Equations for the L-fng secretion model

A system defined by a set of the equations below, without coupling to any neighbours, undergoes oscillations for a wide range of parameters. The mechanism seems to rely primarily on the positive feedback circuit, with the system "firing" a burst of sensitised Notch and L-fng once a threshold has been reached in Notch and L-fng.

The equations for cell $i$ (the index denotes the anterio-posterior position in the PSM) are given by equation 1, where $l_i$ is the quantity of Lunatic fringe protein in cell $i$, $n_i$ the quantity of un-sensitised Notch receptor, $n^{s}_{i}$ the quantity of sensitised Notch receptor, $\vartheta_{a}(i)$ the set of axial (longitudinal) neighbours of oscillator $i$ which are considered to influence it, and $\vartheta_{l}(i)$ the set of lateral neighbours considered to influence it, with $\epsilon ^{a}$ and $\epsilon ^{l}$ measuring the respective effects of L-fng in proportion to the cell-autonomous effects. In the case of a one-dimensional chain and nearest-neighbour coupling, neighbours of cell $i$ this would be cells $i-1$ and $i+1$, except for the first and last oscillators in the chain. $\alpha_3$ is expected to be small, and corresponds to weak activation of unsensitised Notch by Delta. The simulations presented below were performed with coupling extending to the 4 nearest neighbours (2 anterior neighbours and 2 posterior neighbours, except for cells close to the borders).

The coupling function used was

$$\epsilon ^{l}_{i}=\epsilon^{l} \left( 2 + \tanh \left( \frac{-1}{{\left(i/n\right)}^3} + \frac{1}{{\left(1-i/n\right)}^3} \right) \right)$$ (1)

with $n$ the number of cells in the simulated PSM.

The dynamics of Notch sensitisation are taken to be linear in both enzyme and substrate as a first simplification. Conditions matching this approximation are saturating un-sensitised Notch or roughly constant levels of un-sensitised Notch.