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.
Highlights
The derivation of continuum models which represent underlying discrete phenomena is emerging as an important part of mathematical biology: integration between subcellular, cellular and tissue-level behaviour is crucial to understanding tissue growth and mechanics with self-evident application to, for instance, in vitro tissue engineering or the understanding of tumour growth and invasion.It is well known that cell-signalling mechanisms regulate differentiation, cell-fate determination and, tissue and organ development
It has typically been assumed that the analysis of cell signalling mechanisms demands a discrete approach, especially when considering contact-dependent juxtacrine signalling processes (Wearing et al, 2000; Plahte and Øyehaug, 2007) and short-range patterning (Roussel and Roussel, 2004); in our previous work (O’Dea and King, 2011), we have shown how a multiscale method may be employed to analyse the fine-grained patterns of period two generated by the discrete model of Collier et al (1996) in an array of square cells within a continuum formulation
In the case for which the heterogeneous steady states of Delta and Notch activation are of comparable order, the generation of patterns from the homogeneous state and the transition between patterning modes are governed by models of a similar form
Summary
It is well known that cell-signalling mechanisms regulate differentiation, cell-fate determination and, tissue and organ development Such regulation is mediated by the production, transport and binding of intercellular signalling molecules, which may be free to diffuse throughout the tissue or may be anchored in the cell membrane. Lateral inhibition is a juxtacrine pattern-forming mechanism employed by developing tissues to create fine-grained patterns of cell differentiation, in which adjacent or nearby cells diverge to achieve differing cell fates. This mechanism is controlled by a negative feedback loop: receipt of inhibition reduces the ability of a cell to inhibit others, leading to the amplification of differences between cells. While other ligand-receptor interaction-mediated cell signalling mechanisms have been characterised (e.g. the binding of cyclic AMP to Dictyostelium cells (Martiel and Goldbeter, 1987; Dallon and Othmer, 1997) and Transforming Growth Factor-α and Epidermal Growth Factor binding in keratinocytes (Clark et al, 1985; Coffey et al, 1987), here we consider the well-studied Delta-Notch signalling interaction, which provides an ideal model system to illustrate our methodology
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