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

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 (1998, Mathematical modelling of juxtacrine cell signalling. Math. Biosci., 153, 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.