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


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.


Plant morphogenesis Biomechanics Mechanotransduction Cortical microtubules Cellulose microfibrils Numerical simulation 

List of symbols

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

Growth rate tensor


Elastic strain tensor


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 \(\phi \)


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


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 



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


  1. Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover, New YorkzbMATHGoogle Scholar
  2. Allard JF, Wasteneys GO, Cytrynbaum EN (2010) Mechanisms of self-organization of cortical microtubules in plants revealed by computational simulations. Mol Biol Cell 21(2):278–286. CrossRefGoogle Scholar
  3. Ali O, Mirabet V, Godin C, Traas J (2014) Physical models of plant development. Ann Rev Cell Dev Bio 30:59–78. CrossRefGoogle Scholar
  4. Baskin TI (2005) Anisotropic expansion of the plant cell wall. Annu Rev Cell Dev Biol 21(1):203–222. CrossRefGoogle Scholar
  5. Beauzamy L, Louveaux M, Hamant O, Boudaoud A (2015) Mechanically, the shoot apical meristem of Arabidopsis behaves like a shell inflated by a pressure of about 1 MPa. Front Plant Sci 6(6):1038–1038. Google Scholar
  6. Bell GI (1978) Models for the specific adhesion of cells to cells. Science 200(4342):618–627CrossRefGoogle Scholar
  7. Boudaoud A, Burian A, Borowska-Wykret D, Uyttewaal M, Wrzalik R, Kwiatkowska D, Hamant O (2014) FibrilTool, an ImageJ plug-in to quantify fibrillar structures in raw microscopy images. Nat Protoc 9(2):457–463. CrossRefGoogle Scholar
  8. Boudon F, Chopard J, Ali O, Gilles B, Hamant O, Boudaoud A, Traas J, Godin C (2015) A computational framework for 3d mechanical modeling of plant morphogenesis with cellular resolution. PLOS Comput Biol 11(1):e1003,950. CrossRefGoogle Scholar
  9. Bozorg B, Krupinski P, Jönsson H (2014) Stress and strain provide positional and directional cues in development. PLOS Comput Biol 10(1):e1003,410. CrossRefGoogle Scholar
  10. Bozorg B, Krupinski P, Jönsson H (2016) A continuous growth model for plant tissue. Phys Biol 13(6):065,002. CrossRefGoogle Scholar
  11. Braybrook SA, Peaucelle A (2013) Mechano-chemical aspects of organ formation in Arabidopsis thaliana: the relationship between auxin and pectin. PLoS ONE 8(3):57813. CrossRefGoogle Scholar
  12. Burian A, Ludynia M, Uyttewaal M, Traas J, Boudaoud A, Hamant O, Kwiatkowska D (2013) A correlative microscopy approach relates microtubule behaviour, local organ geometry, and cell growth at the Arabidopsis shoot apical meristem. J Exp Bot 64:5753–5767. CrossRefGoogle Scholar
  13. Burk DH, Ye ZH (2002) Alteration of oriented deposition of cellulose microfibrils by mutation of a katanin-like microtubule-severing protein. Plant Cell 14(9):2145–2160. CrossRefGoogle Scholar
  14. Cerutti G, Ali O, Godin C (2017) DRACO-STEM: an automatic tool to generate high-quality 3d meshes of shoot apical meristem tissue at cell resolution. Front Plant Sci 8:353. CrossRefGoogle Scholar
  15. Changeux JP (2012) Allostery and the Monod–Wyman–Changeux model after 50 years. Annu Rev Biophys 41(1):103–133CrossRefGoogle Scholar
  16. Coen E, Rolland-Lagan AG, Matthews M, Bangham JA, Prusinkiewicz P (2004) The genetics of geometry. Proc Natl Acad Sci USA 101(14):4728–4735. CrossRefGoogle Scholar
  17. Corson F, Hamant O, Bohn S, Traas J, Boudaoud A, Couder Y (2009) Turning a plant tissue into a living cell froth through isotropic growth. Proc Natl Acad Sci 106(21):8453–8458. CrossRefGoogle Scholar
  18. Cosgrove DJ (2001) Wall structure and wall loosening. A look backwards and forwards. Plant Physiol 125(1):131–134. CrossRefGoogle Scholar
  19. Cosgrove DJ (2005) Growth of the plant cell wall. Nat Rev Mol Cell Biol 6(11):850–861. CrossRefGoogle Scholar
  20. Cox HL (1952) The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 3(3):72. CrossRefGoogle Scholar
  21. De Gennes PG, Prost J (1995) The physics of liquid crystals. Oxford University Press, OxfordCrossRefGoogle Scholar
  22. Delingette H (2008) Triangular springs for modeling nonlinear membranes. IEEE Trans Visual Comput Graph 14(2):329–341. CrossRefGoogle Scholar
  23. Dixit R, Cyr R (2004) Encounters between dynamic cortical microtubules promote ordering of the cortical array through angle-dependent modifications of microtubule behavior. Plant Cell 16(12):3274–3284. CrossRefGoogle Scholar
  24. Dumais J, Shaw SL, Steele CR, Long SR, Ray PM (2006) An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth. Int J Dev Biol 50(2–3):209–222CrossRefGoogle Scholar
  25. Dupuy L, Mackenzie JP, Haseloff JP (2006) A biomechanical model for the study of plant morphogenesis: coleocheate orbicularis, a 2d study species. In: Proceedings of the 5th plant biomechanics conference, Stockholm, SwedenGoogle Scholar
  26. Dyson RJ, Jensen OE (2010) A fibre-reinforced fluid model of anisotropic plant cell growth. J Fluid Mech 655:472–503. MathSciNetCrossRefzbMATHGoogle Scholar
  27. Dyson RJ, Band LR, Jensen OE (2012) A model of crosslink kinetics in the expanding plant cell wall: yield stress and enzyme action. J Theor Biol 307:125–136. MathSciNetCrossRefzbMATHGoogle Scholar
  28. Emons AMC, Höfte H, Mulder BM (2007) Microtubules and cellulose microfibrils: how intimate is their relationship? Trends Plant Sci 12(7):279–281. CrossRefGoogle Scholar
  29. Erickson RO (1976) Modeling of plant growth. Annu Rev Plant Physiol 27(1):407–434CrossRefGoogle Scholar
  30. Faure F, Duriez C, Delingette H, Allard J, Gilles B, Marchesseau S, Talbot H, Courtecuisse H, Bousquet G, Peterlik I, et al (2012) Sofa: a multi-model framework for interactive physical simulation. In: Soft tissue biomechanical modeling for computer assisted surgery. Springer, pp 283–321Google Scholar
  31. Fozard JA, Lucas M, King JR, Jensen OE (2013) Vertex-element models for anisotropic growth of elongated plant organs. Front Plant Sci 4:233. CrossRefGoogle Scholar
  32. Gelder AV (1998) Approximate simulation of elastic membranes by triangulated spring meshes. J Graph Tools 3(2):21–41. CrossRefGoogle Scholar
  33. Ghanti D, Patra S, Chowdhury D (2016) Theory of strength and stability of kinetochore-microtubule attachments: collective effects of dynamic load-sharing. arXiv preprint arXiv:1605.08944
  34. Goriely A, Amar MB (2007) On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity. Biomech Model Mechanobiol 6(5):289–296. CrossRefGoogle Scholar
  35. Hamant O, Traas J (2010) The mechanics behind plant development. New Phytol 185(2):369–385. CrossRefGoogle Scholar
  36. Hamant O, Heisler MG, Jönsson H, Krupinski P, Uyttewaal M, Bokov P, Corson F, Sahlin P, Boudaoud A, Meyerowitz EM, Couder Y, Traas J (2008) Developmental patterning by mechanical signals in Arabidopsis. Science 322(5908):1650–1655. CrossRefGoogle Scholar
  37. Hervieux N, Dumond M, Sapala A, Routier-Kierzkowska AL, Kierzkowski D, Roeder AHK, Smith RS, Boudaoud A, Hamant O (2016) A mechanical feedback restricts sepal growth and shape in Arabidopsis. Curr Biol 26(8):1019–1028. CrossRefGoogle Scholar
  38. Kennaway R, Coen E, Green A, Bangham A (2011) Generation of diverse biological forms through combinatorial interactions between tissue polarity and growth. PLOS Comput Biol 7(6):e1002,071. MathSciNetCrossRefGoogle Scholar
  39. Kutschera U (1991) Regulation of cell expansion. The cytoskeletal basis of plant growth and form. Academic Press, London, pp 149–158Google Scholar
  40. Landau LD, Lifshitz E (1986) Theory of elasticity, vol 7. Elsevier, New YorkzbMATHGoogle Scholar
  41. Landrein B, Hamant O (2013) How mechanical stress controls microtubule behavior and morphogenesis in plants: history, experiments and revisited theories. Plant J 75(2):324–338. CrossRefGoogle Scholar
  42. Lockhart JA (1965) An analysis of irreversible plant cell elongation. J Theor Biol 8(2):264–275. CrossRefGoogle Scholar
  43. Nicolas A, Geiger B, Safran SA (2004) Cell mechanosensitivity controls the anisotropy of focal adhesions. Proc Natl Acad Sci USA 101(34):12,520–12,525CrossRefGoogle Scholar
  44. Ortega JKE (1985) Augmented growth equation for cell wall expansion. Plant Physiol 79(1):318–320. CrossRefGoogle Scholar
  45. Paredez AR, Somerville CR, Ehrhardt DW (2006) Visualization of cellulose synthase demonstrates functional association with microtubules. Science 312(5779):1491–1495. CrossRefGoogle Scholar
  46. Peaucelle A, Braybrook SA, Le Guillou L, Bron E, Kuhlemeier C, Höfte H (2011) Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis. Curr Biol 21(20):1720–1726. CrossRefGoogle Scholar
  47. Pradal C, Dufour-Kowalski S, Boudon F, Fournier C, Godin C (2008) OpenAlea: a visual programming and component-based software platform for plant modelling. Funct Plant Biol 35(10):751–760. CrossRefGoogle Scholar
  48. Rojas ER, Hotton S, Dumais J (2011) Chemically mediated mechanical expansion of the pollen tube cell wall. Biophys J 101(8):1844–1853. CrossRefGoogle Scholar
  49. Sampathkumar A, Krupinski P, Wightman R, Milani P, Berquand A, Boudaoud A, Hamant O, Jönsson H, Meyerowitz EM (2014a) Subcellular and supracellular mechanical stress prescribes cytoskeleton behavior in Arabidopsis cotyledon pavement cells. eLife 3:e01,967. CrossRefGoogle Scholar
  50. Sampathkumar A, Yan A, Krupinski P, Meyerowitz EM (2014b) Physical forces regulate plant development and morphogenesis. Curr Biol 24(10):R475–R483. CrossRefGoogle Scholar
  51. Sassi M, Ali O, Boudon F, Cloarec G, Abad U, Cellier C, Chen X, Gilles B, Milani P, Friml J, Vernoux T, Godin C, Hamant O, Traas J (2014) An auxin-mediated shift toward growth isotropy promotes organ formation at the shoot meristem in Arabidopsis. Curr Biol 24(19):2335–2342. CrossRefGoogle Scholar
  52. Tindemans SH, Hawkins RJ, Mulder BM (2010) Survival of the aligned: ordering of the plant cortical microtubule array. Phys Rev Lett 104(5):058,103. CrossRefGoogle Scholar
  53. Tsugawa S, Hervieux N, Hamant O, Boudaoud A, Smith RS, Li CB, Komatsuzaki T (2016) Extracting subcellular fibrillar alignment with error estimation: application to microtubules. Biophys J 110(8):1836–1844. CrossRefGoogle Scholar
  54. Uyttewaal M, Burian A, Alim K, Landrein B, Borowska-Wykret D, Dedieu A, Peaucelle A, Ludynia M, Traas J, Boudaoud A, Kwiatkowska D, Hamant O (2012) Mechanical stress acts via katanin to amplify differences in growth rate between adjacent cells in Arabidopsis. Cell 149(2):439–451. CrossRefGoogle Scholar
  55. Williamson R (1990) Alignment of cortical microtubules by anisotropic wall stresses. Funct Plant Biol 17(6):601–613MathSciNetCrossRefGoogle Scholar
  56. Wolf S, Hmaty K, Höfte H (2012) Growth control and cell wall signaling in plants. Annu Rev Plant Biol 63(1):381–407. CrossRefGoogle Scholar

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

Personalised recommendations