Out of more than 100,000 parameter sets tested, about 0.5% met the conditions above, with gradient ranges of up to 34-fold (range average was 4). Remarkably, satisfactory parameter sets spanned the entire sampling range for each individual parameter, even though the ranges were chosen to be very wide, showing that the structure of the system can accommodate a wide variety of kinetic parameters. An example gradient is shown in Figure 2.
For 4.4% of the parameters tested, the concentration of bound receptor did not form a proper gradient, in that the concentration did not steadily decrease as the distance from the source of the morphogen increased. This is illustrated in Figure 3. Figure 4 shows the concentrations of other variables of the same system; the increasing concentration of morphogen is associated with an increasing membrane Dll/Dlp concentration (and also an increasing ratio of membrane to EM Dll/Dlp, as the latter is constant).
To examine the influence of the parameters on the establishment of the desired gradient, satisfactory parameters were plotted pairwise, along with measures of the gradients established (Figure 5). It resulted from this analysis that for all satisfactory bound-receptor gradients, the gradient of Dll/Dlp was extremely shallow (with no more than 3% variation across the field for the EM form, with an average of 0.16%, and no more than 29% variation for the membrane form, with an average of 3%), as was that of Notum (with a maximum variation of 10% and an average of 3%). Diffusion coefficients for Notum were biased towards higher values, and correlated strongly with the amplitude of both Notum and Dll/Dlp gradients.
This suggested that localized production of Notum was not favorable to the establishment of bound receptor gradient.