Modelling of the Hydraulic Architecture of Root Systems: An

Annals of Botany 81 : 213–223, 1998

Modelling of the Hydraulic Architecture of Root Systems: An Integrated

Approach to Water Absorption—Model Description

CLAUDE DOUSSAN*†, LOI$ C PAGE' S‡ and GILLES VERCAMBRE‡

* INRA, UniteU de Science du Sol, Domaine Saint Paul, Site Agroparc, 84914 A�ignon Cedex 9, France

and ‡INRA, UniteU de Recherche en Ecophysiologie et Horticulture, Domaine Saint Paul, Site Agroparc,

84914 A�ignon 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 locallymeasured root hydraulic conductances with data on the root system architecture to give a detailed description ofwater absorption, from the single root level to the entire root system. This is achieved by coupling a three-dimensionalroot system architecture model with laws describing water flow in roots. In addition to water absorption studies, themodel has been developed so that it can be included in a soil water transfer simulator in order to analyse soil–plantinteractions for water uptake. The use of the model in describing water absorption is illustrated for a specific casewhere the hydraulic conductances are considered uniform in the whole root system. In this way, analytical results ofLandsberg and Fowkes (Annals of Botany 42 : 493–508, 1978) are extended from the single root to the root systemlevel. 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 inthe root system are examined. The dynamics of the total conductance of the maize root system with plant growth isalso shown for this case of uniform conductance in the root system. Cases which consider other distributions of theconductance 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 determinedby three main factors : (1) the soil properties (water releaseand hydraulic conductivity of the soil) ; (2) the root systemarchitecture (i.e. extension in space and connections betweenroots), here considered as a network of absorbing organs;and (3) the absorption capability of roots, dependent on thesoil-root interface and the resistance of the root to watertransfer.

The pioneering work of Gardner (1960) emphasized thesoil 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 thecomplexity of a real root system, much work has been doneto define a ‘macroscopic’ function for water uptake. Thisapproach consists of incorporating a root sink term in theDarcy-Richards equation for flow in soils. As stated byMolz (1981), all of the various sink functions proposed inthe literature are more or less empirical and often includeimplicit assumptions on the location of any major resistanceto flow. Generally, in the description of water uptake in soil,the root systems are extremely simplified, and describedonly by the root length density and, sometimes, anhomogeneous root resistance to water flow. But exper-imental data (Smucker and Aiken, 1992, Tardieu, Bruckler

† For correspondence. Fax (33) 490 31 62 44, e-maildoussan!avignon.inra.fr

and Lafolie, 1992) and calculations (Bruckler, Lafolie andTardieu, 1991; Lafolie, Bruckler and Tardieu, 1991; Petrieet al., 1992) show that the spatial arrangement of roots inthe soil (not reflected in the average root length density) hasa substantial influence on water uptake. Moreover, thecapability of roots to absorb water may evolve in space andtime because of the development of an interfacial resistanceto water transport between the soil and the roots (Herkel-rath, Miller and Gardner, 1977) or due to a variation in thephysiological properties along the root (Sanderson, 1983;Varney and Canny, 1993).

Most models describing water uptake consider only alimited number of the features of the water absorptionprocess. If, generally, water transfer in soil is well representedby the microscopic or macroscopic approaches, a detaileddescription of the root system is lacking (Molz, 1981). Whenthe geometrical distribution of roots (Lafolie et al., 1991) orthe root system architecture (Clausnitzer and Hopmans,1994) is taken into account, the hydraulic continuity in andbetween the roots is neglected and hence the water transferin the root system is poorly described. Nevertheless, thexylem water potential has been shown to vary, sometimesquite substantially, along roots (Passioura, 1972; Nobel andLee, 1991; Alm, Cavelier and Nobel 1992; Simmoneau andHabib, 1994). Finally, if some kind of interfacial resistanceis accounted for, the variation in the water uptake capabilityor in the root hydraulic conductance is only poorlyconsidered.

Experimental advances have been made recently in

0305-7364}98}02021311 $25.00}0 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 betweenroots, age and the branching pattern of roots (Page' s andPellerin, 1994; Pellerin and Page' s, 1994). Moreover, it ispossible to perform numerical simulations of root growthand architecture (Page' s, Jordan and Picard, 1989; Lynchand Nielsen, 1996). In this way, realistic and detailedsimulations of root systems can be obtained relatively easilyand the computer can be used as a virtual experiment tostudy the interactions of root systems with their environment(e.g. Clausnitzer and Hopmans, 1994). The physiologicalcharacteristics of water uptake by root segments (axial andradial conductances) can now be examined at the centimetrescale (Steudle, 1994;North andNobel, 1995).Measurementsof the flux into the root (Varney and Canny, 1993) or xylemtension (Balling et al., 1988) as a function of position alongthe root, or of branching order, give new insights into thecharacteristics of water absorption. Recent anatomicalinvestigations (xylem and endoderm maturation, branching)also provide information on the evolution and developmentof water conduction capabilities in roots (McCully, 1995).

This scattered information would provide a betterunderstanding of water absorption if it was brought togetherin a single quantitative global framework. As a consequence,we believe that it is useful to develop, with the current stateof knowledge, a model integrating this localized informationusing physical laws which describe the water transfer in theroot system (‘Hydraulic Tree Model ’ of the root system).The model could simulate both the distribution of fluxesand water potentials in the root system and the resistance towater flow in part of or in the whole of the root system. Sucha model, describing water absorption from the single root tothe root system level, can (1) help to describe moreconsistently water absorption in soil without empiricallydefined sink functions; (2) give new insights into root waterabsorption by integrating processes analysed at the singleroot level (e.g. variation of root water conductance) ; (3)help to identify prominent architectural features influencingwater uptake (such as root lengths, diameters, branchingdensity, type of root system architecture etc) ; and (4) beused to help in the experimental design and measurement oflocal root properties. Above all, it will be possible to assessby sensitivity analysis the influence of variations and}oruncertainty in the architectural or physiological (e.g. rootresistance to water flow) parameters on the water uptake atlocal or global levels in the root system.

Apart from the Ohm’s law analogy (van den Honert,1948), water movement through plant roots has beentheoretically analysed by Landsberg and Fowkes (1978)who derived an analytical solution for the case of a singleroot with a constant hydraulic conductance. Alm et al.(1992) extended the results of Landsberg and Fowkes to thecase where the conductance varies along the root, separatingthe root into segments, each with a homogeneous con-ductance. In this article, we extend the approach of Alm etal. (1992) to the whole root system by coupling the rootsystem architecture with the process of water absorption. Inthe first paper, the fundamentals of the architecture model

of Page' s et al. (1989) are briefly reviewed and the theoreticalfoundations for the ‘Hydraulic Tree Model ’ of the rootsystem are given. The application of the model to a test case,extending the results of Landsberg and Fowkes (1978), ispresented. In an accompanying paper, we focus onexperimental data taken from the literature concerningwater absorption by roots and look at the consequences formodelling water uptake.

MODEL DEVELOPMENT

The root system architecture model

We used the model of Page' s et al. (1989) that simulatesmaize (Zea mays L.) root architecture. The model simulatesthe three-dimensional architecture of the maize root systemin discrete time steps. At each time step (1 d), the rootsystem extends by three basic processes : (1) emergence ofnew root axes (seminal and nodal) from the shoot; (2)growth; and (3) branching. Root system development isrelated to cumulated temperatures on a daily basis, with a6 °C base temperature.

The root system is simulated by a set of segments, eachsegment representing the part of the root that was generatedduring one time step. The spatial coordinates of eachsegment are stored in a data structure, together withinformation on the position of the segment within thearchitecture (i.e. branching order, internode of origin,formation date, connection with other segments).

The emergence of main root axes on the same phytomeris assumed to occur simultaneously and the rank of theactive phytomer is calculated from a linear function ofcumulated thermal units. The length and diameter of eachinternode is taken from experimental data. Roots areassumed to elongate according to the following function ofthermal time:

L¯A(1®e−bT) (1)

where L is the length of the root, T is thermal time fromemergence, A is the final root length and b is a rateparameter. The asymptotic value A is randomly drawn froman estimated distribution for each root emergence. Growthdirections are calculated by combining the effects ofgravitropism 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, andwhose density depends on branching order. The branchingdirection is calculated using a branch angle drawn from anormal distribution and the radial angle randomly drawnfrom a uniform distribution over 0 to 2π.

Equations for flow in the root

Water flow through a root can be characterized by theroot hydraulic conductivity L

p(m s−" MPa−") (Fiscus, 1975) :

J�¯Lp(∆P®σ∆π) (2)

where J� (m$ m−# s−") is the flux density of water entering theroot (i.e. the flow rate divided by the root area), ∆P (MPa)is the difference in hydraulic pressure between the externalmedium 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 oftwo main roots with laterals) showing the nodes for flow calculation in the root system.

osmotic potential between the external medium and thexylem sap, and σ is the reflection coefficient for solutes.

In this study, two hypotheses are formulated: firstly, theinfluence of solutes on flow is neglected, because : (1) duringperiods of active transpiration, the hydrostatic pressuregradient (∆P) rather than the osmotic potential gradient isthe effective driving force for flow [and the relation betweenhydrostatic pressure gradient and flux is linear (Fiscus,1975; Weatherley, 1982)] ; (2) we consider here that the soilwater is a dilute solution as is the sap. The case of a moreconcentrated soil solution can be investigated by usingthe total water potential gradient (∆ψ) and assuming thatσ¯ 1; (3) the presence of solutes in the water flow greatlyincreases the mathematical complexity of the problem.Moreover, as the radial pathway for water in roots is stillnot clear (Huang and Nobel, 1994), models coupling flow ofwater and solutes are still controversial (Fiscus, 1975;Katou and Taura, 1989; Steudle, 1994).

With this first assumption, eqn (2) can be rewritten:

J�¯Lp∆P or J�¯L

p∆ψ (3)

and expressed in terms of hydraulic pressure or total waterpotential.

Secondly, we consider only steady-state flow, that is tosay the capacitive effect of the roots is neglected. Transienteffects in plants are more important when an abrupt change

occurs in the flow environment (water potential or weatherchanges). This is, for example, the case when transpirationcommences, but in soil the variations are more gradual.Moreover, the water stored in the roots is generally smallcompared to transpiration requirements (Simmoneau,1992). However, it will be necessary to incorporate acapacitive term for the roots into the model to examinesmall time scales or woody plant species (Waring andRunning, 1976).

The hydraulic conductivity Lpcan be separated into two

terms, corresponding to water movement into the root andalong the root: the radial conductivity [L

r(m s−" MPa−")] for

flow from the root surface to the xylem and the axialconductance [K

h(m% s−" MPa−")] for flow along the xylem.

These can be defined, following Landsberg and Fowkes(1978), as :

Jh(z)¯®K

h

dψx(z)

dz(4)

Jr(z)¯L

r[ψ

s(z)®ψ

x(z)] (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.


11 1Lr 1

12 2Lr 2

Kh 1

13 3Lr 3

Kh 2

18 8Lr 8

Kh 7

19 9Lr 9

Kh 8

20 10Lr 10

Kh 9

4Kh 3 Kh 4

5 7

14 15 17

6

16

Lr 6

Secondaryroot

Legend

Tertiaryroot

Kh 5

Lr 4 Lr 5

Collar boundary condition

Plant node

Soil node

Axial conductance

Radial conductance

Primary root

Kh 6

Lr 7

F. 2. A simplified discretized root system to show numbering of the soil and plant calculation nodes, and numbering of the conductances (axialand 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 ofthe root architecture model at a given time after germination.Some of the results presented here are derived from graphicaltheory (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). Eachcompartment is connected to its neighbours (same root orbranch roots) by an oriented link which is the axialconductance between the two compartments. In this case,eqn (4) can be approximated by:

Jh(z)¯®K

h

∆ψx

∆z(6)

where ∆z is the distance between the two nodes of thecompartments. Likewise, each root node is connected to asoil node by an oriented link (radial conductivity) for whicheqn (5) is valid.

Compartments are numbered serially in the upstreamdirection, starting with the collar of the root system, andbranch roots being numbered first (Fig. 2). Nodes cor-responding to the soil are then numbered in the same way asthe root nodes. The same is done for the links (axial andradial conductances) oriented in the direction of increasingnode values. The xylem water potentials at the root nodesare unknowns, whereas the matric (or total) water potentialsat the soil nodes are given boundary conditions. At thecollar of the root system, either a known value of xylemsuction (or water potential), or flow rate (transpiration), isimposed. In such a system, the fluid flow laws are expressedusing a matrix notation:

DY ¯®L¬dψMN

(7)

IM¬DY ¯ 0Y (8)

where DY (dimension [Nc®1], where N

cis the number of

compartments) is the flow rate vector through the linkbetween two nodes, dψ

MN[N

c®1] is the pressure difference


vector between two adjacent compartments, L is a diagonalmatrix [N

c®1, N

c®1] whose terms are the axial and radial

conductances (see Appendix). IM is the incidence matrix[N

c, N

c®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 volumetricflow rates is null in a compartment for steady state flow).

Taking into account the fact that :

dψMN

¯ IMt¬ψY (9)

IMt is the transpose of the IM matrix, ψY is the vector [Nc]

which includes firstly the xylem water potential nodes (ψxi

)and secondly the soil water potential nodes (ψ

si

).The combination and simplification of eqns (7), (8) and

(9) results in a simple linear system of equations, in matrixform:

C¬ψx

MN

¯QY (10)

where QY , on the right hand side (dimension Np, where N

pis

the number of plant nodes), contains the soil factors (Qi¯

Lri

¬Si¬ψ

si

, where Siis the surface of the root segment)

and the boundary condition used at the base of the plant. C

is the conductance (squared) matrix [Np,N

p] which takes a

simple form (see Appendix).Taking into account the fact that C is a sparse matrix

which might be large (from 60000 to" 10' equations), thelinear system of equations is solved by a preconditionedconjugate gradient method (Larabi and de Smedt, 1994).Iterations of the conjugate gradient are stopped when theresidual norm between two successive iterations is lowerthan 10−"&. When the solution for the xylem water potentialsis known, flow rates into or up the root can be calculated atany 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 soiland the plant, water efflux will proceed at the same rate asthe absorption flux in the model (i.e. the flow is non-polar).

Analytical–numerical solution comparison for the singleroot case

In the case of homogeneous radial conductivity and axialconductance (L

rand K

h), Landsberg and Fowkes (1978)

derived an analytical expression for the xylem waterpotential �s. distance from the apex (z) for a root immersedin a solution at constant potential :

ψx(z)¯ψ

s

(ψx(L)®ψ

s) cosh (αz)

cosh (αL)(11)

α#¯2πrL

r

Kh

(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. Inthis case K

h¯ 5¬10−"" m s−" MPa−" and L

r¯ 2¬

10−( m% s−" MPa−", which are the values given for maize in

50

1

0Distance from base of root (cm)

Xyl

em s

uct

ion

(M

Pa)

0.2

0.8

0.6

0.4

10 20 30 400

F. 3. Comparison of analytical (——) and numerical (+) solutionsfor the single root case. Imposed water suctions are 1 MPa at the rootcollar and 0 MPa in the outside solution. Axial conductance (K

h) 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 is3 mm, the water potential at the base of the root system is1 MPa, and the soil potential is 0 MPa. Calculation nodesare evenly spaced every 2±5 cm. We can see in Figure 3 thatthe fit is good (mean relative error : 4±4¬10−%, s.d. :2¬10−% MPa) and the accuracy of the water flux is within0±5%. As the distance between two adjacent nodes in theroot architecture model is typically less than 0±5 cm, theaccuracy should be improved further (numerical errors areinversely proportional to the space increment and pro-portional to the second derivative of the xylem pressure �s.distance).

AN EXAMPLE APPLICATION (CONSTANTAXIAL CONDUCTANCE AND RADIALCONDUCTIVITY ALONG THE ROOTS)

We now illustrate the use of the model. To keep thedemonstration simple, we employ only constant axialconductance and radial conductivity along the roots. Thisalso allows us to compare numerical outputs from acomplex root system with the conclusions of Landsberg andFowkes (1978) in the single root case. More complexdistributions of hydraulic conductance in the root systemwill be investigated in the second part of the article. Theinfluences of the type of root system architecture, theevolution of the transpiration during the day, the axialresistance of shoot internodes in the case of adventitiousroots, the non-homogeneous soil water potential and theevolution of the total root system conductance during plantdevelopment will be briefly examined. In the text whichfollows, we will use the units cm$ s−" (10−' m$ s−") for waterflux and cm$ cm−# s−" (10−# m s−") for water flux density.

Most of the examples given are for root system archi-tectures corresponding to field grown maize, according tothe data of Page' s and Pellerin (1994) and Pellerin and Page' s(1994). The homogeneous axial conductance and radial


40

–80

–40Horizontal distance (cm)

Dep

th (

cm)

20

–20

–40

–60

–20 0

0

–1 MPa0

C

40

–80


Dep

th (

cm)

20

–20

–40

–60

–20 0

0

–1 MPa0A

60–120


20

–20

–40

–60

–20 0

0

–1 MPa0

D

30

–60


–10

–20

–40

–20 0

0

–1 MPa0B

2010

40S

oil w

ater

pot

enti

al

–80

–100

F. 4. Examples of the evolution of the water potential in three-dimensional simulated root systems (30-d-old) projected on a vertical plane. Axialconductance 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, imposedroot system collar water potential is ®1 MPa. C, Maize root system where the axial conductance of internodes between the crowns of adventitiousroots is very high (10% times root conductance, i.e. no resistance to axial water flow between internodes), root axial conductance and radialconductivity 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, butwith 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 theright 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 fromFrensch and Steudle’s (1989) experiments with (young)maize roots. The axial conductance (K

h) is set to

5¬10−"" m% s−" MPa−" (the maximum value they found)and the radial conductivity (L

r) to 2±2¬10−( m s−" MPa−"

(constant, except at the tip, in their experiment).


Type of root systems and a general �iew of waterabsorption

Figure 4A and B present the xylem water potential inroots for the case of a maize root architecture (withadventitious roots) and for the case of a hypotheticaltaproot system generated by the architecture model. Axialhydraulic conductance and radial conductivity are the samefor both types of root system and constant in the whole rootsystem. 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 fromthe base and with depth of the root system. The maindifference (for a homogeneous conductance distribution)between the two types of architectures arises from the xylemwater potential evolution in the lateral roots : the taprootsystem shows variations of xylem potential along the lateralroots while the maize does not.

It can also be seen in Fig. 4B that the xylem waterpotential gradients in the laterals are much less marked thanthose in the tap root. This underlines the importance of theaxial conductance of the main root for receiving flow fromlaterals. This effect seems less important for maize becauseadventitious roots act in a parallel way. In this example,total root length of maize is greater than in the case of thetaproot system, but this has little influence on the waterpotential distributions. More important is the ratio of thebranch root length to the main axis length, and the parallelarrangement of the maize main root axes.

Influence of axial internode conductance in the case ofad�entitious roots

Figure 4A and C show water potentials and Fig. 5Aand B present radial inflow �s. depth for a simulated maizeroot system for the cases where (a) the axial conductance isthe same in the whole root system, even in the internodesbetween crowns of nodal roots ; and (b) where the axialconductance is also the same in the whole root system,except in these internodes. Here, the axial conductance ofthe internodes is set much higher than the root conductance(10% times greater) in order to nullify the water potentialgradient between the crowns of nodal roots. In this case,there is almost no restriction to axial flow in the part of thestem where nodal roots emerge. In both cases, the water fluxto the roots is far from homogeneous with depth, althoughthe soil water potential is homogeneous (Fig. 5). Again, therelatively small axial conductance of the roots is reflected bya sharp decrease in the flux and xylem suction with depth orlength of the root. Clearly here, only a part of the maize rootsystem is active in water uptake if the axial conductance isset to the value determined by Frensch and Steudle (1989)for young root segments. The effect of axial conductance isalso illustrated by comparing Figs 4A and C, and 5A andB. In the case of negligible axial resistance of the internodesbetween adventitious roots, the water potential at the baseof the plant may propagate further (Fig. 4) and the waterinflow increases (particularly in the laterals) in the upperpart of the root system (Fig. 5). Here, the total water flux in

8–120

–2Radial inflow (cm s–1 × 106)

Dep

th (

cm)

–20

–40

–60

–80

–100

420 6

0

C

2

–80

0Radial inflow (cm s–1 × 105)

Dep

th (

cm)

–20

–40

–60

10.5 1.5

0

B

–80

Dep

th (

cm)

–20

–40

–60

0

A

F. 5. The distribution of the water flux density for each root in theroot system with depth, for a simulated maize root system architecture(cf. Fig. 4) in the case of homogeneous axial conductance and radialconductivity in the whole root system (A). Soil water potential is0 MPa, imposed root system collar water potential is ®1 MPa. B, Asfor A, but the axial conductance of the internodes between the crownsof adventitious roots is very high. C, As for B, but the soil waterpotential decreases from ®0±1 MPa at 100 cm depth to ®0±74 MPa atthe 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 internodesrepresents 164% of the flux in the homogeneous con-ductance case, for the same water potential at the base of theplant. This shows that the water conducting capacity ofinternodes between the crowns of adventitious roots canalso be a limiting factor in root water uptake.


E�olution of water potentials and flux with transpiration

Figure 6 shows the response of water potential and radialinflow for two roots and two locations on these roots toincreasing and decreasing transpiration rates. Here, theaxial conductance of the internodes is non-limiting to waterflow. The evolution of the water potential and flux followsclosely the transpiration rate, but the response increasestowards both the base of the root and the base of the plant.So, in the case of a homogeneous axial conductance andradial conductivity along the roots, the sites and intensity ofwater absorption in the root system vary in the course of theday.

Non-homogeneous soil water potential

Figures 4D and 5C show the xylem suction and fluxdistribution as a function of depth for the case of anexponential decrease of soil suction with depth. Here, thesoil suction decreases from 0±74 MPa at the soil surfacedown to 0±1 MPa at a 100 cm depth, with a decay rate of0±02 cm −". In contrast to the case of a constant soil waterpotential, almost all parts of the root system are active intaking up water (Fig. 4D). Although the water potential ismost negative in the upper part of the profile, most of thewater is taken up here (Fig. 5C). In the case of aheterogeneous soil water potential, the position of the rootsin the soil space is important. This can be seen in Fig. 5Cwhere negative water fluxes appear (the flow is non-polar inthe model). These exorption fluxes are related to the lateralroots pointing upwards.

As also shown by Landsberg and Fowkes (1978) for thesingle root case, it is found from numerical experiments that(R

het®R

hom)C 1}Q, i.e. the difference between the total

resistance to water flow of the whole root system in theheterogeneous (R

het) and constant (R

hom) soil water potential

cases varies inversely with total outflow (Q). Moreover, ifthe water potential gradient is inverted, i.e. the soil deepdown is drier than that near the surface, and if the waterpotential is below a threshold value (greater than the meansoil water potential), the total outflow may be negative andthe plant loses water to the soil.

E�olution of the total root system conductance duringplant de�elopment

As the root system architecture model also simulatesgrowth of the root system, it is possible to simulate theevolution of the water conducting capacity of the rootsduring the development of the plant for a given set of axialconductances and radial conductivities. This is illustrated inFig. 7 with two examples concerning a maize rootarchitecture. The total root system conductance is calculatedby dividing the total outflow (transpiration) by the differencebetween the water potential in soil and in the xylem of theroot system collar. The root system conductivity is the totalconductance per unit root surface area. The differencebetween the two examples is that in Fig. 7A the axialconductance of internodes between adventitious roots isvery high, 10% times the root conductance (case a), while in

Fig. 7B the axial conductance of internodes is the same asthe roots (case b). We can see that the two examples showvery different behaviour with respect to the total con-ductance of the root system. In case b, the evolution of thetotal conductance presents a relatively complicated patternwhich is related to the development of new crowns ofadventitious roots and the creation of a new internode. Onthe contrary, in case a, the total conductance increases in amonotonous manner with time, each new crown of nodalroots increasing the water conducting capacity of the rootsystem. The difference in water conducting capacity betweencases a and b may be as great as 450%. On the other hand,the root system conductivity shows similar behaviour inboth cases, a more or less exponential decline whichindicates again that only a part of the total root system isactive in water uptake for the test case investigated. Thisalso holds for the total conductance per unit root length orvolume (data not shown). Finally, it is interesting to notethat even if the characteristics of the root systems are drawnfrom statistical distributions and so differ in configurationand total root length, the total root conductance varies littleas shown by the standard deviation in Fig. 7A and B.

DISCUSSION

Water uptake by plant roots is a key process for watertransfer in the soil-plant-atmosphere-continuum. As moredetailed data on the physiology of water absorption androot system architecture become available, the integrationof this information into a quantitative framework isnecessary to give new insights into water absorption at theroot system level. To this end, we have presented a numericalmodel which combines information on locally measuredaxial and radial root conductances with root systemarchitecture data. The root architecture model of Page' s etal. (1989) is coupled with laws which determine the waterflow in roots to obtain a ‘Hydraulic Tree Model ’ of theroot system. For a given distribution of soil water potentialsand either a given flux or water potential at the root systemcollar, fluxes into and along the roots, as well as the xylemwater potentials, can be calculated everywhere in the rootsystem.

If the conductance is uniform along the roots in the wholeroot system for the maize root system architecture, themodel shows that the distribution of fluxes and xylem waterpotentials may be very heterogeneous, differing for eachroot, even if the soil water potential is uniform. Suchbehaviour poses questions of the water uptake models thatassume uniformity throughout the root system. Fluxes andwater potentials intimately depend on the distribution of theroot hydraulic conductance used. Our assumption ofuniform conductance allowed us to present the model assimply as possible and to extend the analytical results ofLandsberg and Fowkes (1978) from the single root to theroot system level.

The integration of the local root conductance at the rootsystem level is important in analysing the total conductance(or conductivity) of the root system. For example, thevalues of total conductance (per unit length or surface)obtained by Newman (1973) with maize are close to the


4

Rad

ial f

low

(cm

3 cm

–2 s

–1 ×

105 )

3

2

1

0

Xyl

em s

uct

ion

(M

Pa)

2

0.5

1

0

0.012

Tra

nsp

irat

ion

(cm

3 s–1

)

0.008

0.004

0.0

12Time (h)

102 4 6 80

C

Phytomer 3

Seminalroot

B

Phytomer 3

Seminalroot

A

1.5

3.6 cm from root base24 cm from root base

calculated ones presented in Fig. 7A [Newman’s results fortotal conductance per unit length or surface (m# s−" MPa−"

and m s−" MPa−") being, respectively, 2±37¬10−"" and2±22¬10−), while the simulation gives 2±09¬1®−"" and1±73¬10−)). When comparing these total conductances withthose of other herbaceous species to locate the zones ofmajor resistance in the roots, Newman assumed a negligibleaxial resistance. In our example, we see that approximatelythe same total conductance can be found, but with, in thiscase, a non-negligible axial resistance to water flow. Thisexample illustrates the difficulty of interpreting global wateruptake back to a singular root without knowing the detailsof the root conductances for integration into an entire rootsystem with the help of a model.

Conversely, the local influence on the root system of theclimatic demand (transpiration) can be investigated. In thecase of a homogeneous conductance, the intensity of wateruptake is shown to vary along the root system in the courseof the day.

The link that this model makes with root architecture isimportant. It can be employed as a new tool to deviseintegrated 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 potentialdistribution have been examined. Moreover, the distributionof the conductance in the root system is probably related tothe growth strategy, the latter implying a definite topologicalstructure (i.e. branching pattern). Thus, the axial con-ductance of the tap root (for taproot system) or of theinternodes between adventitious roots (for adventitiousroot systems) may have dramatic consequences on the totalconductance of the root system. As the growth of the rootsystem is simulated in the model, the development of thewater conducting capacity with the phenological stage maybe investigated.Moreover, as the characteristics of simulatedroot system development and architecture are drawn fromstatistical distributions, the consequences of the variabilityof architecture on water absorption can also be examined.

Besides the integration of experimental conductance data(cf. the accompanying paper), it is worthwhile consideringthe optimal approach to water transfer in the root system asproposed by Fitter (1991) in conjunction with the ‘HydraulicTree Model ’ of the root system. This could be analysedthrough a cost}benefit concept, the cost being the creationof vascular structures for axial transfer, while the benefit(root system outflow) being largely dependent on the‘coherent ’ distribution of the axial conductance in the rootsystem investigated.

Considering the spatial distribution of roots in soil, thecase of an exponential distribution of soil water potentialwas examined. The simulated root system adjusts xylem

F. 6. Evolution of the calculated xylem suction (B) and water flux inroots (C) with variation in transpiration (A) during the course of theday for two roots and two locations on these roots (3±6 cm and 24 cmfrom the base of the root). Simulated maize root system architecture(30-d-old) with constant axial conductance and radial conductivity inthe whole root system except at the internodes between the crowns ofadventitious 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.


60

16

Time after germination (d)

Tot

al c

ondu

ctan

ce (

m3 s

–1 M

Pa–1

× 1

09 )

50

6

14

12

10

8

4

2

10 20 30 400

10–6

10–10

Tot

al c

ondu

ctiv

ity

(m3 m

–2 r

oot

s–1 M

Pa–1

)

10–7

10–8

10–9

ATotal conductance

Conductivity

60

5

Time after germination (d)

Tot

al c

ondu

ctan

ce (

m3 s

–1 M

Pa–1

× 1

09 )

50

4

3

2

1

10 20 30 400

10–6

10–10

Tot

al c

ondu

ctiv

ity

(m3 m

–2 r

oot

s–1 M

Pa–1

)

10–7

10–8

10–9

B

Total conductance

Conductivity

F. 7. Evolution of the conductance and conductivity of the whole root system with time for (A) very high axial conductance of internodesbetween crowns of adventitious roots (10% times root conductance, i.e. no resistance to axial water flow between internodes) and (B) internodeaxial conductance equal to axial conductance of roots. Simulated maize root system architecture with constant axial conductance and radialconductivity 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 ofsoil water. Some kind of hydraulic lift is shown in this case,where water exorption is not regulated (flow is non-polar inthe model, but this could be adjusted). This functionalflexibility of the root system in relation to the heterogeneityof the growth medium is an important aspect of thesimulation, as the soil is an intrinsically heterogeneousmedium (in space and time). Moreover, this heterogeneitymay be reinforced by agricultural practices (surface ortrickle irrigation). If the idea of an optimal conductancedistribution is to be retained, this distribution should bedynamic due to the variability in the growth conditions inthe soil. Indeed, the hydraulic architecture of the plant rootsystem may be able to adapt to new conditions by rootgrowth and decay or evolution of the conductance due tochanges in environmental conditions.

Since maps of xylem water potential and fluxes of the rootsystem are outputs of the model, this should enable rootgrowth to be linked with water availability. The ABA fluxhypothesis could possibly be investigated using this linkage.Future developments of the model should include thecapacitive effect of roots (but fewer experimental data areavailable) and the extension of the plant boundary conditionfrom the collar of the root system to the leaves.

Besides studies on root water absorption, the ultimategoal of the model is its integration into a soil water transfersimulator to study the interactions between the soil and theplant water absorption.

Finally, by using the model in conjunction with someexperimental data of Frensch and Steudle (1989) on theaxial and radial conductances of maize root segments, it isshown that only a part of the root system of maize is activein water uptake. The data of Varney and Canny (1993)show that, in fact, this is not the case. This will be examinedan accompanying paper which focuses on the estimation ofconductances in the root system to match observed patternsof water uptake.

LITERATURE CITED

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and axial conductivities for individual roots : Development and

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measurement of negative pressure in artificial-biological systems.

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and soil-root water transport : II field comparisons. Soil Science

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Clausnitzer V, Hopmans JW. 1994. Simultaneous modelling of transient

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induced flow. Plant Physiology 55 : 917–922.

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Waisel Y, Eshel A, Kafkafi U, eds. Plant roots: the hidden half.

Basel : Marcel Dekker, 3–24.

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roots of maize (Zea mays L.) Plant Physiology 91 : 719–726.

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plants : 1. Divided root experiments. Soil Science Society of

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and soil-root water transport : I model presentation. Soil Science

Society of America Journal 55 : 1203–1212.Landsberg JJ, Fowkes ND. 1978. Water movement through plant

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Lynch J, Nielsen KL. 1996. Simulation of root system architecture. In:

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Molz FJ. 1981. Models of water transport in the soil—plant system: areview. Water Resources Research 17 : 1245–1260.

Newman EI. 1973. Permeability to water of the roots of five herbaceousspecies. The New Phytologist 72 : 547–555.

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Page' s L, Pellerin S. 1994. Evaluation of parameters describing the rootsystem architecture of field grown maize plants (Zea mays L.). IIDensity, length and branching of first order lateral roots. Plant andSoil 164 : 169–176.

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APPENDIX

L is a diagonal matrix (i.e. Lij1 0 if i¯ j) whose terms (L

ii)

represent, firstly, the axial conductances (Khi

}∆zi) of the

links in the plant and secondly the radial conductances(L

ri

¬Si, where S

iis the surface of the root segment)

between the soil and plant.The incidence matrix IM. If N

cis the number of compart-

ments, the dimensions of IM are (Nc,N

c®1) (i.e. number of

compartments, number of links) :IM

ij¯ 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 iIM

ij¯minus 1 if i and j are connected and if the link is

directed from i to jThe conductance matrix C is defined by:

Cij¯ 0 if nodes i and j are not connected,

Cij¯C

ji

Cil¯®K

hl−"

}∆zil, for a node i, if l is connected to i

Cii¯Σl

l&iK

hi−"

}∆zilL

ri

Documents

Modelling of the Hydraulic Architecture of Root Systems: An