Cinquin Lab

A.1 Parameter Values

Table 1: Ranges from which parameters were selected at random for the simulations described in section 3.
Parameters Range Distribution
$ \delta^{m}_{\mathrm{EM}}, \delta^{m}_{\mathrm{mem}}, \delta^{d}_{\mathrm{EM}},... ...}, \delta_{n}, \delta^{r}_{\mathrm{free}}, \delta^{r}_{\mathrm{bound}},\delta_i$ $ 10^{-7}-10^{-4} s^{-1}$ Log-uniform
$ D^{m}_{\mathrm{EM}}, D^{n},D_i$ 5-30 $ \mu m^{2}.s^{-1}$ Uniform
$ \beta$ $ 10^{-8}-1 s^{-1}$ Log-uniform
$ \gamma$ $ 1-10^{6}s^{-1} M^{-1}$ Log-uniform
$ \sigma_{d}, \sigma_{n}$ $ 10^{-8}-10^{-13} M s^{-1}$ Log-uniform
$ \sigma_{r}$ $ 10^{-14}-10^{-8}M s^{-1}$ $ \frac{\sigma_{r}}{\delta^{r}_{\mathrm{free}}}$ log-uniform in $ 10^{-7}-10^{-4}$
$ \alpha_{f},k^{+}$ $ 10^{5}-10^{8} M^{-1} s^{-1}$ Log-uniform
$ \alpha_{r},k^{-}$ $ 10^{-7}-10^{-1} s^{-1}$ $ \frac{\alpha_{f}}{\alpha_r}$, $ \frac{k^{+}}{k^{-}}$ log-uniform in $ 10^5 - 10^8$
$ \nu$ $ 10^{-8}-10^{-13}M \mu m^{-1}$ Log-uniform

The concentrations of morphogens and receptors, as well as their kinetic reaction coefficients, have been precisely evaluated only in a restricted number of cases (for example by Dyson and Gurdon, 1998; see Freeman and Gurdon, 2002 for a review). Relevant concentration ranges for morphogens have been proposed to be around 10-100 pM (Freeman and Gurdon, 2002). Dyson and Gurdon (1998) have shown that cells can respond with as few as 2% of their receptors bound to the morphogen (the total of number of receptors per cell being estimated to 5,000); this gives a maximum 50-fold useful variation range of the concentration of bound receptor (also compatible with the data of Gurdon et al., 1999). Cultured S2 cells show responses to Dpp over a 10-fold range, from 100pM to 1000pM (Shimmi and O’Connor, 2003) (but the presence of Dlp could possibly alter those responses). BMP receptor affinity for its ligand is of the order of nanomolars (Suzuki et al., 1994, Koenig et al., 1994), and that of the FGF receptor picomolars (Nugent and Edelman, 1992).

The association rates of two proteins in solution can be as high as $ 10^{8}$ $ 10^9 M^{-1}s^{-1}$, close to the rate of random encounter given by Smoluchowski’s equation $ k_{\mathrm{on}}=4\Pi DR$, probably thanks to long-range electrostatic interactions (Gabdoulline and Wade, 1997, Northrup and Erickson, 1992). The order of magnitude of binding rates seems however to normally be around $ 10^6 M^{-1}s^{-1}$. It has been proposed that, for BMP-like morphogens, it can be as low as $ 3.10^5 M^{-1}s^{-1}$ (references in Lander et al., 2002). For FGF, the binding rate has been measured at $ 4.10^{6}M^{-1}s^{-1}$ (Nugent and Edelman, 1992); such high values have been used in some simulations (Kruse et al., 2004).

The extracellular diffusion rate of a 50-kDa albumin, in vivo, has been reported to be 16$ \mu$m$ ^2s^{-1}$, and that of a 15-kDa albumin 24$ \mu$m$ ^2s^{-1}$ (Tao and Nicholson, 1996). The molecular weights of processed Wingless and Decapentaplegic dimers are about 36kDa (value from Pubmed protein) and 30kDa (Doctor et al., 1992), respectively. 10$ \mu$m$ ^2s^{-1}$ seems therefore to be a good assumption for the order of magnitude of the extracellular diffusion rates (this is the value used by Kerszberg and Wolpert, 1998, and Lander et al., 2002).