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1.  Computing Competition for Light in the GREENLAB Model of Plant Growth: A Contribution to the Study of the Effects of Density on Resource Acquisition and Architectural Development 
Annals of Botany  2007;101(8):1207-1219.
Background and Aims
The dynamical system of plant growth GREENLAB was originally developed for individual plants, without explicitly taking into account interplant competition for light. Inspired by the competition models developed in the context of forest science for mono-specific stands, we propose to adapt the method of crown projection onto the x–y plane to GREENLAB, in order to study the effects of density on resource acquisition and on architectural development.
The empirical production equation of GREENLAB is extrapolated to stands by computing the exposed photosynthetic foliage area of each plant. The computation is based on the combination of Poisson models of leaf distribution for all the neighbouring plants whose crown projection surfaces overlap. To study the effects of density on architectural development, we link the proposed competition model to the model of interaction between functional growth and structural development introduced by Mathieu (2006, PhD Thesis, Ecole Centrale de Paris, France).
Key Results and Conclusions
The model is applied to mono-specific field crops and forest stands. For high-density crops at full cover, the model is shown to be equivalent to the classical equation of field crop production ( Howell and Musick, 1985, in Les besoins en eau des cultures; Paris: INRA Editions). However, our method is more accurate at the early stages of growth (before cover) or in the case of intermediate densities. It may potentially account for local effects, such as uneven spacing, variation in the time of plant emergence or variation in seed biomass. The application of the model to trees illustrates the expression of plant plasticity in response to competition for light. Density strongly impacts on tree architectural development through interactions with the source–sink balances during growth. The effects of density on tree height and radial growth that are commonly observed in real stands appear as emerging properties of the model.
PMCID: PMC2710279  PMID: 18037666
Functional–structural plant models; GREENLAB; competition for light; Beer–Lambert Law; plant plasticity; dynamical system
2.  Modeling tree crown dynamics with 3D partial differential equations 
We characterize a tree's spatial foliage distribution by the local leaf area density. Considering this spatially continuous variable allows to describe the spatiotemporal evolution of the tree crown by means of 3D partial differential equations. These offer a framework to rigorously take locally and adaptively acting effects into account, notably the growth toward light. Biomass production through photosynthesis and the allocation to foliage and wood are readily included in this model framework. The system of equations stands out due to its inherent dynamic property of self-organization and spontaneous adaptation, generating complex behavior from even only a few parameters. The density-based approach yields spatially structured tree crowns without relying on detailed geometry. We present the methodological fundamentals of such a modeling approach and discuss further prospects and applications.
PMCID: PMC4104424  PMID: 25101095
leaf area density; continuity equation; functional-structural plant model; crown plasticity; competition for light; Beer-Lambert's law
3.  NEMA, a functional–structural model of nitrogen economy within wheat culms after flowering. I. Model description 
Annals of Botany  2011;108(6):1085-1096.
Background and Aims
Models simulating nitrogen use by plants are potentially efficient tools to optimize the use of fertilizers in agriculture. Most crop models assume that a target nitrogen concentration can be defined for plant tissues and formalize a demand for nitrogen, depending on the difference between the target and actual nitrogen concentrations. However, the teleonomic nature of the approach has been criticized. This paper proposes a mechanistic model of nitrogen economy, NEMA (Nitrogen Economy Model within plant Architecture), which links nitrogen fluxes to nitrogen concentration and physiological processes.
A functional–structural approach is used: plant aerial parts are described in a botanically realistic way and physiological processes are expressed at the scale of each aerial organ or root compartment as a function of local conditions (light and resources).
Key Results
NEMA was developed for winter wheat (Triticum aestivum) after flowering. The model simulates the nitrogen (N) content of each photosynthetic organ as regulated by Rubisco turnover, which depends on intercepted light and a mobile N pool shared by all organs. This pool is enriched by N acquisition from the soil and N release from vegetative organs, and is depleted by grain uptake and protein synthesis in vegetative organs; NEMA accounts for the negative feedback from circulating N on N acquisition from the soil, which is supposed to follow the activities of nitrate transport systems. Organ N content and intercepted light determine dry matter production via photosynthesis, which is distributed between organs according to a demand-driven approach.
NEMA integrates the main feedbacks known to regulate plant N economy. Other novel features are the simulation of N for all photosynthetic tissues and the use of an explicit description of the plant that allows how the local environment of tissues regulates their N content to be taken into account. We believe this represents an appropriate frame for modelling nitrogen in functional–structural plant models. A companion paper will present model evaluation and analysis.
PMCID: PMC3189836  PMID: 21685431
Rubisco turnover; remobilization; functional–structural plant model; nitrogen; light acclimation; senescence; wheat; Triticum aestivum; root uptake; common pool
4.  NEMA, a functional–structural model of nitrogen economy within wheat culms after flowering. II. Evaluation and sensitivity analysis 
Annals of Botany  2011;108(6):1097-1109.
Background and Aims
Simulating nitrogen economy in crop plants requires formalizing the interactions between soil nitrogen availability, root nitrogen acquisition, distribution between vegetative organs and remobilization towards grains. This study evaluates and analyses the functional–structural and mechanistic model of nitrogen economy, NEMA (Nitrogen Economy Model within plant Architecture), developed for winter wheat (Triticum aestivum) after flowering.
NEMA was calibrated for field plants under three nitrogen fertilization treatments at flowering. Model behaviour was investigated and sensitivity to parameter values was analysed.
Key Results
Nitrogen content of all photosynthetic organs and in particular nitrogen vertical distribution along the stem and remobilization patterns in response to fertilization were simulated accurately by the model, from Rubisco turnover modulated by light intercepted by the organ and a mobile nitrogen pool. This pool proved to be a reliable indicator of plant nitrogen status, allowing efficient regulation of nitrogen acquisition by roots, remobilization from vegetative organs and accumulation in grains in response to nitrogen treatments. In our simulations, root capacity to import carbon, rather than carbon availability, limited nitrogen acquisition and ultimately nitrogen accumulation in grains, while Rubisco turnover intensity mostly affected dry matter accumulation in grains.
NEMA enabled interpretation of several key patterns usually observed in field conditions and the identification of plausible processes limiting for grain yield, protein content and root nitrogen acquisition that could be targets for plant breeding; however, further understanding requires more mechanistic formalization of carbon metabolism. Its strong physiological basis and its realistic behaviour support its use to gain insights into nitrogen economy after flowering.
PMCID: PMC3189838  PMID: 21685429
Rubisco turnover; remobilization; functional–structural plant model; nitrogen; light acclimation; senescence; wheat; Triticum aestivum; root uptake; common pool
5.  Characterization of the interactions between architecture and source–sink relationships in winter oilseed rape (Brassica napus) using the GreenLab model 
Annals of Botany  2010;107(5):765-779.
Background and Aims
This study aimed to characterize the interaction between architecture and source–sink relationships in winter oilseed rape (WOSR): do the costs of ramification compromise the source–sink ratio during seed filling? The GreenLab model is a good candidate to address this question because it has been already used to describe interactions between source–sink relationships and architecture for other species. However, its adaptation to WOSR is a challenge because of the complexity of its developmental scheme, especially during the reproductive phase.
Equations were added in GreenLab to compute expansion delays for ramification, flowering of each axis and photosynthesis of pods including the energetic cost of oil synthesis. Experimental field data were used to estimate morphological parameters while source–sink parameters of the model were estimated by adjustment of model outputs to the data. Ecophysiological outputs were used to assess the sources/sink relationships during the whole growth cycle.
Key Results
First results indicated that, at the plant scale, the model correctly simulates the dynamics of organ growth. However, at the organ scale, errors were observed that could be explained either by secondary growth that was not incorporated or by uncertainties in morphological parameters (durations of expansion and life). Ecophysiological outputs highlighted the dramatic negative impact of ramification on the source–sink ratio, as well as the decrease in this ratio during seed filling despite pod envelope photosynthesis that allowed significant biomass production to be maintained.
This work is a promising first step in the construction of a structure–function model for a plant as complex as WOSR. Once tested for other environments and/or genotypes, the model can be used for studies on WOSR architectural plasticity.
PMCID: PMC3077979  PMID: 20980324
Biological system modelling; source–sink relationships; ramification; GreenLab model; energetic cost; oleaginous seeds; Brassica napus; winter oilseed rape
6.  Parameter Optimization and Field Validation of the Functional–Structural Model GREENLAB for Maize at Different Population Densities 
Annals of Botany  2007;101(8):1185-1194.
Background and Aims
Plant population density (PPD) influences plant growth greatly. Functional–structural plant models such as GREENLAB can be used to simulate plant development and growth and PPD effects on plant functioning and architectural behaviour can be investigated. This study aims to evaluate the ability of GREENLAB to predict maize growth and development at different PPDs.
Two field experiments were conducted on irrigated fields in the North China Plain with a block design of four replications. Each experiment included three PPDs: 2·8, 5·6 and 11·1 plants m−2. Detailed observations were made on the dimensions and fresh biomass of above-ground plant organs for each phytomer throughout the seasons. Growth stage-specific target files (a description of plant organ weight and dimension according to plant topological structure) were established from the measured data required for GREENLAB parameterization. Parameter optimization was conducted using a generalized least square method for the entire growth cycles for all PPDs and years. Data from in situ plant digitization were used to establish geometrical symbol files for organs that were then applied to translate model output directly into 3-D representation for each time step of the model execution.
Key Results
The analysis indicated that the parameter values of organ sink variation function, and the values of most of the relative sink strength parameters varied little among years and PPDs, but the biomass production parameter, computed plant projection surface and internode relative sink strength varied with PPD. Simulations of maize plant growth based on the fitted parameters were reasonably good as indicated by the linearity and slopes similar to unity for the comparison of simulated and observed values. Based on the parameter values fitted from different PPDs, shoot (including vegetative and reproductive parts of the plant) and cob fresh biomass for other PPDs were simulated. Three-dimensional representation of individual plant and plant stand from the model output with two contrasting PPDs were presented with which the PPD effect on plant growth can be easily recognized.
This study showed that GREENLAB model has the ability to capture plant plasticity induced by PPD. The relatively stable parameter values strengthened the hypothesis that one set of equations can govern dynamic organ growth. With further validation, this model can be used for agronomic applications such as yield optimization.
PMCID: PMC2710275  PMID: 17921525
Functional–structural plant model; GREENLAB; plant architecture; source–sink relationship; plant population density; maize (Zea mays); model parameterization
7.  Rhythms and Alternating Patterns in Plants as Emergent Properties of a Model of Interaction between Development and Functioning 
Annals of Botany  2007;101(8):1233-1242.
Background and Aims
To model plasticity of plants in their environment, a new version of the functional–structural model GREENLAB has been developed with full interactions between architecture and functioning. Emergent properties of this model were revealed by simulations, in particular the automatic generation of rhythms in plant development. Such behaviour can be observed in natural phenomena such as the appearance of fruit (cucumber or capsicum plants, for example) or branch formation in trees.
In the model, a single variable, the source–sink ratio controls different events in plant architecture. In particular, the number of fruits and branch formation are determined as increasing functions of this ratio. For some sets of well-chosen parameters of the model, the dynamical evolution of the ratio during plant growth generates rhythms.
Key Results and Conclusions
Cyclic patterns in branch formation or fruit appearance emerge without being forced by the model. The model is based on the theory of discrete dynamical systems. The mathematical formalism helps us to explain rhythm generation and to control the behaviour of the system. Rhythms can appear during both the exponential and stabilized phases of growth, but the causes are different as shown by an analytical study of the system. Simulated plant behaviours are very close to those observed on real plants. With a small number of parameters, the model gives very interesting results from a qualitative point of view. It will soon be subjected to experimental data to estimate the model parameters.
PMCID: PMC2710268  PMID: 17715304
Rhythms; plasticity; plant growth model; GREENLAB; interactions; branching system; fructification; emergent properties
8.  Quantitative Genetics and Functional–Structural Plant Growth Models: Simulation of Quantitative Trait Loci Detection for Model Parameters and Application to Potential Yield Optimization 
Annals of Botany  2007;101(8):1243-1254.
Background and Aims
Prediction of phenotypic traits from new genotypes under untested environmental conditions is crucial to build simulations of breeding strategies to improve target traits. Although the plant response to environmental stresses is characterized by both architectural and functional plasticity, recent attempts to integrate biological knowledge into genetics models have mainly concerned specific physiological processes or crop models without architecture, and thus may prove limited when studying genotype × environment interactions. Consequently, this paper presents a simulation study introducing genetics into a functional–structural growth model, which gives access to more fundamental traits for quantitative trait loci (QTL) detection and thus to promising tools for yield optimization.
The GREENLAB model was selected as a reasonable choice to link growth model parameters to QTL. Virtual genes and virtual chromosomes were defined to build a simple genetic model that drove the settings of the species-specific parameters of the model. The QTL Cartographer software was used to study QTL detection of simulated plant traits. A genetic algorithm was implemented to define the ideotype for yield maximization based on the model parameters and the associated allelic combination.
Key Results and Conclusions
By keeping the environmental factors constant and using a virtual population with a large number of individuals generated by a Mendelian genetic model, results for an ideal case could be simulated. Virtual QTL detection was compared in the case of phenotypic traits – such as cob weight – and when traits were model parameters, and was found to be more accurate in the latter case. The practical interest of this approach is illustrated by calculating the parameters (and the corresponding genotype) associated with yield optimization of a GREENLAB maize model. The paper discusses the potentials of GREENLAB to represent environment × genotype interactions, in particular through its main state variable, the ratio of biomass supply over demand.
PMCID: PMC2710265  PMID: 17766844
Plant growth model; GREENLAB; genetics; QTL; breeding; yield optimization; genetic algorithm; Zea mays

Results 1-8 (8)