A Mechanical Model to Interpret Cell-Scale Indentation Experiments on Plant Tissues in Terms of Cell Wall Elasticity and Turgor Pressure

International audienceMorphogenesis in plants is directly linked to the mechanical elements of growing tissues, namely cell wall and inner cell pressure. Studies of these structural elements are now often performed using indentation methods such as atomic force microscopy. In these methods, a probe applies a force to the tissue surface at a subcellular scale and its displacement is monitored, yielding force-displacement curves that reflect tissue mechanics. However, the interpretation of these curves is challenging as they may depend not only on the cell probed, but also on neighboring cells, or even on the whole tissue. Here, we build a realistic three-dimensional model of the indentation of a flower bud using SOFA (Simulation Open Framework Architecture), in order to provide a framework for the analysis of force-displacement curves obtained experimentally. We find that the shape of indentation curves mostly depends on the ratio between cell pressure and wall modulus. Hysteresis in force-displacement curves can be accounted for by a viscoelastic behavior of the cell wall. We consider differences in elastic modulus between cell layers and we show that, according to the location of indentation and to the size of the probe, force-displacement curves are sensitive with different weights to the mechanical components of the two most external cell layers. Our results confirm most of the interpretations of previous experiments and provide a guide to future experimental work

Multifarious Hierarchies of Mechanical Models for Artist Assigned Levels-of-Detail

International audienceWe present a new framework for artist driven level of detail in solid simulations. Simulated objects are simultaneously embedded in several, separately designed deformation models with their own independent degrees of freedom. The models are ordered to apply their deformations hierarchically, and we enforce the uniqueness of the dynamics solutions using a novel kinetic filtering operator designed to ensure that each child only adds detail motion to its parent without introducing redundancies. This new approach allows artists to easily add fine-scale details without introducing unnecessary degrees-of-freedom to the simulation or resorting to complex geometric operations like anisotropic volume meshing. We illustrate the utility of our approach with several detail enriched simulation examples

A mechanical model to interpret cell-scale indentation experiments on plant tissues in terms of cell wall elasticity and turgor pressure

Morphogenesis in plants is directly linked to the mechanical elements of growing tissues, namely cell wall and inner cell pressure. Studies of these structural elements are now often performed using indentation methods such as atomic force microscopy. In these methods, a probe applies a force to the tissue surface at a subcellular scale and its displacement is monitored, yielding force-displacement curves that reflect tissue mechanics. However, the interpretation of these curves is challenging as they may depend on not only the cell probed, but also on neighboring cells, or even on the whole tissue. Here, we build a realistic three-dimensional model of the indentation of a flower bud using SOFA (Simulation Open Framework Architecture), in order to provide a framework for the analysis of force-displacement curves obtained experimentally. We find that the shape of indentation curves mostly depend on the ratio between cell pressure and wall modulus. Hysteresis in force-displacement curves can be accounted for by a viscoelastic behavior of the cell wall. We consider differences in elastic modulus between cell layers and we show that, according to the location of indentation and to the size of the probe, force-displacement curves are sensitive with different weights to the mechanical components of the two most external cell layers. Our results confirm most of the interpretations of previous experiments and provide a guide to future experimental work

Simulation bio-informatique de la structure des plantes pour la caractérisation de leurs propriétés mécaniques au niveau cellulaire

Morphogenesis in plants is an active field of research. While genetics influence on plant shape has been extensively studied, rheology through scales: from cells to the whole organism through various tissues or organs, is still poorly analysed. Nevertheless, it should lead to the understanding of plants evolving, since it directly drives the organism growth.This work aims at understanding the mechanical caracteristics of plant cells, through several realistic and physically based models of various sub-domains of our plant archetype: Arabidopsis thaliana. In this project, a systematic approach has been developped, where the structures underlying such models rely on experimental data (images of different plant organs), describing precisely the inner structure of A. thaliana, which allows the use of realistic meshesfor our simulations.Then physical modelling allows us to retrieve, through the deformation field of different plant sub-domains, the mechanical properties underlying each type of structure described, which is a typical inverse problem form of a complex biological system. We apply this optimization methodology to several plant organs, beginning with embryonic stem, then with roots and finally with meristems, which constitute the zone where cells can divide and growthtypically takes place.Then, we develop a theoretical framework, on which biologists may rely, describing a realistic model of plant sub-system, typically a meristem. We hope that this conceptual framework will help experimentalists to validate some hypothesis regarding plant manipulations that are still subject to debate, as the use of Atomic Force Microscopy to experimentaly extract mechanical parameters from various plant tissues.Finally, we present a new approach coupling a coarse physically based simulation to a more detailed one : the Multifarious Hierarchiy of Mechanical Models. MHMMs are eclectics as they combine arbitrarily any type of physically based simulation (meshless, modal physics, Finite Elements, ...). Moreover, they are flexible as they allow the modularity of the various domainscontaining the underlying models. Finally MHMMs are much faster than full Finite Element simulation, at the same level of detail. This should allow the development of fast algorithm for local detailed simulation, as was the case for the numerical Atomic Force Microscope in previous part.Dans le cadre de l'étude de la morphogénèse des plantes, bien que la génétique soit intensivement étudiée, le concept clé de rhéologie à différentes échelles : cellulaire, tissulaire ou de l'organisme entier, qui gouverne directement la croissance des organismes et devrait permettre la connaissance de l'évolution de la plante reste encore très pauvrement analysé.Ce travail porte ainsi sur la description des paramètres mécaniques des cellules de plantes, à travers la réalisation de modèles numériques physiques et réalistes de différents sous-domaines de l'organisme Arabidopsis thaliana. Dans ce projet, une approche systématique sera développée pour la réalisation de ces modèles où nos structures s'appuieront sur des données expérimentales (images des différents organes de la plante) décrivant précisément la structure interne d'A. thaliana, ce qui permettra l'obtention de structures et maillages réalistes.Ensuite l'étape de modélisation permettra d'une part de retrouver, à travers le champs de déformation de l'organe décrit, les caractéristiques mécaniques sous-jacentes au modèle, ce qui constitue une forme de problème inverse biologique complexe. Nous appliquerons cette méthodologie à différents sous-domaine de notre plante archétype, commençant par les jeunes tiges, puis les racines et enfin le meristème qui constitue la zone première de croissance de la plante.D'autre part, nous développerons un cadre théorique, sur lequel pourront s'appuyer les biologistes, décrivant un modèle d'organe d'A. thaliana réaliste, typiquement le meristème, et qui permettra de valider ou d'infirmer certaines hypothèses de manipulations encore sujettes à débat, notamment dans le cadre d'utilisation de microscope à force atomique afin d'extraire expérimentalement les propriétés mécaniques des tissus.Enfin, nous présenterons une nouvelle approche mécanique hiérarchique couplant des modèles de simulation à différentes échelles : Multifarious Hierarchy of Mechanical Models. Cette nouvelle approche est à la fois éclectique, car elle permet de coupler des modèles physiques variés (sans maillages, physique modale, éléments finis...), mais aussi flexible, car elle autorise la modularité des différents domaines des modèles sous-jacents, et enfin efficace par l'obtention de gains de performance et de temps face à des simulations éléments finis s'appuyant sur le même niveau de détail. Cette approche devrait permettre le développement d'algorithmes rapides pour les simulations physiques où un niveau de détail local est requis. Par exemple, elle pourrait s'appliquer aux simulations de microscope à force atomique décrites dans la partie précédente

Computational modeling of internal mechanical structure of plant cells

Dans le cadre de l'étude de la morphogénèse des plantes, bien que la génétique soit intensivement étudiée, le concept clé de rhéologie à différentes échelles : cellulaire, tissulaire ou de l'organisme entier, qui gouverne directement la croissance des organismes et devrait permettre la connaissance de l'évolution de la plante reste encore très pauvrement analysé.Ce travail porte ainsi sur la description des paramètres mécaniques des cellules de plantes, à travers la réalisation de modèles numériques physiques et réalistes de différents sous-domaines de l'organisme Arabidopsis thaliana. Dans ce projet, une approche systématique sera développée pour la réalisation de ces modèles où nos structures s'appuieront sur des données expérimentales (images des différents organes de la plante) décrivant précisément la structure interne d'A. thaliana, ce qui permettra l'obtention de structures et maillages réalistes.Ensuite l'étape de modélisation permettra d'une part de retrouver, à travers le champs de déformation de l'organe décrit, les caractéristiques mécaniques sous-jacentes au modèle, ce qui constitue une forme de problème inverse biologique complexe. Nous appliquerons cette méthodologie à différents sous-domaine de notre plante archétype, commençant par les jeunes tiges, puis les racines et enfin le meristème qui constitue la zone première de croissance de la plante.D'autre part, nous développerons un cadre théorique, sur lequel pourront s'appuyer les biologistes, décrivant un modèle d'organe d'A. thaliana réaliste, typiquement le meristème, et qui permettra de valider ou d'infirmer certaines hypothèses de manipulations encore sujettes à débat, notamment dans le cadre d'utilisation de microscope à force atomique afin d'extraire expérimentalement les propriétés mécaniques des tissus.Enfin, nous présenterons une nouvelle approche mécanique hiérarchique couplant des modèles de simulation à différentes échelles : Multifarious Hierarchy of Mechanical Models. Cette nouvelle approche est à la fois éclectique, car elle permet de coupler des modèles physiques variés (sans maillages, physique modale, éléments finis...), mais aussi flexible, car elle autorise la modularité des différents domaines des modèles sous-jacents, et enfin efficace par l'obtention de gains de performance et de temps face à des simulations éléments finis s'appuyant sur le même niveau de détail. Cette approche devrait permettre le développement d'algorithmes rapides pour les simulations physiques où un niveau de détail local est requis. Par exemple, elle pourrait s'appliquer aux simulations de microscope à force atomique décrites dans la partie précédente.Morphogenesis in plants is an active field of research. While genetics influence on plant shape has been extensively studied, rheology through scales: from cells to the whole organism through various tissues or organs, is still poorly analysed. Nevertheless, it should lead to the understanding of plants evolving, since it directly drives the organism growth.This work aims at understanding the mechanical caracteristics of plant cells, through several realistic and physically based models of various sub-domains of our plant archetype: Arabidopsis thaliana. In this project, a systematic approach has been developped, where the structures underlying such models rely on experimental data (images of different plant organs), describing precisely the inner structure of A. thaliana, which allows the use of realistic meshesfor our simulations.Then physical modelling allows us to retrieve, through the deformation field of different plant sub-domains, the mechanical properties underlying each type of structure described, which is a typical inverse problem form of a complex biological system. We apply this optimization methodology to several plant organs, beginning with embryonic stem, then with roots and finally with meristems, which constitute the zone where cells can divide and growthtypically takes place.Then, we develop a theoretical framework, on which biologists may rely, describing a realistic model of plant sub-system, typically a meristem. We hope that this conceptual framework will help experimentalists to validate some hypothesis regarding plant manipulations that are still subject to debate, as the use of Atomic Force Microscopy to experimentaly extract mechanical parameters from various plant tissues.Finally, we present a new approach coupling a coarse physically based simulation to a more detailed one : the Multifarious Hierarchiy of Mechanical Models. MHMMs are eclectics as they combine arbitrarily any type of physically based simulation (meshless, modal physics, Finite Elements, ...). Moreover, they are flexible as they allow the modularity of the various domainscontaining the underlying models. Finally MHMMs are much faster than full Finite Element simulation, at the same level of detail. This should allow the development of fast algorithm for local detailed simulation, as was the case for the numerical Atomic Force Microscope in previous part

A Mechanical Model to Interpret Cell-Scale Indentation Experiments on Plant Tissues in Terms of Cell Wall Elasticity and Turgor Pressure

Morphogenesis in plants is directly linked to the mechanical elements of growing tissues, namely cell wall and inner cell pressure. Studies of these structural elements are now often performed using indentation methods such as atomic force microscopy. In these methods, a probe applies a force to the tissue surface at a subcellular scale and its displacement is monitored, yielding force-displacement curves that reflect tissue mechanics. However, the interpretation of these curves is challenging as they may depend not only on the cell probed, but also on neighboring cells, or even on the whole tissue. Here, we build a realistic three-dimensional model of the indentation of a flower bud using SOFA (Simulation Open Framework Architecture), in order to provide a framework for the analysis of force-displacement curves obtained experimentally. We find that the shape of indentation curves mostly depends on the ratio between cell pressure and wall modulus. Hysteresis in force-displacement curves can be accounted for by a viscoelastic behavior of the cell wall. We consider differences in elastic modulus between cell layers and we show that, according to the location of indentation and to the size of the probe, force-displacement curves are sensitive with different weights to the mechanical components of the two most external cell layers. Our results confirm most of the interpretations of previous experiments and provide a guide to future experimental work