#### Canonical modelling

Canonical nonlinear models, such as GMA-systems or S-systems, 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 *f*_{2} originates from the *x*_{1} compartment and is influenced by the *x*_{3} compartment, so it would be written as

- ( Eqn 1)

where α_{2} is the rate constant parameter and *k*_{21} and *k*_{23} 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 *x*_{1} would be

- (Eqn 2)

- ( Eqn 3)

which is normally written in the abbreviated form

- (Eqn 4)

The other two differential equations representing the model would be

- (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 *f*_{i} 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

- ( Eqn 7)

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 *x*_{d} defined to be equal to *x*_{max}–*x*, where *x*_{max} 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:

- ( 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 *x*_{d} 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 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 *f*_{i}. Environmental variables could also directly affect other model variables. For example, the maximum size of the plant *x*_{max} 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 (*x*_{u}) and an additional structural-allocation flux (*f*_{u}), as shown in Fig. 4 (left-hand panel), for the unbounded growth model. The resulting equations representing the system are

- ( 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 (*x*_{u}), one for the above-ground (shoot) parts of the plant (*x*_{s}) and one for the below-ground (root) parts of the plant (*x*_{r}), linked as shown in Fig. 4 (right-hand panel). The corresponding set of equations is then

- ( Eqn 11)

- (Eqn 12)

- ( 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 (*x*_{u}), another for fruits (*x*_{f}), and a third for other shoot components (*x*_{s}). It also includes a Boolean switch variable *x*_{b}, which is defined to be equal to zero when *x*_{u} is below a certain threshold, and one when *x*_{u} is above this threshold. Assuming that the fruit flux function is written in the standard way,

- ( Eqn 14)

this means that fruiting is ‘turned on’ when enough free resources are available. The model also includes a supplementary variable *x*_{p} (defined to be equal to *x*_{f} + 1) which influences the shoot allocation flux. The shoot flux function is then written in the standard way

- ( Eqn 15)

and the parameter *k*_{sp} is constrained to be less than zero. When there is no fruit (*x*_{f}= 0 and *x*_{p} = 1), shoot allocation is ‘normal’, but when there is fruit (*x*_{f} > 0 and *x*_{p} > 1), shoot allocation will be reduced according to the amount of fruit, with the value of *k*_{sp} 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.

#### 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 and testing are available elsewhere (Thornby *et al*., 2003; Renton, 2004; Thornby, 2004).