Ordered hexagonal patterns via notch–delta signaling

Abstract

Many developmental processes in biology utilize notch-delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, exploiting this 'juxtacrine' property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high delta at their center, as opposed to those with central high notch levels (referred to as anti-hexagons). We show that outside this range of parameters, in particular for low cis-coupling, a novel kind of pattern is produced, where high delta cells have high notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.

Stochastic fluctuations promote ordered pattern formation of cells in the Notch-Delta signaling pathway

“…Several models have been proposed to model the Notch-Delta switch [ 13 , 15 , 16 , 18 , 34 , 35 ]. In particular, to build the stochastic single cell model, we generalize the Notch-Delta switch developed by Boareto and collaborators [ 12 ] to include stochastic fluctuations on the Notch receptor and Delta ligand to account for the multicell environment.…”

“…This framework, however, could be generalized in the future to inject prior knowledge on a greater range of biological questions. For instance, epithelial cells often organize in nearly hexagonal geometries that exhibit a lowest energy state where Senders are surrounded by six Receivers [ 18 ], as seen for instance in the patterning of bristles during fruit fly development [ 38 ]. Also, the ‘standard’ Notch-Delta signaling can be modified by additional mechanisms such as long-range interactions via diffusible ligands [ 51 ], variable cell-cell contact area [ 52 ], and intracellular heterogeneity arising from spatial diffusion [ 49 ].…”

“…While Notch-Delta lateral inhibition is generally accepted as a qualitative model of alternate cell patterning, the precise mechanism by which cells can enforce robust patterning in a noisy cellular environment remains poorly understood. While several theoretical frameworks have been proposed to model Notch-Delta lateral inhibition, multicellular pattern formation has mostly been studied with deterministic models [12,13,[15][16][17][18][19], despite the importance of noise in the Notch-Delta pathway in creating stable pattern formation [20][21][22].…”

Role of Delta-Notch signalling molecules on cell–cell adhesion in determining heterogeneous chemical and cell morphological patterning

“…56,57 However, a careful stability analysis of similar models indicate that in certain parameter regimes anti-hexagonal patterns in which Notch rich cells are surrounded by Delta cells are also feasible. 62 Moreover, it has also been proposed that the presence of optimal strength of spatiotemporal 'noise' in the system can help formation of defect-free, ordered, checkerboard patterns. 62,63 Nevertheless, the checkerboard patterns due to lateral inhibition based Delta-Notch signalling can be smoothed to spatially homogeneous hybrid patterns due to lateral induction arising from Jagged.…”

“…62 Moreover, it has also been proposed that the presence of optimal strength of spatiotemporal 'noise' in the system can help formation of defect-free, ordered, checkerboard patterns. 62,63 Nevertheless, the checkerboard patterns due to lateral inhibition based Delta-Notch signalling can be smoothed to spatially homogeneous hybrid patterns due to lateral induction arising from Jagged. 64 On the other hand, some models demonstrate that when the contacts are created between non-adjacent neighbours due to cellular protrusions, then the dynamics of components associated with these contacts can give rise to a wider range of Delta-Notch patterns in the tissue.…”