Results

Simulations were run with successive sets of parameters sampled at random from the ranges detailed in Table 1. A set of parameters was considered to be suitable if the following conditions on the gradient of morphogen-bound receptor were met:

• about 3 hours after the start of the simulation, the concentration of morphogen-bound receptor was sufficiently close to linearity (as per a measure described in the appendix)
• after a further 3 hours of simulation, the gradient stayed with 30% of its original values
• the range of the gradient was greater than 2-fold
• the concentration of bound-receptor at the high end was greater than 30nM (corresponding to 100 bound receptor molecules, following the calculations in Lander et al., 2002), and more than 2% of the receptors were bound to morphogen 75$\mu$m into the field.

Figure 2: Example of a gradient of bound receptor which meets the criteria described in section 3.1 10000s after the start of the simulation.

Figure 3: Example of a non-monotonous gradient of bound receptor.

Figure 4: Concentrations of other elements of the system, for the same parameter values as in Figure 3, at 10000s. The curves were scaled with their highest value so their variations would be visible on the same graph.

Figure 5: Pairwise plot of parameters for which the model with localized Notum synthesis gives rise to a gradient of bound receptor meeting the conditions set out in section 3.1. 1: $\frac{r_\mathrm{bound}(0)} {r_\mathrm{bound}(100)}$, 2: $\frac{{d}_{\mathrm{EM}}(0)}{{d}_{\mathrm{EM}}(100)}$, 3: $\frac{{d}_{\mathrm{mem}}(0)}{{d}_{\mathrm{mem}}(100)}$, 4: $\frac {n(0)} {n(100)}$, 5: ${d}_{\mathrm{EM}}(0)$, 6: $\frac{{d}_{\mathrm{EM}}(0)} {{d}_{\mathrm{mem}}(0)}$, 7: $\frac{{d}_{\mathrm{EM}}(100)} {{d}_{\mathrm{mem}}(100)}$. Ranges are given in section A.6. Scales are logarithmic except for $D^{m}_{\mathrm{EM}}$, $D_n$, and $\frac{r_\mathrm{bound}(0)} {r_\mathrm{bound}(100)}$.

Figure 6: Pairwise plot of parameters for which the model with global Notum synthesis gives rise to a gradient of bound receptor meeting the conditions set out in section 3.1. Number correspondences are the same as in Figure 5.