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Using the canonical modelling approach to simplify the simulation of function in functional–structural plant models Blackwell Publishing, Ltd.
Michael Renton1, Jim Hanan2, Kevin Burrage2 1
UMR BEPC, INRA, Montpellier, France; 2ACMC, University of Queensland, Brisbane, Australia 4072
Summary Author for correspondence: Jim Hanan Tel: +61 7 3365 6132 Fax: +61 7 3365 4325 Email:
[email protected] Received: 28 September 2004 Accepted: 23 November 2004
• Functional–structural plant models that include detailed mechanistic representation of underlying physiological processes can be expensive to construct and the resulting models can also be extremely complicated. On the other hand, purely empirical models are not able to simulate plant adaptability and response to different conditions. In this paper, we present an intermediate approach to modelling plant function that can simulate plant response without requiring detailed knowledge of underlying physiology. • Plant function is modelled using a ‘canonical’ modelling approach, which uses compartment models with flux functions of a standard mathematical form, while plant structure is modelled using L-systems. • Two modelling examples are used to demonstrate that canonical modelling can be used in conjunction with L-systems to create functional–structural plant models where function is represented either in an accurate and descriptive way, or in a more mechanistic and explanatory way. • We conclude that canonical modelling provides a useful, flexible and relatively simple approach to modelling plant function at an intermediate level of abstraction. Key words: canonical modelling, functional–structural plant modelling, L-systems, nonlinear modelling, plant function, power law, simulation. New Phytologist (2005) 166: 845–857 © New Phytologist (2005) doi: 10.1111/j.1469-8137.2005.01330.x
Introduction The morphogenetic process is key to the diversity of structure seen in plants, both among species and between individuals of the same species in different environments. One way of trying to understand plant growth and morphogenesis is to construct architectural plant models based on the classification and quantification of growth patterns using analysis of architectural data (Hallé et al., 1978; Hayes et al., 1990; Godin et al., 1999; Suzuki, 2000). Such models, which are designed to describe and analyse plant growth without modelling underlying physiological processes, can be called descriptive or empirical models (Thornley & Johnson, 1990; Haefner, 1996; Prusinkiewicz, 1998). Another way of using modelling to understand plant growth is to build computational models that represent both
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the function and structure of a plant, often termed functional– structural (FS) plant models (Room et al., 1996; de Reffye et al., 1997; Kurth & Sloboda, 1997; Fournier & Andrieu, 1998, Lacointe, 2000; Godin et al., 2004). Such models can provide a theoretical framework for experimental investigations aimed at deepening our understanding of plant growth. The general approach underlying many FS models is to represent the plant as a large number of interconnected components (such as internodes and leaves). The physical, chemical and physiological processes (such as light interception, photosynthesis and nutrient transport) that take place within and between these components are then modelled separately (Perttunen et al., 1998; Lacointe, 2000; Sievänen et al., 2000; Sinoquet & Le Roux, 2000; Le Roux et al., 2001). Models based on this approach can therefore be called process-based or deeply mechanistic models (Thornley, 1990; Haefner,
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1996; Prusinkiewicz, 1998). Unlike empirical models, mechanistic models can simulate the growth of a plant in situations and conditions beyond those already observed and represent causal hypotheses for explaining observed patterns of growth and response (Fisher, 1992; Sachs & Novoplansky, 1995; Prusinkiewicz, 1998; Marcelis, 2001; Thornby et al., 2003). However, the amount of experimental research required to provide all of the physiological information needed to construct a full process-based FS model for a single species is huge indeed. Even if complete and detailed physiological information is available, a model incorporating all of this knowledge is likely to be extremely complex, and a simpler model may often be more useful (Grimm, 1994). For example, we may want to construct an adaptive FS plant model to act as a relatively abstract and simple ‘submodel’ within a more complex and larger-scale simulation. Or we may wish to build a FS plant model to act as a ‘theoretical framework’ to help in initial investigations into a particular physiological mechanism that is not yet well understood. In these situations, we would want to construct a model that does not require in-depth experimental investigations or a high level of model complexity, yet is capable of capturing the most important, interesting or relevant aspects of the function and structure of the plant. We would need a model that is adaptable and able to represent causal hypotheses (unlike a purely descriptive or empirical model), yet is simpler and easier to construct than the detailed process-based models normally referred to as FS plant models. A FS modelling approach that tried to find this balance between the advantages of empirical and mechanistic modelling could be called an ‘intermediate-level’ approach. Recent attempts to develop such intermediate-level FS modelling approaches include linking a relatively abstract cotton crop model with structural representations of individual plants (Hanan & Hearn, 2002); incorporation of a small number of local-scale mechanistic hypotheses within a generally empirical model of a whole cotton plant (Thornby et al., 2003); and determining bud fate in frangipani trees using generalised probability functions and abstract representation of local resource levels (Renton et al., 2003). Another recently developed intermediate-level FS approach (Renton, 2004) is based on integrating canonical models (Savageau, 1969; Savageau, 1976; Voit, 2000) of plant function with L-system (Lindenmayer, 1968; Lindenmayer & Prusinkiewicz, 1990) simulations of plant structural development. A previous presentation of this integrated approach (Renton et al., 2005) focussed on how a structural L-system model could be developed and linked to an existing canonical model (Kaitaniemi, 2000) of plant function. In contrast, the focus here is on how canonical modelling can be used to simulate plant function in FS models. The following Methods section presents an introduction to canonical nonlinear modelling and a discussion of how this approach could be used to represent many aspects of plant function, including resource acquisition, growth, allocation,
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storage and suppression. This is followed by brief introductions to L-systems and how they can be linked to canonical models, and to the experimental background and data for two modelling examples. In the Results section, we demonstrate that canonical modelling can be used to represent plant function in both an accurate and descriptive way, or in a more mechanistic and explanatory way, through two examples. We also illustrate the possibility of creating FS plant models by linking such canonical models to structural L-system representations. The paper concludes with a discussion of the potential role of canonical modelling in providing a needed intermediate-level approach to representing plant function in FS models.
Methods Canonical modelling Canonical nonlinear models, such as GMA-systems or Ssystems, have been employed to model a variety of complex biological systems (Torres, 1996; Voit & Sands, 1996a; Martin, 1997; Kaitaniemi, 2000) in a mathematically standard way (Voit, 1991; Voit, 2000). The diagrammatic representation of a canonical model consists of ‘compartments’, ‘fluxes’ and ‘influences’. Fig. 1 provides a simple example where compartments are drawn as circles, fluxes as solid arrows and influences as dashed arrows. Compartments are associated with a variable that usually represents some real-world quantity. Fluxes represent flows moving into, out of, and between compartments. An influence drawn between a compartment and a flux indicate that the magnitude of the movement represented by the flux depends on the magnitude of the quantity represented by the compartment variable. When the diagrammatic (or compartment) representation of the model has been formulated, fluxes are then represented as ‘canonical’ (or standardised) functions of all compartment variables that influence that flux (usually including the compartment that they originate from). The standard canonical power-law function consists of a ‘rate constant’ multiplied by the product of all influencing compartment variables, each raised to a constant ‘kinetic order’ power. In the example in Fig. 1, flux f2 originates from the x1 compartment and is influenced by the x3 compartment, so it would be written as
Fig. 1 Example of a canonical model.
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f 2 (x 1(t ), x 3(t )) = α 2x 1(t )k21 x 3(t )k23 ,
Eqn 1
where α2 is the rate constant parameter and k21 and k23 are the kinetic order parameters (with subscript pairs indicating the flux being operated on followed by the compartment variable identifier). With the GMA-system formalism, the rate of change of a compartment variable is written as the sum of all influxes, minus all outfluxes. Therefore, in this example the differential equation for x1 would be d x1(t ) = f 1(x 2 (t )) − f 2 (x1(t ), x 3(t )) − f 3(x1(t )) dt
Eqn 2
= α1x 2 (t )k12 − α 2x 1(t )k21 x 3(t )k23 − α 3x 1(t )k31 ,
Eqn 3
which is normally written in the abbreviated form k
k
k
k
˛ 1 = α1x 212 − α 2x 121x 323 − α 3x 131.
Eqn 4
The other two differential equations representing the model would be k
k
˛ 2 = α 2x 1 21x 323 , k
˛ 3 = α 3x 1 31 .
Eqn 5 Eqn 6
Canonical modelling of plant function The procedure for using the canonical power-law modelling approach to simulate an aspect of plant function can be summarised in five steps, as follows. Firstly, the important or significant quantities must be identified, and associated with a compartment and variable. Secondly, the fluxes or flows into, out of, and between these compartments must be chosen. Thirdly, the modeller must decide which quantities affect which of these fluxes, and include these ‘influences’ in the model. At this point, we have formulated a compartment
model, such as that shown in Fig. 1, which represents qualitative mechanistic assumptions or hypotheses. Until this point, the modelling is mechanistic in nature and, in general, the inclusion of more compartments and connections (fluxes and influences) corresponds to a more mechanistic model. Fourthly, a canonical form, such as the power-law form, is used to represent each flux as a function of the influencing variables. Finally, values for the parameters of these flux functions are estimated, using one or more of a number of possible approaches, which include eliminating unnecessary parameters; recognising constraints on parameters implied by the modelled system; reformulating equations to give parameters a clearer significance; scaling variables; using ad hoc manipulation of parameter values with visual feedback of model output; flux-based estimation and using computational algorithms to fit model output at the global scale (Voit, 2000; Renton, 2004). At this point, a model of function has been created, which can be tested and compared against data, and modified as necessary. The simplest way to model growth with a canonical model is with just one compartment and one flux, as shown in Fig. 2. In this case, the compartment and its associated variable x represents the size of the whole plant, which could be interpreted as the total biomass of the plant, the total leaf area, the total height of the plant, or any other reasonable, desirable or useful measure of plant size. The one incoming flux fi represents the processes of resource acquisition through photosynthesis, and water and nutrient uptake by the roots. The basic assumption behind the model is that the amount of resources acquired depends on the size of the plant. The influence of size on resource acquisition represents the fact that photosynthesis and water and nutrient uptake will be affected by leaf area and root mass. Representing this system using the normal power-law flux representation would then give the differential equation ˛ = fi = α x k.
Eqn 7
Fig. 2 Unbounded growth model. Canonical equations are derived from the compartment representation (left-hand panel) to give different model output behaviour (right-hand panel) depending on model parameter values (shown in legend).
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Fig. 3 Bounded growth model. Canonical equations are derived from the compartment representation (left-hand panel) to give different model output behaviour (right-hand panel) depending on model parameter values (shown in legend).
Fig. 4 Compartment models including representation of storage (left-hand panel) and root/shoot allocation (right-hand panel).
Integrating this equation gives a range of unbounded growth behaviours, depending on the values of the parameters α and k, as shown in Fig. 2. This model can then be adapted to simulate bounded growth in a number of ways (Savageau, 1980; Voit, 1991). One possibility is to add an outflux that is influenced by the size of the x compartment. Another is to add a supplementary variable xd defined to be equal to xmax – x, where xmax is the maximum size of the plant, and let this variable also influence the flux, as shown in Fig. 3. This compartment representation can then be rewritten in the normal way: ˛ = fi = α x k x dh = αx k (x max − x )h ,
Eqn 8
where α, h and k are all positive parameters. Integrating this equation gives a range of bounded growth behaviours, depending on the values of the parameters, as illustrated in Fig. 3. In this example, the variable xd represents the remaining potential growth, which may correspond biologically to a structural limitation, for example. Note also that the standard logistic equation, which is commonly used to model bounded growth, is a special case of Equation 8 where the values of k and h are fixed to equal one. Adding more compartments, fluxes and influences can result in more interesting functional models that are less
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abstract and hence capture more details. For example, environmental influences can be included in the canonical models (Renton, 2004). This could be done by adding an environmental variable representing temperature or nutrient level, and letting this variable influence a flux, such as the resource acquisition flux fi. Environmental variables could also directly affect other model variables. For example, the maximum size of the plant xmax could be defined as a function of an environmental variable representing nutrient level. The process of storage can be modelled in a very simple way by adding a compartment and variable representing stored or unallocated resources (xu) and an additional structuralallocation flux ( fu), as shown in Fig. 4 (left-hand panel), for the unbounded growth model. The resulting equations representing the system are ˛ u = α i x ki − α u x uku , ˛ = α u x uku .
Eqn 9 Eqn 10
To simulate both storage and allocation to different parts of the plant, we can consider a model with three compartments, one for the unallocated resources (xu), one for the above-ground (shoot) parts of the plant (xs) and one for the below-ground (root) parts of the plant (xr), linked as shown in Fig. 4 (right-hand panel). The corresponding set of equations is then
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f s = α sx uksu x psp ,
Fig. 5 Compartment model including representation of shoot/fruit allocation, fruiting triggering and suppression.
˛ u = α i x rkir x ks is − α sx uksu − α r x ukru ,
Eqn 11
˛ r = α r x ukru ,
Eqn 12
˛ s = α sx uksu
Eqn 13
Hypotheses regarding suppression of growth can be included in canonical models by adding influences to compartment models, while noncontinuous changes in allocation patterns can be represented by adding Boolean ‘switch’ variables. For example, consider Fig. 5. This model includes a variable representing stored or unallocated resources (xu), another for fruits (xf), and a third for other shoot components (xs). It also includes a Boolean switch variable xb, which is defined to be equal to zero when xu is below a certain threshold, and one when xu is above this threshold. Assuming that the fruit flux function is written in the standard way, k
f f = α f x ukfu x bf b ,
Eqn 14
this means that fruiting is ‘turned on’ when enough free resources are available. The model also includes a supplementary variable xp (defined to be equal to xf + 1) which influences the shoot allocation flux. The shoot flux function is then written in the standard way
Eqn 15
and the parameter ksp is constrained to be less than zero. When there is no fruit (xf = 0 and xp = 1), shoot allocation is ‘normal’, but when there is fruit (xf > 0 and xp > 1), shoot allocation will be reduced according to the amount of fruit, with the value of ksp controlling the strength of suppression. In this example, triggering and suppression are related to ‘internal’ variables, but they could also be related to environmental variables, such as temperature (daily or day-degree sum), light levels (photoperiod), nutrient levels, damage, pollution levels, lack of water, or time. L-systems and linking Formally, an L-system consists of an alphabet of symbols (which can represent parts of a plant), an axiom (or starting point) and productions or rewriting rules. Figure 6 gives a simple example of how symbols and re-writing rules, combined with a graphical translation, can be used to represent plant development (Lindenmayer & Prusinkiewicz, 1990). The basic strategy in linking a canonical model of function to a structural L-system model is to make state variables in the canonical model control or affect the L-system rules for growth (change in size of existing plant components) and /or development (the addition of new plant components). There are various ways in which this control or influence can be defined (Renton, 2004). For example, there may be a variable in the canonical model representing some continuous global characteristic of the plant. Increases in this variable can then be ‘shared out’ to existing components in the L-system model according to certain allocation rules. Another possibility is that canonical model variables correspond directly to individual component characteristics (‘leaf length’, for example). A third possibility (Renton et al., 2005) is that canonical model variables correspond to a discrete quantity of plant components, such as the number of new annual shoots. In this case, the indicated number of new components must be added to the L-system model at each time-step, again according to some kind of allocation rule.
Fig. 6 Using L-systems to represent plant components and development with symbols. In this case the axiom is I[L]A, where I represents an internode, L stands for a leaf, A is an apical meristem. The square brackets enclose branching structures. At each step the A is rewritten as I[L]A and in the corresponding graphical translation the small triangle representing an apical meristem is replaced with a new internode, a leaf and a new apical meristem.
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Fig. 7 Left-hand panel: output from two optimally fitted canonical models and the optimally fitted logistic model showing the simulated height of the second amaranth plant, compared with data points. The output of the two canonical models is virtually indistinguishable, despite a significant difference in parameter values. Right-hand panel: Output from leaf length models showing the simulated lengths of all leaves of the second amaranth plant changing with time. Output for leaves 1, 15 and 17 is indicated for clarity.
Data collection for modelling examples The first modelling example concerns the construction of a descriptive FS model of an amaranth plant. The length of the main stem of two particular amaranth plants were measured five times over a period of 70 d during their growth, starting approximately 20 d after the appearance of the seedling (Renton, 2004). Data is shown in Fig. 7 in comparison to model output. The time between appearance of new leaves (the plastochron) was estimated by dividing the length of the observation period by the number of leaves produced in that time, resulting in a figure of 3 d. Data was also collected on the lengths of the leaves of the amaranth plants on the last day of measurement. The length of one particular leaf was also measured frequently during elongation (every 2 to 4 d). The second modelling example concerns the construction of an explanatory FS model of the growth of the cotton plant and its response to defoliation. The model was based on an experiment that investigated the effects of defoliation in cotton (Thornby, 2004). The cotton experiment included two treatments (‘upper’ and ‘lower’, each involving different amounts of defoliation) and a control. The available raw data, including structural information such as internode and leaf lengths measured at regular intervals, were treated to give estimates of leaf and stem biomass for an average plant in each of the three treatments, at various times (Renton, 2004; Thornby, 2004). Data is shown in Figs 10, 12 and 13 (later) in comparison to model output. A major finding of this experiment was that cotton plants were largely able to compensate for significant amounts of defoliation by significantly increasing their rate of leaf biomass production after leaves had been removed. Full details on experimental procedure; data collection and treatment; and model construction, parameterisation
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and testing are available elsewhere (Thornby et al., 2003; Renton, 2004; Thornby, 2004).
Results Descriptive model This example will illustrate how this new FS modelling approach can be used in constructing a descriptive model. The specific modelling aim was to construct two empirical FS models that describe two particular plants as accurately as possible, based on the limited data available. The potential uses of such a descriptive model could be to act as a starting point for a more explanatory model, to communicate experimental results, or to act as a submodel within a broader computational model that required an accurate description of plant structural development. We began by constructing three canonical models, one for the height of each of the amaranth plants and one for the elongation of the frequently measured leaf. In each of these three cases, the data suggested a bounded growth model, and so the canonical model of bounded growth represented by Fig. 3 and Equation 8 was chosen. The model was then parameterised separately for each of the three data sets. Initial estimates for the parameters of the models were found using a combination of flux-based estimation, logarithmic transformation and ad hoc manipulation (Voit, 2000; Renton, 2004), and these estimates were refined using a computational optimisation algorithm (Renton, 2004), so that the model output matched the measured data for the two amaranth plants as accurately as possible. For each model, the parameter fitting procedure was carried out a number of times with different initial values, in order to evaluate the uniqueness
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Fig. 8 Output from the FS model of the growth of an amaranth plant.
of the resulting parameter value set. For comparison, a logistic model was also fitted to the measured data in the same way. Different initialisations resulted in different but similar fits for the canonical model, but the optimal parameter values for the logistic model were always the same. The canonical model always gave a better fit than the logistic model, and for the height data for the second plant the fit was significantly better (r 2 = 0.9998 vs r 2 = 0.9884). Fig. 7 (left-hand panel) gives a visual comparison of the fit of two canonical models with different parameters and the logistic model. The same model of elongation (Equation 8) is used for all leaves on the plants, with the values of three model parameters (a, h and k) taken directly from the model of the frequently measured leaf. However, each leaf model has a unique value for the fourth parameter, xmax, which is set equal to the measured final length of the individual leaf. Using these models and the estimated plastochron of 3 d, the length of each leaf in the plant can be estimated at any time, as illustrated in Fig. 7 (right-hand panel). A structural L-system model of the amaranth plant was designed. Every 3 d, the apex produces a new internode and leaf, together with a record of their time of production. The age of a component can then be calculated as the global time minus the production time of that individual component. The length of each leaf is calculated at each time-step directly from the corresponding canonical model, based on the leaf ’s current age. The height of the plant is also calculated at each time-step from the appropriate canonical model. The difference between this height and the sum of the lengths of all existing internodes is calculated, and a third of this difference is added to the length of each of the three youngest internodes. This ensures that the sum of the internode lengths in the L-system model is always equal to the height variable in the corresponding canonical model. The resulting FS model is illustrated in Fig. 8 and in Animation 1, available online as supplementary material.
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Fig. 9 The basic compartment representation of the canonical model of cotton growth.
Explanatory model This example illustrates how the new FS modelling approach can be used to construct an explanatory model. The purpose of the model was to act as a conceptual framework for understanding how cotton plants were able to compensate for defoliation and to suggest new experiments to investigate the validity of these hypotheses. We first constructed a basic functional cotton model using the general approaches to modelling growth, storage and allocation. This model consists of a compartment for unallocated or reserve resources (xu), a compartment for leaf (xl) and a compartment for stem (xs), as shown in Fig. 9. The flux fi represents resource acquisition under the standard conditions in which the plants were grown, fl and fs represent allocation of reserve resources to leaf and stem, respectively, and fo represents the allocation of resources to all other parts of the plant, or resources that are used or lost in other ways. This diagrammatic representation was translated in the standard way into a system of canonical equations, which was then parameterised to fit the data for the control treatment. This basic cotton growth model (Fig. 9) was able to simulate the observed data for the control treatment plants, as shown by Fig. 10. This figure illustrates how different parameterisations were able
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Fig. 10 Output of the basic cotton growth model for two different parameterisations compared with the same data points.
Fig. 11 Compartment representations of canonical models of growth and compensation in cotton, based on the hypotheses that compensation following defoliation is caused by the presence of a hormone (upper panel), or the absence of a hormone (lower panel). In these models, xh represents the amount of the hormone in the plant, xc represents the concentration of the hormone and xb is a Boolean trigger variable that is ‘turned on’ following defoliation for an amount of time that depends on the level of defoliation. The flux fp represents the production of the hormone and fd represents its degradation.
to fit the data equally well, with each parameterisation representing a different hypothesis about the level of reserve resources. Defoliation can then be modelled by reduction of the leaf compartment value at the appropriate time. In order for compensation to occur, there must be an additional mechanism that increases leaf production rates following defoliation. We formulated two alternative explanatory hypotheses regarding this mechanism. One hypothesis is that the defoliation causes the production of some hormone in the plant, and it is the presence of this hormone that causes increased allocation of resources to leaf production. The other hypothesis is that
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leaves constantly produce some hormone so that the concentration of this hormone within the plant remains relatively constant unless defoliation occurs, causing production of the hormone to drop. According to this hypothesis, it is the drop in concentration caused by the absence of this hormone that, in turn, causes the increased rate of leaf growth. Based on the original cotton growth model (Fig. 9), we developed the two compartment models shown in Fig. 11 to represent these two alternative hypotheses. These were translated into canonical equations in the standard way and parameterised to fit the observed data for the two defoliation treatments. These two alternative models of compensation were both able to simulate
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Fig. 12 Simulating compensation for 58% defoliation at day 79 with the presence model. The timing and amount of the simulated defoliation corresponds to the ‘upper’ treatment in the original experiment, as do the data points on the plot. The two plots show the results of different parameterisations of the model, resulting in behaviours that differ at the level of the hidden controlling mechanism, although both match the data points. In the first, the hormone is produced in a small quantity and breaks down slowly, while in the second, it is produced in a larger quantity and breaks down faster.
Fig. 13 Simulating compensation for defoliation with the absence model. In the plot on the left, the simulated 58% defoliation at day 79 corresponds to the ‘upper’ treatment in the original experiment, as do the data points. In the plot on the right, the defoliation also occurs at day 79, but only 21% of the leaf area is removed, which corresponds to the ‘lower’ treatment. The data points in this plot also correspond to the ‘lower’ treatment.
the observed compensation behaviour, as illustrated in Figs 12 and 13. To construct the FS model, we began with the canonical model described above and a existing ‘template’ structural Lsystem model of cotton development (Thornby et al., 2003; Hanan, 2004; Thornby, 2004). Because we were interested primarily in defoliation and leaf biomass compensation, the models are linked using the total leaf biomass variable of the
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canonical model. Apart from this link, the two submodels run in parallel, with development being controlled by the original L-system model completely independently of the canonical model. The canonical submodel is integrated with a very small (approximately continuous) time-step and the structural submodel is ‘updated’ with a daily time-step. At the end of each day, the ‘potential growth’ of each leaf in the structural submodel is calculated using a function that first rises, then
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Fig. 14 Output of the FS model of the cotton plant’s growth and response to defoliation (three leaves removed between images 3 and 4).
falls with the age of the leaf. The ‘total potential growth’ is calculated as the sum of all potential growths for all leaves in the structure, the ‘total actual growth’ is set to be the amount of new leaf growth indicated by the canonical model for that day, and the ‘growth proportion’ is set equal to the ‘total actual growth’ divided by the ‘total potential growth’. The actual growth of each individual leaf is then calculated as this ‘growth proportion’ multiplied by the original potential growth of that leaf. This ensures that the total leaf area in the structural model is the same as that indicated by the canonical model at the end of the day, that each leaf grows according to a sigmoidal function in ideal conditions, and that the rate of individual leaf growth slows when many leaves are growing at the same time. The user of the FS model is able to specify if and when a particular leaf is removed from the structural L-system model; at the same time, the equivalent amount of defoliation is simulated in the canonical model. The response to this defoliation (simulated by the canonical model according to the presence or absence hypothesis) will involve a boost
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to leaf biomass allocation in the canonical model, leading to an increase in the rate at which new leaves will grow in the L-system model. The FS model can thus be used to predict individual leaf biomass (and hence area), and the change in individual leaf biomass as a result of defoliation. Output from the FS model is shown in Fig. 14 and in Animation 2, available online as supplementary material.
Discussion Canonical modelling limitations There appear to be a number of limitations to the canonical modelling approach and its application to simulating plant function. One main limitation concerns the depth of mechanism that can be usefully represented. In many situations, a modeller may be interested in the physiological processes underlying plant growth, and the interactions between them, rather than the higher-level, observable patterns of plant growth. Any model that attempted to model a whole plant at
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this level of mechanism would require a very large number of variables. If a canonical model was used, it would therefore involve a large number of compartments, fluxes, influences, and therefore parameters, thereby becoming very difficult to construct (parameterise) and understand, and thus losing many of the original advantages of canonical modelling. If a modeller is interested in simulating plant growth at a detailed and process-based level, then canonical modelling would probably not be the appropriate overall approach. Another limitation concerns the data needed to construct a model. The variables and fluxes in a canonical model may represent quite abstract quantities and processes, which may be difficult to measure directly. For example, in the cotton model, the variables representing leaf and stem biomass are clearly defined, and the corresponding data can thus be measured reasonably easily. However, the variable representing unallocated reserve resources is less clearly defined and represents a quantity that is hidden from direct observation. In this case, the model was parameterised by making hypotheses about the level of stored resources as a proportion of observed biomass. This points to the fact that, even though canonical modelling can allow us to model processes that are not understood in detail at a higher level of abstraction and thus reduce the need for very detailed data and experimentation, we still need appropriate data and experimentation to construct and validate the model. A final limitation concerns the adaptation of canonical modelling to simulating plant function. The canonical modelling approach was originally applied to biochemical modelling applications (Savageau, 1969; Voit, 1991, 2000), where the dynamics of interest are how the system behaves when perturbed from a steady state. In this context, the relationship between the biochemical processes and the form of the canonical power-law flux functions and their rateconstant and kinetic-order parameters has been clearly demonstrated (Voit, 1991, 2000), as has the way that this canonical form can facilitate parameterisation. It has also been shown, in this paper and elsewhere (Voit & Sands, 1996a; Kaitaniemi, 2000; Renton, 2004), that the approach can be used to model systems that exhibit growth behaviour, such as plants, where we are interested in the longer term dynamics of how the system changes from one state to another. However, in extending and adapting canonical modelling to simulate plant function, the relationship between the biological processes and the form and parameters of the functions may become less clear, or at least more abstract. It may also reduce the usefulness of some of the parameterisation techniques developed for biochemical canonical modelling (such as flux-based estimation) (Voit, 2000). Canonical modelling strengths and uses The main strength of the canonical modelling approach is that it can be used to model plant function at an ‘intermediate’
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level of abstraction, between that of mechanistic models aiming for a high degree of realism, and empirical models aiming simply to describe observed patterns. Like more detailed process-based models, canonical plant models include a representation of underlying processes, mechanisms and interactions, and therefore share some of the advantages that mechanistic models have over empirical models. Firstly, the process of constructing the model requires the modeller to construct a set of causal hypotheses regarding the observed behaviour and represent these hypotheses in a ‘formalised’ and precise way. Secondly, the finished model can have explanatory value; the model is capable of explaining observed behaviour in terms of underlying mechanisms. Thirdly, the model can be used to predict behaviour in a wider range of conditions and situations beyond those in which the original data used to construct them was collected (Kaitaniemi, 2000). Finally, because of these first three attributes, the model can function as a theoretical framework for experimental investigations. However, unlike the detailed process-based approach, the canonical modelling approach can be used when detailed knowledge or data regarding underlying physiological or chemical processes is not available. This is because the compartment model can be formulated at a relatively abstract level, the flux functions are of a standard power-law form, and the parameters of these functions can be found empirically by fitting model output to relatively easily observed global variables, rather than by direct measurement of physiological processes (Voit & Sands, 1996a; Kaitaniemi, 2000). Both the examples presented in this paper were constructed without detailed knowledge of physiological processes. The cotton model is based on hypotheses regarding causal mechanisms and physiological interactions, but these were formulated and represented at a general and nonquantitative level. In a situation such as this, where there seems to be little conclusive evidence for what physiological processes are involved in compensation after defoliation (Thornby, 2004), a more abstract style of modelling, such as the canonical modelling approach, may be the only option. Even if the eventual aim is to construct a more detailed and process-based model, a canonical model can act as a ‘placeholder’ for modelling parts of the system that are not yet well understood, while less abstract modelling can be used to represent those parts for which a more detailed understanding already exists. Canonical models can be used to produce accurate output. When parameters are fitted using data regarding the whole plant, canonical models will tend to be more accurate at the global level than more mechanistic process-based models with parameters fitted at the level of the underlying processes. The accuracy with which the canonical cotton model output fits the observed average data is illustrated in Figs 10, 12 and 13. By adding a stochastic element, the same model was also successfully adapted to accurately simulate the general variability in growth patterns between the plants observed (Renton,
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2004). Furthermore, because the canonical form of the flux functions is relatively flexible, the models will tend to be more accurate than less flexible empirical models. This is evidenced by the fact that the canonical models of bounded growth gave a better fit than the logistic model, according to the comparison of r 2 and the outputs shown in Fig. 7. We have shown that canonical modelling can be used directly, or adapted, to simulate a range of aspects of plant function, such as resource acquisition and growth, limits on growth, storage, allocation and suppression, as well as noncontinuous changes in behaviour, such as the triggering of fruiting. Other studies have shown how canonical modelling can be used or adapted to simulate environmental influence (Voit, 1993; Renton, 2004); prioritised allocation of resources (Renton, 2004); conversion between continuous biomass quantities and discrete numbers of shoots at the end of a season (Kaitaniemi, 2000); and thinning dynamics (Voit, 1988) and biomass budgets and growth (Voit & Sands, 1996a,b) in tree stands. Canonical modelling can be used to represent plant function at different levels of mechanism, resulting in FS models with a range of different uses, as illustrated by the two examples given in this paper. The amaranth model illustrates the empirical end of this spectrum, with the causal hypotheses represented by the model being very simple and very general (‘The rate of change of the height of the plant is related to the height of the plant’, for example). A descriptive/empirical FS model like this might be used to visualise, describe or communicate the results of an experiment, or alternatively, it could be used in a broader theoretical investigation of light interception patterns changing with the growth of the plant, or a simulation of patterns of insect movement through the structure of the plant over time. The cotton model illustrates the more mechanistic end of the spectrum, with the underlying causal hypotheses being more complex and specific (‘Defoliation causes the production of some substance that, in turn, triggers an increased allocation of available resources to leaf biomass production’, for example). The ‘presence’ and ‘absence’ variations of the model can be seen as alternative hypotheses explaining the observed compensation for defoliation in cotton. An explanatory/mechanistic FS model like this could be used as the basis for further experiments in an ongoing process of investigation, by using the alternative explanatory hypotheses to generate alternative predictions, and designing further experiments to test these predictions. If this process of testing and refining the model was continued, against a wider range of experimental results, they the model would eventually have the potential to be used for management applications, such as decision-making, prediction or control. The canonical modelling approach can be used to construct intermediate-level models of plant function, which can then be linked to structural plant models to produce both empirical and mechanistic FS plant models. Future research could further develop the usefulness of the approach by showing how the approach can be applied to a wider range of
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experimental and modelling situations; simplifying and possibly automating the process of parameter estimation; and exploring in greater depth and eventually formalising the process of linking the canonical model to the structural model.
Supplementary material The following material is available as supplementary material at http://www.blackwellpublishing.com/products/journals/ suppmat/NPH/NPH1330/NPH1330sm.htm Animation 1 Output from the FS model of the growth of the amaranth plant. Animation 2 Output from the FS model of the cotton plant’s growth and response to defoliation. The output of the underlying canonical model of function is shown on the right, while the linked structural representation is on the left. Removed leaves are shown briefly in red, removal sites are shown as small orange spheres, individual leaf biomass is shown to the right of each leaf, and total days and total leaf biomass are shown at the base of the plant.
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