Annals of Botany 85: 869±886, 2000
doi:10.1006/anbo.2000.1149, available online at http://www.idealibrary.com on
MassFlowDyn I: A Carbon Transport and Partitioning Model for Root System
Architecture
L . P. R . B I D E L * {, L . PA GEÁ S {, L . M . R IV IEÁ R E {, G . P E L LO U X } and J . Y. LO R E N DE A U }
{I.N.R.A., 42 rue Georges Morel, B.P. 57, 49071 BeaucouzeÂ, France, {I.N.R.A., Unite d'Ecophysiologie
et Horticulture, Domaine Saint-Paul, Site Agroparc, 84914 Avignon Cedex 9, France and }I.N.R.A.,
Laboratoire d'Automatique et de MicroÐInformatique d'Avignon, Domaine Saint-Paul, Site Agroparc,
84914 Avignon Cedex 9, France
Received: 16 September 1999 Returned for revision: 24 February 2000
Accepted: 28 February 2000
Carbon partitioning is important for understanding root development but little is known about its regulation. Existing
models suggest that partitioning is controlled by the potential sink strength. They cannot, however, simulate hierarchical uptake other than by using absolute priorities. Moreover, they cannot explain that the changes in photoassimilate partitioning result from changes in photosynthesis. In this paper we present a model of phloem sieve
circulation, based on the model of Minchin et al. (Journal of Experimental Botany 44: 947±955, 1993). The root system
was represented by a network of segments to which meristems were connected. The properties of the segments were
determined by the di€erentiation stage. Photoassimilate import from each organ was assumed to be limited by a
metabolic process and driven by Michaelis±Menten kinetics. The axial growth was proportional to meristem respiration, which drives the ¯ux of new cells required for root elongation. We used the model to look at trophic apical
dominance, determinate and indeterminate root growth, the e€ect of the activity of a root on competition with its
neighbours, and the e€ect of photoassimilate availability on changes in partitioning. The simulated phloem mass ¯ow
yielded results of the same order of magnitude as those generally reported in the literature. For the main well
vascularized axis, the model predicted that one single apical meristem larger than its neighbouring laterals, was enough
to generate a taprooted system. Conversely, when the meristem of laterals close to the collar had a volume similar to
that of the taproot, the predicted network became ®brous. The model predicted a hierarchical priority for organ
photoassimilate uptake, similar to that described in the literature, during the decline in photosynthetic activity. Our
model suggests that determinate growth of the ®rst laterals resulted from a local shortage of photoassimilate at their
meristem, as a result of the limited transport properties of the developed roots. # 2000 Annals of Botany Company
Key words: MuÈnch theory, phloem transport model, photoassimilate-partitioning, root growth, root system
architecture, translocation.
I N T RO D U C T I O N
Carbon partitioning plays an important role in determining
the architecture of a plant, particularly the root system
(Nielsen et al., 1994). Photoassimilate availability during
plant growth a€ects the number of root primordia
(Bingham and Stevenson, 1993; Bingham et al., 1997),
and consequently the number of roots and their size
(Aguirrezabal et al., 1993; Thaler and PageÁs, 1996a,b),
altering the overall architecture of the plant and therefore
its ability to intercept resources from the environment.
Complex antagonistic interactions for assimilate partitioning have been observed between shoots and roots
(Dickson, 1991), and within the root system itself (Atzmon
et al., 1994). Depending on their activity, the needs and
sensitivity to carbohydrate availability will di€er with the
type and growth stage of plant organs (Wardlaw, 1990;
Dickson, 1991). Furthermore, similar roots at di€erent
locations in the root system react di€erently to carbohydrate availability; this is probably related to the distance
* For correspondence. Fax ‡33 2 41 22 56 35, e-mail bidel@angers.
inra.fr
0305-7364/00/060869+18 $35.00/00
from the source (Coutts, 1987). Photoassimilate partitioning is an important factor in the general functioning of
plants, partly conditioning their responses to a wide variety
of environmental constraints such as temperature (Farrar
and Williams, 1991; Williams et al., 1991), water stress or
mineral availability (Philipson and Coutts, 1977).
To consider the complex interactions within the plant
network, involving a spatial dimension (distances, topological relationships) and a temporal dimension (organ
growth dynamics), it is necessary to use a model to understand the whole system. There are three main groups of
models designed to simulate architectural development of
root systems depending on how carbon assimilate partitioning is approached. However, all of these models have
limitations.
The ®rst group of root models focuses on formalizing the
results of morphogenetic programmes with the help of a set
of production rules (Diggle, 1988; PageÁs and Aries, 1988;
Nielsen et al., 1994; Jourdan and Rey, 1997; Lynch et al.,
1997). These models do not explain the dependence of
morphogenesis and growth on photoassimilates. Individual
roots are assumed to grow following a predetermined
# 2000 Annals of Botany Company
870
Bidel et al.ÐA Carbon Transport and Partitioning Model
average relation derived from experimental data regardless
of the photoassimilate ¯uxes. When roots followed relations
that described responses to environmental constraints, they
acted independently of each other. Although these models
reproduce organ systems with topologies corresponding to
those observed in plants, they do not consider competition
for carbohydrate.
In the second group, photoassimilate sources feed a pool
which supplies the di€erent sinks (Clausnitzer and
Hopmans, 1994; Thaler and PageÁs, 1998)Ðthese models
are similar to those of the shoot (Jones et al., 1991;
Tabourel-Tayot and Gastal, 1998). A large number of
experiments have revealed apparent priorities between roots
(Atzmon et al., 1994; Thaler and PageÁs, 1996a), and such
pre®xed priorities have been imposed in these models
(Marcellis, 1993). When the environmental conditions are
carefully controlled, these models are able to take carbon
partitioning into account in global compartments. However,
the models' rules have proved to be unsatisfactory when
environmental conditions vary (Lacointe et al., 1995). These
models do not make it possible to simulate more local
morphogenetic responses. For example when the growth of
a root axis is blocked, photoassimilate does not redistribute
equally among all the axes in the system, yet the youngest
meristems and those nearest to the blocked apex seem to
bene®t more from the surplus of assimilate (PageÁs et al.,
1992).
The third group of models is mechanistic, and describes
partitioning as a result of sink strength and phloem transfer
properties at the organ level (e.g. Thornley, 1991; Minchin
et al., 1993). However, in these models, the organ network
is simpli®ed to the extreme, so much so that it does not
represent an evolving source/sink network.
In this paper we model the growth of the root system as
limited by photoassimilate supply along the phloem. Sink
strength of the meristems is determined by their size and
their respiratory potential. The concentration of photoassimilate at each meristem depends on the phloem hydrodynamic properties, the distance from the source, and the
competition from other segments and meristems. The
model is used to simulate the gradients in carbohydrate
concentration in the root system, and to show how di€erences in meristem sink strength and phloem conductivity
can control the morphology of the root system.
THE MODEL
The model consists of one photoassimilate source, which
represents the shoot, connected to a network of compartments describing roots of di€erent transport and sink
properties. Each meristem of the network produces a
segment of variable length daily, corresponding to the
meristem consumption of photoassimilates. The photoassimilates move by mass ¯ow along phloem vessels of
de®ned transport properties.
Representation of plant root architecture
The root system was composed of axes (Figs 1 and 2).
Each axis was divided into segments a few millimetres long,
corresponding to the daily root growth. Segments were
connected in series with nodes, each axis ending with the tip
meristem. To simulate rami®cation, the meristem-carrying
segment was divided into two sub-segments interconnected
with a node into which a lateral primordium was introduced.
A network of compartments, with their topological and
morphogenetic properties and energy consumption characteristics, represented the entire root system. Phloem hydrodynamic properties (number and average radius of sieve
tubes) were simulated to mimic a network of capillary tubes,
with uniform and constant properties in space. In time,
however, these properties were varied according to the type
of axis and the ontogenic di€erentiation stages. Each node,
i, was characterized by ( for units, see Appendix):
the solute photoassimilate concentration Ci within the
sieve tubes
the water potential Csieve tubei and its components Cpi
(hydrostatic), Cpi (osmotic), Cgi (gravitational), Cmi
(matric)
the passive axial ¯ux of water Fwi
the passive axial ¯ux of photoassimilates Fi
the passive e‚ux of water qwi
the photoassimilate unloading e‚ux Fi (and derived
parameters).
This representation o€ers discrete spatial compartmentalization for modelling. In order to simplify calculations, the
sieve tube elements of each segment were assumed to be
non-permeable along their lateral surfaces, the supply of
photoassimilates and water being restricted only to the
node (Fig. 2).
The photosynthetic source of photoassimilates was not
described in the present model. Consequently, the photoassimilate source function was modelled only by varying the
solute concentration C0 at the collar level and, therefore,
changing its concentration at this level simulated a change
in photosynthetic activity.
Basic assumptions
According to the theory developed by MuÈnch (1930),
carbon partitioning between sources and sinks is caused by
a convective bulk ¯ow resulting from the osmotic pressure
gradient generated by photoassimilate accumulation within
the sieve tube at the leaf level and its consumption at the
sink level.
Water potential of photoassimilates in the phloem
According to Nobel (1991), water potential on either side
of the sieve tube plasmalemma can be considered equal
because of high membrane hydraulic conductivity:
Capoplasti ˆ Csieve tubei
…1†
The sieve tube network is usually surrounded by an
apoplast at di€erent water potential values along the path
between the source and the sink (Nobel, 1991). The
simpli®cation of Minchin et al. (1993) was used and the
Bidel et al.ÐA Carbon Transport and Partitioning Model
871
F I G . 1. Simulated root system at 15 d. Each yellow segment corresponds to 1 d of root growth. Red cones are meristems, nodes are black.
sieve tube network was hypothesized to be in a pure water
solution in equilibrium with atmospheric pressure:
Csieve tubei ˆ Capoplasti ˆ 0
…2†
At any location in the sieve tube network, water potential is
given by:
Csieve tubei ˆ Cgi ‡ Cmi ‡ Cpi ‡ Cpi ˆ 0
…3†
Both gravitational potential gradient …DCgi =Dz ˆ
00098 MPa m ÿ1 † and matric potential Cmi were assumed
to be negligible, such simpli®cations having already been
introduced by Minchin et al. (1993). Thus:
Cpi ‡ Cpi ˆ 0
…4†
The Van't Ho€ law describes the osmotic potential as:
Cpi ˆ ÿRT
X
Ci;k
…5†
k solutes
where Ci,k is the concentration of the (i, k)th solute within
the ith phloem element.
On a molar basis, it is assumed that photoassimilates
generally represent over 80 % of solutes present in the
phloem sieve (Milburn and Kallarackal, 1989), with
concentrations ranging between C0 ˆ 50 and 1000 mol
sucrose equivalents m ÿ3. Therefore, in line with previous
models (Christy and Ferrier, 1973; Tyree et al., 1974;
Minchin et al., 1993), the osmotic components other than
872
Bidel et al.ÐA Carbon Transport and Partitioning Model
segment i
Φwi−1
i−1
Ψpi−1
qwi
node
i
Ri
Φi−1
Fi
Ψpi
Φi_k
Φwi
Ri + 1
i+1
Φi
Φwi_k
Ψpi+1
Rk
k
Ψpk
meristem k
F I G . 2. Discrete spatial compartmentalization for photoassimilate transport and partitioning. Symbols are de®ned in the Appendix.
those resulting from solute photoassimilates were omitted.
Hence
Cpi ÿRTCi
…6†
Cpi RTCi
…7†
continuous straight pipes (apparent length tube ˆ actual
length tube ˆ li), as commonly accepted (Young et al.,
1973; Tyree et al., 1974; Tyree and Dainty, 1975; Minchin
et al., 1993).
so that:
Photoassimilate transport between source and sinks
through the phloem. Passive solution ¯ow Fwi from the ith
element to the (i ‡ 1)th element of a sieve tube was
assumed to be governed by the irreversible thermodynamic
equation describing volume ¯ux across a membrane
(Katchalsky and Curran, 1965) as used already in modelling (Christy and Ferrier, 1973; Tyree et al., 1974).
Therefore, at the ith node:
F wi ˆ
‰…Cpi ÿ Cpi ‡ 1 † ‡ ssucrose …Cpi ÿ Cpi ‡1 †Š
Ri
…8†
(see Fig. 2) where Li ˆ 1=Ri is the water conductivity of the
sieve tube and plate. The re¯ection coecient ssucrose was
assumed to be zero for the sieve plate, that is nondiscriminating between solute and water (Christy and
Ferrier, 1973). These simpli®cations are also used by
Thornley and Johnson (1990) and Minchin et al. (1993),
and eqn (8) reduces to a Darcy law when ssucrose ˆ 0:
F wi ˆ
…Cpi ÿ Cpi ‡ 1 †
Ri
…9†
It is well established that ¯uid ¯ow within the sieve tube is
laminar under steady-state conditions (Christy and Ferrier,
1973; Tyree et al., 1974; Goeschl and Magnuson, 1986).
Axial resistance, Ri , to water movements along the phloem
was then calculated using the Poiseuille law, assuming
perfectly circular cross-sections of sieve tubes forming
Ri ˆ
8li Zsieve
pni r4i
…10†
where Zsieve ˆ 1.15 10 ÿ9 MPa s is the solution viscosity at
293.15 K, ni is the number of sieve tubes in the ith segment
and ri is the average radius of sieve tubes in the ith segment.
Sieve plate resistance resulted in additional head losses,
which may be estimated using the Poiseuille law (Tyree
et al., 1974). It was not taken into account at ®rst, but
incorporated into the model later (see below). The
di€erentiation rate of protophloem and metaphloem was
taken into account by assuming that the number of sieve
tubes ni of the ith segment increased linearly with time, t, up
to some maximum value nmax, i :
8
< ti 5 tM
:
ti 5 tM
ni ˆ nmax;i
ti
tM
…11†
ni ˆ nmax;i
where tM is the age of complete maturation of sieve tubes.
In order to simulate the secondary phloem vascularization, 40 sieve tubes of 10 mm radius were added on the 20th
day of growth.
As mean ¯uid speed is known to be high in translocation
processes, di€usion ¯uxes of photoassimilate along the
sieve tube were neglected (Tyree and Dainty, 1975).
Photoassimilate ¯ux Fi was made proportional to the
water ¯ux Fwi with:
Fi ˆ Fwi C i
…12†
Bidel et al.ÐA Carbon Transport and Partitioning Model
Phloem unloading at the sinks. Unloading from the sink
region was assumed to supply organs exhibiting photoassimilate catabolism limited by an enzymatic or transport
process (Ho et al., 1989). Their consumption, Fi , was
described by Michaelis±Menten kinetics (Goesch et al.,
1976; Magnuson et al., 1979; Minchin et al., 1993)
depending on photoassimilate concentration Ci :
F i ˆ P i Vi
Ci
Kmi ‡ Ci
Fi ˆ qwi Ci
1. Defining initial phloem potential conditions
Ψcollar = RTCcollar
Ψi = Ψcollar
…13†
where Pi is potential sink strength and Vi the volume of the
ith consuming segment.
Translocation by mass ¯ow processes implied that
carbohydrate unloading at each node occurred through
water ¯ow. Photoassimilate e‚ux Fi induced a concurrent
osmotic water e‚ux qwi , which was assumed to be not
limited by the permeability of the plamalemma. Furthermore, the volume contribution from sucrose was ignored.
873
4. Calculation of the water
flux and tree potential
Φi−1 RT
Φwi−1 = _______
Ψpi−1
Φi−1 Ri RT
Ψpi = Ψpi−1 − __________
Ψpi−1
2. Calculation of the
consumption tree of all organs
Ψp
___i
RT
Fi = FPPi Vi ________
Ψpi
Kmi + ___
RT
3. Calculation of the
carbohydrate flux tree
Φi−1 = Fi + Φi
Σ
Φk
k laterals
…14†
Hydrostatic potential Cpi at the ith phloem node and axial
water ¯ux Fwi ÿ 1 supplying this node were deduced from the
de®nition of both water and photoassimilate ¯uxes [eqns (9)
and (12)]. They were calculated from the status of the
(i ÿ 1)th previous element (Cpi ÿ 1 and Fiÿ1):
8
Fiÿ1 RT
>
>
(15)
> Fwi ÿ1 ˆ
<
Cpi ÿ 1
>
Fiÿ1 Ri RT
>
>
: Cpi ˆ Cpi ÿ1 ÿ
Cpi ÿ1
(16)
Axial photoassimilate ¯ux Fiÿ1 was deduced from the mass
conservation principle for water and photoassimilates
under steady state conditions:
X
8
Fi;k
(17)
Fiÿ1 ˆ Fi ‡ Fi ‡
>
>
<
i;k laterals
X
>
>
Fwi;k
(18)
: Fwi ÿ1 ˆ qwi ‡ Fwi ‡
i;k laterals
Algorithm to calculate photoassimilate supply and
consumption
The iterative algorithm of Fig. 3 was used to calculate
hydrostatic potential (Cpi ), photoassimilate concentration
(Ci) and photoassimilate consumption (Fi) of all compartments in the network, referred here onwards as the potential
tree, concentration tree, and consumption tree, respectively.
For the ®rst iteration, consumption was calculated assuming
all segments to have the initial concentration C0 . This choice
was arbitrary, but it was veri®ed that this initial concentration had no impact on the algorithm convergence
towards the same ®nal values. A decrease in the length of
the basic segments (i.e. from 5.0 to 0.1 mm) did not change
the predicted ¯uxes, velocities, or hydrostatic potential
gradients. If the unloading rate (sink strength) exceeded the
supply rate, the model predicted negative concentrations
5. Numerical convergence at
10−20 MPa (on potential)
for all compartments
– consumption tree
– photosynthate flux tree
– phloem water flux tree
– phloem water potential tree
F I G . 3. Calculation algorithm for steady state energetic calculations.
At each time step, a ®rst consumption tree is calculated (2) with the
collar potential value. Photoassimilate ¯uxes on the whole tree are then
deduced (3). These ¯uxes make it possible to calculate the new values
of potential that generate them until algorithm convergence.
within the given compartment. Consequently, the consumption rate of i ‡ 1; : : : ; n downhill segments was set at zero,
thereby resulting in root stunting.
Modelling root growth as a€ected by photoassimilate supply
At the start of each simulation, the taproot meristem was
adjacent to the collar node in which photoassimilate concentration was set at a constant level for the entire simulation. Thereafter, at each time step the calculation of the
energetic variables, using the algorithm described in Fig. 3,
made it possible to compute the consumption for each
organ. Axis growth was assumed to be linearly related to
apical meristem respiration (Bret-Harte and Silk, 1994). In
each time step the growth of the axis was calculated from the
meristem activity, and if this resulted in an elongation
greater than 0.1 mm, then an additional segment was
inserted just behind the meristem. Otherwise, growth was
neglected. Since the model assumed that the diameters of the
meristem and primary root segments were always constant,
they became model parameters. As a general rule, each 3-dold segment initiated a lateral primordium emerging on the
sixth day and beginning a new axis. This time interval was
taken from the average time length for meristem appearance
in Prunus persica L. Batsch grown in a rhizotron (data not
874
Bidel et al.ÐA Carbon Transport and Partitioning Model
shown). Meristem size at emergence was an input variable in
the model.
Input parameters measured by experiment
Model input parameters were determined from a previous
experimental study describing the architecture and respiration of four root types of Prunus persica L. Batsch exhibiting
contrasting growth behaviour (Bidel et al., 1999, 2000).
When the taproot and early ®rst order laterals have large
meristems and develop vascularization, they exhibit indeterminate growth. Moreover, the remaining ®rst and second
order laterals have smaller meristems and sieve tubes with
smaller average radius. They exhibit determinate growth of
variable duration. In this paper, each root type has been
represented in the model with its own characteristics
deduced from an histological study (Table 1). Each
meristem was given a diameter, a volume and a potential
respiration. The volume was calculated as a half-ellipsoid
where the focus is 1.5 times the diameter. The potential
respiration rate within the meristem was calculated using an
oxygen di€usion model (Bidel et al., 2000). Based on the
histological database, the average number and size of the
primary structure of the sieve tubes were set as being
volume-dependent, and used as input variables for the four
types of axes (Table 1).
The present model was developed using an Oriented
Object Analysis following the Uni®ed Modelling Language
(Muller, 1997) and was implemented in C‡‡ with
Visual C‡ ‡ 5.0 (Microsoft) on a personal computer
(Pentium 133).
Input parameters obtained by simulation
The potential respiration of a root segment calculated by
Bidel et al. (2000) was 2.5 10ÿ3 mol O2 m ÿ3 of tissue s ÿ1,
corresponding to 2.08 10 ÿ4 mol of sucrose equivalents
m ÿ3 of tissue s ÿ1. Using the common assumption that
respiration may represent one-third of photoassimilate
consumption (Lambers et al., 1996), the potential sink
strength of the primary structure was therefore set at
Pi ˆ 6.24 10 ÿ4 mol of sucrose equivalents m ÿ3 of tissue
s ÿ1, and applied to all rami®cations. This Pi value was
consistent with the values of Farrar and Williams (1992) for
Triticum aestivum L., and the values of Brouquisse et al.
(1992) for Zea mays L.
Phloem resistance
An increase in phloem resistance (by multiplying
Poiseuille resistance by a coecient ranging from 1.0 to
5.0) was tested to simulate the fact that Poiseuille resistance
underestimates axial phloem resistance because of head
losses due to sieve plates (Fig. 4). If resistance increased to
5.0, this increased the concentration gradient and reduced
the total ¯uxes consumed by the axis. It is unlikely that this
resistance would exceed 2 Poiseuille's resistance, since the
potential gradients needed to sustain the respiratory ¯uxes
MPa
A
0.26
0.22
0.18
The source ¯ux (F0) was set at variable concentrations
between C0 ˆ 0 and 1000 mol m ÿ3 similar to Minchin et al.
(1993). The model requires the Michaelis±Menten kinetics
and the phloem resistance parameters to describe carbon
partitioning of the whole root system. Because of their very
low tissue volume, meristems had a negligible contribution
to the whole ¯ux in the network at any given time and the
segment properties controlled the consumption level of the
root system at all levels of source activity.
0.14
Kinetics of photoassimilate uptake by meristems and
segments
0.18
A Km value of 100 mol m ÿ3 (Goeschl and Magnuson,
1986) was given to segments of over 1-d-old, which
made it possible to reach 90 % of the potential sink
strength at C0 ˆ 1000 mol m ÿ3 (Km ˆ 50 mol m ÿ3, 95 %
at 1000 mol m ÿ3). Many authors have observed that, with
low photosynthate supply, taproot meristem activity and
sub-apical growth continue, at the expense of the activity of
other organs (Schulz, 1994). This situation is possible only if
the meristem dominates the rest of the axis, i.e. it possesses
higher photoassimilate anity and maintains itself in the
elongating tissues before rapidly decreasing in adult tissues.
Thus, meristem anity was set at 50 mol m ÿ3 and the 1-dold elongating sub-apical segment at 75 mol m ÿ3.
0.14
0.1
nr, R=1 x P
r, R=1 x P
nr, P=1.5 x P
nr, P=1.5 x P d=1
r, P=1.5 x P
rf, P=1.5 x P
nr, R=2 x P
0
100
r, R=2 x P
200
300
400
MPa
mm
B
0.26
0.22
0.1
nr, d=1
nr, d=3
nr, d=4
nr, d=5
0
100
r, d=1
r, d=3
r, d=4
r, d=5
200
300
400
mm
F I G . 4. A, Phloem hydrostatic potential pro®le of the taproot for
di€erent segment resistivities, in the rami®ed (r) and the non-rami®ed
(nr) case, for a protophloem di€erentiation duration of 1 or 5 d. Curve
(r) was predicted for a taprooted system with meristem diameters of
760, 280 and 100 mm for the three orders. Curve (rf) was predicted for
a ®brous root system, with meristem diameters of 760, 700 and 100 mm
for the three orders. B, Phloem hydrostatic potential pro®le of the
taproot for di€erent numbers of functional sieve tube in both the
rami®ed (r) and the non-rami®ed (nr) case. This number was
dependent upon the protophloem di€erentiation duration (1 to 5 d).
I
±
II
III
IV
Taproot ( primary structure)
(secondary structure)
Early ®rst lateral order
Late ®rst lateral order
Second lateral order
Type
140
100
50
380
±
Radius
(mm)
Kmi
(mol m ÿ3)
50
±
50
50
50
Size
(m3)
229.8 10 ÿ12
±
17.2 10 ÿ12
6.3 10 ÿ12
0.8 10 ÿ12
Meristem
11.25 10 ÿ3
15.37 10 ÿ3
28.13 10 ÿ3
4.30 10 ÿ3
±
Pi
(mol O2 m ÿ3 s ÿ1)
0.8
0.7
0.6
1.1
1.5
Diameter
(mm)
8(3.0) ‡ 16(7.5)
4(3.0) ‡ 8(6.0)
4(3.0)
100
Kmi
(mol m ÿ3)
Segment
12(3.0) ‡ 20(7.5)
12(3.0) ‡ 20(7.5) ‡ 20(10.0)
Sieve tubes number
(radius mm)
T A B L E 1. Input parameters of the simulation model
6.24 10 ÿ4
Pi
(mol m ÿ3 s ÿ1)
5d
Protophloem
maturation
Bidel et al.ÐA Carbon Transport and Partitioning Model
875
876
Bidel et al.ÐA Carbon Transport and Partitioning Model
would be unrealistic (higher than 1.0 MPa m ÿ1). Therefore,
in the simulations it was assumed that sieve tubes had a
resistance corresponding to 1.5 Poiseuille's resistance, to
integrate estimations made by Sheehy et al. (1995), obtained
for dicots.
M O D E L S I M U L AT IO N S
Firstly, photoassimilate concentration pro®les along the
root axes were studied by running the model with the parameter values listed in Table 1. We compared the photoassimilate supply and the growth of the main root axis,
which has many large phloem vessels, with ®rst and second
order laterals axes that have fewer thinner vessels. Secondly,
we focused on competition for photoassimilates between
roots in di€erent topological locations. Finally, we studied
the in¯uence of photoassimilate availability on the root
architecture.
Simulations were analysed using ®ve variables: sap ¯ux
Fwi (m3 s ÿ1), average apparent water speed vi (m s ÿ1),
soluble photoassimilate concentration in sieve tubes Ci (mol
of sucrose equivalents m ÿ3), axial mass ¯ux for photoassimilate Fi and soluble photoassimilate ¯ux density (often
referred to as speci®c mass transfer, SMTi , in mol of
sucrose equivalents s ÿ1 m ÿ2 of phloem sieve tube) which is
the product of sap concentration Ci and average speed vi .
R E S U LT S
Photoassimilate concentration pro®les along root axes
As described previously, when the axis grew, meristem
activity led to the creation of additional competing segments, driving the meristem further from the source and
increasing total phloem resistance to the source. Because of
the concentration gradient along the axis, the model
predicted decreasing consumption towards the apex. The
concentration and ¯ux pro®les along the axis varied greatly
depending on the assumed phloem transfer properties and
segment uptake kinetics.
If all segments were given the same uptake kinetics
(Kmi ˆ 100 mol m ÿ3 and Pi ˆ 6.02 10 ÿ4 mol of sucrose
equivalents m ÿ3 of tissue s ÿ1), the model suggested that the
duration of protophloem di€erentiation is the main factor
a€ecting the apical concentration pro®le (Fig. 4) and
conditioning meristem activity. When protophloem maturation duration was set at 5 d, lateral meristems were maintained at low photoassimilate concentration for a few days,
thus altering their initial growth (Fig. 4). Resistance level
had a major in¯uence on concentration gradient and
therefore on consumption.
assimilate consumption was essentially controlled by source
concentration, C0 . When C0 approached the Km of the
segments, their consumption changed markedly, a€ecting
total photoassimilate consumption. A gradual change from
a source to a sink-limiting situation was observed, as C0
increased. The whole axis then behaved almost as a
Michaelis sink.
When potential sink strength, Pi , was given a value of
6.24 10 ÿ4 mol m ÿ3 of tissue s ÿ1, it took a metre of
primary structure for the meristem to stop initiating
segments, when photoassimilate concentration decreased
to 0. When this primary structural axis (20 sieve tubes of
7.5 mm in diameter) reached a length of 40 cm, average
speed v0 and SMTi exceeded those of the most intense sinks
known (e.g. seeds with an SMT of about 0.049 mol
equivalent sucrose m ÿ2 of sieve tube s ÿ1, corresponding to
6 g cm ÿ2 of sieve tube h ÿ1). Such a value is unrealistic for a
weak sink. Conversely, if cambial growth appeared on the
20th day (40 sieve tubes of 20 mm diameter added to the
protophloem vascularization), the number of large diameter
phloem tubes became large enough to reduce segment
resistance by an order of magnitude. This resulted in small
concentration decreases along the woody structures and,
consequently, primary axis growth was maintained. In this
case, SMTi and average speed v0 , as well as the concentration gradient, had realistic values (Fig. 5). Axis resistance
did not have the same in¯uence for a weak or a strong source.
When C0 ˆ 900 mol m ÿ3, a 1 mm reduction in tube diameter
from 8 to 7 resulted in a 0.12 to 0.35 MPa m ÿ1 increase in
gradient concentration, a 2.3 to 6.5 10 ÿ4 m s ÿ1 increase
in speed, and a very slight decrease in total photoassimilate
consumption. Conversely, at C0 ˆ 100 mol m ÿ3, the same
reduction markedly reduced total consumption while speed
v0 remained unchanged.
Photoassimilate supply and growth in lateral roots
(with fewer, thin phloem vessels)
In the ®rst (type III) and second (type IV) order lateral
axes formed from meristems smaller than the taproot, both
the number of vascular strands and the tube diameter
imposed a much higher resistance than that of the taproot
(more than an order of magnitude for types III and IV;
Table 2). Consequently, this resulted in a rapid drop in
photoassimilate concentration, leading to axis growth
stunting 3 to 15 d later, depending on the set number of
phloem vessels. The rapid initial growth decreased even
faster as the phloem concentration in the meristem dropped
below its Km value. According to the model, determinate
root axis growth would therefore be the result of a local
shortage of photoassimilates at the meristem level.
Photoassimilate supply and growth in main root axes
(with many thick phloem vessels)
Formation of ®brous and taproot system depends on relative
meristem size
Photoassimilate concentration gradient along the main
type I axes remained relatively low ( from 0.01 to
0.5 MPa m ÿ1 of root length, depending on Pi), each
segment consuming a rather stable fraction of its potential
sink strength along the axis (Table 2). Hence, photo-
The concentration gradient along the non-rami®ed axis
was uniform. The introduction of laterals resulted in a rapid
drop in concentration and pronounced gradients at speci®c
locations, linked to a large photoassimilate uptake by the
®rst laterals (Fig. 4). The gradients between these locations
Bidel et al.ÐA Carbon Transport and Partitioning Model
877
C0 = 300 mol m−3
0.10
0.30
0.50
0.70 MPa
F I G . 5. Phloem hydrostatic potential at 30 d, with the beginning of the woody structure meristem diameter (760, 500, 280 mm). Meristems are
more depleted in the taproot due to its large consumption and in the second order laterals due to the lower phloem transport capacities inducing
large gradients. Introduction of secondary phloem maintained high photoassimilate concentration in the older third part of the taproot
(coloured red).
were quite constant. The predicted gradients were of the
same orders of magnitude as those reported in the literature
(Hocking, 1980; Milburn and Kallarackal, 1989).
While the large meristems had lower respiration rates
than small meristems (Bidel et al., 2000), their greater
volume made them larger sinks for photoassimilates, and
therefore allowed them to form longer segments between
each time step (Fig. 6). Even if the taproot resistance value
assigned to ®rst and second order laterals was the same, the
simulations showed that the size di€erences between meristems were large enough to generate an apical dominance
pattern with an indeterminate ®rst order lateral growth
(Fig. 6B). The smaller the lateral meristem at emergence,
the more pronounced the apical dominance (Fig. 6A).
Conversely, when lateral meristems of a volume similar to
that of the taproot were formed at the base of the taproot,
they rapidly generated taproot competing axes consuming a
large amount of photoassimilates. As the taproot nourished
all the axes, its longitudinal gradient became more pronounced than for the non-rami®ed taproot. Therefore, its
apical meristem had a lower photoassimilate concentration
than the early competing laterals (Fig. 4). This was more
pronounced if the early laterals had equivalent vascularization levels. In this case, early lateral growth became equivalent to indeterminate taproot growth.
More realistically, small meristems appearing on the
taproot had reduced vascularization. Therefore, phloem
resistance along these axes led to a greater decrease in
concentration than that of the taproot. This resulted in
determinate growth giving more strength to apical dominance. The model suggested that apical dominance was
mainly linked to the size of the apical meristem relative to
that of the laterals.
Apical dominance became more pronounced as source
activity became less intense, because lateral meristems
rapidly reached low photoassimilate concentrations.
878
Bidel et al.ÐA Carbon Transport and Partitioning Model
A
B
F I G . 6. Two 20-d-old root systems obtained with the same energetic parameters for di€erent meristem diameters: A, taprooted root system with
meristem diameters of 760, 280 and 100 mm and a taproot 34 cm in length. B, Fibrous root system with meristem diameters of 300, 300 and
300 mm and a taproot 10 cm in length.
Simulations revealed that ®rst order lateral axis growth was
greatly a€ected, and the second order laterals stopped
growing prematurely 2 to 3 d after emergence. For example,
comparison between two simulations where C0 was
decreased from 600 to 50 mol m ÿ3 (but assuming identical
Km ˆ 50 mol m ÿ3 for all meristems), yielded contrasting
results. Thus, on the 20th day of growth, the development
of the taproot meristem decreased from 250 to 170 mm,
that of the initial ®rst laterals from 53 to 30 mm, and that of
the initial secondary laterals from 20 to 5 mm.
Activity of a single ®rst order lateral can in¯uence
photoassimilate distribution in the root network
To investigate the e€ect on photoassimilate distribution
of the activity of a single lateral root axis, we compared the
Bidel et al.ÐA Carbon Transport and Partitioning Model
C0 = 100 mol m−3
879
C0 = 50 mol m−3
0.00
0.0
1.7
2.7
1.7
2.3
3.6
2.7
2.7
2.3
6.5
6.5
5.6
0.14
0.19
Simulation
FPP = 10 FPP
0.5
6.6
0.00
0.7
0.9
0.9
0.9
3.6
0.9
5.9
4.3
2.7
0.10
0.00
5.9
5.7
6.5
0.9
5.9
6.5
6.5
0.7
5.0
5.6
5.7
6.6
6.6
6.6
0.24 MPa
F I G . 7. Phloem hydrostatic potential of a 20-d-old taprooted system
(input parameters of Table 1). The taproot is 36 cm long.
same architecture with only one ®rst order lateral di€ering
in sink strength (Figs 8 and 9). It appeared that varying the
activity of a given lateral axis of type II had much more
in¯uence on the network downstream than upstream. The
consumption of the last segment located upstream of the
taproot was a€ected to a greater extent than that of the
taproot itself because of its anity for photoassimilates.
For the organs located upstream of the modi®ed axis, their
in¯uence increased as they were in close proximity and
showed a weak anity for photoassimilates. A series of
simulations showed that the segment activity had a greater
e€ect on the energy balance when it was closer to the collar
source, had a high anity for photoassimilates or was
supplied through a low resistance pathway. Consumption
by early laterals (type II) close to the source was slightly
a€ected by changes in consumption by later laterals (type
III) (Fig. 8). Conversely, the taproot was strongly a€ected
by consumption in the early lateral axes (Fig. 9). This could
have occurred because these early laterals always had higher
photoassimilate concentrations (Figs 5 and 7) and their
consumption tended towards saturation. Alternatively, this
was also due to the fact that sinks located in parallel are
always less a€ected than sinks located in series. In terms of
energy, the ®rst order lateral roots were quite independent
with respect to the taproot.
5.6
F I G . 8. In¯uence of a lateral root on the rest of the root network. The
number represents the percentage decrease in the consumption of an
organ when the lateral (in black) increases its potential sink strength by
a factor 10. For example the taproot meristem reduces its consumption
by 5.6 %.
It should be noted that, as the total amount of available
photoassimilates brought to the root system remained
unlimited, the removal or addition of a lateral (type III)
had little e€ect on the rest of the root network activity.
E€ect of photoassimilate source concentration on root
branching
When a concentration C0 ˆ 50 mol m ÿ3 was applied to a
root network built with a C0 ˆ 600 mol m ÿ3, a negative
photoassimilate concentration was observed at the extremity of the ®rst and second order laterals, provided that a
nil consumption rate was not applied. This clearly indicated
that weak photoassimilate concentrations at the collar level
could not ensure a full supply to the network under the
potential gradient of C0 ˆ 50 mol m ÿ3 ( for root system
formed at C0 ˆ 600 mol m ÿ3), because of high resistance of
the protophloem. When the source activity was varied from
0 to 800 mol m ÿ3 (0.00 to 0.24 MPa of osmotic pressure),
the ratio of photoassimilate ¯uxes feeding the two
rami®cations linked to the same node changed markedly
and di€erently, according to the considered root sub-system
(Fig. 10). Results showed that, for a similar anity for
photoassimilates, the taproot with the largest sink strength
880
Bidel et al.ÐA Carbon Transport and Partitioning Model
C0 = 50 mol m−3
0.00
1.0
11.8
2.1
19.2
31.8
28.1
99.5
4.1
44.9
35.4
86.0
3.6
4.1
4.8
5.0
5.0
72.8
78.7
Simulation
FPP = 10 FPP
2.0
6.8
3.2
24.1
29.8
5.3
5.8
Are the model predictions realistic?
21.7
6.1
35.4
3.1
Cheeseman hypothesis (1993), where the commonly
observed functional interrelationship is the result of shared
emerging properties, depending on biophysical principles
determined at the organ level. The proposed representation
of architecture can be extended to the aerial part. For an
hydrodynamic simulation of carbon partitioning throughout the whole plant, it will be necessary to add loading
function to the circulation and unloading rules.
4.0
4.4
7.5
8.5
5.6
8.5
F I G . 9. In¯uence of an early lateral root on the rest of the root network. The number represents the percentage decrease in the consumption of an organ when the lateral (in black) increases its potential sink
strength by a factor 10. For example the taproot meristem reduces its
consumption by 8.5 %.
was better fed than its laterals at low photoassimilate
concentrations (with parameters of Table 1).
DISCUSSION
The proposed representation of root architecture makes it
possible to characterize the behaviour of di€erent organs
and test their in¯uence throughout the whole plant. It is a
conceptual framework to simultaneously study the photoassimilate pathway within the root system and the in¯uence
of its availability on root growth. The model is based on
hierarchical ordering of the di€erent root sinks based on the
The model was only applied to roots with a primary
structure known to have a low photoassimilate reserve, as
opposed to secondary structured roots. Therefore, the
model neglected photoassimilate storage along the root
axis. Model predictions for photoassimilate concentrations
were of the same order of magnitude as those published
elsewhere. For a consumption of Pi ˆ 6.24 10 ÿ4 mol
sucrose equivalents m ÿ3 of tissue s ÿ1, the concentration
gradient along the axis ranged from 0.01 to 1.0 MPa m ÿ1,
depending on the root type. In comparison, Milburn and
Kallarackal (1989) reported concentration gradients ranging from 0.1 to 1.5 MPa m ÿ1. The predicted sieve speed
decreased from vi ˆ 2.0 10 ÿ4 m s ÿ1 at the collar level to
nearly zero at the apex, which was close to the average
speed range described by Milburn and Kallarackal (1989).
Flux estimates also appeared reasonable. The photoassimilate ¯ux density SMTi ranged between 0.0001 and
0.040 mol of sucrose equivalents m ÿ2 of sieve tube s ÿ1.
On the 20th growth day at C0 ˆ 100 mol m ÿ3, SMTi at the
elongating taproot apex was 0.0053 mol sucrose equivalents
m ÿ2 of sieve tube s ÿ1. In comparison, Schulz (1997)
estimated this ¯ux to be 0.0070 for the seminal root apex
of corn. Speed and speci®c ¯ux pro®le were consistent with
predictions of published models of phloem circulation in a
non-rami®ed root network. These models describe the
phloem sieve tube more realistically, but require far more
parameters (Tyree et al., 1974; Magnuson et al., 1979;
Smith et al., 1980; Goeschl and Magnuson, 1986). The
model proposed in this paper successfully simulated a
marked decrease in net photoassimilate consumption by
younger root segments, linked to a decrease in phloem
photoassimilate concentration and a higher apex consumption rate.
T A B L E 2. Hydrodynamic phloem characteristics of axes at the 20th iteration for C0 ˆ 100 mol m ÿ3
Axes
Parameters
Vascularization
Resistivity on fully di€erentiated segment
Average speed at the collar node
SMT at the collar node
Average potential gradient on
metaphloem structure
Potential gradient on protophloem
structure at the apex
Average root growth rate
Units
Type I
Type II
Type III
Type IV
number (radius mm)
m ÿ4 s ÿ1 MPa
m h ÿ1
mol sucrose m ÿ2 of
sieve tube s ÿ1
MPa m ÿ1
12(3.0) ‡ 20(7.5)
1.6 107
1.740
0.049
8(3.0) ‡ 8(7.5)
39.9 107
0.570
0.048
8(3.0) ‡ 4(6.0)
75.3 107
0.345
0.032
4(3.0) ‡ 0
2711.5 107
0.053
0.018
0.07
0.13
0.26
0.87
MPa m ÿ1
1.11
0.38
0.19
0.19
5.3
3.5
1.1
mm d ÿ1
14.2
Bidel et al.ÐA Carbon Transport and Partitioning Model
881
C0 ranges from 0 to 1000 mol m−3
F2/F1 RATIO
A
0.5
0.4
0.3
F2
0.2
0.1
F1
0
0
1
2 MPa
F3
F3/F4 RATIO
B
0.22
0.2
0.18
0.16
0.14
0.12
0.1
F4
0
1
2
MPa
F I G . 10. In¯uence of the source activity on the partitioning ratio between highlighted (black) lateral roots and the main root axis. A, For 1 d the
new root axis (F2) was given a higher anity for photoassimilate than the main axis (F1). Therefore, when the source photoassimilate
concentration was dilute, relatively more photoassimilate was partitioned to the highlighted root (F2). B, Conversely, for the taproot (F4), which
has a low anity for photoassimilates but could consume larger amounts, the ratio (F3/F4) increased with photoassimilate availability.
The shape of the consumption pro®le depended largely
on axial resistance and the kinetics of photoassimilate
import by the sinks, and therefore on the basic assumptions
made in the model. These basic assumptions suggested that
phloem unloading involved a single carbohydrate form via
a Michaelis±Menten kinetic process, therefore ignoring
other carbohydrate forms and neglecting carbohydrate
compartmentalization. Import kinetics were also ignored
as Kmi and Pi were assumed to be uniform along the whole
axis but not at the apex. However, since metabolic activity
evolves during successive di€erentiation stages (Gahan and
Carmignac, 1989), the most limiting process for photoassimilate import probably evolves over time. Schulz (1994)
showed a gradual decrease in 14C-sucrose unloading along
the taproot in pea seedlings (Pisum sativum L.), suggesting
sink strength evolution. Our own data agree with this
hypothesis (Bidel et al., 1999). Photoassimilate anity
probably varies gradually along the axis. If apex growth
was much less sensitive than the rest of the axis to a
reduction in photoassimilate availability (Bingham and
Stevenson, 1993), Km would probably be much lower in that
zone.
It is possible that the total segment consumption evolved
di€erently from that of respiration (e.g. exudation and
mineral uptake). It would be very useful to better describe
the evolution of the potential sink strength of segments,
taking into account not only respiration, but also total
potential consumption, with biomass accumulation (Sharp
882
Bidel et al.ÐA Carbon Transport and Partitioning Model
et al., 1990) and exudation (Minchin and McNaughton,
1984). The axial phloem resistance model assumed that the
number of sieve tubes increased linearly with time up to the
total number determined on the primary structure. Sieve
tube resistance was arbitrarily set at 1.5 Poiseuille resistance, according to the calculations of Sheehy et al. (1995).
Since the sieve tubes on the laterals were short and small in
diameter, their head losses due to sieve plates are also
probably greater. The use of this model will remain qualitative, if this resistance cannot be described more accurately.
The model does not include plasmodesmata, the occurrence
and geometry of which evolve together with the formation
of the walls they pass through (Patrick, 1997; Kragler et al.,
1998). This additional resistance will also modify the
distribution of photoassimilate consumption and its concentration pro®le along the axes. Despite the fact that
translocation processes were described very crudely, the
model predicted sound phloem translocation and photoassimilate uptake patterns. Here are some of the dominant
features.
activity greatly decreased the photoassimilate concentration
in the primordium-forming zone (Figs 5 and 7). Apex
activity was then able to complete lateral primordia growth
and, according to our simulations, the smaller these
primordia were at the time of their formation, the more
the apex dominated. Simulations for 20 d of growth showed
that maintaining a higher meristem volume was necessary
for the taproot to maintain its dominance. Auxins would
favour this, because of their role in maintaining the organization of the quiescent zone of the meristem (Kerk and
Feldmann, 1995). Cytokinins are known to inhibit lateral
primordia development and enhance apical dominance
(Wightman et al., 1980; Torrey, 1986). Elongation of the
sub-apical zone was assumed to be non-limiting in our
model. However, growth may be impaired by biomechanical constraints. Hence, a more realistic description of
sub-apical growth might be attained using the equation of
Lockhard (1965). Indeed, Cosgrove (1993) considered that
photoassimilate import into the apex zone was the result of
biomechanical growth control.
Root apical dominance is mainly determined by relative
meristem size
Fibrous root systems
The model showed that a carbon distribution pattern,
characteristic of taprooted development, is predicted when
axis growth is proportional to meristem respiration. In
Prunus persica L. Batsch seedlings, with a taproot meristem
substantially larger than that of laterals, this mechanism
may control taproot apical dominance. This dominance
would be reinforced if lateral meristems develop reduced
phloem vascularization, hence rapidly reducing photoassimilate concentration along these axes. Simulations
showed that the form and magnitude of dominance largely
depend on axis resistance pro®le and uptake kinetics (Kmi
and Pi) along the axis, as described above. The model
ignored resistance resulting from rami®cations (Tyree and
Alexander, 1993) in order to simulate pressure losses
associated with xylem connections between the rami®cation
and the main axis. The latter would reinforce apical dominance and further cleave the architecture in term of energy,
since it would reduce the photoassimilate concentrations of
the lateral meristems. Unequal resistance between axis types
was enough to generate hierarchical uptake for the di€erent
sinks lacking photoassimilates, even when all the meristems
of di€erent orders had the same anity for photoassimilates. In other words, the proportion of photoassimilates
consumed for taproot growth increased when source
activity diminished. These simulations were in line with
previously published results (Minchin et al., 1993). Photoassimilate partitioning models are unable to predict
whether a variation in source activity modi®es the partitioning within the root network (Lacointe et al., 1995).
However, if two organ sub-networks do not have the same
metabolic activity or the same response to environmental
factors, there is no reason why the partition coecient
should be constant.
When the apex containing the sub-apical elongation zone
had high photoassimilate anity as well as pronounced
sink strength, the simulations clearly showed that intense
Due to their proximity to the carbohydrate source, lateral
meristems located at the taproot base had the advantage of
being able to maintain their consumption closer to
saturation compared to other axes. Simulations showed
that ®rst and second order laterals of the same diameter and
meristem volume always grew better when located close to
the taproot, because of their proximity to the photoassimilate source (Fig. 6B). These simulations agree with the
experimental results of Aguirrezabal et al. (1994). According to the MuÈnch ¯ux theory, they probably have a good
supply of nitrogenous compounds and growth substances.
Abundant root distribution in surface horizons is generally
attributed to favourable surface conditions (temperature,
gas exchange, mechanical impedance, etc.). The model also
showed that this distribution was a consequence of photoassimilate translocation. Simulations showed that, with
limiting source activity, the activity of ®rst order laterals
a€ected that of the taproot. Conversely, consumption by
the network located downstream had little impact on their
photoassimilate concentration, and hence on their activity
(Fig. 8). This relative autonomy described by the model is
probably more obvious in woody plants, because of the
speci®c vascular network existing between low branches
and ®rst laterals.
Determinate root growth
Varney et al. (1991) showed that axis meristems with
determinate growth developed parenchyma down to the tip.
To our knowledge, the conditions leading to the determinate or indeterminate nature of root growth have never
been clearly identi®ed (McCully, 1987). The simulations
suggested that determinate growth might result from a local
shortage of photoassimilates at the meristem level. It is
hypothesized that meristem activity ceases at a threshold
photoassimilate concentration at the protophloem extremity. This threshold is most likely triggered when the water
Bidel et al.ÐA Carbon Transport and Partitioning Model
potential of the apoplast goes below the value at which
plasmolysis occurs in the sieve tubes. Recent studies on Zea
mays L. suggested that de®ciencies may occur in the seminal
meristems of corn plantlets either in vitro (James, 1994) and
in vivo after 1 week of shading (Strosser, 1996). In fact,
determinate growth is usually accompanied by decreases in
meristem size and parallel growth and it is also enhanced by
shading and/or intensive shoot growth duration (Thaler
and PageÁs, 1996b). This decreased meristem volume,
ignored in this model, may slow down predicted growth
reduction if the number and size of the formed sieved tubes
are maintained. Conversely, if they diminish, axial resistance increases, reducing the axis growth period. The model
predicts that at least one protophloem tube is perfectly
di€erentiated at the meristem interface. However, photoassimilate transfer probably occurs at a di€usional distance,
which is dicult to evaluate, of over 1 mm for the seminal
root meristem (Warmbrodt, 1987) or even several mm (e.g.
Zea mays L.; Bret-Harte and Silk, 1994). Recently, studies
carried out by Eleftheriou (1996) on protoxylem di€erentiation showed that, for wheat apex, only seven to nine cells
at the end of each thread were at a di€erentiation stage. The
®rst element observed was located at about 300 mm from
the apex (Eleftheriou, 1989), and the ®rst mature element at
about 550 mm, 16 to 21 h after being created. Additional
resistance upstream of the meristem should therefore be
incorporated into our model, to simulate axis growth
stunting of poorly vascularized axes. Neither meristem
volume variations, nor the existence of a di€usion zone at
the protophloem extremity can slow down the process
leading to determinate growth. On the contrary, they are
likely to speed it up.
Does the translocation pathway control carbon partitioning?
In the absence of a simple representation of transfer
resistance and of simple non-disturbing measurement
methods, phloem resistance has, to date, been neglected in
carbon partitioning models (Jones et al., 1991 for
TOMGRO; Heuvelink, 1996; Tabourel-Tayot and Gastal,
1998). Photoassimilate distribution was assumed to be
controlled mainly by the sink strength of the di€erent
organs (Farrar, 1992; Lacointe et al., 1995). Patrick (1988)
and Wardlaw (1990) suggested that transfer may be limiting
only in zones where phloem di€erentiation is incomplete,
particularly at the root apex level. The following discussion
focuses on primary root structure. Taking hydrodynamic
considerations into account, sap has to take preferential
¯ow pathways of lowest resistance, even though all the
pathways have a low resistance. Moreover, Farrar (1992)
explained that a sink could overcome the disadvantage of
being far from the source by increasing its sink strength; our
model con®rmed this observation. In our model, axis
transfer properties controlled phloem ¯ow to a greater
extent, as source activity was low, in close agreement with
Patrick (1991).
Integrating axis transfer properties enabled us to simulate
experimental conditions on the e€ect of organ proximity
and their apparent priority for imports in a photoassimilatelimiting situation. Partitioning models using a common
883
photoassimilate pool have been poor at describing this
situation. By introducing both resistance and Michaelis±
Menten kinetics in a three-compartment model, Minchin
et al. (1993) showed that the apparent photoassimilate
uptake priority of the di€erent organs evolved with photoassimilate availability without modifying the properties of
the root network. This result could explain the di€erential
growth behaviour observed in Hevea brasiliensis MuÈll. Arg.
in shaded and light conditions (Thaler and PageÁs, 1996a, b).
Our simulations also suggest that the apparent partitioning
coecient between organs evolves daily for plants exhibiting a marked diurnal variation in photoassimilate concentration, as shown for Nicotiana tabacum Graph. (Hocking,
1980).
Our simulations agree with three major facts reported for
14
C-labelled assimilate partitioning by Cook and Evans
(1976, 1978) for Triticum aestivum L. (1) Proximity to the
source conferred a marked advantage on meristems, and a
greater sink size was particularly important in securing
assimilates from a distant source. (2) Larger sinks received
more than their pro rata share of 14C-labelled assimilates
from the ¯ag leaf, and the bias in favour of the larger sink
increased with its relative size. (3) The advantage of the
larger sink size was much more apparent as the distance
between the sink and source increased. Our calculations
supported the hypothesis of photoassimilate partitioning
within the root system and allowed us to investigate its
development kinetics.
CO N C L U S I O N S
To understand photoassimilate distribution within the root
system, we represented it by an organ network containing
meristems and segments at successive di€erentiation stages.
Photoassimilate import by each meristem and segment was
assumed to be limited by a metabolic process in order to
follow Michaelis±Menten kinetics. A model of phloem sap
¯ow, extended from that of Minchin et al. (1993), made it
possible to describe photoassimilate partitioning throughout the whole root network. Root axis growth was proportional to meristem respiration producing the new cells.
Simulated phloem ¯ow was of the same order as that
measured elsewhere (Canny, 1973; Milburn and Kallarackal, 1989) and consistent with analytical results from
more complex models derived from the MuÈnch theory
(Magnuson et al., 1979; Goeschl and Magnuson, 1986).
Our results showed that, for axes with a vascularized
primary structure wall, the volume of apical meristems has
to be greater than that of the laterals to generate a taprooted system. Conversely, the volume of lateral meristems
close to the collar has to be of a similar size to that of the
taproot meristem to generate a ®brous root system.
Theoretical calculations supported the hypothesis that the
determinate growth of axes with a primary structure results
from a photoassimilate storage at their meristem level. With
respect to their photoassimilate consumption, the phloem
transport properties of most ®rst and second order laterals
of the primary structure were insucient to enable indeterminate growth. The model simulated an hierarchical
priority of organ uptake similar to that described by Thaler
884
Bidel et al.ÐA Carbon Transport and Partitioning Model
and PageÁs (1998). Following this model, transport properties of axes act indirectly in photoassimilate partitioning by
creating concentration gradients along the axes. They controlled phloem ¯ow to a greater extent, as source activity
was low, in close agreement with Patrick (1991). Their e€ect
di€ered depending on the organ activity response curve at
the photoassimilate level.
The design of this model may help understand the
development of di€erent root networks, separate the
speci®c in¯uence of trophic and hormonal factors in root
apical dominance and determinate growth, and identify the
in¯uence of vascularization produced by cambial growth
mechanisms. The partitioning of other nutrients could also
be modelled using the mechanism described in this article.
Finally, the approach could help simulate growth behaviour under changing environmental factors (temperature,
moisture, etc.).
AC K N OW L E D G E M E N T S
This study was carried out in both the Ecophysiology and
Horticulture UnitÐINRA, Avignon (France) and the
Agronomy UnitÐINRA, Angers (France). Dr G. Pelloux
and J. Y. Lorendeau (LAMA UnitÐINRA, Avignon) are
acknowledged for their contributions and help in using and
structuring the object oriented analysis and in developing
the architecture editor. We thank Dr J. Caron and Dr J. Le
Bot for their advice and help throughout the preparation of
the manuscript, Dr A. G. Bengough (SCRI, Dundee, UK)
for its critical reading and improvements, and A-M. Wall
for reviewing the English version of the manuscript.
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Bidel et al.ÐA Carbon Transport and Partitioning Model
APPENDIX
List of symbols
Symbol
De®nition
Unit
R
T
Ci
Capoplast,i
Csieve tube,i
Cp
Cpi
Cgi
Cmi
Fwi
Fi
qwi
vi
SMTi
Fi
Pi
Vi
Km i
ssucrose
Ri ˆ 1/Li
Zsieve
li and ri
ni
nmax,i
ti
tM
Gas law constant
Absolute temperature
Solute photoassimilate concentration within the sieve tubes (sucrose equivalents)
Water potential of apoplast surrounding the ith sieve tube element
Water potential at the ith node
Hydrostatic potential
Osmotic potential
Gravitational potential
Matric potential
Passive axial ¯ux of water at the ith node
Passive axial ¯ux of photoassimilates
Passive e‚ux of water at the ith node
Average water speed at the ith node
Axial mass ¯ux density for photoassimilates (often referred to as speci®c mass transfer)
Photoassimilate unloading e‚ux at the ith node
Maximum unloading rate of the ith sink potential sink strength (sucrose equivalents)
Volume of the ith consuming segment
Michaelis constant of the ith sink
Re¯ection coecient for the sieve plate
Axial hydraulic resistance of the phloem
Solution viscosity at 293.15 K
Length and average radius of sieve tubes in the ith segment
Number of functional sieve tubes in the ith segment
Number of sieve tubes in the ith segment
Age of the ith segment
Age of complete maturation of sieve tubes
m3 MPa mol ÿ1 K ÿ1
K
mol m ÿ3
MPa
MPa
MPa
MPa
MPa
MPa
m3 s ÿ1
mol s ÿ1
m3 s ÿ1
m s ÿ1
mol m ÿ2 of sieve tube s ÿ1
mol s ÿ1
mol m ÿ3 of tissue s ÿ1
m3
mol m ÿ3
No units
m ÿ4 s MPa m of root ˆ m ÿ3 s MPa
MPa s
m
No units
No units
d
d