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The Derivation of Sink Functions of Wheat Organs

using the GREENLAB Model

MENGZHEN KANG

1,3,

*, JOCHEM B. EVERS

2

,JANVOS

2

and PHILIPPE DE REFFYE

3,4,5

1

Capital Normal University, 100037, BeiJing, China,

2

Crop and Weed Ecology, Plant Sciences Group, Wageningen

University, 6709 RZ, Wageningen, The Netherlands,

3

LIAMA, Institute of Automation, Chinese Academy of Sciences,

100080, Beijing, China,

4

Projet DigiPlante, INRIA Rocquencourt, France and

5

CIRAD, Montpellier, Cedex 5, France

Received: 1 March 2007 Returned for revision: 23 April 2007 Accepted: 11 July 2007 Published electronically: 27 November 2007

† Background and Aims In traditional crop growth models assimilate production and partitioning are described with

empirical equations. In the GREENLAB functional–structural model, however, allocation of carbon to different kinds

of organs depends on the number and relative sink strengths of growing organs present in the crop architecture. The

aim of this study is to generate sink functions of wheat (Triticum aestivum) organs by calibrating the GREENLAB

model using a dedicated data set, consisting of time series on the mass of individual organs (the ‘target data’).

† Methods An experiment was conducted on spring wheat (Triticum aestivum, ‘Minaret’), in a growth chamber from,

2004 to, 2005. Four harvests were made of six plants each to determine the size and mass of individual organs,

including the root system, leaf blades, sheaths, internodes and ears of the main stem and different tillers. Leaf

status (appearance, expansion, maturity and death) of these 24 plants was recorded. With the structures and mass

of organs of four individual sample plants, the GREENLAB model was calibrated using a non-linear least-

square-root ﬁtting method, the aim of which was to minimize the difference in mass of the organs between measured

data and model output, and to provide the parameter values of the model (the sink strengths of organs of each type,

age and tiller order, and two empirical parameters linked to biomass production).

† Key Results and Conclusions The masses of all measured organs from one plant from each harvest were ﬁtted sim-

ultaneously. With estimated parameters for sink and source functions, the model predicted the mass and size of indi-

vidual organs at each position of the wheat structure in a mechanistic way. In addition, there was close agreement

between experimentally observed and simulated values of leaf area index.

Key words: Wheat, Triticum aestivum ‘Minaret’, tiller, GREENLAB, organ mass, functional–structural model, model

calibration, multi-ﬁtting, source– sink.

INTRODUCTION

Functional–structural crop models aim at simulating plant

development and growth. An architectural model for

wheat based on L-systems has been developed for winter

wheat (Fournier et al., 2003) and spring wheat (Evers

et al., 2005, 2007). With accurate description of organ

size and geometry, the resulting 3-D architecture is very

realistic, and can be used to compute the light regime

within the wheat canopy. As yet in these L-system based

models, the functions describing organ size (sheath

length, blade length, blade area, etc.) in relation to phyto-

mer rank follow a predeﬁned pattern, because they are

ﬁtted directly to data using empirical functions, i.e.

without consideration of the underlying processes that

give rise to the ﬁnal size of the organs. Current research

is addressing the implementation of carbon gain and parti-

tioning from which the distribution of organ mass and size

along phytomer rank can be derived. The resulting sink

functions may be used in the L-system-based wheat

model to explore the response of tillering to light quality

and light quantity (Evers et al., 2005, 2007).

In this context, the current paper presents calibration of the

GREENLAB model (de Reffye and Hu, 2003; Yan et al.,

2004; Kang and de Reffye, 2007) for the spring wheat culti-

var ‘Minaret’. Data from four dates throughout the life cycle

of the plant, describing the mass of individual leaf blades,

sheaths, internodes and ears of main stem and tillers and of

the root system as a whole, were ﬁtted simultaneously to

the GREENLAB model, yielding one single set of model

parameters, including those deﬁning sink functions. The

application of sink functions in L-system-based models

allows, in principle, the size and the number of organs,

especially leaf blades, to become an emergent property of

the simulation, rather than being prescribed in advance.

Previous calibration of the GREENLAB model has been

done on single-stem wheat (Zhan et al., 2000) and on maize

from a ﬁeld experiment (Guo et al., 2006), using the gener-

alized non-linear least-square method presented in Zhan

et al. (2003). Model validation has been made on maize

with independent data from other years (Ma et al., 2006).

Therefore to date, all model ﬁtting and validation has

been performed for monoculm plants. Here we introduce

a new feature to the GREENLAB capabilities: calibration

on a branching source– sink system, exempliﬁed by tillering

wheat.

MATERIAL AND METHODS

Experiment set-up

An experiment was conducted in a growth chamber in

Wageningen, the Netherlands, from October, 2004 to

* For correspondence. E-mail mzkang@liama.ia.ac.cn

# The Author 2007. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved.

For Permissions, please email: journals.permissions@oxfordjournals.org

Annals of Botany 101: 1099–1108, 2008

doi:10.1093/aob/mcm212, available online at www.aob.oxfordjournals.org

February, 2005. Spring wheat plants, Triticum aestivum cul-

tivar ‘Minaret’, were grown in eight containers. These con-

tainers held a layer of medium-textured sandy soil of

approximately 35 cm depth; seeds were sown at 100

plants m

22

in a regular grid of 10 10 cm, at a depth of

approximately 5 cm. The soil was enriched with fertilizer

resulting in 15 g N m

22

to ensure a non-limited development,

and appropriate biocides were sprayed to control pests and

diseases. Day length of the growth chamber was set at 15 h,

with light intensity at around 425 mmol m

22

s

21

at the

level of the top of the canopy. The light was emitted by

400 W SON– T Agro Philips lamps and 400 W HPIT Plus

Philips lamps (3

.

5 lamps m

22

). Temperature was 17 8C

(0900– 2100 h), and 11 8C (2100– 0900 h); relative humi-

dity was 75 %.

Measurements

Six plants per harvest were sampled destructively,

excluding plants that were at the edge of a container.

Measured data included the dimensions and fresh and dry

mass of leaf blades, sheaths, internodes, ears and roots.

There were four harvests in total: at maturity (i.e. ligule

appearance) of the fourth main-stem leaf, at maturity of

the seventh main-stem leaf, at the ﬂowering stage, and

during grain ﬁlling. ‘Leaf states’ were monitored two or

three times per week for the main stem and all tillers of

six plants. The considered ‘leaf states’ included tip appear-

ance, leaf blade expansion, maturity and death. These six

monitored plants were the ones that were used for the

fourth harvest. Prior to harvests, incoming radiation was

measured at the top of the canopy using a Sunﬂeck cepto-

meter (Delta-T devices).

Naming system for leaves and tillers

The main stem arises from the embryonal axis and gives

birth to the ﬁrst-order branches, or ‘primary tillers’, from its

axillary buds in the leaf axils. ‘Secondary tillers’ are borne

from the buds in the leaf axils on the primary tillers, includ-

ing the buds in the axil of the prophyll at the base. Tillers

and leaves are distinguished here using the same naming

method as Evers et al. (2005), where a wheat phytomer is

deﬁned as a set of an internode with a tiller bud at the

bottom, a node above the internode, a sheath inserted on

the node, and a leaf blade. This deﬁnition is botanically

correct, but is different from the one presented by

Klepper et al. (1998), where the tiller bud is in the axil of

the leaf on the node. Leaves are numbered acropetally,

beginning with the ﬁrst foliar leaf, L1. The coleoptile

tiller is named T1, instead of T0 in Klepper et al. (1982).

Tiller leaves and secondary tillers are associated with the

name of the primary tillers. For example, the ﬁrst leaf on

T1 is L11. Similar to the coleoptile tiller, tillers arising

from the axil of the prophyll have a ‘1’ for the second

digit. For example, the tiller that appears from the axis of

the prophyll of T2 is T21.

Description of a potential and real wheat topology

In the current study, the main stem consisted of eight

phytomers below the ear. The ﬁrst four-to-ﬁve phytomers

showed internodes that did not elongate (‘short’ inter-

nodes), and the rest of the phytomers produced internodes

that did elongate (‘long’ internodes). Tillers emerged

only from the short internodes located at the bottom of

the parent shoot. The potential tillering pattern, which

is related to plant type, has been deﬁned in Bos and

Neuteboom (1998). Following this deﬁnition, a complete

potential pattern of tillers up to secondary order is illus-

trated for the current experiment in Fig. 1A, with the

example of ﬁve short internodes (concerning summed phy-

tomer number, see below) being present per axis. The

number of phytomers of primary tillers decreased acrop-

etally from T1 to T5. In addition, for the secondary tillers

the number of phytomers decreased with an increase in

FIG. 1. Illustration of wheat topology including primary and secondary tillers: (A) a potential pattern and (B) a real example. The white rectangles

represent short internodes, and the shaded ones represent long internodes. A circle represents an ear.

Kang et al. — Sink Functions of Wheat in GREENLAB1100

tiller position, which was associated with the decrease in

the number of short internodes (for example, 4 to 0 from

T1 to T5). The number of long internodes was the same

for all tillers, resulting in a similar height for all shoots.

The potential pattern of plant structure helps in understand-

ing the behaviour of the plant.

In reality, however, in a plant stand the topology per

plant is not so regular: the number of tillers, as well as

the number of phytomers in tillers, is often lower than the

potential pattern. Figure 1B shows an example of an

actual wheat topology observed in this experiment, in

which some tiller buds remained dormant. The potential

pattern can be regarded as the upper boundary of actual

wheat topology.

Summed phytomer number

In the main stem, phytomers are numbered or ranked

acropetally without ambiguity, i.e. one to eight for the

main stems in Fig. 1. For tillers, two kinds of phytomer

number can be distinguished. One is to regard the ﬁrst phy-

tomer in the tiller as rank one. According to this criterion, in

Fig. 1A, the top phytomer on T1 is of rank seven. Another

way is to count the ‘summed phytomer number’, from the

base of the plant to the phytomer in question (Bos and

Neuteboom, 1998). In that case, all tillers in the ideal

potential tiller pattern as in Fig. 1A have the same

summed number of phytomers (eight) as the main stem.

Modelling wheat development in GREENLAB

In the GREENLAB model, the different types of stem

can be represented by integer numbers, using the concept

of ‘physiological age’ (PA) (Barthe

´

le

´

my and Caraglio,

2007). PA is estimated a posteriori by analysing the mor-

phological, anatomical and/or functional attributes of a

botanical entity (Barthe

´

le

´

my and Caraglio, 2007). In the

GREENLAB model for wheat, the PA of phytomers on

the main stem is given as ‘1’, primary tillers are of PA

‘2’, and so on. The maximum PA of a plant is P

m

. As neg-

ligibly few tillers of higher order were observed in this study,

P

m

is set to 3. Another concept, ‘chronological age’ (CA) of

an organ or a tiller, is the number of growth cycles (see

below) that it has experienced since it was born.

Simulation of plant development is based on growth

cycles (GCs). One GC corresponds to the thermal time it

takes to generate a phytomer. In the current application of

GREENLAB to an annual plant such as wheat, a growth

cycle is equal to a phyllochron, the time between the

moments of appearance of two successive main stem

leaves (Klepper et al., 1982). The GREENLAB organogen-

esis model simulates plant topology with the automaton

concept (Zhao et al., 2001). The initial state of a stem is

a seed (for the main stem) or a bud (for a tiller). New phy-

tomers are produced at the top of the stem at each growth

cycle until the development of the stem ﬁnishes. The

maximum number of phytomers in a stem of a given PA

is another parameter of the automaton, being eight for the

main stem and tillers in the sense of summed phytomer

number. When the tip of the main stem transforms into a

ﬂower, the tillers become reproductive as well.

Computing the number of phytomers in the model

The number of growing organs is an important variable

for computing plant substrate demand in GREENLAB

(see below). For the potential pattern in Fig. 1A, the

number of organs produced can be computed with the sub-

structure method as presented in Courne

`

de et al. (2006). A

substructure is a component of a plant grown from a bud.

Let S

q, p

n, i

be the number of phytomers of PA p and CA i

in a tiller of PA q (q p) and CA n (n i). S

q, p

n, i

can be

computed recurrently from the tillers of higher order born

from the short internodes, as in eqn (1):

S

q; p

n;i

¼

1

; i . maxð0; n 8Þ; 0; otherwise p ¼ q

P

minð5;nÞ

j ¼ 1

S

qþ1;p

njþ1;i

p . q

8

<

:

ð1Þ

*1 is taken only when n2 iþ1 is smaller than 8.

The number of phytomers of PA p and CA i at plant age

n is N

p

(n, i) ¼ S

1,p

n, i

. The total number of phytomers in a

plant at CA n is sum of those of each PA and CA:

T

n

¼

X

P

m

p¼1

X

n

i¼1

S

1;p

n;i

Using eqn (1) for the potential pattern in Fig. 1A, from

cycle 1 to 8 the total number of phytomers at each cycle

is T ¼ [1, 3, 7, 14, 25, 41, 62, 87].

Given the number of organs per phytomer, the number of

organs, O,ofPAp and CA i at plant age n, N

p

O

(n, i), can be

computed in a similar manner; O indicating organ type:

B ¼ blade, S ¼ sheath, I ¼ internode, F ¼ ear (fruit in

general), R ¼ root. This variable is important in giving

the total demand and leaf area (see below).

Biomass production and partitioning in GREENLAB

At each growth cycle, the assimilates available for

growth, called ‘biomass’ in the GREENLAB environment,

are supposed to be located in a virtual common pool

(Heuvelink, 1995). From the common pool, biomass is par-

titioned dynamically among individual organs according to

their number, age and relative sink strength. The relative

sink strength for organs of given type O and PA p is

denoted as P

p

O

, which is a dimensionless variable indicating

the ability of different kinds of organs in competing for

biomass. The relative sink strength of all leaf blades in

the main stem is set to 1 as a reference value, i.e. P

1

B

¼ 1.

The growth rate of an individual organ can change from

its ﬁrst appearance through to full expansion. In the model,

for an organ O of CA i and PA p, its actual sink strength

varies with time in GCs, as expressed in eqn (2):

P

O

ði; pÞ¼P

p

O

f

O

ðiÞð2Þ

Kang et al. — Sink Functions of Wheat in GREENLAB 1101

where f

O

is an organ-type-speciﬁc function indicating the

pattern of change in sink strength in each cycle before the

organ stabilizes in mass. It can be any empirical function

that is suitable. In GREENLAB, because of its ﬂexible

form, a normalized discrete Beta function (Abramowitz

and Stegun, 1972) was chosen for f

O

, as in eqn (3):

f

O

ðiÞ¼

ði 0 5Þ

a

O

1

ðt

O

i þ 0 5Þ

b

O

1

=M

O

1 i t

O

0 otherwise

ð3Þ

where

M

O

¼

X

t

O

i¼1

ði 0 5Þ

a

O

1

ðt

O

i þ 0 5Þ

b

O

1

is a normalizing factor. t

O

is the organ expansion duration in

cycles. The form of f

O

is controlled by the parameters a

O

and b

O

. Different combinations of values of a

O

and b

O

result in bell-shaped, J-shaped, or U-shaped forms. A bell-

shaped curve means quick growth in the middle period and

small increments at the beginning and end of its growth

procedure.

Relative source and sink functions are computed in

GREENLAB. The model does not compute (gross) photo-

synthesis and net growth after subtraction of carbon losses

due to growth respiration and maintenance respiration.

The omission of respiration is a justiﬁable simpliﬁcation

as long as the gain in mass per unit substrate utilization

does not differ too much between the different organ

types; this is the case for wheat. Such differences become

large when some of the plant’s organs accumulate relatively

greater proportions of fat, oil or protein compared with the

rest of the organs (Penning de Vries, 1974). Assimilate

‘demand’ of an organ is thus derived from its net gain in

mass and is not augmented with the substrates that are

used in respiration. Hence, in cycle n, the appearing and

growing organs compose the total plant demand for

biomass, as expressed in eqn ( 4):

DðnÞ¼ð1 þ P

R

Þ

X

O¼B;S;I;F

X

n

i¼1

X

P

m

p¼1

p

O

ði; pÞN

p

O

ðn; iÞð4Þ

N

p

O

(n, i) is the number of organs of type O,PAp and CA i

present in the plant at cycle n. P

R

is the relative sink

strength of the root system, being the ratio between the

sink strength of the root system and the aerial parts.

As outlined in previous papers on GREENLAB (Yan

et al., 2004; Guo et al., 2006), growth per time step

(referred to as ‘biomass production’) is assumed to be

proportional to leaf area per unit ground area and plant

transpiration, which in turn depends on weather variables.

In GREENLAB the effects of light, water and temperature

are accounted for in a function E(i)(Wuet al., 2004).

Hence, the biomass production for a plant in cycle n, Q(n)

(g plant

21

GC

21

), is calculated using the Beer–Lambert

Law to account for a diminishing contribution per unit

leaf area as the latter increases, as in eqn (5):

QðnÞ¼EðnÞ

S

P

rk

1 exp k

S

Plant

ðnÞ

S

P

ð5Þ

S

P

(cm

2

) is the ground area available to a plant, computed

as the inverse of the population density. r and k are two

empirical parameters to be estimated with inverse model-

ling (i.e. ﬁnding the parameter set that reduces the differ-

ence between model output and measured data, listed in

the target data; see below, ‘Model calibration’): k is analo-

gous to the extinction coefﬁcient in the Beer– Lambert

Law. S

Plant

/S

P

gives the leaf area index (LAI), where S

Plant

is the total functioning leaf area of a plant, being the sum

of the individual leaf areas, as in eqn (6):

S

Plant

ðnÞ¼

X

P

m

p¼1

X

t

p

a

i¼1

S

p

B

ðn; iÞN

p

B

ðn; iÞð6Þ

In eqn (6), t

p

a

is the life span of a leaf blade of PA p from

appearance to senescence. For some plants this value

changes according the position of the leaves in the plant

(Vos and Biemond, 1992). S

p

B

(n, i) is the surface area of

a leaf blade of PA p and CA i at plant age n, computed

according to its dry mass, q: S

p

B

(n, i) ¼ q

B

(n, i)/e

p

. e

p

is

the speciﬁc leaf weight (the ratio between dry mass and

surface area of individual leaf blade, in g cm

22

), an input

variable of the model, obtained from observation.

A growing organ gains mass at each cycle proportional to

its sink strength and the ratio between the biomass supply

from the last cycle and current demand, as shown in eqn

(7):

Dq

p

O

ðn; iÞ¼p

O

ði; pÞ

Qðn 1Þ

DðnÞ

ð7Þ

The organ mass is the accumulation of its increment in

previous cycles, as in eqn (8):

q

p

O

ðn; iÞ¼

X

i

k¼1

Dq

p

O

ðn i þ k; kÞð8Þ

From the mass, the length and diameter of internodes are

obtained using an allometric relationship (Yan et al.,

2004), as well as the area of the leaf blade using speciﬁc

leaf weight. The mass and size of organs are therefore the

result of source– sink dynamics. The most important

output is the individual leaf area, which is used to

compute biomass production for the next cycle.

In summary, the growth of the plant is represented with a

set of recurrent mathematical formulae, based on several

assumptions regarding botany and ecophysiology (Yan

et al., 2004; Guo et al., 2006). The size and mass of organs

are calculated in a mechanistic way, based on modelling

their competition within the developing plant structure.

Kang et al. — Sink Functions of Wheat in GREENLAB1102

Nevertheless, parts of the model are still empirical, for

example eqn (5) and the expansion law, f, for each

organ type.

Model calibration

Several parameters controlling sink and source values are

difﬁcult to measure directly, including the relative sink

strength, P

p

O

, in eqn (2), the parameters a

O

and b

O

of the

expansion function eqn (3), and r and k in eqn (5). These

parameters are estimated by ﬁtting the measured organ

mass (the target) with the model output. The aim of

model calibration is to ﬁnd a set of parameters such as to

minimize the difference between the model output and

the measured data. The goodness of ﬁt is expressed in the

root-mean-squared error (RMSE) between the target data,

y, and the corresponding model output y

0

(

P

Q

), being func-

tions of model parameters (

P

Q

), as shown in eqn (9).

Jð

P

Q

Þ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

T

X

T

i¼1

y

i

y

0

i

ð

P

Q

Þ

2

v

u

u

t

ð9Þ

where J(

P

Q

) is RMSE. The generalized non-linear

least-square method (Press et al., 1992) was used for the

ﬁtting process. The Levenberg– Marquardt algorithm was

adopted by using the MINPACK library (Argonne

National Laboratory, Argonne, USA). The software for

model simulation, calibration and visualization was deve-

loped by the authors, and is freely available online (www.

greenscilab.org).

The target data, y, are the mass of individual organs from

one stage (in a single ﬁtting procedure) or several stages (in

a multi-ﬁtting procedure: data from several dates in one

analysis); the latter giving a better accuracy of parameter

estimation (Guo et al., 2006). For single-stem crops

without much variation in total phytomer number,such as

maize (Guo et al., 2006; Ma et al., 2006), the target data

can be the average mass of organs of the same phytomer

rank. For the wheat in current study, however, the plant top-

ology varies from one plant to another, especially for the

tillers. This prevents the use of the average data of the six

samples. Therefore, rather than calculating the average

plant, one individual plant with median tillering level

from each of the four sampling dates was taken for building

the target ﬁle.

According to eqn (4), the number of organs in the sample

plant needs to be known to be able to allocate biomass to a

speciﬁc structure. That number is derived from the observed

topological structure, which is coded in the target ﬁle that

can be read by the software for calibration. For each

tiller, its position, parent shoot and possible sub-tillers are

recorded. The historical development of this plant also

needs to be known in order to obtain the number of

organs in previous cycles. This was deduced from the

recorded structure. For example, there are four phytomers

in T3 at plant age eight in Fig. 1A: thus there were three,

two, one, zero phytomers in T3 at plant age seven, six,

ﬁve, four respectively. Each tiller is processed this way.

An important assumption in GREENLAB is that organs

appearing in the same cycle compete for assimilates from a

common pool. Therefore, those organs of the same CA and

PA and of the same type are regarded as being of the same

mass – noted as q

p

O

(i, j ) in eqn (8). To construct the corre-

sponding target, the data for organs of the same PA and

CA in a plant were averaged. For example, in Fig. 1B, the

top phytomers of all primary tillers are supposed to appear

together at cycle 8, thus the average mass of their leaf

blades is noted as q

2

B

(8, 1) in the target. Similarly, all q

p

B

(8, i)

(1 p 3, 1 i 8) were computed.

RESULTS

Time duration per growth cycle

With base temperature T

b

¼ 0 8C (Gallagher, 1979), phyto-

mer production per unit thermal time was 0

.

011 8Cd

21

for

the main stem (Fig. 2, slope of the line for main stem).

According to this development speed, the thermal time

per GC was 87

.

7 8Cd. With an average day and night temp-

erature of 14 8C in this experiment, the number of days per

GC was 6

.

15. At the four measurement dates (thermal time

399, 698, 1254 and 1553 8Cd), the corresponding ages of

plants were 5, 8, 14 and 18 GCs, respectively.

Expansion duration of organs

Expansion duration of an organ (t

O

in eqn 3) is an input

parameter that can be directly estimated from observations.

According to the main stem data (Fig. 2), the thermal time

for the full expansion of a leaf blade was 126 8Cd (i.e. from

tip appearance to ligule appearance), which corresponds to

nearly two GCs for main stem and tillers. The thermal time

it takes for full expansion of leaves was constant along the

stem (Fig. 2). This differed from the case of maize in Guo

et al. (2006), in which leaf expansion duration increased

acropetally before becoming stable.

Internode elongation occurs at the end of sheath exten-

sion (deﬁned by collar appearance) of the same phytomer

FIG. 2. Number of leaves that have appeared and matured on the main

stem in wheat as a function of thermal time.

Kang et al. — Sink Functions of Wheat in GREENLAB 1103

(Fournier et al. , 2003). It can be seen from the data for

internodes at cycle 8 (circles in Figs 3 and 4) that the

mass of the third internode from the top continued to

increase after cycle 8, which means that its expansion had

not ﬁnished at CA 3. An expansion duration of four GCs

is taken for internodes. The sequence of growth (ﬁrst leaf

blade, followed by the sheath and then the internode) was

simulated by calibration of the parameters that control the

expansion. For the ear, it was observed that the dry mass

did not vary during the last two measurements, so an expan-

sion duration of seven GCs is sufﬁcient. The root system is

regarded as a whole here, and it is supposed to have a con-

stant relative sink strength, P

R

, compared with the aerial

part, as in eqn (4), with unlimited expansion duration.

Functioning duration of leaves

The leaf blade stayed green for several cycles before

senescing. The leaf functioning duration t

p

a

is one of the para-

meters that inﬂuence source activity (eqn 6). According to

observations, for main stem ( p ¼ 1) t

p

a

was set to eight

GCs for the leaves associated with short internodes, and

was increased to 11 GCs from phytomer rank six to eight

(elongated ones). For the primary tillers, t

p

a

was set to

seven GCs for the leaves from short internodes, and then

took values 8, 9, 10 for the top three leaves, respectively.

For secondary tillers it was set to three.

Speciﬁc leaf weight

In GREENLAB the leaf blade area is computed from

leaf mass using the speciﬁc leaf weight. Values were

0

.

0037 g cm

22

for leaves of the main stem, 0

.

0035 g cm

22

for leaves of primary tillers, and 0

.

0033 g cm

22

for leaves

of secondary tillers.

Climate effects

According to the recorded data over the canopy, the light

intensity was stable during the experiment, with an average

416 mmol m

22

s

21

during the day. As the other climate

conditions (temperature, humidity) were set to be constant

during the experiment, E in eqn (5) was constant during

growth cycles. Without losing generality, it was set to 1,

and the effect of the actual value of E is captured in the

empirical parameter r in eqn (5).

Results of model calibration

At the ﬁrst measurement date (growth cycle, GC ¼ 5) the

internodes had not yet started to elongate, and only the mass

of leaf sheaths and blades were measured for the upper part

of thewheat plants. At the second measurementdate (GC ¼ 8),

the leaves on the main stem had ﬁnished their development,

and the long internodes had begun to elongate. On the last

two measurement dates most leaves had senesced, and only

productive tillers were present. As more and more organs

had appeared over time, multi-ﬁtting, i.e. one analysis on

FIG. 3. Mass of organs at each phytomer rank from four measurement

dates for the main stem (PA 1), from measurement (symbols) and simu-

lation (lines).

Kang et al. — Sink Functions of Wheat in GREENLAB1104

pooled data from all dates (Guo et al. , 2006), was necessary

in order to catch the full growth process.

In total, 19 parameters needed to be calibrated: para-

meters b

O

(eqn 3) that control the shape of the sink

change function, f, for leaf blade, sheath, internode and

ear (values of a

O

are set to 1); the relative sink strength

P

p

O

(eqn 2) of these four types of organs for each PA,

except that the sink strength of the leaf blade of the main

stem was set to a reference value of 1. The root system

was regarded as a whole, thus only one sink value P

R

(eqn 4) needed to be estimated. Other parameters are

the initial biomass, Q

0

, the leaf blade resistance, r, and

the light extinction efﬁcient, k. Prior knowledge about the

plants can be of great value in the ﬁtting process (for

example the general pattern of sink-strength change of

organs over time, or the ratio of sink strength between

different values of PA). Suitable initial values of the para-

meters decrease the risk of falling into a local minimum

in the ﬁtting process (in which case ﬁtting is not achieved).

Table 1 shows the set of parameters that give the ﬁtting

results displayed in Figs 3– 6. In contrast to the initial

biomass, Q

0

(0

.

013 g), the parameters in Table 1 are

dimensionless.

Figures 3– 6 show the target data and the model output

using the parameter values listed in Table 1. In total, 195

data are ﬁtted simultaneously, being organ type P

m

number of phytomer in axis (5, 8, 8, 8, respectively,

for each stage) and root system, excluding missing data

(e.g. fallen leaves). The total J(

P

Q

)) computed with eqn

(9) is 0

.

15 g. Note that the four target plants have different

topologies. Therefore, curves for individual organs dis-

played in Figs 3 and 4, computed for the four dates with the

same parameter set, do not exactly fall on top of each other,

although the old organs have ﬁnished expansion long

before the later dates and thus should keep the same mass

as on earlier dates. The main stem data was the most

regular, as the observed number of phytomers produced

was stable, while those from PA 2 had more experimental

variation. No ear was produced on tillers of PA 3 in these

plants. The mass of the root system was not measured at

the last sampling date. Thus only three data points were

available in the target data, as shown in Fig. 6.

TABLE 1. Calibrated parameters of wheat, ﬁtted with the least-

square-root method

(A) P

B

, P

S

, P

I

, P

F

and P

R

are sink strength of organs (blade, sheath,

internode, ear and root system), see eqn (2)

Sink strength PA 1 PA 2 PA 3

Blade (P

B

)1 0

.

56 0

.

3

Sheath (P

S

)0

.

47 0

.

32 0

.

14

Internode (P

I

)0

.

67 0

.

40 0

.

20

Ear (P

F

)0

.

57 0

.

23 0

Root (P

R

)0

.

10

.

10

.

1

(B) b

B

, b

S

, b

I

and b

F

are parameters controlling the form of sink function

of organs, see eqn (3). r and k are linked to source function, see eqn (5)

b

B

b

S

b

I

b

F

rk

0

.

53 2 0

.

45 0

.

6110

.

52

FIG. 4. Mass of organs at each summed phytomer rank from four

measurement dates for primary tillers (PA 2), from measurement

(symbols) and simulation (lines).

Kang et al. — Sink Functions of Wheat in GREENLAB 1105

The phytomer number in Figs 3–5 is the summed

phytomer number. It can be seen from the ﬁtted curve for

the blade in Fig. 3 that at GC 18 only the last leaf was

present. Moreover, the blade mass at GC 18 was lower

than at GC 14. This may have been caused by assimilate

remobilization due to grain ﬁlling. Here, this phenomenon

is not taken into consideration, i.e. the organs do not lose

biomass after they have reached maturity; the changing

role of organ from sink to source would otherwise have to

be made explicit in the model.

Sink functions

Using the parameters for sink-change function, f,aspre-

sented in Table 1, the normalized change over GCs of rela-

tive sink strength is shown in Fig. 7. The x-axis is the age of

organs in cycles, and the y-axis indicates the pattern of

change of the relative sink strength (see eqn 3) of each

type of organ; the value of which is zero at an organ age

older than the expansion duration, t

O

. Each type of organ

has its own speciﬁc pattern of expansion and related sink

strength, P

O

. Together with the developmental sequence

of organs, the growth process of a shoot can be simulated.

For phytomers with short internodes at the base of the plant,

where leaves are the only organs, a new leaf the blade

(possibly hidden) starts expansion growth a bit earlier

than its sheath, as the value of f

B

(1) is greater than f

S

(1).

Both organs ﬁnish expansion in two GCs. For phytomers

with long internodes, the duration of internode growth

extends over more CGs (i.e. 5) than those of leaf blades

and leaf sheaths. The top phytomer represents the whole

ear, which has a long period of growth. While the develop-

ment model gives the number and age of sink organs

during plant growth, the competition of organs, resulting

FIG. 6. Mass of the root system per plant from measurement (symbols)

and simulation (line) plotted against plant age, expressed in growth

cycles (GC).

FIG. 5. Mass of organs at each summed phytomer rank from four

measurement dates for secondary tillers (PA 3), from measurement

(symbols) and simulation (lines).

FIG. 7. Sink-change function of organs (blade, sheath, internode and ear),

expressed against growth cycles (GC).

Kang et al. — Sink Functions of Wheat in GREENLAB1106

in speciﬁc shares of allocation of resources, is simulated

according to sink functions.

Simulation with the calibrated model

Using the simulated leaf area, the simulated leaf area

index was compared to the leaf area index at the four

sampling dates. They were closely related, with the

RMSE being 0

.

598. The corresponding 3-D wheat plants

with potential tiller pattern were drawn (Fig. 8) to visualize

the results of this functional –structural model.

The accumulated biomass increment per plant at each

cycle, computed with eqn (5), was compared to the real

data (Fig. 9). According to the increment value (data not

shown), leaf blade and sheath were the dominant sinks in

the vegetative state, followed by the internode and ﬁnally

the ear. The pattern of biomass increment of the plant is

in line with that of PAR interception of existing wheat

growth models (Porter et al., 1993).

DISCUSSION AND CONCLUSIONS

This paper presents a calibration exercise using the

GREENLAB model for a spring wheat crop grown in a

climate chamber. Both the aerial part and the underground

part were included in the ﬁtting target, and the aerial part

consisted of vegetative and reproductive organs, with

tillers up to the secondary order. Therefore the work here

gives a complete example of applying the GREENLAB

model to branching cereal crops including the grain-ﬁlling

stage. From Figs 3–9, one can conclude that although the

model is simple in principle, it captures most of the features

of plant development and growth.

GREENLAB assumes all organs of a particular type have

the same sink properties for a given PA (equivalent to tiller-

ing order in the case of wheat), and consequently change in

organ mass along the shoot is the result of competitive

interactions among growing organs. In cereals, very pro-

nounced relationships can be seen between properties of

phytomers on tillers and on the main stem, expressed by

relative phytomer number (RPN) (Evers et al, 2005).

GREENLAB can reproduce similar patterns because

organs of the same type and RPN expand over the same

duration with the same expansion law, f

O

. This can be

seen if one plots the mass of organs of the same type and

different PA (Figs 3 –5) in one graph.

In this study, we have ﬁtted to detailed data from a

branching structure, where the topology of the plant

needed to be recorded. The procedure of transforming

from original data to the target data is thus tedious.

Moreover, although multiple plants were taken in this

study, the data from the rest of the plants were not fully

used; as a result, the parameters may be not representative

enough for the population. An alternative choice is to ﬁt

an average (in the sense both of topology and organ

mass) plant instead of individual ones.

The current work was done for wheat plants from a ﬁxed

environment. Our next work is modelling and veriﬁcation

of different wheat tillering levels and production under

different population densities, using a similar approach

to the one in this article with extension to stochastic

modelling.

ACKNOWLEDGMENTS

This work is supported in part by LIAMA (Sino-French

Laboratory in Information, Automation and Applied

Mathematics), Natural Science Foundation of China

(60073007), Chinese 863 program (2006AA10Z229) and

C.T. de Wit Graduate School for Production Ecology and

Resource Conservation (PE&RC).

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