New Root growth models: towards a new generation of continuous...

Journal of Experimental Botany, Vol. 61, No. 8, pp. 2131–2143, 2010doi:10.1093/jxb/erp389 Advance Access publication 27 January, 2010

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Root growth models: towards a new generation ofcontinuous approaches

Lionel Dupuy*, Peter J. Gregory and A. Glyn Bengough

Scottish Crop Research Institute, Invergowrie, Dundee DD2 5DA, UK

* To whom correspondence should be addressed. E-mail: [email protected]

Received 24 August 2009; Revised 24 November 2009; Accepted 8 December 2009

Abstract

Models of root system growth emerged in the early 1970s, and were based on mathematical representations of root

length distribution in soil. The last decade has seen the development of more complex architectural models and the

use of computer-intensive approaches to study developmental and environmental processes in greater detail. There

is a pressing need for predictive technologies that can integrate root system knowledge, scaling from molecular to

ensembles of plants. This paper makes the case for more widespread use of simpler models of root systems based

on continuous descriptions of their structure. A new theoretical framework is presented that describes the dynamics

of root density distributions as a function of individual root developmental parameters such as rates of lateral root

initiation, elongation, mortality, and gravitropsm. The simulations resulting from such equations can be performedmost efficiently in discretized domains that deform as a result of growth, and that can be used to model the growth

of many interacting root systems. The modelling principles described help to bridge the gap between continuum and

architectural approaches, and enhance our understanding of the spatial development of root systems. Our

simulations suggest that root systems develop in travelling wave patterns of meristems, revealing order in otherwise

spatially complex and heterogeneous systems. Such knowledge should assist physiologists and geneticists to

appreciate how meristem dynamics contribute to the pattern of growth and functioning of root systems in the field.

Key words: Architecture, deformable model, development, dynamics, meristem, root, wave.

Introduction

Root system architecture is important for the plant to

access soil resources. There is now considerable evidence

linking root architecture with water and nutrient acquisition

efficiency (Hodge et al., 1999; White et al., 2005; Lynch,

2007). The structure and dynamics of root system architec-

ture, however, are complex, and architectural modelling has

resulted from the need to incorporate space into models andto take account of the biophysical interactions taking place

between roots and their environment. Simple spatial models

of root growth and development have been available for

a long time (Gerwitz and Page, 1974), but the advent of

more powerful computers and the availability of software

and simple programming languages has facilitated the

development of more complex plant architectural models

(Jourdan and Rey, 1997)—some examples of model typesare given in Table 1.

Currently, most plant root architectural models use

computer simulations to reproduce the developmental pro-

cesses of root apical meristems and to construct virtual root

architectures: single roots are assembled incrementally

through the growth of a set of virtual apical meristems

whose activity is determined at each time step of the

simulation. The resulting root architecture is a complexdata structure describing the geometrical properties of the

different roots and the topology of connection they

establish with each other. Usually the same morphogenetic

rules are used to prescribe the behaviour of whole sets of

meristems, and complex architectures arise as emergent

properties of these simple rules (Prusinkiewicz, 2004).

The last 10 years have seen the development of a variety of

models integrating important biological and physical pro-cesses of growth, a discipline now referred to as ‘functional

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structural plant modelling’ (Godin and Sinoquet, 2005).

Root architectural models have been used to describe water

and nutrient transfer from soil to plants (Doussan et al.,

1998; Dunbabin et al., 2002). Elaborate models of light

interception by shoots and canopies have been developed

(Birch et al., 2003), and whole plant architectural models

that simulate carbon and mineral elements are beingconstructed (Drouet and Pages, 2007). Architectural models

have also proven to be particularly powerful in analysing

plant–environment interactions, incorporating various

forms of mechanical (Fourcaud et al., 2003; Dupuy et al.,

2005a) and hydraulic (Somma et al., 1998) processes.

Models incorporating links to genetic data (Tardieu, 2003;

Letort et al., 2008) and processes regulating development

are now under active development (Lucas et al., 2008).The success of models incorporating root system archi-

tecture has also created a need for different types of

models and approaches (de Dorlodot et al., 2007), and also

for realistic parameter values for a range of crops (Tsegaye

et al., 1995). For example, the models needed to assist

breeding strategies (Hammer et al., 2006) or population

dynamics involve simulating large numbers of single plants

simultaneously (Podlich and Cooper, 1998; Grimm, 1999).

This is still a huge computational challenge for architec-

tural models because the functioning of all organs is

computed explicitly (e.g. Dunbabin, 2007). Effective mod-

elling of plant–soil interactions also requires coupling

discrete structures, i.e. roots, to continuous descriptionsof the environments (e.g. soil mechanics and hydraulic

models). Other forms of spatial models that could provide

efficient solutions to these problems, continuous-based

approaches in particular, have not developed at a compa-

rable rate.

The purpose of this paper is to review existing, continu-

ous, spatial approaches for the modelling of root systems,

set them in a historical context, and to examine theirpotential to provide new insight into the mechanisms of

plant growth and development. Recent advances in the field

are presented and the case will be made that further

development of these techniques could lead to a new

generation of accurate and computationally efficient whole

root system models.

Table 1. Examples of root system models, parameters, assumptions, and applications

Model type anddescription

Main parameters Underlying assumptions Application examples Advantages anddisadvantages

Root depth models:

continuous models that give

distribution of root length

densitya

Maximum rooting depth.

Variation of root

distribution with depth

Representative volume

sufficiently big for continuity

in space and time. Empirical

root distribution with depth,

and advance of rooting front

with time

Describing root distribution with

depth.b Predicting root distribution

to model crop growth or uptakec

Good simple descriptions of root

distribution with depth. Not

explicitly related to many biological

parameters. May neglect factors

such as root clustering in

structured soils that can have

major effects on uptaked

Architectural models: discrete

models are based about an

explicit description of root

growth and branching: 1-D,

2-De, or 3-Df

For each order of

root—elongation rate;

branching interval;

branching angle; growth

tortuosity; gravitropism

Growth parameters are valid

for conditions and types of

roots simulatedg

Predicting efficacy of root

measurement methods.h Predicting

effect of gravitropism on phosphate

uptake.i Predicting effects of plant

competition on uptake.j Predicting

effect of architecture and exudates

on root uptakek

Good explicit descriptions and

visualization of root systems,

especially in 3-D. Require large

numbers of input parameters that

may be difficult to measure.

Spatial models of root growth

dynamics: continuum models

for describing the 2-D or 3-D

patterns of growth using root

length density, meristem

density

Elongation rate, and

gravitropism as coefficient

of differential equations

Representative volume is

sufficiently big for continuity

in space and time. Growth

parameters represent

average root behaviour

Describing the dynamics of

exploration of the soil domain.l

Potentially to develop a better

understanding of spatial and

temporal patterns of root growth

and development. Potentially to

bridge the gap between

architectural and continuous

function models

May provide good spatial

description of root growth, with

fewer parameters than full

architectural models. Model

parameterization requires

development, and issues such as

representative volume need further

consideration

a Gerwitz and Page (1974); Mulia and Dupraz (2006).b Gerwitz and Page (1974); Jackson et al. (1996).c Greenwood et al. (1982); Hillel and Talpaz (1975).d Passioura (1991); Tardieu et al. (1992).e Jakobsen and Dexter (1987).f Diggle (1988); Lynch et al. (1997); Pages et al. (1989); Jourdan and Rey (1997); Wu et al. (2007).g Tsegaye et al. (1995).h Bengough et al. (1992); Grabarnik et al. (1998).i Lynch and Brown (2002).j Dunbabin (2007); Rubio et al. (2002).k Dunbabin et al. (2004, 2006); Fitter and Stickland (1991).l Bastian et al. (2008); Heinen et al. (2003); Dupuy et al. (2010).

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A brief history of root models

Root system models were first developed in the 1970s when

Hackett and Rose (1972) proposed, for the first time,

a mechanistic model describing the extension and branching

of a cereal root system. They showed that the increase in

root length and the number of root tips for each branching

order could be derived from simple measurements of only the

number and length of root members, together with simple

assumptions of root branching interval and elongation rates.

Further work showed that these developmental analyses

could be used to model the vertical extension of the roots

and their proliferation in a uniform soil profile (Lungley,

1973; Rose, 1983). When combined with shoot developmen-

tal data, physiologically plausible, dynamic models of plant

growth were obtained (Porter et al., 1986). Simultaneously,

Gerwitz and Page (1974) analysed 107 published and un-

published data sets of root length measurements with depth

in the soil profile for many crops and found that a simple

exponential model of the change of root length density

distribution with depth described >80% of the variance in 71

of the data sets. This model was simple to use on field data,

had good predictive power, and its utility has been exploited

widely in the crop science and ecological communities

(Greenwood et al., 1982; Gale and Grigal, 1986; Jackson

et al., 1996). Since then, root density models have improved

gradually, combining developmental information and obser-

vations of growth mechanisms, and incorporating elements

of plant physiology and interaction with the environment

(Robertson et al., 1993; King et al., 2003).

The success of Gerwitz and Page’s (1974) approach led, in

turn, to a generalized description of the growth of root

systems based on diffusive processes. Page and Gerwitz (1974)

demonstrated the utility of this approach for isolated plants

(lettuce), a row crop (onions), and a sward (ryegrass). This

theoretical framework was refined further by several groups

of researchers with, for example, soil physicists developing

spatial models using reactive diffusion systems to describe the

development of the root zone (Hillel and Talpaz, 1975;

Hayhoe, 1981). Convection terms (Acock and Pachepsky,

1996) and anisotropic coefficients (de Willigen et al., 2002)

were added to incorporate tropism into the development of

the root zone, and soil heterogeneity (Heinen et al., 2003).

A review of these approaches was undertaken by Reddy

and Pachepsky (2001). In general these diffusion-type

models have proved most popular with soil physicists

studying water distribution and uptake, and have not been

used widely by plant physiologists—possibly due to the

difficulty in attributing explicit biological meaning to the

parameters.

With the advent of modern computers, computational

approaches have come to dominate the field of root

modelling. The migration was initiated by Diggle (1988) with

the ‘ROOTMAP’ simulator, shortly followed by the Pages

model for maize (Pages et al., 1989). In such models, the

architecture of the root system was made explicit, and new

types of analyses became possible. For example, Ho et al.

(2004) have employed architectural models of the common

bean root system to explore the optimum architecture to

capture phosphate; root gravitropic traits differ substantially

between genotypes and lines. Fitter et al. (1991) introduced

graph theory concepts to analyse uptake efficiency, and

fractals have been used as a tool to extract global descriptors

of root branching structures and to simulate their growth

(van Noordwijk et al., 1994). The role of architecture in the

transport of water and nutrients to plants has become moreapparent with the inclusion of water conductance models

(Doussan et al., 1998) allowing couplings between growth

and soil hydraulic processes to be revealed (Clausnitzer and

Hopmans, 1994). Models that encode the regulatory mecha-

nisms of development are now under active development

(Chavarrıa-Krauser and Schurr, 2004 ; Lucas et al., 2008). A

more extensive review of current models of root systems can

be found in Tobin et al. (2007).Currently, architectural models are the preferred ap-

proach when the development and physiology of the plant

are the main focus of study. The models can incorporate

explicitly the nature of interactions between the plant

components and its environment, and detailed predictions

of the evolution of such systems can be made. However,

root architectural models are difficult to parameterize.

Continuous root distribution models, on the other hand,can be used when only limited information about growth

can be collected. Such models are easy to parameterize, but

they do not permit plant developmental processes to be

integrated, and this has been a major limitation in the past.

The following sections show that continuous models can be

modified to incorporate root developmental processes

explicitly. This could provide new areas of application for

models of the root system.

Root systems as density distributions

Density-based models of root growth, as in many other

fields of the physics of continua, are based on two essentialconcepts. First, root distribution at a given point in space

can be prescribed in a representative volume, the minimum

volume above which stochastic effects become negligible.

This representative volume will depend upon both crop and

soil type, and may not be trivial to define. Initially it seems

reasonable to define it as a soil volume typically used in root

system measurement—perhaps between 100 cm3 and 200 cm3.

In a representative volume, therefore, various measures suchas length density, specific area, or volume fractions are good

local approximations of the state of the system. Secondly,

variations in space and time in the representative volume are

assumed to be smooth. This additional assumption allows

the determination of analytical solutions to the governing

equations (Tinker and Nye, 2000), although numerical

approaches can now address discontinuity in numerous

ways (Leveque, 2002; Heinen et al., 2003).Continuous models deal with heterogeneity by upscaling

processes and morphological properties so that the root–soil

system can be represented as a continuum. The mathemat-

ical objects that allow this to be achieved are root densities.

Root architectures are complex structures characterized by

Continuous root growth models | 2133 at N




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both their geometrical and topological properties (Danjon

et al., 1999), and representing their morphologies accurately

requires the traditional definition of density to be extended.

First, densities must provide a suitable representation of the

geometry/topology duality (Fig. 1). This can be expressed by

the use of root length density (qn, cm�2) and root branching

density (qb, cm�3). Individual roots have cylindrical-like

geometries, and root length density provides good descrip-tions of the local geometrical properties of root systems to

model root–soil interactions (Bengough, 1997). The topology

of root architecture, the set of connections between roots,

can be equivalently described using branching density (Dupuy

et al., 2005c). Both root length density and root branching

density evolve with time as a result of the functioning of

root apical meristems so that the root meristem density

(qa, cm�3) is a third crucial object linking the dynamics of

root length and root branching density.

Roots also have different morphologies (e.g. inclination,

diameter, branching order) and this affects their functioning

and interaction with the environment (Waldron, 1977;

Porter et al., 1994). For densities to incorporate these

properties, the space in which densities are defined needs tobe generalized to spaces of larger than the three physical

dimensions (Grabarnik et al., 1998). For example, densities

may vary according to distance from the row, depth, the

angle of inclination of the roots, and time (respectively x, z,

a, and t). The root meristem density qa(x,z,a,t) is such thatR a2a1

R z2z1

R x2x1qaðx; z; a; tÞ:dx:dz:da represents the number of

root apices at time t, at a distance from the row between x1and x2, between depths z1 and z2, and whose growthdirection is contained between a1 and a2.

Theoretical basis for modelling the dynamicsof root distribution in space

Continuous models have seen little conceptual development

since diffusion models were first developed. In particular,

continuous models have failed to propose theories equiva-

lent to those underlying architectural-based models,

whereby explicit measurable and meaningful individual root

developmental processes, for example branching angle, root

elongation rate, or gravitropic rate, are encoded in themathematics of the model. This makes it more difficult to

envisage the physical basis of the model, and may deter

many biologists from using this approach. In the next

sections, modifications to continuous models that make the

description of developmental processes more explicit are

described. It may be useful conceptually to consider growth

of parts of a root system in known representative volumes

of soil. Root system growth is simply the sum of rootproduction rate, for example elongation, cessation, or

death, generated by each root meristem. The number of

meristems present increases by the birth of new lateral roots

and by new roots entering a particular soil volume, and

decreases by root death and roots leaving that soil volume.

Conservation law for meristem dynamics

Conservation laws are mathematical relationships describ-

ing the variations of properties such as meristem density,

which remains constant within the system in the absence of

external sources or sinks (e.g. branching or meristem

mortality). In the case of root densities, such equations

need to be defined in terms of both the physical coordinates,

for example x and z, and the angles defining the direction ofgrowth, for example a. In such generalized domains, the

conservation of meristems can be written as:

@qa@t

þ @qag@a

þ @qaecosa@x

þ @qaesina@z

¼ b� d ð1Þ

This equation shows that the number of meristems

defined at a given time t, whose spatial coordinates are

Fig. 1. Representing the geometry–topology duality using density

distributions. Branching density is required to encode the topology

of the root system. In a previous study (Dupuy et al., 2005c) the

root systems of maritime pine were digitized (A) and maps of the

root systems branching density were extracted (B). Assuming that

roots were growing as straight lines, it was possible to reconstruct

realistic root architecture using solely the branching densities (C).

Because roots have more complex trajectories and have no pre-

defined length, the complete root system needs both root length

and root branching density to characterize its structure fully.

Adapted, with permission, from Dupuy et al. (2005c).





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(x,z) and angle a, is changing with time (@/@t) as a result of

meristems moving in and out of the neighbourhood (@/@x,@/@y) via elongation rate e and according to the direction of

growth (cos a, sin a) but also due to reorientation (@/@a) ofmeristems as a result of gravitropism g (� d�1) and the

creation of new meristems due to branching b (cm�3 d�1)

and loss of mersitem caused by mortality d (Dupuy et al.,

2009). Generalizing the space in which root distribution isdefined allows the root developmental parameters to appear

explicitly in the same equation. e represents the elongation

rate (cm d�1) of individual root meristems, g is the re-

orientation rate (change of root angle per unit time, � d�1).

b is the number of new meristems created per unit volume

(cm�3 d�1) and can be expressed as a function of individual

root branching rate (cf. following paragraph), and d is the

number of meristems being destroyed per unit of time (cm�3

d�1).

Branching and root length densities result in turn from

the sum of the activity of meristems with time (e.g. see

Fig. 2). This can be expressed mathematically as:

@qb@t

¼ b ð2Þ

@qn@t

¼ eqa ð3Þ

Equation 2 states that the number of roots being created

per unit of time in a unit volume (volumetric branching

rate, b) equals the change with time of the branching

density. In the next section, we describe how b can be

expressed as a function of root meristem density to describe

lateral root initiation. Equation 3 states that the increase in

root length per unit of time in a unit volume equals the

meristem density multiplied by the elongation rate (e).

Equations 1–3 together describe the dynamics of the root

architecture.

An interesting property of root architectures can befound by integrating Equation 1 with time: if the gravi-

tropic term is small in comparison with other terms, then

branching density is the sum of the derivative of the root

length density in the direction u (cos a, sin a) and meristem

density.

qb � qa þ@qn@u

ð4Þ

Equation 1 is a classic conservation law, similar to those

found in many areas of physical science, and is hyperbolicin nature. The solutions form travelling waves of meristem

density. The root length density and branching densities,

determined as the integration with time of the meristematic

activity (Equations 2 and 3), result from the morphology of

this wave. This is illustrated in the following section.

Modelling root system development

The coefficients e, g, and b defined in Equation 1 are needed

to describe the complexity of root development. These

coefficients are therefore generally not constant but more

Fig. 2. Root system dynamics in a known volume. (A and B) Region of a barley root system growing in a thin transparent container,

imaged 2.3 h apart (root diameter is ;0.4 mm). (C) Change in root length between A and B. (D and E) Idealized representation of root

growth showing new root length associated with existing meristems, newly initiated meristems, and meristems that have grown into the

rooting volume from immediately above.





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sophisticated growth functions. For example, the coefficient

g describing gravitropism is needed to describe root

responses that are proportional to the angle from the

vertical. One possible function for that could be (Dupuy

et al., 2009):

g¼g0ðaþp=2Þ ð5Þ

Lateral root initiation at a branching rate b0 and at

a branching angle of p/2 can be modelled as a source term

of the form:

b¼b0½qaðx; z; aþp=2; tÞþqaðx; z; a�p=2; tÞ�=2 ð6Þ

In this equation, the creation of new roots of a given

growth direction u is due to roots having inclination angle

a+p/2 and branching on the right in addition to the roots

having inclination angle a–p/2 and branching on the left

(Fig. 3A). Following the same reasoning, dichotomous

branching would require a different form of source term:

b¼b0½qaðx; z; aþp=6; tÞþqaðx; z; a�p=6; tÞ�=2�qaðx; z; aÞð7Þ

The first part of the mathematical expression here is

similar to a herringbone type, with a branching angle of p/6,and the second part of the expression incorporates the death

of the mother root at branching (Fig. 3B). More complex

systems of equations can also be derived from Equation 1,

in order to simulate the behaviour of different types of

roots. Modelling the dynamics of roots of different branch-ing orders is a good example. It is known, for example, that

roots of higher branching orders grow more slowly and are

less gravitropic (Drew, 1975; Rose, 1983). This can be

represented using a system of equations of the type:

Fig. 3. Modelling development with continuous models. Branching processes can be modelled through the source term b in Equation 1.

In herringbone types (A), the quantity of roots produced at angle a is half the branching rate of roots with angle a+p/2 (those branching

on the right) plus half the branching rate of roots with angle a–p/2 (those branching on the left). Dichotomous types can be modelled by

incorporating the mortality of mother roots (B). Here bd represents the branching rate density with contours representing their distribution

at 3, 6, and 9d (respectively blue, green, and red). ld is the root length density and the red line schematizes the equivalent structure.

These models shows that the fundamental difference between herringbone and dichotomous root systems is the spatial pattern of

branching activity in meristems. Architectural responses to environmental factors can be represented through spatially varying

coefficients (C). The meristem density distribution (md) and root length density distribution are shown resulting from an increase in

elongation rate and branching rate in a patch of nutrient. Simulations in (A–C) were carried out using a finite volume upwind scheme. All

simulations were initiated in the top left corner of the soil volume (x,z¼0,0). The densities of root meristems in boundary elements outside

the simulated soil volume were defined as the mirror image of the boundary elements inside the soil volume to preserve a vertical plane of

symmetry. Fluxes on the remaining faces are prevented. See Dupuy et al. (2009) for more details on the numerical approach developed.





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@qia@t

þ @qiagi@a

þ @qiaeicosa@x

þ @qiaeisina@z

¼ bi; i 2 f1; ng ð8Þ

Here, i represents the branching order, and the branchingterm bi is a function of the root meristem density at

branching order i–1, for example Equations 6 and 7. The

branching term bi therefore couples the different equations

together. The system allows roots of different types to have

a distinct elongation rate (ei) and gravitropic rate (gi).

Root mortality is also an important process in the

development and growth of root systems, with total root

production exceeding the final root system produced by upto 80% (Steingrobe et al., 2001). Current methods for

estimating root mortality using rhizotron or ingrowth core

techniques have not allowed the mechanisms of mortality to

be identified (Smit and Zuin, 1996; Steingrobe et al., 2001).

The simplest way to integrate mortality is to assume that

the mortality of root tips is uniformly spread throughout

the root system (Robertson et al., 1993), and to decrease

root length density accordingly:

@qn@t

¼ eqa � d�qn ð9Þ

Mortality can also vary significantly between roots of

different classes or according to soil conditions (Wells and

Eissenstat, 2001). More sophisticated models have yet to be

developed to account for the complexity of the processes of

root mortality where, for instance, a whole root axis dies

together with its associated lateral branches.

Feedbacks with the environment

The models presented so far assume uniform and steady

growth conditions. To incorporate the effects of 3-D soilheterogeneity into such a model, the coefficients e, g, and

b must be considered as a function of local environmental

variables h (Fig. 3C), such as volumetric water content, soil

strength, or solute concentration:

@qa@t

þ @qagðhÞ@a

þ @qaeðhÞcosa@x

þ @qaeðhÞsina@z

¼ bðhÞ � dðhÞ

ð10Þ

Environmental variables are also modified as a result of

growth processes and this has been the subject of numerousstudies, for example radial uptake of water by roots

(Gardner, 1960). Upscaling mechanisms from root to root

systems, however, can be considerably more difficult than

simply summing up individual effects (Baldwin et al., 1972).

In the past, this has been approached through generic

functions incorporating group effects, for example the

concept of volume fraction available for uptake by roots

(King et al., 2003).When root environmental processes occur at larger scales,

for example water depletion in root systems (Garrigues

et al., 2006), growth must be coupled to additional equa-

tions such as transport of water and nutrients (Mmolawa

and Dani, 2000). Such a system of equations needs more

sophisticated numerical methods to be developed (Hayhoe,

1981). Alternatively, it is simpler to assume that growth is

slow in comparison with the transport of mobile nutrients

and therefore to implement density-dependent growth of

root meristems (Brugge, 1985; Chen and Lieth, 1993).

Bastian et al. (2008), for example, proposed a simplified

function of the form:

e¼kðqmaxn �qnÞ ð11Þ

In these cases the velocity and branching are linked

directly to root length density through an arbitrary

parameter (k) that requires experimental characterization.

Calibration of density-based models

The choice of a model is often motivated by the nature of

the data that can be collected from experimental systems.

For that reason, density-based models have been particu-

larly popular for root studies. This is due to the fact that

density can be estimated from excavated cores of soil, for

which numerous experimental techniques are available (doRosario et al., 2000; van Noordwijk et al., 2000). For

example, exponential models can be fitted to vertical cores

of soil obtained by auger sampling (Barraclough and Leigh,

1984) and 2-D diffusion models have used pinboard

techniques to obtain mappings of root length distribution

in space (Heinen et al., 2003). Considerable progress has

been made in the measurements of individual root growth

parameters (Lecompte and Pages, 2007; Danjon andReubens, 2008; Pages et al., 2009). However, parameterizing

architectural models is still difficult and often requires

partial excavation of root systems.

In the case of the models presented herein, spatial

distributions of both root length density and meristem

density are required. These data may be estimated from

rhizotron or minirhizotron window observations (e.g. Fig.

2), or from washed root samples augered from the soil asmentioned above. Image analysis programs such as Winrhizo

(Regent Instruments Inc.) can provide the number of root

ends in a washed root sample, although in samples where

there are many broken roots this would substantially

overestimate meristem density. One possible approach

would be to develop an image analysis procedure that

combined root end estimation with the use of stains for

identifying dividing cells in the meristem. To verify anycalibration or parameterization procedure it would be

possible to simulate the procedure using explict 3-D

architecture models, for example by simulating the mea-

surement of root length density and meristem density in

particular regions of the root system, and determining

whether this gave appropriate parameters for the continu-

ous model. For example, if the gravitropism term was

relatively small, this may facilitate parameterization of themodel (e.g. Equation 4).

Partial differential equation (PDE) models (Equation 1)

usually need to be solved using numerical approaches and

the estimation of model parameters requires specific proce-

dures to be developed. This has been solved in the past by





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using non-linear optimization algorithms (Heinen et al.,

2003). With this approach, the model is run using different

sets of parameter values incorporating small variations in

each parameter. A minimization criterion is used to measure

the difference between the observed root length density and

the predicted root length density. The model parameters are

then updated in the direction of the greatest rate of decrease

of the criterion. These steps are repeated until the criterionno longer decreases. This method has been further refined to

allow derivation of the standard error of the model param-

eters and to test their significance (Acock and Pachepsky,

1996). This allows model complexity to be explored.

Models that are constructed on mechanistic principles,

such as in Equation 1, define root developmental processes

explicitly. Therefore, the optimization of their parameters,

e.g. g0 (Equation 5), b0 (Equations 6 and 7), or ei (Equation8), could prove particularly useful in the future to ‘invert’

root developmental programs and to reveal root growth

parameters from the measurement of root length density

patterns. Individual root properties are not required as

inputs for density-based models. More parameters are

required for more complex models, and to account for

environmental influences on root growth—this will also

affect the uncertainties associated with estimates of theseparameters. The complexity of any given model must be in

balance with the amount and quality of data on root

distribution in relation to soil properties.

Root system development as waves ofmeristems

In previous work (Dupuy et al., 2009), it was shown using

simple growth functions that a one-dimensional analytical

approximation of Equation 1 can be obtained, using the

method of characteristics (Leveque, 2002):

qa¼ð1þbtÞN0

�z�et�

ea2�e�2gt � 1

�4g

; a

�ð12Þ

The function N0(z,a) is the initial root meristem distribu-

tion which is assumed to be the normal distribution. Theexpression z–et in this model therefore implies that root

systems tend to propagate and amplify the initial peak of

meristem density, and that such spatial peaks travel

approximately at the speed of the average root elongation

rate e.

Developmental processes, through the coefficients e, g,

and b (Equations 1 and 8), have a strong influence on the

propagation of the waves of meristems and the resultingshapes of root systems. The branching patterns (e.g. angle,

frequency and mortality) determine the topology, for

example herringbone and dichotomous architectures in Fig.

3A, B. Systems possessing roots of several branching orders

growing at different rates produce the superposition of

waves of meristems of different branching order (Fig. 4).

This superposition of waves is largely the reason for the

exponential profiles of root length density distribution withdepth (Gerwitz and Page, 1974). Root–soil interactions can

also influence root developmental properties locally and can

generate irregular patterns of root length distribution in

space (Fig. 3C).

From a biological perspective, the waves of meristems

are due to the polarity of the development of root

apical meristems. Newly created tissues from a single meri-

stem expand and become increasingly rigid during

Fig. 4. The development of root systems represented as waves of meristems. Preliminary experimental results indicate that meristem

density may form travelling peaks of activity. A time lapse minirhizotron experiment (Dupuy et al., 2009) carried out on root systems of

barley seedlings was used to build spatial maps of root length density (A) and the equivalent meristem activity in (B) (volumetric root

length production rate, cm cm�3 d�1) obtained from Equation 3. More complex root system morphologies can be modelled using the

superposition of meristematic waves of different types. For example, roots of a certain branching order have an average elongation rate

and form travelling peaks of meristem density during the growth of the root system. Because root systems have roots of different

branching order (here first order in blue long dashed–dotted lines and second order in green dashed lines), the system can be modelled

as the superposition of travelling waves of different velocities (C).





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differentiation. During this process, root tips are pushed by

expanding cells and tend to travel away from their branching

point. At the scale of the root system, this mechanism induces

travelling spatial peaks of meristem density, and the mathe-

matical evidence for this lies in the nature of Equation 1.

The concept of a wave of meristems still needs to be

tested experimentally. Similar behaviour has been observed

in the propagation of the soil drying front (Davies et al.,

1990). In particular, it was found that spatial maxima in the

water extraction rate were formed and propagated through

the soil. More recently, Garrigues et al. (2006) have used X-

ray imaging and computer simulation to characterize in

great detail the formation and propagation of these spatial

maxima when following water uptake by narrow leaf lupin

(Doussan et al., 2006; Garrigues et al., 2006). Theoretical

studies also suggest that resource-dependent growth of

branching systems, such as blood vessels, fungal mycelia, or

hairy roots, induces the formation of waves of tip activity in

response to chemical signals (Anderson and Chaplain, 1998;

Boswell et al., 2003; Bastian et al., 2008).

Regions of high density of root apical meristems are of

particular importance for the uptake of water and

nutrients, and raise many interesting questions. Depletion

around the root increases as a function of the distance

from the tip (Claassen and Barber, 1974), while living

root hairs are distributed in the proximity of root tips

(Gilroy and Jones, 2000) and sensing for environmental

signals has been shown to be located in the root tips

(Zhang and Forde, 1998; Wilkinson and Davies, 2002;

White et al., 2005; Hodge, 2009). Are water extraction

fronts the result of travelling maxima of meristematic

activity? Have plants evolved to proliferate their root

meristems in regions less likely to be depleted? These

questions remain unresolved at this stage. Further studies

are required to characterize the nature of the dynamics of

meristems in soil.

Deformable domains, the next generation ofdensity-based models?

It is not always possible to obtain analytical solutions from

the equations developed in previous sections. In such cases,

numerical approaches must be developed to generate

approximate solutions. Numerical solutions are obtained

by computing the values of the root densities on a mesh,

using finite element, or finite volume methods (Zienkiewicz

and Taylor, 1998; Leveque, 2002).

Traditionally, fixed meshes were used to model soil and

to assign root length density values to elements of the mesh

(referred to as control volumes or elements). The simulation

uses the system variables, defined at control volumes, to

compute fluxes of roots entering and leaving the control

volume. The dynamics of root distribution are then built

incrementally from the initial conditions of the system

(Heinen et al., 2003). These ‘soil-centred’ approaches,

referred to as Eulerian, are appropriate when tracking of

the motion of matter is not required or achievable, for

example fluid dynamics (Ashgriz and Mostaghimi, 2002).

A promising new type of ‘plant-centred’ model could be

constructed using deformable meshes that assign control

volumes to domains of moving meristems and growing root

systems (Fig. 5A). Such models are referred to in physics as

Lagrangian and are used commonly in the field of solid

mechanics (Zienkiewicz and Taylor, 1998). Deformable

plant models could have a range of new applications for

models of root systems. Deformable models represent

explicitly the domain occupied by the root system of

a plant and are naturally adapted to model interactions

with other plants. Algorithms inspired from the field of

contact modelling (Dupuy et al., 2005b) can be developed

to identify neighbouring plants and implement density-

dependent growth (Fig. 5B). Such models could have

application in ecology to improve individual-based,

Fig. 5. Continuous deformable domains to represent the development of the root system. (A) Schematic representation of the principle

of continuous deformable domains: the distribution of meristems in the root system is encoded in a discretized domain (blue). The

growth equations (Equations 1–3) are then used to deform the domain and update the meristem distribution in the domain and the root

length distribution (red). (B) Population-based simulation of the model of barley seedlings developed by Dupuy et al. (2009) and

implementing a density-dependent growth of meristems. Initial conditions consisted of 81 seeds placed randomly on the plane and

initiation of root–root interaction. The mesh shows the deformed domains resulting from the growth of interacting root systems.





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population models (Begg et al., 2006). Also, deformable

meshes use less control volumes: 16 subdivisions have been

used to decompose the deformable domains as opposed to

40 in Dupuy et al. (2009). Deformable models would have

numerous applications where ensembles of root systems

need to be simulated efficiently.

Concluding remarks

Models have become important tools for characterizing the

growth and functioning of plants. Pioneering work in the

1970s used continuous approaches to model root systemgrowth. However, models of this type are based on spatial

mathematics that often requires significant expertise in

modelling. Perhaps for this reason, root architectural

models, where the shape and structure of roots are

represented explicitly within simulation programs, have

now come to dominate the field (Godin and Sinoquet,

2005). Based on the simulation of an ensemble of identical

modules, i.e. virtual meristems, complex mechanisms caneasily be encoded as morphogenetic rules and emergent

properties can be simulated using existing platforms such as

L-studio, AMAP, or Lignum (Pertunen et al., 1996; Allen

et al., 2005; Barczi et al., 2008). New algorithms are being

developed to improve the efficiency of computation (Renton

et al., 2005; Cournede et al., 2006) and facilitate the

estimation of model parameters (Letort et al., 2008).

Existing models are so sophisticated that one may evenquestion the need for alternative forms of models.

Mathematical approaches, however, offer compelling

alternatives to architectural models. Analytical models,

exact or approximate solutions to growth equations, can be

obtained in the form of mathematical functions. These

functions provide insight into the development of the root

system as a whole, and can be parameterized relatively

easily from field data. The use of continuous variables toaggregate root morphological properties also facilitates the

coupling of growth with environmental and physical pro-

cesses, which is useful for a wide range of plant environ-

mental studies (Watanabe et al., 2004). For example, soil

mechanics and transport models use partial differential

equations to describe soil mechanics, and diffusion and

mass flow in soil (Bengough, 1997; van Beek et al., 2005;

Doussan et al., 2006), and such models would communicatemore naturally with continuous descriptions of plant root

structure. Finally, representing root systems as continua

allows more efficient computational models to be devel-

oped. The continuous deformable domain approach pre-

sented herein is a first step toward a next generation of

continuous root system models. The principle is likely to

have applications when the growth of numerous plants

needs to be simulated. For example, most individual-basedpopulation models involve the definition of the plant’s

zone of influence, where acquisition or competition for

resources occurs (Begg et al., 2006). These zones have been

represented classically as domains contained in envelopes

(Cescatti, 1997). The growth of these regions and the

evolution of their structure have generally been imple-

mented through empirical, arbitrary functions (Weiner and

Damgaard, 2006) and could be replaced by more accurate

mechanistic principles.

Continuous-based approaches for modelling root system

growth are still at an early stage of development. Such

models could have a wide range of applications, particularly

where classic architectural models cannot provide predic-tions due to computational limitations, for example pop-

ulation models or breeding scenarios, or due to the

difficulty in calibrating the models efficiently. There are

still, however, considerable challenges to develop suitable

analytical methods and to optimize the numerical analysis

of such systems, notably for the meshing and approxima-

tion of solutions on deformable domains. Fortunately, there

is abundant literature on the subject that indicates thepotential for improvements.

Acknowledgements

The Scottish Crop Research Institute receives financial

support from the Scottish Government Rural and Environ-

ment Research and Analysis Directorate (RERAD, Work-

package 1.7).

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