The isolation of spatial patterning modes in a mathematical model of juxtacrine cell signalling

O'Dea, Reuben D.; King, John R.

Abstract

Juxtacrine signalling mechanisms are known to be crucial in tissue and organ development, leading to spatial patterns in gene expression. We investigate the patterning behaviour of a discrete model of juxtacrine cell signalling due to Owen \& Sherratt (\emph{Math. Biosci.}, 1998, {\bf 153}(2):125--150) in which ligand molecules, unoccupied receptors and bound ligand-receptor complexes are modelled. Feedback between the ligand and receptor production and the level of bound receptors is incorporated. By isolating two parameters associated with the feedback strength and employing numerical simulation, linear stability and bifurcation analysis, the pattern-forming behaviour of the model is analysed under regimes corresponding to lateral inhibition and induction. Linear analysis of this model fails to capture the patterning behaviour exhibited in numerical simulations. Via bifurcation analysis we show that, since the majority of periodic patterns fold subcritically from the homogeneous steady state, a wide variety of stable patterns exists at a given parameter set, providing an explanation for this failure. The dominant pattern is isolated via numerical simulation. Additionally, by sampling patterns of non-integer wavelength on a discrete mesh, we highlight a disparity between the continuous and discrete representations of signalling mechanisms: in the continuous case, patterns of arbitrary wavelength are possible, while sampling such patterns on a discrete mesh leads to longer wavelength harmonics being selected where the wavelength is rational; in the irrational case, the resulting aperiodic patterns exhibit `local periodicity', being constructed from distorted stable shorter-wavelength patterns. This feature is consistent with experimentally observed patterns, which typically display approximate short-range periodicity with defects.

Citation

O'Dea, R. D., & King, J. R. (2013). The isolation of spatial patterning modes in a mathematical model of juxtacrine cell signalling. Mathematical Medicine and Biology, 30(2), https://doi.org/10.1093/imammb/dqr028

• Journal Article Type: Article
• Acceptance Date: Nov 11, 2011
• Online Publication Date: Dec 8, 2011
• Publication Date: Jun 1, 2013
• Deposit Date: Jun 11, 2015
• Publicly Available Date: Jun 11, 2015
• Journal: Mathematical Medicine and Biology
• Print ISSN: 1477-8599
• Electronic ISSN: 1477-8602
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• Volume: 30
• Issue: 2
• DOI: https://doi.org/10.1093/imammb/dqr028
• Public URL: https://nottingham-repository.worktribe.com/output/1002031
• Publisher URL: http://imammb.oxfordjournals.org/content/30/2/95
• Additional Information: This is a pre-copyedited, author-produced PDF of an article accepted for publication in Mathematical Medicine and Biology following peer review. The version of record R. D. O'Dea and J. R. King, The isolation of spatial patterning modes in a mathematical model of juxtacrine cell signalling,; Math Med Biol 2013 30: 95-113 is available online at http://imammb.oxfordjournals.org/content/30/2/95.
• Contract Date: Jun 11, 2015