2000 18 1005 ± 85 Degree days after which the cell wall extensibility decreased in 2000ddini (°C day) in 2001 18 686 ± 108 Degree days after which the cell wall extensibility decreased in 2001

Table A1. Parameter values of the empirical model of organic acids, minerals and sugars composition of the mango cv. ‘Cogshall’ fruit flesh:propsolute = δ1 + δ2dd + δ3DMfl + δ4(dd)DMfl, where dd are degree days since full bloom and DMfl is dry mass of fruit flesh. All parameters (exceptδ1 for sucrose, δ4 for oxalic acid, Na+ and fructose concentrations) were significantly different from zero (P < 0.05). The linear relationship foreach solute was highly significant (P < 0.01).

Solutes (g g DM– 1 ) δ1 (10– 4 g g DM

– 1 ) δ2 (10– 6 g g DM– 1 dd–1) δ3 (10– 4 g g DM

– 2 ) δ4 (10– 6 g g DM– 2 dd–1) R2

Malic acid 662.06 –53.88 –24.64 2.41 0.43Citric acid 1625.02 –64.07 39.06 –4.78 0.59Pyruvic acid 6.90 1.61 0.51 –0.07 0.22Oxalic acid 47.51 –2.11 –0.30 0.0 0.62K+ 139.50 –5.24 –2.88 0.27 0.38Mg2+ 11.56 –0.79 –0.23 0.02 0.62Ca2+ 15.89 –0.66 –0.23 0.015 0.62NH4

+ 2.46 0.37 0.25 –0.03 0.25Na+ 1.28 0.08 –0.015 0.0 0.11Glucose 807.41 –63.26 –11.62 1.16 0.47Fructose 497.22 96.60 –10.79 0.0 0.42Sucrose 0.0 176.95 –72.49 9.03 0.66

by guest on May 14, 2011

treephys.oxfordjournals.orgD

ownloaded from