Modélisation par éléments finis de la croissance du tube pollinique.

RESUME Dans le domaine de la biologie vegetale, la reproduction sexuelle des plantes par pollinisation est un sujet d’interet presentant plusieurs enjeux humains et economiques. Le tube pollinique est une cellule vegetale indispensable a la pollinisation des fleurs. La croissance de la cellule vegetale se fait par une deformation de la paroi l’entourant sous l’effet de la pression hydrostatique interne, la turgescence. Simultanement, de nouveaux materiaux doivent etre livres vers la paroi pour empecher qu’elle n’eclate a cause de l'amincissement cause par l'etirement. Chez le tube pollinique, l'expansion de surface est localisee a l’apex de la cellule et le resultat du processus est une protuberance de forme cylindrique. Ce mode de croissance combine au systeme biologique particulier du tube pollinique en fait un modele privilegie pour les etudes mecaniques des cellules vegetales en croissance. De nombreux modeles portant sur la croissance apicale ont ete construits afin d’etudier les parametres et processus impliques. Ces modeles presentent differentes manieres d’aborder le probleme que ce soit des approches geometriques, mathematiques ou mecaniques mais dans chacun des constantes demeurent comme l’axisymetrie du tube. La methode des elements finis permet de representer des formes complexes ou de changer facilement les modeles mathematiques utilises pour les calculs. Ce potentiel est particulierement adapte pour l’etude de structures biologiques amenees a changer de forme ou de proprietes mecaniques. Cette approche de modelisation par elements finis a ete privilegiee dans ce projet. Le premier objectif de ce projet de maitrise visait a developper un modele par elements finis representant la croissance du tube pollinique permettant de changer facilement les parametres mecaniques, geometriques et de chargement du tube pollinique. Le modele a ete construit en se basant sur l’observation experimentale, soit un long cylindre avec un rayon de 6 μm, termine par un apex de la forme d’un spheroide prolate. La paroi, d’une epaisseur moyenne fixee a 50 nm, a ete modelisee par des elements de type SHELL 181 via un maillage structure. Ces elements possedent un comportement de type «coque» adequat pour representer la paroi et pourraient etre modifies dans le futur pour integrer la viscoplasticite. Le tube a cependant ete soumis a un comportement lineaire elastique dans un premier temps. Le coefficient de Poisson utilise etait alors fixe a 0.3. Les principaux parametres du modele incluaient entre autres les modules d’Young de la paroi cellulaire ainsi que leur gradient le long du tube, dans le but de tenir compte----------ABSTRACT In the field of plant biology, sexual reproduction through pollination is a topic of interest since it affects human life and economic issues. The pollen tube is a plant cell that is indispensable for the pollinisation of flowers. Plant cell growth occurs by a deformation of the surrounding wall under the effect of an internal hydrostatic pressure, the turgor pressure. At the same time, new material has to be delivered to the wall in order to prevent bursting caused by the thinning of the wall that in turn is a result of the turgor induced stretching. In the pollen tube, surface growth is located at the apex of the cell and it leads to the formation of a cylindrical protuberance. This type of growth, combined to the particular biological system of the pollen tube, provides an excellent model for mechanical studies of growing plant cells. Numerous models for apical growth have been used to study the parameters and processes involved. These models present different methods to approach the problem whether it is geometrical, mathematical or mechanical. However, most of these models depend on the axisymmetry of the tubular structure. The finite element method allows one to represent complex shapes or to easily change the mathematical models used for calculation. This potential is particularly suited for the study of biological structures that are able to change shape or to manipulate their mechanical properties. The finite element modeling approach has been adopted in this project. The first objective of this project was to develop a finite element model representing the growth of the pollen tube that would allow to easily change mechanical parameters, geometry and loading. The model was constructed based on experimental observation. It consisted in a long cylinder with a radius of 6 μm, terminated by an apex shaped as a prolate spheroid. The wall, with an average thickness fixed at 50 nm was modeled by elements of type SHELL 181 via a structured mesh. These elements behave as a thin "shell" and are thus an appropriate choice to represent the relatively thin wall of the pollen tube. Although they offer the possibility to incorporate viscoplastic material properties, we have chosen a linear elastic behavior for this first version of the model. The value for Poisson’s coefficient was fixed at 0.3. The main parameters of the model included the Young’s modulus of the cell wall and its gradient along the tube, in order to take into account the changing biochemical composition of the cell wall between the apex and the cylindrical region. To cover a large number of possible cases, different values for