Modeling of spatial structure and development ofplantsPrzemyslaw Prusinkiewicz

Abstract

Developmental models of plant structure capture the spatial arrangement of plant componentsand their development over time. Simulation results can be presented as schematic or realisticimages of plants, and computer animations of developmental processes. The methods ofmodel construction combine a variety of mathematical notions and techniques, fromregression analysis and function fitting to Markov processes and Lindenmayer systems(L−systems). This paper presents an overview of the wide range of spatial model categories,including both empirical and causal (mechanistic) models. An emphasis is put on L−systemsand their extensions, viewed as a unifying framework for spatial model construction.

Reference

Przemyslaw Prusinkiewicz: Modeling of spatial structure and development of plants. Scientia Horticulturaevol. 74, pp. 113−149.

Modeling of spatial structure and development ofplants

Przemyslaw PrusinkiewiczDepartment of Computer Science

The University of CalgaryCanada

Abstract

Developmental models of plant structure capture the spatial arrangement ofplant components and their development over time. Simulation results can be pre-sented as schematic or realistic images of plants, and computer animations of de-velopmental processes. The methods of model construction combine a variety ofmathematical notions and techniques, from regression analysis and function fittingto Markov processes and Lindenmayer systems (L-systems). This paper presentsan overview of the wide range of spatial model categories, including both empir-ical and causal (mechanistic) models. An emphasis is put on L-systems and theirextensions, viewed as a unifying framework for spatial model construction.

1 IntroductionMathematical models in botany correspond to various levels of plant organization (Fig-ure 1). In this paper, we focus on spatial organization of individual plants. A plant is

organelle

molecule

organ

cell

crop

plant

biome

ecosystem

Figure 1: A hierarchy of levels of plant organization. One objective of modeling is topredict and understand phenomena taking place at a given level on the basis of modelsoperating at lower levels. Adapted from [170, 175].

1

apical meristem

bud (lateral)

apex

internode

leaf

inflorescence

flowers

metamerorshoot unit

branch

apical segment

Figure 2: Selected modules and groups of modules (encircled with dashed lines) usedto describe plant structure. From [130].

viewed as a configuration of discrete constructional units or modules, which developover time. Modules represent repeating components of plant structure, such as flowers,leaves, and internodes, or groupings of these components, such as metamers (singleinternodes with an associated leaf and lateral bud) and branches (Figure 2) [10, 72,119, 173]). (A different meaning of the term “module” is also found in the litera-ture [4, 65, 155].) The modeling task is focused on the description of plant structureand development as the integration of the development and functioning of individualmodules.

As with other models of nature, computer models of plant structure and develop-ment can be divided into empirical (descriptive) or causal (mechanistic, physiologi-cally based). The distinction between these two classes is described authoritatively byThornley and Johnson [170]. Summarizing their point, empirical models represent theacquired data in a more convenient form, and are useful in making practical predic-tions based on the interpolation of these data. In the context of the modeling hierarchyshown in Figure 1, “the modeler attempts to describe level i behavior (observational

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data) in terms of level i attributes alone, without regard to any biological theory. Theapproach is primarily one of examining the data, deciding on an equation or set ofequations, and fitting these to the data.” In contrast, mechanistic models follow thetraditional reductionist method of the natural sciences, in which phenomenon at leveli is described and understood in terms of processes at levels i − 1 and lower. Thesemodels provide explanations and responses that integrate the underlying mechanisms,thus contributing to our understanding of the processes under study. Potentially, theirpredictive value is not limited to the interpolation of the collected data, and includesextensive possibilities for asking “what if?” type questions. In practice, many plantmodels combine empirical and mechanistic aspects. Although such models cannot beunequivocally categorized, it is useful to categorize their individual features, in orderto fully understand the status and the predictive power of the model.

The distinction between empirical and causal models parallels the relation betweenthe top-down (analytic) and bottom-up (synthetic) approaches to modeling. In the top-down case, the construction of the model is based on the analysis of empirical data.In the bottom-up case, the model synthesizes known or postulated mechanisms of de-velopment. The emphasis is on the properties of the whole model that emerge fromthe interactions between individual components. In the most abstract form, construc-tion of models with emergent properties crosses the line dividing biology and artificiallife [94, 168].

The spatial modeling of plants is a highly interdisciplinary area. Botany and ap-plied plant sciences are at the roots of many approaches to model construction, and arean important domain for model applications. Nevertheless, many visually convincingplant models were created within computer graphics (for instance, see [15, 141, 174]).The underlying modeling and visualization techniques are important from a biologicalperspective, because realistic presentation adds credibility to the models and facilitatestheir validation based on visual comparisons with nature [126]. In addition, computergraphics has contributed methods for calculating light reflectance and distribution insimulated environments [47]. They are important in the modeling of plants taking intoaccount their local light conditions (Section 5.4) and in the application of spatial mod-els to remote sensing [17, 59, 60, 96].

Lindenmayer systems (L-systems) are another interdisciplinary component of ar-chitectural plant modeling. They originated within theoretical biology [101] and wereextensively studied by mathematicians and computer scientists [77] before they becamean effective modeling tool [136] (for a historical perspective see [125]). L-systems be-long to the class of mathematical formalisms known as rewriting systems or formalgrammars [157]. To a biologist they offer a conceptual framework for constructingdevelopmental plant models and expressing them in special-purpose modeling lan-guages [68, 91, 136]. The use of a modeling language makes it possible to simulatethe development of a variety of organisms, from algae to herbaceous plants to trees,using the same simulation program with different input files. This approach offers thefollowing benefits:

• The programming effort needed to develop L-system models of specific plants is

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significantly reduced in comparison to the effort needed to implement the samemodels in a general-purpose programming language, such as Fortran or C1.

• The models can be easily modified during experimentation. These modificationsare not limited to the values of numerical parameters, but may also involve fun-damental changes in model definition and operation.

• The L-system language makes it possible to document models in a compact andprecise manner (for example, in publications).

• The expression of models in the same language facilitates their comparisons.

Reflecting these advantages, all models illustrating this paper have been created or re-produced from the original publications using the L-system-based simulation programcpfg, included in the Virtual Plant Laboratory [121].

2 Data acquisition

2.1 Qualitative description of plant architectureData acquisition is the starting point for constructing all plant models, yet the type ofdata used may vary greatly. On the most qualitative end of the spectrum one finds thearchitectural unit, introduced by Edelin [38] (see also [8]) to characterize plants withinthe conceptual framework of architectural models proposed by Halle, Oldeman, andTomlinson [66]. The morphological characteristics incorporated into an architecturalunit can be directly observed or estimated without an extensive use of measuring in-struments. They include, among others: the orientation of branches (e.g. orthotropic orplagiotropic), type of branching (monopodial or sympodial), persistence of branches(indefinite, long or short), degree of lateral shoot development as a function of theirposition on the mother branch (acrotony, mesotony or basitony), type of meristematicactivity (rhythmic or continuous), number of internodes per growth unit, leaf arrange-ment (phyllotaxis), and position of reproductive organs on the branches (terminal orlateral). An authoritative description of these and other notions used to specify plantarchitecture is given by Bell [10], and Caraglio and Barthelemy [18]. The architec-tural unit is a set of these characteristics, given for all branch orders. For examples ofarchitectural description of specific trees in terms of architectural units see [2, 120].

Plant architecture is a dynamic concept, in the sense that the observed structuralfeatures reflect plant development over time. As stated by Halle et al., “The idea ofa form implicitly contains also the history of such a form” [66]. Correspondingly,the architectural unit may be viewed as a sequence of branch types created over time,rather than merely a set of branch types. “In this sequence, leading from axis 1 to theultimate axes following the specific branching pattern, each branch is the expression

1The programming effort needed to develop specific models may also be reduced using appropriate soft-ware libraries, shared between various models. Such libraries have been created within the framework ofL-systems [64, 67] and outside of it [118].

4

Figure 3: Example of a plant model (Lychnis coronaria) based on a qualitative descrip-tion of the architecture [152]. Parameter values have been set interactively using thecontrol panel on the right to achieve proper appearance of the visual model.

of a particular state of meristematic activity and the branch series as a whole can beconsidered to be tracking the overall activity” [8].

By itself, qualitative characterization is insufficient to construct a spatial model ofa plant. Nevertheless, interactive computer graphics makes it possible to incorporatethe lengths of internodes, the magnitudes of branching angles, and other quantitativeaspects into a model, even if these characteristics were not explicitly measured. Theobserved architectural features form the basis of a model, in which the quantitative as-pects are parametrized. The parameter values are manipulated interactively to achieveproper appearance of the plant. This technique can be traced to the first computer treemodels devised by Honda [80]. It was also applied by Prusinkiewicz, Lindenmayerand Hanan to model various types of inflorescences [68, 136] (Figure 3). An extendedgraphical interface, which makes it possible to manipulate parameters of the model aswell as its underlying topological structure using graphical operations on the screen,has been recently proposed by Deussen and Lintermann [34].

2.2 Description of plant topologyA description of a plant in terms of its architectural unit assumes a generalization per-formed before the observed features are recorded. These features do not characterizespecific branches of a specific plant, but represent general characteristics of branches

5

I1

I3I2

B1

I7 I4

I6

I5

B2

F1

F2

I1[L1][I2B1]I3[L2][I4[I5F1]I6B2]I7F2

L1

L2

Figure 4: A hypothetical branching structure and the description of its topology usingthe bracketed string notation. I: internode, L: leaf, B: bud, F: flower.

of a given order in any specimen of a given species.Plant maps [26, 111] can be considered the first step towards characterizing the

structure of particular plants. This description captures the branching topology, that isthe aspects of the arrangement of branches, organs, and other features that do not de-pend on the structure’s geometry (the lengths of internodes and the magnitudes of thebranching angles). Plant maps can be recorded using various notations. For example,Hanan and Room [71] adapted for this purpose the bracketed string notation intro-duced by Lindenmayer [101]. The essence of this notation is illustrated in Figure 4. Adifferent notation is presented by Rey et al. [148].

A refinement of the topological description of plants has been proposed by Godinand Caraglio [56]. Their formalism, called multiscale tree graphs, makes it possible tospecify plant topology at different scales and levels of detail, and incorporate temporalaspects into the descriptions. Multiscale tree graphs form the basis of a coding lan-guage implemented in AMAPmod, an interactive program for analyzing the topologi-cal structure of plants [57]. The need for multiscale representation of plant architectureis also discussed by Remphrey and Prusinkiewicz [147].

2.3 Measurement of plant geometryThe processes of plant measurement and modeling influence each other. The initial,hypothetical architectural model guides the first phase of data acquisition. Shortcom-ings of the model that results from the incorporation of these data reveal the areas in

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which more data are needed. The cyclic process of model refinement continues un-til the desired characteristics of the model have been reached. The choice of featuresto be measured depends on the nature of the model (empirical or causal), its desiredaccuracy, spatial scale, level of detail, time scope, and resolution [147].

Geometric features of small plants or plant parts can be measured using calipersand a protractor. Unfortunately, this process is slow and costly. A faster method isto use a three-dimensional digitizer, which records positions of selected features (forexample, the nodes of the branching structure) pointed to by the operator using a hand-held probe. Digitizers used in the practice of plant measurement operate on a varietyof principles, including the measurement of the angles between the joints of an articu-lated arm [93], propagation time of (ultra)sound between the probe and a set of micro-phones [154, 160], and the distribution of a magnetic field around the probe [161, 162](see [112] for a review). Proper software makes it possible to enter measured datadirectly into a database, complement them with additional information identifying thefeature being measured (e.g. position of a node, leaf, flower, or fruit), and schemat-ically represent the measured plant on the computer screen for visual feedback anderror checking [70, 71]. The application of three-dimensional digitizers is limited bythe volume within which the measuring devices operate (currently not exceeding sev-eral cubic meters), and the participation of a human operator. Circumventing the vol-ume limitation, Ivanov et al. describe the use of images obtained by a pair of cameras(stereovision) to collect structural data of maize plants in a field [85, 86]. This tech-nique is limited to the top elements of the canopy (visible from both cameras) and,at present, requires manual identification and digitization of matching leaves in bothimages by an operator. The elimination of the operator’s involvement using computervision techniques is an open research problem.

Models of plant development are based on the observation and measurement ofplants over time. The frequency of these actions may vary greatly. For example, aninterval of one minute may be appropriate for time-lapse photography of opening flow-ers. On the other hand, developmental models of trees and shrubs may be based onmeasurements made at yearly intervals [143].

At a conceptual level, methods for expressing plant geometry at different levelsof spatial detail are needed as a natural extension of the multiscale tree graphs forrepresenting plant topology. The general problem of representing geometric objectsat different levels of spatial detail (multiresolution representations) was addressed incomputer graphics (e.g. citeHoppe1997vdr), but the proposed solutions have not yetbeen adapted to highly branching plant structures. A related issue is the reconciliationof numerical data at different levels of resolution (for example, the measured lengthof an entire branch with the sum of the length of the internodes), which may exhibitinconsistencies due to numerical errors. This issue has not yet been addressed in theliterature.

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3 Empirical models of plant structure

3.1 Reconstruction modelsReconstruction of three-dimensional plant structure based directly on the raw data ob-tained by plant measurement can be considered the extreme case of empirical plantmodeling. The resulting reconstruction models can be created dynamically, during themeasurement process [71], or after all the measurements have been completed [58,161, 162, 163]. Smith at al. [163, 164] indicate the value of reconstruction mod-els in the analysis of spatial distribution of plant organs in the plant canopy (fruit inkiwifruit vines). Reconstruction models may also be useful in such applications ascomputer-assisted landscape and garden design, inclusion in plant registries, and adver-tising of plant varieties to customers (Campbell Davidson, personal communication).Recent effort towards standardization of the format for representing and transferringthree-dimensional models over the Internet (the Virtual Reality Modeling Language, orVRML [75]) may popularize this type of models. Nevertheless, reconstruction modelshave several drawbacks:

• they incorporate a large amount of raw data, and consequently are representedby relatively large data files.

• they represent features of a single plant specimen and cannot be manipulated toobtain other representatives of the same species,

• they have no predictive value (although they may assist in data analysis leadingto predictions).

3.2 Curve fittingStatistical methods of data analysis (regression analysis in particular) make it possi-ble to overcome these drawbacks by fitting empirical curves to the measured data.For example, this approach was applied by Remphrey and Powell to create statisti-cal reconstruction models of Larix laricina saplings [144, 145, 146]. Further exam-ples of statistical plant analysis at the structural level are presented by Davidson andRemphrey [30, 31, 142]. As categorized by Remphrey and Prusinkiewicz [147], aspatial model constructed by curve fitting may be deterministic (representing the bestfit, and ignoring the variance of parameters) or stochastic (allowing for the variationof the parameters values consistent with their distribution obtained through statisti-cal analysis). For early examples of the construction of stochastic models see deR-effye1981,Nishida1980.

3.3 Paracladial relationshipsAn interesting aspect of empirical model construction is the unraveling of similaritiesbetween an entire branch and its parts. To capture these similarities at the topological

8

B

A

x

y

a b c

Figure 5: Definition of the overhanging part x of mother branch A with respect to thedaughter branch B (a), and schematic branching structures with paracladial formulasy = x (b) and y = 1

2x (c). Based on [53, 103].

level, Frijters and Lindenmayer used the notion of a paracladium [53]. This notion wasoriginally introduced by Troll [171], who described an inflorescence as “a system con-sisting of the main florescence and of paracladia. Paracladia are branches which repeatthe florescence of the main axis and which on their turn can give rise to paracladia ontheir own” (quoted from [53, 103]). For example, the arrangement of the second-orderbranches along the axis of a first-order branch of a lilac inflorescence (Syringa vul-garis) approximately repeats the arrangement of the first-order branches on the mainaxis of the same inflorscence. Thus, each branch can be considered a small version —a paracladium — of the entire structure. The presence of paracladia is closely relatedto the process of plant development. “The ‘program’ which is responsible for the de-velopment of the main axis (the mother branch) is followed repeatedly in each of theparacladia. Furthermore, since the main axis can produce paracladia we must assumethat paracladia of first order can in turn give rise to paracladia of second order and soforth” [103].

Studying compound structures corresponding to this description, Frijters and Lin-denmayer observed that the number of internodes in the daughter branch (y) and in theoverhanging part of the mother branch (x, see Figure 5a) are often related by a linearfunction, which they called a uniform paracladial relationship. For example, in fig-ure Figure 5b the number of internodes in each lateral branch is equal to the numberof internodes in the entire overhanging part of the main axis (y = x). On the otherhand, in figure Figure 5b each (even-numbered) branch has one half the number ofinternodes found in the overhanging part of the main axis 9y = 1

2x). A structure that

exhibits a linear paracladial relationship can be modeled in a particularly concise man-ner, because the topology of each branch is determined by its position on the motherbranch. Non-linear paracladial relationships are also possible, but have not yet beeninvestigated.

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3.4 Self-similarityThe repetitive hierarchical organization may be reflected not only in the topology of abranching structure, but also in its geometry. Specifically, we say that a geometric ob-ject (such as a branching structure) is self-similar if its parts are geometrically similarto the whole. Self-similarity is the distinctive feature of fractals [43, 110]. A key theo-rem characterizing self-similar objects has been obtained by Hutchinson [83]. It statesthat if an object can be decomposed into a finite number of reduced copies of itself, itis completely described by the set of transformations that map the whole object onto itsparts. This set of transformations is called an Iterated Function System or IFS [6] andcan be used to reconstruct the original object using a number of computational meth-ods (for example, see [74, 76]). Barnsley and Demko extended Hutchinson’s result toobjects that are only approximately self-similar, and gave it the name of collage theo-rem [6]. The collage theorem was applied to construct extremely compact IFS modelsof highly self-similar structures, in particular fern fronds [5, 6]. Subsequent recurrentextension of Iterated Function Systems [7] made it possible to characterize structuressatisfying a relaxed condition of self-similarity (parts of the structure may result fromthe transformations of other parts, rather than the transformations of the entire struc-ture). Examples of branching structures modeled using recurrent IFS and their relationto L-systems were presented by Prusinkiewicz and Hammel [127, 128]. To date, ap-plications of iterated function systems to the modeling of plants have been investigatedmainly from the computer graphics perspective. Their relevance to biology is yet to bedetermined.

4 Simulation of plant development: empirical modelsDevelopmental models introduce two new elements, compared to models of staticstructures: the emergence of new modules during development, and the growth of in-dividual modules over time. These processes can be conveniently characterized withinthe framework of L-systems [101, 102]. L-systems can be used to express both empir-ical and mechanistic models. In this section we focus on empirical models, leaving thediscussion of mechanistic models to Section 5.

L-systems are typically described using the terminology of formal language the-ory [77, 157]. We give instead their intuitive description, which emphasizes the bio-logical interpretation of all terms.

4.1 L-systemsThe essence of development at the modular level can be regarded as a sequence ofevents, in which predecessor or parent modules are replaced by configurations of suc-cessor or child modules. The rules of replacement are called productions. It is assumedthat the number of different module types is finite, and all modules of the same typebehave in the same manner. Consequently, the development of a large structure (con-

10

Figure 6: Developmental model of a compound leaf, modeled as a configuration ofapices (thin lines) and internodes (thick lines). The inset represents production rulesthat govern this development. From [130].

figuration of modules) can be characterized by a finite set of rules. An L-system issimply a specification of the set of all module types that can be found in a given organ-ism, the set of productions that apply to these modules, and the initial configuration ofmodules (the axiom), from which the development begins.

For example, Figure 6 shows the development of a stylized compound leaf includ-ing two types of modules, the apices and the internodes. An apex yields a structurethat consists of two internodes, two lateral apices, and a replica of the main apex. Aninternode elongates by a constant scaling factor. The developmental sequence beginswith a single apex, and yields an intricate branching structure in spite of the simplicityof this L-system.

This description leaves open the questions of what notation to use to specify theproductions, and what data structures to choose to represent a growing plant in a simu-lation program based on L-systems. Lindenmayer addressed these questions by intro-ducing the bracketed string notation [101], which was described above in the contextof plant mapping (Figure 4). A branching structure is represented by a string of sym-bols corresponding to plant modules, with the branches enclosed in square brackets.Originally, the bracketed strings were intended to capture only the topology of the de-scribed structures. Nevertheless, subsequent extensions made it possible to describeplant geometry as well (turtle interpretation [122, 123]). Measurable quantities, suchas the length and width of internodes, and the magnitude of the branching angles, arecharacterized as numerical parameters associated with the individual modules (para-metric L-systems [68, 133, 136]). Productions are specified using the standard notationof formal language theory, with a reserved symbol (an arrow) separating the predeces-sor from the successor [102]. For example, an L-system model of the compound leafin Figure 6 is written as follows:

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ω : A(1)p1 : A(s) → I(s)[−(45)A(s)][+(45)A(s)]I(s)A(s)p2 : I(s) → I(2 ∗ s)

(1)

The growing structure consists of two module types, apices A and internodes I . Theparameter s determines the size (length) of any of the modules. The initial structureω is an apex A of unit length. Production p1 specifies that an apex yields a structureconsisting of two internodes I , two lateral apices A, and a replica of the main apexA. The symbols + and − do not represent components of the growing structure, butindicate that the lateral apices are positioned at an angle of +45◦ and−45◦ with respectto their supporting mother branch. A list of commonly used L-system symbols witha predefined geometric interpretation is given in [136]. Production p2 states that aninternode elongates by a factor of two in each simulation step.

A central observation underlying L-system simulations is that the application ofproductions, that is the replacement of predecessor modules by their successors, can becarried out at the level of their bracketed string representation. For example, the firststeps of a simulation using L-system (1) yield the following sequence of structures:

Initial structure : A(1)Step 1 : I(1)[−(45)A(1)][+(45)A(1)]I(1)A(1)Step 2 : I(2)[−(45)I(1)[−(45)A(1)][+(45)A(1)]I(1)A(1)]

[+(45)I(1)[−(45)A(1)][+(45)A(1)]I(1)A(1)]I(2)I(1)[−(45)A(1)][+(45)A(1)]I(1)A(1)

...

(2)

The bracketed strings are easily manipulated by a computer program, and formthe key data structure on which most simulation programs using L-systems are based(see [132, Appendix A] for the listing of a program using non-parametric L-systems,and [68] for details concerning their parametric extension). The conversion of strings tothree-dimensional graphical objects has been extensively documented [122, 130, 136,137]; for recent extensions see [91, 138]. Other data structures may also be used inconjunction with L-systems. For example, Hammel describes an implementation inwhich structures are represented as linked lists with tree topology [67].

The inherent capability of L-systems to describe the development of plants, ratherthan just their static structure, is illustrated by a number of developmental plant mod-els. Models with mainly empirical characteristics have been proposed for variousracemose and cymose inflorescences [51, 136, 137], green ash shoots [139], younggreen ash trees [147], Norway spruce trees [92], cotton plants [153], bean plants [69],maize shoots [50], maize root systems [159], and seaweed [27, 158]. A novel use ofL-systems has been recently proposed by Battjes and Bachmann [9], who related L-system parameter values to genetic variation between modeled plants (four species ofMicroseris, a herbaceous plant in the aster family).

The L-systems outlined above belong to the simplest, context-free class. This termmeans that a production can be applied to a module irrespective of its adjacent modules

12

a

*

B

AI[L]

I[F]

I[F]

IF

b

*

B

A0.9

0.8

0.1

0.2

c

Figure 7: A racemose inflorescence (a), the state transition diagram illustrating theprogression of apical states (b), and a sample Markov chain corresponding to this pro-gression (c).

(neighbors in the tree structure). In causal models it is often convenient to use context-sensitive L-systems, in which the applicability or outcome of a production dependsnot only on the module being replaced, but also on its neighbors. Such L-systems arediscussed in Section 5.

4.2 The process view of developmentWe can emphasize the dynamic aspect of plant modeling using the concept of a process,defined as a sequence of events ordered in time [90]. At any moment, a process ischaracterized by a number of parameters, called its state. Over time, the process maychange its state, create new processes, become inactive, be reactivated, or cease toexist. These notions, with a very practical meaning in computer science [167], canalso be applied to characterize plant modules, in particular the apices [3, 32]. Forinstance, an active apex may grow and repetitively create internodes, leaves, and buds.Production of a bud may initiate a new sequence of events, and therefore be regardedas a creation of a new active process. On the other hand, a bud may be dormant andremain inactive until it is activated by another process. Finally, an apex may die andcease to exist. Examples of paths of development involving elaborate state changeshave been presented by Bell [12].

As an example of the process point of view, let us consider the changes in apicalactivity taking place during the development of a closed racemose inflorescence. Fig-ure 7a indicates that the apex first produces a sequence of internodes with leaves, thenswitches to the production of lateral flowers, and eventually creates a terminal flower.We can describe this progression of activities using a state transition graph (a finite au-tomaton or finite state machine in the automata theory) in which nodes represent states,the short arrow distinguishes the initial state, and arcs (directed edges) represent statetransitions and the activities associated with them (Figure 7b). For instance, the twoarcs originating at node A indicate that an apex in state A may produce an internode

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and a leaf and remain in state A, or produce an internode and a lateral flower and switchto state B. As there is no edge from node B to node A, the apex cannot revert to theproduction of leaves once the production of flowers has begun.

In causal models, the switch from the vegetative to the flowering condition may betriggered by some external event, such as an environmental influence or the arrival ofa signal propagating through the plant body [136]. In contrast, empirical models oftenassume a stochastic mechanism, in which transitions are selected according to someprobabilities. A state diagram with associated probabilities is called a discrete Markovprocess, a Markov chain, or a probabilistic automaton [108]. For example, the Markovchain shown in Figure 7c indicates that, after producing a leaf, the apex will remain instate A with probability 0.9, and switch to state B with probability 0.1.

A key issue in the use of Markov chains for modeling is the inference of the setof states, transitions, and probabilities from empirical data. Godin et al. [58] describea software package which makes it possible to define and manipulate Markov chains(and their variations) interactively, until their behavior matches the observed proba-bility distributions. For further discussion and examples of the application of Markovprocesses to plant modeling see [3, 29, 28, 63, 109, 177].

A model based on a Markov chain can be expressed using a stochastic L-system [40,178] (see also [123, 136]). A probabilistic L-system includes several productions withthe same predecessor. A particular successor is selected according to the probabilitiesassociated with the productions. For example, the following probabilistic L-systemgenerated the sample racemose structure shown in Figure 7a according to the Markovchain in Figure 7c.

ω : A

p1 : A0.9−→ I [L]A

p2 : A0.1−→ I [F (1)]B

p3 : B0.8−→ I [F (1)]B

p4 : B0.2−→ IF (1)

p5 : F (s) −→ F (s ∗ 1.1)

(3)

This example shows that the formalisms of Markov processes and L-systems canbe used jointly, with a Markov process providing a stochastic description of the fatesof the individual apices, and the L-system integrating the development of these apicesand other plant components into a comprehensive model of the entire developing plant.

4.3 Discrete-time simulation of developmentAccording to the original definition of L-systems, productions are applied in parallel,with all modules being replaced simultaneously in every simulation step [101, 107]).The parallel operation of L-systems reflects the fact that all parts of a plant develop atthe same time, and that the time separating consecutive simulation steps can often beconveniently interpreted as a plastochron, which has a well defined biological meaning(the time separating production of consecutive nodes [41]). Nevertheless, Hogeweg

14

M1

M2

M3

M4

M5

. . .

. . .

. . .

time0 tα tβ

Figure 8: Fragment of the lineage tree of a hypothetical modular structure. From [131].

pointed out [78, 79] that the assumption of synchronous replacement of all modules bytheir successors at precisely the same time is too strong, and does not have a biologicaljustification. Consequently, she proposed to simulate development within the frame-work of discrete-event simulation, in which individual events may occur at arbitrarypoints in time [90]. An ordered sequence of events to be simulated is then organizedinto a data structure called the event queue, managed by a simulator component calledthe scheduler. The scheduler advances time to the next event and initiates actions as-sociated with it; these actions may, in particular, insert new events into the queue. Anapplication of this principle to the simulation of plant development was described indetail by Blaise [14].

4.4 Continuous-time simulation of developmentNeither synchronous nor discrete-event simulation capture the continuous develop-mental processes that take place between the events. To overcome this limitation,Prusinkiewicz, Hammel and, Mjolsness introduced a continuous-time extension of L-systems called differential L-systems [67, 131], which is based on the paradigm ofcombined discrete-continuous simulation [42, 90]. A module is created and ceases toexist in discrete events, as in the case of discrete-event simulation. Between the events,parameters of a module change in a continuous manner. An event is triggered if afunction of these parameters reaches a threshold value.

The combined discrete-continuous view of development is illustrated in Figure 8.Module M2 is created at time tα as one of two descendants of the initial module M1.It develops in the interval [tα, tβ), and ceases to exist at time tβ , giving rise to two newmodules M4 and M5. These modules will develop over some time, create new modulesat the end of their existence, and so on.

Parameters involved in module development may represent features inherent in amodule, such as the length of an internode, the size of a bud, or the magnitude of abranching angle, or external factors, such as time, temperature, temperature sum, daylength, or light sum. For examples of the application of differential L-systems to theanimation of plant development see [131].

15

x01

x02

x03

xmax

0.0 1.0 2.0

x

t

6

- xmin

xmax

T

x

t

∆x

6

-

6

?

Figure 9: Examples of sigmoidal growth functions. a) A family of logistic functionsplotted using r = 3.0 for different initial values x0. b) A cubic function x(t) extendedby constant function x = xmax for t > T . From [131].

4.5 Growth functionsGradual changes of parameter values may be specified using differential or algebraicfunctions of time called growth functions [84, 140, 151]. In empirical models, growthfunctions are found by fitting mathematical functions to data using statistical meth-ods (for a general methodological discussion see [82]). A wide range of functionswith different degrees of biological justification have been proposed in the literature;see [150, 165, 179] for reviews. Specifically, parameters representing geometric fea-tures of a module, such as internode length, leaf size, and branching angles, oftenincrease according to sigmoidal functions, which means that they initially increasein value slowly, then accelerate, and eventually level off near or at the maximumvalue [169]).

A popular example of a sigmoidal function is Velhurst’s logistic function (c.f. [39,page 212]), defined by the equation:

dx

dt= r

(

1 −

x

xmax

)

x (4)

with a properly chosen initial value x0 (Figure 9a). Another example is the cubicfunction:

x(t) = −2∆x

T 3t3 + 3

∆x

T 2t2 + xmin, (5)

which increases sigmoidally from xmin to xmax when t changes from from 0 to T

(Figure 9b). Both functions were employed in a developmental model of Fraxinuspennsylvanica shoots and leaves described in [139] (Figure 10). Logistic functions,presumed to have a more sound biological justification, were chosen to approximategrowth functions on the basis of measured data (the expansion of rachis segments andleaflets). The coefficients were estimated using regression analysis. In contrast, thecubic function was applied where detailed data were not available (specifically, to sim-ulate the gradual increase of branching angles over time). In this case, function pa-rameters were manipulated interactively to match the developmental patterns observedin the field and recorded on photographs of developing shoots. The straightforward

16

Figure 10: Simulation of the expansion of a Fraxinus pennsylvanica shoot over thecourse of 35 days. Adapted from [139].

interpretation of function parameters and their link with interactive computer graphicstechniques for curve specification [47, Chapter 11.2] facilitated this manipulation.

The logistic function does not have enough parameters to allow for flexible fitting toempirical data. Several other growth curves, such as the monomolecular function andthe Gompertz function, are limited in the same way [149]. To overcome this limitation,Richards [149] proposed a family of growth functions

x(t) = (A1−m− βe−kt)

1

1−m (6)

with four parameters A, m, β, and k. Although polynomial functions of the third orhigher degree may provide comparable or better approximations [172], the Richardsfunction has been widely used in modeling practice. For example, Berghage and Heinsapplied it to model the elongation of internodes in poinsettia [13], Lieth and Carpenterto model stem elongation and leaf unfolding in Easter lily [98], and Larsen and Liethto model shoot elongation in chrysanthemum [95].

5 Simulation of plant development: causal models

5.1 Information flow in growing plantsCommunication between modules plays a crucial role in the control of developmentalprocesses in plants. Lindenmayer distinguished two forms of communication: lin-eage (also called cellular descent), which represents information transfer from a parent

17

module to its children, and interaction, which represents information transfer betweencoexisting modules [101, 104]. In the latter case, the information exchange may beendogenous (between adjacent modules of the structure, as defined by its topology), orexogenous (through the space embedding the structure) [11, 124]. The flow of water,hormones, or nutrients through the vascular system of a plant are examples of endoge-nous information transfer, whereas the shading of lower branches by upper ones is aform of exogenous transfer. We will use this classification to organize our discussionof causal models.

5.2 Development controlled by lineageBy definition, development is controlled by lineage if the fate of each module is de-termined by its own history [100]. Within the formalism of L-systems, developmentcontrolled by lineage is expressed using the formalism of context-free L-systems (OL-systems) [102] and their parametric and differential extension, which were outlined inSection 4. It is difficult to formulate a clear dividing line between empirical modelsand causal models controlled by lineage; the key criterion appears to be the degree towhich the model can be justified in terms of fundamental mechanisms of development.For example, we may regard the leaf model given in Equation 1 as a causal model con-trolled by lineage, if we view production p1 as a manifestation of the cyclic nature ofthe activities of the apex.

Progress of time is the force that drives models controlled by lineage; consequently,time-related variables, such as delays and and rates of growth, play an important role.For example, in the model of Lychnis coronaria shown in Figure 3 one lateral branchalways develops ahead of the other branch supported at the same branching point [137,136], which explains the asymmetry observed in the plant structure [152].

A key question that has to be asked when constructing a causal model is whetherthe postulated control mechanism is powerful enough to simulate an observed devel-opmental sequence. The theory of L-systems offers many results that characterizedifferent classes of control mechanisms [77, 106]. Although most results are ratherabstract, some do apply to practical model construction. For example, Frijters and Lin-denmayer observed that acrotonic structures are difficult to obtain in models controlledby lineage [53] (see also [135]). A similar difficulty occurs when modeling basipetalflowering sequences [87, 105]. Simple models of these phenomena can be obtained,however, assuming endogenous control mechanisms.

5.3 Development controlled by endogenous mechanismsEndogenous control mechanisms rely on the flow of control information through thestructure of the growing plant. The information exchanged between the modules mayrepresent discrete signals (for example, the presence of a hormone triggering the trans-formation of a bud to a flower), or quantifiable values (for example, the concentrationof photosynthates produced by leaves). Endogenous information flow can be conve-niently captured using the formalism of context-sensitive L-systems. In the context-

18

I

J J

J

context

predecessor successor

Figure 11: Development of an inflorescence controlled by an acropetal signal. Theinsert shows the context-sensitive production that propagates the signal acropetally.Adapted from [132].

sensitive case, a production can be applied to a particular module only if this modulehas some specific neighbors. For example, Figure 11 shows a hypothetical model of agrowing inflorescence, in which flowering is induced by an upward-moving (acropetal)signal. The context-sensitive production describing signal propagation states that if amodule I is not yet reached by a signal, and it is situated immediately above a moduleJ already reached by a signal, then I will be transformed into J in the next simulationstep. Another context-sensitive production transforms an apex reached by a signal intoa flowering bud, leading to a flower. As shown and analyzed by Janssen and Linden-mayer [87, 105] (see also [132, 136]), the flowering sequence in this model dependson the relationship between the plastochron of the main axis, plastochron of the lat-eral branches, and the respective propagation rates of the flower-inducing signal. Inparticular, the model is capable of generating basipetal flowering sequences.

In the above example, discrete information was transferred between the modulesof a developing structure. In nature, however, developmental processes are often con-trolled in a more modulated way, by the quantity of substances (resources) exchangedbetween the modules. An early developmental model of branching structures makinguse of quantitative information flow was proposed by Borchert and Honda [16]. Be-low we outline an extension of this model, which captures the partitioning of resourcesbetween the shoot and the root [129, 130].

Borchert and Honda postulated that the development of a branching structure iscontrolled by a flow or flux of substances, which propagate from the base of the struc-ture towards the apices and supply them with materials needed for growth. When theflux reaching an apex exceeds a predefined threshold value, the apex bifurcates andinitiates a lateral branch; otherwise it remains inactive. At branching points the flux isdistributed according to the types of the supported internodes (straight or lateral) and

19

7 7

17 17

38 38

73 73

134 134

7 7

6 17

7 26

17 29

43 38

3 5 7 9 11

Figure 12: Application of Borchert and Honda’s model to the simulation of a completeplant. The development of an undamaged plant (top row) is compared with the devel-opment of a plant with a damaged shoot (bottom row). The numbers of live apices inthe shoot and root are indicated above and below the ground level. The numbers at thebase of the figure indicate the number of completed developmental cycles. Adaptedfrom [130].

the numbers of apices in the corresponding branches.In the case shown in Figure 12, two structures representing the shoot and the root of

a plant are generated simultaneously. The flux penetrating the root at the beginning of adevelopmental cycle is assumed to be proportional to the number of apices in the shoot;reciprocally, the flux penetrating the shoot is proportional to the number of apices inthe root. These assumptions form a crude approximation of plant physiology, wherebythe photosynthates produced by the shoot fuel the development of the root, and waterand mineral compounds gathered by the root are required for the development of theshoot. The model also captures an increase of internode width over time, and a gradualassumption of the position of a straight segment by its sister lateral segment, after thestraight segment has been lost. The developmental sequence shown in the top row ofFigure 12 represents the development of an undamaged plant. The shoot and the rootdevelop in concert. The bottom row illustrates development affected by damage to theshoot. The removal of a shoot branch slows down the development of the root; on theother hand, the large size of the root, compared to the remaining shoot, fuels a fastre-growth of the shoot. Eventually, the plant is able to redress the balance between the

20

size of the shoot and the root.Applications of context-sensitive L-systems are not limited to the simulation of

endogenous information flow in plants. For example, the same formalism has beenused to simulate movements of an insect foraging on a plant [129, 130].

5.4 Development of plants interacting with their environmentThe incorporation of interactions between a plant and its environment is one of themost important issues in the domain of plant modeling [44, 155, 156]. Its solution isneeded to construct predictive models suitable for applications ranging from computer-assisted landscape and garden design to the determination of crop and lumber yields inagriculture and forestry.

Using characteristics of the information flow between a plant and its environment asthe classification key, we can distinguish three forms of interaction and the associatedmodels of plant-environment systems:

1. The plant is affected by global properties of the environment, such as day lengthcontrolling the initiation of flowering [52] and daily minimum and maximumtemperatures modulating the growth rate [69].

2. The plant is affected by local properties of the environment, such as the presenceof obstacles controlling the spread of grass [1] and directing the growth of treeroots [62], geometry of support for climbing plants [1, 61, 130], soil resistanceand temperature in various soil layers [35], and predefined geometry of surfacesto which plant branches are pruned [134].

3. The plant interacts with the environment in an information feedback loop, wherethe environment affects the plant and the plant reciprocally affects the environ-ment. Specific models capture:

• competition for space (including collision detection and access to light) be-tween segments of essentially two-dimensional schematic branching struc-tures [11, 25, 48, 49, 81, 88, 91];

• competition between root tips for nutrients and water transported in soil [23,97] (this mechanism is related to competition between growing branches ofcorals and sponges for nutrients diffusing in water [88]);

• competition for light between shoots of herbaceous plants [61] and branchesof trees [20, 21, 22, 33, 81, 89, 166].

Although some phenomena belong quite naturally to one of these groups, the classifi-cation of others may depend on the level of abstraction. For example, an approximatemodel may consider temperature as a global property of the environment, a more de-tailed one may express temperature locally as a function of distance from the ground,and a yet more detailed model may take into account the changes of temperature de-termined by the distribution of radiative energy between plant parts. Thus, the above

21

Plant Environment

Internal processes

Reception

Response

Internal processes

Reception

Response

Figure 13: Conceptual model of plant and environment treated as communicating con-current processes. From [115].

classification is useful primarily from the modeling perspective, since different tech-niques are required to capture phenomena in each class.

The first comprehensive approach, introducing a methodology for modeling plantsinteracting with the environment, was developed by Blaise [14]. Another approach,based on the notion of L-systems, has been proposed by Mech, Prusinkiewicz, andJames [115, 134]. Below we summarize this second approach.

As described by Hart [73], every environmentally controlled phenomenon can beconsidered as a chain of causally linked events. After a stimulus is perceived by theplant, information in some form is transported through the plant body (unless the site ofstimulus perception coincides with the site of response), and the plant reacts. This re-action reciprocally affects the environment, causing its modification that in turn affectsthe plant. For example, roots growing in the soil can absorb or extract water (dependingon the water concentration in their vicinity). This initiates a flow of water in the soiltowards the depleted areas, which in turn affects further growth of the roots [23, 55].

According to this description, the interaction of a plant with the environment canbe conceptualized as two concurrent processes that communicate with each other, thusforming a feedback loop of information flow (Figure 13). The plant process performsthe following functions:

• reception of information about the environment in the form of scalar or vectorvalues representing the stimuli perceived by specific organs;

• transport and processing of information inside the plant;

• generation of response in the form of growth changes (e.g. development of newbranches) and direct output of information to the environment (e.g. uptake andexcretion of substances by a root tip).

Similarly, the environmental process includes mechanisms for the:

• perception of the plant’s actions;

• simulation of internal processes in the environment (e.g. the diffusion of sub-stances or propagation of light);

22

Figure 14: A two-dimensional model of a root interacting with water in soil. Back-ground colors represent concentrations of water diffusing in soil. From [115].

• presentation of the modified environment in a form perceivable by the plant.

For the purposes of simulation the environment is represented as a scalar or vectorfield. Modules of a growing plant can test values of this field at points of interest,and send values that affect the field at specific locations. Sample models constructedaccording to this scheme are shown below. For their full description see [115].

Figure 14 shows a two-dimensional model of a root seeking water in the soil duringits development. The initial water distribution formed an S-shaped zone of high waterconcentration. The growing tips of the main root and rootlets absorb water that diffusesin the soil. The decreased water concentration is indicated by dark areas surroundingthe root system. In the areas with insufficient water concentration the rootlets cease togrow before they have reached their potential length.

Figure 15 shows a three-dimensional extension of this model based on the work ofClausnitzer and Hopmans [23]. Water concentration is visualized by a semi-transparentiso-surface surrounding the roots. As a result of competition for water, the main rootsgrow away from each other. This behavior is an emergent property of the model.

Figure 16 shows a model of a horse chestnut tree inspired primarily by the work ofTakenaka [166]. The branches compete for light from the sky hemisphere. Clusters ofleaves cast shadows on branches further down. An apex in shade does not produce newbranches. Products of photosynthesis are transported from the leaves towards the baseof the tree. If the amount of photosynthates reaching the base of a branch is below athreshold value, the branch is considered a liability and is shed from the tree. Thus, thedistribution of branches in the crown is controlled by their competition for light.

Figure 17 further illustrates the impact of competition for light on tree growth. Thesimulation reveals essential differences between the shape of the crown in the middleof a stand, at the edge, and at the corner. In particular, the tree in the middle retains

23

Figure 15: A three-dimensional extension of the root model. Water concentration isvisualized by semi-transparent iso-surfaces [176] surrounding the roots. As a result ofcompetition for water, the roots grow away from each other. The divergence betweentheir main axes depends on the spread of the rootlets, which grow faster on the left thenon the right. From [115].

only the upper part of its crown. Simulations of this type may assist in choosing anoptimum distance for planting trees, where self-pruning is maximized (reducing knotsin the wood and the amount of cleaning that trees require before transport), yet spacebetween the trees is sufficient to allow for unimpeded growth of trunks in height anddiameter.

The next example illustrates a plant’s reaction to the quality, rather than quantity,of the incoming light. The simulation reproduces an experiment with the recumbentplant Portulaca oleracea described by Novoplansky, Cohen, and Sachs [116]. A plantwas surrounded with a plastic rim, one half of which was gray, and the other green.The green half changed the red / far red ratio in the spectral composition of transmittedand reflected light in a manner similar to real plants. The plant surrounded by the rimavoided growing in the direction of the green half. A simulated Portulaca seedling,which developed in a manner consistent with this observation, is shown in Figure 18.The distribution of the radiative energy was calculated using a Monte Carlo method [19,60], extended to account for the energy of specific wave lengths in the light spectrum.

The general framework for simulating interactions between plants and their envi-ronment depicted in Figure 13 may also be applied in cases where the developmentof a plant is affected by the environment, but the reciprocal flow of information fromthe plant to the environment can be neglected. A simple example of such a situation is

24

Figure 16: A tree model with branches competing for access to light, shown withoutthe leaves. From [115].

illustrated in Figure 19, which shows the growth of roots around mechanical obstacles(rocks) in the soil. The root model used in this simulation is based on the work ofDiggle [35, 36].

The response of trees to pruning is another important phenomenon in which thedevelopment of a plant is affected by external factors [134]. As described, for example,by Halle et al. [66, Chapter 4] and Bell [10, page 298], during the normal developmentof a tree many buds remain dormant and do not produce new branches. These budsmay be subsequently activated by the removal of leading buds from the branch system,which results in an environmentally-adjusted tree architecture. The model depictedschematically in Figure 20 represents the extreme case of this process, where buds areactivated only as a result of pruning. The developing structure is confined to a square,and the apices test whether they are within or outside this area. During the initial phaseof development the apex of the main axis creates a sequence of internodes and dormantbuds. After crossing the bounding square the apex is pruned and a basipetal signal is

25

Figure 17: Relationship between tree form and its position in a stand. From [115].

Figure 18: A model of Portulaca sensitive to the red / far red ratio. The left half of therim is green, the right one is gray.

26

Figure 19: A model of roots growing around the obstacles.

sent to activate the nearest dormant bud. The activated bud initiates a lateral branch,which grows in the same manner as the initial structure (traumatic reiteration). Aftercrossing the bounding square, the apex of the reiterated branch is also pruned, and thebud-activating signal is generated again. The final structure results from the repetitionof this process.

A three-dimensional extension of the above model is shown in Figure 21. In thiscase, some of the newly created buds initiate new branches spontaneously, yielding atree structure. Pruning constrains the outline of the growing tree to a bounding box andactivates dormant buds, which increases the density of branches and leaves near thebox boundaries.

Figure 22 applies the same modeling principle to simulate the effect of pruninga tree to a more elaborate, spiral form. This form can be found in the Levens Hallgarden in England, laid out at the beginning of the 18th century, and considered themost famous topiary garden in the world [24, pages 52–57]. Figure 23 combines treespruned to a variety of shapes into a synthetic image of a larger part of the Levens Hallgarden. For other models of topiary trees see [134].

The scope of this survey does not allow for a detailed description of individualmodels, and a complete account of phenomena that have been captured, at various

27

Figure 20: A simple model of a tree’s response to pruning. Top row: simulation steps6,7,8, and 10; middle row: steps 12, 13, 14, and 17; bottom row: steps 20, 40, 75, and94. Small black dots indicate dormant buds, the circles indicate the position of signalS. From [134].

levels of accuracy, by causal models developed to date. It is important to realize, how-ever, that the technology for creating complex mechanistic models already exists andcan be applied to capture a variety of phenomena of relevance to botany, agriculture,horticulture, and forestry.

6 Applications of architectural plant modelsThe modeling of plant architecture is an active research area. In horticulture, someempirical models have already achieved the predictive value needed in practical appli-cations. Examples include:

• A model of flowering rose shoot development as a function of air tempera-ture [117], intended to predict the timing of harvest in commercial greenhouserose production.

• Models of stem elongation, leaf unfolding [98] and flower bud elongation [46]in Easter lily; these models are intended to precisely control the flowering dateby manipulating greenhouse air temperature.

• Models of stem elongation in poinsettia as a function of temperature [13] andphotoperiod treatments (manipulation of the short-day date) [45]; these modelsare intended to use various treatments as a means for controling plant height.

28

Figure 21: Simulation of tree response to pruning. From [134].

29

Figure 22: Trees pruned to a spiral shape. From [134].

• Models of shoot elongation retardation in chrysanthemum caused by the appli-cation of daminozide [95, 99]; these models are intended to predict the finalshoot length reduction resulting from single or multiple daminozide applicationat various dates.

The above models capture selected topological and geometrical features of the modeledplants. They have not been accompanied by realistic visualization of the modeledplants, which would require collection of more comprehensive data.

The architectural modeling of entire plants is at the point where the methodologyof model construction is relatively well understood, well calibrated empirical modelsof selected plants are being developed, and mechanistic models are subject of activeresearch. In this context, Room et al. listed the following prospective application areasof the architectural plant models (cited with minor modifications from [154]):

Horticulture: Identification of horticultural treatments (pruning and pinching, tem-perature and day length manipulation, application of chemicals, etc.) aimed atthe optimization of plant size, shape, quality, and timing of flower production.

Agronomy: Exploration of competition for space and resources at the level of singleshoots, roots, and individual plants, both intraspecific and interspecific (relevantto intercropping, sowing rates, and control of weeds).

Forestry: Identification of optimal strategies for pruning and spacing trees.

Landscape architecture: Simulations of interactions between trees and structures;pruning strategies to minimize tree contact with power lines; interplanting forcontinuous flower displays.

30

Figure 23: A model of the topiary garden at Levens Hall, England. From [130].

31

Management of pathogens: Improved understanding of disease dynamics through sim-ulation of pathogen deposition and growth in the microclimates produced by de-veloping plants.

Biological control of weeds: Identification of combinations of weed architecture andtypes of damage caused by herbivores or pathogens which interact particularlyeffectively to limit weed populations.

Management of pests: Improved definition of action thresholds through simulationof interactions between plant architecture, pesticide deposition, insect movementand feeding, and compensatory growth of plants.

Grazing: Identification of optimal strategies for the timing and intensity of grazing tomaximize compensatory growth of pasture plants.

Plant breeding and genetic engineering Specification of target “designer plants” byidentification of architectures optimal for interception of light, harvestability,damage compensation, aesthetic appeal, etc.

Remote sensing: Improved interpretation of images through exploration of effects ofarchitecture and leaf arrangement on reflectance.

Developmental biology: Exploration of hypotheses relating physiology of plant de-velopment and information in genes to integrated 3D structures.

Paleobotany: Realistic modeling of extinct plants for research and educational pur-poses

Entomology: Improved understanding of insect behavior through simulation of insectmovement and feeding on growing plants.

Education: Use of plant models to illustrate and explore developmental processes inplants and the relationships between plants and their environment.

Entertainment: Use of plant models as components of games and films.

Art: Exploration of the aesthetics of growth forms, realistic and imaginary.

While the computer science techniques involved in the specification and visualiza-tion of the models seem to be relatively mature, the development of well calibratedempirical models of specific plants remains a labor-intensive task, and construction offaithful mechanistic models is a current research problem. Fortunately, the value of ar-chitectural models is not limited to support of decision making processes, and extendsto the process of model construction itself. This point of view was clearly stated byBell [11]:

32

The very process of constructing computer simulations to reproduce aparticular branching structure can be a useful experience in its own right,even without proceeding to the use of such a simulation to test an hypoth-esis. Either the morphology of the organism must be recorded in consid-erable detail or the underlying features of its developmental architecturefully appreciated ... Shortcomings of the model will soon become appar-ent as ‘mistakes’ which are readily identifiable qualitatively but are notalways easy to quantify.

7 Concluding remarksIn contrast to crop models, which describe plants in global terms such as biomass, yield,and number of flowers and fruits, architectural plant models attempt to capture spatialarrangement of plant components and their development over time. Recent progress inthe methodology of model construction forms the base on which well calibrated modelsof specific plants have begun to be built. Their availability will allow for the realizationof the anticipated practical applications of architectural models.

This survey has been focused on the modeling of branching architecture of plants.Quantitative characteristics of plant organs can be described using growth functions ina manner similar to plant architecture (Section 4.5). Examples include leaf area [113,114] and fruit size [54]. These characteristics, however, do not suffice to reproducedetails of organ shape and development. Consequently, visual representations of plantorgans have been created with standard (interactive) modeling techniques used in com-puter graphics [34, 131, 136, 139]). A biologically better motivated approach, based onthe distribution of growth rates in the growing surface or volume is a topic of currentresearch [37].

On the conceptual plane, the relationship between L-systems and process-basedmodels deserves a more detailed study than has been possible within the limits of thissurvey (Section 4.2). In particular, the incorporation of complex state transitions de-scribed by Bell [12] into developmental plant models may offer a framework that willfacilitate construction of causal models.

8 AcknowledgmentsThe seminal work of the late Professor Aristid Lindenmayer lies at the origin of the pre-sented perspective of spatial plant modeling. The insightful comments by ChristopheGodin, Jim Hanan, Chris Prusinkiewicz, and the anonymous referees were incorporatedinto the final form of this paper. I have also benefited from discussions with BrunoAndrieu, Jean-Francois Barczi, Frederic Blaise, Paul Boissard, Campbell Davidson,Claude Edelin, Mark Hammel, Winfried Kurth, Radomır Mech, Philippe de Reffye,Bill Remphrey, and Peter Room. The support by research, equipment, and infrastruc-ture grants from the Natural Sciences and Engineering Research Council of Canada,

33

and by a Killam Resident Fellowship is gratefully acknowledged.

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