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Journal of Mathematical Biology

, Volume 78, Issue 3, pp 625–653 | Cite as

Regulation of plant cell wall stiffness by mechanical stress: a mesoscale physical model

  • Hadrien Oliveri
  • Jan Traas
  • Christophe GodinEmail author
  • Olivier AliEmail author
Article

Abstract

A crucial question in developmental biology is how cell growth is coordinated in living tissue to generate complex and reproducible shapes. We address this issue here in plants, where stiff extracellular walls prevent cell migration and morphogenesis mostly results from growth driven by turgor pressure. How cells grow in response to pressure partly depends on the mechanical properties of their walls, which are generally heterogeneous, anisotropic and dynamic. The active control of these properties is therefore a cornerstone of plant morphogenesis. Here, we focus on wall stiffness, which is under the control of both molecular and mechanical signaling. Indeed, in plant tissues, the balance between turgor and cell wall elasticity generates a tissue-wide stress field. Within cells, mechano-sensitive structures, such as cortical microtubules, adapt their behavior accordingly and locally influence cell wall remodeling dynamics. To fully apprehend the properties of this feedback loop, modeling approaches are indispensable. To that end, several modeling tools in the form of virtual tissues have been developed. However, these models often relate mechanical stress and cell wall stiffness in relatively abstract manners, where the molecular specificities of the various actors are not fully captured. In this paper, we propose to refine this approach by including parsimonious biochemical and biomechanical properties of the main molecular actors involved. Through a coarse-grained approach and through finite element simulations, we study the role of stress-sensing microtubules on organ-scale mechanics.

Keywords

Plant morphogenesis Biomechanics Mechanotransduction Cortical microtubules Cellulose microfibrils Numerical simulation 

List of symbols

\(\varvec{L}_{\text {g}}\)

Growth rate tensor

\(\varvec{E}\)

Elastic strain tensor

\(\varvec{S}\)

Stress tensor

\(\Phi \)

Cell wall extensibility

\(\tau \)

Cell wall yield strain

\(\mathbb {C}_{\text {w}}\)

Cell wall stiffness tensor

\(\mathbb {C}_{\text {g}}\)

Stiffness tensor associated with the wall’s isotropic matrix

\(\mathbb {C}_{\text {f}}\)

Stiffness tensor associated with microfibrils

\(Y, \nu \)

Wall matrix reduced Young’s modulus and Poisson’s ratio

\(Y_{\text {f}}\)

Microfibril reduced Young’s modulus

\(\theta \)

Angle parameter in the wall tangential plane

\(\varvec{e}_{\theta } \)

Unit vector oriented by \(\theta \)

\(\varvec{\Theta }\)

Projector on \({{\mathrm{span}}}\left( \varvec{e}_{\theta } \right) \)

\(\rho \left( \theta \right) \)

Angular density of microfibrils

\(\phi \left( \theta \right) \)

Angular density of microtubules

\(f\left( \theta \right) \)

Angular density of force (per unit surface)

\(\hat{\rho }_n, {\rho }_n, \tilde{\rho }_n\)

Complex, even and odd Fourier coefficients of \(\rho \)

\(\hat{\phi }_n\)

Complex Fourier coefficients of \(\phi \)

\(\hat{f}_n\)

Complex Fourier coefficients of \(f\)

\(\alpha _{\rho }\)

Anisotropy of microfibrils

\(\alpha _{\phi }\)

Anisotropy of microtubules

\(\alpha _{f}\)

Anisotropy of forces

\(k_{\rho },k'_{\rho }\)

Microfibril polymerization and depolymerization constants

\(k_{\phi }\)

Microtubule polymerization constant

\(k'_{\phi }{^0}\)

Inverse of stress-free microtubule half-life

\(\gamma \)

Coupling coefficient of the stress-induced microtubule stabilization

\(k'_{\phi }\left( \theta \right) = k'_{\phi }{^0}e^{-\gamma f\left( \theta \right) }\)

Angular microtubule depolymerization coefficient

\(c_0\)

Total concentration of tubulin

\(K_{\rho },K_{\phi }\)

Equilibrium constants of the microfibril/microtubule kinetics

\(\eta \)

Measure of the relative stiffness between the gel and the fiber

Mathematics Subject Classification

92B05 74F25 

Notes

Acknowledgements

The authors would like to thank Guillaume Cerutti for assistance with the visualization tool TissueLab (github.com/VirtualPlants/tissuelab). Funding was provided by Inria Project Lab Morphogenetics and European Research Council (Grant No. 294397).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Reproduction et Développement des PlantesUniv Lyon, ENS de Lyon, UCB Lyon 1, CNRS, INRA, InriaLyonFrance

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