Details of the model

In the following, proteins and their concentrations are denoted in the same way.

• Two forms of the Notch receptor are considered, a regular one ($n$), and another one ($n^s$), which has been greatly sensitised to Delta signalling by L-fng ($l$). L-fng makes Notch more sensitive to Delta signalling and less sensitive to Serrate signalling, which could make the outcome of L-fng modification of Notch uncertain if both ligands were expressed, but Serrate has been shown not to be expressed in the chick embryo up to the first somite stage (Caprioli et al., 2002). Notch is supposed to be translated at a constant rate (in the PSM, no oscillations in Notch mRNA have been shown). Because there is no evidence for oscillations in the levels of Notch-binding proteins in the mouse or the chick, Notch signalling is taken to be proportional to the quantities of Notch protein, with a weighting factor on unsensitised Notch to account for its lower efficiency. This approximates the time for Notch cleavage, for intra-cellular fragment migration to the nucleus, and for degradation of the transcriptionally-active intra-cellular fragment, as being small compared to the period of the somitogenesis clock, so that the intensity of signalling is at a quasi-equilibrium for both sensitised and unsensitised Notch.

• Cell-cell coupling is accounted for by the action of secreted L-fng on neighbouring cells. To keep the model simple, diffusion is not explicitly considered, being replaced by a weighting factor on L-fng from neighbours (if L-fng is indeed secreted, it can probably not travel over distances covering more than a few cells). The strength of coupling is discussed in more detail in section 2.4.

• Notch drives the transcription of L-fng, with a cooperativity which has been chosen to be 3 (a minimum cooperativity is required for oscillations; 3 is not the minimum, as oscillations are also observed for example with a cooperativity coefficient of 2.7). Notch-dependent expression of L-fng is intricate, but two binding-sites for CBF1, which directly mediates Notch signalling, have been identified in the L-fng regulatory region (Morales et al., 2002); what's more, experimental data suggests that other such binding sites could be present (Morales et al., 2002). There is thus a basis for cooperative action of Notch on the L-fng gene.

• All three proteins undergo exponential decay. This corresponds to degradation which is either spontaneous or mediated by proteases functioning far from saturation.

The graph of the interactions between the components of the model is shown in Figure 1. Corresponding equations are given in Appendix A.

Figure 1: Interaction graph of the Lunatic fringe secretion model. For clarity, the influence of cell i on cells i-1 and i+1 is not shown.

Although the general mathematical description of the model is given in two dimensions, only a linear chain of coupled cells will be considered in the following. Results for simulations in 2 dimensions could be in agreement with the experimentally-observed chevron of clock-gene expression (Freitas et al., 2001, Jouve et al., 2002), if one supposes that oscillations are initiated in a laterally-restricted posterior region, and that lateral coupling is weaker then longitudinal coupling. One reason for the lateral coupling to be weaker than the longitudinal coupling (in or close to the primitive streak, not in the PSM), could be that only a proportion of cells are competent to oscillate; if the lateral density of such cells is lower than the longitudinal density, weaker coupling could be a crude approximation of the decreased efficiency of lateral diffusion of L-fng from one cell to another (since competent cells are more distant laterally than longitudinally). This will be addressed in a later study.