A canonical toolkit for modeling plant function

Transcription

1 F S P M 0 4 A canonical toolkit for modeling plant function M. Renton, J. Hanan, K. Burrage ACMC, University of Queensland, Australia Introduction From seeds, forms emerge, growing and evolving and interacting with each other and with their environment to produce what we see when we walk into a garden, a field or a forèst. As these plants provide resources such as fard for us to eat, timber for us to build with and fibre to clothe ourselves with, it is important to understand how they grow. Computational mortels can provide a theoretical framework for experimental investigations aimed at deepening our understanding of plant growth. However, the construction of a detailed mortel of a plant's physiological processes may require a large amount of expensive and time consuming experimentation. Such mortels can potentially become very complex and as diflicult to understand as the systems they represent. On the other hand, if the functional aspect is modelled at a very simple level, the resulting mortel may net be capable of representing the causal processes that we are interested in. Canonical modelling [9] provides an intermediate-level approach to modelling plant function capable of simulating plant function in a way that is less complex, yet capable of representing mechanistic aspects of plant function at a range of levels of abstraction. Canonical models of plant function can also be linked to structural and visual representations such as L-systems in a number of ways to give functional-structural (FS) plant mortels [2, 3, 4]. This paper will describe a basic trot kit for building such canonical mortels. Canonical Modelling Basics The term canonical mortel [8, 9] is used to signify a compartment or mass flow model [1] (Figure 1). Rather than representing a flux or flow between compartments based on knowl-edge or assumptions about the actual processes, the expressions of the flux function are ail of a standard 'canonical' form. The canonical mathematical form of the equations can facilitate the processes of parameter estimation and mortel analysis [8, 9]. The general nature of the equations makes the models flexible, and able to simulate a wide range of behaviour. 1 2 A well established canonical form for fluxes is the non-linear power-law form f = αx k 1 x k 2..., where the x i variables are the state variables of the mortel that influence the flux f, and α and the k i values are constant parameters [8, 9]. The most important variant of these approaches for the purpose of plant modelling is the generalised mass action (GMA) system. A GMA-system is a compartment and flux model with each flux represented separately by an equation of the form above. The mechanistic aspect of canonical modelling is primarily in the qualitative formulation of the compartment/flux representation of the system, such as deciding which compartments, fluxes and influences to include. A detailed quantitative understanding of the underlying physiological mechanisms is net required, because the flux functions are of a standard canonical form, and the parameters are determined empirically, by fitting model output to observeddata. Estimation of parameters can be facilitated by using methods particular to canonical modelling, such as reformulation of equations and variable scaling [2] and flux-based estimation [5, 8, 2], in conjunction with more general techniques, such as computational optimisation algorithms and ad hoc fitting [l, 2]. 4th International Workshop on Functional-Structural Plant Models, 7-11 june 2004 Montpellier, France Edited by C. Godin et al., pp

2 A canonical toolkit for modeling plant function 227 Figure 1: Examples of compartment models. Compartments are represented as circles, fluxes as solid arrows and influences by dotted arrows. Each compartment is associated with a variable representing the size of the compartment. In the one compartment canonical model of unbounded growth (left), the variable x represents the size of the plant. In the three compartment canonicai model of resource storage, allocation, and growth (right), the variables a, s and r represent allocable resources, stem biomass and root biomass respectively. In both models the flux f i represents resource acquisition, while in the second model f r and f a represent allocation to root and stem respectively. In canonical models fluxes are assumed to always have a positive value. This means that the values of the α parameters (known as the rate constant parameters in biochemical applications) are always greater than zero. However, the k parameters (known as kinetic order parameters) may be either positive or negative. When positive, they indicate that the variable to which they are attached has a positive or promotional effect on the size of the flux. When negative, they indicate that the variable to which they are attached has a negative or inhibiting effect on the size of the flux. Resource Acquisition and Growth The simplest way to model growth with a canonical model is with just one compartment and one flux, as shown in Figure 1 (left). In this case, the compartment and its associated variable x represents the size of the whole plant, which coula 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 effected by leaf area and root mass. Representing this system using the power-law flux representation gives the differential equation k x& = f i = α x (l) Figure 2 (left) shows some possible solution curves for equation 1 for different parameter values. It is apparent that a range of different unbounded growth behaviours can be modelled with this simple representation, including increasing growth rates, decreasing growth rates and constant growth rates. In reality, plants do net grow without bound (net even the rampant cucumber vines in my garden), and so we need to consider ways of placing bounds on growth. One simple way to represent bounded growth with a power-law form uses a single flux f i, and a supplementary variable x d. We can define x& = f = α i k x ( x max k h x xd x) h, (2) = α where α, x max, h and k are ail positive parameters. This system clearly has a stable fixed point at x = that corresponds to an upper bound on growth. Figure 2 (right) shows a range of possible x max solution curves for equation 2 for different parameter values. Notice that the logistic equation, which is commonly used to mortel bounded growth, is a special case of equation 2.

3 228 M. Renton et al. Figure 2: Different behaviours displayed by the canonical unbounded (left) and bounded (right) growth model with diflerent combinations of values for the parameters (as shown in the legends). Storage and Allocation By adding more compartments and fluxes, canonical models can be used tu mortel processes of storage and allocation [2]. For example, we can consider a model with three compartments, une for the stored allocable resources (represented with an a for 'allocable'), une for the above-ground parts of the plant (represented with an s for 'shoot') and une for the below ground parts of the plant (represented with an r for 'root'), linked as shown in Figure 1 (right). By varying parameter values, this model can be used to represent a wide range of different allocation and resource storage strategies. Other Tools Simple canonical tools such as these have been used to represent a range of other plant growth phenomena [2]. With the addition of negative influences from compartments to allocation fluxes, they can represent suppression, where the presence of one part of the plant suppresses the growth of another part. By using piecewise defined flux functions, they can model priority allocation scenarios, where allocation to some types of plant component (fruits, for example) is prioritised over other types (leaves and stem, for example). They have been used to simulate the maintenance of a functional equilibrium, such as root-shoot ratio, through adding influences between compartments and allocation fluxes. Triggering of a change of resource allocation patterns, corresponding to fruiting for example, can also be represented using boolean variables within canonical models. Finally, environmental influences such as temperature, nutrient levels, stress, or damage have also been modelled by using independent variables which influence fluxes and actions on model variables. Amaranth Example Canonical modelling was used to represent the elongation of the leaves and main stem of an amaranth plant [2]. The bounded growth equation 2 was used for both the leaves and the main stem length (the plant is determinate). We also considered a logistic model for the purpose of comparison. The canonical and logistic mortels were fitted as accurately as possible to the same measured data. We found that the relative error (1 - r 2 ) for different components rangea from about 40% to 98% less for the canonical mortels than the logistic model. The canonical models of leaf and main stem elongation were then linked to a structural L-system representation, to create a relatively empirical FS model [2], as shown in Figure 3. SESSION 5 ORAL PRESENTATIONS

4 A canonical toolkit for modeling plant function 229 Figure 3: Descriptive structural and visual model of the growth of an arnaranth plant. Cotton Example Canonical modelling was also used to investigate a number of hypotheses regarding the way that cotton plants compensate for defoliation [7, 2]. For example, the two compartment models shown in Figure 4 represent two alternative hypotheses regarding the causal mechanisms of defoliation. These were translated into canonical equations, parameterised to fit observed data, and linked to a structural L-system representation [6, 7], to create a FS model that expresses a range of mechanistic hypotheses regarding resource acquisition and growth, allocation of resources, and compensation for defoliation [2] (Figure 5). Figure 4: Compartment models of growth and compensation in cotton expressing alternative hypotheses that compensation following defoliation is triggered by the absence of some substance (left), or the presence of some substance (right). Conclusions The strength of the canonical modelling approach is that it can be used to model plant function at an 'intermediate' level of abstraction, between that of deeply mechanistic models aiming for a high degree of realism, and empirical models aiming simply to describe observed patterns. Because parameters are fitted at the level of the observed data and the canonical form of the functions is very flexible, the models will tend to be more accurate than both less flexible empirical models and more mechanistic models with parameters fitted at the level of the underlying processes. The approach can also be used when detailed knowledge of these underlying processes is not available. On the other hand, the possibility of including different numbers of compartments allows the models to represent mechanistic hypotheses at a range of levels. This means that canonical models can act both as a theoretical framework for experimental investigation into causal mechanisms [2, 7, cotton model] and as a more empirical way of representing the way that plants respond to different conditions [2, 3, amaranth model].

5 230 M. Renton et al. Figure 5: Model of a cotton plant's growth and response to defoliation. The plot on the right shows the state variables of the canonical model changing over time. These state variables control a structural model, which is then translated into the visual representation on the left. The numbers between the plot and the plant structure are the sizes of the individual main stem leaves. References [1] James W. Haefner. Modeling Biological Systems: Principles and Applications. Chapman and Hall, New York, [2] M. Renton. Form, Function and fihangipanis: Modelling the Pattems of Plant Growth. PhD thesis, University of Queensland, to be submitted January [3] M. Renton, J. Hanan, and P. Kaitaniemi. The inside story: including physiology in structural plant models. Proceedings of the international conference on computer gmphics and interactive techniques in Australia and South-East Asia (Graphite 2003), pages , [4] M. Renton, P. Kaitaniemi, and J. Hanan. Functional-structural tree modelling combining canonical modelling of function, architectural analysis and L-systems. Ecological Modelling. (submitted). [5] P. J. Sands and E. O. Voit. Flux-based estimation of parameters in S-systems. Ecological Modelling, 93:75-88, [6] D. Thornby. Using New Computational Tools ta Inuestigate the Responses of Cotton Plants (Gossypium Hirsutum L. ) to Defoliation. PhD thesis, University of Queensland, [7] D. Thornby, M. Renton, and J. Hanan. Using computational plant science tools to investigate morphological aspects of compensatory growth. Lecture Notes in Computer Science, 2660: , [8] Eberhard O. Voit. Canonical Nonlinear Modeling: S-System Approach ta Understanding Complezity. Van Nostrand Reinhold, New York, [9] Eberhard O. Voit. Computational Analysis of Biochemical Systems. Cambridge University Press, UK, SESSION 5 ORAL PRESENTATIONS