Dong Li

Véronique Letort

Yan Guo

Philippe De Reffye

Zhigang Zhan
email: zhigang.zhan@cau.edu.cn

Modeling branching effects on source-sink relationships of the cotton plant

Keywords: Functional-structural model, Parameter optimization, Source-sink relationship, Simulation

Compared with classical process-based models, the functional-structural plant models provide more efficient tools to explore the impact of changes in plant structures on plant functioning. In this paper we investigated the effects of branches on the sourcesink interaction for the cotton plant (Gossypium hirsutum L.) based on a two-treatment experiment conducted on cotton grown in the field: the singlestem plants and the plants with only two vegetative branches.

It was observed that the branched cotton had more organs for the whole plant but the organs on the trunk were smaller than those on the single-stem cotton. The phytomer production of the branches was four or five growth cycles delayed compared with the main stem. The organs on the trunk had similar dynamics of expansion for both treatments. Effects of branches were evaluated by using the functionalstructural model GREENLAB. It allowed estimating the coefficients of sink strength to differentiate the biomass acquisition abilities of organs between different physiological ages. We found that the presence of the two vegetative branches increased the ground projection area of plant leaves and had led to slight changes on the directly measured parameters; the potential relative sink strengths of organs were found similar for the two treatments.

Introduction

The functional-structural models simulate the architectural development and physiological functioning of plants. Most of them describe the source-sink relationships among the plant organs, see for instance L-PEACH for peach tree [START_REF] Allen | Using L-systems for Modeling Source-Sink Interactions, Architecture and Physiology of Growing Trees: the L-PEACH Model[END_REF], LIGNUM for pine [START_REF] Perttunen | LIGNUM: A Tree Model Based on Simple Structural Units[END_REF][START_REF] Sievänen | Evaluation of Importance of Sapwood Senescence on Tree Growth Using the Model Lignum[END_REF].

GREENLAB is one of these models. It aims at simulating and predicting the growth and structural development of plants [START_REF] Zhan | Fitting A Functional-Structural Growth Model with Plant Architectural Data[END_REF][START_REF] Yan | A Dynamic, Architectural Plant Model Simulating Resource-Dependent Growth[END_REF]. The functional parameters that control the processes of biomass production and allocation are estimated in parallel from data of organ biomass. GREENLAB has been calibrated on some unbranched plants, such as maize [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF][START_REF] Ma | Parameter Stability of the Functional-Structural Plant Model GREENLAB as Affected by Variation within Populations, among Seasons and among Growth Stages[END_REF][START_REF] Ma | Parameter Optimization and Field Validation of the Functional Structural Model GREENLAB for Maize at Different Population Densities[END_REF], single-stem cotton [START_REF] Zhan | Study on the Effects of Defoliation on the Growth of Cotton Plant Using the Functional Structural Model GREENLAB[END_REF], sunflower [START_REF] Guo | Modeling of Biomass Acquisition and Partitioning in the Architecture of Sunflower[END_REF], and tomato [START_REF] Dong | Does the Structure Function Model GREENLAB Deal with Crop Phenotypic Plasticity Induced by Plant Spacing? A Case Study on Tomato[END_REF]. The procedure of parameter estimation thus provides new insights about the plant functioning as it allows us to trace back the dynamics of the source-sink relationship within the plant [START_REF] Li | Improving the Calibration Process of GREENLAB Model on the Cotton Plant[END_REF]. A few studies have been undertaken to analyze the growth of branched plants with GREENLAB [START_REF] Kang | The Derivation of Sink Functions of Wheat Organs Using the GreenLab Model[END_REF][START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-Source Relationships in Chinese Pine Saplings[END_REF][START_REF] Letort | Parametric Identification of A Functional-Structural Tree Growth Model and Application T1 T2 to Beech Trees (Fagus Sylvatica)[END_REF]. However, these studies were performed on trees grown in open field area that did not belong to any experimental setting. Therefore no pruning strategies were applied on these individuals.

However, it is of great interest to use functionalstructural models to analyze the influence of plant structure on its functional processes. This is precisely the reason why such kind of models has been developed. In this paper we aim at understanding the impact of the structure of cotton plants (Gossypium hirsutum L.) on their physiology in terms of biomass production and allocation. Previous models developed to simulate the growth of cotton are mainly process-based models (e.g. GOSSYM [START_REF] Reddy | Application of GOSSYM to Analysis of The Effects of Weather on Cotton Yields[END_REF] and OZCOT [START_REF] Hearn | OZCOT, A Simulation Model for Cotton Crop Management[END_REF]), therefore the effect of the pruning strategy on the source-sink balance has not yet been studied. There are already some functional-structural models to simulate the branched plant, such as L-OZCOT for cotton [START_REF] Hanan | Linking Physiological and Architectural Models of Cotton[END_REF], but little attention is put to the effect of branches.

In this paper, we present an experiment on pruned cotton with two different pruning treatments. In treatment 1, all branches were pruned (single-stem cotton); in treatment 2, only the two vegetative branches were kept. The goals of this study are [START_REF] Allen | Using L-systems for Modeling Source-Sink Interactions, Architecture and Physiology of Growing Trees: the L-PEACH Model[END_REF] to observe the effects of vegetative branch presence on cotton development and growth; (2) to simulate the two treatments using the GREENLAB model; (3) to test whether the GREENLAB model can simulate the growth of cotton plants submitted to the two different treatments with a single set of parameters and, in the opposite case, to find which parameters are different.

Materials and methods

Field experiment

The field experiments were conducted at Quzhou experiment station (36°52' N, 115°1' E) in North China Plain, from 6 May to 25 August 2006. The cotton cultivar DP99B (Gossypium hirsutum L.) was used. The seeds were sown at a spacing of 0.8 m×0.8 m. Fertilizer inputs and irrigation were conducted so as to avoid any mineral or water limitations to plant growth.

Two pruned treatments were conducted in the course of plant growth. For the first treatment (T1), all the branches were removed immediately after their initiation [START_REF] De Reffye | Calibration of A Hydraulic Architecture-Based Growth Model of Cotton Plants[END_REF]. The resulting plants were singlestemmed having only leaves kept on the main stem. For the other treatment (T2), all the branches were removed except the two vegetative branches on the 5th and 6th node were kept on the main stem, and both branches were pruned as the main stem. An example of the resulting plant was shown in Fig. 1.

Both non-destructive observations in the field and destructive measurement in the laboratory were conducted. We marked 10 single-stem cottons and every two days we measured the length of blade rib, petiole and internode to determine the expansion time of the organs.

Destructive samples were taken every two weeks. Each time, four plants per treatment were measured on dry weight and area for leaf blades, dry weight, length and diameter for internodes and petioles. Blade area was measured using a LI-COR 3100 leaf area meter (Lincoln, NB, USA). Roots were not taken into account. The thermal time was computed as the sum of mean daily air temperature minus a base temperature (15 ℃ in this study) after emergence. The climate data were from the climate station located near the experimental field. The measured organ dry weight and size data were input in a target file according to the topological structure which was subsequently used to estimate the model parameters. The target file format and optimization procedures for single-stem plants were described in [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF]. For branched cotton, the position of the two vegetative branches was recorded and their biomass data were described the same way as for the main stem. The Digiplante software [START_REF] Cournede | Structural Factorization of Plants to Compute Their Functional and Architectural Growth[END_REF] was used to fit the GREENLAB model for the two treatments respectively. For each treatment, four-stage data were used as the target to fit the parameters.

The GREENLAB model and new features

The functional-structural GREENLAB model has been previously described in detail for non-branching plants [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF][START_REF] Ma | Parameter Stability of the Functional-Structural Plant Model GREENLAB as Affected by Variation within Populations, among Seasons and among Growth Stages[END_REF][START_REF] Ma | Parameter Optimization and Field Validation of the Functional Structural Model GREENLAB for Maize at Different Population Densities[END_REF][START_REF] Zhan | Study on the Effects of Defoliation on the Growth of Cotton Plant Using the Functional Structural Model GREENLAB[END_REF]. There are some developments with the model for the branched plants [START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-Source Relationships in Chinese Pine Saplings[END_REF][START_REF] Letort | Parametric Identification of A Functional-Structural Tree Growth Model and Application T1 T2 to Beech Trees (Fagus Sylvatica)[END_REF]. We will present here the main principles of this model for branched cotton.

We define Growth Cycle (GC), which is the time elapsed between emergence of two successive phytomers expressed in thermal time, as the simulation time step. Dates of events such as phytomer appearance, as well as biomass production T1 T2 and partitioning, leaf expansion time and longevity, are expressed in GC. At each GC except the first one, GREENLAB calculates the biomass production for the plant from the total functional blade area by an empirical nonlinear function representing intercepted light. It is assumed that the biomass for the first GC comes from the seed. Biomass partitioning among the growing organs is according to their relative sink strengths. Organs can expand during several GCs and their weight corresponds to the accumulative biomass they received. Their chronological age is defined as the time elapsed since their initiation, expressed in GC. Geometric dimensions of organs, e.g. blade area and internode length, are computed according to specific allometric rules that define organ shapes.

The pruned cotton produces simple plant architectures. Each phytomer consists of a few components: a node, an internode, and a leaf (except the first phytomer with two cotyledons). Nodes are difficult to split from internodes, so the studied organs are internodes, blades and petioles.

For cotton with two vegetative branches, the two branches are assumed to have the same physiological age (denoted by PA2) while the physiological age of the main stem is set to PA1 [START_REF] Barthélémy | Plant Architecture: A Dynamic, Multilevel and Comprehensive Approach to Plant Form, Structure and Ontogeny[END_REF]. So there is only PA1 for single-stem cotton. In the model, organs of the same type (internodes, blades, petioles) have the same shape of sink variation along their life while they can have different coefficients of the potential sink strength according to their PA. Biomass acquisition at GC(i) is computed by Eqn. (1), based on the formalism of the Beer-Lambert's law for light interception in an homogeneous medium :

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = ∑ ∑ = = p 1 ) , ( 1 b p b ) , ( exp 1 ) ( ) ( S PA k S R S i E i Q m PA PA PA i n k (1)
where Q(i) is the biomass production during GC(i); E(i) is the average potential of biomass production during GC(i) depending on environmental factors (e.g. light, temperature, wind) and it is assumed constant through all the growth cycles in this study; PA is the physiological age of the axis and PA m is the largest PA for the whole plant ( PA m = 1 for T1, and PA m = 2 for T2); and n b is the number of functional leaves at the beginning of GC(i) with certain PA and it is relevant with blade functional time T f ; S b is the functional blade area (including leaves on the main stem and on the two branches); R is an empirical resistance parameter of the plant; S p is the ground projection area of the leaves, which takes into account their inclination [START_REF] Zhan | Study on the Effects of Defoliation on the Growth of Cotton Plant Using the Functional Structural Model GREENLAB[END_REF].

Biomass feeds all growing organs (sinks) through a globally shared reserve pool. The amount of biomass partitioned to individual organs is proportional to its relative sink strength which is a function of organ chronological age k in terms of GCs, with respect to the type of organ o:

) ( ) ( , , k f P C k D o o PA o PA o = (2)
where the subscript o denotes organ type: b, leaf blade; p, petiole; e, pith (In the model, internode is considered as an organ made up of pith, which elongates for a few GCs, and layers added during the secondary growth, which lasts during the whole growth period.); D o,PA is the sink strength of the organ; P o is the potential sink strength (for leaf blade P b =1 is set as reference, therefore P o is a relative value); C o,PA is the coefficient of sink strength to differentiate the biomass acquisition abilities of organs between different physiological ages (for PA1, C o,1 =1 is set); f o (k) is an organ type specific function of sink variation in their developing period with one GC as time step, taking the beta-like form:

( ) ( ) ⎩ ⎨ ⎧ > ≤ ≤ = o o o o o T k T k M k g k f 0 1 / ) ( ) ( (3)
where The secondary growth for cotton covers all the existing internodes along the whole stem for each GC similarly to the process of ring growth of trees. In this study, virtual rings were defined to model secondary growth for cotton. The ring growth rate was found tightly dependent of the number of functional leaves left on the plants. Let P c be the potential sink strength of the ring compartment. Its sink strength, denoted by D c (i), is assumed to be proportional to the number of functional leaves, i.e. the sum of n b (i,PA) (the number of functional leaves during cycle GC(i) of a certain PA; for branched cotton, the sum includes the functional blades on branches):

1 1 5 . 0 1 5 . 0 ) ( - - ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = o o o o o T k T k k g β and { } o o o T k k g M ≤ ≤ = 1
∑ = = m PA PA PA i n P i D 1 b c c ) , ( ) ( (4)
One new ring is added to each phytomer at each growth cycle. The biomass allocated to internode ring is proportional to its length. The total biomass produced at GC(i) is distributed among all the organs according to the ratio of their relative sink to the total demand of the plant. And the organ biomass is cumulated GC by GC. At the end of GC(i), the weight of an organ with chronological age k is calculated as:

) 1 , 1 ( ) ( ) ( ) ( ) , ( , , , - - + = k i Q i D k D i Q k i Q PA o t PA o PA o (5)
where D t (i) is the total sink strength of all the organs including blades, petioles, piths and rings in GC(i).

As plant grows, the plant functional organs expand in mass according to their relative sink values and to the total biomass, which in turn depends on the functional blade area. This retroactive loop allows expressing the model in the form of dynamic system.

We set that internode pith and layers have the same density ρ whatever their position in the plant is (branches or trunk). Volumes of internodes can be obtained from their weights.

The geometric shape for the pith is considered as a cylinder. For the volume of pith (V e ), the corresponding length l and cross section σ of the pith are then computed according to the relation with two allometric coefficients b e and a e [START_REF] Zhan | Study on the Effects of Defoliation on the Growth of Cotton Plant Using the Functional Structural Model GREENLAB[END_REF][START_REF] De Reffye | Calibration of A Hydraulic Architecture-Based Growth Model of Cotton Plants[END_REF], i.e., and σ = V e / l . At each growth cycle, a new ring is added to the internode, so its diameter increases every GC. To summarize, fifteen crop parameters that are not accessible from direct measurement are optimized by the fitting procedures. Four potential sink strengths and their coefficients for PA2 corresponding to the organ types (i.e., P b and C b,2 , P p and C p,2 , P e and C e,2 , P c for leaf blade, petiole, pith, and ring, respectively); For single-stem cotton there is only PA1, so the number of parameters is 12. Six coefficients of three beta function parameters are used to define the sink strength variation (i.e., α b and β b , α p and β p , α e and β e for leaf blade, petiole and pith, respectively); two parameters (i.e., R and S p ) for biomass acquisition. The allometric parameters including ε (specific leaf weight), b e and a e (two shape coefficients of pith), as well as Q s (seed weight), T o (organ expansion time), n b (number of functional blades), T f (blade functional time) were directly measured and input in the model.

Results

Direct observation

The pruned cotton grew taller, had larger blades but less total biomass than the regular-growth cotton. And the blades of the pruned cotton were prone to twist when they grew larger (Fig. 1). The first and second branch was observed to emerge averagely six and five GCs after their bearing internode, respectively.

The organ size in two-vegetative branch cotton is smaller than single-stem cotton.

The directly measured parameters and their values were shown in Table 1. The organ expanding times were assumed the same for both treatments, i.e., 8 GCs for pith, and 15 GCs for blade and petiole. The functioning time was set to 22 for all the blades.

The blades in branched cotton were thinner than single-stem cotton but they were only about 5% difference between the two treatments for the main stem. The shapes of the pith for the two treatments were similar for the main stem and the branches were more different. And the internode densities of the two treatments were very similar, too. ε is specific leaf weight; b e and a e are coefficients of the allometric relationship for pith; ρ is internode density. T1 denotes single-stem cotton and T2 denotes branched cotton.

Fitting results

The fitting target consisted of four-stage data of cotton for both treatments. For branched cotton, data were from the main stem and two branches. Fig. 2 showed fitting results on main stem for both treatments, and on the first branch for T2. Fitting on the second branch was similar as branch 1 (not shown). The model fitted the data well in both cases, especially for biomass of blade and petiole. The fitted internode biomass at the early stages was larger than the observation, and the fitted internode length was smaller.

The potential sink parameters and biomass production parameters optimized by the fitting process were shown in Table 2. The sink parameter values were nearly the same for the two treatments. The different parts were not shown (coefficients of PA2 for branched cotton, i.e., C b,2, C p,2, C e,2, for blade, petiole and pith, respectively. Their values were less than 1.0). The value of S p for branched cotton was twice larger than that for single-stem cotton. Parameter P b , P p , P e , and P c are potential sink strength for blade, petiole, pith, and ring, respectively (P b is set to 1 as reference); R is coefficient for plant resistance to transpiration; S p is light interception parameter for Beer-Lambert's law. Data in brackets were standard errors of the estimated parameter values; T1 and T2 are specified as in Table 1.

Compared with single-stem cotton, all the sink variation functions for branched cotton were asymmetrically shrunk, and the time when the maximum growth rate occurred for both blade and petiole were delayed (Fig. 3). The branches influence on sink variation for pith was the least.

The 3D representation of cotton plants at four stages was also generated (Fig. 4).

Discussion and conclusions

The presence of branches not only decreased organ (blade, petiole and internode) size on the trunk (Fig. 2), but also changed the organ expansion function (Fig. 3). Branches can change plant trunk in a few aspects, for example, the specific leaf weight of main stem for cotton of two vegetative branches was smaller than single-stem cotton; the internode allometric parameters were different. The size and shape of organs on branches were different from those on the trunk, and accordingly parameters varied in GREENLAB with PAs. The model can simulate the growth of both treatments well, so the fitted values of parameters may reflect the influence of branch presence.

The potential sink strength (P o ) was very close for the two treatments (Table 2). Because the beta function is very flexible, the sink variation is similar between the two treatments (Fig. 3) despite of the differences of the parameters, especially when it shifts one cycle since the value of the initial stage is very small. In fact, we can see that the fitting results of two-vegetative branch cotton can be improved and maybe more similar organ expansion function can be obtained as the single-stem cotton. It indicates that there is no significant change in their expansion type, just as shown in the results part. So for the sink parameters, we can use the same values to simulate the trunk growth for the two treatments.

The ground projection area (S p ) was acquired by the fitting optimization procedure. The value range of this parameter can also be estimated from the observed data [START_REF] Li | Improving the Calibration Process of GREENLAB Model on the Cotton Plant[END_REF]. For branched cotton, the projection area was made up of contributions of leaves both on trunk and on branches. So the existence of branches may increase the value of S p . Because it is difficult to separate the pith from the ring, the allometry of pith was not precisely obtained. Based on these parameters, the fitted internode length was a bit smaller than measured. The allometric parameters may be improved by optimization procedures.

For a first approximation, we used a constant projection area (S p ) for all plant growth stages. A further study is undergoing to take into account the dynamics of this parameter. 1.

Figure 1 .

 1 Figure 1. Two treatments on cotton plants. T1 is the single-stem cotton having no branches. T2 is the cotton plant with two vegetative branches on the 5th and 6th node of the main stem.

  α o and β o are parameters associated with the organ type o. The six parameters (three organ types, blade, pith and petiole) can be obtained simultaneously by fitting the model to measured data, as well as other parameters (R, S p , C o,PA and P o ). T o is organ expansion time which can be obtained from the observed data.

Figure 2 .Figure 3 .

 23 Figure 2. Fitting GREENLAB model to four-stage data of cotton plants. Letters a, b, c and d denote single-stem cotton; a1, b1, c1, d1 denote main stem of branched cotton; a2, b2, c2 and d2 denote the first branch. The target items include biomass and length of internode (a, d), biomass of leaf blade (b) and petiole (c). D1, D2, D3, and D4 denote four growth stages. The observed data are the samples which are nearest to the mean of four replications.

Figure 4 .

 4 Figure 4. The 3D representation of cotton plants at four stages of 18, 25, 30, 35 GCs, respectively. T1 and T2 are specified as in Table1.

Table 1 . The directly measured parameters and their values

 1

			Treatment
		T1	T2
	Parameter	PA1	PA1	PA2
	ε (10 -3 g cm -2 )	6.33	5.98	5.59
	b e	12.66	14.87	20.80
	a e	0.047	-0.028 -0.0038
	ρ (g cm -3 )	0.26	0.25	0.25

Table 2 . The optimized parameters and their values of the two treatments

 2

		Treatment
	Parameter	T1	T2
	P b	1	1
	P p	0.37 (0.05)	0.36 (0.04)
	P e	0.36 (0.04)	0.31 (0.04)
	P c	0.27 (0.02)	0.23 (0.01)
	R	192.9 (10.1)	223.1 (6.8)
	S p (cm 2 )	2209 (161)	5475 (257)

Acknowledgements

This research has been partly supported by the Hi-Tech Research and Development (863) Program of China (2006AA10Z229). The authors are grateful to Chuntang Gong, Yufei Li and Zhifei Li for their kind help on the experiment measurements, and to Paul-Henry Cournède for allowing the use of the Digiplante software developped at the Laboratory of Applied Mathematics in Ecole Centrale of Paris.


Hong Guo
email: hongguo@caf.ac.cn

Veronique Letort
email: letort@mas.ecp.fr

Xiangdong Lei
email: xdlei@caf.ac.cn

Yuanchang Lu

Philippe De Reffye
email: philippe.de_reffye@cirad.fr

Applying GreenLab Model to Adult Chinese Pine Trees with Topology Simplification

This paper applied the functional structural model GreenLab to adult Chinese pine trees (pinus tabulaeformis Carr.). Basic hypotheses of the model were validated such as constant allometry rules, relative sink relationships and topology simplification.

To overcome the limitations raised by the complexity of tree structure for collecting experimental data, a simplified pattern of tree description was introduced and compared with the complete pattern for the computational time and the parameter accuracy. The results showed that this simplified pattern was well adapted to fit adult trees with GreenLab.

Introduction

Functional structural tree models (FSTMs) were developed to understand the interactions and feedbacks between the morphological structure and physiological processes of individual tree growth [START_REF] Sievanen | Components of Functional -structural Tree Models[END_REF]. Thus, structure design is very important in FSTMs [START_REF] Godin | Functional-structural Plant Modeling[END_REF].Tree architectural structure has been described at several spatial scales in different models ranging from accurate descriptions of each organ to coarse descriptions of branching systems at the individual tree level. Most of functional structural models use a very detailed representation of tree architecture based on the description of individual organs. Such detailed organ description entail complex data bases and huge computation and measurement time, which in turn leads to difficulties in applications of FSTMs to forest management.

As a generic functional structural model, GreenLab was developed to simulate plant growth at organ scale and on the basis of architectural rules and organogenesis' processes as described in details in [START_REF] De Reffye | Calibration of Hydraulic Growth Model on the Architecture of Cotton Plants[END_REF][START_REF] Guo | Parameter Optimization and Field Validation of the Functional-structural Model GREENLAB for maize[END_REF]. The model has been successfully applied to agricultural crops, Chinese pine saplings and beech trees [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-structural Model GREENLAB for maize[END_REF][START_REF] Dong | Tomato Growth Modeling Based on Interaction of its Structure-Function[END_REF][START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-source Relationships in Chinese Pine Saplings[END_REF][START_REF] Letort | Parametric Identification of a Functionalstructural Tree Growth Model and Application to Beech Trees (Fagus sylvatica)[END_REF]. However, adult trees have much more complex architecture which makes model fitting very difficult with real topology data. The main obstacles for designing experimental protocols aiming at describing adult trees at phytomer scales are the high number of organs (several thousands) and the complexity of their topological organization. Moreover computational performance is often a limitation for any algorithm of parametric identification of tree growth models as they require running many simulations. Aiming at simplifying tree structure appropriate for different demands in forestry, the paper introduced two structural patterns and fitted them using GreenLab based on data collected on two adults Chinese pine trees (pinus tabulaeformis carr.). The two patterns were compared in terms of experimental work needed, of fitting results and of computational time. The simplified pattern was found adequate for adult trees in GreenLab.

Materials and Method

Model Structure

The GreenLab tree model was described in details in [START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-source Relationships in Chinese Pine Saplings[END_REF]. Here, the main equations and parameters are listed as follows.

GreenLab is a discrete dynamic model simulating tree growth with a time step called growth cycle (GC) and equal to one year. Carbon production Q(i) during one GC i is calculated as:

                     p p S i S k r S i E i Q ) ( exp 1 ) ( ) (
Where r is defined as the hydraulic resistance of the plant to transpiration, E(i) is the average potential environmental factor during GC i. S p is the total [START_REF] Sievanen | Components of Functional -structural Tree Models[END_REF] ground projection area available for the plant, as defined in [START_REF] Wilson | Analysis of Light Interception by Single Plants[END_REF], and where k is the Beer-Lambert extinction coefficient related to leaf angular deviation [START_REF] Nilson | A Theoretical Analysis of the Frequency of Gaps in Plant Stands[END_REF]. At each GC, the carbon production is partitioned between all growing parts of the tree according to a model of sink competition. The sink of new organs P o (k) depend on their branching order (k) hereafter called physiological age (PA) in reference to previous works [START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-source Relationships in Chinese Pine Saplings[END_REF]. The biomass allocated to organ o of PA k at GC i is proportional to the ratio of available biomass Q(i-1) to the plant total demand D(i) which is the sum of demand of new organs and of demand of rings D rg (i):

         e a o rg o o o o i D i k N k P i Q k P i k q , ) ( ) , ( ) ( ) 1 ( ) ( ) , (
where Q(i-1) is the total biomass production at GC i-1, D rg (i) the demand of rings at GC i and o represents indices for organ types (needles: a; internodes: e). N o (k, i) is the number of organs belonging to category o at GC i and physiological age (PA) k. Axes are grouped into categories according to physiological age which was related to branching order for pinus tabuleformis [START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-source Relationships in Chinese Pine Saplings[END_REF].

The biomass allocated to the radial growth is proportional to the ring demand D rg (i) which is calculated according to the ratio of biomass supply to demand as follows:

D(i) 1) Q(i P P (i) D 1 O rg    
where P 0 is the constant sink (dimensionless) of the ring compartment, P 1 is the corresponding slope per mass unit (g -1 ), D(i) is the total demand of the plant at GC i. The supply to demand ratio is assumed to represent the state of trophic competition within the tree and therefore to drive the relative allocation of biomass to rings.

Biomass available for ring growth is partitioned into every living phytomer according to their position in the tree topological structure. As presented in [START_REF] Guo | Adaptation of the GreenLab Model for Analyzing Sink-source Relationships in Chinese Pine Saplings[END_REF], the modeling of radial growth is based on a generalisation of the Pressler law [START_REF] Assmann | Vorläufige Fichten-Ertragstafel für Bayern[END_REF] that states that annual ring increment of a phytomer is proportional to the foliage quantity located above it. The biomass allocated to ring growth of a phytomer of PA k and position p at GC i is calculated as follows:

(i) Q i) l(k, (k) p ) (i) D p) i, (k, λN (i) D λ 1 ( p) i, (k, q rg rg pressler a pool rg            m PA k rg i pool i k l k p i k N i D 1 ) , ( ) ( ) , ( ) (        m k PA k rg P p a pressler i k l k p p i k N i D 1 ) , ( ) ( ) , , ( ) (
Where PA m is the maximal PA in the tree, l(k,i) is the length of the N i (k,i) internodes of PA k on plant of GC i and p rg (k) is the linear density of sink of rings for internodes of PA k. N a (k,i,p) is the number of leaves at GC i above the considered internodes P k is the set of possible positions p of metamers of PA k in the whole plant of age i. The parameter λ (0 ≤ λ ≤ 1) drives the relative influence of the number of leaves on the cambial growth: if λ=1, the equation ( 4) is similar to the Pressler lax while if λ=0, the cambial growth is uniform whatever the position of the phytomer relatively to foliage is.

Structural Patterns

Two structural patterns for tree topological description at different levels of details were defined in the paper with considerations of measurement feasibility and potential applications in forestry. They are pattern 1 at organ level, called substructure model and pattern 2 including compartment for all branches. The two patterns are described in figure 1. Pattern 1: Organ level (substructure model)

In GreenLab, branches of the same PA appeared at the same GC are simulated as identical: they have same structure, biomass and geometry. Therefore the whole tree structure can be represented by a recursive equation (see [START_REF] Cournède | Structural Factorization of Plants to Compute their Functional and Architectural Growth[END_REF]) and the computation of the whole tree can be highly enhanced by factorizing its elements (i.e. similar structures are computed only once). This substructure algorithm speeds up more than 1000 times of computation time needed to simulate the complex tree architecture and biomass repartition [START_REF] Cournède | Structural Factorization of Plants to Compute their Functional and Architectural Growth[END_REF].

Pattern 2: Compartment level for branches

In pattern 2 the biomass of needles and internodes for all branches are aggregated (Figure 1b). For this pattern, stem was described internode by internode using internode length, internode radius, internode mass and needle mass, and branches were represented by total mass of needles and internodes. (2)

(4) (5)
(6)

Tree materials

Two Chinese pine trees were sampled including one 18 years old measured in autumn of 2008 and the other 31 years old measured in spring of 2006. Both were from the nursery garden located in open area in Changping forest farm, Beijing. In these experiments, the generative organs were not considered. The 3 year old needles were very scarce and most of them almost dead. So we set functioning span time for needles as 2 cycles. For the sake of simplicity, the segment of branch generated during one year was considered as a single internode although it is not true botanically speaking. At each whorl and topological level, we empirically selected one average branch which was measured internode by internode. These selected branches were described for each metamer by the following measurements: length, diameter, fresh woody weight and fresh needle weight. For other branches, we measured the total fresh weight of internodes and needles.

Data Analysis

Concerning our Chinese pine saplings, several types of input parameters listed by table 1 have been calculated directly from the measurements.

Parameters calibration

Hidden parameters showed in table 2 were estimated with nonlinear generalized least squares method [START_REF] Zhan | Architectural Modeling of Wheat Growth and Validation Study[END_REF] by Digiplante software [START_REF] Cournède | Structural Factorization of Plants to Compute their Functional and Architectural Growth[END_REF].

Fitting results used substructure data of the two trees respectively. Figure 2 shows that the fitting results for stem internodes biomass and cumulated internodes length are consistent with the observations. Then data formatted according to pattern 2 were fitted. The computation time and error were listed in Table 3.

Results and conclusion

A potential challenge for Functional structural tree model is to face a bottleneck due to complex topological structure of adult trees [START_REF] Godin | Functional-structural Plant Modeling[END_REF]14]. Substructure factorization has been introduced in detail in [START_REF] Zhan | Architectural Modeling of Wheat Growth and Validation Study[END_REF]. This paper applied substructure model and other compartment pattern to adult Chinese pine trees for comparing the results of simplification.

Firstly, based hypothesis was test for one 41 years old tree and one 18 years old tree.

We found strong adequation of the internodes allometries shape to the data for the two trees (R 2 ranged from 0.73 to 0.95).

Secondly, from table 2, we found that for 18 years old tree parameter λ is equal to 0.01 which means that radial growth is nearly uniform along axes: it does not depend on the number of leaves located about each phytomer. In contrast, for 41 years old tree λ=0.40, which means that the partitioning model for radial growth is closer to the Pressler law: the relative foliage position influences for 40% of the radial growth of each phytomer. We can come to a conclusion that our general model that allows turning the relative influence of foliar biomass on ring growth is well adapted to fit Chinese pine trees of different ages.

This paper tried to find simplified topology structure to fit GreenLab model to save computation From table 3, the final error of pattern 2 is lower than that of pattern 1 because the number of data is lower. Pattern 2 saved 8 times to 89 times time than Pattern 1. Thus, we can collect only compartment data to fit GreenLab model and according to different application and demand, we select different simplified way such as pattern 2. In a future work, a third pattern will be introduced at an intermediate level between pattern 1 and pattern 2. This pattern 3 would account for more detailed measurement on first order branches (internode by internode) and biomass data will be aggregated by compartment (wood/leaves) for branches of second order only. The generalization to any branching order will be considered. Thus several levels of tree description will be available in order to adapt the fitting process to flexible experimental protocols and formats of target data files.

( 3 )Figure 1

 31 Figure 1 Three structural patterns (a. organ level b. compartment level )

Figure 2

 2 Figure 2 Calibration of 2 trees

Table 1 Directly calculated parameters and the definitions in GreenLab

 1

	Name	Definition
	P e P a b e	Internode sink Blade sink Internode allometry
	β	Internode allometry
	ε	Needle SLW

Table 2 Estimated parameter values after fitting the data of Tree 1 and Tree 2

 2

	Name	Value Tree 1 CV(%) Value CV(%) Tree 2
	r	1.79	0.46	7.44	0.14
	P 1 λ	0.54 0.01	2.6 6.26	0.033 0.40	0.30 0.35
	p rg (2) Sp	0.89 3.04 m 2	4.76 0.49	0.99 78 m 2	0.3 1.0

Table 3 Three patterns comparison

 3

Acknowledgements

The study was funded by Central Basic R & D Special Fund for Public Welfare Institutes of China (RIFRIGTZGZ2007008), Natural Science Foundation of China (30872022), and Chinese 948 project (2008-4-63).


Meng Feng Wang

Zhen Kang

Hui Han

Qi Lu

Véronique Letort

Yan Guo

Philippe De Reffye

Baoguo Li

Mengzhen Kang

Veronique Letort

Calibration of topological development in the procedure of parametric identification: application to the stochastic GreenLab

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The ecosystem in the arid and semiarid region of North China is very fragile because of hostile climate and sterile soil. Moreover, human-induced vegetation destruction and soil degradation further intensify the problem. Vegetation protection and optimal water and soil management are an important pathway to improving the function of local ecosystem services. The selection of appropriate species for restoring vegetation is a key point to preventing desertification.

Mongolian Scots pine tree (Pinus sylvestris var. mongolica) is one of the principal species of the windbreak and sand-fixing forest in the arid and semiarid area. It possesses many desirable properties, such as trunk straightness, valuable wood material and especially high tolerance to environmental stresses (e.g. coldness, drought, infertility, salinity) [START_REF] Kang | Natural distribution of Pinus sylvestris var. mongolica on sandy land and its cultivation as an exotic species[END_REF], which make it fit to grow in sandy land. Therefore, it is widely planted in North China to combat desertification. However, some degenerating phenomena such as top withering, slow growth and stems death have occurred in the early Mongolian pine plantations since the later 1980s [START_REF] Zhu | Comparison of stand structure and growth between artificial and natural forests of Pinus sylvestiris var. mongolica on sandy land[END_REF]. Modelassisted analysis on the canopy architecture development and function of Mongolian Scots pine tree are valuable for better understanding its role and behavior.

Functional-structural plant models (FSPMs) are effective tools to simulate the growth and development of plants by integrating threedimensional (3D) plant architecture with ecophysiological processes [START_REF] Vos | Functional-Structural Plant Modelling in Crop Production[END_REF]. Plant architecture in this type of models is usually represented as a collection of interconnected elementary units [START_REF] Godin | Functional-structural plant modelling[END_REF]. FSPM simulates physiological and geometrical variation of every elementary unit during plant growth and development [START_REF] Sievänen | Components of functional-structural tree models[END_REF]. Early FSPMs focused more on the dynamic changes of plant function in fixed tree architecture than on architectural development, e.g., ECOPHYS [START_REF] Rauscher | ECOPHYS: An ecophysiological growth process model for juvenile poplar[END_REF], EMILION [START_REF] Bosc | EMILION, a tree functional-structural model: presentation and first application to the analysis of branch carbon balance[END_REF] and SIMWAL [START_REF] Balandier | SIMWAL: a structuralfunctional model simulating single walnut tree growth in response to climate and pruning[END_REF]. One of representative FSPMs is the LIGNUM model [START_REF] Perttunen | LIGNUM -A Tree Model Based on Simple Structural Units[END_REF], which initially applied to Scot pines [START_REF] Perttunen | LIGNUM: a model combining the structure and the functioning of trees[END_REF], then to Jack Pine [START_REF] Lo | Adaptation of the LIGNUM model for simulations of growth and light response in Jack pine[END_REF] and Sugar Maple [START_REF] Perttunen | Application of the Functional-Structural Tree Model LIGNUM to Sugar Maple Saplings (Acer saccharum Marsh) Growing in Forest Gaps[END_REF]. LIGNUM simulates the relationships between plant architecture and biomass with a time scale of one year. It integrates the pipe model theory [START_REF] Shinozaki | A Quantitative Analysis of Plant Form --The Pipe Model Theory I. Basic Analyses[END_REF] with a simplified representation of tree architecture [START_REF] Perttunen | Incorporating Lindenmayer systems for architectural development in a functional-structural tree model[END_REF]. Another FSPM appeared recently was L-PEACH [START_REF] Lopez | Integrating simulation of architectural development and source-sink behaviour of peach trees by incorporating Markov chains and physiological organ function submodels into L-PEACH[END_REF][START_REF] Allen | Using L-systems for modeling source-sink interactions, architecture and physiology of growing trees: the L-PEACH model[END_REF], which describes plant architecture using Lsystems [START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF], and the assimilate movements was driven by concentration gradients across resistive pathways based on the "transport -resistance" model.

Derived from the powerful plant architecture model AMAP, GreenLab is a generic plant growth model that integrates various morphological features with biomass production and allocation at organ scale [START_REF] Yan | A Dynamic, Architectural Plant Model Simulating Resource-dependent Growth[END_REF], called GL1. The plant architecture is built based on an automaton which is analogous to writing Lsystem rules, and the biomass acquisition and allocation are based on source-sink relationship. GreenLab model is able to simulate plant's phenotypic plasticity that results from feedbacks among growth (biomass acquisition), differentiation (phenology, morphogenesis) and the physiological condition of the organism (e.g. stresses) [START_REF] Mathieu | A dynamic model of plant growth with interactions between development and functional mechanisms to study plant structural plasticity related to trophic competition[END_REF]. At present, GreenLab has been applied to crops, e.g., cotton [START_REF] De Reffye | Calibration of a hydraulic architecture-based growth model of cotton plants[END_REF], maize [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF], wheat [START_REF] Kang | The Derivation of Sink Functions of Wheat Organs using the GREENLAB Model[END_REF], tomato [START_REF] Dong | Does the Structure-Function Model GREENLAB deal with crop phenotypic plasticity induced by plant spacing? A case study on tomato[END_REF], and trees, e.g., Chinese pine saplings [START_REF] Guo | Adapation of the GreenLab Model for Analyzing Sink-Source Relationships in Chinese Pine Saplings[END_REF] and beech trees [START_REF] Letort | Parametric identification of a functionalstructural tree growth model and application to beech trees (Fagus sylvatica)[END_REF]. However, these calibration processes were performed with deterministic architecture that represents an average topological structure from several samples. No appropriate method was developed to include topological information in the set of target data used for the procedure of parametric identification. Moreover a deterministic topological development is not realistic with regards to the high variability of branching patterns observed in tree stands. To get realistic simulations of tree growth, it is important to consider the randomness of topological development. This motivated the development of a stochastic version of GreenLab, called GL2 [START_REF] Kang | Analytical study of a stochastic plant growth model: Application to the GreenLab model[END_REF]. Beside simulation of stochastic plant sample, the theoretical mean and variance of numbers of organs at any growth stage can be calculated.

The aim of this study is to describe the plant development and growth of Mongolian Scots pine tree, taking into account the variations in plant topology and biomass. The benefits of these mathematical properties were taken to fit the parameters driving the topological development, as well as functional processes. We present the calibration process of GL2 based on experiment data of Mongolian Scots pine trees. The stochastic 3D plant architectures were simulated and visualized.

Materials and methods

Model description

GL2 model is a stochastic functional-structural model, in which bud probability is introduced in plant architecture development, including survival probabilities of buds, branching probabilities of buds, growth probability of apical buds and appearance probability of metamers. Theoretical mean and variance of organ number can be calculated in an analytical way [START_REF] Kang | Analytical study of a stochastic plant growth model: Application to the GreenLab model[END_REF]. Another component of the model is on biomass production and allocation, the same principle as in [START_REF] Yan | A Dynamic, Architectural Plant Model Simulating Resource-dependent Growth[END_REF]. General description of this submodel can be found in [START_REF] Yan | A Dynamic, Architectural Plant Model Simulating Resource-dependent Growth[END_REF], [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF] and [START_REF] Letort | Parametric identification of a functionalstructural tree growth model and application to beech trees (Fagus sylvatica)[END_REF].

Wang et al. [START_REF] Wang | Modelling three-dimensional architecture of pine tree (Pinus sylvestris Linn. var. mongolica Litv.) in a semiarid area[END_REF] defined the topological parameters of the dual-scale automaton [START_REF] Zhao | Dual scale automaton model for virtual plant[END_REF] specifically for Mongolian Scots pine tree. The plant architecture description is hierarchically organized using botanical notion of physiological age (PA) [START_REF] Barthelemy | Plant Architecture: A Dynamic, Multilevel and Comprehensive Approach to Plant Form, Structure and Ontogeny[END_REF]. Each bud of Mongolian Scots pine tree develops to a growth unit (GU) (includes one metamer only) each year. In that case, PA is equivalent to branching order: one for trunk, two for 1 st order branches, three for 2 nd order branches, and four for 3 rd order branches, which is the maximum PA for the Mongolian Scots pine trees in this study. The death or dormant probability of apical bud is negligible for young trees according to observation. Therefore the growth probability is set to one and death probability is set to zero. The variability of Mongolian Scots pine trees architecture is mainly represented by the number of branches at different order. Thus in following study, only the branching probability is taken into consideration.

Let p k be the probability that an axillary bud of PA k can develop to a branch, called branching probability of PA k. Suppose that there are potentially n k axillary buds at a node. The model assumed that buds of the same PA have the same branch probability, regardless their position inside plant structure. The number of appeared branches on a node then follows a binomial law (n k , p k ), the mean M B and standard deviation V B of branches on such a GU is:

(1 )

B k k B k k k M n p V n p p          (1)
Total number of GUs in a plant is a function of branching probability of each order. Using compound law, their mean and variance can be computed recurrently from highest PA in an analytical way: ' ' '

p p S G B S G B G B G M M M V V M M V M             (2)
In Eqn. (2), S p M and p S V are mean and variance of numbers of GUs in a substructure of PA p. When p = 1, it is the plant structure. M G and V G are mean vector and covariance matrix of number of branches along the axis of substructure p, their element computed from Eqn. [START_REF] Kang | Natural distribution of Pinus sylvestris var. mongolica on sandy land and its cultivation as an exotic species[END_REF]. For V G , their diagonal elements are zeros supposing the numbers of branches per node are independent. M B and V B are mean vector and covariance matrix of number of GUs in branches along the axis. Started from the simplest branch of maximum PA, Eqn (2) gives recursively the mean and variance of number of GUs. It follows also a binomial distribution as it is a compound law of binomial ones.

Based on the mean of organ number, an average plant can be calculated using sink-source parameters as in GL1 model [START_REF] Yan | A Dynamic, Architectural Plant Model Simulating Resource-dependent Growth[END_REF].

Model calibration

The model parameters were estimated by model inversion with an adequate set of experimental data with the simulation output. In GL2, the calibration was done in two steps. The first step is on the automaton, aiming at quantifying the branching probability for branches of each order that the organogenesis model produces the closest number of organs to the observations. In observations, the numbers of branches were counted, from which numbers of GUs were calculated. By fitting the observed and calculated number of GUs, the branching probabilities were estimated. Notice that the aim of fitting is to find bud probabilities that give closest organ number with real data, and it is not guaranteed that the probability is the same as the actual ones observed from a certain node.

The second step is to resolve sink-source parameters by fitting the organ biomass of the average plant. As number of internodes and needles (consider all needles in one metamer as whole) are equal to that of GUs, the number of sink and source organs of an average plant is known. Following the same fitting process as for GL1 [START_REF] Guo | Parameter Optimization and Field Validation of the Functional-Structural Model GREENLAB for Maize[END_REF] , the sink-source parameters were identified. GreenScilab toolbox (http://www.scilab.org) for GL2 was used for model calibration and simulation. Based on the calibration, the architecture and biomass of Mongolian Scots pine trees was stochastically simulated with Monte Carlo methods.

Experiments

The experimental site is located at the plantations station affiliated with Liaoning Sand Fixation and Afforestation Institute (122º22'E, 42º43'N), in Zhanggutai, Liaoning of China. This region is adjacent to Kerqin sandy land, which is one of the four biggest sandy lands in China. The average annual precipitation, potential evaporation, temperature and wind speed are 505.9 mm, 1755.3 mm, 6.0°C and 4.2 m s -1 respectively. Soil type is aeolian sandy soil.

Two experiments for plant topological structure and organ biomass were carried out in November, 2006 and August, 2007. Two observations on number of branches were done in the field on 100 Mongolian Scots pine trees of 4, 5 and 6 years old. The numbers of different order branches at each node of trees were recorded. For organ biomass, samples were taken from 1, 2, 3, 5, 6-year-old young trees with four replications for each tree age. To prevent water loss during measurements, plants were dug out with roots and soil and transported to the laboratory for measurement (length, diameter and fresh biomass of every internodes and needles fresh biomass).

Results

Model parameters from direct observations

The 2 nd order branches were not observed for 4year-old trees. A maximum branching order of three was observed on the 6-year-old trees. The maximal numbers of branch per node was 15 for 1 st order branch (n 2 =15), six for 2 nd order branch (n 2 =6) and four for 3 rd order branch (n 3 =4). There is a decrease with higher branching order.

Average specific leaf weight (SLW) and internode allometry scales ratio were estimated directly from experiment data (Table 1). The high correlation coefficient showed the close relationship between internode mass and length for each order branch (Fig. 1). These allometric parameters were used to calculate organ size from their biomass.

Fitting results

Fig. 2 shows the mean and variance of number of the GUs of trunk and branches of different order. Both the mean and variance were well fit simultaneously by the model, verifying the hypothesis that the number of GU follows a binomial distribution. Trunk grows one GU every year, thus the mean number of GUs of trunk increased linearly with tree age, and the variance is zero. The mean number of GUs in 2 nd order branch (PA3) increased most rapidly with tree age, so do its variance. This is reasonable as both n 2 and p 2 are higher for PA 2. The identified bud probabilities are shown in Table 1. It can be seen that they decrease with higher branching order. The average numbers of branches per node for 1st, 2nd, 3rd order branch were 3.98, 1.33 and 0.52 respectively from equation [START_REF] Zhu | Comparison of stand structure and growth between artificial and natural forests of Pinus sylvestiris var. mongolica on sandy land[END_REF]. Using the calibrated automaton, the next step is to calibrate the sinksource parameters by fitting biomass and size of organs. Fig. 3 shows the fitting results on fresh biomass of internodes and needles for trunk and branches of all orders in the 6-year-old Mongolian Scots pine tree. Sink-source parameters are shown in Table 1. It can be seen that the fresh biomass of trunk internode was much more than those of branches. Take a 6-year-old Mongolian Scots pine tree for example, average internode fresh biomass of PA1, PA2, PA3, PA4 were 50.7, 1.9, 0.3 and 0.04 g respectively. Actually, the sink strength of trunk internode is significantly higher than that of the branches (Table 1). Moreover, the secondary sink strength for branches are very low, which means the most of biomass for ring growth was allocated to the trunk. The fresh biomass of needles in each GU decreased also with increasing branch order, being 6.5, 2.7, 1.0, and 0.27 g from PA1 to PA 4. Naturally, the relative sink strength of branch needles are lower compared to those in trunk. It can be seen from Fig. 3 (b), leaf functioning time was three years, and leaves older than 3-year-old fell down.

Fig.3 Fitting results of single internodes and needles fresh biomass of trunk and different order branches of a 6-year-old Mongolian Scots pine tree

PA1: Trunk; PA2: 1 st -order branch; PA3: 2 nd -order branch; PA4: 3 rd -order branch, from measured data (symbols) and model output (lines) Fig. 4 shows the fitting results on length and diameter of each GU in the 6-year-old Mongolian Scots pine tree. Tree height and ground diameter were 82.9 cm and 3.1 cm respectively. Good estimation of trunk size is useful to predict precisely tree height and wood volume. Compared with the fitting on GU fresh biomass of branches as in Fig. 2(a) , that of internode length and diameter were less good. This may be due to the ongoing growth of internodes when the measurements were done in August. This result showed that it is important to set the date of measurment for obtaining good allometric rules.

Fig. 5 shows the simulated and measured organ compartment weight and total fresh biomass. They are the sum of the weights of individual organs they are composed of. The fresh biomass of internodes compartments decreased from trunk to higher order branch. However, 1 st order branch bore the highest fresh biomass of needles compartment, because of its high number of organs (Fig. 1a) and relatively more biomass compared with the internodes (Fig. 2). The fresh biomass of internodes and needles compartment of 3 rd order branch was lowest because the number of branches was very small.

The simulations of a 6-year-old Mongolian Scots pine tree were conducted after model calibration. The visualization of five simulated Mongolian Scots pine trees with stochastic features is shown (Fig. 6).

Rank vs Target_Bld_PA1

Rank vs Model_Bld_PA1 Rank vs Target_Bld_PA2

Rank vs Model_Bld_PA2 Rank vs Target_Bld_PA3

Rank vs Model_Bld_PA3 Rank vs Target_Bld_PA4

Rank vs Model_Bld_PA4 1)

branching probability reflects that higher order branches are more vulnerable to environmental stress, reflecting the adaption of plant architecture to drought condition [START_REF] Maillette | Structural Dynamics of Silver Birch. I. The Fates of Buds[END_REF] . On the other hand, losing the 2 nd or 3 rd order branches will not significantly decrease plant growth, as their contribution on leaf area is small (Fig. 5b). As most of wood biomass is reserved in trunk and primary branches (Fig. 5b), the production is kept. Due to high topological complexity and tremendous number of organs in trees, most functional-structural tree models only used less experimental data to indicate the functional nature of plant growth and development, and lacked full calibration and validation [START_REF] Lopez | Integrating simulation of architectural development and source-sink behaviour of peach trees by incorporating Markov chains and physiological organ function submodels into L-PEACH[END_REF] [START_REF] Eschenbach | Emergent properties modelled with the functional structural tree growth model ALMIS: Computer experiments on resource gain and use[END_REF] .The fitting results of this paper show that considering mean and variance of number of organs to calibrate the GL2 model of topological development was reasonable and acceptable for Mongolian Scots pine trees. The GL2 model can simulate plant morphological plasticity and organ biomass change fairly well based on stochastic dual-scale automaton and source-sink relationships.

By introducing branching probabilities into the GreenLab model, the GL2 version allows reproducing the variability of branching structures that can be observed within plant communities.

The functional and structure tree models by Sterck et al. [START_REF] Sterck | Performance of trees in forest canopies: explorations with a bottom-up functional-structural plant growth model[END_REF] and Renton et al. [START_REF] Renton | Functionalstructural plant modelling using a combination of architectural analysis, L-systems and a canonical model of function[END_REF] also accounted for stochastic phenomena of plant growth. However, the parameter values driving apical and axillary buds breaks in their models are set intuitively. The branch probabilities in GL2 were estimated from experimental data of mean and variance of numbers of organs, which reflected the observed stochastic characters of Mongolian Scots pine trees stand in this region of North China. In the stochastic architecture model presented here, the resulting plant architecture is not influenced by biomass production. In nature, the plant architecture is the result of both genotype and environment influence. The numbers of branches are strongly influenced by plant physiological activities [START_REF] Buck-Sorlin | Models of crown architecture in Quercus petraea and Q. robur: Shoot lengths and bud numbers[END_REF]. It can be expected that the branching probability is dependent on the water content of the soil. The biomass is allocated into determined number and type of sinks in GL2, which omits the effects of source variations on sink numbers and types [START_REF] Dingkuhn | Environmental and genetic control of morphogenesis in crops: towards models simulating phenotypic plasticity[END_REF]. In the experiments, we found the number of 2 nd and 3 rd order branch increased with plant age. Thus plant buds behavior might be highly related to plant biomass production. A new perspective is to integrate these observations into the new version of GreenLab model based on the previous work [START_REF] Mathieu | Rhythms and Alternating Patterns in Plants as Emergent Properties of a Model of Interaction between Development and Functioning[END_REF]. Thus the variation of topological structure depending on dynamic sink-source relationships of trees can be simulated.

Fig. 1

 1 Fig. 1 The allometric relationship of GU of trunk and different order branches for Mongolian Scots pine tree (a) Trunk, y = 8.74x 0.38 (R 2 = 0.98, n = 19, p < 0.0001); (b) 1 st -order branch, y = 12.78x 0.35 (R 2 = 0.84, n = 31, p < 0.0001); (c) PA3: 2 nd -order branch, y = 14.06x 0.40 (R 2 = 0.81, n = 17, p < 0.0001); (d) PA4: 3 rd -order branch, y = 18.92x 0.57 (R 2 = 0.84, n = 6, p = 0.0023). Measured data (symbols) and regression line (lines)

Fig. 2

 2 Fig.2 Mean and variance of number of GUs in trunk and branches of different orderPA1: Trunk; PA2: 1 st -order branch; PA3: 2 nd -order branch; PA4: 3 rd -order branch, from measured data (symbols) and model output (lines)

Fig. 4 Fitting 4 DiscussionsFig. 5 FittingFig. 6

 4456 Fig. 4 Fitting results of single internodes length and diameter of trunk and different order branches of a 6-year-old Mongolian Scots pine treePA1: Trunk; PA2: 1 st -order branch; PA3: 2 nd -order branch; PA4: 3 rd -order branch, from measured data (symbols) and model output (lines)

Table 1 The description and values of the main parameters used in GL2 model for Mongolian Scots pine trees

 1

	Parameters	Description(unit)	Value
	n k	max number of branches per node (PA = 2, 3, 4)	15, 6, 4
	p k	branching probability(PA = 2, 3, 4)	0.27, 0.22, 0.13
	ε	specific leaf weight(g cm -2 )	0.035
	b	scale coefficient of single internodes (PA=1, 2, 3, 4)	76.4, 163.3, 197.7, 358.0
	β	shape coefficient of single internodes (PA=1, 2, 3, 4)	-0.24, -0.30, -0.20, 0.14
	p e	relative sink of "internode"(PA=1, 2, 3, 4)	0.45, 0.24, 0.07, 0.02
	p a	relative sink strength of needles (PA=2, 3, 4)	0.46, 0.17, 0.05
	p c	relative sink strength of ring demand	7.16
	R p	secondary sink for ring repartition (PA=2, 3, 4) (cm -1 )	0.05, 0.01, 0.001
	r p	biomass production resistance	0.92

Acknowledgements

This study was supported by Hi-Tech Research and Development (863) Program of China (2006AA10Z229 and 2008AA10Z218) and the National Program for Key Science and Technology Projects of China (No. 2005BA517A, 2006BAD26B01). We are grateful to Bin Wang, Xuefeng Bai and Lu Feng for assisting in the experiments. We greatly appreciate Rui Qi for help on GreenSciLab.


Véronique Letort
email: letort@mas.ecp.fr

Paul-Henry Cournede
email: cournede@mas.ecp.fr

Philippe De Reffye
email: dereffye@cirad.fr

Impact of topology on plant functioning : a theoretical analysis based on the GreenLab model equations

A growing interest emerges for the application of plant functional-structural models to study the influence of topological development on plant growth. This kind of study is classically done through virtual experiments on simulated plants to analyze the effects of architectural changes on the model outputs. In this paper, we take advantage of the mathematical formalism developped for describing plant structure and growth in the GreenLab model to perform a theoretical analysis of the importance of topological development for plant functioning. Using the basic formulation of the model, it is possible to solve analytically the equation giving the limit biomass production under stable environment and to estimate the sensitivity of this limit value to the parameters of topological development. Reciprocally, we analyze the conditions under which plants with different architectures can have the same trajectory of biomass production in models based on Beer-Lambert-like formulations.

Introduction

It is a current scientific challenge to understand and analyze the potentialities of the different architectural models of plant structure [START_REF] Fourcaud | Plant growth modelling and applications: the increasing importance of plant architecture in growth models[END_REF]. The ongoing development of functionalstructural tree models (FSTM) [START_REF] Sievänen | Components of a functionalstructural tree model[END_REF], [START_REF] Prusinkiewicz | Modeling plant growth and development[END_REF] opens new ways to study plant growth and more particularly to investigate the complex interactions between tree structure and physiological processes. Most of them represent the tree topology or geometry as a set of basic units, phytomers [START_REF] Perttunen | LIGNUM: a tree model based on simple structural units[END_REF].

These models are useful tools to integrate the current biological knowledge at different scales, even for organisms as complex as trees (see for instance [START_REF] Godin | A multiscale model of plant topological structures[END_REF], [START_REF] Kurth | Specifications of morphological models with Lsystems and relational growth grammars[END_REF]). Virtual experiments can be performed to investigate the effects of changes in plant architecture on different phenomena (e.g. simulations of horticultural strategies, such as pruning fruit trees [START_REF] Lopez | Integrating simulation of architectural development and source-sink behaviour of peach trees by incorporating Markov chains and physiological organ function submodels into L-PEACH[END_REF], [START_REF] Costes | MApple-T: simulation of apple tree development using mixed stochastic and biomechanical models[END_REF], interception of solar radiation [START_REF] Perttunen | LIGNUM: a model combining the structure and the functioning of trees[END_REF]- [START_REF] Cieslak | Quasi-monte carlo simulation of the light environment of plants[END_REF], plant-pathogens interactions [START_REF] Robert | Coupling a 3D virtual wheat plant model with a Septoria tritici epidemic model (Septo3D): a new approach to investigate plant-pathogen interactions linked to canopy architecture[END_REF],...). However these models are often formulated as complicated flowcharts with a high number of parameters, including many processes and interactions. On the other hand, models such as the GreenLab model [START_REF] Yan | A dynamic, architectural plant model simulating resourcedependent growth[END_REF], [START_REF] Cournède | Structural factorization of plants to compute their functional and architectural growth[END_REF] consider only simplified versions of biological processes so that they can be formulated in the form of dynamic systems.

Although writing a model under a mathematical form is sometimes done at the expense of its biological relevance (some phenomena are neglected), it offers some possibilities compared to mere simulation black boxes. In his review of structure modelling, Prusinkiewicz [START_REF] Prusinkiewicz | Modeling of spatial structure and development of plants : a review[END_REF] presents the advantages of representing the topological development of plants using a language, and more particularly L-systems, to reduce the programming effort, to describe the plant structure in a compact and precise manner and to facilitate model comparisons. A further step is to exploit the properties that naturally stem from the use of L-systems rules, such as structural factorization [START_REF] Cournède | Structural factorization of plants to compute their functional and architectural growth[END_REF], computation of moments of numbers of organs [START_REF] Kang | A stochastic language for plant topology[END_REF] and generating functions [START_REF] Loi | Generating functions of stochastic L-systems and application to models of plant development[END_REF]. Kurth [START_REF] Kurth | Some new formalisms for modelling the interactions between plant architecture, competition and carbon allocation[END_REF] presents additional arguments in favour of the use of an explicit formalism to describe plant architectures. Equations are more elegant and condensed than a complicated algorithm or flowchart that must be analyzed to understand the model. The model gets a higher level of transparency and of universality when an exact specification language, designed for that purpose, is used. It allows applying general theories that have been previously developed in other disciplines.

In this approach, there is a search for simplicity and for extracting the minimal set of variables necessary to simulate the growth of plants at the organ scale. In GreenLab, the minimal set of processes that has been identified represents the plant development, biomass production and biomass allocation to each organ. The associated variables are linked through recurrence equations. In the current versions of the model, biomass production is computed according to the active blade surface through an equation of the form of the Beer-Lambert law. In this study, we take advantage of this formalism to study theoretically the influence of topology on biomass production in GreenLab. In particular, we show that the biomass production stabilizes as time goes to infinity and we determine the conditions under which this limit value is non-zero. We define an index based on the topological parameters of the model to evaluate the potentialities of different architectural models to reach high values of limit biomass production. We show that under particular conditions, plants with different topological developments can have exactly the same trajectory of biomass production over time. This last property helps bridging the gap between process-based models and functional-structural models, as it shows that under the appropriate assumptions, the topological development has no influence on the plant prodution.

Limit production of biomass according to its topological model

The limit behaviour of a model is defined as the limit of its state variables when times tends to infinity with constant input parameters and under stable control variables. In this paper, we focus more particularly on the limit biomass production when the number of growth cycles increases to infinity.

The growth equation of the GreenLab model

Botanical analyses of tree architecture have shown that trees exhibit a hierarchical organization and repetitive sequences of the same structures [START_REF] Barthélémy | Architecture, gradients morphogénétiques et âge physiologique chez les végétaux[END_REF]. Therefore, axes in a plant can be sorted according to their morphological or functional propoperties and indexed thanks to the notion of physiological age of the apical meristem. In the GreenLab model, plant development is simulated using a dual-scale automaton [START_REF] Zhang | A process-based growth model for young red pine[END_REF] whose states and transition rules are based on that index of physiological age. As shown in [?], [START_REF] Kang | A stochastic language for plant topology[END_REF], this method is analogous to writing a growth grammar, as done in simulations based on L-Systems [START_REF] Prusinkiewicz | The Algorithmic Beauty of Plants[END_REF]. The rules defining the plant development can be either deterministic [START_REF] Yan | A dynamic, architectural plant model simulating resourcedependent growth[END_REF], stochastic [START_REF] Kang | Stochastic 3D tree simulation using substructure instancing[END_REF] or mechanistic [START_REF] Mathieu | A dynamic model of plant growth with interactions between development and functional mechanisms to study plant structural plasticity related to trophic competition[END_REF]. The alphabet of GreenLab is introduced in [START_REF] Cournède | Structural factorization of plants to compute their functional and architectural growth[END_REF], [START_REF] De Reffye | The dynamic equations of the tree morphogenesis GreenLab model[END_REF]: it is used to recursively describe tree structure (concept of structural factorization) and to define morphisms representing operations on the plant, e.g. the growth morphism. Another interesting application is the possibility to analytically compute the number of organs appearing at each growth cycle in the deterministic and mechanistic versions of GreenLab [START_REF] Cournède | Structural factorization of plants to compute their functional and architectural growth[END_REF] and the moments or generating functions of numbers of organs in the stochastic case [?], [START_REF] Kang | A stochastic language for plant topology[END_REF].

As is the case for development, the evolution of the physiological variables can be expressed using dynamics equations. As the model considered here is based on discrete time steps (called growth cycles), the computation of biomass production at a given cycle can be explicitly written according to the state variables of the model at the previous cycles.

Assuming that the resistivities of internodes and petioles are negligible, the biomass production is supposed to only depend on the plant total blade area. The plant development influences the value of S(t), as blades have to compete with other organs to get their biomass. Detailed presentations of the GreenLab model can be found in [START_REF] Yan | A dynamic, architectural plant model simulating resourcedependent growth[END_REF], [START_REF] Cournède | Structural factorization of plants to compute their functional and architectural growth[END_REF], [START_REF] Mathieu | A dynamic model of plant growth with interactions between development and functional mechanisms to study plant structural plasticity related to trophic competition[END_REF], [START_REF] Guo | Parameter optimization and field validation of the functional-structural model GreenLab for maize[END_REF]. We present here the generic recurrence equation of GreenLab, linking the production of biomass Q(t) at cycle t to the state variables of the system at the previous growth cycles:

∀t ≥ t a , Q(t) = E(t)µS p   1 -exp   -k eS p ta i=1 N a t-i+1 i j=1 φ a j Q(t -i + j -1) D(t -i + j -1)     (1)
where E(t) is the environmental factor during cycle t, e denotes specific blade mass, µ is a parameter of biomass conversion efficiency, k is the Beer coefficient of light extinction, S p is a characteristic surface related to the ground projection of the plant, t a is the duration of blade photosynthetic activity and N a (t -i + 1) is the number of leaves appearing at cycle t-i+1. Blade sinks are defined by their sink variation functions φ a (j). For the sake of clarity, we present here the case when blades of every physiological age have the same sink strength, equal to 1. Note that if different sink values P a k for blades have to be taken into account, the term N a (t -i + 1) simply needs to be replaced by Pm k=1 N a k (t -i + 1)P a k , P m being the maximal index of physiological age in the plant and N a k (t -i + 1) the number of blades of physiological age k appeared at cycle t -i + 1. The domain of definition of the sink variation function of every organ o, φ o : j → φ o (j), is {0, . . . , t a } although the expansion duration t exp can be shorter than its time of activity. The function takes the value 0 in the interval {t exp , . . . , t a }.

This expression shows that the plant development influences the production through the demand (that includes numbers of organs appearing at each growth cycle) and the numbers of blades.

The limit production of the model is defined as the production of the plant when the number of growth cycles t goes to infinity. We investigate under which conditions the plant production can reach a stable non-zero phase. Of course the notion of infinite number of growth cycles is purely theoretical but in practice the stable regime can be reached after a phase of growth establishment. In this stable regime, the plant production is equal (or tends to) the limit production. If the limit production is zero, it means that the simulated plant does not survive. Therefore it is interesting to compare the performances of different architectural models.

Calculation of limit biomass production

Due to the exponential term, the production is always bounded:

∀t, Q(t) ≤ E(t)µS p (2)
.

Let us first consider the case of a plant with a fixed deterministic development (GL1) with no or negligible ring compartment growing under constant environmental conditions or at least under conditions that tend to stabilize (lim t→∞ E(t) = E ∞ ). Thus the notion of topological development only refers to the number of organs appearing at each cycle. According to the fixed point theorem, the iterated function sequence, if it converges, tends to the solution of the equation:

x = A • 1 -e -B•x (3)
where:

A = E ∞ µS p B = -k eS p ta i=1 i j=1 φ a (j) N a D ∞ with N a D ∞ = lim t→∞ N a (t) D(t)
. Its value depends on the plant development. It can be proved that this limit exists in the deterministic case [START_REF] Letort | Adaptation of the greenlab growth model to plants with complex architectures and multi-scale analysis of sourcesink relationships for parametric[END_REF], [START_REF] Chauveheid | Formalisation et étude d'un système complexe dans la théorie des systèmes dynamiques[END_REF] but in the mechanistic case (GL3), there can be emergence of periodic oscillations of this ratio [START_REF] Mathieu | Rhythms and alternating patterns in plants as emergent properties of a model of interaction between development and functioning[END_REF] so only majorations of plant production can be written. Hereafter we consider that B converges and reaches its limit.

The function f : R → R as: f (x) = A(1 -e -Bx ) admits a non zero solution if and only if f (0) > 1, i.e. AB > 1. Therefore, the condition to reach a positive limit value of biomass production is:

E ∞ µk e ta i=1 i j=1 φ a (j) N a D ∞ > 1
Corless et al [START_REF] Corless | On the Lambert W function[END_REF] (see also [START_REF] Grange | GreenLab mathematical model behavior analysis on Roux architectural model[END_REF]) has proved that in that case, the solution of this equation is:

Q ∞ = A + 1 B • W -AB • e -AB ( 4
)
where W is the Lambert function. The Lambert function, also called the omega function, is the inverse function of h defined by h(w) = we w . It means that W (x) is solution of the equation:

W (x)e W (x) = x ( 5
)
As h is non injective, W is multivalued: on the interval [-1 e ; 0), W (x) takes two values, as shown in Figure 1.

As AB > 1 ⇒ -AB • e -AB ∈ (-1 e ; 0), there are two possible values for W in equation 4. One of these solutions is W (x) = -AB, wich is lower than -1 and gives the solution Q ∞ = 0. The other solution is given by the second value of W (x), that must be read on its principal branch, denoted as W 0 [START_REF] Corless | On the Lambert W function[END_REF]. It is the non-dotted part of the curve represented in Figure 1. Therefore the unique strictly positive solution f (x) = x is given by:

Q ∞ = A + 1 B • W 0 -AB • e -AB (6)
Moreover, Q ∞ is an attractive fixed point of f :

|f (Q ∞ )| = W 0 -AB • e -AB < 1 since -AB • e -AB ∈ (- 1 e ; 0) for AB > 1.
As, in addition, f is continuously differentiable, this proves that there is linear convergence of the sequence

Q(t) to Q ∞ .
The value of this fixed point belongs to the following interval:

0 < A - 1 B < Q ∞ < A
Good approximations of the values of W 0 (x) can be computed thanks to its Taylor serie [START_REF] Corless | A sequence of series for the Lambert W function[END_REF], converging for |x| < 1 e (d'Alembert ratio test):

W 0 (x) = ∞ n=1 (-n) n-1 n! x n (7)

2.3.

A criterion to assess the potential of an architectural model for limit production Q ∞ is an increasing function of A. This is coherent with the interpretation of A (= E ∞ µS p ): increasing the environmental variable E, the coefficient of biomass conversion efficiency µ or the maximal ground surface projection of blades naturally results in increasing the limit production.

Using the first term of this development of W 0 , we can see that Q ∞ is also an increasing function of B. Therefore calculating the value of B for different topological parameters provides a criterion to evaluate the performance of different architectural models.

In application, we give the expression of B for two particular architectural models: Corner and Leeuwenberg models, as represented in Figure 2. The case of the Roux model can be found in [START_REF] Grange | GreenLab mathematical model behavior analysis on Roux architectural model[END_REF]. No fruits are considered and all organs are of the same physiological ages. P a and P i represent respectively blade and internode sink strength while φ a and φ i are sink variation functions.

• Corner model: there is one more metamer at each growth cycle.

∀t ≥ 1, N a (t) = N i (t) = 1 ∀t ≥ t exp , D(t) = texp j=1 P a • φ a (j) + •P i • φ i (j) B = kP a eS p ta n=1 n j=1 φ a (j) texp j=1
P a φ a (j) + P i φ i (j)

If t a = 1, B does not depend on the sink variation coefficients and the condition for a strictly positive limit production (AB > 1) can be further simplified to:

E ∞ µk e P a P a + P i > 1 A numerical value of the limit production is given in the example of Figure 3.

• Leeuwenberg model: each apical bud gives birth to M new metamers:

∀t ≥ 1, N a (t) = N i (t) = M t-1 ∀t ≥ t exp , D(t) = texp j=1
M t-j+1 P a φ a (j) + P i φ i (j)

B = kP a eS p ta n=1 n j=1 φ a (j) texp j =1
M j-j (P a φ a (j ) + P i φ i (j )) 1. Numerically (eq. 7), the limit productions are:

Q ∞ Corner = 33.79 and Q ∞ Leeuwenberg = 40.16
Table 1. Parameter values of simulations of Fig. 3.

ta=3 Sp = 500 E=1 k=1 e=0.06 µ=0.1 texp=3 P a , P i = 1 φ a , φ i = 1 It gives: A = 50 B Corner = k • P a e • Sp 2 P a + P i = 0.033 B Lee = kP a e • Sp(P a + P i ) 3M 2 + 2M + 1 M 2 + M + 1 = 0.04
In the illustration presented in Figures 2 and3, the limit production for the Leeuwenberg model is higher than that of the Corner model. In that case (T exp = 3), it can be shown that as soon as its number of buds per metamer is M > 1, the limit production of a Leeuwenberg model is necessary higher than that of a Corner model. As all functional parameters are chosen identical, this difference is purely related to the architectural model that develops. However, this trend would be likely to change if hydraulic resistances of axes were considered as there are more internodes in the Leeuweberg tree. Note that the seed biomass (initial condition of the system) has no effect neither on the limit production nor on the condition (AB > 1) to reach a stable positive value.

Invariances of biomass production

To analyze the importance of architecture on plant functioning, it is also interesting to study the cases when, contrary to the cases presented above, topological development has no influence on biomass production. In fact, it can be shown that under some conditions, plants with different topologies have the same biomass production at every cycle of their growth (not only the limit production of the steady state).

Hereafter we consider the case when all organs have the same physiological age (P m = 1) and there is no ring compartment in the demand.

Case 1: immediate expansion of organs

Under these assumptions (T exp = 1), the recurrence equation of biomass production (Eq 1) is:

Q(t) = EµS p 1 -exp - kP a eS p ta i=1 N a t-i+1 Q(t -i) D(t -i) (8
) In that case, there is a linear relationship between the numbers of organs and leaves with the proportionality constant k o . Thus the demand of the shoot part is the sum of the demand of new organs only (internodes with sink P i and fruits or flowers with sink P f ):

D(t) = (P a + k i • P i + k f • P f ) • N a (t + 1)
where N a (t + 1) is the number of new leaves at GC t + 1 (the demand is calculated at end of GC so D(t) account for the new organs appearing at the following GC). As the number of blades N a (t + 1) is a factor of this equation, the ratio in equation 8 simplifies. Therefore the recurrent equation giving the biomass production at GC t, Equation (1), can be written as :

Q(t) = EµS p 1 -exp - k eS p P a • ta i=1 Q(t -i) P a + k i P i + k f P f (9)
Thus, if the initial condition (biomass of the seed Q 0 ) is fixed, the sequence (Q t ) t∈N is fixed. It means that two plants verifying these assumptions and having leaves with the same functioning duration produce exactly the same biomass amount at each GC, whatever their topologies are. Figure 4 shows some simulation illustrations with Corner and Leeuwenberg models (GL1) and with a plant simulated with the GL3 version. The same arguments remain valid if there are several physiological ages in the plant (P m > 1) but with a constant proportionality between sinks of organs of each PA, i.e. under the following condition:

∃k o , ∀p ∈ 1 . . . P m , P o p • N o p (t) = k o • P a p • N a p (t)
The same calculations apply also for the case (purely theoretical) where there is immediate expansion (T exp = 1) of all organs with only one physiological age (P m = 1) and ring demand depending on the number of leaves with a constant number of leaves all along its life.

Case 2: expansion duration of several G.C. (t exp ≥ 1) with same shape of sink variation function for all organs

Again, the number of internodes is proportional to the number of leaves so we can write:

D(t) = (P a + k i • P i ) M in(t,texp) i=1 N a (t -i + 1)φ a (i)
Intuitively, it would seem possible to simplify the recurrent equation for the biomass production, as the same sink variation coefficients and numbers of leaves appear in the ratio of the exponential factor. In fact it is not always true (and this can be checked from the examples of Figure 2). More precisely, after several rearrangements, the above equation can be written under the form:

Q t = EµS p 1 -exp -kP a eS p (P a + k i P i ) • C with C = ta i=1 Qn-i ta-i+1 j=1 N a (t -i -j + 2)φ a (j) texp j=1 N a (t -i -j + 2)φ a (j) (10)
One can see that the ratio does not simplify as the number of terms in the sums are different. The residual terms are due to the fact that when a leaf dies, its whole biomass ceases activity regardless of the time it was allocated to the leaf.

3.2.1.

Residual term due to leaf senescence. An alternative possibility would be to consider progressive senescence of leaf. The amount of leaf biomass allocated at GC t would cease activity at GC t + t a . In that case, equation 10 simplifies to:

Q(t) = EµS p 1 -exp -kP a ta i=1 Q(t -i) eS p (P a + k i P i ) ( 11
)
Figure 5. Leaf senescence (in white): in the current version of GreenLab, leaf activity ceases abruptly: it generates a residual term in the recurrent equation of biomass production. An alternative choice would be to consider that leaf senescence is progressive: every unit of leaf biomass would remain active during t a growth cycles.

Thus with this modelling of leaf senescence, it is possible to find plants with different topological development but similar sequences of biomass production. Note that Equation 11is the same than Equation 9: expansion durations and sink variation shapes have no influence on the sequences of biomass production.

Case of infinite functioning duration of leaves.

Apart from this question, the simplification can also be done when the functioning duration of leaves (t a ) is infinite. Indeed in this case, leaves never become senescent. Figure 6 presents some example of simulated plant growth with the same biomass production and allocation to leaf and internode compartments. Expansion is immediate or infinite. For each plant, all organ kinds have the same shape of sink variation.

To conclude, similar production sequences can be found for plants with different topological developments as soon as they respect one of the following conditions:

• Immediate expansion (t exp = 1), one physiological age (P m = 1), no rings (or ring demand proportional to the number of leaves and a constant number of leaves) • Expansion duration for several cycles (t exp ≥ 1), one physiological age (P m = 1) and same sink variation shape for all organs, provided that one of the following conditions is verified:

-Leaf functioning duration is infinite (at least equal to the plant chronological age) -Leaf senescence is progressive: every unit of leaf biomass remains active during exactly t a growth cycles. In other cases, there is no simplification of the ratio of number of leaves to the total demand and therefore the topological development of the plant influences its production.

Discussion and conclusion

This paper presents a limit state and behaviour analysis of the GreenLab model. It has highlighted the interest of models to assess the performances of different topological developments to maximize the plant potential production. An index defining the plant potential performance was presented and would allow comparing different strategies of development. It seems logical that the number of organs appearing at each cycle has an effect on plant biomass production during the establishment phase. However, once biomass production stabilizes, the increase of blade surface has no impact on the production. Therefore it was not clear whether the influence of architecture remained important. This question is particularly crucial for process-based models where only compartments of biomass are usually considered [START_REF] Sievänen | Fitting process-based models with stand growth data: problems and experiences[END_REF]- [START_REF] Schwalm | A process-based model of forest ecosystems driven by meteorology[END_REF]. The results of our analysis show that in most cases, the number of organs is determinant for the value of limit biomass production. The conditions under which it is possible to get similar sequences of biomass productions with different topological developments revealed themselves to be mainly theoretical and are hardly encountered in nature. Indeed, as soon as the root compartment is considered, the simplifications of the equations presented above no longer hold. Moreover, similar study should be done including the underground part: for instance, Fitter et al [START_REF] Ftter | Architectural analysis of plant root systems 1. architectural correlates of exploitation efficiency[END_REF] analysed the influence of root architecture for soil exploitation efficiency using simulations. However, using sensitivity analysis to further study the distance between different plants as far as their potential limit production is concerned would bring more detailed insights.

The limit state analysis can provide clues to discriminate between several modelling choices. For instance in [START_REF] Letort | Adaptation of the greenlab growth model to plants with complex architectures and multi-scale analysis of sourcesink relationships for parametric[END_REF], the effect of the calculation mode for ring demand on limit biomass production in GreenLab was studied. It was shown that one of the expressions proposed (ring demand proportional to total biomass production) induces lower limit production values than the two other ones (ring demand proportional to the number of leaves or ring demand proportional to the ratio of biomass supply to demand). By comparison with the qualitative biological knowledge, these properties can be criteria to guide modeller's choice.

Studying invariants in a model can be interesting before beginning the procedure of parametric identification. Here we showed that several architectures can produce the same trajectory of biomass production. Although the conditions found to get invariant biomass production imply that these cases are purely theoretical, it is likely that it exists real cases when the difference engendered by two different sequences of topological development is low. Therefore if the only data available is biomass, there might be several appropriate solutions when one aims at estimating the parameters driving the topological development [START_REF] Letort | Parameter identification of a functionalstructural tree growth model and application to beech trees (Fagus sylvatica)[END_REF]. This case could be encountered if data collected at the level required for processbased models (i.e. at compartment level) were used to fit a functional-structural model. In that case, more a priori knowledge should be incorporated: for instance, a default topology can be defined based on the botanical study of the corresponding species. Then parameters can be fitted to drive the variations of this potential topology according to the plant internal state of trophic competition and in relation with environmental constraints.

Figure 1 .

 1 Figure 1. Lambert function for x ≥ -1 e . There are two branches on the interval [-1 e ; 0) [29]. The continuous line represents the principal branch W 0 .

Figure 2 .

 2 Figure 2. Corner and Leeuwenberg architectural models: comparison of biomass productions. The simulated values are consistent with the numerical results found after solving eq. 3 as shown in fig. 3.

Figure 3 .

 3 Figure 3. Graphical resolution of Eq.3: Q ∞ = A(1e -B•Q∞ ) for Corner and Leeuwenberg (M=2) architectural models. The parameter values are given in table1. Numerically (eq. 7), the limit productions are: Q ∞ Corner = 33.79 and Q ∞ Leeuwenberg = 40.[START_REF] Kang | A stochastic language for plant topology[END_REF]

Figure 4 .

 4 Figure 4. Simulation of plant growth with different topologies but same biomass production (Case 1: t exp = 1, P m = 1, D ring = 0). The functional parameters are: t a = 5, P a = 1, P i = 1.5, µ = 0.1, e = 0.06, S p = 500.

Figure 6 .

 6 Figure 6. Examples of simulated plants with same biomass production and allocation to blade and internode compartments. Leaf functioning time is infinite.Expansion is immediate or infinite. For each plant, all organ kinds have the same shape of sink variation.

Acknowledgment

The authors thank V. Le Chevalier for his corrections on this manuscript.