Figure 4. Partial Differential Equations Describing a “Minimal” Model of Gradient Formation Driven by Diffusion and Receptor-Mediated Ligand Degradation.

L = BMP ligand (e.g., Dpp); R = receptor; S = Sog; T = Tsg; V_L, V_S, and V_T are production rates and D_L, D_S, and D_ST; D_T and D_LST are diffusion coefficients for their subscripted species; and association, dissociation, and degradation rate constants for L-R binding are represented by k_on, k_off, and k_deg, respectively. The assembly of LST complexes is modeled as initial S-T binding (with association and dissociation rate constants j_on and j_off) followed by L-ST binding (with association and dissociation rate constants of n_on and n_off), but analysis and calculations show that the output of the model is substantially the same for other assembly orders. In the absence of any data showing that free Sog and Tsg are degraded over the time course of embryonic patterning, these processes were generally not included (the effect of including degradation of free Sog [Srinivasan et al., 2002] was explored preliminarily, and unless very rapid, did not substantially influence the model’s behavior). Tolloid-mediated cleavage of LST complexes was represented by a single first-order rate constant, τ.