N

For all (p, q) such that 1 < p < P, p < q < P, {ufi (t)) t and (bpq(t))t are sequences ofintegers that are characteristic of the plant organogenesis; upq(t) corresponds to the number of metamers mpq in growth units of physiological age p appearing at growth cycle t; bpq(t) is the number of axillary substructures of physiological age q in growth units of physiological age p that appeared at growth cycle t. These sequences can be deterministic or stochastic.

In eqn [1], substructure Sp (n) is decomposed into:

• its oldest growth unit, called base growth unit:

• the lateral substructures borne by the base growth unit (they are one cycle younger):

• the substructure grown from the apical bud of the base growth unit (also one cycle younger):

Its decomposition is illustrated on S1(2) in Figure 12. If we add geometric rules (internode lengths, branching angles, phyllotaxy) to the construction equations, we obtain the 3-D architecture of a geometrical tree.

The mathematical equations of plant functioning

Biomass acquisition

The practical ecophysiological concepts implemented in PBMs, such as modeling of source-sink interactions and fit

Figure 12 Factorization of the plant development and inductive construction of the substructures for a plant with three physiological ages.

the common pool hypothesis, are adapted to the individual plant, at organ level. We consider two steps at each growth cycle: first, the neat biomass production (dry or fresh), and then, its repartition among all growing organs according to their sink strengths.

For high crop density, eqns [1] and [2] of resource acquisition per unit surface area (m2) can be summarized by:

I denotes the light radiation and the intercepted light is computed by the Beer-Lambert law. A is the light-use efficiency, which is the conversion coefficient of light interception to biomass.

If d is the plant density and Sj = 1 /d, Sj represents the potential surface area available for each plant. We thus have LAI = Sf (n)/Sj, with Sf (n) the total surface area of nonsenescent leaves for the individual plant. Moreover, we can integrate eqn [5] on the whole growth cycle. Thus, we obtain the equation giving at growth cycle n the neat biomass production Q(n) for the individual plant:

For lower densities, we define a virtual surface area S

p linked to Sd by a relevant empirical relationship Sp = F(Sd), and eqn [6] is rewritten:

Biomass distribution

As explained above, biomass allocation is driven by the sink strengths of all expanding organs. The sink value of an organ depends on its physiological age and on its type o (o = b, p, i, c, f for leaf blade, petiole, internode, secondary growth rings on stems, and fruits or flowers, respectively) and its chronological age j. It is denoted by p0 (i) and usually modeled by a parametric function (density function of beta law for example). From the organogenesis equations, we can easily deduce N0 (j) the number of organs of type o and of chronological age j and thus write D(n) the total plant demand at growth cycle n:

Consequently, biomass allocated to an organ of type o and of chronological age j is proportional to its sink and to the incremental pool of biomass for distribution, divided by the total plant demand. If we do not consider reserve allocation or remobilization, it is given by:

The sum of all the biomass increments for a given organ gives it weight. Using the SLW e, we can retrieve leaf surface area, which will be used to compute resource acquisition in eqn [7]. If we want to build the plant geometry, we need to compute the fresh weights, from which we can deduce organ volumes, dimensions, and shapes, thanks to allometric rules.

The general equation of plant growth

If we consider a constant SLW at the growth cycle scale, and if Ty denotes the number of growth cycles before a leaf get senescent (also supposed constant for all leaves), we can replace Sf (n) in eqn [7] by its expression as a function of the successive biomass productions and demands, to obtain the general inductive equation [9] in which development and growth are implicit:

An important variable emerges from eqn [9] - the ratio of available biomass to demand (Q/D value). It represents the global source-sink balance and the trophic competition during plant growth. It can be used as a key variable to control interactions between growth and development, such as fruit set, bud break, or vigor.

For applications that do not necessitate plant geometrical representation, this equation coupled with the allocation equations can be sufficient to describe the whole plant growth. Such a mathematical formalism provides interesting advantages for estimation of model parameters from experimental data, optimization, and optimal control problems, which makes it suitable for applications in agriculture, forestry, or ecology. The GreenLab model, developed jointly in China, the Netherlands, and France, is based on this set of equations.

Estimation of model parameters from experimental data: From architecture to functioning

The key step for model applications is its validation on real plants, by estimating the specific parameters of a given plant variety, from experimental data. Some model parameters can be measured, while some cannot be directly assessed from experimental observations. It concerns both development (when the rules of organoge-netic grammars are too complex) and growth (like the parameters of the sink and source functions). These hidden parameters have to be estimated by model inversion. Since plant growth can be modeled in the form of a dynamical system, classical methods of parameter p estimation can be used, based on the maximum likelihood criteria and Newtonian methods of optimization. The model outputs from which this identification can be achieved are the organ masses, as they can be easily measured on real plants and as they result from the plant functioning and thus keep track of the whole history of source-sink balances (Figure 13).

If we consider a monospecies population, several plants, at different ages, can be used simultaneously to form the observation vector. Complications can be induced if the population has strong intraspecific genetic variability and environmental variability. The amount of data collected is a compromise between the statistical accuracy of estimation and the heaviness of the measurements.

In most cases, the simplifications used in the physiological model are justified by the confrontation of the model to real plants, since a very small number of model parameters are sufficient to predict a large number of data. Even though all the complex phenomena underlying plant growth and development are not accounted for, the prediction ability of such models remains quite good. The reason is that the simple theoretical plant given by the model is such that its architectural trajectory is very close to that of the real complex plant.

Compared to Figure 9, the simulations of Figure 14 are more accurate, because the sizes of organs depend on the biomass production and the biomass partitioning and do not directly result from empirical data sets.

Applications of functional structural models

Functional structural models aiming at simulating detailed physiological functioning, thanks to sophisticated algorithms, are still far from current applications on crops. On the contrary, the ones that are based on dynamical equations share the same applicative philosophy and the same knowledge as PBMs and can easily be used in agriculture or forestry. Their advantage is that populations of organs instead of compartments are considered. More information about the history of growth is thus collected on the plant architecture, which allows more accurate model identification and growth description. This is particularly true when considering plant plasticity:

• Sink values or organ allometry may be modified during plant growth in response to an environmental stress. These effects can be measured and integrated in the dynamical equations quite easily, thus allowing the consideration of plant-environment interaction and stresses (light, temperature, water, density).

• Organogenesis can be strongly modified by growth as a result of growth and development interaction. Plant development may be quantitatively strongly modified and 15-year-old beech trees in the shade or in open field conditions can, for instance, respectively, be 15 cm high, without branch, or 15 m high with a vigorous crown (Figure 15)! According to Q/D value (source/sink ratio), axillary stems may develop or organs may abort, thus allowing a faithful modeling and simulation of plant architectural plasticity.

Optimized source function of leaf

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Age of organ Optimized sink function of leaf

Age of organ

Growth cycle: 8

Figure 13 Several growth stages of the same plant are fitted simultaneously at the organ biomass level with an optimized common set of parameters that insure biomass production and biomass partitioning. The organs of different types are fitted together because they share the same plant demand and production. Eventually the plant growth dynamics are controlled by a small set of constant parameters. From Zhan Zhigang, LIAMA.

Age of organ Optimized sink function of leaf

Optimized sink function of internode

Age of organ

Growth cycle: 8

Figure 13 Several growth stages of the same plant are fitted simultaneously at the organ biomass level with an optimized common set of parameters that insure biomass production and biomass partitioning. The organs of different types are fitted together because they share the same plant demand and production. Eventually the plant growth dynamics are controlled by a small set of constant parameters. From Zhan Zhigang, LIAMA.

Figure 14 Simulations of 3-D plant architectures including growth and development (GreenLab model: developed at INRIA, ECP, LIAMA, CIRAD): (a) Arabidopsis plant, LEPSE (Digiplante software: ECP); (b) beetroot plant, Institut Tecnique de la Betterave (Digiplante software); (c) wheat plant, Wageningen University (GreenScilab software: LIAMA); (d) maize plant, Chinese Agricultural University (software CAU); (e) sunflower plant: INRA/ LEPSE (Digiplante software); (f) Chrysanthemum plant, Wageningen University (GreenScilab software); (g) pine tree, Chinese Academy of Forestry (Digiplante software); (h) coffee tree, CIRAD (Digiplante software); (i) cucumber plant, CAU (Digiplante software); (j) tomato plant, CAU (CornerFit software: LIAMA).

Figure 14 Simulations of 3-D plant architectures including growth and development (GreenLab model: developed at INRIA, ECP, LIAMA, CIRAD): (a) Arabidopsis plant, LEPSE (Digiplante software: ECP); (b) beetroot plant, Institut Tecnique de la Betterave (Digiplante software); (c) wheat plant, Wageningen University (GreenScilab software: LIAMA); (d) maize plant, Chinese Agricultural University (software CAU); (e) sunflower plant: INRA/ LEPSE (Digiplante software); (f) Chrysanthemum plant, Wageningen University (GreenScilab software); (g) pine tree, Chinese Academy of Forestry (Digiplante software); (h) coffee tree, CIRAD (Digiplante software); (i) cucumber plant, CAU (Digiplante software); (j) tomato plant, CAU (CornerFit software: LIAMA).

Figure 15 Interaction between plant growth and plant development. In different incident light conditions, the tree architecture can be strongly modified. Parameters for development can be modified depending on the evolution of the ratio supply/demand during the growth process (Amelie Mathieu, Digiplante: ECP).

Figure 15 Interaction between plant growth and plant development. In different incident light conditions, the tree architecture can be strongly modified. Parameters for development can be modified depending on the evolution of the ratio supply/demand during the growth process (Amelie Mathieu, Digiplante: ECP).

For applications in agronomy, forestry, or ecology, a key question is raised by the transition from individual-based (plant) models to population-based (crop) models. Competition for space and resources are modeled and the dynamic equations can be written per unit surface area (per square meter), similar to PBMs. Optimization and control of cultivated systems can thus be achieved, like determination of optimal irrigation, fertilization, or densities.

Another important application consists in linking plant growth models with genetic models, since the stable endogenous parameters of the dynamical model are good candidates for quantitative trait loci detection.

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