The L-PEACH plant model is expressed in terms of modules that represent individual plant organs. An organ may be represented as one or more elementary sources or sinks of carbohydrates. The whole plant is modeled as a branching network of these sources and sinks, connected by conductive elements. Our model extends the approach proposed by Thornley & Johnson (1990), and further developed by Minchin et al. (1993), according to which the flux of carbohydrates is proportional to osmotically generated differences in hydrostatic pressure, and inversely proportional to a resistance to transport. As (for a given temperature) osmotic pressure is proportional to the concentration of carbohydrates, the fluxes can be related directly to the differences in concentration. In contrast to the previous models, we allow the branching networks representing plants to be arbitrarily large. In general, all elements of the network may have a nonlinear and time-dependent behavior.
The plant model is interfaced with a model of the light environment, which calculates the distribution of light in the canopy using a quasi-Monte Carlo method. This interface is implemented using the formalism of open L-systems (Mech & Prusinkiewicz, 1996). Simulation proceeds in user-defined time-steps that can correspond to real time (e.g. days). In each step the local distribution of light in the canopy is computed, and is a factor influencing production of carbohydrates by the leaves. The plant model is also sensitive to the amount of available water, which influences both the production of carbohydrates by the leaves and the uptake of carbohydrates by various sinks. In contrast to the architecturally detailed model of carbon assimilation, transport and partitioning, tree water use and water stress are calculated at the whole-canopy level. The water demand for each individual leaf is a function of light exposure, and all the individual leaf demands are summed to determine the whole-canopy water demand. The ability of the root system to provide water is determined by root system structural biomass, the soil volume available to the tree, a user-defined soil moisture release curve, and the user-defined irrigation schedule. The ratio of canopy water demand and root water supply capability provides an index of the water stress in the tree at any given time – as the value of this ratio goes down, the impact of water stress on tree growth and photosynthesis increases.
The L-PEACH model is developmental, with the buds producing new stem segments, leaves, fruit, etc. Each simulated growing season is initiated with bud break. The growth of organs initiated during the previous season (preformed) as well as the subsequent initiation of new organs (neo-formed) is controlled by the amount of available carbon. If the carbon supply is insufficient for growth and/or maintenance, organs (fruits, leaves or branches) are shed by the tree. Thus the development and growth of the branching plant structure (topology and geometry) are closely coupled with the production and partitioning of carbohydrates.
The L-PEACH simulation algorithm
The formalism of L-systems automatically couples the tree structure with the topology and parameters of the carbohydrate-supply network that represents the sources, sinks and conductive elements. At the heart of this coupling lies the notion of context-sensitive L-systems (Lindenmayer, 1968; Prusinkiewicz & Lindenmayer, 1990), which provides a convenient means of capturing connections between elements of a growing structure at each stage of its development. Given this information, L-systems are used to compute the distribution of carbohydrate, its concentrations, and fluxes at each step of the simulation. Efficient implementation of this computation is the main methodological innovation of the L-PEACH model.
Within L-PEACH, the plant is modeled as a growing network comprised of elements that represent individual organs such as leaves, stem segments, fruit, buds and roots. The behavior of each type of organ is given by a set of user-defined functions. For example, a mature leaf is characterized by its source strength which, in turn, depends on the amount of mobilizable carbohydrates that have been accumulated in the leaf as a result of photosynthesis. During each time-step, these accumulated carbohydrates can flow into the various sinks within the tree (roots, fruit, etc.). Stem segments, in addition to being potential sources or sinks, act as conduits for the fluxes throughout the tree. The magnitude of these fluxes depends on the differences in carbohydrate concentrations between sources and sinks, and the resistances of the intervening paths. All elements may exhibit nonlinear behavior, meaning that the resistances may depend on concentrations.
In general, the network representing a growing plant has a dynamically changing structure (its topology changes over time); is nonstationary (the values of parameters associated with the various organs change over time); and is nonlinear (the resistance associated with a given sink depends on the potential at the sink's attachment point). L-systems are used to ‘develop’ the plant (development here denotes changes in the network topology that result from the addition of new segments by buds, or the loss of organs through shedding), and to solve the set of equations defined by the network at any given point in time. These equations are solved numerically, by taking advantage of the branching topology of the network.
In order to calculate the accumulation, flow and partitioning of carbohydrates between the individual components of this network, we rely on an analogy to electric circuits and employ equations developed in linear circuit theory. The underlying correspondence between biological and electrical quantities is summarized in Table 1. The fundamental concept is to identify the amount (mass) of mobilizable carbohydrates with an electric charge. Other correspondences are a straightforward consequence of this identification. The only nonintuitive notion is the source/sink strength, the analogue of electromotive force. It can be thought of as the concentration of carbohydrates inherent in an organ (a source or a sink), as it would be measured in the absence of flow through resistive conductive elements associated with that organ.
Table 1. Correspondence between biological and electric quantities employed in the model
|Biological quantity||Electric counterpart||Symbol|
|Amount (mass) of carbohydrate||Charge||q|
|Rate of photosynthesis||–||dq/dt|
|Source/sink strength||Electromotive force||e|
|Resistance (to concentration-driven flow)||Resistance||r|
There are two types of connection between the elements of the modeled tree: a serial connection between two consecutive elements, or a parallel connection that occurs at a branch point. Using standard rules for combining components in an electric circuit, it is then possible to reduce any two connected elements into an equivalent single element. For serial connections, the combination is done as shown in Fig. 1a, where r and e (on the right side of the figure) are the resistance and the source/sink strength of the network that results from the combination of the two separate elements shown on the left side of the figure. Following the same labeling conventions, the combination of two parallel elements is shown in Fig. 1b.
Figure 1. Schematic representation of how elements of the model are added, in series (a) or parallel (b). Rectangles, resistances; circles, sources or sinks.
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Given these two rules, context-sensitive L-system productions are used to calculate fluxes and concentrations at the nodes (i.e. the connection points between elements), given the source/sink strengths and resistances to carbohydrate flow. The calculation is performed in two phases, which we refer to as the folding and unfolding of the network. In the folding phase, information is passed from the tips of branches to the base of the tree. The branching network is then gradually simplified by combining the elements into a sequence of ever more inclusive equivalent networks. For example, let us consider the branching axis comprised of a series of elements, shown in Fig. 2a. Working from right to left in this figure, network elements are combined, or ‘folded’, into equivalent simpler networks. In the first step, the elements shown in white (Fig. 2b) are combined into a single element. The resulting simplified topology is shown in Fig. 2c, where e′ is the source/sink strength of the network equivalent to the combined white elements from Fig. 2b, and r′ is the combined resistance. Proceeding in this way, the entire axis can be sequentially simplified to the network shown in Fig. 2d.
Figure 2. Schematic representation of the method used for the simplification of a long series of elements. As the process moves from right to left, a simple equivalent circuit is gradually created to represent the entire starting series.
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As there is nothing to the left of the resistance rs1 (Fig. 2d), no carbohydrate will flow through it. Therefore the carbohydrate concentration at the point of the 90° bend (v1) will be equal to the sink/source strength . Now that this concentration is known, the network can be ‘unfolded’, and the fluxes can be determined for the entire branching structure. The first step of this process is shown in Fig. 3a. Proceeding to the right, we gradually unfold the entire network into its original form, and calculate values for the concentrations and fluxes in the network (Fig. 3b). While this example deals with a simple nonbifurcating ‘branch’, structures that include bifurcations can be handled in a similar way.
Figure 3. Schematic representation of the method used to ‘unfold’ the simplified network resulting from the process shown in Fig. 2 into its original form. As this process moves from left to right, concentrations and carbohydrate fluxes throughout the modeled tree are calculated.
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If the relationship between the carbohydrate flux into a sink and the carbohydrate concentration at the point where that sink is attached were linear for each of the elements in the model, the computation of the fluxes flowing to each element of the network would be completed after the unfolding phase. However, as mentioned above, elements in the model may exhibit a nonlinear behavior, which can be thought of as the dependence of resistances and source/sink strengths on concentrations at the attachment points. Because of this, the distribution of concentrations and fluxes in the network is calculated iteratively, using an L-system implementation of the Newton–Raphson method (Press et al., 1992). First, the functional characteristics of sinks and sources are linearized for the values of concentrations obtained in the previous iteration. The resulting source/sink strengths and resistances are then used to compute new values of concentrations in the next iteration of folding and unfolding. Once this process converges on a solution, fluxes out of each source and into each sink are assigned and accounted for, and the simulation advances one time-step.