Parameters for the L-fng secretion model

Parameters used for simulations are shown in Table 1. They were chosen such that the period of the oscillations is about 120 minutes, as in the mouse PSM. Protein concentrations are dimensionless.

The robustness of the model regarding parameter variation was investigated by varying individually each of the parameters in Table 1 and equation 2, the other parameters and the initial conditions being kept similar (it was too computationaly costly to attempt to find "good" initial conditions for each set of parameters, and the results below thus give a lower bound on robustness). Parameters were deemed satisfactory when a wave spent more than 3 times as much time in the anterior PSM as in the posterior PSM (meaning that oscillations in the posterior PSM are much more synchronous than in the anterior PSM).

The system is least sensitive to parameters governing L-fng ( $\alpha_0,\ \alpha_1,\ \alpha_2,$ and $\alpha_3$), the strength of coupling ( $\epsilon ^{l}$), and the coupling differential between anterior and posterior PSM (given by equation 2), which all can be individually varied 5-fold around the values given in Table 1, with the system preserving its behaviour (the oscillation period can be affected). It is slightly more sensitive to the parameters governing the formation of sensitised Notch and its degradation (respectively $\gamma_1$ and $\gamma_2$), which can be varied 3-fold around the values given in Table 1. The most sensitive parameters are those governing Notch synthesis and degradation (respectively $\beta_1$ and $\beta_2$), which can however be varied by 50%.

The parameters in Table 1 correspond to lifetimes of about 3 minutes for L-fng and the sensitised Notch receptor, and of about 64 minutes for the unsensitised Notch receptor. It is normal for the lifetime of the sensitised receptor to be much shorter than that of the sensitised receptor, as it is much more likely to be bound by Delta and cleaved (a process which is not explicitly taken into account by the model). The sensitised receptor is assumed to be about 30 times more efficient at signalling (parameter $\alpha_3$); thus, if intrinsic stabilities were the same, there should be a 30-fold difference in lifetimes. Since there is only a 20-fold difference, the model assumes that sensitisation makes the receptor more stable than the unsensitised form. Biological data corresponding to these parameters is lacking. The lifetime of L-fng takes into account not only spontaneous or proteolytic degradation, but also diffusion away from the secreting cell, which explains its low value.

Numerical details of the simulations are given in Appendix C.

Table 1: Parameters used for simulations.

Parameter	Value
$\alpha_0$	1.08 min$^{-1}$
$\alpha_1$	4.0
$\alpha_2$	0.217 min$^{-1}$
$\alpha_3$	0.03
$\beta_1$	0.0217 min$^{-1}$
$\beta_2$	0.05 min$^{-1}$
$\gamma_1$	15.2 min$^{-1}$
$\gamma_2$	0.217 min$^{-1}$
$\epsilon ^{l}$	0.3