Cinquin Lab

A.5 Equations for Shh-like oligomerization

$\displaystyle \dot{x}_1 =$ $\displaystyle D_1 \Delta x_1 -\delta_1 x_1 + \Sigma_{j>1} k^{-}_{1,j-1}x_j +k^{-}_{1,1}x_{2} - \Sigma_{j=1..n-1}k^{+}_{1,j}x_1 x_j$
$\displaystyle - \alpha_{f} r_\mathrm{free} x_1 + \alpha_{r} r_\mathrm{bound}$
$\displaystyle \dot{x}_{i,1<i\le n}=$ $\displaystyle D_i \Delta x_i -\delta_i x_i + \Sigma_{j+k=i,j>k} \left( k^{+}_{j,k}x_j x_k - k^{-}_{j,k} x_i \right)$
$\displaystyle + \Sigma_{j>i} k^{-}_{i,j-i}x_j +k^{-}_{i,i}x_{2i} - \Sigma_{j=1..n-i}k^{+}_{i,j}x_i x_j - 1_{2i\le n} k^{+}_{i,i}x_{i}^{2}$
$\displaystyle \dot{r}_\mathrm{free}=$ $\displaystyle - \delta^{r}_{\mathrm{free}} r_\mathrm{free} - \alpha_{f} r_\mathrm{free} x_1 + \alpha_{r} r_\mathrm{bound}+ \sigma_{r}$
$\displaystyle \dot{r}_\mathrm{bound}=$ $\displaystyle - \delta^{r}_{\mathrm{bound}} r_\mathrm{bound} + \alpha_{f} r_\mathrm{free} x_1 - \alpha_{r} r_\mathrm{bound}$
$\displaystyle \frac{\partial x_1}{\partial x} \left( x=0 \right) =$ $\displaystyle - \nu$
$\displaystyle {r}_\mathrm{free}(t=0)=$ $\displaystyle \sigma_r/ \delta^{r}_\mathrm{free}$

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}$.