Finite Element Modeling of Shape Changes in Plant Cells

Abstract

Mechanical modeling of plant cells using finite element methods serves to simulate the behavior of complex cell shapes with the aim to understand biological functioning

Figure 1.

A, A closed cylindrical shell with hemispherical caps generated by the rotation of a line (orange) around a symmetry axis (yellow). The thin-shelled closed vessel is constrained on its right half by two nondeforming rigid, flat plates. B, The cylinder is meshed using three-dimensional quadrilateral shell elements (curved shell). The image on the left shows the first-order elements defined by four nodes (purple) used to discretize the geometry. Additional nodes (blue) would formulate the second-order elements. The elements can be regularly shaped or skewed. Excessively skewed element shapes are to be avoided. C, Boundary conditions are applied to the model. The rigid plates are fixed to prevent their rotation or displacement. Displacement boundary conditions are applied to prevent the cylinder moving freely in the space. The turgor pressure is applied uniformly to the internal surfaces. A sliding frictionless contact property is defined between the rigid plates and the deformable cylinder to prevent the penetration of one body into the other, while allowing their relative displacement. D, The isotropic closed cylinder deforms toward a spherical shape where it is not constricted by the plates. The heat map represents the von Mises stress distribution. E, First-order (left) and second-order (right) elements used around a discontinuity. F, Graph depicting the results obtained in a mesh convergence study. A value such as stress in a critical region is plotted against the total number of elements representing the structure to verify the independence of results from the mesh quality and the number of elements.

Figure 2.

A, A pollen tube modeled as a hollow shell with uniform thickness. The apical dome is divided into subregions, allowing for the elastic properties to be adjusted in each region independently. B, Several key points are followed on the FE model upon each loading cycle and remeshing to mimic growth. C, The stiffness gradient predicted by the FE model to produce a self-similar tube closely matches the deesterification pattern of pectin. Images were adopted from Fayant et al. (2010).

Figure 3.

A, Epidermal cells on the adaxial surface of an Arabidopsis leaf feature three cell types: trichomes (brown), stomatal guard cells (red), and pavement cells (green). B, Development of the trichome branch embodies reduction of the tip radius of curvature, while the radius at the base of the branch remains constant. L, branch length; RT, branch radius at the tip; RB, branch radius at the base. C, The growth and thickness of the cell wall in a trichome branch are correlated and exhibit a gradient toward the tip of the branch. Microtubules and CESA trajectories are oriented transversely to the long axis of the branch, while the tip exhibits a microtubule-depleted zone. Image redrawn after Yanagisawa et al. (2015).

Figure 4.

A, FE model suggesting a stress pattern in pavement cells circumferential to the site of laceration. B, Fluorescence-tagged microtubules demonstrate hyperbundling and a seemingly circumferential pattern around the wound site. White arrowheads show examples of noncircumferential microtubules in cells away from the site of laceration. The white arrow shows that, even in cells adjacent to laceration, there seems to be a local competition between the cell shape-dictated microtubule organization and the putative circumferential reorientation of microtubules due to tissue-level stress. C, FE model suggesting a stress pattern in pavement cells circumferential to removed cells. D, Microtubules demonstrate a change in bundling and orientation upon the small-scale wound. However, their orientations seem longitudinal to cell axes rather than being circumferential to the site of the wound. White arrows indicate examples of cells with microtubules oriented parallel to the long cell axis, inconsistent with the hypothesized circumferential orientation. The star indicates ablated cells. Numbers indicated individual cells. All these images are reprinted from Sampathkumar et al. (2014) with permission from the authors. Bars = 25 µm (B) and 50 µm (D).

Figure 5.

A, Cross-sectional view of guard cells composed of ventral wall (VW), dorsal wall (DW), inner wall (IW), and outer wall (OW). Ra and Rb refer to the horizontal and vertical radii of the elliptical cross-section, respectively. Only the outer ledge (OL) is shown. Inflation of the guard cells causes a change of the elliptical cross-section to circular and then to an ellipse with the major axis perpendicular to the plane of the leaf. B, Confocal micrograph of guard cells in an Arabidopsis cotyledon, stained with Calcofluor White to reveal cellulose. Bar = 15 µm.