Pattern Formation by Lateral Inhibition with Feedback: a Mathematical Model of Delta-Notch Intercellular Signalling

Abstract

In many developing tissues, adjacent cells diverge in character so as to create a fine-grained pattern of cells in contrasting states of differentiation. It has been proposed that such patterns can be generated through lateral inhibition--a type of cell-cell interaction whereby a cell that adopts a particular fate inhibits its immediate neighbors from doing likewise. Lateral inhibition is well documented in flies, worms and vertebrates. In all of these organisms, the transmembrane proteins Notch and Delta (or their homologues) have been identified as mediators of the interaction--Notch as receptor, Delta as its ligand on adjacent cells. However, it is not clear under precisely what conditions the Delta-Notch mechanism of lateral inhibition can generate the observed types of pattern, or indeed whether this mechanism is capable of generating such patterns by itself. Here we construct and analyse a simple and general mathematical model of such contact-mediated lateral inhibition. In accordance with experimental data, the model postulates that receipt of inhibition (i.e. activation of Notch) diminished the ability to deliver inhibition (i.e. to produce active Delta). This gives rise to a feedback loop that can amplify differences between adjacent cells. We investigate the pattern-forming potential and temporal behaviour of this model both analytically and through numerical simulation. Inhomogeneities are self-amplifying and develop without need of any other machinery, provided the feedback is sufficiently strong. For a wide range of initial and boundary conditions, the model generates fine-grained patterns similar to those observed in living systems.

Mathematical Modelling of Juxtacrine Patterning

“…In particular, Collier et al (1996) studied a discrete model for Delta-Notch signalling during development. Their model is considerably different from ours because of the par-ticular details of the Delta-Notch system; the model includes lateral inhibition of neighbouring cells via a negative feedback loop involving two variables-in contrast to our three-variable model which has positive feedback.…”

“…There are numerous extensions which could be carried out to the present work. A natural step would be to consider the problem in two dimensions for varying geometrical structures; Collier et al (1996) investigated pattern formation on a hexagonal cellular network. The model itself could be extended to incorporate other biologically relevant features, such as cell movement and cell polarization; the latter could arise from receptors moving on the cell surface while remaining bound within the cell membrane.…”

“…In the latter case, the relative rates of cleavage and decay of the membrane-bound form determine the relative importance of paracrine and juxtacrine signalling modes. Collier et al (1996) were the first to consider explicit mathematical modelling of juxtacrine communication when they investigated the pattern-forming potential of Delta-Notch signalling during Drosophila development. Their model incorporates lateral inhibition; a type of cell-cell interaction whereby a cell fated in a particular way inhibits its neighbours from developing similarly.…”

Oscillations and patterns in spatially discrete models for developmental intercellular signalling

“…The first model to consider juxtacrine signalling was formulated in terms of the activity of a protein and its receptor [4], incorporating a feedback loop termed lateral inhibition. The signalling molecule in this model is a protein called Delta which binds to the receptor Notch -this interaction is known to be important in early animal development [20,21,42].…”

“…In the Delta-Notch system, for example, high Delta expression in a cell downregulates Delta in its neighbours, via the receptors Notch on their cell surface [16,20]. This lateral inhibition has been shown to be a robust mechanism for the formation of spatial patterns -provided that the inhibition is sufficiently strong, small differences between neighbouring cells are self-amplifying, leading to the generation of fine grained patterns [4]. Detailed analysis, however, showed that in all cases the scale of the predicted patterns does not extend beyond a wavelength of two cells (for a linear array) or three cells (for a hexagonal array).…”