Plant Soil (2010) 332:177–192DOI 10.1007/s11104-010-0284-7

REGULAR ARTICLE

A dynamic root system growth model based on L-SystemsTropisms and coupling to nutrient uptake from soil

Daniel Leitner · Sabine Klepsch ·Gernot Bodner · Andrea Schnepf

Received: 30 May 2009 / Accepted: 7 January 2010 / Published online: 23 January 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract Understanding the impact of roots andrhizosphere traits on plant resource efficiency isimportant, in particular in the light of upcomingshortages of mineral fertilizers and climate changewith increasing frequency of droughts. We de-veloped a modular approach to root growth andarchitecture modelling with a special focus on soilroot interactions. The dynamic three-dimensionalmodel is based on L-Systems, rewriting systemswell-known in plant architecture modelling. Weimplemented the model in Matlab in a way thatsimplifies introducing new features as required.Different kinds of tropisms were implemented asstochastic processes that determine the position ofthe different roots in space. A simulation study

Responsible Editor: Philippe Hinsinger.

D. Leitner (B) · S. Klepsch · A. SchnepfDepartment of Forest- and Soil Sciences,University of Natural Resources and AppliedLife Sciences, Vienna, Austriae-mail: daniel.leitner@boku.ac.at, d_leitner@gmx.at

S. KlepschHealth & Environment Department, AIT AustrianInstitute of Technology, 2444 Seibersdorf, Austria

G. BodnerDepartment of Applied Plant Sciences and PlantBiotechnology, University of Natural Resourcesand Applied Life Sciences, Vienna, Austria

was presented for phosphate uptake by a maizeroot system in a pot experiment. Different sinkterms were derived from the root architecture,and the effects of gravitropism and chemotropismwere demonstrated. This root system model is anopen and flexible tool which can easily be coupledto different kinds of soil models.

Keywords Root system development ·Phosphate uptake · Tropism · L-System ·Mathematical modelling · Sink term

Introduction

Increasing plant water and nutrient useefficiencies is a major challenge that must bemet in order to respond to the rising demand forsufficient food supply of an ever growing humanpopulation. Plant productivity is governed bymany environmental factors such as radiationinterception and water and nutrient availabilityin soils. Thus, a fundamental understanding ofthe key processes determining plant growth isnecessary in order to improve cropping systemsand cultivars for resource limited environments(de Dorlodot et al. 2007). Understanding theimpact of roots and rhizosphere traits on plantresource efficiency is important, in particular inthe light of sustainable production with reducedfertilizer input, potential shortages and increasing

178 Plant Soil (2010) 332:177–192

costs of fertilizers and climate change withincreasing frequency of droughts. In this context,root architecture is a fundamental aspect of plantproductivity (Lynch 1995) and thus needs tobe accurately considered when describing rootprocesses.

Existing root system models can be dividedinto pure root growth models, which focus ondescribing the root system’s morphology, andmore holistic models, which include several root–environment interaction processes, e.g. water andnutrient uptake.

The first descriptive three-dimensional rootsystem models were presented by Diggle (1988)(RootMap) and Pagès et al. (1989). These mod-els described a herringbone topology and definedroot properties for every topological order inthe root system. Based on these ideas, Lynchet al. (1997) and Spek (1997) developed new rootgrowth models (SimRoot and ArtRoot) with astrong focus on visualisation. Pagès et al. (2004)presented a root system model in which thedifferent types of roots composing the root systemtopology are not strictly related to the topologicalorder (Root Typ).

These dynamic three-dimensional root growthmodels have provided a basis to couple rootgrowth to plant and soil interaction models inorder to simulate environmental processes such aswater and nutrient uptake. A root system modeldeveloped for such a process oriented analysis waspresented by Fitter et al. (1991) to study the ex-ploration efficiency of root systems in dependenceof root architecture. It could be demonstratedthat both root topology and link lengths stronglyinfluence nutrient uptake efficiency (Fitter andStrickland 1991). Dunbabin et al. (2006, 2002)predicted water and nutrient uptake, in the pres-ence of exudates (Dunbabin et al. 2006), by inte-grating RootMap into a simulation environment.Dunbabin et al. (2004) analysed the effect ofdifferent root system architectures on nitrate up-take efficiency. Walk et al. (2006) used SimRootto assess the trade off effects of different rootsystem morphologies on phosphorus acquisition.Based on the model of Pagès et al. (1989), amodel of water conduction within roots has beenproposed by Doussan et al. (1998) and combinedwith a model for water transport in soil (Doussan

et al. 2006). Clausnitzer and Hopmans (1994) de-veloped a new root growth model and analysedwater flow in the soil root zone which was fur-ther extended by Somma et al. (1998) for nu-trient uptake. Javaux et al. (2008) combined themodel of Somma et al. (1998) and the model ofDoussan et al. (1998) to comprehensively describewater flow between the soil and plant root do-main. Roose and Fowler (2004) derived sink termsfor nutrient and water uptake from a continuousroot system growth model which can be solvedanalytically.

Beside the effects of root growth dynamics andarchitectural traits on the soil (water and nutri-ent uptake), a major challenge of root modellingis the dynamic interactions of soil properties onthe root growth and architecture itself. Wateravailability, nutrient concentration as well as me-chanical impedance have been shown to stronglyinfluence plant root traits (e.g. Hodge 2004; Eapenet al. 2005; Bengough et al. 2006). Integrating rootplasticity in response to the growth environmentinto root architecture models remains a majorchallenge for an accurate model based analysisof plant growth strategies in different environ-ments (Tsutsumi et al. 2003). Root responses toresource heterogeneity in the soil are subject toongoing empirical research concerning phytohor-mone mediated signalling pathways and speciesspecific morphological and physiological reac-tions (e.g. Forde and Lorenzo 2001; Callawayet al. 2003; Hodge 2004; Zhang et al. 2007; Peretet al. 2009). This requires a flexible modellingapproach for continuous incorporation of newknowledge.

Processes and interactions in the soil–plant–atmosphere continuum are often described bypartial differential equations. The most importantequations are the Richards equation for waterflow (Jury and Horton 2004) and the convec-tion diffusion equation for solute transport insoil (Barber 1995). With increasing computationalpower, it is now possible to solve these equa-tions numerically in three dimensions using finitedifference, finite volume or finite element meth-ods. The complexity of the root–soil system, how-ever, requires an accurate and detailed descriptionnot only of each subsystem (e.g. root growth,root architecture, water and solute transport), but

Plant Soil (2010) 332:177–192 179

also of their mutual linkage and influence. Thus,a modelling approach with a modular structurewould be most appropriate to join developmentsin modelling the individual components of the rhi-zosphere. The powerful numerical solvers avail-able today (e.g.: Comsol Multiphysics, FlexPDE),combined with techniques such as operator split-ting, yield a detailed description of the singlesubsystems and subsequently enable these to bejoined into a comprehensive system analysis.

This study presents such a modular approachfor root growth, root architecture and root–soil interactions based on L-Systems. L-Systemsare often used in plant architecture modelling(Prusinkiewicz 2004) and have also been appliedto root systems (Prusinkiewicz 1998). We providean implementation of our L-system based rootgrowth and architecture model in Matlab and theintegration of the root system description into aplant–soil interaction model. This approach con-siders the plasticity of root growth and branchingstrategies as influenced by local soil properties.From the root system architecture model, parame-ters such as density distributions can be derivedand used in soil models. Matlab makes it is easy tolink the model to existing simulation codes (e.g.:Java, C, Fortran) or to apply external numericalsolvers. The new approach is exemplified in asimulation study of maize root growth and phos-phate uptake in a pot experiment. The effects ofdifferent sink terms derived from the modelledroot architecture and the role of gravitropism andchemotropism is demonstrated. Our main objec-tive is to provide a mathematically sound and pub-licly accessible dynamic root growth model with afocus on a modular approach to integrate varioustypes of interactions between root architectureand the soil environment.

Model description

Introduction to L-Systems

L-Systems are rewriting systems based on stringswhere each character stands for a specific en-tity (Prusinkiewicz and Lindenmayer 1990). Everycharacter of an initial string ω is replaced accord-ing to its corresponding production rule. This is

performed iteratively n times to achieve an L-System of the nth generation which can be inter-preted graphically. For example the initiator ωand the production rule for the character X,

ω = X,X → F [ + X ] [ −X ] F X, (1)

produce Fig. 1. In this example X denotes thetips and F represents the segment that has beenproduced. The characters +, −, [ and ] are used forthe graphical interpretation using turtle graphics.In Appendix we briefly describe turtle graphicsand provide a list of turtle commands.

L-Systems provide a compact description forbranched geometries. However, in root growthsimulations, we need to describe continuousgrowth in time. For this reason, we use parametricL-Systems, which include a special parameter tdenoting the local age of an L-System charac-ter. Additionally, every production rule is appliedfor a certain time step �t. We indicate this bywriting the time step as a superscript of the L-System character. For example, a character thatis replaced after a predetermined time tend by asuccessive L-System string Ns (delay rule D�t) isgiven by

D�t(t) →{

t + �t ≥ tend : Nt+�t−tendsotherwise : D(t + �t), (2)

where the parameter t is the local age and tendis the time at which the character is replaced,�t is the time which passes during the applica-tion of the rule. If the local time is longer thantend the character is replaced by its successor Ns.The production rule of Ns is applied with thetime t + �t − tend, which is the corresponding timeoverlap. Otherwise, the local time is increased tot + �t, which is the new local time of the char-acter D.

The production rules have to be designed suchthat the result does not depend on the discreti-sation of the overall simulation time. The delayrule utilises two different kinds of parameters.The local age t changes when the production ruleis applied, while tend and Ns are predetermined

180 Plant Soil (2010) 332:177–192

Fig. 1 The 2nd–5th

generation of anL-System (reproducedfrom Prusinkiewicz andLindenmayer 1990)

initial values. We use a syntax in which localparameters are given in front of the semicolon andinitial parameters are given after the semicolon,i.e., D(t) := D(t; tend, N).

The following sections describe the basic pro-duction rules of the L-System model for rootgrowth. A major concern is that all productionrules are based on biological mechanisms. Ourproduction rule for elongation of individual rootsis based on the continuous root growth model ofRoose et al. (2001). If required, it can be replacedby other models.

Axial growth

We first present a production rule for the growthof a single root without branching according toa continuous growth function λ(t). In our appli-cation, λ(t) is chosen for every root type i. Rootelongation follows a negative exponential func-tion (following Pagès et al. 2004).

λi(t) = ki(1 − e− riki t), (3)

where ki is the maximal length of the root and ri isinitial growth speed. In the simplest case, the roottype corresponds to the topological order of theroot within the root system.

The production rule G�t for the growth of asingle root is given by

G�t(t, l)

→{

l+�x < λ(t+�t) : R F�x G�t(t, l+�x)otherwise : G(t+�t, l).

(4)

The parameter t represents the local age of theroot tip, l is an approximation of the actual rootlength λ(t), �t is the time step and �x is thespatial discretisation. The character R denotesa rotation that describes root deflection and thecharacter F�x denotes a segment of the root withlength �x. The choice of the rotation R describingthe root tip deflection will be addressed in Section“Tropisms and root tip deflection”. The first ex-pression of Eq. 4 produces segments R F�x until thepredetermined length λ(t) is approximated. Thisis done recursively, thus, in one step numerous

Plant Soil (2010) 332:177–192 181

segments can be produced. Otherwise the localtime is increased to t + �t. In this way the rootaxis is described by segments with length �x.The growth function λ(t) and the discretisationalong the root �x are initial values, thus G(t, l) :=G(t, l; λ, �x).

In this section, we have described a produc-tion rule for the growth of an individual root.In the next section, we present an L-System rulefor lateral branching and summarise the root pa-rameters that are needed for a single root withbranches.

Lateral branching

In a root system, every root of a certain orderproduces lateral branches. A root is thereforedivided into three zones: the basal- and apicalzones near the base and the tip of the root, re-spectively, where no branches are produced, andthe branching zone where new roots of successiveorder are created. The self similar structure of theroot system is used in the model description.

Growth of basal and apical zones as well as thegrowth between the branches, are described withthe section growth rule S�t given by Eq. 5, whichis a generalisation of the axial growth rule givenby Eq. 4. In our implementation, new branchescan lie at an arbitrary position that is independentof the discretisation �x. Thus, Eq. 4 needs to bemodified such that it can produce segments oflength �x or less:

S�t(t, l)

→

⎧⎪⎨⎪⎩

A & (l+�x < ls) : R F�x S�t(t, l+�x)A & (l+�x ≥ ls) : R Fls−l Nt+�t−(tend−t0)sotherwise : S(t+�t, l),

(5)

A := λ(t0) + l + �x ≤ λ(t0 + t + �t),

where t0 and tend are the times at which the growthstarts and ends; λ(t0) and λ(tend) are the respectivepositions on the root axis and ls := λ(tend) − λ(t0) isthe length of the section. The successive L-Systemstring Ns represents the next section.

The first expression of the rule corresponds tothe first expression of Eq. 4 with the additionalconstraint that growth does not exceed length ls.The second expression describes the case if thelength exceeds ls. In that case, a segment with theremaining length Fls−l is produced and the succes-sor, Ns is applied with the correct time overlap t +�t − (tend − t0). Otherwise, local time is increased.The following parameters and initial values areneeded: S(t, l) := S�t(t, l; t0, tend, Ns, λ, �x). Inthis way, a section is described by a number ofsegments with lengths equal to or less than �x.Using the section growth rule, the basal zone issucceeded by the branching zone which is thensucceeded by the apical zone.

Within the branching zone, a predeterminednumber of branches are created. The spacing be-tween the branches is determined by the sectiongrowth rule. The rule allows that branches canoccur at any point along the root axis and not onlyat segment edges with fixed segment length. Whennew branches are created, they begin to grow assoon as the apical zone has reached its requiredlength. The branching zone produces L-Systemstrings Nb , which represent branches of the nexttopological order; these are then followed by asuccessive L-System string Ns, which representsthe apical zone. The production rule for branchingB�t is given by

B�t(c) →

⎧⎪⎪⎪⎨⎪⎪⎪⎩

c

182 Plant Soil (2010) 332:177–192

Eq. 5, and is followed by the branching rule withincreased counter c. The second expression statesthat after the last branch is created, the finalsection is followed by the successive string Ns,which produces the apical zone. All parame-ters and initial values of the branching rule aregiven by B�t(c; Nb ,Ns, n, {t0,i}, {tend,i}, {di}, λ,�x),where i = 1 . . . c.

When a new branch is created, first the tur-tle state, position, heading, width and colour isstored. The local root tip axis performs a radialrotation by a uniform random angle β between−π and π . Then the tip axially rotates by a pre-determined angle α (following Pagès et al. 1989),see Fig. 2. The new branch is created with a newlocal axis and, finally, the original turtle state isretrieved. Thus, the string Nb has the followingstructure:

Nb = [ Rβ Rα b ], (7)where b describes the new branch. The produc-tion rule of the character b sets up the basal,apical and branching zone and the delays d. Thelocal times t0 and tend, which are needed for rule(6), are calculated from the growth function λ(t)i,which is given by Eq. 3 and is determined by theparameters ri and ki for every root type i.

We summarise the required model parametersneeded for a root type in Table 1. All parametersare given by their mean value with a standard

.3.3

Fig. 2 The change of root tip heading due to the rotationsRα and Rβ

Table 1 Parameters for a root with lateral branches

Mean SD∗ Description Unitr rs Initial growth speed [cm/day]lb lbs Length of basal zone [cm]la las Length of apical zone [cm]ln lns Spacing between branches [cm]n ns Number of branches −δ δs Angle between root [degree]

and predecessora as Radius of the root [cm]C Colour [RGB]Nb Successive branch∗ Standard deviation

deviation. For roots with no lateral branches theparameters lb , ln, and n are not needed.

Root systems

In order to describe different kinds of root systemslike tap root systems or fibrous root systems, anadequate initial L-System string must be created.In the case of a tap root system, the initial stringconsists of a single root tip of a zero order root. Inthe case of fibrous root systems, the initial stringis created as follows: We start with a root collarusing a predefined number of root tips n0. Weassume that the roots initially grow away fromthe stem in a conical way. The initial angle isdetermined by assuming an even distribution ofroots on a cone’s base with radius r0 and height1 cm. The cone’s tip is positioned at the centre ofthe root collar. Thus, the initial parameters of theL-System model are the number of initial rootsn0, and the cone’s radius r0. To achieve uniformdistribution of the roots on the cone’s base, theinitial axial insertion angles α and initial radialinsertion angle β must be chosen as follows,

α = arctan(√X r0), (8)

with random X uniformly distributed between 0and 1. Random β is uniformly distributed between−π and π .

To include effects that arise from gravitationand soil heterogeneities, more mechanisms haveto be included, which are discussed in the follow-ing section.

Plant Soil (2010) 332:177–192 183

Tropisms and root tip deflection

The previous explanation shows that the root sys-tem is described by a large number of segmentsof length less than or equal to �x. In front ofevery segment F�x is a rotation R which is createdwhen the root axis grows, see Eqs. 4 and 5. In themodel, we specify R by two rotations (followingDiggle 1988). The root tip axis rotates radially byan angle β and then axially by an angle α, seeFig. 2. In reality, the direction of root growth isinfluenced by mechanical as well as plant physi-ological properties. We include this in the modelby specifying different ways to choose α and β.In this way, we can simulate root tip response tomechanical soil heterogeneities as well as varioustypes of tropisms like gravitropism, hydrotropismor chemotropism. The specific growth behaviourcan be chosen for every root type.

To choose the angles α and β in relation toglobal parameters, the root tip position xp andlocal axis given by hx, hy, and hz are needed.Therefore, an additional set of parameters is in-troduced for L-System characters describing roottips. If a root tip is replaced by any production rule

tip → . . . RF�x︸ ︷︷ ︸transformation T

tip, (9)

the set of parameters is updated corresponding tothe transformation T.

A simple approach to model the effect of soilparticles on the direction of root growth is to mod-ify the growth direction of the root tip randomly.In this case we do not consider the positions of thesoil particles explicitly but merely describe theirinfluence on growth by random variation of thegrowth direction. Therefore, we choose a randomradial angle β uniformly distributed from −π toπ and a random axial angle α that is normallydistributed with mean 0 and standard deviationσdx. The standard deviation σdx is dependent onthe length of the next segment dx. It can be cal-culated from a global parameter σ that controlshow strong the root’s direction changes per 1 cmgrowth. The standard deviation σdx is calculatedfrom σ using Gaussian error propagation where

the number of trials is 1/dx (segments per cm).The standard deviation σdx is given by

σdx =√

dx σ. (10)

In this way, the expected change of the root axisdoes not depend on the spatial resolution. Theroot axis is described by segments. The segmentslengths dx are equal to or less than the spatial dis-cretisation �xi, whereby the spatial discretisationcan be chosen for every root type i.

Tropisms

An easy way to implement gravitropism is to ran-domly calculate several rotations and pick out therotation that leads to a downward movement ofthe root tip. Therefore, we introduce the para-meter N denoting the number of trials to findthe optimal angles α and β for the rotation Rin order to achieve a downward movement. Theangle α is a normally distributed random numberwith mean 0 and standard deviation σdx, the angleβ is uniformly distributed from −π to π . Figure 3demonstrates how the choice of parameters N andσ influences a fibrous root system, whereby σdescribes the expected change of root tip headingper 1 cm root growth and N controls the trend togrow downwards. The parameter N can be a realnumber. N=1.5 means that the number of trialscan be either one or two.

This approach has two main advantages:Firstly, the effect of gravitropism does not dependon the spatial resolution along the root axis. Sec-ondly, the objective function can be freely chosen;thus, different tropisms can be realised using thisconcept. In the following we give some examplesof objective functions to be minimised:

Gravitropism f = (hx)3Plagiotropism f = |(hx)3|Chemotropism, Hydrotropism,Thermotropism f =−s(xp+dx hx)

(11)

The point xp denotes the position of the roottip, see Eq. 9. The root tip heading hx is thedirection of growth of the local axis after apply-ing the rotation R(α, β), see Fig. 2. The scalarfield s(x) contains nutrient concentration, water

184 Plant Soil (2010) 332:177–192

Fig. 3 The influence ofthe parameters N and σon the extent ofgravitropism in a fibrousroot system (n0 = 30,r0 = 5)

content, pressure head or temperature. The value(hx)3 denotes the z-coordinate of the vector hx.Minimising the z-component (hx)3 yields a vec-tor hx pointing preferably downwards (with anegative z-component) and therefore describinggravitropism. Plagiotropism is obtained by min-imising the absolute value |(hx)3|, which leads tolarger values of |(hx)1| and |(hx)2| and therefore,horizontal growth. Chemotropism is described bymaximising s(xp + dx hx); thus, we multiply by(−1) to obtain a minimisation problem. Objec-tive functions can be freely combined (e.g.: bylinear combination). In this way, different kindsof tropisms can be realised for each root type.

Root tip def lection

In many experiments like pot or rhizobox experi-ments, root growth is spatially bounded (Doussanet al. 2006). We can bound our root growth simu-lations by an arbitrary geometry which is given im-plicitly by a signed distance function. The signeddistance function determines how close a givenpoint is to a boundary and returns a negative valueif the point is outside the boundary. Additionally,

this provides a way to include obstacles in ourmodel.

The following algorithm takes the spatialboundaries into account. In a first step, the rota-tion angles α and β are derived as described inSection “Tropisms”. If the new root tip positiondoes not lie within the geometric boundaries, thena new pair (α, β) is chosen as follows: First, onlyβ is chosen uniformly random between −π andπ while α is left unchanged. If, after a maximalnumber of trials nβ , no new valid pair α and βhas been found, α is increased for a small fixedangle dα and the procedure for finding an angleβ is started again. This simple approach leads to arealistic root behaviour at the boundaries, wherethigmotropism can be observed.

Coupling to a soil model

In many applications, it is important to couple aroot growth model to soil models because theymutually affect each other (Pierret et al. 2007).For example, root growth influences the nutrientdistribution and water content of soil, while rootgrowth itself is dependent on these parameters.

Plant Soil (2010) 332:177–192 185

Fig. 4 Plant and soilinteraction by couplingthe root growth modelwith a soil model

Coupling of the models is achieved by operatorsplitting, i.e. by calculating both models alter-nately with a sufficiently small time step �t. Theinteraction between root growth and soil proper-ties is illustrated in Fig. 4.

In our model, the dynamic root growth can beinfluenced by soil properties in different ways:firstly, by specific tropisms (e.g.: chemotropism,hydrotropism), secondly, by the influence of soilproperties on the root elongation rate and thirdly,by changing branching strategies. These threemechanisms describe the key root responses tosoil properties (Hodge 2004; Fitter and Stickland1991). The soil properties are the dynamic outputof a soil model. The interaction is implemented inthe model as follows.

– A specific tropism can be chosen which de-pends on soil properties (e.g. water content,nutrient concentration or temperature). De-pending on which plant and soil interactionmodel is described, combinations of differenttropisms such as hydrotropism, chemotropismor thermotropism can be considered, seeEq. 11.

– The root growth function λ(t) can be relatedto soil properties, which alters the roots’ elon-gation rate.

– The branching strategy can depend on soilproperties. In this way the density of lateralbranches can vary or the root type of thebranches can change.

The root growth model supplies the time de-pendent three-dimensional geometry of the rootsystem and resulting global parameters such astotal length, surface or spatial distributions, seeright part of Fig. 4. Generally it is computationallytoo expensive to directly use the explicit three-dimensional geometry even for the static case. Forthis reason, density distributions are used to createa sink term in the plant and soil interaction model,which is often described by partial differentialequations and numerically solved by the finiteelement method (Doussan et al. 2006; Javaux et al.2008). The development of a suitable sink term iscrucial and several upscaling techniques are avail-able (Roose and Schnepf 2008). However, thesemethods often use simplifying assumptions, e.g.that roots are evenly distributed, that soil is homo-geneous or that roots are functionally equivalent.Therefore, in general, sink terms need an accuratevalidation on an experimental basis.

We emphasise that the Matlab code for the rootgrowth model is freely available.1 Every produc-tion rule has a Matlab file with a correspondingname. This allows to adapt existing productionrules and to easily extend the model for newproduction rules.

1The Matlab files can be downloaded at www.boku.ac.at/marhizo.

http://www.boku.ac.at/marhizohttp://www.boku.ac.at/marhizo

186 Plant Soil (2010) 332:177–192

The case of phosphate uptake

In the following, we introduce a simple phosphateuptake model to which the root growth model canbe coupled. Neglecting convection, we describethe solute transport by the diffusion equation(Barber 1995):

(θ + b)∂c∂t

= ∇ · (Dl fθ∇c) − f (s, t) (12)

where c is the phosphate concentration, t is thetime, f (s) is a sink term for phosphate uptakeby roots, s is the root surface area per volume ofsoil, Dl is the diffusion coefficient in water, f isthe impedance factor, θ is the volumetric watercontent and b is the buffer power.

For simplicity, and since the aim of this paperis to demonstrate the coupling of the L-Systemroot growth- to a soil model, we assumed that thevolumetric sink term is given by

f (s) = s FmcKm + c , (13)

where Fm is the maximal influx into the root andKm is the Michaelis Menten constant. This simplesink term will often overestimate uptake. Firstly,root uptake capacity may decrease with root age.Assuming an exponential decrease, the sink term(13) is modified such that

f (s, t) = s FmcKm + c exp(−k tage), (14)

where k is the capacity decay constant and tage isthe average root surface age. Secondly, in the caseof sparingly soluble nutrients such as phosphate,depletion zones around the root limit uptake.Roose and Fowler (2004) presented a sink termfor nutrient uptake which considers the creationof depletion zones around individual roots. Thesethree cases will be analysed in our simulations.

At the boundaries representing a pot we applya no-flux condition:

∇c · n = 0, (15)where the unit normal n is pointing inside the pot.The initial phosphate concentration is given by

c(x, 0) = c0. (16)

Results

We first present the results of the root growthmodel only, exemplified for maize. We comparedour L-System root growth model with the continu-ous model of Roose et al. (2001) where no stochas-tic processes are included. Then we simulate theinteraction between root system and soil, exem-plified for phosphate uptake. We investigated spa-tial properties of a maize root system confined in apot and analysed the effect of chemotropism andgravitropism on phosphate uptake by coupling theroot growth model to a soil model. Finally, sinkterms with different complexities were derivedfrom the simulated root architecture and theireffect on phosphate uptake was analysed.

Root growth and architecture

The continuous root growth model of Roose et al.(2001), that is based on a population growthmodel, was used as a benchmark problem. Formodel comparison, we parametrised the L-Systemmodel such that it had exactly the same behaviourbut neglected root mortality. The total root lengthof every root order over 100 days is given in Fig. 5.The dynamic behaviour of the L-System model isexactly the same as in the growth model given byRoose et al. (2001).

Next we added spatial parameters determin-ing the angle between branch and successivebranch δ0 = 63◦, δs,0 = 15, δ1 = 68◦ and δs,1 = 3

0 20 40 60 80 1000

2000

4000

6000

8000

10000

12000

14000

time (days)

leng

th (

cm)

2nd order

1st order

0th order

total

Fig. 5 Total root length of each order of a maize rootsystem with 25 initial roots

Plant Soil (2010) 332:177–192 187

(Shibusawa 1994). To create a more realisticthree-dimensional representation of the root sys-tem, we added a standard deviation of 10% toall root system parameters (except for δ). Wealso included gravitropism, choosing N0,1 = 1.5,N2 = 0 and σ0 = 20◦ cm−1, σ1,2 = 40◦ cm−1, seeSection “Tropisms and root tip deflection”. Thefibrous root system was created with 25 ini-tial roots, n0 = 25, starting to grow into a conewith base radius, r0 = 5, at the same time, seeSection “Root systems”. The resulting root systemafter 25 days is presented in Fig. 6a. This three-dimensional representation of the 25 days oldroot system was analysed for root length fractionand root surface densities, see Fig. 6b, c. Figure6c shows the mean root surface area density involume elements of 1 cm3 along the y-axis. Suchdensities can be the basis for coupling root growthto a soil model.

Phosphate uptake from a pot

Having obtained the root architectural traits fromour root system model, these results were inte-grated into a soil model to analyse phosphateuptake. The soil model is given by Eqs. 12–16. Theroot system grew in a pot with a bottom radiusof 3 cm, a top radius of 5 cm and a height of10 cm. Model parameters were taken from liter-

ature and represent typical values for phosphateuptake from soil by maize (Tinker and Nye 2000;Föhse et al. 1991). The initial concentration c0 wasassumed to be 10−4 μmol cm−3, impedance fac-tor f = 0.3, water content θ = 0.4, buffer powerb = 100, diffusion coefficient for phosphate Dl =10−5 cm2 s−1, maximal influx Fm was assumed tobe 2.76 · 10−7 μmol cm−2 s−1 and the MichaelisMenten constant Km = 4 · 10−4 μmol cm−3.

The phosphate uptake was dependent on theroot surface area s per volume of soil. In everytime step �t of 1 day the values of s were deter-mined from the root growth in volume elementsof 0.125 cm3. The overall simulation time was50 days. We used the finite difference method toobtain a numerical solution of the solute transportmodel.

The first set of simulations studied the effect ofdifferent tropisms on phosphate uptake. Chemo-tropism was included as described in Section“Tropisms” with N0,1=5, σ0=20◦, σ1=40◦. Gravit-ropism was considered with the same parametersas in Fig. 6a. Uptake was calculated with thesimple sink term given in Eq. 13. In order tocreate a suitable root system for the pot size, initialroot system parameters were n0 = 5 and r0 = 2.The resulting root systems are shown in Fig. 7a,and c. The nutrient concentrations in the pot aregiven in Fig. 7b, d. The root system including

(a) Maize root systema fter 25 days (b) Depth dependent root length frac-tion

(c) Mean root surface density along they-axis

Fig. 6 Analysis of a simulated maize root system

188 Plant Soil (2010) 332:177–192

Fig. 7 Maize growing in apot with differenttropisms: Rootdistribution (a) andcorresponding phosphateconcentration (b) in thecase of chemotropism(N0,1 = 5, σ0 =20◦,σ1 =40◦). Rootdistribution (c) andcorresponding phosphateconcentration (d) in thecase of gravitropism(N0,1 =1.5, σ0 =20◦,σ1 =40◦)

chemotropism was more evenly distributed due toinitially homogeneous phosphate concentration.This results in even depletion of the soil in the pot.The root system with pronounced gravitropismhad a higher root density at the bottom of the pot.Hence the soil was less depleted at the top than atthe bottom.

The cumulative phosphate uptake by the tworoot systems is given in Fig. 8. The root systemincluding chemotropism had an 82% higher cumu-lative uptake after 50 days under the chosen setsof model assumptions. This result is in agreementwith Jackson and Caldwell (1996) who estimatedan increase in phosphate uptake due to root pro-liferation in nutrient rich patches up to 70%.

The second set of simulations studied the effectof the different sink terms described in Section“Coupling to a soil model” on phosphate uptakein the case of chemotropism. The resulting cu-mulative phosphate uptake is shown in Fig. 9. Asexpected, the simple sink term yielded the largest

uptake. When root uptake capacity decreasedwith age, cumulative uptake was 88% of the firstcase. We chose the capacity decay constant k to

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

4

time (days)

cum

ulat

ive

upta

ke (µ

mol

)

gravitropismchemotropism

Fig. 8 Cumulative phosphate uptake from a pot as effectedby chemotropism and gravitropism due to the simple sinkterm given in Eq. 13

Plant Soil (2010) 332:177–192 189

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

4

time (days)

cum

ulat

ive

upta

ke (

µmol

)

simple sink termage dependent root uptakeindividual depletion zones

Fig. 9 Cumulative phosphate uptake from a pot as effectedby chemotropism comparing three different sink termsdescribed in Section “Coupling to a soil model”

be 0.5 days−1 such that uptake approaches 0 after10 days (Ernst et al. 1989). When the individualdepletion zones around roots were considered us-ing the model by Roose and Fowler (2004), up-take was 79% of the first case. This indicates thatthe consideration of individual depletion zonesis important, in particular for sparingly mobilenutrients.

Discussion

L-Systems are a common tool in plant architec-ture modelling (Prusinkiewicz and Lindenmayer1990). They have also been used for root archi-tecture modelling, although mainly in the contextof visualisation (Prusinkiewicz 1998). We realisedan interface in our L-System model which en-ables coupling the three-dimensional root growthmodel to any arbitrary soil model. Furthermore,the modular implementation in Matlab allows todefine new production rules as required. Mostdynamic root system models are based on sim-ple production rules that include emergence ofnew main axes, growth of the axes and branch-ing (Doussan et al. 2003). L-System formalism ismost suitable to describe such production rules.We explicitly stated our production rules and pro-vided the corresponding Matlab files to increasereproducibility.

The way tropisms were implemented in ourmodel differs from previous implementations. Themost common approach is to compute the newgrowth direction by adding vectors denoting theinitial growth direction, mechanical constraintsand gravitropism (Pagès et al. 1989). Tsutsumiet al. (2003) compute the new growth direction intheir two-dimensional model based on differencesof elongation rates at the elongation points (lo-cated opposite to each other just behind the roottip). In our implementation, the new direction wascomputed by random minimisation of an objec-tive function. Tropisms were realised by choosingappropriate objective functions. The advantagesof this approach are that tropisms can be easilydescribed, see Eq. 11, and that results do notdepend on the spatial resolution.

We used a maize root architecture to demon-strate basic features of the proposed model. Maizeroot lengths found in literature strongly vary de-pending on maize cultivar and environmental con-ditions. The presented results lie well in this rangeand do for example quantitatively correspond tothe nitrate inefficient maize breed Wu312 de-scribed in Peng et al. (2010).

Root water and nutrient uptake based on three-dimensional root architecture models is com-monly included in soil models via sink terms (e.g.Clausnitzer and Hopmans 1994; Somma et al.1998; Dunbabin et al. 2002, 2006; Doussan et al.2006; Javaux et al. 2008). Those sink terms de-pend on root length or root surface densities. Thespatial resolution of the soil model for which thedensities are computed is often 1 cm3. In our sim-ulations we used half the edge length, i.e. the gridsize was 0.125 cm3. However, in our model, thespatial resolution can be freely chosen by the userdepending on the scale of the problem. Ge et al.(2000) use the SimRoot model (Lynch et al. 1997)to simulate phosphate acquisition efficiency in de-pendence on gravitropism. Contrary to our model,this root architecture model is not coupled to adynamic soil model. It uses the diffusion lengthof phosphate in soil, L = 2√Det, where De is theeffective diffusion coefficient, in order to estimatethe depletion volume for a root system. No othermechanisms such as root responses to phosphateconcentration are considered. This approach pro-vides an estimate for the potential phosphate

190 Plant Soil (2010) 332:177–192

uptake under the assumption that diffusion is thedominant mechanism in the soil. Somma et al.(1998) base their sink term for nutrient uptakeon the root surface density and the local averagednutrient concentration. Depletion around individ-ual roots is neglected. We used a similar approachin the ‘simple’ and ‘age dependent’ sink termbut also provided an example where individualdepletion zones were taken into account. In ourexample water transport was neglected. However,our L-System model was designed in such a waythat it can be coupled to models including watertransport, e.g. based on the Richards equation asused by Somma et al. (1998). This would enableto study hydrotropism. Advances with regards toadditional rhizosphere traits have been made byDunbabin et al. (2006) by simulating the effectsof phospholipid surfactants on water and nutri-ent uptake. In our example, we did not considerexudation. However, due to the flexibility of ourapproach, the root growth model can easily becoupled to soil models including exudation. Thesink term for nutrient uptake in Dunbabin et al.(2002, 2006) is based on plant demand as wellas nutrient concentration at the root surface ac-cording to the Baldwin, Nye and Tinker equa-tion. Thus, depletion around individual roots isapproximated with a steady rate solution. Thisis a similar approach like our sink term whichconsidered individual depletion zones based onan approximate analytic solution to the dynamicmodel (Roose et al. 2001).

In this work we exemplified the coupling ofthe root system to a soil model with a model forphosphate uptake by a maize root system froma pot. We considered root response via chemo-tropism as described in Section “The case of phos-phate uptake” and via root responses to barriersas described in Section “Root tip deflection”. Wecompared three different sink terms for nutrientuptake that were based on the root growth model.We showed that neglecting the development ofdepletion zones around individual roots is likelyto overestimate uptake, in particular for nutri-ents with low mobility such as phosphate. Thecomparison of the different sink terms indicatedthat it may be difficult to distinguish between ageand depletion effects experimentally in soil. Fur-thermore, we showed that chemotropism strongly

increased phosphate uptake from a pot which isin good agreement with Jackson and Caldwell(1996). As demonstrated by Ge et al. (2000), thisis due to overlapping depletion zones in the caseof gravitropism.

A comparison to measurement results fromliterature on phosphorus uptake of maize in potexperiments required consideration of the soil vol-ume, as well as the growing duration of the maizecrop. We converted the results from literatureusing the pot volume of 509.5 cm3 as taken for oursimulation. Data range from 28.74 to 51.66 μmolafter 32 days (El Dessougi et al. 2003), and 33.90to 83.31 μmol after 45 days (Mujeeb et al. 2008)phosphate root uptake depending on fertiliza-tion level and fertilizer type. These amounts arehigher, but in a similar order of magnitude, ascompared to our model outputs. A lower uptakeis anticipated considering that we described a lowphosphate scenario.

The main benefit of our approach in Matlabis that the L-System model can be easily cou-pled to arbitrary soil models. Thereby variousrhizosphere traits such as nutrient and water up-take and exudation can be modelled in responseto root system development and vice versa. Thepossibility to let the root system grow in aconfined environment will be useful for compar-ison to experimental data from pot or rhizotronexperiments.

Conclusions

We presented a new dynamic root architecturemodel based on L-Systems. The objective of thisnew model was to realistically represent a three-dimensional root system that was subsequentlycoupled to a soil model in order to describe thedynamic interactions of both the root system withsoil processes (water, solute transport) as well aslocal soil properties with the rooting pattern (rootplasticity, tropism). L-systems provide a compactdescription of dynamic root system models. Ourimplementation of the model into Matlab makesit accessible to a wider public and encourages itsdevelopment with new features. Furthermore itfacilitates coupling to different soil models.

Plant Soil (2010) 332:177–192 191

We exemplified the proposed root growthmodel and its interaction with a soil model bysimulating phosphate uptake of a maize plantin a confined growth environment (pot exper-iment). This example demonstrated that ourapproach provides a convenient tool to createfeedback loops between root system and soil.Thus it is possible to analyse both root effects onsoil processes via a sink term derived from thearchitecture model as well as soil property impactson the root architecture dynamics via tropisms.We implemented chemotropism and were able toclearly reveal substantial effects of root architec-ture and its interaction with the soil environmenton plant nutrient use. Chemotropism increasedphosphate uptake by as much as 82% compared toa root system governed by gravitropism only. Wecompared three different sink terms and analysedtheir effect on calculated phosphate uptake andfound that considering individual depletion zonesaround each root reduces calculated uptakesignificantly.

In contrast to existing root growth models, ourmodel can be freely coupled to any soil modeland new sink terms can be easily implemented asrequired. With this approach, we hope to facilitatemodel development and increase reproducibilityof simulation studies.

Acknowledgements This work was supported by theVienna Science and Technology Fund (WWTF, Grant No.:MA07-008) and the Austrian Science Fund (FWF, GrandNo.: T341-N13).

Open Access This article is distributed under the termsof the Creative Commons Attribution Noncommercial Li-cense which permits any noncommercial use, distribution,and reproduction in any medium, provided the originalauthor(s) and source are credited.

Appendix

Turtle graphics provides an easy way to createvector graphics using a relative cursor knownas turtle. An L-System string is graphically rep-resented using turtle graphics. Some charactersof the string represent graphical commands. Theknown commands are given in Table 2, othercharacters are ignored. The turtle state consists ofthe turtle’s position xp, the heading or local axis

Table 2 Turtle commands (after Prusinkiewicz andLindenmayer 1990)

Letter Description Parameters

F Draw a segment Length# Width of the following segments WidthC Colour of the following segments ColorRα or +, − Turn left or right DeltaRβ or \, / Roll left or right Delta&, ˆ Pitch down or up Delta| Turn around −[, ] Stores and retrieves the turtle −

state

of the turtle hx, hy, hz, a line colour and the linewidth.

References

Barber SA (1995) Soil nutrient bioavailability: a mechanis-tic approach. Wiley, New York

Bengough AG, Bransby MF, Hans J, McKenna SJ, RobertsTJ, Valentine TA (2006) Root responses to soil phys-ical conditions; growth dynamics from field to cell. JExp Bot 57(2 SPEC. ISS.):437–447

Callaway RM, Pennings SC, Richards CL (2003) Pheno-typic plasticity and interactions among plants. Ecology84(5):1115–1128

Clausnitzer V, Hopmans JW (1994) Simultaneous model-ing of transient three-dimensional root growth and soilwater flow. Plant Soil 164(2):299–314

de Dorlodot S, Forster B, Pagès L, Price A, Tuberosa R,Draye X (2007) Root system architecture: opportuni-ties and constraints for genetic improvement of crops.Trends Plant Sci 12(10):474–481

Diggle AJ (1988) Rootmap—a model in three-dimensionalcoordinates of the growth and structure of fibrous rootsystems. Plant Soil 105(2):169–178

Doussan C, Pages L, Vercambre G (1998) Modelling ofthe hydraulic architecture of root systems: an inte-grated approach to water absorption—model descrip-tion. Ann Bot 81(2):213–223

Doussan C, Pags L, Pierret A (2003) Soil exploration andresource acquisition by plant roots: an architecturaland modelling point of view. Agronomie 23(5–6):419–431

Doussan C, Pierret A, Garrigues E, Pags L (2006) Wateruptake by plant roots: II—modelling of water trans-fer in the soil root-system with explicit account offlow within the root system—comparison with experi-ments. Plant Soil 283(1–2):99–117

Dunbabin V, Rengel Z, Diggle AJ (2004) Simulating formand function of root systems: efficiency of nitrate up-take is dependent on root system architecture andthe spatial and temporal variability of nitrate supply.Funct Ecol 18(2):204–211

192 Plant Soil (2010) 332:177–192

Dunbabin VM, Diggle AJ, Rengel Z, Van Hugten R (2002)Modelling the interactions between water and nutri-ent uptake and root growth. Plant Soil 239(1):19–38

Dunbabin VM, McDermott S, Bengough AG (2006) Up-scaling from rhizosphere to whole root system: mod-elling the effects of phospholipid surfactants on waterand nutrient uptake. Plant Soil 283(1–2):57–72

Eapen D, Barroso ML, Ponce G, Campos ME, CassabGI (2005) Hydrotropism: Root growth responses towater. Trends Plant Sci 10(1):44–50

El Dessougi H, Zu Dreele A, Claassen N (2003) Growthand phosphorus uptake of maize cultivated alone, inmixed culture with other crops or after incorpora-tion of their residues. J Plant Nutr Soil Sci 166(2):254–261

Ernst M, Romheld V, Marschner H (1989) Estimation ofphosphorus uptake capacity by different zones of theprimary root of soil-grown maize (Zea mays l.). J PlantNutr Soil Sci 152:21–25

Fitter AH, Stickland TR (1991) Architectural analysis ofplant root systems. 2. influence of nutrient supply onarchitecture in contrasting plant species. New Phytol118(3):383–389

Fitter AH, Stickland TR, Harvey ML, Wilson GW (1991)Architectural analysis of plant root systems. 1. archi-tectural correlates of exploitation efficiency. New Phy-tol 118(3):375–382

Föhse D, Claassen N, Jungk A (1991) Phosphorus effi-ciency by plants. Plant Soil 132:261–272

Forde B, Lorenzo H (2001) The nutritional control of rootdevelopment. Plant Soil 232(1–2):51–68

Ge Z, Rubio G, Lynch JP (2000) The importance of rootgravitropism for inter-root competition and phospho-rus acquisition efficiency: results from a geometricsimulation model. Plant Soil 218(1–2):159–171

Hodge A (2004) The plastic plant: root responses toheterogeneous supplies of nutrients. New Phytol162(1):9–24

Jackson RB, Caldwell MM (1996) Integrating resource het-erogeneity and plant plasticity: modelling nitrate andphosphate uptake in a patchy soil environment. J Ecol84(6):891–903

Javaux M, Schröder T, Vanderborght J, Vereecken H(2008) Use of a three-dimensional detailed modelingapproach for predicting root water uptake. VadoseZone J 7(3):1079–1088

Jury W, Horton R (2004) Soil physics. Wiley, New YorkLynch J (1995) Root architecture and plant productivity.

Plant Physiol 109(1):7–13Lynch JP, Nielsen KL, Davis RD, Jablokow AG (1997)

Simroot: Modelling and visualization of root systems.Plant Soil 188(1):139–151

Mujeeb F, Hannan R, Maqsood M (2008) Response ofmaize to diammonium phosphate and farmyard ma-

nure application on three different soils. Pak J AgricSci 45:13–18

Pagès L, Jordan MO, Picard D (1989) A simulation-modelof the 3-dimensional architecture of the maize root-system. Plant Soil 119(1):147–154

Pagès L, Vercambre G, Drouet JL, Lecompte F, Collet C,Le Bot J (2004) Root typ: a generic model to depictand analyse the root system architecture. Plant Soil258(1–2):103–119

Peng Y, Niu J, Peng Z, Zhang F, Li C (2010) Shoot growthpotential drives n uptake in maize plants and corre-lates with root growth in the soil. Field Crops Res115(1):85–93

Peret B, Larrieu A, Bennett MJ (2009) Lateral root emer-gence: a difficult birth. J Exp Bot 60(13):3637–3643

Pierret A, Doussan C, Capowiez Y, Bastardie F, Pages L(2007) Root functional architecture: a framework formodeling the interplay between roots and soil. VadoseZone J 6(2):269–281

Prusinkiewicz P (1998) Modeling of spatial structure anddevelopment of plants: a review. Sci Hortic 74:113–149

Prusinkiewicz P (2004) Modeling plant growth and devel-opment. Curr Opin Plant Biol 7(1):79–83

Prusinkiewicz P, Lindenmayer A (1990) The algorithmicbeauty of plants. Springer, Berlin

Roose T, Fowler AC (2004) A mathematical model forwater and nutrient uptake by plant root systems. JTheor Biol 228(2):173–184

Roose T, Fowler AC, Darrah PR (2001) A mathematicalmodel of plant nutrient uptake. J Math Biol 42(4):347–360

Roose T, Schnepf A (2008) Mathematical models of plant–soil interaction. Philos Trans R Soc A 366:4597–4611

Shibusawa S (1994) Modelling the branching growth frac-tal pattern of the maize root system. Plant Soil165(2):339–347

Somma F, Hopmans JW, Clausnitzer V (1998) Transientthree-dimensional modeling of soil water and solutetransport with simultaneous root growth, root waterand nutrient uptake. Plant Soil 202(2):281–293

Spek LY (1997) Generation and visualization of root-likestructures in a three-dimensional space. Plant Soil197(1):9–18

Tinker PB, Nye PH (2000) Solute movement in the rhi-zosphere. Oxford University Press, New York

Tsutsumi D, Kosugi K, Mizuyama T (2003) Root-systemdevelopment and water-extraction model consideringhydrotropism. Soil Sci Soc Am J 67(2):387–401

Walk TC, Jaramillo R, Lynch JP (2006) Architecturaltradeoffs between adventitious and basal roots forphosphorus acquisition. Plant Soil 279(1–2):347–366

Zhang H, Rong H, Pilbeam D (2007) Signalling mecha-nisms underlying the morphological responses of theroot system to nitrogen in Arabidopsis thaliana. J ExpBot 58(9):2329–2338

A dynamic root system growth model based on L-SystemsAbstractIntroductionModel descriptionIntroduction to L-SystemsAxial growthLateral branchingRoot systemsTropisms and root tip deflectionTropismsRoot tip deflection

Coupling to a soil modelThe case of phosphate uptake

ResultsRoot growth and architecturePhosphate uptake from a pot

DiscussionConclusionsAppendixReferences

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /Warning /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 150 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 600 /MonoImageMinResolutionPolicy /Warning /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice