Microfibril organization modes in plant cell walls of variable curvature: a model system for two dimensional anisotropic soft matter (original) (raw)
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Mechanical model for fiber-laden membranes
Continuum Mechanics and Thermodynamics, 2011
An integrated mechanical model for fiber-laden membranes is presented and representative predictions of relevance to cellulose ordering and orientation in the plant cell wall are presented. The model describes nematic liquid crystalline self-assembly of rigid fibers on an arbitrarily curved fluid membrane. The mechanics of the fluid membrane is described by the Helfrich bending-torsion model, the fiber self-assembly is described by the 2D Landau-de Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane interactions (inspired by an extension of the 2D Maier-Saupe model to curved surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic (curvature-avoiding) effects. Analysis of the free energy reveals three fiber orientation regimes: (a) along the major curvature, (b) along the minor curvature, (c) away from the principal curvatures, according to the competing curvo-philic and curvo-phobic interactions. The derived shape equation (normal stress balance) now includes curvaturenematic ordering contributions, with both bending and torsion renormalizations. Integration of the shape and nematic order equations gives a complete model whose solution describes the coupled membrane shape/fiber order state. Applications to cylindrical membranes, relevant to the plant cell wall, shows how growth decreases the fiber order parameter and moves the fibers' director from the axial direction towards the azimuthal orientation, eventually leading to a state of stress predicted by pure membranes. The ubiquitous 54.7 o cellulose fibril orientation in a cylindrical plant cell wall is shown to be predicted by the preset model when the ratio of curvo-phobic and curvo-philic interactions is in the range of the cylinder radius.
Journal of Non-Newtonian Fluid Mechanics, 2010
This paper presents a study of the structure and dynamics of rigid fiber-laden deformable curved fluid membranes based on an viscoelastic model that integrates the statics of anisotropic membranes, the planar nematodynamics of fibers and the dynamics of isotropic membranes. Fiber-laden membranes arise frequently in biological systems, such as the plant cell wall and in protein-lipid bilayers. Based on the membrane's force and torque balance equations and the fiber's balance of molecular fields, a viscoelastic anisotropic model that provides the governing equations for the membrane's velocity and curvature and the fiber structure (fiber orientation and order) is found. A Helmholtz free energy that incorporates the tension/bending/and torsion membrane elasticity, the Landau-de Gennes fiber ordering, and fiber order-membrane curvature interactions is used to derive elastic moments, torques, and stresses. The corresponding viscous stresses and moments include the Boussinesq-Scriven contributions as well as bending, torsion, and rotational dissipation. A spectral decomposition leads to the main viscoelastic material functions for anisotropic fluid membranes. Applications of the rheological model to cylindrical growth and cylindrical axial stretching show that competing curvo-phobic, curvo-philic interactions under extensional flow predict transitions between axial and azimuthal fiber arrangements, of interest to cellulose fiber orientation in plant morphogenesis.
Journal of Theoretical Biology, 2021
Plant morphology emerges from cellular growth and structure. The turgor-driven diffuse growth of a cell can be highly anisotropic: significant longitudinally and negligible radially. Such anisotropy is ensured by cellulose microfibrils (CMF) reinforcing the cell wall in the hoop direction. To maintain the cell's integrity during growth, new wall material including CMF must be continually deposited. We develop a mathematical model representing the cell as a cylindrical pressure vessel and the cell wall as a fibre-reinforced viscous sheet, explicitly including the mechano-sensitive angle of CMF deposition. The model incorporates interactions between turgor, external forces, CMF reorientation during wall extension, and matrix stiffening. Using the model, we reinterpret some recent experimental findings, and reexamine the popular hypothesis of CMF/microtubule alignment. We explore how the handedness of twisting cell growth depends on external torque and intrinsic wall properties, and find that cells twist lefthandedly 'by default' in some suitable sense. Overall, this study provides a unified mechanical framework for understanding left-and right-handed twist-growth as seen in many plants.
A fibre-reinforced fluid model of anisotropic plant cell growth
Journal of Fluid Mechanics, 2010
Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation.
Micromechanical understanding of the cell-wall structure
Comptes Rendus Biologies, 2004
For improving properties of pulp fibres, a better understanding of the relationships between its macroscopic mechanical properties, fibre ultrastructure, and properties of the wood polymers is important. This paper discusses such relations between elastic properties of fibres, their matrix structure and the wood polymer elastic constants. It is argued that an orientation of all of the wood polymers in the direction of the cellulose microfibrils is most likely. The elastic longitudinal modulus of cellulose is well described by the value of 134 GPa dominating the longitudinal fibre properties. In the transverse direction the amorphous polymers play a more important role. To cite this article: L.
On the effect of substrate curvature on cell mechanics
Biomaterials, 2009
Cell movement on a substrate or within the extracellular matrix is the phenomenological response to a biochemical signals' cascade transcripted into biophysical processes and viceversa. The process is complex in nature, including different length scales from the whole cell to organelle and protein levels, where substrate/ECM curvature has been shown to play an important role on cell's behavior. From a macroscopic perspective, the cytoskeleton may be modeled as a continuum body unbalanced by internal protein motors. In this work, we propose a cell constitutive model to simulate cell attachment on curved substrates, activated by contractile forces. We first analyze a single fiber bundle composed by microtubules, actin filaments and myosin machinery. Then, the model is macroscopically extended to the cytoskeletal level using homogenization. Substrate curvature has two implications in our model: (i) it forces fibers to work in a curved (bent) position and (ii) it eventually creates a pre-deformation state in the cytoskeleton. Interestingly, the model shows higher contractile force inhibition as curvature increases when implemented over different substrate morphologies, being this consistent with experimental results. The presented model may result useful in many new regenerative medicine techniques, miniaturized experimental tests, or to analyze cell behavior on manufactured nanoscaffolds for tissue engineering.
Modelling Cell Orientation Under Stretch: The Effect of Substrate Elasticity
Bulletin of Mathematical Biology
When cells are seeded on a cyclically deformed substrate like silicon, they tend to reorient their major axis in two ways: either perpendicular to the main stretching direction, or forming an oblique angle with it. However, when the substrate is very soft such as a collagen gel, the oblique orientation is no longer observed, and the cells align either along the stretching direction, or perpendicularly to it. To explain this switch, we propose a simplified model of the cell, consisting of two elastic elements representing the stress fiber/focal adhesion complexes in the main and transverse directions. These elements are connected by a torsional spring that mimics the effect of crosslinking molecules among the stress fibers, which resist shear forces. Our model, consistent with experimental observations, predicts that there is a switch in the asymptotic behaviour of the orientation of the cell determined by the stiffness of the substratum, related to a change from a supercritical bifu...
A conserved cellular mechanism for cotton fibre diameter and length control
in silico plants, 2022
Highly polarized cotton fibre cells that develop from the seed coat surface are the foundation of a multi-billiondollar international textile industry. The unicellular trichoblast emerges as a hemispherical bulge that is efficiently converted to a narrower and elongated shape that extends for about 2 weeks before transitioning into a cellulose-generating machine. The polarized elongation phase employs an evolutionarily conserved microtubule-cellulose synthase control module that patterns the cell wall and enables highly anisotropic diffuse growth. As the multi-scale interactions and feedback controls among cytoskeletal systems, morphologically potent cell wall properties, and a changing cell geometry are uncovered, opportunities emerge to engineer architectural traits. However, in cotton, such efforts are hampered by insufficient knowledge about the underlying control mechanisms. For example, fibre diameter is an important trait that is determined during the earliest stages of development, but the basic growth mode and the mechanisms by which cytoskeletal and cell wall systems mediate fibre tapering are not known. This paper combines multiparametric and multiscale fibre phenotyping and finite element computational modelling of a growing cell to discover an evolutionarily conserved tapering mechanism. The actin network interconverts between two distinct longitudinal organizations that broadly distributes organelles and likely enables matrix secretion patterns that maintain cell wall thickness during growth. Based on plausible finite element models and quantitative analyses of the microtubule cytoskeleton, tapering and anisotropic growth is programmed by a constricting apical microtubule depletion zone and highly aligned microtubules along the fibre shaft. The finite element model points to a central role for tensile forces in the cell wall to dictate the densities and orientations of morphologically potent microtubules that pattern the cell wall.
Elongation and shape changes in organisms with cell walls: A dialogue between experiments and models
The Cell Surface, 2018
The generation of anisotropic shapes occurs during morphogenesis of almost all organisms. With the recent renewal of the interest in mechanical aspects of morphogenesis, it has become clear that mechanics contributes to anisotropic forms in a subtle interaction with various molecular actors. Here, we consider plants, fungi, oomycetes, and bacteria, and we review the mechanisms by which elongated shapes are generated and maintained. We focus on theoretical models of the interplay between growth and mechanics, in relation with experimental data, and discuss how models may help us improve our understanding of the underlying biological mechanisms.