An expanded Notch-Delta model exhibiting long-range patterning and incorporating MicroRNA regulation

Abstract

Notch-Delta signaling is a fundamental cell-cell communication mechanism that governs the differentiation of many cell types. Most existing mathematical models of Notch-Delta signaling are based on a feedback loop between Notch and Delta leading to lateral inhibition of neighboring cells. These models result in a checkerboard spatial pattern whereby adjacent cells express opposing levels of Notch and Delta, leading to alternate cell fates. However, a growing body of biological evidence suggests that Notch-Delta signaling produces other patterns that are not checkerboard, and therefore a new model is needed. Here, we present an expanded Notch-Delta model that builds upon previous models, adding a local Notch activity gradient, which affects long-range patterning, and the activity of a regulatory microRNA. This model is motivated by our experiments in the ascidian Ciona intestinalis showing that the peripheral sensory neurons, whose specification is in part regulated by the coordinate activity of Notch-Delta signaling and the microRNA miR-124, exhibit a sparse spatial pattern whereby consecutive neurons may be spaced over a dozen cells apart. We perform rigorous stability and bifurcation analyses, and demonstrate that our model is able to accurately explain and reproduce the neuronal pattern in Ciona. Using Monte Carlo simulations of our model along with miR-124 transgene over-expression assays, we demonstrate that the activity of miR-124 can be incorporated into the Notch decay rate parameter of our model. Finally, we motivate the general applicability of our model to Notch-Delta signaling in other animals by providing evidence that microRNAs regulate Notch-Delta signaling in analogous cell types in other organisms, and by discussing evidence in other organisms of sparse spatial patterns in tissues where Notch-Delta signaling is active.

Figure 1. Core Notch-Delta signaling pathway.

Figure 2. Wild-type sensory neuron pattern in the Ciona larval PNS.

A representative transgenic embryo expressing an ASH::RFP reporter in midline cells. Cilia (green) have been detected with an anti-acetylated tubulin antibody; ESN cilia (arrows). Coupled with DAPI staining (blue), these markers facilitated counting the number of ESNs and the number of midline cells between ESNs.

Figure 3. Expanded Notch-Delta model.

(A) Monte Carlo simulations show that our expanded model produces ESN numbers and spacings that match with experimentally determined values. (B) The distributions for the number and spacing of ESNs, including the minimum/maximum and variances of the distributions, are all very similar between model and experiment. For the top graphs, the y-axis shows the number of midlines with the given number of ESNs. For the bottom graphs, the y-axis shows the number of times a given ESN spacing occurs. (C) Schematic showing the intra- and inter-cellular interactions between Notch and Delta. The squiggle arrow represents cis-inhibition of Notch by Delta. Note that for clarity only two cells are shown, but the interactions extend over a linear array of cells. (D) The general form of the ordinary differential equations of our expanded model for Cell , with addition of a Notch activity gradient term indicated in red. (E–F) Shown are the equilibrium values of Delta after a typical run of our expanded model in comparison with the original core model , .

Figure 4. Stability analysis of the ESN spacing, P.

(A–B) The top graphs show the equilibrium values for the Delta and Notch levels. (C–D) The bottom graphs examine the unstable case and show the time varying oscillations (left) of Cell 13 for and Cell 12 for and the variation from the equilibrium for all cells (right).

Figure 5. Stability analysis of the parameter m.

(A) for (blue) and as varies. (B) The equilibrium levels of Delta (top) and Notch (bottom) for our cases (blue) and as varies. (C) Extreme close-up of the graph in (B), showing the supercritical Hopf bifurcation with the stable oscillating periodic orbit.

Figure 6. Stability analysis of the parameter .

(A) for (blue) and (red) as varies. (B) Equilibrium levels of Delta (top) and Notch (bottom) for our cases (blue) and (red) as varies. Note that Delta is shown as a semilog-y plot to show the change in Delta. (C–D) Equilibrium levels for Delta and Notch across all midline cells as we cross .

Figure 7. Parameter sensitivity analysis.

(A) The values of are shown for all the parameters of the model with variations of % in each of the parameters. The ordering of the parameters shows which parameters had the largest increase in the eigenvalues for either a % change with the largest on the left. The dotted line indicates the value of for the original set of parameters used in Table 1. (B) The change in equilibrium value for after % change in parameter values. The y-axis shows the ratio change of the equilibrium with the new parameter value divided by the original equilibrium value . The ordering of the parameters shows which parameters had the largest increase in the magnitude of for either a % change with the largest increase on the left. Since the equilibrium is unstable for , oscillates 17% above and 14% below the equilibrium marked with a . (C) The change in equilibrium value for after % change in parameter values. The y-axis shows the ratio change of the equilibrium with the new parameter value divided by the original equilibrium value . The ordering of the parameters shows which parameters had the largest increase in the magnitude of for either a % change with the largest increase on the left. Since the equilibrium is unstable for , oscillates 13% above and 10% below the equilibrium marked with a .

Figure 8. The microRNA miR-124 modulates the Notch decay rate.

(A) Representative embryo for miR-124 titration experiments. (B) Magnified region of the embryo in (A) shows adjacent ESNs; DAPI staining shows pairs of small nuclei belonging to the ESN pairs. (C–D) Comparison of the distribution of ESN counts and spacing between miR-124 titration experiments and our model. Based on this data and our previous results , , we propose that the Notch decay rate, , is modulated by miR-124.

Figure 9. Canonical target sites for miRNAs expressed in sensory cell types throughout bilaterians.

Notch signaling regulates the differentiation and patterning of each of the sensory cell types shown. Sensory cell expression for each of the miRNAs listed was shown previously (see text for references). miRNA canonical target sites in the UTRs of Notch and Hes homologs were found using our previously described target prediction algorithm . Some of these targets have been experimentally verified (V) (human: ; sea squirt: ; fruit fly: [9]). In Drosophila, miR-2, miR-6 and miR-7 overexpression (O) have all been previously shown to cause a phenotype indicative of suppressed Notch signaling activity and loss of lateral inhibition, such as increased density or clustering of microchaetes . In human airway epithelial tissue, knockdown (K) of miR-449 has been shown to cause a decreased rate of ciliated cells indicative of Notch signaling gain-of-function .