The derivation of sink functions of wheat organs using the GREENLAB Model

Abstract

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').
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).
The masses of all measured organs from one plant from each harvest were fitted simultaneously. With estimated parameters for sink and source functions, the model predicted the mass and size of individual 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.

Full text

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
njþ1;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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