Functional Structural Models and the Simulation of Plant Growth and Development

Two different strategies are explored in the development of functional structural models. The first one focuses mainly on the ability of computers to simulate complex systems and seems more attractive for physiologists and computer scientists. The attempt here is to imitate as closely as possible the details of the plant architectural development, including the parallel functioning of meristems, and the plant physiological functioning, including the biomass circulation in the plant topological and geometrical structure. Eventually the 3-D geometric representation of the plant is obtained as a result of its functioning. In such models, the plant physiology is sometimes described down to the cell level (e.g., stomata functioning).

To cope with such a level of detail, complex simulation software is developed. These models integrate in detail the components of plant morphogenesis and since they consider smaller scales, they do not take into account the relevant simplifications underscored by agronomists at plant scale (simplifications recalled in the previous sections, like the concept of common pool). A bottleneck of this approach is hit when the geometrical and topolo-gical structures become too complex, as in trees, since each meristem is dealt with individually to determine organogenesis and the whole plant structure and geometry have to be explored to compute light interception and matter circulation.

Although these type of models are still far from providing practical applications for agronomy, forestry, or ecology, they remain interesting for plant physiologists as tools of knowledge capitalization for teaching and research.

The second strategy endeavors building mathematical models, solving inverse problems, and developing optimization methods, and is of particular interest for agronomists and automation specialists. It aims at extrapolating PBMs at organ level by merging the botanical knowledge on plant development with the functional equations. It has led to mathematical models of plant growth and development. Contrary to the simulation models, the computation of the plant physiological functioning bypasses the exploration of the whole plant structure. And yet, if necessary, the complete plant architecture can be built.

The mathematical formalism has obvious advantages for applications in agronomy, forestry, and ecology as it makes it possible to study the system behavior, to estimate

Factorization of plant architecture

Depending on the type of plants, phytomers are set in place rhythmically or continuously. In the rhythmic case, the plant grows by successive shoots of several phytomers produced by buds. The apparition of these shoots defines the architectural growth cycle. A growth unit is the set of phytomers built by a bud during a growth cycle. These can be of different kinds and ordered according to botanical rules, like acrotony. Plant growth is said to be continuous when meristems keep on functioning and generate phytomers one by one. The number of phyto-mers on a given axis (that is to say generated by the same meristem) is proportional to the sum of daily temperatures received by the plant, as detailed before. The growth cycle is defined as the thermal time unit necessary for a meristem to build a new phytomer.

In both continuous and rhythmic cases, the chronological age of a plant (or of an organ) is defined as the number of growth cycles it has existed for, and the organogenesis is used as the time step to model the plant as a discrete dynamical system.

Quite naturally, owing to the simultaneous functioning of the mersitems, parallel rewriting grammars introduced by Lindenmayer (L-systems) are most widely used to model plant organogenesis, both for simulation and mathematical models, but in different ways.

At each growth cycle, cohorts of similar organs are created. Simulation models handle each of them individually, which may lead to cumbersome computation in the case of tree growth simulations, as the number of organs may exceed several millions. However, it is not usually necessary to consider local environmental conditions at the organ level. Thus, we can suppose that all organs of the same kind, created at the same growth cycle, behave identically. From a modeling point of view, it leads to a powerful structural factorization of the plant, based on botanical instantiations derived from the concept of physiological age (see above). Compact inductive equations of organogenesis can thus be deduced.

Let P be the maximum number of physiological ages in the plant. It is generally very small (P<5). From the botanical description of the plant, we know that phytomers and buds are the elementary bricks of the plant structure. At growth cycle t, a metamer is characterized by its physiological age p, the physiological age of its axillary branches q (with q > p), and its chronological age n. It is denoted by mi (n). These three indices p, q, and n are sufficient to describe all the phytomers and their number grows linearly with '. A bud is only characterized by its physiological age p and is denoted by Sp.

The terminal bud of a plant axis produces different kinds of metamers bearing axillary buds of various physiological ages. These buds themselves give birth to axillary branches and so on. A substructure is the complete plant structure that is generated after one or several cycles by a bud. In the deterministic case, all the substructures with the same physiological and chronological ages are identical if they have developed at the same moment in the tree architecture. At cycle t, a substructure is thus characterized by its physiological age p and its chronological age n. It is denoted by Sp (n). Since the physiological age of the main trunk is 1, at growth cycle t, the substructure of physiological age 1 and of chronological age t, S| (t), represents the whole plant. Figure 12 illustrates the way substructures are organized. The total number of different substructures in a plant of chronological age t is very small, usually less than 30, even if the total number of organs is high. Substructures and phytomers are repeated a lot of times in the tree architecture, but they need to be computed only once for each kind (Figure 12).

We use the concatenation operator to describe the organization of plant phytomers and substructures and deduce their construction at growth cycle t by induction, as follows:

• substructures of chronological age zero are buds:

• if all substructures of chronological age n - 1 are built, we deduce the substructures of chronological age n:

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