Equations for Shh-like oligomerization

$$\dot{x}_1 = D_1 \Delta x_1 -\delta_1 x_1 + \Sigma_{j>1} k^{-}_{1,j-1}x_j +k^{-}_{1,1}x_{2} - \Sigma_{j=1..n-1}k^{+}_{1,j}x_1 x_j$$

$$- \alpha_{f} r_\mathrm{free} x_1 + \alpha_{r} r_\mathrm{bound}$$

$$\dot{x}_{i,1<i\le n}= D_i \Delta x_i -\delta_i x_i + \Sigma_{j+k=i,j>k} \left( k^{+}_{j,k}x_j x_k - k^{-}_{j,k} x_i \right)$$

$$+ \Sigma_{j>i} k^{-}_{i,j-i}x_j +k^{-}_{i,i}x_{2i} - \Sigma_{j=1..n-i}k^{+}_{i,j}x_i x_j - 1_{2i\le n} k^{+}_{i,i}x_{i}^{2}$$

$$\dot{r}_\mathrm{free}= - \delta^{r}_{\mathrm{free}} r_\mathrm{free} - \alpha_{f} r_\mathrm{free} x_1 + \alpha_{r} r_\mathrm{bound}+ \sigma_{r}$$

$$\dot{r}_\mathrm{bound}= - \delta^{r}_{\mathrm{bound}} r_\mathrm{bound} + \alpha_{f} r_\mathrm{free} x_1 - \alpha_{r} r_\mathrm{bound}$$

$$\frac{\partial x_1}{\partial x} \left( x=0 \right) = - \nu$$

$${r}_\mathrm{free}(t=0)= \sigma_r/ \delta^{r}_\mathrm{free}$$

where $x_i$ is an i-mer of Shh, $n$ is the maximum number of Shh proteins which can associate into a single oligomer, $k^{+}_{i,j}$ and $k^{-}_{i,j}$ are respectively the association and dissociation rates in the reaction $x_i + x_j \leftrightarrow x_{i+j}$, and the other parameters are the same as previously. Unspecified boundary conditions are zero-flux conditions, and unspecified initial conditions are 0.

The diffusion rates for oligomers were scaled from the rate for the monomer, according to the Stokes-Einstein law for spherical particles: $D_i=D_1*i^{-1/3}$.