Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T17:35:07.342Z Has data issue: false hasContentIssue false
  • English
  • Français

A fibre-reinforced fluid model of anisotropic plant cell growth

Published online by Cambridge University Press:  05 July 2010

R. J. DYSON  and
O. E. JENSEN
R. J. DYSON*
Affiliation:
Centre for Plant Integrative Biology, School of Biosciences, University of Nottingham, Sutton Bonington LE12 5RD, UK
O. E. JENSEN
Affiliation:
Centre for Plant Integrative Biology, School of Biosciences, University of Nottingham, Sutton Bonington LE12 5RD, UK School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 655 , 25 July 2010 , pp. 472 - 503
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, C. T., Carroll, A., Akhmetova, L. & Somerville, C. 2010 Real time imaging of cellulose reorientation during cell wall expansion in arabidopsis roots. Plant Physiol. 152, 787–796.Google Scholar
Aris, R. 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall (reprinted 1989 Dover).Google Scholar
Baskin, T. I. 2001 On the alignment of cellulose microfibrils by cortical microtubules: A review and a model. Protoplasma 215, 150–171.Google Scholar
Baskin, T. I. 2005 Anisotropic expansion of the plant cell wall. Annu. Rev. Cell Dev. Biol. 21, 203–222.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.CrossRefGoogle Scholar
Batchelor, G. K. 1971 The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813–829.CrossRefGoogle Scholar
Ben Amar, M. & Cummings, L. J. 2001 Fingering instabilities in driven thin nematic films. Phys. Fluids 13 (5), 1160–1166.CrossRefGoogle Scholar
Bruce, D. M. 2003 Mathematical modelling of the cellular mechanics of plants. Phil. Trans. R. Soc. Lond. B 358, 1437–1444.Google Scholar
Carpita, N. C. & Gibeaut, D. M. 1993 Structural models of primary-cell walls in flowering plants – consistency of molecular-structure with the physical-properties of the walls during growth. Plant J. 3 (1), 1–30.Google Scholar
Chaplain, M. A. J. 1993 The strain-energy function of an ideal plant-cell wall. J. Theor. Biol. 163 (1), 77–97.CrossRefGoogle Scholar
Chavarria-Krauser, A., Jager, W. & Schurr, U. 2005 Primary root growth: A biophysical model of auxin-related control. Func. Plant Biol. 32 (9), 849–862.CrossRefGoogle ScholarPubMed
Cosgrove, D. J. 2005 Growth of the plant cell wall. Nat. Rev. Mol. Cell Bio. 6 (11), 850–861.CrossRefGoogle ScholarPubMed
Cowdrey, D. R. & Preston, R. D. 1966 Elasticity and microfibrillar angle in wood of sitka spruce. Proc. R. Soc. B-Biol. Sci. 166 (1004), 245–272.Google Scholar
Davies, G. C., Hiller, S. & Bruce, D. M. 1998 A membrane model for elastic deflection of individual plant cell walls. J. Texture Stud. 29 (6), 645–667.CrossRefGoogle Scholar
Derbyshire, P., Findlay, K., McCann, M. C. & Roberts, K. 2007a Cell elongation in Arabidopsis hypocotyls involves dynamic changes in cell wall thickness. J. Exp. Bot. 58 (8), 2079–2089.CrossRefGoogle ScholarPubMed
Derbyshire, P., McCann, M. C. & Roberts, K. 2007 b Restricted cell elongation in Arabidopsis hypocotyls is associated with a reduced average pectin esterification level. BMC Plant Biol. 7, 31–43.CrossRefGoogle ScholarPubMed
Dolan, L. & Davies, J. 2004 Cell expansion in root. Curr. Opin. Cell Biol. 7, 33–39.CrossRefGoogle Scholar
Dumais, J., Shaw, S. L., Steele, C. R., Long, S. R. & Ray, P. M. 2006 An anisotropic–viscoplastic model of plant cell morphogenesis by tip growth. Intl J. Dev. Biol. 50, 209–222.Google Scholar
Ericksen, J. L. 1960 Transversely isotropic fluids. Colloid Polym. Sci. 173 (2), 117–122.Google Scholar
van de Fliert, B. W., Howell, P. D. & Ockendon, J. R. 1995 Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292, 359–376.Google Scholar
Geitmann, A. 2006 Plant and fungal cytomechanics: quantifying and model cellular architecture. Can. J. Bot. 84, 581–593.Google Scholar
Gertel, E. T. & Green, P. B. 1977 Cell growth pattern and wall microfibrillar arrangement: Experiments with nitella. Plant Physiol. 60 (2), 247–254.CrossRefGoogle ScholarPubMed
Goriely, A., Robertson-Tessi, M., Tabor, M. & Vandiver, R. 2008 Elastic growth models. In Mathematical Modelling of Biosystems, pp. 1–44. Springer.Google Scholar
Green, P. B. 1960 Multinet growth in the cell wall of Nitella. J. Cell Biol. 7 (2), 289–296.Google Scholar
Green, P. B. 1968 Growth physics in Nitella: A method for continuous in vivo analysis of extensibility based on a micro-manometer technique for turgor pressure. Plant Physiol. 43 (8), 1169–1184.CrossRefGoogle ScholarPubMed
Green, J. E. F. & Friedman, A. 2008 The extensional flow of a thin sheet of incompressible, transversely isotropic fluid. Euro. J. Appl. Math. 19 (3), 225–257.Google Scholar
Hamant, O., Heisler, M. G., Jonsson, H., Krupinski, P., Uyttewaal, M., Bokov, P., Corson, F., Sahlin, P., Boudaoud, A., Meyerowitz, E. M., Couder, Y. & Traas, J. 2008 Developmental patterning by mechanical signals in arabidopsis. Science 322 (5908), 1650–1655.CrossRefGoogle ScholarPubMed
Hand, G. L. 1962 A theory of anisotropic fluids. J. Fluid Mech 13 (1), 33–46.Google Scholar
Hettiaratchi, D. R. P. & O'Callaghan, J. R. 1978 Structural mechanics of plant-cells. J. Theor. Biol. 74 (2), 235–257.CrossRefGoogle ScholarPubMed
Holzapfel, G. A., Gasser, T. C. & Ogden, R. W. 2000 A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity 61 (1–3), 1–48.CrossRefGoogle Scholar
Howell, P. D. 1998 Models for thin viscous sheets (vol 7, 32, 1996). Euro. J. Appl. Math. 9 (1), 93.Google Scholar
Jarvis, M. C. 1984 Structure and properties of pectin gels in plant-cell walls. Plant Cell Environ. 7 (3), 153–164.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102 (715), 161–179.Google Scholar
Kerstens, S. & Verbelen, J. P. 2003 Cellulose orientation at the surface of the Arabidopsis seedling. Implications for the biomechanics in plant development. J. Struct. Biol. 144 (3), 262–270.CrossRefGoogle ScholarPubMed
Kutschera, U. 2008 The growing outer epidermal wall: Design and physiological role of a composite structure. Ann. Bot.-Lond. 101 (5), 615–621.Google Scholar
Lee, M. E. M. & Ockendon, H. 2005 A continuum model for entangled fibres. Euro. J. Appl. Math. 16, 145–160.CrossRefGoogle Scholar
Lindenmayer, A., Prusinkiewicz, P., Hanan, J. S., Fracchia, F. D., Fowler, D. R., de Boer, M. J. M. & Mercer, L. 1996 The Algorithmic Beauty of Plants. Springer-Verlag.Google Scholar
Lipscomb, G. G., Denn, M. M., Hur, D. U. & Boger, D. V. 1988 The flow of fibre suspensions in complex geometries. J. Non-Newton. Fluid Mech. 26 (3), 297–325.CrossRefGoogle Scholar
Lloyd, C. & Chan, J. 2004 Microtubules and the shape of plants to come. Nat. Rev. Mol. Cell Bio. 5 (1), 13–22.Google Scholar
Lockhart, J. A. 1965 An analysis of irreversible plant cell elongation. J. Theor. Biol. 8, 264–275.Google Scholar
Marga, F., Grandbois, M., Cosgrove, D. J. & Baskin, T. I. 2005 Cell wall extension results in the coordinate separation of parallel microfibrils: Evidence from scanning electron microscopy and atomic force microscopy. Plant J. 43 (2), 181–190.CrossRefGoogle ScholarPubMed
McQueen-Mason, S. J. & Cosgrove, D. J. 1995 Expansin mode of action on cell-walls – Analysis of wall hydrolysis, stress-relaxation, and binding. Plant Physiol. 107 (1), 87–100.Google Scholar
Ortega, J. K. E. 1985 Augmented growth equation for cell-wall expansion. Plant Physiol. 79 (1), 318–320.Google Scholar
Passioura, J. B. & Fry, S. C. 1992 Turgor and cell expansion – Beyond the Lockhart equation. Aust. J. Plant. Physiol. 19 (5), 565–576.Google Scholar
Pitt, R. E. & Davis, D. C. 1984 Finite-element analysis of fluid-filled cell response to external loading. Trans. ASAE 27 (6), 1976–1983.Google Scholar
Probine, M. C. 1963 Cell growth and the structure and mechanical properties of the wall in internodal cells of Nitella opaca. III. Spiral growth and cell wall structure. J. Exp. Bot. 14 (1), 101–113.Google Scholar
Probine, M. C. & Preston, R. D. 1961 Cell growth and the structure and mechanical properties of the wall in internodal cells of Nitella opaca. I. Wall structure and growth. J. Exp. Bot. 12 (2), 261–282.Google Scholar
Probine, M. C. & Preston, R. D. 1962 Cell growth and the structure and mechanical properties of the wall in internodal cells of Nitella opaca. II. Mechanical properties of the walls. J. Exp. Bot. 13 (1), 111–127.CrossRefGoogle Scholar
Proseus, T. E., Ortega, J. K. E. & Boyer, J. S. 1999 Separating growth from elastic deformation during cell enlargement. Plant Physiol. 119 (2), 775–784.Google Scholar
Proseus, T. E., Zhu, G. L. & Boyer, J. S. 2000 Turgor, temperature and the growth of plant cells: Using Chara corallina as a model system. J. Exp. Bot. 51 (350), 1481–1494.Google Scholar
Richmond, P. A., Metraux, J. P. & Taiz, L. 1980 Cell expansion patterns and directionality of wall mechanical-properties in Nitella. Plant Physiol. 65 (2), 211–217.CrossRefGoogle ScholarPubMed
van Sandt, V. S. T., Suslov, D., Verbelen, J.-P. & Vissenberg, K. 2007 Xyloglucan endotransglucosylase activity loosens a plant cell wall. Ann. Bot.-Lond. 100, 1467–1473.CrossRefGoogle ScholarPubMed
Scheres, B., Benfey, P. & Dolan, L. 2002 Root development. In The Arabiposis Book. American Society of Plant Biologists.Google Scholar
Schopfer, P. 2006 Biomechanics of plant growth. Am. J. Bot. 93 (10), 1415–1425.Google Scholar
Sellen, D. B. 1983 The response of mechanically anisotropic cylindrical cells to multiaxial stress. J. Exp. Bot. 34 (143), 681–687.CrossRefGoogle Scholar
Smith, L. G. & Oppenheimer, D. G. 2005 Spatial control of cell expansion by the plant cytoskeleton. Annu. Rev. Cell Dev. Biol. 21, 271–295.Google Scholar
Somerville, C., Bauer, S., Brininstool, G., Facette, M., Hamann, T., Milne, J., Osborne, E., Paredez, A., Persson, S., Raab, T., Vorwerk, S. & Youngs, H. 2004 Toward a systems approach to understanding plant-cell walls. Science 306 (5705), 2206–2211.CrossRefGoogle Scholar
Spencer, A. J. M. 1972 Deformations of Fibre-Reinforced Materials. Oxford University Press.Google Scholar
Suslov, D. & Verbelen, J.-P. 2006 Cellulose orientation determines mechanical anisotropy in onion epidermis cell walls. J. Exp. Bot. 57 (10), 2183–2192.Google Scholar
Swarup, R., Kramer, E. M., Perry, P., Knox, K., Leyser, H. M. O, Haseloff, J., Beemster, G. T. S, Bhalerao, R. & Bennett, M. J. 2005 Root gravitropism requires lateral root cap and epidermal cells for transport and response to a mobile auxin signal. Nat. Cell Biol. 7 (11), 1057–1065.CrossRefGoogle ScholarPubMed
Swarup, R., Perry, P., Hagenbeek, D., der Straeten, D. Van, Beemster, G. T. S., Sandberg, G., Bhalerao, R., Ljung, K. & Bennett, M. J. 2007 Ethylene upregulates auxin biosynthesis in arabidopsis seedlings to enhance inhibition of root cell elongation. The Plant Cell 19 (7), 2186–2196.CrossRefGoogle ScholarPubMed
Taiz, L. 1984 Plant cell expansion: Regulation of cell wall mechanical properties. Ann. Rev. Plant. Physiol. 35, 585–657.CrossRefGoogle Scholar
Taiz, L. & Zeiger, E. 2002 Plant Physiology, 3rd edn. Sinauer Associates.Google Scholar
Tanimoto, E., Fujii, S., Yamamoto, R. & Inanaga, S. 2000 Measurement of viscoelastic properties of root cell walls affected by low pH in lateral roots of Pisum sativum L. Plant Soil 226 (1), 21–28.Google Scholar
Thompson, D. S. 2001 Extensiometric determination of the rheological properties of the epidermis of growing tomato fruit. J. Exp. Bot. 52 (359), 1291–1301.Google Scholar
Thompson, D. S. 2005 How do cell walls regulate plant growth. J. Exp. Bot. 56 (419), 2275–2285.Google Scholar
Tomos, D. 2000 The plant cell pressure probe. Biotech. Lett. 22 (6), 437–442.CrossRefGoogle Scholar
Trouton, F. T. 1906 On the coefficient of viscous traction and its relation to that of viscosity. Proc. Roy. Soc. 77A, 426–440.Google Scholar
Veytsman, B. A. & Cosgrove, D. J. 1998 A model of cell wall expansion based on thermodynamics of polymer networks. Biophys. J. 75, 2240–2250.CrossRefGoogle Scholar
Zhu, G. L. & Steudle, E. 1991 Water transport across maize roots – simultaneous measurement of flows at the cell and root level by double pressure probe technique. Plant Physiol. 95 (1), 305–315.Google Scholar