| 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 mum, 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. To model growth, an internal turgor pressure, varying between 0.1 and 0.4 MPa, was simulated inside the tube. In order to reduce computation time, only one quarter of the tube was represented and the boundary conditions corresponded to the fixation of radial and longitudinal edges. The application of this internal pressure coupled with boundary conditions resulted in a deformation of the wall and represented one growth cycle. After achieving equilibrium, the geometry of the model was updated for this state of large deformation and internal pressure was applied again. This process was repeated for the required number of cycles to produce a significant amount of growth.;To obtain a unidirectional growth for which the shape of the apex remains constant, three sets of simulations have been performed in order to cover a large number of combinations. The first set of 64 simulations was based on variations of the two coefficients mL and mT and of the angular distribution of the seven parameterized domains along the tube. Visual assessment of the results suggested that a cell wall consisting of an isotropic material represents pollen tube growth better than a wall made from orthotropic material. Subsequently, a second set of 32 combinations was tested, including only simulations with an isotropic wall (mT=1, variation of mL and angle distribution). Visual assessment of these two set of simulations showed that an increase in mL or m T caused a gradual decrease of the tube diameter at each step and vice versa.;This work has allowed us to make conclusions on the distribution of mechanical properties in the wall of the pollen tube. The combination of parameters that produced a perfectly cylindrical tube and that was self-similar in time, was characterized by a gradient of isotropic Young's modulus whose shape corresponded to the distribution of material in the wall. The pollen tube is surrounded by a polysaccharide layer rich in pectin. While at the apex the pectin has a high degree of methyl-esterification, these polymers are de-methyl-esterified moving towards the cylindrical part. The de-esterification of pectin causes a change in the rigidity of this material and the spatial location of the change in chemical configuration was found exactly where the most pronounced change in the rigidity of the wall was predicted to be.;Furthermore, the study of the influence of different parameters determined, that the pressure, the radius of the tube or the thickness of the wall (as long as it is modified uniformly) have minimal impact on the qualitative change in the shape of tube growth. Among the mechanical and geometrical parameters that were studied, only the gradient of Young's modulus resulted in shape changes. The result of such manipulation of mechanical properties has already been demonstrated in biological experiments. (Abstract shortened by UMI.);The second objective of this project was to geometrically validate the finite element model developed for the case of unidirectional growth. This validation was based on a comparison with experimental results. Two methods were developed for this purpose. The first validation method determined the degree of temporal self-similarity of the growth process, i.e. the ability of the simulation to produce a tube with unchanged shape at the apex and increasing length of the cylindrical region. The second validation method used experimental results showing the displacement of tracers placed on the surface of the tube. These trajectories were marked by fluorescent beads deposited on the wall and monitored during growth. These trajectories were reproduced in the simulations and compared with experimental data. Together these methods enabled us to identify a parameter combination that generates a growth pattern similar to that produced by a real pollen tube. |
