Annals of Botany 81 : 213–223, 1998
Modelling of the Hydraulic Architecture of Root Systems : An Integrated
Approach to Water Absorption—Model Description
C L A U D E D O U S S AN*†, L O I$ C P A G E' S‡ and G I L L E S V E R C A M B R E‡
* INRA, UniteU de Science du Sol, Domaine Saint Paul, Site Agroparc, 84914 AŠignon Cedex 9, France
and ‡INRA, UniteU de Recherche en Ecophysiologie et Horticulture, Domaine Saint Paul, Site Agroparc,
84914 AŠignon Cedex 9, France
Received : 14 July 1997
Returned for revision : 25 August 1997
Accepted : 21 September 1997
A numerical model simulating water uptake by root systems is presented. This model can combine the locally
measured root hydraulic conductances with data on the root system architecture to give a detailed description of
water absorption, from the single root level to the entire root system. This is achieved by coupling a three-dimensional
root system architecture model with laws describing water flow in roots. In addition to water absorption studies, the
model has been developed so that it can be included in a soil water transfer simulator in order to analyse soil–plant
interactions for water uptake. The use of the model in describing water absorption is illustrated for a specific case
where the hydraulic conductances are considered uniform in the whole root system. In this way, analytical results of
Landsberg and Fowkes (Annals of Botany 42 : 493–508, 1978) are extended from the single root to the root system
level. The influence of the type of root system architecture, axial conductance between crowns of maize nodal roots,
transpiration in the course of the day, and non-homogeneous soil water potential on fluxes and water potentials in
the root system are examined. The dynamics of the total conductance of the maize root system with plant growth is
also shown for this case of uniform conductance in the root system. Cases which consider other distributions of the
conductance in the root system are presented in an accompanying paper.
# 1998 Annals of Botany Company
Key words : Water, uptake, root system, model, architecture, hydraulic conductance, Zea Mays L.
INTRODUCTION
Water absorption by plant roots from the soil is determined
by three main factors : (1) the soil properties (water release
and hydraulic conductivity of the soil) ; (2) the root system
architecture (i.e. extension in space and connections between
roots), here considered as a network of absorbing organs ;
and (3) the absorption capability of roots, dependent on the
soil-root interface and the resistance of the root to water
transfer.
The pioneering work of Gardner (1960) emphasized the
soil properties in water uptake and led to the so-called
‘ microscopic approach ’ of water absorption. More recently,
faced with the difficulty in extending this approach to the
complexity of a real root system, much work has been done
to define a ‘ macroscopic ’ function for water uptake. This
approach consists of incorporating a root sink term in the
Darcy-Richards equation for flow in soils. As stated by
Molz (1981), all of the various sink functions proposed in
the literature are more or less empirical and often include
implicit assumptions on the location of any major resistance
to flow. Generally, in the description of water uptake in soil,
the root systems are extremely simplified, and described
only by the root length density and, sometimes, an
homogeneous root resistance to water flow. But experimental data (Smucker and Aiken, 1992, Tardieu, Bruckler
† For correspondence.
doussan!avignon.inra.fr
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and Lafolie, 1992) and calculations (Bruckler, Lafolie and
Tardieu, 1991 ; Lafolie, Bruckler and Tardieu, 1991 ; Petrie
et al., 1992) show that the spatial arrangement of roots in
the soil (not reflected in the average root length density) has
a substantial influence on water uptake. Moreover, the
capability of roots to absorb water may evolve in space and
time because of the development of an interfacial resistance
to water transport between the soil and the roots (Herkelrath, Miller and Gardner, 1977) or due to a variation in the
physiological properties along the root (Sanderson, 1983 ;
Varney and Canny, 1993).
Most models describing water uptake consider only a
limited number of the features of the water absorption
process. If, generally, water transfer in soil is well represented
by the microscopic or macroscopic approaches, a detailed
description of the root system is lacking (Molz, 1981). When
the geometrical distribution of roots (Lafolie et al., 1991) or
the root system architecture (Clausnitzer and Hopmans,
1994) is taken into account, the hydraulic continuity in and
between the roots is neglected and hence the water transfer
in the root system is poorly described. Nevertheless, the
xylem water potential has been shown to vary, sometimes
quite substantially, along roots (Passioura, 1972 ; Nobel and
Lee, 1991 ; Alm, Cavelier and Nobel 1992 ; Simmoneau and
Habib, 1994). Finally, if some kind of interfacial resistance
is accounted for, the variation in the water uptake capability
or in the root hydraulic conductance is only poorly
considered.
Experimental advances have been made recently in
bo970540
# 1998 Annals of Botany Company
214
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
understanding the structure and functions of plant roots,
with an improved reliability and a better spatial resolution.
This is the case for root system architecture dynamics,
including the shape of the root system, connections between
roots, age and the branching pattern of roots (Page' s and
Pellerin, 1994 ; Pellerin and Page' s, 1994). Moreover, it is
possible to perform numerical simulations of root growth
and architecture (Page' s, Jordan and Picard, 1989 ; Lynch
and Nielsen, 1996). In this way, realistic and detailed
simulations of root systems can be obtained relatively easily
and the computer can be used as a virtual experiment to
study the interactions of root systems with their environment
(e.g. Clausnitzer and Hopmans, 1994). The physiological
characteristics of water uptake by root segments (axial and
radial conductances) can now be examined at the centimetre
scale (Steudle, 1994 ; North and Nobel, 1995). Measurements
of the flux into the root (Varney and Canny, 1993) or xylem
tension (Balling et al., 1988) as a function of position along
the root, or of branching order, give new insights into the
characteristics of water absorption. Recent anatomical
investigations (xylem and endoderm maturation, branching)
also provide information on the evolution and development
of water conduction capabilities in roots (McCully, 1995).
This scattered information would provide a better
understanding of water absorption if it was brought together
in a single quantitative global framework. As a consequence,
we believe that it is useful to develop, with the current state
of knowledge, a model integrating this localized information
using physical laws which describe the water transfer in the
root system (‘ Hydraulic Tree Model ’ of the root system).
The model could simulate both the distribution of fluxes
and water potentials in the root system and the resistance to
water flow in part of or in the whole of the root system. Such
a model, describing water absorption from the single root to
the root system level, can (1) help to describe more
consistently water absorption in soil without empirically
defined sink functions ; (2) give new insights into root water
absorption by integrating processes analysed at the single
root level (e.g. variation of root water conductance) ; (3)
help to identify prominent architectural features influencing
water uptake (such as root lengths, diameters, branching
density, type of root system architecture etc) ; and (4) be
used to help in the experimental design and measurement of
local root properties. Above all, it will be possible to assess
by sensitivity analysis the influence of variations and}or
uncertainty in the architectural or physiological (e.g. root
resistance to water flow) parameters on the water uptake at
local or global levels in the root system.
Apart from the Ohm’s law analogy (van den Honert,
1948), water movement through plant roots has been
theoretically analysed by Landsberg and Fowkes (1978)
who derived an analytical solution for the case of a single
root with a constant hydraulic conductance. Alm et al.
(1992) extended the results of Landsberg and Fowkes to the
case where the conductance varies along the root, separating
the root into segments, each with a homogeneous conductance. In this article, we extend the approach of Alm et
al. (1992) to the whole root system by coupling the root
system architecture with the process of water absorption. In
the first paper, the fundamentals of the architecture model
of Page' s et al. (1989) are briefly reviewed and the theoretical
foundations for the ‘ Hydraulic Tree Model ’ of the root
system are given. The application of the model to a test case,
extending the results of Landsberg and Fowkes (1978), is
presented. In an accompanying paper, we focus on
experimental data taken from the literature concerning
water absorption by roots and look at the consequences for
modelling water uptake.
M O D E L D E V E L O P M E NT
The root system architecture model
We used the model of Page' s et al. (1989) that simulates
maize (Zea mays L.) root architecture. The model simulates
the three-dimensional architecture of the maize root system
in discrete time steps. At each time step (1 d), the root
system extends by three basic processes : (1) emergence of
new root axes (seminal and nodal) from the shoot ; (2)
growth ; and (3) branching. Root system development is
related to cumulated temperatures on a daily basis, with a
6 °C base temperature.
The root system is simulated by a set of segments, each
segment representing the part of the root that was generated
during one time step. The spatial coordinates of each
segment are stored in a data structure, together with
information on the position of the segment within the
architecture (i.e. branching order, internode of origin,
formation date, connection with other segments).
The emergence of main root axes on the same phytomer
is assumed to occur simultaneously and the rank of the
active phytomer is calculated from a linear function of
cumulated thermal units. The length and diameter of each
internode is taken from experimental data. Roots are
assumed to elongate according to the following function of
thermal time :
(1)
L ¯ A(1®e−bT)
where L is the length of the root, T is thermal time from
emergence, A is the final root length and b is a rate
parameter. The asymptotic value A is randomly drawn from
an estimated distribution for each root emergence. Growth
directions are calculated by combining the effects of
gravitropism and mechanical stresses (Page' s et al., 1989).
Branching occurs after a constant time lag from initiation,
which is assumed to take place just behind the apex, and
whose density depends on branching order. The branching
direction is calculated using a branch angle drawn from a
normal distribution and the radial angle randomly drawn
from a uniform distribution over 0 to 2π.
Equations for flow in the root
Water flow through a root can be characterized by the
root hydraulic conductivity Lp(m s−" MPa−") (Fiscus, 1975) :
JŠ ¯ L (∆P®σ∆π)
(2)
p
m−# s−") is the flux density of water entering the
where JŠ (m$
root (i.e. the flow rate divided by the root area), ∆P (MPa)
is the difference in hydraulic pressure between the external
medium and the xylem, ∆π (MPa) is the difference in
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
215
F. 1. Example of a three-dimensional simulated architecture of the maize root system projected on a vertical plane (with an enlarged view of
two main roots with laterals) showing the nodes for flow calculation in the root system.
osmotic potential between the external medium and the
xylem sap, and σ is the reflection coefficient for solutes.
In this study, two hypotheses are formulated : firstly, the
influence of solutes on flow is neglected, because : (1) during
periods of active transpiration, the hydrostatic pressure
gradient (∆P) rather than the osmotic potential gradient is
the effective driving force for flow [and the relation between
hydrostatic pressure gradient and flux is linear (Fiscus,
1975 ; Weatherley, 1982)] ; (2) we consider here that the soil
water is a dilute solution as is the sap. The case of a more
concentrated soil solution can be investigated by using
the total water potential gradient (∆ψ) and assuming that
σ ¯ 1 ; (3) the presence of solutes in the water flow greatly
increases the mathematical complexity of the problem.
Moreover, as the radial pathway for water in roots is still
not clear (Huang and Nobel, 1994), models coupling flow of
water and solutes are still controversial (Fiscus, 1975 ;
Katou and Taura, 1989 ; Steudle, 1994).
With this first assumption, eqn (2) can be rewritten :
JŠ ¯ Lp ∆P or JŠ ¯ Lp ∆ψ
(3)
and expressed in terms of hydraulic pressure or total water
potential.
Secondly, we consider only steady-state flow, that is to
say the capacitive effect of the roots is neglected. Transient
effects in plants are more important when an abrupt change
occurs in the flow environment (water potential or weather
changes). This is, for example, the case when transpiration
commences, but in soil the variations are more gradual.
Moreover, the water stored in the roots is generally small
compared to transpiration requirements (Simmoneau,
1992). However, it will be necessary to incorporate a
capacitive term for the roots into the model to examine
small time scales or woody plant species (Waring and
Running, 1976).
The hydraulic conductivity Lp can be separated into two
terms, corresponding to water movement into the root and
along the root : the radial conductivity [Lr(m s−" MPa−")] for
flow from the root surface to the xylem and the axial
conductance [Kh(m% s−" MPa−")] for flow along the xylem.
These can be defined, following Landsberg and Fowkes
(1978), as :
Jh(z) ¯®Kh
dψx(z)
dz
Jr(z) ¯ Lr[ψs(z)®ψx(z)]
(4)
(5)
where Jh(z) (m$ s−") is the flux up the root in the xylem at
distance z from the apex, Jr(z) (m$ m−# s−") is the flux into the
root from the soil per unit area, ψs(z) is the water potential
in the soil and ψx(z) is the xylem water potential.
216
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
Lr 1
11
1
16
Kh 1
Lr 2
12
Lr 6
2
Tertiary
root
6
Secondary
root
Kh 2
Kh 5
Lr 3
13
Kh 3
3
Kh 4
4
5
7
Kh 6
Kh 7
Lr 8
18
Lr 5
Lr 4
8
14
Lr 7
15
17
Kh 8
Legend
Lr 9
19
9
Collar boundary condition
Plant node
Kh 9
Soil node
Lr 10
10
20
Axial conductance
Radial conductance
Primary root
F. 2. A simplified discretized root system to show numbering of the soil and plant calculation nodes, and numbering of the conductances (axial
and radial) considered as oriented links. The soil nodes are given boundary conditions (water potential), whereas the water potentials of the plant
nodes are the unknowns. Either the water potential or the total outflow at the root system collar is a known boundary condition.
Coupling the flow and architecture
The simulated root system used is the numerical output of
the root architecture model at a given time after germination.
Some of the results presented here are derived from graphical
theory (Pelletier, 1982 ; Simmoneau, 1992). As in Alm et al.
(1992), this root system is divided into small compartments,
the centre of which are nodes of the simulated root system
(Fig. 1), with a specified diameter (root diameter). Each
compartment is connected to its neighbours (same root or
branch roots) by an oriented link which is the axial
conductance between the two compartments. In this case,
eqn (4) can be approximated by :
Jh(z) ¯®Kh
∆ψx
∆z
(6)
where ∆z is the distance between the two nodes of the
compartments. Likewise, each root node is connected to a
soil node by an oriented link (radial conductivity) for which
eqn (5) is valid.
Compartments are numbered serially in the upstream
direction, starting with the collar of the root system, and
branch roots being numbered first (Fig. 2). Nodes corresponding to the soil are then numbered in the same way as
the root nodes. The same is done for the links (axial and
radial conductances) oriented in the direction of increasing
node values. The xylem water potentials at the root nodes
are unknowns, whereas the matric (or total) water potentials
at the soil nodes are given boundary conditions. At the
collar of the root system, either a known value of xylem
suction (or water potential), or flow rate (transpiration), is
imposed. In such a system, the fluid flow laws are expressed
using a matrix notation :
MN
DY ¯®L¬dψ
(7)
IM¬DY ¯ 0Y
(8)
where DY (dimension [Nc®1], where Nc is the number of
compartments) is the flow rate vector through the link
MN
between two nodes, dψ [Nc®1] is the pressure difference
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
MN
dψ ¯ IMt¬ψY
(9)
IMt is the transpose of the IM matrix, ψY is the vector [Nc]
which includes firstly the xylem water potential nodes (ψx )
i
and secondly the soil water potential nodes (ψs ).
i
The combination and simplification of eqns (7), (8) and
(9) results in a simple linear system of equations, in matrix
form :
MN
(10)
C¬ψx ¯ QY
where QY , on the right hand side (dimension Np, where Np is
the number of plant nodes), contains the soil factors (Qi ¯
Lr ¬Si¬ψs , where Si is the surface of the root segment)
i
i
and the boundary condition used at the base of the plant. C
is the conductance (squared) matrix [Np, Np] which takes a
simple form (see Appendix).
Taking into account the fact that C is a sparse matrix
which might be large (from 60 000 to " 10' equations), the
linear system of equations is solved by a preconditioned
conjugate gradient method (Larabi and de Smedt, 1994).
Iterations of the conjugate gradient are stopped when the
residual norm between two successive iterations is lower
than 10−"&. When the solution for the xylem water potentials
is known, flow rates into or up the root can be calculated at
any location by eqns (4) and (5).
Owing to the symmetry and linearity of the flow equations,
if the water potential gradient is inverted between the soil
and the plant, water efflux will proceed at the same rate as
the absorption flux in the model (i.e. the flow is non-polar).
Analytical–numerical solution comparison for the single
root case
In the case of homogeneous radial conductivity and axial
conductance (Lr and Kh ), Landsberg and Fowkes (1978)
derived an analytical expression for the xylem water
potential Šs. distance from the apex (z) for a root immersed
in a solution at constant potential :
(ψx(L)®ψs) cosh (αz)
cosh (αL)
ψx(z) ¯ ψs­
α# ¯
2πrLr
Kh
(11)
(12)
where L and r are the root length and radius, respectively.
Figure 3 presents a comparison between the analytical
and numerical calculations for a 50 cm long root. In
this case Kh ¯ 5¬10−"" m s−" MPa−" and Lr ¯ 2¬
10−( m% s−" MPa−", which are the values given for maize in
1
Xylem suction (MPa)
vector between two adjacent compartments, L is a diagonal
matrix [Nc®1, Nc®1] whose terms are the axial and radial
conductances (see Appendix). IM is the incidence matrix
[Nc, Nc®1] which represents the structure of the root system
in matricial form (cf. Appendix).
With these notations in mind, eqn (7) represents the law
for flow into and up the roots [eqns (4) and (5)], while eqn
(8) is the Kirchoff law (i.e. the algebraic sum of volumetric
flow rates is null in a compartment for steady state flow).
Taking into account the fact that :
217
0.8
0.6
0.4
0.2
0
0
10
20
30
40
Distance from base of root (cm)
50
F. 3. Comparison of analytical (——) and numerical (+) solutions
for the single root case. Imposed water suctions are 1 MPa at the root
collar and 0 MPa in the outside solution. Axial conductance (Kh) is
5 10−"" m% s−" MPa−" and radial conductivity (Lr) is 2 10−( m s−" MPa−".
the review by Huang and Nobel (1994). The root radius is
3 mm, the water potential at the base of the root system is
1 MPa, and the soil potential is 0 MPa. Calculation nodes
are evenly spaced every 2±5 cm. We can see in Figure 3 that
the fit is good (mean relative error : 4±4¬10−%, s.d. :
2¬10−% MPa) and the accuracy of the water flux is within
0±5 %. As the distance between two adjacent nodes in the
root architecture model is typically less than 0±5 cm, the
accuracy should be improved further (numerical errors are
inversely proportional to the space increment and proportional to the second derivative of the xylem pressure Šs.
distance).
A N E X A M P L E A P P L I C A T I ON (C O N S T A N T
AXIAL CONDUCTANCE AND RADIAL
C O N D U C T I V I T Y A L O N G T H E R O O TS)
We now illustrate the use of the model. To keep the
demonstration simple, we employ only constant axial
conductance and radial conductivity along the roots. This
also allows us to compare numerical outputs from a
complex root system with the conclusions of Landsberg and
Fowkes (1978) in the single root case. More complex
distributions of hydraulic conductance in the root system
will be investigated in the second part of the article. The
influences of the type of root system architecture, the
evolution of the transpiration during the day, the axial
resistance of shoot internodes in the case of adventitious
roots, the non-homogeneous soil water potential and the
evolution of the total root system conductance during plant
development will be briefly examined. In the text which
follows, we will use the units cm$ s−" (10−' m$ s−") for water
flux and cm$ cm−# s−" (10−# m s−") for water flux density.
Most of the examples given are for root system architectures corresponding to field grown maize, according to
the data of Page' s and Pellerin (1994) and Pellerin and Page' s
(1994). The homogeneous axial conductance and radial
Depth (cm)
218
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
0
0
–20
–20
–40
–40
–60
–60
0
0
–1 MPa
–40
–20
20
0
Horizontal distance (cm)
–1 MPa
A
–80
B
40
0
–30
C
–20
10
–10
0
Horizontal distance (cm)
20
0
30
D
–20
–20
–40
–60
–60
Soil water potential
Depth (cm)
–40
–80
–100
–80
0
0
–1 MPa
–1 MPa
–120
–40
20
–20
0
Horizontal distance (cm)
40
–40
–20
0
20
Horizontal distance (cm)
40
60
F. 4. Examples of the evolution of the water potential in three-dimensional simulated root systems (30-d-old) projected on a vertical plane. Axial
conductance and radial conductivity are constant in the whole root system in A and B. A, Maize root system architecture with adventitious roots.
Soil water potential is 0 MPa, imposed root system collar water potential is ®1 MPa. B, Taproot system. Soil water potential is 0 MPa, imposed
root system collar water potential is ®1 MPa. C, Maize root system where the axial conductance of internodes between the crowns of adventitious
roots is very high (10% times root conductance, i.e. no resistance to axial water flow between internodes), root axial conductance and radial
conductivity are the same for all roots. Soil water potential is 0 MPa, imposed root system collar water potential is ®1 MPa. D, As for C, but
with an exponential decrease of soil water potential from ®0±1 MPa at 100 cm depth to ®0±74 MPa at the soil surface. The coloured line on the
right of the figure represents the soil water potential. The scale of colours varies from 0 (black) to ®1 MPa (white) except in C where it varies
from ®0±1 (black) to ®1 MPa (white).
conductivity employed in all simulations are taken from
Frensch and Steudle’s (1989) experiments with (young)
maize roots. The axial conductance (Kh) is set to
5¬10−"" m% s−" MPa−" (the maximum value they found)
and the radial conductivity (Lr) to 2±2¬10−( m s−" MPa−"
(constant, except at the tip, in their experiment).
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
0
Type of root systems and a general Šiew of water
absorption
Depth (cm)
–20
A
–80
0
–20
–40
–60
–80
B
0
0.5
1
1.5
Radial inflow (cm s–1 × 105)
2
0
–20
Depth (cm)
Influence of axial internode conductance in the case of
adŠentitious roots
Figure 4 A and C show water potentials and Fig. 5 A
and B present radial inflow Šs. depth for a simulated maize
root system for the cases where (a) the axial conductance is
the same in the whole root system, even in the internodes
between crowns of nodal roots ; and (b) where the axial
conductance is also the same in the whole root system,
except in these internodes. Here, the axial conductance of
the internodes is set much higher than the root conductance
(10% times greater) in order to nullify the water potential
gradient between the crowns of nodal roots. In this case,
there is almost no restriction to axial flow in the part of the
stem where nodal roots emerge. In both cases, the water flux
to the roots is far from homogeneous with depth, although
the soil water potential is homogeneous (Fig. 5). Again, the
relatively small axial conductance of the roots is reflected by
a sharp decrease in the flux and xylem suction with depth or
length of the root. Clearly here, only a part of the maize root
system is active in water uptake if the axial conductance is
set to the value determined by Frensch and Steudle (1989)
for young root segments. The effect of axial conductance is
also illustrated by comparing Figs 4 A and C, and 5 A and
B. In the case of negligible axial resistance of the internodes
between adventitious roots, the water potential at the base
of the plant may propagate further (Fig. 4) and the water
inflow increases (particularly in the laterals) in the upper
part of the root system (Fig. 5). Here, the total water flux in
–40
–60
Depth (cm)
Figure 4 A and B present the xylem water potential in
roots for the case of a maize root architecture (with
adventitious roots) and for the case of a hypothetical
taproot system generated by the architecture model. Axial
hydraulic conductance and radial conductivity are the same
for both types of root system and constant in the whole root
system. The water potential at the base of the stem is
®1 MPa, while the soil potential is uniformly 0 MPa.
Plants are at the 420 °C d stage (30-d-old). In both cases,
xylem suction decreases along the roots with distance from
the base and with depth of the root system. The main
difference (for a homogeneous conductance distribution)
between the two types of architectures arises from the xylem
water potential evolution in the lateral roots : the taproot
system shows variations of xylem potential along the lateral
roots while the maize does not.
It can also be seen in Fig. 4 B that the xylem water
potential gradients in the laterals are much less marked than
those in the tap root. This underlines the importance of the
axial conductance of the main root for receiving flow from
laterals. This effect seems less important for maize because
adventitious roots act in a parallel way. In this example,
total root length of maize is greater than in the case of the
taproot system, but this has little influence on the water
potential distributions. More important is the ratio of the
branch root length to the main axis length, and the parallel
arrangement of the maize main root axes.
219
–40
–60
–80
–100
C
–120
–2
0
2
4
6
Radial inflow (cm s–1 × 106)
8
F. 5. The distribution of the water flux density for each root in the
root system with depth, for a simulated maize root system architecture
(cf. Fig. 4) in the case of homogeneous axial conductance and radial
conductivity in the whole root system (A). Soil water potential is
0 MPa, imposed root system collar water potential is ®1 MPa. B, As
for A, but the axial conductance of the internodes between the crowns
of adventitious roots is very high. C, As for B, but the soil water
potential decreases from ®0±1 MPa at 100 cm depth to ®0±74 MPa at
the soil surface. The evolution of fluxes in long axile roots is a
continuous curved line on the figure.
the case of a negligible axial resistance in the internodes
represents 164 % of the flux in the homogeneous conductance case, for the same water potential at the base of the
plant. This shows that the water conducting capacity of
internodes between the crowns of adventitious roots can
also be a limiting factor in root water uptake.
220
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
EŠolution of water potentials and flux with transpiration
Figure 6 shows the response of water potential and radial
inflow for two roots and two locations on these roots to
increasing and decreasing transpiration rates. Here, the
axial conductance of the internodes is non-limiting to water
flow. The evolution of the water potential and flux follows
closely the transpiration rate, but the response increases
towards both the base of the root and the base of the plant.
So, in the case of a homogeneous axial conductance and
radial conductivity along the roots, the sites and intensity of
water absorption in the root system vary in the course of the
day.
Non-homogeneous soil water potential
Figures 4 D and 5 C show the xylem suction and flux
distribution as a function of depth for the case of an
exponential decrease of soil suction with depth. Here, the
soil suction decreases from 0±74 MPa at the soil surface
down to 0±1 MPa at a 100 cm depth, with a decay rate of
0±02 cm −". In contrast to the case of a constant soil water
potential, almost all parts of the root system are active in
taking up water (Fig. 4 D). Although the water potential is
most negative in the upper part of the profile, most of the
water is taken up here (Fig. 5 C). In the case of a
heterogeneous soil water potential, the position of the roots
in the soil space is important. This can be seen in Fig. 5 C
where negative water fluxes appear (the flow is non-polar in
the model). These exorption fluxes are related to the lateral
roots pointing upwards.
As also shown by Landsberg and Fowkes (1978) for the
single root case, it is found from numerical experiments that
(Rhet®Rhom) C 1}Q, i.e. the difference between the total
resistance to water flow of the whole root system in the
heterogeneous (Rhet) and constant (Rhom) soil water potential
cases varies inversely with total outflow (Q). Moreover, if
the water potential gradient is inverted, i.e. the soil deep
down is drier than that near the surface, and if the water
potential is below a threshold value (greater than the mean
soil water potential), the total outflow may be negative and
the plant loses water to the soil.
EŠolution of the total root system conductance during
plant deŠelopment
As the root system architecture model also simulates
growth of the root system, it is possible to simulate the
evolution of the water conducting capacity of the roots
during the development of the plant for a given set of axial
conductances and radial conductivities. This is illustrated in
Fig. 7 with two examples concerning a maize root
architecture. The total root system conductance is calculated
by dividing the total outflow (transpiration) by the difference
between the water potential in soil and in the xylem of the
root system collar. The root system conductivity is the total
conductance per unit root surface area. The difference
between the two examples is that in Fig. 7 A the axial
conductance of internodes between adventitious roots is
very high, 10% times the root conductance (case a), while in
Fig. 7 B the axial conductance of internodes is the same as
the roots (case b). We can see that the two examples show
very different behaviour with respect to the total conductance of the root system. In case b, the evolution of the
total conductance presents a relatively complicated pattern
which is related to the development of new crowns of
adventitious roots and the creation of a new internode. On
the contrary, in case a, the total conductance increases in a
monotonous manner with time, each new crown of nodal
roots increasing the water conducting capacity of the root
system. The difference in water conducting capacity between
cases a and b may be as great as 450 %. On the other hand,
the root system conductivity shows similar behaviour in
both cases, a more or less exponential decline which
indicates again that only a part of the total root system is
active in water uptake for the test case investigated. This
also holds for the total conductance per unit root length or
volume (data not shown). Finally, it is interesting to note
that even if the characteristics of the root systems are drawn
from statistical distributions and so differ in configuration
and total root length, the total root conductance varies little
as shown by the standard deviation in Fig. 7 A and B.
D I S C U S S I ON
Water uptake by plant roots is a key process for water
transfer in the soil-plant-atmosphere-continuum. As more
detailed data on the physiology of water absorption and
root system architecture become available, the integration
of this information into a quantitative framework is
necessary to give new insights into water absorption at the
root system level. To this end, we have presented a numerical
model which combines information on locally measured
axial and radial root conductances with root system
architecture data. The root architecture model of Page' s et
al. (1989) is coupled with laws which determine the water
flow in roots to obtain a ‘ Hydraulic Tree Model ’ of the
root system. For a given distribution of soil water potentials
and either a given flux or water potential at the root system
collar, fluxes into and along the roots, as well as the xylem
water potentials, can be calculated everywhere in the root
system.
If the conductance is uniform along the roots in the whole
root system for the maize root system architecture, the
model shows that the distribution of fluxes and xylem water
potentials may be very heterogeneous, differing for each
root, even if the soil water potential is uniform. Such
behaviour poses questions of the water uptake models that
assume uniformity throughout the root system. Fluxes and
water potentials intimately depend on the distribution of the
root hydraulic conductance used. Our assumption of
uniform conductance allowed us to present the model as
simply as possible and to extend the analytical results of
Landsberg and Fowkes (1978) from the single root to the
root system level.
The integration of the local root conductance at the root
system level is important in analysing the total conductance
(or conductivity) of the root system. For example, the
values of total conductance (per unit length or surface)
obtained by Newman (1973) with maize are close to the
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
A
Transpiration (cm3 s–1)
0.012
0.008
0.004
3.6 cm from root base
24 cm from root base
0.0
2
B
Phytomer 3
Xylem suction (MPa)
1.5
Seminal
root
1
0.5
0
C
Radial flow (cm3 cm–2 s–1 × 105)
4
Phytomer 3
3
Seminal
root
2
1
0
0
2
4
6
8
Time (h)
10
12
221
calculated ones presented in Fig. 7 A [Newman’s results for
total conductance per unit length or surface (m# s−" MPa−"
and m s−" MPa−") being, respectively, 2±37¬10−"" and
2±22¬10−), while the simulation gives 2±09¬1®−"" and
1±73¬10−)). When comparing these total conductances with
those of other herbaceous species to locate the zones of
major resistance in the roots, Newman assumed a negligible
axial resistance. In our example, we see that approximately
the same total conductance can be found, but with, in this
case, a non-negligible axial resistance to water flow. This
example illustrates the difficulty of interpreting global water
uptake back to a singular root without knowing the details
of the root conductances for integration into an entire root
system with the help of a model.
Conversely, the local influence on the root system of the
climatic demand (transpiration) can be investigated. In the
case of a homogeneous conductance, the intensity of water
uptake is shown to vary along the root system in the course
of the day.
The link that this model makes with root architecture is
important. It can be employed as a new tool to devise
integrated studies between the root activity and architecture.
Some of the consequences of the type of branching system
(adventitious or taproot) on the water fluxes and potential
distribution have been examined. Moreover, the distribution
of the conductance in the root system is probably related to
the growth strategy, the latter implying a definite topological
structure (i.e. branching pattern). Thus, the axial conductance of the tap root (for taproot system) or of the
internodes between adventitious roots (for adventitious
root systems) may have dramatic consequences on the total
conductance of the root system. As the growth of the root
system is simulated in the model, the development of the
water conducting capacity with the phenological stage may
be investigated. Moreover, as the characteristics of simulated
root system development and architecture are drawn from
statistical distributions, the consequences of the variability
of architecture on water absorption can also be examined.
Besides the integration of experimental conductance data
(cf. the accompanying paper), it is worthwhile considering
the optimal approach to water transfer in the root system as
proposed by Fitter (1991) in conjunction with the ‘ Hydraulic
Tree Model ’ of the root system. This could be analysed
through a cost}benefit concept, the cost being the creation
of vascular structures for axial transfer, while the benefit
(root system outflow) being largely dependent on the
‘ coherent ’ distribution of the axial conductance in the root
system investigated.
Considering the spatial distribution of roots in soil, the
case of an exponential distribution of soil water potential
was examined. The simulated root system adjusts xylem
F. 6. Evolution of the calculated xylem suction (B) and water flux in
roots (C) with variation in transpiration (A) during the course of the
day for two roots and two locations on these roots (3±6 cm and 24 cm
from the base of the root). Simulated maize root system architecture
(30-d-old) with constant axial conductance and radial conductivity in
the whole root system except at the internodes between the crowns of
adventitious roots where the axial conductance is very high (here,
phytomer 3 is the second crown of the nodal roots on the stem). Soil
water potential is 0 MPa.
10–6
A
14
Total conductance
10–7
12
10
8
Conductivity
10–8
6
10–9
4
2
0
10
20
30
40
50
Time after germination (d)
10–10
60
5
10–6
B
Total conductance
4
10–7
3
10–8
2
10–9
1
0
Conductivity
10
20
30
40
50
Time after germination (d)
10–10
60
Total conductivity (m3 m–2 root s–1 MPa–1)
16
Total conductance (m3 s–1 MPa–1 × 109)
Doussan et al.—Modelling of the Hydraulic Architecture of Root Systems
Total conductivity (m3 m–2 root s–1 MPa–1)
Total conductance (m3 s–1 MPa–1 × 109)
222
F. 7. Evolution of the conductance and conductivity of the whole root system with time for (A) very high axial conductance of internodes
between crowns of adventitious roots (10% times root conductance, i.e. no resistance to axial water flow between internodes) and (B) internode
axial conductance equal to axial conductance of roots. Simulated maize root system architecture with constant axial conductance and radial
conductivity along the whole root system except at internodes for case (A). Dotted lines are ³ one s.d. of four replicate numerical experiments.
water potentials and fluxes in response to the availability of
soil water. Some kind of hydraulic lift is shown in this case,
where water exorption is not regulated (flow is non-polar in
the model, but this could be adjusted). This functional
flexibility of the root system in relation to the heterogeneity
of the growth medium is an important aspect of the
simulation, as the soil is an intrinsically heterogeneous
medium (in space and time). Moreover, this heterogeneity
may be reinforced by agricultural practices (surface or
trickle irrigation). If the idea of an optimal conductance
distribution is to be retained, this distribution should be
dynamic due to the variability in the growth conditions in
the soil. Indeed, the hydraulic architecture of the plant root
system may be able to adapt to new conditions by root
growth and decay or evolution of the conductance due to
changes in environmental conditions.
Since maps of xylem water potential and fluxes of the root
system are outputs of the model, this should enable root
growth to be linked with water availability. The ABA flux
hypothesis could possibly be investigated using this linkage.
Future developments of the model should include the
capacitive effect of roots (but fewer experimental data are
available) and the extension of the plant boundary condition
from the collar of the root system to the leaves.
Besides studies on root water absorption, the ultimate
goal of the model is its integration into a soil water transfer
simulator to study the interactions between the soil and the
plant water absorption.
Finally, by using the model in conjunction with some
experimental data of Frensch and Steudle (1989) on the
axial and radial conductances of maize root segments, it is
shown that only a part of the root system of maize is active
in water uptake. The data of Varney and Canny (1993)
show that, in fact, this is not the case. This will be examined
an accompanying paper which focuses on the estimation of
conductances in the root system to match observed patterns
of water uptake.
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APPENDIX
L is a diagonal matrix (i.e. Lij 1 0 if i ¯ j) whose terms (Lii)
represent, firstly, the axial conductances (Kh }∆zi) of the
i
links in the plant and secondly the radial conductances
(Lr ¬Si, where Si is the surface of the root segment)
i
between the soil and plant.
The incidence matrix IM. If Nc is the number of compartments, the dimensions of IM are (Nc, Nc®1) (i.e. number of
compartments, number of links) :
IMij ¯ 0 if compartments i and j are not connected
IMij ¯ 1 if i and j are connected and if the link is directed
from j to i
IMij ¯minus 1 if i and j are connected and if the link is
directed from i to j
The conductance matrix C is defined by :
Cij ¯ 0 if nodes i and j are not connected,
Cij ¯ Cji
Cil ¯®Kh }∆zil, for a node i, if l is connected to i
l−"
Cii ¯ Σll&i Kh }∆zil­Lr
i−"
i