Glycoprotein phase-repartition with global Notum

In order to test this, the same simulation was run, but with Notum synthesis over the whole field, rather than over the first 10 $\mu$m. Pairwise plots of satisfactory parameters are shown in Figure 6. About 1.7% of the parameter tested (out of about 90,000 tested) met the conditions, i.e. 3-fold more than with localized Notum synthesis, with ranges of up to 30-fold (range average was 4.6). Only in 0.9% of the cases was the gradient reversed as in Figure 3.

Comparison of Figures 5 and 6 shows that global Notum synthesis relaxes restriction on the relative values of the morphogen diffusion rate $D^{m}$ and $\beta$, the rate of exchange of the morphogen between the membrane and EM phases (the former must be sufficiently high compared to the latter, if Notum is synthesized locally), as well as the restriction on the relative values of $D^{m}$ and the activity of Notum $\gamma$ (again, the former tends to be high compared to the latter, if Notum is synthesized locally).

Notable restrictions on parameter sets common to the models with localized or global Notum synthesis are that the product of $\delta^{m}_{EM}$, the degradation rate of the morphogen in the extracellular matrix and $\beta$, as well as that of the activity of Notum $\gamma$ and its synthesis rate $\sigma_n$, must be sufficiently high.

Interestingly, the rate of association of receptor and morphogen is not skewed in the $10^5 - 10^8$ range; the proposed model thus does not share the restriction of that of Lander et al. (2002). Also, to investigate the role of bound-receptor downregulation, the corresponding parameter $\delta^{r}_{\mathrm{bound}}$ was set to 0, and the same analysis as previously carried out. The results were roughly identical (data not shown), showing that receptor downregulation is not an important feature of this model.