Shh-like oligomerization

Figure 7: Example of a suitable gradient of bound receptor for the Shh-like oligomerization model, with formation of up to 10-mers and no scaling of forward reaction rates, after 15625s of simulation. (A) Concentrations of the monomer $x_1$, the maximal-size multimer $x_{10}$, and free and bound receptors; note the different scale for $x_{10}$ (B) Concentrations of intermediary multimer forms.

Different simulations were run, varying the number of oligomer forms considered (from 3 to 38), the form under which Shh was injected (as a monomer or a multimer), and the rate constants for the oligomerization reaction $x_i + x_j \leftrightarrow x_{i+j}$, where $x_p$ is a p-mer of Shh. The laws tried out for the forward rate constants were $k^{+}_{i,j}=k^{+}$ (i.e. the same rate for all reactions), $k^{+}_{i,j}=k^{+}/(i+j)$ (i.e. "mild" scaling down of the rates according to the size of the product), and $k^{+}_{i,j}=k^{+}/\left( i+j \right)^2$ (stronger scaling down of the rates). Backward reactions were considered to happen all at the same rate.

No suitable gradients were identified with 18-mer Shh injection at the boundary (rather than monomer Shh), or with strong scaling of the oligomerization rates. It was easiest to identify suitable parameter sets for a system with Shh forming up to 10-mers, with no scaling of forward reaction rates, and for systems with Shh forming up to 8-mers or 18-mers, with mild scaling of forward reaction rates.

The low number of parameter sets identified (about 60) does not warrant a detailed parameter study. An example gradient is shown in Figure 7.