Neoformation is the process by which organs not preformed in a bud are developed on a growing shoot, generally after preformation extension. The aim of this study was to apply a deconvolution method to estimate the neoformation distributions (rather than simply the mean number of neoformed organs) of shoots of five woody species known to develop both preformed and neoformed organs. For each of eight data sets, the distribution of the number of neoformed organs was estimated from two independent samples, one corresponding to the distribution of the total number of organs (in fully extended shoots of the first sample), and the other to the distribution of the number of preformed organs (in dissected buds of the second sample). It was shown that neoformation contributed more than preformation to explain full-size differences between shoots developed in different positions within the architecture of each tree species and that, for a given position, the dispersion of the number of neoformed organs is higher (due to differences of growth cessation times) than the dispersion of the number of preformed organs .

The elongation of leafy axes is influenced by environmental conditions such as for instance rainfall. In the context of plant growth follow-up, measurement is made of the number of newly elongated leaves during successive observation periods. These count data are often underdispersed with reference to the Poisson distribution. Incorporating the rainfall covariate within a statistical model requires to take into account both the delayed response of the plant to a water stress and the difference in time step between the rainfall covariate (daily data) and the plant response (observation period of several days). Statistical models studied for analysing these longitudinal count data belong to the classes of generalized linear mixed models. We are studying particular models based on weighted Poisson distributions where weighted Poisson distributions define a two-parameter exponential family.

The objective of this study was to model the changes in growth unit branching structures during tree ontogeny. A single statistical model was estimated from around 700 branching sequences measured on two six-year-old apple trees, cultivar "Fuji". This statistical model is a hidden semi-Markov chain were the underlying semi-Markov chain represents the succession and the lengths of branching zones along the growth units while the observation distributions represent the branching type composition within each zone. The main output of this study was to show that, while the succession and the lengths of branching zones change during ontogeny, the compositions of branching zones remain unchanged. In particular the composition of the flowering shoot zone is not affected by the alternation of flowering and non-flowering years .

Observed growth, as given for instance by the length of successive annual shoots along a tree trunk, is mainly the result of two components: an ontogenetic component and an environmental component. An open question is whether the ontogenetic component along an axis at the growth unit or annual shoot scale takes the form of a trend or of a succession of phases. Various methods of analysis ranging from exploratory analysis (symmetric smoothing filters, sample autocorrelation functions) to statistical modeling (multiple change-point models, hidden semi-Markov chains and hidden hybrid model combining Markovian and semi-Markovian states ) were applied to extract and characterize both the ontogenetic and environmental components using contrasted examples. This led us in particular to favor the hypothesis of an ontogenetic component structured as a succession of stationary phases and to highlight phase changes of high magnitude in unexpected situations (for instance when growth globally decreases). These results shed light in a new way on botanical concepts such as "morphogenetic gradient" and "physiological age of meristem".

Incorporating both the influence of explanatory variables and inter-individual heterogeneity in a hidden Markovian model is a challenging problem. We are studying different families of Markov switching models including Markov switching linear mixed models, i.e. models that combine linear mixed models in a Markovian manner and Markov switching generalized linear mixed models. In the case of a Markov switching linear mixed model, the underlying Markov chain represents the succession of growth phases while the linear mixed models attached to each state of the Markov chain represent both the trend, the influence of the explanatory variables (mainly climatic variables which are time-dependent explanatory variables measured at a scale different from the scale of the plant response) and the inter-individual heterogeneity within a given growth phase. The EM algorithm which is the standard algorithm for estimating hidden Markovian models cannot be applied for estimating Markov switching mixed models. We are thus investigating alternative solutions relying either on numerical techniques or on algorithms for simulating state sequences and random effects (see Figure ).

Once a hidden Markovian model has been estimated, it is generally of interest to understand the hidden state sequence (or tree) structure underlying each observed sequence (or tree). Questions of interest are:

Methods for exploring the state sequences that explain a given observed sequence for a known hidden Markovian model may be divided into three categories: (i) enumeration of state sequences, (ii) state profiles i.e. state sequences summarized in a J× array where Jis the number of states and the length of the sequence, (iii) computation of a global measure of the state sequence uncertainty (entropy of the state sequence that explains an observed sequence for a known hidden Markovian model). Various methods belonging to these three categories have been developed for different families of hidden Markovian models including hidden semi-Markov chains, hidden hybrid model combining Markovian and semi-Markovian states and different categories of hidden Markov tree models. In particular, we propose a new type of state profiles that can be viewed as the superposition of all the state sequences from the less probable to the most probable . Due to their multidimensional nature, state profiles are difficult to visualize and interpret on trees. Hence, we propose to compute in a first stage a unidimensional profile of conditional entropies that summarize for each vertex the state uncertainty. In a second stage, the usual state profiles are visualized on selected paths of interest within the tree.

The retrospective or off-line change-point detection problem is addressed. From an algorithmic point of view, this research theme is closely connected to the preceding theme restricted to hidden semi-Markov chains. The studied algorithms share a common structure with similar algorithms recently proposed for exploring the state sequence space structure for hidden semi-Markov chains . The separability of the computations at change points corresponds to conditional independence at jump times for hidden semi-Markov chains. A key difference between a hidden semi-Markov chain and a multiple change-point model lies in the fact that the hidden semi-Markov chain parameters are fixed parameters (previously estimated) for state sequence or state profile computation while change-point model parameters (for instance mean and variance for each segment in the Gaussian case) are "contextual" parameters that depend on the change points. This explains the specific behaviour of the proposed diagnostic tools in connection with the determination of the optimal number of change points. Methods for exploring the segmentation space structure for a fixed number of segments may be divided into two categories: (i) enumeration of segmentations, (ii) summary of the possible segmentations in change-point or segment profiles. Concerning the first category, a forward dynamic programming algorithm for computing the top Lmost probable segmentations and a forward-backward algorithm for sampling segmentations are studied. Concerning the second category, a forward-backward dynamic programming algorithm and a smoothing-type forward-backward algorithm for computing two types of change-point and segment profiles are studied. The proposed methods are mainly useful for exploring the segmentation spaces for successive numbers of segments. Change-point and segment profiles may help to predict supplementary change points, highlight overestimation of the number of segments and a specific type of change-point profiles summarizes the uncertainty concerning the location of change points.

Plant architecture is the result of repetitions that occur throughout growth and branching processes. During plant ontogeny, changes in the morphological characteristics of plant entities such as growth units or annual shoots, are interpreted as the indirect effect of the meristems being in different physiological states. Thus, connected entities can exhibit either similar or very contrasted characteristics. We designed a statistical model to reveal and characterise homogeneous zones and transitions between zones within tree-structured data: the hidden Markov tree (HMT) model. This model enables to assign the entities to classes sharing a same "physiological state" . The ability of hidden Markov tree models (HMTs) to discriminate classes of entities that make biological sense was investigated using diverse temperate and tropical forest species.

We are investigating multitype branching processes with dependences as a new analysis tool for plant structures. This research theme is closely connected to the preceding one concerning hidden Markov tree models since, in most cases, the types are the states restored for each entity using a previously estimated hidden Markov tree model. Our objective is twofold: First, we are testing various sub-families of branching processes (with different parameterizations and dependencies) using a set of reference data sets (apple tree, Aleppo pine, Symphonia globulifera) in order to determine the most useful ones. Second, branching processes focus on an unusual way of studying plant development and this generates new biological questions concerning the rules governing the generation of offspring entities from a parent entity.

Self-similarity of plants has attracted the attention of biologists for at least 50 years , yet its formal treatment is rare, and no measure for quantifying the degree of self-similarity of plants was defined. To fill this gap, we recently introduced a formal definition and corresponding measures of self-similarity, tailored to axial branching plant structures (branching structures having a trunk). These measures are based on edit distance algorithms that makes it possible to compute topological distances between branching systems which are not isomorphic. The formalism was illustrated using theoretical axial branching systems, and applied to analyze self-similarity in two sample plant structures: inflorescences of Syringa vulgaris(lilac) (Figure ) and shoots of Oryza sativa(rice) . From a biological perspective, we may observe that this approach of self-similarity enables us to set up a link between the macroscopic world of branching structures and the microscopic world of meristems (from which these structures were supposedly produced). This is made possible through the following simplifying, though fundamental, scaling hypothesis: If two branching structures in a plant are similar, they were probably produced by meristems with similar state and context.

An extension of this first approach has been undertaken to define self-similarity for a more general class of branching structures (including structures that does not have a well defined trunk) with sound theoretical fundation. This extension is based on the systematic identification of isomorphic structures in trees. This makes it possible to define the class of self-similar trees as a class of trees with particular distribution of isomorphic subtrees. A polynomial algorithm able to compute, for any given input tree, the closest tree in the class of self-similar trees has been designed. This optimal closest self-similar tree (CST) can be viewed as a maximally compressed version of the input tree. On the one hand, this opens the way for designing new compression techniques of trees in various application domains (e.g. computer graphics). On the other hand, under the above scaling hypothesis, the CST structure is expected to provide biologically significant hierarchies of meristem states.

Methods have been developed in the past decade to digitize plant architecture in 3D. These methods are based on direct measurements of position and shape of every plant organ in space. Although they provide accurate results, these methods are particularly time consuming. In this work, we design a method to reconstruct the 3D leaf density of the plant based on horizontal photographs of the plant crown and on foliage aggregation assumptions. The problem is formulated as an inverse problem were the model is based on different variants of Beer-Lamber model for light interception. This results in a large number of non-linear equations, which depends on several structural parameters (discretization of the image, discretization of the canopy, number of "black zones" on the image, etc.). The huge set of equations is solved using the optimization algorithm L-BFGS-B with simple bounds. This algorithm is based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function. Comparisons were made between the actual leaf foliage density of plants digitized leaf by leaf and results of our methods. They show that optimal experimental parameter can be defined depending on the type of plant crown (number of photographs, size of the image zones used for discretization, size of the voxels, ...).

Plant geometry is a key factor for the modeling of plant ecophysiological interaction with the environment. To capture and analyse the irregular nature of plant shapes we develop tools based on fractal geometry. For this purpose, a range of fractal methods is applied to various 3D digitized tree databases and on theoretical plants generated from fractal rules (Iterated Function Systems). We determined the fractal dimension of tree crowns using two different methods (see Figure ). The first one is based on the classical box-counting estimator, adapted here to 3D scenes. The method was extensively analysed and various features and drawbacks were discussed. Possible variants of the box counting estimators of the fractal dimension with better properties have been investigated . Other estimators of the fractal dimension were also considered, based on the so called "two-surface" method and on mass estimators . These latter estimators are connected with the notion of lacunarity, used in fractal geometry to characterize the texture of the studied object as a function of scale. Here, we defined a notion of "centered" lacunarity that can be used to characterise the typical size of gaps at different scales in the object.

Light models for vegetation canopies based on the turbid medium analogy are usually limited by the basic assumption of random foliage dispersion in the canopy space. The objective of this work was to assess the effect of three possible sources of non-randomness in tree canopies on light interception properties. For this purpose, four three-dimensional (3D) digitized trees and four theoretical canopies - one random and three built from fractal rules - were used to compute canopy structure parameters and light interception, namely the sky-vault averaged STAR (Silhouette to Total Area Ratio). STAR values were computed from (1) images of the 3-D plants, and (2) from a 3-D turbid medium model using space discretization at different scales. For all trees, departure from randomness was mainly due to the spatial variations in leaf area density within the canopy volume. Indeed STAR estimations, based on turbid medium assumption, using the finest space discretization were very close to STAR values computed from the plant images. At this finest scale, foliage dispersion was slightly clumped, except for one theoretical fractal canopy, which showed a marked regular dispersion. Taking into account a non-infinitely small leaf size, whose effect is theoretically to shorten self-shading, had a minor effect on STAR computations. STAR values computed from the 3-D turbid medium were very sensitive to plant lacunarity, a parameter introduced in the context of fractal studies to characterize the distribution of gaps in porous media at different scales. This study shows that 3-D turbid medium models based on space discretization are able to give correct estimation of light interception by 3-D isolated trees, provided that the 3-D grid is properly defined, that is, discretization maximizes plant lacunarity.

Plant development is controlled by the combined effect of gene activity and environmental constraints. At a given date, a plant architecture is thus the outcome of this combination. The question of identifying the genetic and environmental components of plant development can be addressed by studying the heritability of architectural traits. We started to investigate this issue in the context of two agronomic applications, respectively on coffee and apple tree ( ; ). In these studies, architectural parameters were used to predict target traits for breeding programs: (i) yield capacity for coffee trees; (ii) adequate forms for an easy and low cost training in the field and regular bearing behaviour, for apple trees. Observation protocols for describing the architecture were applied to six clones of Coffea canephora in a comparative trial on the one hand, and on an apple progeny whose parents were chosen for their contrasted architecture, on another hand. Architectural traits, including both topological and geometric traits, were collected at different scales (trees, branching systems, axes and nodes) and were included in architectural databases which were explored using the V-Plantssoftware (formely AMAPmod). Several traits exhibited high heritability values, for instance branching, internode length and branch orientation in apple tree. In the case of coffee tree, some of the traits displayed strong genetic correlations with cumulated yield over two cycles (14 years). In the apple tree progeny, since a genetic map was built in UMR GenHort in Angers, correlations between the phenotypic variation of a given trait and allelic variations observed in the population are currently under investigation to seek for quantitative trait loci (QTL).

The problem of formulating lumpability hypotheses for a Markov chain is addressed. In the classical approach to lumpability, this problem can be formulated as the determination of an appropriate state space (smaller than the original state space) such that the lumped chain defined on this state space retains the Markov property. We propose a different perspective on lumpability where the state space is fixed and the partitioning of this state space is represented by a one-to-many probabilistic function within a two-level stochastic process. Three nested classes of lumped processes can be defined in this way as sub-classes of first-order Markov chains. These lumped processes enable parsimonious reparameterizations of Markov chains that help to reveal relevant partitions of the state space. Characterizations of the lumped processes on the original transition probability matrix are derived. Different model selection methods relying either on hypothesis testing or on penalized log-likelihood criteria are investigated and extensions to lumped processes constructed from high-order Markov chains are proposed. These lumped processes enables to highlight differences between intronic sequences and gene untranslated region sequences in human DNA sequences .

Individual or stand-level biomass is not easy to measure. The current methods employed, based on cutting down a representative sample of plantations, make it possible to assess the biomasses for various compartments (bark, dead branches, leaves, ...). However, this felling makes individual longitudinal follow-up impossible. In this context, we propose a method to evaluate individual biomasses by compartments when these biomasses are taken as ordinals. Biomass is measured visually and observations are therefore not destructive. The technique is based on a probit model redefined in terms of latent variables. A generalization of the univariate case to the multivariate case is then natural and takes into account the dependency between compartment biomasses. These models are then extended to the longitudinal case by developing a Dynamic Multivariate Ordinal Probit Model. The performance of the MCMC algorithm used for the estimation is illustrated by means of simulations built from known biomass models. The quality of the estimates and the impact of certain parameters, are then discussed .

A first integration of stochastic and mechanistic approaches has been carried out to model the development of apple trees (over the first six years of the growth). The simulation was based on two submodels: hidden semi-Markov chains described the branching patterns (type of axillary production at each node of a branch) and a biomechanical model was used to simulate the bending of branches under fruit and branch weight. The model thus attempts to capture in an integrated, developmental framework both the apple tree's topology (the branching patterns) and its form (the shape of the branches, as determined dynamically by the gravity and the wood properties). The core simulation is specified using a L-system implemented with the Ł+C language (Prusinkiewicz & Karwowski, University of Calgary) with which the stat module of VPlants has been coupled. The statistical analysis was done using the the statistic module of VPlants( VPstats) . This work is the first attempt worldwide to integrate advanced stochastic descriptions of topology and mechanistic processes. This new type of structure-function model demonstrates that such an integration at tree level can now be done quantitativelly and faithfully with respect to observed tree architectures.

We developed a general protocol for observing and reconstructing in 3D and 4D the surface of shoot apical meristems from confocal microscopy at cell scale. This includes: acquisition of images, preprocessing of the images, image segmentation, three- and four-dimensional reconstruction of the meristem surface and quantitative analysis. This protocol uses both classical techniques from image processing (topological closure, watersheds) and specially designed algorithms . We intend now to adapt this methodology We are currently working on an algorithm to automate the detection of cell lineage throughout time in temporal series corresponding to meristem development. These tools are being integrated in the software platform OpenAleafor plant modeling and analysis and are freely available for the scientific community.

The active transport of the plant hormone auxin plays a major role in the initiation of organs at the shoot apex. Polar localized membrane proteins of the PIN1 family facilitate this transport and recent observations suggest that auxin maxima created by these proteins are at the basis of organ initiation. This hypothesis is based on the visual, qualitative characterization of the complex distribution patterns of the PIN1 protein in Arabidopsis Thaliana. To take these analyzes further, we investigated the properties of the patterns using computational modeling and extensive sensitivity analysis of the model. The simulations reveal novel aspects of PIN1 distribution (see Figure ). In particular they suggest an important role for the meristem summit in the distribution of auxin and the emergence of phyllotactic patterns. We confirm these predictions by further experimentation and propose a detailed model for the dynamics of auxin fluxes at the shoot apex .

We integrated the previous results (organization of the PIN transporters, auxin fluxes, accumulation of auxin in the meristem center) in a dynamic model of the meristem development. The meristem structure is represented by a Voronoï diagram and the changes over time of the meristem structure were described using declarative rules using the MGSlanguage (language dedicated to the modeling of dynamic systems with dynamic structures (DS)2, developed at the University of Evry by J.L. Giavitto and O. Michel). The complete model relies on i) a model of cell growth and division, ii) a spring-mass description of the mechanical interaction between cells, iii) a magnetic polarisation of PIN transporters towards the primordia and the center iv) assumptions for auxin transport identical to the static case. Based on these models and assumptions, we showed that phyllotactic patterns emerge from cell growth and cell-cell interactions . This cell-centered model is a first step towards the development of a 3D mechanical and dynamic model of the Arabidopsis Thalianameristem based on cell-cell interaction.

From a mechanical point of view, a meristem may be represented by a set of sheets (the walls) on which pressure is acting (called turgor pressure). In the special case of a transversal cut of the meristem, the sheets appear as segments whose mechanical properties may be described using a set of springs of stiffness Kand rest length l₀. That kind of model allows the computation of the shape of the meristem as the arrangement in space of the cells that minimizes mechanical constraints or deformations. The previously mentioned chemical molecules interact with the system by changing Kvalues. Growth is modeled by increasing l₀proportionally to the wall deformation above some threshold. Replacing walls by springs is sufficient to model a slide of the meristem or its external shape. This approximation reaches its limits for structures with great constraints. In that case, an explicit representation of cells as 3D volumes is required. This representation (see Figure ) use tensor mechanics to compute the shape of the meristem as a minimum of energy. The problem must be solved using finite elements methods, thus increasing computer simulation time and complexity.

In this work, our aim is to understand the genetic and physiological determinants of axillary root initiation in Arabidopsis Thaliana. As in the case of primordia initiation of the shoot apical meristem, auxin plays a crucial role in root initiation. Organ initiation is correlated to high concentrations of auxin. However, in both systems, organ initiation is very different. In particular, in the root development, lateral organs initiation is observed at a long distance from the root meristem, with no apparent spatial or temporal structure. Our main assumption is that this system is governed by a mechanism of competition for auxin while the root system is developing. To understand this complex dynamic interaction, we develop a sink-source dynamic model of the root system development in which the transport of auxin is controlled by active carriers of the PIN family (a first prototype has been developed using L-Systems). First experimental protocols were carried out on Arabidopsis Thaliana, with accuracy ranging from macroscopic to cellular levels. We obtained complete data about the developmental sequence in the primary root and its laterals. This extensive database is explored using tools for sequence analysis (distance between sequences, different types of Markovian models). First results show structures and correlations between positions and stages of development of lateral organs which can now be exploited for the design of the dynamical model.