title: Continuum limits of pattern formation in hexagonal-cell monolayers.
abstract: Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O'Dea and King (Math Biosci 231(2):172-185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.
authors: Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. reuben.odea@ntu.ac.uk
authors: R D
authors: RD
authors: O'Dea
authors: J R
authors: JR
authors: King
references: Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Feb;69(2 Pt 1):021910
references: 14995494
references: Math Med Biol. 2010 Mar;27(1):39-74
references: 19617301
references: Curr Biol. 2000 May 4;10(9):491-500
references: 10801437
references: J Math Biol. 2004 Apr;48(4):444-76
references: 15052506
references: J Theor Biol. 1996 Dec 21;183(4):429-46
references: 9015458
references: Wilehm Roux Arch Dev Biol. 1983 Mar;192(2):62-74
references: 28305500
references: Science. 1978 Jan 20;199(4326):259-70
references: 413193
references: Nature. 1987 Aug 27-Sep 2;328(6133):817-20
references: 2442615
references: J Math Biol. 2001 Nov;43(5):411-45
references: 11767205
references: Bull Math Biol. 2000 Mar;62(2):293-320
references: 10824431
references: Philos Trans R Soc Lond B Biol Sci. 1997 Mar 29;352(1351):391-417
references: 9134569
references: Curr Opin Genet Dev. 2007 Aug;17(4):309-13
references: 17618111
references: Proc Natl Acad Sci U S A. 1985 Dec;82(24):8374-8
references: 3001700
references: Dev Biol. 1990 Nov;142(1):13-30
references: 2227090
references: Biophys J. 1987 Nov;52(5):807-28
references: 19431710
references: Bull Math Biol. 2008 May;70(4):971-91
references: 18338214
references: Exp Cell Res. 1999 Feb 1;246(2):312-8
references: 9925746
references: Math Biosci. 1998 Nov;153(2):125-50
references: 9825636
references: Prog Biophys Mol Biol. 2004 Sep;86(1):113-60
references: 15261527
references: BMC Dev Biol. 2001;1:13
references: 11495630
references: Math Biosci. 2011 Jun;231(2):172-85
references: 21385590
references: Development. 1998 Dec;125(23):4637-44
references: 9806913