Annals of Botany 114: 813– 827, 2014 doi:10.1093/aob/mcu155, available online at www.aob.oxfordjournals.org

PART OF A SPECIAL ISSUE ON FUNCTIONAL –STRUCTURAL PLANT MODELLING

An approach to multiscale modelling with graph grammars Yongzhi Ong*, Katarı´na Streit, Michael Henke and Winfried Kurth Department of Ecoinformatics, Biometrics and Forest Growth, University of Go¨ttingen, Bu¨sgenweg 4, D-37077 Go¨ttingen, Germany * For correspondence. E-mail [email protected]

† Background and Aims Functional–structural plant models (FSPMs) simulate biological processes at different spatial scales. Methods exist for multiscale data representation and modification, but the advantages of using multiple scales in the dynamic aspects of FSPMs remain unclear. Results from multiscale models in various other areas of science that share fundamental modelling issues with FSPMs suggest that potential advantages do exist, and this study therefore aims to introduce an approach to multiscale modelling in FSPMs. † Methods A three-part graph data structure and grammar is revisited, and presented with a conceptual framework for multiscale modelling. The framework is used for identifying roles, categorizing and describing scale-to-scale interactions, thus allowing alternative approaches to model development as opposed to correlation-based modelling at a single scale. Reverse information flow (from macro- to micro-scale) is catered for in the framework. The methods are implemented within the programming language XL. † Key Results Three example models are implemented using the proposed multiscale graph model and framework. The first illustrates the fundamental usage of the graph data structure and grammar, the second uses probabilistic modelling for organs at the fine scale in order to derive crown growth, and the third combines multiscale plant topology with ozone trends and metabolic network simulations in order to model juvenile beech stands under exposure to a toxic trace gas. † Conclusions The graph data structure supports data representation and grammar operations at multiple scales. The results demonstrate that multiscale modelling is a viable method in FSPM and an alternative to correlation-based modelling. Advantages and disadvantages of multiscale modelling are illustrated by comparisons with singlescale implementations, leading to motivations for further research in sensitivity analysis and run-time efficiency for these models. Key words: Functional–structural plant modelling, FSPM, L-system, multiscale, graph grammar, relational growth grammar, XL, GroIMP, Fagus sylvatica, beech, metabolic network, ozone, Monte Carlo, crown growth, shikimate pathway.

IN T RO DU C T IO N Multiscale modelling attempts to integrate structures and dynamics at different spatial and temporal scales. There are various reasons for adopting such an approach, one of which is the existence of scientific data at multiple scales. In the context of functional – structural plant models (FSPMs), Godin and Caraglio (1998) have introduced the representation of individual plant structures at different scales and created the corresponding Multiscale Tree Graph (MTG) encoding format. Multiscale formulation of problems is addressed in various domains dealing with the modelling of complex biological systems. Seidl et al. (2012) presented a hierarchical multiscale framework in ecology that integrates functional, structural and spatial processes and where ecosystems at landscape level are built from the scale of individual trees. Generation of plant ecosystems, albeit for image synthesis purposes, using a multiscale approach, was addressed by Deussen et al. (1998) and Lane and Prusinkiewicz (2002). Boudon et al. (2003) demonstrated with examples of bonsai trees how to generate tree models through step by step down-scaling of a global tree envelope. Da Silva et al. (2008) presented a framework for efficient modelling of light interception of individual trees at different scales.

Multiscale modelling was also considered in models dealing with specific plant organs, such as apple fruits (Band et al., 2012) or roots (Ho et al., 2011), modelled by coupling a hierarchy of scales, starting from cellular or even sub-cellular levels, respectively. In systems biology, multiscale methods are of great importance to study and integrate biological functions at several temporal and spatial scales (Dada and Mendes, 2011). Recent FSPMs try to integrate several scales in models as well. Courne`de et al. (2008) have combined a competition model based on point patterns in a plane with an individual-tree FSPM to simulate three-dimensional (3-D) forest structures. By taking into account individual 3-D tree structures and competition in a mixed stand, Lintunen et al. (2011) have successfully simulated the crown structures of trees. Buck-Sorlin et al. (2005) and Han et al. (2010) have created models with genetic and hormonal processes integrated into FSPM. There exist other examples of research in FSPM involving multiscale data that cannot be listed exhaustively (cf. Sieva¨nen et al., 2013). Data representation alone is not the only reason to adopt a multiscale approach. A second motivation arises from the dynamic aspects of modelling. Considering the hierarchy of physical models (ranging from continuum models to quantum mechanics), the interest of a modeller usually resides in a predominant scale.

# The Author 2014. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved. For Permissions, please email: [email protected]

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Received: 24 March 2014 Returned for revision: 11 June 2014 Accepted: 16 June 2014 Published electronically: 17 August 2014

814

Ong et al. — An approach to multiscale modelling with graph grammars graph formalisms to simulate multiscale structures. Example 2 shows ‘bottom-up’ coarse-scale dependency on fine scale; and example 3 shows an extensive range of scales with nested dependencies. Finally, the Discussion revisits the original motivations to raise advantages, problems, comparisons and potential future work. METHODS Multiscale graph model and grammar

A suitable data structure is important for multiscale modelling. On one hand, the data structure and its operations should take into account compliance with existing multiscale data, such as MTG-encoded plant structures. On the other hand, the restrictions imposed by them on the modelling approach should be minimal. The primary restriction a data structure imposes on the modelling approach is the type of refinement ordering it supports. We briefly describe two types of refinement orderings. Conventionally, the notion of plants as modular organisms (Harper et al., 1986) has been used as a reference to relate and construct orderings for the modularities of plants (Godin and Caraglio, 1998), i.e. for the types of repeatedly occurring morphological entities. Orderings represent decomposition relationships between modularities. In the context of this paper, the terms refinement and decomposition are used as synonyms. A set of pairwise intercomparable modularities yields a linear, i.e. strict, ordering (Fig. 1A). When a set of modularities has no decomposition relationships among its elements and is regarded as representing the plant’s constituents at a certain spatial resolution, we call it a scale. A multiscale model can potentially transcend the scope of a single plant. For example, the focus of a modeller may be fixated on a population of plants without disregarding structures of individual plants or even molecular processes. In such scenarios, the reference to incomparable modularities in individual plants, as perceived from the population scale, necessitates the aggregation of finer modularities into common coarse representations (Fig. 1B). The dissolution of coarse modularities to common fine representations (Fig. 1C) may occur when microscopic models are taken into account. Furthermore, arbitrary measurements or data in 3-D space compartments (e.g. layers or boxes) that cannot be classified as plant modularities can thus be easily included in the refinement ordering as properties referring to specific scales. The distinction between scales and modularities is intentional to highlight scales as sets of modularities as well as the transcendence beyond the scope of single organisms. If incomparable scales exist, a linear ordering is unsuitable for representation. In such cases, a generalized multiscale graph (Godin and Caraglio, 1998) or a structure-of-scales (Ong and Kurth, 2012) can represent the scales as a partially ordered set. Such refinement orderings (as seen in Fig. 2A) are potentially useful for models with an extensive range of scales. The structure-of-scales is one of the three algebraic structures proposed by Ong and Kurth (2012) that form a three-part graphbased data structure (Fig. 2). The second structure is the multiscale type graph with refinement (hereafter referred to as type graph for simplicity); and the third is the multiscale typed graph (hereafter referred to as instanced graph for simplicity). The rest of this sub-section presents a summary of this data structure to provide a bridge between its formal definitions and the modelling framework proposed in the next sub-section.

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

If the scale of interest is macroscopic, the effects of inputs to the microscopic model are usually modelled by some constitutive correlations at the macro-scale because representative models at the micro-scale often pose computational or analytical problems (E, 2011). These correlations are usually obtained empirically. However, a correlation-based model is soon loaded with parameters with obscure meaning as it becomes increasingly complex with more microscopic inputs. For example, consider constructing a model for the structural growth of trees under the effects of an atmospheric component. One possible way could be an experimental observation of structural growth in relation to gas concentrations, followed by a construction of a mathematical model after data interpretation. It would be computationally impractical to model particular responsive metabolic networks in each cell of the trees. Despite their success in many applications, the extension of such correlation-based approaches to complex scenarios has proven to be difficult, often requiring complicated mathematical functions. E (2011) illustrated this argument with extensions of the Navier– Stokes equation (which commonly uses an empirically obtained stress tensor parameter) for complex fluids such as polymeric fluids (Bird et al., 1987). The quantum mechanics –molecular mechanics (QM-MM) model of chemical reactions (Warshel and Levitt, 1976) and the first-principle-based molecular dynamics (Car and Parrinello, 1985) are two examples of successful multiscale applications that overcame these difficulties. Subsequently, several general frameworks for multiscale modelling in mathematical physics such as the heterogeneous multiscale method (HMM) by E and Engquist (2003) have been developed. These frameworks from domains of science relatively distant from FSPM nevertheless offer a concise overview of multiscale concepts. They address a fundamental difficulty in mathematical modelling inherent to correlation-based approaches. The L-system formalism introduced by Lindenmayer (1968) is used commonly for describing plant architectural development. Variants of this formalism have emerged as a result of increasingly complex requirements in FSPM. A non-exhaustive list includes stochastic L-systems, context-sensitive L-systems, parametric L-systems (Hanan, 1992), environmentally sensitive L-systems (Prusinkiewicz et al., 1997) and relational growth grammars (Kurth et al., 2005). As indicated by Hanan (2013), one of the current challenges in FSPM is the extension of formalisms to support multiscale modelling. Corresponding to the notion of decomposition in L-systems (Prusinkiewicz et al., 2001) and multiscale representation of plant structures, recent developments have been made by Boudon et al. (2012) to allow string-based grammar operations on MTG data structures and by Ong and Kurth (2012) to allow the operation of rules on multiscale typed graphs. The manipulation of multiscale structures addresses only the first motivation towards multiscale modelling. This study attempts to embed this technique into a more general approach to multiscale FSPM, utilizing established graph grammar concepts to organize interactions between multiple spatial scales in the model and thus avoiding complete correlation-based modelling. In the Methods section, a graph model and grammar based on previous work as well as a systematic framework is described. These methods address both motivations of multiscale modelling. In the subsequent Results, three examples illustrate the methods. Example 1 is a concise demonstration of the extended

Ong et al. — An approach to multiscale modelling with graph grammars A

B

C

Plant

Population

Axis

Plant

Organ

Voxel

Tissue

Crown and trunk

Growth unit Internode

815

Voxel

Metabolic species

Modularity

Coarse to fine refinement

F I G . 1. Refinement orderings: (A) linearordering with intercomparable modularities; (B) fine modularities with common encoarsements; and (C) coarse modularities with common refinement.

A

Individuals

B

I

Tree

I Root

Axis A

Axes A Voxel space

O

Voxel S

S

Growth units G

Growth unit G

Flower O

Organs O

Fruit O

Leaf

O

Internode O

Aerial

C

O G

A

O G

I

Roots

A O

O

O

G

O

G

O

A O

O

A G

O

O

S

O

G

O

S

O

S

Scale: I Individuals

A Axes

G Growth units

S Voxel space

O Organs

F I G . 2. An illustration of the three-part graph data structure (Ong and Kurth, 2012) comprising a structure-of-scales (A), a type graph (B) and an instanced graph (C). The structure-of-scales (A) shows incomparability between the axes and the voxel space, but both refine to a common organ scale. The type graph (B) shows the node types of each scale and the edge connections allowed. Here, the dashed arrows at the bottom indicate that internodes can be connected by successor or branching edges to flowers, fruits and leaves (and, implicitly, also to internodes). The instanced graph (C) shows topology for the roots of a plant and the aerial organs.

The structure-of-scales (Fig. 2A) is a finite partially ordered set (S, ≤), i.e. a generalization of the type of refinement ordering underlying an MTG or a generalized multiscale graph. It establishes the scales as well as an ordering for their refinement relationships with each other in an abstract representation. In this

structure, one scale is a refinement of, an encoarsement of or incomparable with another scale. Graph re-writing and rule-based operations are generally defined for types rather than scales. For example, one scale may have leaf and internode types, while another, coarser scale

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Radial crown section

Axis

816

Ong et al. — An approach to multiscale modelling with graph grammars The problem categorization component adopts the two categories of multiscale problems introduced by E (2011). Both problem categories are characterized by significant disparities of simulation output from expected output. One category (hereafter referred to as ‘local’) is characterized by disparities at localized regions in the domain of the coarse scale. The fine scale is employed to resolve the disparities occurring at these regions. As a hypothetical example, an FSPM of tree growth at the organ scale may be integrated with a micro-scale biophysical model of xylem vessels to predict sudden embolisms that can cause a catastrophic dysfunction of the water supply system in a branch or in a whole crown part (cf. Cochard and Tyree, 1990). The second category (hereafter referred to as ‘global’) of problems requires the fine scale to overcome disparities throughout the domain of the coarse scale. For example, the production of new internodes and buds may be an extrapolation of meristem cell differentiation throughout the vegetation period. This form of categorization is helpful for identifying and describing the aim of a multiscale model. Contrasting with the problem types by E (2011), our framework does not mandate the occurrence of disparities only in the coarse scale, i.e. disparities in the fine-scale domain can also be resolved with feedback from coarse-scale models. The notion of scale dependency is closely related to problem categorization. The scale with disparities, either locally or throughout its entire domain, is said to be scale dependent on the scale providing information that alleviates the disparities. The scale that provides information to alleviate disparities is said to be scale independent. Scale integration serves as a template for scale-to-scale interactions during simulation. It comprises three steps between two comparable scales that are executed in the order of initialization, evolution and extrapolation (Fig. 3 shows a schematic representation of the steps). Suppose X is a scale dependent on scale Y. If Y is a finer scale than X, it is usually simulated at spatial and time scales much smaller in magnitude than the model in X. While correlation-based approaches seek to avoid simulation at scale Y due to computational or analytical impracticality, our approach attempts to perform a feasible simulation at scale Y and estimates the state of X using the results. In initialization, the state of X may

X

Initialization Y

Evolution

Y

Multiscale modelling framework

The three-part graph data structure offers only an infrastructure for modelling. To justify its utility, we propose a multiscale modelling framework inspired by the extended multi-grid method (Brandt, 2002), the equation-free approach (Kevrekidis et al., 2003) and the heterogeneous multiscale method (HMM; E and Engquist, 2003; E, 2011). This approach consists of three components, namely problem categorization, scale dependency and scale integration.

X Y

Extrapolation

F I G . 3. Scale integration steps – initialization, evolution and extrapolation. X is a dependent scale and Y is an independent scale. Dotted arrows are directed at scales with state modification (excluding preparatory modifications of X in initialization).

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

may have an axis type. In order to ensure a strict modellers’ control of the possible relationships between scales, we adopt a ‘typed’ approach from languages such as Java (in which types are called classes) and require a definition of all types and their potential relationships before the transformation rules are specified in the code. Thus, in order to utilize scale relationships in rules, types required in the model need to be specified and related to their respective scales in a type graph (Fig. 2B). Mathematically, a type graph over a structure-of-scales (S, ≤) is a labelled, directed graph with node set N (containing the types), label sets LN and LE, edge set E # N × LE × N, node labelling function fN : N  LN, scale function fS : N  S, and with a distinguished edge label r [ LE (‘refinement’) which fulfils (u, r, v) [ E ⇒ fS(u) . fS(v), i.e. which lifts the partial ordering from the scalesto the types. One can thus say that the refinement relationships between the types are controlled or restricted by the structureof-scales. [Some additional constraints are demanded for type graphs; see Ong and Kurth (2012) for a complete definition.] A type graph may contain circles (undirected closed paths in the sense of graph theory). The line diagram of the structure-of-scales is a homomorphic image of the type graph, with the underlying homomorphism preserving the scales (for definitions of algebraic notions in connection with partially ordered sets and graphs, see, for example, Ganter, 2005; Knauer, 2011). Each type is represented by a node in the type graph. Refinement relationships between types are represented by edges labelled by the unique refinement edge label. The appearance of other edges, such as successor and branching edges, between two nodes a and b in a type graph means that edges with these labels are basically allowed between instances of types a and b (not that they must always exist). Analogous to the creation of object instances from classes in the object-oriented programming paradigm, instances of types can be created as nodes in an instanced graph (Fig. 2C). The type graph is again a homomorphic image of this graph, with the underlying homomorphism preserving node types and edge labels (cf. Ong and Kurth, 2012). This means that the type graph controls or restricts the architecture of the instanced graph. If two types a and b are, for example, not connected by a successor edge in the type graph, such an edge is forbidden between instances of a and b. Once the type graph is given, less information is necessary for a full specification of an instanced graph. The syntax of our programming language will make use of this feature. The instanced graph is used for modelling virtual entities such as biological organisms, cells, forest stands, etc., which are represented as nodes, and their relationships as edges. This graph is the subject of rule-based transformations. Successor and branching edges in instanced graphs are interpreted in the same way as the conventional successor and branching relations in L-systems.

Ong et al. — An approach to multiscale modelling with graph grammars first undergo preparatory modifications. Subsequently, the state of Y is initialized with information from the observable state of the model, including the state of X. Evolution refers to the simulation of a model in Y following the initialized parameters. Finally, the state of X is modified based on the evolved state of Y in extrapolation.

A Tree T

B

817 Tree

T

Axis A

Axis A

Bud Organ O

Internode

O

O

The programming language XL

C Tree T Axis A

Bud O Scale: T

Tree

Refinement

R E S U LT S In this section, we first present an example that illustrates the use of rules to generate multiscale structures with the three-part graph model and grammar. The next two examples demonstrate different applications of the multiscale framework. We begin each example by describing first the static structure-of-scales and type graph specifications. The dynamic aspects are then recounted first with simulation initialization, followed by an overview of processes within each simulation step. Lastly, individual processes are conveyed along with key rule-based operations. The applications of problem categorization, scale dependency and scale interaction in examples 2 and 3 are highlighted. Example 1: specifying and generating a multiscale plant structure

A structure-of-scales (Fig. 4A) is made to represent the tree, axis and organ scales. It is not explicitly declared in source code, but can be derived from the type graph. The model’s type graph (Fig. 4B) consists of types Tree, Axis, Internode and Bud. The Internode and Bud types belong to the organ scale, while the Axis and Tree types belong to the axis and tree scales, respectively. The types in the organ scale are pairwise interconnected by branching and successor edges to allow these relationships between them. The type graph is constructed in XL by: ¼ ¼. . ^ /. TypeRoot /. Tree /. Axis /. {# Internode Bud #}; Because nodes at the organ scale (i.e. Internode, Bud) can be connected to one another by successor or branching edges, a clique (complete graph; syntax {# . . . #}) is established for these node types in the type graph such that the pair is completely (in graph terminology) interconnected by branching and successor edges (cf. Fig. 4B). To initialize simulation, the model’s instanced graph is created with a Tree node refined to an Axis node that is further refined to a Bud node (Fig. 4C): Axiom ¼ ¼. Tree /. Axis /. Bud;

A

Axis Successor

O Organ Branching

F I G . 4. An illustration of the structure-of-scales (A), type graph (B) and initial instanced graph (C) for modelling a multiscale plant structure.

Each simulation step consists of parallel applications of a rule that replaces a bud with a new internode, lateral axis, lateral bud and apical bud: Bud ¼ ¼ . Internode [Axis Bud] Bud; Axis is the only node that belongs to a coarser scale in this rule. Without it, the rule appears like classical (single-scale) L-system rules. When a bud in the instanced graph is matched to the left-hand side of the rule, the query recognizes its axis and tree encoarsements at the same time, following the relationships specified in the type graph. Subsequently, organ scale nodes produced by the right-hand side of the rule, except for the lateral axis and bud, are automatically refined from the same encoarsements. For the new lateral Axis specified in the right-hand-side production statement, the framework establishes a branching edge from the existing parent axis node and a refinement edge from the tree node automatically. The new lateral Bud branches from the newly produced internode and refines from the new lateral Axis. Figure 5 illustrates the modification of the initial instanced graph after one application of this rule. A detailed account of grammar operations at multiple scales is given by Ong and Kurth (2012). The resultant data structure is a multiscale graph (in this case, also an MTG) representing the plant topology. Example 2: crown generation – coarse-scale dependency on fine scale

In this example, we demonstrate the dependency of an artificial light-sensitive tree on fine-scale organ developments. Light sources are directed at a growing tree from equal distances – vertically above, diagonally around from an elevated height and horizontally around from ground level. The model is catered to produce coarse-scale outputs such as tree height and crown dimensions (e.g. radii in different directions).

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

The three-part graph data structure and multiscale grammar operations are implemented as part of the programming language XL (Kniemeyer, 2008). XL is a multi-paradigm language with syntax designed for both object-oriented and rule-based programming. In addition, our work takes advantage of the existing implementations of geometrical classes and visualization functionalities coupled with XL in the open-source software GroIMP (www.grogra.de; Kniemeyer et al., 2007). XL allows queries in the global instanced graph that represents the current status of the modelled scene. See Supplementary Data Table S1 for a selection of XL code syntax and descriptions used herein.

818

Ong et al. — An approach to multiscale modelling with graph grammars

Graph before rewrite

TA B L E 1. Multiscale problem categorization, scale dependency and scale integration for example 2 (crown generation).

Graph after rewrite

T

T

Problem category

A

A

Dependency

A

Integration

O

O

O

Evolution

O

Internode

Extrapolation

[ Axis bud ]

Bud ;

Scale: T

A

Tree

O Organ

Axis

Refinement

Successor

Branching

F I G . 5. An illustration of the initial instanced graph modified by an XL rule. The initial Bud node is removed and replaced by a new internode, lateral axis, lateral bud and apical bud.

A

B

Tree

Trunk

CrownLayer

T

T

T Foliage

Organ O

Bud O

O Branch

O

Marker O

CrownLayer

C T CrownLayer

T

CrownLayer T

T

CrownLayer

T Trunk

Scale: T

Tree Refinement

O Organ Successor/Branching

F I G . 6. An illustration of the structure-of-scales (A), type graph (B) and initial instanced graph (C) for modelling the growth of four crown layers based on the development of a sample of organs.

These outputs are derived from organ developments extrapolated from finer to coarser spatial and time scales. The structure-of-scales in this example has a tree scale refined to an organ scale (Fig. 6A). Several types defined in the type graph (Fig. 6B) make up the model of the growing tree. At the tree scale, a CrownLayer type represents a vertical quartile of the tree crown and a Trunk type represents the growing tree trunk. At the organ scale, a Bud type represents buds, a Branch type represents branch segments, and a Foliage type emulates leaves to reduce light intensity. In addition, the organ scale has

a Marker type to mark initial bud sample positions in crown layers. The types at the organ scale are interconnected by branching and successor edges to allow these relationships between them. An instanced graph (Fig. 6C) is created with one node for the trunk and four nodes for the crown layers of the growing tree. Light sources are included as nodes in the graph but are excluded from figures for simplicity. The XL code for creating the type graph is similar to that in the previous example. Tree growth is modelled by a series of simulation steps. In each simulation step, crown growth is formulated as a multiscale problem where the tree scale is entirely scale dependent on the organ scale, i.e. crown dimensions are determined by the developments of light-sensitive buds and branches. An overview of the scale integration, in this case also a Monte Carlo method (Kalos and Whitlock, 2008) application, is shown in Table 1. In initialization, the age of the tree determines the number of buds. From the finite population of buds, a representative sample is created. A wide range of sampling methods can be used. In this example, simple random sampling with finite population correction (Lohr, 2009) is employed. The mean number of the buds forming the height and maximum radii in each direction of the crown layers is assumed to be 20 % of the bud population. The sample buds are independently distributed to each crown layer following proportions given as inputs to the model. Each sample bud is assigned a random position within its crown layer (Fig. 7A). The sampled and positioned buds are refined from their respective crown layers. For example, the XL code to establish the refinement is: cl [createBud(cl)]; where createBud(cl) is a method returning a new Marker graph node representing a single bud from the samples in the crown layer represented by cl. The two nodes are connected by a refinement edge, although createBud is contained within square brackets, because of the refinement relationship defined in the type graph. Spheres are placed in each crown layer to emulate leaves for self-shading within the crown (similar to the approach by Chiba et al., 1994). The volumes of these spheres are proportional to the volume of their respective crown layer. The rule-based code for refining foliage graph nodes from crown layer nodes is similar to the refinement of crown layers to buds. In evolution, a photon-tracing technique (Hemmerling et al., 2008) is used to obtain irradiance of each sampled bud from each light source. The final growth direction of branches from each bud is determined by a weighted sum of the direction

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

==>

Bud

Dependent Independent Initialization

Crown growth always based on bud and branch developments. Tree (crown layers) Organ (branches and buds) Bud positioning — Based on vertical distribution and crown dimensions, a representative number of buds are positioned. Branch growth — Rule-based branch elongations over short time scale. Crown estimation – Bounding boxes of branches scaled up and aggregated into new crown dimensions.

Ong et al. — An approach to multiscale modelling with graph grammars A

B

C

819

CrownLayer

CrownLayer

T

T Marker

Bud O

O Branch O Branch Branch O

O

O

O

O

O

Bud

Bud

Bud

Bud Scale: T Tree Refinement

O Organ Successor/Branching

F I G . 8. Modification of instanced graph at the organ scale. After two iterations, the bud produces three branches. The branches with paths from the same marker node are eventually used to create a bounding box at the organ scale.

F I G . 7. Illustration of crown generation using our multiscale modelling framework. Black spheres represent samples of buds (sizes intentionally increased for visibility). Green spheres represent random foliage. Green frustums represent crown layers. (A) Random positioning of bud samples and foliage spheres in initialization. (B) Up-scaled bounding boxes of fine organs shown in blue. (C) Result of extrapolation. Crown layers are updated to reflect new tree height and crown dimensions containing the bounding boxes in (B).

towards the light source providing maximum irradiance and a default direction for the crown layer given as model input [similar to the method by Palubicki et al. (2009) but tropism is accounted for in the default direction]. Rules generate branch segments for a fine-scale (i.e. short) time frame. Figure 8 illustrates an example of the graph modifications when rules are executed to create branch segments from buds. In extrapolation, minimal bounding boxes enclose branches originating from each Marker object. The dimensions of these bounding boxes are scaled up based on the ratio of the macro-scale simulation time to the micro-scale simulation time (Fig. 7B). Finally, existing crown dimensions and the up-scaled bounding boxes from each bud node are aggregated into new crown dimensions (Fig. 7C). Sample buds are discarded at the end and the three processes (initialization, evolution and extrapolation) repeat for each step. Example 3: Fagus sylvatica stands under ozone exposure

European beech (Fagus svlvatica) is one of the most important tree species in central Europe (Scalfi et al., 2004). Models and observations indicate increasing concentrations of tropospheric

ozone in Europe since 1996 (Denby et al., 2010). Tropospheric ozone triggers oxidative stress responses in the enzymes of the shikimate pathway (Betz et al., 2009) as well as in protein levels related to the Calvin cycle (Kerner et al., 2011) in beech trees. These responses lead to structural (e.g. leaf lesions) and functional (e.g. photosynthetic capacity) depreciations. A structure-of-scales (Fig. 9A) is designed to incorporate the effects of ozone on beech tree stands. The stand scale consists of ozone concentrations and light sources. The tree scale describes the positional information of trees in a stand and aggregated attributes of individual beech trees. It is further refined into two incomparable scales: the axis scale and the crown scale. The crown scale contains collective information of tree crowns and is refined to a crown layer scale. The layers are unique vertical height sections that divide the tree crown one-dimensionally. The axis scale is refined to the growth unit (GU) scale. The crown layer scale is additionally decomposed into an organ scale that is a refinement of the GU scale concurrently. Lastly, the crown layer scale refines to a metabolic network scale. Here we deploy the simplifying model assumption that the metabolic network dynamics do not significantly differ between organs of the same crown layer. The model’s type graph (Fig. 9B) illustrates the types and relationships utilized in each scale. The stand, tree, crown, crown layer, axis and GU scale each consists of only one representative node type. In a straightforward manner, they are the stand type, tree type, crown type, crown layer type, axis type and GU type, respectively. Leaf, bud, internode, root and marker types are the basic building organs of the modelled beech trees. The internode type acts in place of tree segments, while the marker type is used as a virtual and invisible marker in the topological structure for self-pruning. The root type is used for basal nodes for consolidation in the carbon transportation model. At the finest scale, the metabolic species types represent constituents of the shikimate

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

2 Ruleiterations

820

Ong et al. — An approach to multiscale modelling with graph grammars A

B

Stand S Tree

Axis

A

Stand

S

Tree T

T

C Crown CrownLayer L

Axis A

C Crown GU L CrownLayer

Organ O

M Metabolic network

G

Leaf O PEP H2O Bud O

DHQ NADPH

Internode O Root O

Shikimate S3P Chorismate

Marker O

DHQS SD EPSPS

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

E4P DAHP DHS H ATP EPSP DAHPS DHQD SK CS

Scale: S Stand

T Tree

A Axis

L CrownLayer

G Growth unit

Refinement (coarse to fine)

C Crown

O Organ

M Metabolic network

Successor/Branching

F I G . 9. An illustration of the structure-of-scales (A) and type graph (B) for modelling three beech stands under ozone exposure.

pathway each accounting for quantity. The node types at organ scale are interconnected by branching and successor edges to allow these relationships between the instantiated nodes of these types. (Here we permit more than needed, since in reality, for example, a leaf will not be succeeded by a bud. However, there is no necessity to include all possible restrictions in the type graph.) An identical interconnection by branching and successor edge types is also specified for the types at the metabolic network scale. Aside from their role in the type graph, each type is declared as a module (in terms of the programming language XL) (Kniemeyer, 2008) with attribute values that contribute to simulation. Simulation initialization begins with a procedure to create three stands with ozone AOT40 (accumulated ozone exposure of 40 parts per billion) of 10 000, 25 000 and 40 000 (mg m – 3) h – 1. For each stand, the refinements to seven individual beech trees are created each with a single axis, a GU, a bud and a root node. The crown decomposition of each tree is created using a crown node and a series of crown layer nodes. Every crown layer node is decomposed into the 20 species of the shikimate pathway, connected in a specific order closely resembling the reaction sequence. The type graph is specified using the following XL code: ^ /. TypeRoot /. Stand /. Tree [/. Crown /. cl:CrownLayer

/. {# PEP E4P . . . #} ] /. Axis /. GU /. {# Bud Internode Root l:Leaf Marker #}, cl /. l; The last line cl /. l establishes the refinement of crown layers (CrownLayer, cl) to leaves (Leaf, l) in addition to the refinement from growth units (GU) to leaves (Leaf, l). Figure 10 depicts a condensed instanced graph of the model upon initialization. Multiple steps simulate growth of the beech trees in the stands per year. Table 2 gives an overview of the steps in order. Of particular interest for our results are the application of (multiscale) grammar formalisms in step 10 and steps 2 – 4 that endorse the multiscale framework. Figure 11 summarizes the problem categories, scale dependencies and scale interactions for steps 2 – 4. A photon tracing technique following the implementation by Hemmerling et al. (2008) is performed to obtain irradiance for single leaves. The photosynthetic production for each leaf is multiplied by a normalized capacity factor obtained from the crown layer in which it resides. This interscale dependency is categorized as a multiscale problem requiring feedback from the macro-scale globally (i.e. scale feedback is required not only at the localized domain) and illustrated as Framework A in Fig. 11. The organ (leaf ) scale takes the role of dependent scale and the crown

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Growth G unit

Ong et al. — An approach to multiscale modelling with graph grammars Stand 1 S

Stand 2 S

*7 C Crown

Tree T

*50

A GU

Stand 3 S

*7

Axis G

L

O

O

Root

Bud

*7 C Crown

Tree T Axis

*50

A

CrownLayer *20 M Species

GU

821

G

L

Tree

O Bud

*50

A

CrownLayer *20

GU

M

O Root

C Crown

T

Axis

Species

L

G

O

O

Root

Bud

CrownLayer *20 M Species

S Stand

T Tree

A Axis

L CrownLayer

G Growth unit

Refinement (coarse to fine)

C Crown

O Organ

M Metabolic network

Successor/Branching

F I G . 10. An illustration of the initial instanced graph for modelling three beech stands under ozone exposure. Edges labelled with an asterisk (*) and number represent multiple connections to the specified number of distinct nodes.

TA B L E 2. Overview of the steps executed within one simulation step for modelling the three beech stands under ozone exposure No.

Step description

1 2* 3*

Light model, ray tracing and irradiance of leaves Mean irradiance in crown layers Photosynthetic depreciation in crown layers and metabolic network simulation Photosynthesis and carbon assimilation Transportation, allocation and distribution Secondary growth Primary growth (segment elongation) Branch fall and bending Update of refinement relationships between crown layers and leaves Aggregation of data in tree and stand nodes

4* 5 6 7 8 9 10*

Steps with an asterisk (*) indicate usage of the multiscale approach.

scale is the independent scale. Scale integration begins with initialization (Framework A) by computing the mean irradiance of each crown layer using the individual irradiance of leaves within: layer.lightMean ¼ mean((* layer Leaf *) .lightIntercepted); Notice that the query for Leaf follows layer without any edge specification due to the pre-defined refinement relationship in the type graph. This corresponds to step 2 of the simulation steps (see Table 2). In evolution (Framework A), maximum photosynthetic depreciation in each crown layer is estimated from the chorismate concentration. This dependency is again formulated as a multiscale problem requiring feedback from the metabolic network scale globally. It is an application of the multiscale framework nested within Framework A, illustrated by Framework B in Fig. 11. The dependent scale is the crown scale while the independent scale is the metabolic network scale. In initialization (Framework B), ozone AOT40 is used to determine the concentrations of enzymes raised by hypersensitivity to ozone. For example, the concentration for DAHPS (3-deoxy-D-arabino-

heptulosonate-7-phosphate synthase) is specified by the rule-based statement: s:Stand (/.)* CrownLayer [dahps:DAHPS][dhqd:DHQD] [sd:SD][epsps:EPSPS] ::. { dahps.con ¼ DAHPS.conMin + (s.ozone * DAHPS.conRange); . . .//other 3 enzymes } No specific edge specification between CrownLayer and the metabolic species is required due to the pre-defined refinement relationships in the type graph. The rest of the enzymes have their concentration values reset. In evolution (Framework B), the metabolic reactions are simulated. First, the rates of all reactions are determined. The concentrations are then updated using the computed rates. In extrapolation (Framework B), the chorismate concentration is expressed as a percentage of a maximum chorismate concentration. This percentage is interpreted as the maximum photosynthetic depreciation for the crown layer. Returning to extrapolation (Framework A), mean irradiance and maximum photosynthetic depreciation estimate the normalized capacity factor in the crown layer. The photosynthetic production of each leaf is reduced with a multiplication by the capacity factor of the crown layer it resides in. An example of the code is: cl:CrownLayer lf:Leaf ::. { lf.carbonAssimilated ¼ calculatePS(lf ); lf.carbonAssimilated *¼ cl.photosynCapacity;} where calculatePS computes the photosynthetic output from leaf lf and the operator *¼ multiplies the output by the normalized capacity factor ( photosynCapacity) from the crown layer. Carbon assimilation, transport and allocation are implemented at the organ scale with the topological structure of individual trees using the methods of Kang and de Reffye (2007) and Strobel (2004). Allocated carbon is used for secondary growth, i.e. incrementing the diameter attribute values of Internode nodes in the graph.

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Scale:

822

Ong et al. — An approach to multiscale modelling with graph grammars Problem category

Scale dependency

Photosynthetic capacity in leaves always based on depreciation factor in crown layer

Dependent

Independent

Organ (Leaf)

Crown (CrownLayer)

Initialization – Simulation Step: 2 Compute mean irradiance in crown layer from individual leaf irradiance values.

Maximum photosynthetic depreciation always based on chorismate concentration

Scale dependency Dependent

Independent

Crown (CrownLayer)

Metabolic network

Initialization Ozone concentration in stand determines enzyme concentrations.

Evolution Shikimate pathway simulation.

(Application of multiscale) Framework B

Problem category

Extrapolation Chorismate concentration estimates maximum photosynthetic depreciation in crown layer.

Extrapolation – Simulation Step: 4 Estimate photosynthetic depreciation factor in crown layer using mean irradiance and maximum depreciation. Photosynthetic output of each leaf multiplied by depreciation factor. F I G . 11. Schematic diagram for the application of the multiscale framework for determining photosynthetic capacity and production of leaves at their respective crown layers. The first application (Framework A) estimates photosynthetic production of each leaf using a capacity factor obtained from the crown layer. The nested application (Framework B) estimates the maximum photosynthetic depreciation at each crown layer using ozone concentrations.

The number of primordial leaves and, consequently, the number of internodes is computed for each bud based on the diameter of the nearest internode (Cochard et al., 2005). Based on empirical measurements by Schober (1995), the length of internodes based on the age of the tree is determined. Primary growth demonstrates the application of multiscale grammar rules, establishing relationships from the tree to the organ scale in a concise statement. For example, the code t:Tree a:Axis g:GU Bud ¼ ¼. t a g for(1:numInternode) (Internode [Axis GU Bud]) GU Bud; creates a number of internodes, lateral buds and leaves specified by numInternode. Figure 12 illustrates an iteration of this sample code for the production of two internodes from a bud.

After a simplified simulation of branch fall and bending, the refinement from crown layers to leaves is updated based on the vertical height of leaves. Finally, data are aggregated from organ to tree scale and from tree to stand scale. Ozone AOT40 values in the three stands are updated using trends proposed by Denby et al. (2010). Figure 13 shows the stands after 15 simulation years. DI S C U S S IO N A ND CO NC L US I ON S Multiscale vs. single-scale rules

Example 1 generates a multiscale representation of plant topology. We have implemented an alternative model with single-scale rules to achieve the same goal. In this model, coarse representations (not scales since the model is not formulated with a structure-of-scales or equivalent) of organs are implemented using an object-oriented

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Compute maximum photosynthetic depreciation in crown layer.

(Application of multiscale) Framework A

Evolution – Simulation Step: 3

Ong et al. — An approach to multiscale modelling with graph grammars Graph before rewrite

823

Graph after rewrite T

A

T A

A

A

G

G

G

G

O

O

O

G

G G

O

O

O

t:Tree a:Axis g:GU Bud ==>t a g for(1 : 2)(Internode[Axis GU Bud]) GU Bud

Scale: T Tree

A Axis

G Growth unit

Refinement (coarse to fine)

O Organ Successor

Branching

F I G . 12. Example of an execution of a multiscale rule in XL. The instanced graph is shown on the top and the rule in XL code is shown in the middle. The bottom shows a corresponding geometrical representation of the instanced graph. In this rule, a bud is queried on the left-hand side of the rule along with the chain of encoarsements up to the tree scale. On the right-hand side of the rule, light grey graph nodes (with light grey XL syntax and geometrical picture) represent the two lateral buds produced, along with their chain of encoarsements up to the axis scale.

F I G . 13. Screenshot of the three beech stands after 16 simulation years. The left stand is exposed to the least ozone, the middle one to moderate ozone and the right one to the most ozone.

(O-O) approach. To represent the axes and the organs in O-O code, a Java class Axis and a Java interface Organ are implemented in separate files. An Axis object has references to other Axis objects to which branching relationships exist, as well as references to its fine representations at organ scale. In the main modelling file, an array is defined to contain all axes in the plant. XL modules for internodes and buds are defined with an index reference to the array, identifying their respective encoarsements: module Internode(int axis) implements Organ; module Bud(int axis) implements Organ;

The rule to generate the same multiscale structure from example 1 is then specified as: b:Bud ¼¼ . { removeOrganFromAxis (b.axis, b); int latAxis ¼ createAxis (b.axis);} i:Internode(b.axis, 1) [bl:Bud(latAxis)] ba:Bud(b.axis) {addOrganToAxis (b.axis, i);. . .}

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

O

O

824

Ong et al. — An approach to multiscale modelling with graph grammars

The production statement is sandwiched between O-O code blocks (between curly brackets) that invoke methods to maintain and create refinement relationships between axes and organs. The method removeOrganFromAxis is invoked to remove the matched Bud node from its axis encoarsement. After the specification of the production graph (consisting of a new internode, lateral bud and apical bud), the method addOrganToAxis is invoked multiple times to add references of the newly created organs into the axis objects. By comparing this implementation with example 1, some advantages and disadvantages of the three-part graph model and grammar are identified. One significant advantage of the multiscale rules is the implicit handling of refinement relationships once a type graph is specified. An index or reference-based implementation of scales together with classical single-scale rules, as shown in our alternative implementation, requires the modeller explicitly to maintain scale relationships with method invocations, introducing more room for error. Another advantage of the multiscale rules over index-based implementations is the implicit type constraint, i.e. rules referencing the type graph are less likely to generate erroneous refinement relationships. In contrast, such constraints need to be manually handled by a modeller who chooses an index-based implementation approach, since indices are not programmatically tied to specific arrays. Comparing the length of the implementation code files reveals a third advantage of the multiscale rules. For generating a simple multiscale structure, multiscale rules require ,30 % of the code (ignoring empty and commentary lines) required by single-scale rules (see source code files ‘Example1MultiScale’ and ‘Example1SingleScale’ in Supplementary Data Zip File). These advantages currently come with a price of slower executions in comparison with single-scale rules. This occurs at least for the existing implementation in GroIMP, which does not utilize possible speed-up techniques for graph matching. Another disadvantage of multiscale rules is the introduction of new syntax and semantics, deviating from classical L-systems and resulting in a steeper learning curve.

B

To concretize a comparison of example 2 against single-scale approaches of plant modelling, we implement a corresponding model that contains complete plant topology and light conditions (Palubicki et al., 2009). Four plant architectures with contrasting apical dominance and tropism (cf. figs 7 and 12 in Palubicki et al., 2009) are implemented using both models. Figure 14 shows that the model in example 2, i.e. the multiscale framework, is capable of replicating selected architectural shapes with specific parameters. No crown dimensions or tree-scale aggregates are created from the single-scale implementation for comparisons because aggregates constitute dependencies that make a model multiscale in nature. A correlation-based crown development model, on the other hand, cannot offer fine light sensitivity for comparisons. Due to the self-replicating characteristic of plants, the numbers of branches and buds can increase exponentially during the juvenile growth phase, often causing computational limitations in single-scale approaches (one of the earlier mentioned motivations of multiscale modelling). Example 2 overcomes this while retaining fine sensitivity to light by sampling the bud population. Juxtapositions of bud count and simulation time for the multiscale and single-scale implementations are shown in Figs 15 and 16. The above advantages are attributed to the multiscale framework and not to the sampling method. An attempt to utilize bud sampling for the single-scale model requires a consideration of bud positions in relation to the crown’s geometric space, i.e. only buds near the boundary of the crown should be sampled, deeming the model a multiscale model. Moreover, with sampling, light model mechanisms in the single-scale model still operate for a large number of branches and buds. A salvaging attempt to use coarse representations of woody and foliage objects would constitute, once again, a multiscale model. Despite the advantages, the implementation of example 2 requires additional procedures which are absent in the singlescale model to up-scale organ developments to the tree crown. More specifically, the scaling and aggregation of bounding boxes to crown layer dimensions lengthens the simulation

C

D

F I G . 14. Black trees depict the results of the single-scale model (SS) with complete topological data. Green crown layers show the results of the corresponding multiscale model (MS) parameterized from example 2. Each architectural type is described with distinguishing source code parameters values. ALLOC_L and apical are parameters indicating apical dominance. PLANT_WT_TROPISM is a parameter indicating tropism in branch development. budAngleRL consists of the minimum and maximum branching angles in the four crown layers. O_LEN is a parameter for branch elongation length. (A) Low apical dominance and low tropism. SS parameters: ALLOC_L ¼ 0.49, PLANT_WT_TROPISM ¼ 0.3. MS parameters: apical ¼ 0.6, budAngleRL ¼ {{0,20},{30,45},{40,75},{50,85}}, O_LEN ¼ 0.071. (B) Low apical dominance and high tropism. SS parameters: ALLOC_L ¼ 0.49, PLANT_WT_TROPISM ¼ 1.0. MS parameters: apical ¼ 0.6, budAngleRL ¼ {{0,20},{15,25},{20,30},{20,30}}, O_LEN ¼ 0.071. (C) High apical dominance and low tropism. SS parameters: ALLOC_L ¼ 0.54, PLANT_WT_TROPISM ¼ 0.3. MS parameters: apical ¼ 0.78, budAngleR L ¼ {{0,20},{30,45},{40,75},{50,85}}, O_LEN ¼ 0.2. (D) High apical dominance and high tropism. SS parameters: ALLOC_L ¼ 0.54, PLANT_WT_TROPISM ¼ 1.0. MS parameters: apical ¼ 0.78, budAngleRL ¼ {{0,20},{15,25},{20,30},{20,30}}, O_LEN ¼ 0.2.

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

A

The multiscale framework vs. single-scale approaches

Ong et al. — An approach to multiscale modelling with graph grammars 3·5

Single-scale Multiscale

2·5 2·0 1·5 1·0 0·5 0 5

10 Step

15

20

F I G . 15. Logarithm of the number of buds in each simulation step for the singlescale model and the multiscale model (example 2).

15

Single-scale Multiscale

The notion of plant modularities and MTG topology is shown in the structure-of-scales containing plant, axis, growth units and organs in example 3. Two forms of static spatial division are illustrated. A continuous concept of 3-D space is used for the random positioning of individual beech trees in a stand area. The MTG modularities and stand space are additionally merged with a discrete spatial division employed by the crown layers. Dynamic spatial segregation is shown in example 2 that has an evolving tree crown space. Network-based systems such as the metabolic network in example 3 are successfully embedded in the graph data structure. A single graph data structure is used to represent both geometric spaces and topological structures in the examples. This allows the use of rules to access and modify relationships between the spaces, e.g. relating a leaf to a canopy layer. Moreover, in this manner, topological changes directly impose a change in the graph data structure, reducing the need to synchronize changes to the geometric space. For example, the removal of a graph node (e.g. simulating the death of a leaf ) imposes the removal of all edges connected with it, including the edge from the graph node representing the canopy layer containing the leaf. If the two spaces are maintained in separate data structures, an explicit procedure to remove the reference to the leaf from the geometric space (canopy layer) is required. On the other hand, the representation of geometry and topology in a single data structure (which is also a feature of classical L-systems with turtle interpretation) leads to a ‘dense’ code where both aspects are somewhat interlaced. This can be seen as a disadvantage when the modeller desires a strict separation of the aspects.

Time (s)

10

Scale dependencies

5

0 5

10 Step

15

20

F I G . 16. Simulation time taken at each simulation step for the single-scale model and the multiscale model (example 2).

pipeline. In addition, example 2 produces output at tree scale, i.e. at coarse spatial resolutions, limiting its practicality to cases where fine-scale outputs can be ignored. Additional coarse-scale inputs such as the vertical distribution of buds and spatial distribution of foliage in crown layers are also not required in the single-scale approach (see source code files ‘Example2MultiScale’ and ‘Example2SingleScale’ in Supplementary Data Zip File). Representation of spatial, topological and metabolic data

The three-part graph data structures used in the examples have demonstrated support for various data representations.

A comparison between the scale dependencies in examples 2 and 3 is made. A ‘bottom-up’ dependency is utilized in both examples. Example 2 shows the adoption of manageable micro-scale simulation by the macro-scale model. Instead of constructing a light-sensitive crown model based on empirical information (i.e. a correlation-based approach), response to light at the organ scale by sampling is adopted. This allows the inclusion of light sensitivity at fine spatial resolutions in the coarser crown layers. Example 3 attempts to formulate another computational problem into a multiscale problem. A simulation of each beech tree at the metabolic scale would be ideal but computationally prohibitive. In example 3, generalization of the metabolic behaviour to individual crown layers (Framework B, Fig. 11) allows a feasible inclusion of ozone effects to photosynthesis as opposed to a direct morphology – ozone concentration correlation model. This formulation is based on the vertical distribution of canopy light (Lemoine et al., 2002) and the different response of sun leaves and shade leaves to oxidative stress from ozone (Olbrich et al., 2010). To introduce more inputs, the metabolic network model can be extended and modified, avoiding the scenario of adding parameters at the organ scale for re-calibration in a correlation-based approach. A ‘top-down’ dependency is at times useful in modelling to impose restrictions or constraints to fine-scale models. In example 3, the effects of ozone are conveyed to individual leaves via the coarse crown layers (Framework A, Fig. 11). To propagate the output of the metabolic – crown layer sub-model

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Log (number of buds)

3·0

825

826

Ong et al. — An approach to multiscale modelling with graph grammars

(Framework B, Fig. 11) earlier in the pipeline to leaves, this dependency results from the available data, i.e. photosynthetic capacity factor, in each crown layer. Similar scenarios may occur in the design of other models. Particularly in the context of forestry, where macro-scale tree data are easy to obtain and considered as reliable, there arises the wish to utilize them to superimpose restrictions to finer models of crown architecture and branching (cf. Lintunen et al., 2011; Scho¨n, 2014), e.g. for fine-scale calculations of light interception and scattering. The correct overall shape of the crown and a realistic fine-scale structure of branching can thus be guaranteed in a pragmatic way.

With the introduction of a multiscale framework, the processes in the examples (e.g. the simulation steps in Table 2) become formulated in a manner distinctively showing the category, dependency and integration of scales (Table 1, Fig. 11). By comparing the extrapolation component in the examples, different ways of extrapolating or estimating data from independent scales are observed. In example 2, data in the dependent scale are aggregated, i.e. crown layers are generated as bounding volumes, based on feedback from the independent scale. In example 3, data in the dependent scale ( photosynthetic product) are multiplied by feedback ( photosynthetic capacity in crown layer) from the independent scale. However, a model can utilize different extrapolation forms for the same problem category and scale dependency specification; for example, in example 2, the crown dimensions are not necessarily a complete aggregation of the bounding boxes. They can be computed from an empirical model and geometrically scaled to match the directional growth bias from the organ scale. In this case, the extrapolation form is then a multiplication, similar to that in example 3. Not only can different extrapolation forms be used, but different approaches exist for the same extrapolation form. For example, in example 2, the crown dimensions are not necessarily an aggregation of geometrically up-scaled bounding boxes from the finer scale but could be a geometrical up-scale of an aggregated bounding box at the finer scale. These two approaches to aggregation reflect the different time-scaling approaches used in the extended multi-grid method (Brandt, 2002) and the equationfree approach (Kevrekidis et al., 2003). An optimal choice of extrapolation form, and consequently the dependent-scale’s model, remains an open problem both in our approach and in mathematical physics (E, 2011). In conclusion, the three-part graph model and the multiscale framework form an encapsulating approach that addresses both motivations for multiscale modelling. Results reveal some insights worth considering when deciding between multiscale and classical model designs. Considering the advantages (such as acceleration, concise code, integrated data representation, etc.) and disadvantages (such as complexity in terms of inputs, resolution of outputs, run-time performance, etc.), factors influencing model design are manifold and the methods herein are alternatives for but not entirely replacements of classical techniques. Potential future works are the addition of sensitivity analysis (Kleijnen, 1992; Wu et al., 2012) concepts to our multiscale approach and improvements in the run-time efficiency of the graph grammar operations.

Supplementary data are available online at www.aob.oxford journals.org and consist of the following. Table S1: a selection of XL commands and syntax descriptions used in the text. Zip File: source code files of the examples and corresponding comparative models. ACK N OW L E DG E M E N T S The authors thank Tim Ritter and Reinhold Meyer for their valuable assistance. Furthermore, we thank two anonymous reviewers for constructive comments. This work was supported by the German Academic Exchange Service (Deutscher Akademischer Austausch Dienst, DAAD), grant no. A/10/98126. LIT E RAT URE CITED Band LR, Fozard JA, Godin C, et al. 2012. Multiscale systems analysis of root growth and development: modeling beyond the network and cellular scales. The Plant Cell 24: 3892–3906. Betz GA, Knappe C, Lapierre C, et al. 2009. Ozone affects shikimate pathway transcripts and monomeric lignin composition in European beech (Fagus sylvatica L.). European Journal of Forest Research 128: 109– 116. Bird RB, Armstrong RC, Hassager O. 1987. Dynamics of polymeric liquids, 2nd edn. New York: John Wiley. Boudon F, Prusinkiewicz P, Federl P, Godin C, Karwowski R. 2003. Interactive design of bonsai tree models. Computer Graphics Forum 22: 591– 599. Boudon F, Pradal C, Cokelaer T, Prusinkiewicz P, Godin C. 2012. L-Py: an L-System simulation framework for modeling plant development based on a dynamic language. Frontiers in Plant Science 3: 76. Brandt A. 2002. Multiscale scientific computation: review 2001. In: Barth TJ, Chan T, Haimes R, eds. Multiscale and multiresolution methods: theory and applications. Berlin: Springer, 3– 96. Buck-Sorlin G, Kniemeyer O, Kurth W. 2005. Barley morphology, genetics and hormonal regulation of internode elongation modelled by a relational growth grammar. New Phytologist 166: 859–867. Car R, Parrinello M. 1985. Unified approach for molecular dynamics and density-functional theory. Physical Review Letters 55: 2471– 2474. Chiba N, Ohkawa S, Muraoka K, Miura M. 1994. Visual simulation of botanical trees based on virtual heliotropism and dormancy break. Journal of Visualization and Computer Animation 5: 3 –15. Cochard H, Tyree MT. 1990. Xylem dysfunction in Quercus: vessel sizes, tyloses, cavitation and seasonal changes in embolism. Tree Physiology 6: 393– 407. Cochard H, Coste S, Chanson B, Guehl J, Nicolini E. 2005. Hydraulic architecture correlates with bud organogenesis and primary shoot growth in beech (Fagus sylvatica). Tree Physiology 25: 1545–1552. Courne`de P-H, Mathieu A, Houllier F, Barthe´le´my D, de Reffye P. 2008. Computing competition for light in the GREENLAB model of plant growth: a contribution to the study of the effects of density on resource acquisition and architectural development. Annals of Botany 101: 1207– 1219. Da Silva D, Boudon F, Godin C, Herve´ S. 2008. Multiscale framework for modeling and analyzing light interception by trees. Multiscale Modeling and Simulation 7: 910– 933. Dada JO, Mendes P. 2011. Multi-scale modelling and simulation in systems biology. Integrative Biology 3: 86– 96. Denby B, Sundvor I, Cassiani M, de Smet P, de Leeuw F, Hora´lek J. 2010. Spatial mapping of ozone and SO2 trends in Europe. Science of the Total Environment 408: 4795– 4806. Deussen O, Hanrahan P, Lintermann B, Meˇch R, Pharr M, Prusinkiewicz P. 1998. Realistic modeling and rendering of plant ecosystems. In: Cunningham S, Bransford W, Cohen MF, eds. SIGGRAPH ‘98 Proceedings of the 25st Annual Conference on Computer Graphics and Interactive Techniques, Orlando, Florida. New York: ACM Press, 275– 286. E W. 2011. Principles of multiscale modeling. Cambridge: Cambridge University Press.

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

Extrapolation methods

S U P P L E M E N TARY D ATA

Ong et al. — An approach to multiscale modelling with graph grammars

Graphics Interface 2002, Calgary, Alberta, May 2002. Canadian Human–Computer Communications Society, 69– 80. Lemoine D, Cochard H, Granier A. 2002. Within crown variation in hydraulic architecture in beech (Fagus sylvatica L): evidence for a stomatal control of xylem embolism. Annals of Forest Science 59: 19–27. Lindenmayer A. 1968. Mathematical models for cellular interaction in development. Journal of Theoretical Biology 18: 280– 315. Lintunen A, Sieva¨nen R, Kaitaniemi P, Perttunen J. 2011. Models of 3D crown structure for Scots pine (Pinus sylvestris) and silver birch (Betula pendula) grown in mixed forest. Canadian Journal of Forest Research 41: 1779– 1794. Lohr SL. 2009. Sampling: design and analysis, 2nd edn. Boston: Cengage Learning. Olbrich M, Gerstner E, Bahnweg G, et al. 2010. Transcriptional signatures in leaves of adult European beech trees (Fagus sylvatica L.) in an experimentally enhanced free air ozone setting. Environmental Pollution 158: 977–982. Ong Y, Kurth W. 2012. A graph model and grammar for multi-scale modelling using XL. In: Gao J, Dubitzky W, Wu C, et al., eds. 2012 IEEE International Conference on Bioinformatics and Biomedicine Workshops, Philadelphia, USA, October 2012. USA: IEEE Computer Society, 1– 8. Palubicki W, Horel K, Longay S, et al. 2009. Self-organizing tree models for image synthesis. ACM Transactions on Graphics 28(3): 58:1– 58:10. Prusinkiewicz P, Hammel M, Hanan J, Meˇch R. 1997. Visual models of plant development. In: Rozenberg G, Salomaa A, eds. Handbook of formal languages. Heidelberg: Springer, 535–597. Prusinkiewicz P, Muendermann L, Karwowski R, Lane R. 2001. The use of positional information in the modeling of plants. In: Fiume E, ed. SIGGRAPH ‘01 Proceedings of the 28th Annual Conference on Computer Graphics And Interactive Techniques, Los Angeles, California. New York: ACM Press, 289–300. Scalfi M, Troggio M, Piovani P, et al. 2004. A RAPD, AFLP and SSR linkage map, and QTL analysis in European beech (Fagus sylvatica L.). Theoretical and Applied Genetics 108: 433–441. Schober R. 1995. Ertragstafeln wichtiger Baumarten. Frankfurt am Main: J. D. Sauerla¨nder’s Verlag. Scho¨n M. 2014. Structural– functional concepts in forest modeling applicable for higher resolution of forest ecosystem simulations. PhD Thesis, Technical University in Zvolen, Slovak Republic. Seidl R, Rammer W, Scheller RM, Spies TA. 2012. An individual-based process model to simulate landscape-scale forest ecosystem dynamics. Ecological Modelling 231: 87– 100. Sieva¨nen R, Nikinmaa E, Godin C, Lintunen A, Nygren P. 2013. Proceedings of the 7th International Conference on Functional– Structural Plant Models. Finland: Finnish Society of Forest Science, Finnish Forest Research Institute, Department of Forest Sciences, University of Helsinki. Strobel J. 2004. Die Atmung der verholzten Organe von Altbuchen (Fagus sylvatica L.) in einem Kalk- und einem Sauerhumusbuchenwald. PhD Thesis, University of Go¨ttingen, Germany. Warshel A, Levitt M. 1976. Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. Journal of Molecular Biology 103: 227– 249. Wu Q-L, Courne`de P-H, Mathieu A. 2012. An efficient computational method for global sensitivity analysis and its application to tree growth modelling. Reliability Engineering and System Safety 107: 35– 43.

Downloaded from http://aob.oxfordjournals.org/ at University of Georgia Libraries, Serials Department on June 29, 2015

E W, Engquist B. 2003. The heterogeneous multiscale methods. Communications in Mathematical Sciences 1: 87–132. Ganter B. 2005. Contextual attribute logic of many-valued attributes. In: Ganter B, Stumme G, Wille R, eds. Formal concept analysis: foundations and applications. Berlin: Springer, 101– 113. Godin C, Caraglio Y. 1998. A multiscale model of plant topological structures. Journal of Theoretical Biology 191: 1 –46. Han L, Gresshoff PM, Hanan J. 2010. A functional –structural modelling approach to autoregulation of nodulation. Annals of Botany 107: 855 –863. Hanan J. 1992. Parametric L-systems and their application to the modelling and visualization of plants. PhD Thesis, University of Regina, Canada. Hanan J. 2013. Functional –structural modelling with L-systems: where from and where to. In: Sieva¨nen R, Nikinmaa E, Godin C, Lintunen A, Nygren P, eds. Proceedings of the 7th International Conference on Functional– Structural Plant Models, Saariselka¨, Finland, June 2013. Finland: Finnish Society of Forest Science, Finnish Forest Research Institute, Department of Forest Sciences, University of Helsinki, 1 –3. Harper JL, Rosen BR, White J. 1986. The growth and form of modular organisms. London: The Royal Society. Hemmerling R, Kniemeyer O, Lanwert D, Kurth W, Buck-Sorlin G. 2008. The rule-based language XL and the modelling environment GroIMP illustrated with simulated tree competition. Functional Plant Biology 35: 739– 750. Ho QT, Verboven P, Verlinden BE, et al. 2011. A three-dimensional multiscale model for gas exchange in fruit. Plant Physiology 155: 1158– 1168. Kalos MH, Whitlock PA. 2008. Monte Carlo methods, 2nd edn. Weinheim: Wiley-VCH. Kang M-Z, de Reffye P. 2007. A mathematical approach estimating source and sink functioning of competing organs. In: Vos J, Marcelis LFM, de Visser PHB, Struik PC, Evers JB, eds. Functional–structural plant modelling in crop production. Berlin: Springer, 65– 74. Kerner R, Winkler J, Dupuy J, et al. 2011. Changes in the proteome of juvenile European beech following three years exposure to free-air elevated ozone. iForest 4: 69– 76. Kevrekidis IG, Gear CW, Hyman JM, Kevrekidis PG, Runborg O, Theodoropoulos C. 2003. Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Communications in Mathematical Sciences 1: 715–762. Kleijnen JPC. 1992. Sensitivity analysis of simulation experiments: regression analysis and statistical design. Mathematics and Computers in Simulation 34: 297–315. Knauer U. 2011. Algebraic graph theory: morphisms, monoids and matrices. Berlin: Walter De Gruyter. Kniemeyer O. 2008. Design and implementation of a graph grammar based language for functional – structural plant modelling. PhD Thesis, University of Technology at Cottbus, Germany. Kniemeyer O, Buck-Sorlin G, Kurth W. 2007. GroIMP as a platform for functional–structural modelling of plants. In: Vos J, Marcelis LFM, de Visser PHB, Struik PC, Evers JB, eds. Functional–structural plant modelling in crop production. Dordrecht: Springer, 43–52. Kurth W, Kniemeyer O, Buck-Sorlin G. 2005. Relational growth grammars – a graph rewriting approach to dynamical systems with a dynamical structure. In: Banaˆtre J-P, Fradet P, Giavitto J-L, Michel O, eds. Unconventional programming paradigms. Lecture Notes in Computer Science 3566. Berlin: Springer, 56–72. Lane B, Prusinkiewicz P. 2002. Generating spatial distributions for multilevel models of plant communities. In: McCool MD. ed. Proceedings of

827

An approach to multiscale modelling with graph grammars.

Functional-structural plant models (FSPMs) simulate biological processes at different spatial scales. Methods exist for multiscale data representation...
1MB Sizes 0 Downloads 6 Views