Abstract
Apical constriction is one of the fundamental mechanisms by which embryonic tissue is deformed, giving rise to the shape and form of the fully-developed organism. The mechanism involves a contraction of fibres embedded in the apical side of epithelial tissues, leading to an invagination or folding of the cell sheet. In this article the phenomenon is modelled mechanically by describing the epithelial sheet as an elastic shell, which contains a surface representing the continuous mesh formed from the embedded fibres. Allowing this mesh to contract, an enhanced shell theory is developed in which the stiffness and bending tensors of the shell are modified to include the fibres’ stiffness, and in which the active effects of the contraction appear as body forces in the shell equilibrium equations. Numerical examples are presented at the end, including the bending of a plate and a cylindrical shell (modelling neurulation) and the invagination of a spherical shell (modelling simple gastrulation).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bement WM, Mooseker MS (1996) The cytoskeleton of the intestinal epithelium: components, assembly, and dynamic rearrangements. In: Hesketh JE, Pryme IF (eds) The cytoskeleton: a multi-volume treatise. vol 3: cytoskeleton in specialized tissues and in pathological states, Elsevier, pp 359–404, doi:10.1016/S1874-6020(96)80015-2
Brodland GW, Wiebe CJ (2004) Mechanical effects of cell anisotropy on epithelia. Comput Methods Biomech Biomed Eng 7(2): 91–99. doi:10.1080/1025584042000209369
Burnside B (1971) Microtubules and microfilaments in newt neurulation. Dev Biol 26(3): 416–441. doi:10.1016/0012-1606(71)90073-X
Clausi DA, Brodland GW (1993) Mechanical evaluation of theories of neurulation using computer simulations. Development 118(3): 1013–1023
Cummings FW (1990) A model of morphogenetic pattern formation. J Theor Biol 144(4): 547–566. doi:10.1016/S0022-5193(05)80089-X
Davidson LA (2008) Integrating morphogenesis with underlying mechanics and cell biology. Curr Top Dev Biol 81: 113–133. doi:10.1016/S0070-2153(07)81003-9
Davidson LA, Koehl MAR, Keller R, Oster GF (1995) How do sea urchins invaginate? Using biomechanics to distinguish between mechanisms of primary invagination. Development 121(7): 2005–2018
Davidson LA, Oster GF, Keller RE, Koehl MAR (1999) Measurements of mechanical properties of the blastula wall reveal which hypothesized mechanisms of primary invagination are physically plausible in the sea urchin Strongylocentrotus purpuratus. Dev Biol 209(2): 221–238. doi:10.1006/dbio.1999.9249
Dawes-Hoang RE, Parmar KM, Christiansen AE, Phelps CB, Brand AH, Wieschaus EF (2005) Folded gastrulation, cell shape change and the control of myosin localization. Development 132(18): 4165–4178. doi:10.1242/dev.01938
Dervaux J, Ciarletta P, Ben Amar M (2009) Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Föppl–von Kármán limit. J Mech Phys Solids 57(3): 458–471. doi:10.1016/j.jmps.2008.11.011
Fristrom D (1988) The cellular basis of epithelial morphogenesis. A review. Tissue and Cell 20(5): 645–690. doi:10.1016/0040-8166(88)90015-8
Gierer A (1977) Physical aspects of tissue evagination and biological form. Q Rev Biophys 10(4): 529–593. doi:10.1017/S0033583500003218
Gilbert SF (2006) Developmental biology, 8th edn. Sinauer Associates, Sunderland
Gordon R, Brodland GW (1987) The cytoskeletal mechanics of brain morphogenesis. Cell Biochem Biophys 11(1): 177–238
Graebert KS, Bauch H, Neumüller W, Brix K, Herzog V (1997) Epithelial folding in vitro: studies on the cellular mechanism underlying evagination of thyrocyte monolayers. Exp Cell Res 231(1): 214–225. doi:10.1006/excr.1996.3456
Hardin JD, Cheng LY (1986) The mechanisms and mechanics of archenteron elongation during sea urchin gastrulation. Dev Biol 115(2): 490–501. doi:10.1016/0012-1606(86)90269-1
Hardin J, Keller R (1988) The behaviour and function of bottle cells during gastrulation of Xenopus laevis. Development 103(1): 211–230
Hutson MS, Ma X (2008) Mechanical aspects of developmental biology: perspectives On Growth and Form in the (post)-genomic age. Phys Biol 5(1): 015,001. doi:10.1088/1478-3975/5/1/015001
Jones GW, Chapman SJ (2009) Modelling growth in biological materials. SIAM Review (submitted)
Keller R, Davidson LA, Shook DR (2003) How we are shaped: the biomechanics of gastrulation. Differentiation 71(3): 171–205. doi:10.1046/j.1432-0436.2003.710301.x
Koiter WT (1966) On the nonlinear theory of thin elastic shells. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (Royal Dutch Academy of Sciences) 69: 1–54
Lanir Y (1983) Constitutive equations for fibrous connective tissues. J Biomech 16(1): 1–12. doi:10.1016/0021-9290(83)90041-6
Lee JY, Harland RM (2007) Actomyosin contractility and microtubules drive apical constriction in Xenopus bottle cells. Dev Biol 311(1): 40–52. doi:10.1016/j.ydbio.2007.08.010
Lewis WH (1947) Mechanics of invagination. Anat Rec 97(2): 139–156. doi:10.1002/ar.1090970203
Lim CT, Zhou EH, Quek ST (2006) Mechanical models for living cells—a review. J Biomech 39(2): 195–216. doi:10.1016/j.jbiomech.2004.12.008
Martin AC, Kaschube M, Wieschaus EF (2009) Pulsed contractions of an actin–myosin network drive apical constriction. Nat 457(7228): 495–499. doi:10.1038/nature07522
Mittenthal JE (1987) The shaping of cell sheets: an application of mechanics in developmental biology. In: Akkas N (eds) Biomechanics of cell division, NATO ASI Series A: life sciences, vol 132. Plenum Press, New York, pp 327–346
Mittenthal JE, Mazo RM (1983) A model for shape generation by strain and cell–cell adhesion in the epithelium of an arthropod leg segment. J Theor Biol 100(3): 443–483. doi:10.1016/0022-5193(83)90441-1
Morrill JB (1986) Scanning electron microscopy of embryos. Methods in cell biology, chap 15, vol 27. Elsevier, pp 263–293. doi:10.1016/S0091-679X(08)60353-2
Muñoz JJ, Barrett K, Miodownik M (2007) A deformation gradient decomposition method for the analysis of the mechanics of morphogenesis. J Biomech 40(6): 1372–1380. doi:10.1016/j.jbiomech.2006.05.006
Niordson FI (1985) Shell theory. North-Holland, Amsterdam
Odell GM, Oster G, Alberch P, Burnside B (1981) The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Developmental Biology 85(2): 446–462. doi:10.1016/0012-1606(81)90276-1
Pilot F, Lecuit T (2005) Compartmentalized morphogenesis in epithelia: from cell to tissue shape. Dev Dyn 232(3): 685–694. doi:10.1002/dvdy.20334
Ramasubramanian A, Taber LA (2008) Computational modeling of morphogenesis regulated by mechanical feedback. Biomech Model Mechanobiol 7(2): 77–91. doi:10.1007/s10237-007-0077-y
Schöck F, Perrimon N (2002) Molecular mechanisms of epithelial morphogenesis. Annu Rev Cell Dev Biol 18: 463–493. doi:10.1146/annurev.cellbio.18.022602.131838
Stern CD (2004) Gastrulation: from cells to embryos. Cold Spring Harbor Laboratories Press, New York
Suiker ASJ, Chang CS (2000) Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mech 142 (1–4): 223–234. doi:10.1007/BF01190020
Sweeton D, Parks S, Costa M, Wieschaus E (1991) Gastrulation in Drosophila: the formation of the ventral furrow and posterior midgut invaginations. Development 112(3): 775–789
Thomas GH (2001) Spectrin: the ghost in the machine. BioEssays 23(2): 152–160. doi:10.1002/1521-1878(200102)23:2<152::AID-BIES1022>3.0.CO;2-1
Wiebe C, Brodland GW (2005) Tensile properties of embryonic epithelia measured using a novel instrument. J Biomech 38(10): 2087–2094. doi:10.1016/j.jbiomech.2004.09.005
Wright NA, Alison M (1984) The biology of epithelial cell populations. Oxford University Press
Zinemanas D, Nir A (1992) A fluid-mechanical model of deformation during embryo exogastrulation. J Biomech 25(4): 341–346. doi:10.1016/0021-9290(92)90253-W
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jones, G.W., Chapman, S.J. Modelling apical constriction in epithelia using elastic shell theory. Biomech Model Mechanobiol 9, 247–261 (2010). https://doi.org/10.1007/s10237-009-0174-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-009-0174-1