Section: Application Domains
Calibration of GreenLab model on real cultivated plants
The plant architecture is a target for the mathematical model, and it is the visual result of the growth process. The hidden parameters of source and sink functions must be optimised in order to fit the best the weights and the sizes of all the organs produced by the plant development at each stage of growth. Theoretically speaking, this inverse method should be able to assess also the effect of the environment (climate and density), the leaf biomass production and the biomass partitioning in each organ from the plant architecture during the growth. The fitting can be done upon the following conditions:
The plant development must be entirely known. This includes the organ numbers, their functioning and expansion durations, their weights and sizes. Moreover allometric parameters that control the organ shape have to be assessed. It is not necessary to have the complete recording of each organ weight and size in a plant. Sparse data from the samples can be sufficient. But to be efficient, the number of measured data must be bigger enough than the number of hidden parameters.
The growth cycle must be defined according to the thermal time. This needs to follow the plant development on several stages of growth to set up the phyllochron. The average value of the environment efficiency En must be known at each G.C.. If no information is available about climate (that is often the case), the value is supposed to be a constant. Slight variations of En usually have no important effect compared to a constant climate, because they are smoothed by the successive organ expansions.
Generalized Least Square Method was used for parameter optimisation of the model. The application of this method to GreenLab was described by Zhan et al.  and Guo et al.  . Advantages of this method are that it provides rapid convergence and the standard error linked to the parameter values thus indicating the accuracy of the solution. Fitting process means to compare the observed organ weights and sizes, to the model prediction values, so it is not simply curve-fitting. Each class of organs (leaves, internodes, fruits) is a different output of the model corresponding to a set of hidden parameters. In a given class for a given plant age, the variation of the organ age controls its behaviour during the growth.
Fitting can be done on a single architecture (single fitting), or on several stages of growth to follow the trajectory of the dynamical process (multi-fitting). This second case is more accurate. In both cases all the data are fit in the same time by the same parameters set. If Data on root system are available they can be taken into account.
The Chinese Agriculture University (CAU) has a tight collaboration with Digiplante and its associate team in Liama, for developing, testing and using GreenLab model. Calibration experiments have been undertaken successfully in CAU on several plants (Wheat, Cotton, Maize, Tomato, ...) and other are in progress ( Rice, Soybean, Pine tree, ...). Here, as a good example we present the Maize case (see Guo et al. 2006 for details). The measurements have been carried out on several stages of growth (8,12,16,21,27,30 G.C.), so multi-fitting is possible. But the plants have to be sacrificed for the measurements at each stage. This introduces noises in the data, linked to different local environments. Nevertheless we can accept this drawback if the plantation is homogeneous. The fitting is done on maize that has a finite development with 21 metamers for the Chinese cultivar. The architecture begins with metamers that have short internodes and is ended by the tassel. The cob location is on the 15th internode. The growth still continues and the expansion of organs acts until GC 33. It is obvious that the cob gets a big sink. The parameter E here is chosen to be the average potential transpiration ETO during the GC. So the resistance r to water transpiration is linked to the water use efficiency. The problem was to compute the functioning of this plant from the multiple growth stages and to solve the biomass production and the biomass partitioning at each GC.
Here it is obvious from Figure 11 , that the GreenLab model works well. We need to compute 12 parameters belonging to the source and sink functions for the calibration, meanwhile the number of data to fit are about 400. The number of organs is few: one kind of leaf, sheath, cob, tassel, and two kinds of internodes (short and long). The accuracy on the parameters that control the sink function is necessarily less for the cob than for the leaf, because there is only one cob and there are twenty leaves on the Maize plants. Here we are sure that a same set of constant parameters controls the plant growth, because the trajectory of the dynamical process is captured thank to 6 intermediate stages of growth.
Biomass Production and Biomass Partitioning. Once the problem of assessing the hidden parameters is done, the problem of biomass production and biomass partitioning is fulfilled. The model gives the amount of biomass fabricated by the plant at each stage of growth and how it is shared into different compartments (figure 12 ).
Simulating 3D. Simulations of the 3D architectures are shown Figure 13 . The 3D organs come from digitalisation and their sizes are related to their weights thank to their allometric rules.
The excellent results obtained on Maize in CAU are similar on other plants like Tomato, Rice and Cotton, and they are to be published in 2006. The model seems really to be versatile.
Generalization of the sources and sinks concepts in a plant
Participant : V. Letort.
Plants with simple architectures as Maize or Sunflower are not often encountered. In such plants all metamers can be measured for sizes and weights at any growth stages. Usually plants have more or less complex branching patterns that make the recording of the plant structure quite tedious. Therefore it is relevant to simplify the measurements using the substructure formalism that allows transforming a substructure in a meta-organ.
The meta-organ is both source and sink, and its functioning is the result of the sum of the functioning of underlying organs. GreenLab model allows computing the emergent properties at the level of the meta-organ. Several levels of aggregation are possible that needs adapted strategies for plant measurements and Data processing. This generalisation is the subject of V Letort PHD at ECP. It should lead if successfully, to analyse complex trees architectures.
Theoretical issues on plant fitting with the generalized least square method
Applying the GLSQM on the equations of GreenLab model, needs several statistical studies . What about several minima, or what about the sensibility related to the parameters?, or how to compare two plants from their parameters sets? Such study is carried out by PH Cournede and F. Houllier of Inra.