Towards a dynamic model of the Arabidopsis meristem

Transcription

1 F S P M 0 4 Towards a dynamic model of the Arabidopsis meristem Pierre Barbier de Reuille 1, Jan Traas 2, Christophe Godin 3 1 INRA, UMR AMAP Montpellier 2 INRA, Laboratoire de biologie cellulaire de Versailles 3 INRIA, UMR AMAP Montpellier Introduction Shoot apical meristems (SAMs) are populations of dividing, undifferentiated cells that generate organs at the tips of stems and branches throughout the life of the plant. As they define the number, type and position of lateral organs, meristems are the basis of plant architecture, allowing plants to adapt their development to their environment. In particular during the last decade, an impressive body of knowledge concerning shoot apical meristem function has been generated. This concerns information on the genes involved, their expression patterns, cell differentiation, cell division patterns, etc. The complexity of these data is such that an integrated view of meristem function is not yet possible. Therefore, adapted mathematical and informatics approaches are now required to integrate the knowledge in such a way that it can advance the level of understanding in the field. To formulate and test hypotheses on spatial and temporal aspects such as flows of signalling molecules between cells, strain within tissues, and the role of gene products in the spatial control of cell proliferation, we are creating a virtual meristem that will integrate as much spatial, dynamic and quantitative information as possible. Material and methods Meristems were observed using confocal microscopy. The protocol output is a series of 2D optical sections regularly spaced along the z axis. We typically obtain several stacks of 2D images taken during 2 to 4 days. The method hereafter describes how the 3D structure of these meristems can be reconstructed from these raw images. In this initial analysis, we focussed on the reconstruction of outer cell layer (L1) of the apical dome and disregarded the meristem internal cells. All computer tools needed at each stage of the meristem reconstruction and analysis were gathered into a software dedicated to virtual meristem analysis. Plant material and growth conditions Wild type plants of the WS ecotype were used in this study. As the meristem is usually hidden by the leaves and flowers, the plants were first grown on a medium with NPA (Naphtyl Phtalamic Acid) to inhibit organ production [14]. In the presence of the drug, these plants produce a naked stem, without flower primordia. When these treated plants are put on medium without NPA, they spontaneously reform new organ primordia. The regenerating meristems can then be easily viewed in the microscope. For this purpose, they are stained using a fluorescent membrane specific dye and subsequently observed in a confocal microscope for several days. The protocol and the plants are completely described in [11]. 3D meristem digitizing The experimental protocol produces a sequence of 3D meristem images of the same meristem at different dates, each 3D image consisting of a stack of 2D images. Each 2D image of a stack represents a picture of a slice of the meristem at some altitude z (the slices are 0.4µm thick and are taken every 2µm). On each image, the information coming exclusively from the meristem surface was extracted. To this end, we computed for each image a transparency mask (also called alpha mask), defining pixels that most probably belong to the surface. The intensity of pixels in the final image was computed as a combination of pixels from the different transparency masks. To determine the altitude of the pixels in the resulting image, we selected the altitude of the highest pixel in the stack with intensity close 4th International Workshop on Functional-Structural Plant Models, 7-11 june 2004 Montpellier, France Edited by C. Godin et al., pp

2 372 P. Barbier de Reuille et al. enough to the one in the final image. Figure 1 gives an example of a stack of images and the resulting reconstructed 2D image corresponding to a top view projection of the meristem. From this reconstructed image of the meristem surface, the cellular structure of layer L1 was extracted by vectorizing the cell walls. This digitizing operation was done by hand. The cell digitizing output is a graph of cells with the geometry and topology (adjacency relationships) of each cell in 3D. Reconstruction of dynamic sequences of meristems At each date, a graph representing the cells of layer L1 was obtained similarly. A sequence of meristem images typically contained 4 to 8 images, taken at 12 to 24 hours time intervals. The cell lineage between the successive graphs at each date was then reconstructed by defining manually the lineage relationships between cells of consecutive images. Since the analysis of a time sequence of meristem structures is particularly complex, we created movies of meristem growth by computing intermediate images based on a linear interpolation of each cell growth between consecutive dates. Data analysis and visualisation These virtual 4D reconstructions of observed meristems were analyzed using a software specially developed for this purpose. This software was intended to ease manual digitizing operations and to enable the analysis of meristem complex structures in space and time. Tools contained in the virtual meristem software (VIM) include: - Tools to analyse their geometry: surface (2D, 3D), curvature (3D), connectivity (2D, 3D) of the cells (example illustrated in Figure 3), etc. - Statistical analysis tools using the R module under the python language. - 2D visualisation tools: the Qt toolkit augmented by home-made functions and GnuPlot for the data analysis visualisation, depending on which is the easier to use. - 3D visualisation tools developed in our laboratory: PlantGL (formerly GEOM, see [2]). Currently, PlantGL is used as an external program: VIM generates PlantGL files which can be opened by the PlantGL viewer. Design of a virtual meristem Our general objective is to design a virtual meristem that would enable us to test in silico various hypotheses about the interaction between the physiological processes that drive the cell growth and division, and the emergence of a macroscopic form (apparition of primordia, phyllotaxy) were these physiological processes take place. In this paper, we describe two major problems that we had to address to achieve this goal : i) define a simplified representation of the meristem at cellular resolution, ii) find a formalism to express the physical and physiological interaction of the cells within the meristematic dome. Simplified representation of the virtual meristem The simulation of a meristem growth requires that simplifications are introduced in the meristem representation to express growth using simple rules. Our approach is based on a method initially tested by Hisao Honda on 2D representations of animal cells ([12],[13]). This approach uses Voronoi complexes (named Dirichlet domains in 2D). In this model, each cell is described by a single 3D point and the connectivity and dimensions of the cells are derived by the construction of the Voronoi complex (see [1] for the construction method and the properties of Voronoi complexes). This approach greatly simplifies the modelling problem, as the development of the meristematic tissue is reduced to the description of the development of a set of 3D points throughout time. Growth formalism Meristem structures represented as Voronoi complexes correspond to particular types of dynamical systems. In such systems, not only the value of the system state changes throughout time but also its structure. To emphasize this unusual property, Giavitto et al. [16] denoted this type of structure a dynamical system with dynamic structure (DS 2 ). The modelling of such systems requires special types of modelling paradigm. L-systems are an example of such a paradigm successfully applied to developing sequences or branching patterns [17]. To tackle the development of more complex SESSION 7 ORAL PRESENTATIONS

3 Towards a dynamic model of the Arabidopsis meristem 373 structures, Giavitto and Michel introduced a new type of rewriting system, that can be applied to the development of more general structures like arrays, Voronoi complexes or graphs, and designed a special language for it. This language is called MGS ([9], [10]). In this study, we considered the use of MGS to express the biological knowledge about the meristem growth. MGS allows us to express directly the time variation of a Delaunay graph 1 with simple expressions taking into account the topology of the meristem structure. The language for instance allows us to express cell growth, cell division and exchange of substances between cells using simple declarative rules. A first prototype of a simple virtual meristem (a moss-like meristem) using this approach was designed in order to assess the overall adequation of MGS to the modelling of developmental structures like meristems (see Figure 4). Based on these first encouraging results, we envisage to apply this modelling framework to Arabidopsis meristems, represented as Voronoi complexes. We thank Jean-Louis Giavitto and Olivier Michel for their help in the dynamical part of the study, especially with the modelling of the simple meristem shown in Figure 4. References [1] Jean-Daniel Boissonnat and Mariette Yvinec. Géométrie algorithmique. Ediscience international, [2] Frédéric Boudon, Christophe Nouguier, and Christophe Godin. GEOM Module manual: I User guide. GEOM Module manual: II Developper guide. CIRAD, March [3] Dorota Kwiatkowska and Jacques Dumais. Growth and morphogenesis at the vegetative shoot apex of Anagallis arvensis L.. Journal of Experimental Botany, Vol. 54, No. 387, pp. 1585±1595, June 2003 [4] Louis Bravais and Auguste Bravais. Essai sur la disposition des feuilles curvisériées. Annales des Sciences Naturelles, 7: , [5] Louis Bravais and Auguste Bravais. Essai sur la disposition symétrique des inflorescences. Annales des Sciences Naturelles, 8: 11 42, [6] Stéphane Douady and Yves Couder. Phyllotaxis as a dynamical self organizing process. part i: The spiral modes resulting from time-periodic iterations. Journal of theoretical Biology, (178): , [7] Stéphane Douady and Yves Couder. Phyllotaxis as a dynamical self organizing process. part ii: Tthe spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns. Journal of theoretical Biology, (178): , [8] Stéphane Douady and Yves Couder. Phyllotaxis as a dynamical self organizing process. part iii: The simulation of the transient regimes of ontogeny. Journal of theoretical Biology, (178): , [9] Jean-Louis Giavitto and Olivier Michel. Mgs: a programming language for the transformations of topological collections. Technical report, LaMI, May [10] Jean-Louis Giavitto and Olivier Michel. Mgs: a ruled-based language for complex objects and collections. Electronic Notes in Theoretical Computer Science, 59(4), [11] Oliver Grandjean, Teva Vernoux, Patrick Laufs, Katia Belcram, Yuki Mizukami, and Jan Traas. In vivo analysis of cell division, cell growth and differentiation at the shoot apical meristem in arabidopsis. The Plant Cell, in press. [12] Hisao Honda. Descrption of cellular patterns by dirichlet domains: The two-dimensional case. Journal of theoretical Biology, (72): , [13] Hisao Honda. Geometrical models for cells in tissues. International review of cytology, 81: , [14] K. Okada, J. Ueda, M.K. Komaki, C.J. Bell, and Y. Shimura. Requirement of the auxin polar transport system in early stages of arabidopsis floral bud formation. Plant Cell, 3: , A Delaunay graph express the connectivity relationship in a Voronoi complex

4 374 P. Barbier de Reuille et al. [15] Luc Vincent and Pierre Soille. Watersheds in digital spaces: An efficient algorithm based on immersion simulations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(6): , June [16] Giavitto, J.L., Godin, C., Michel, O. et Prusinkiewicz, P., Computational models for integrative and developmental biology. In: Actes du Colloque Modélisation et simulation de processus biologiques dans le contexte de la génomique, Autrans, France 17-21/03/2002; pp. 43 [17] Prusinkiewicz, P., Modeling of spatial structure and development of plants: a review. Scientia Horticulturae, 74: Figure 1: Construction of the projection. The left-hand side mosaic shows the different optical sections as given by the confocal microscope. The top left image is the most apical one the bottom right image is the most basal one. The right-hand side image shows the reconstructed projection of the meristem surface. Figure 2: 3D reconstruction of the meristems. The colour varies with the position of the cell on the z axis, blue for the most basal positions, red for highest, green for median. Figure 3: Connexity visualisation on the 3D meristems. Red cells have a high number of neighbours when blue ones have very few neighbours. We see, in that meristem, disconnected highly connected cells. SESSION 7 ORAL PRESENTATIONS

5 Towards a dynamic model of the Arabidopsis meristem 375 Figure 4: Modelling principle of the growth of a 3D virtual meristem using the MGS language. In this simple system, the blue cell represents the only meristematic cell. It divides on one side then on the other regularly. Its daughters divide two more times so that each division gives 3 non-meristematic cells. The daughter cells grow continuously afterward. Rules are of the following form: c / dividing( c) left( c) child_left( c), child_center( c) c / dividing( c) right( c) child_center( c),child_right( c) c / dividing( c) grow( c)