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Fig. 2. Spatial and temporal evolution of the Shh signal. (A) Reaction-diffusion equations for the core Shh signaling network. `Promoter' and `basal' terms have been previously defined (Lai et al., 2004). (B) Three classes of steady-state behavior in the core single cell Shh model as described previously (Lai et al., 2004). Two time-invariant levels of concentrations (steady states) corresponding to a gli1 `on' and a gli1 `off' state can exist, as we have previously described (Lai et al., 2004). There are three distinct regimes controlled by extracellular Shh concentration: only the `off' state is stable, only the `on' state is stable, and an intermediate bistable regime where either state is stable. (C) Sensitivity analysis of parameters in the single cell Shh network. To determine which parameters most strongly control the response of single cells to Shh, we performed a sensitivity analysis, i.e. we varied each parameter while holding others constant and observed changes in the on/off switch. This graph shows the changes in steady-state behavior as particular parameters are varied 100-fold above and below the best available literature value. Ranges of parameter values at which single cell behavior falls into the three classes schematically represented and shaded in B are shown. k_perturb, value of parameter for which the behavior of the model is plotted; k_lit, value of parameter in literature. Notice that several parameters need to be controlled within a narrow range of values (e.g. k_Gmax) and several can vary over a wide range (e.g. k_Pout). (D) The time for a single cell to switch from a V3 to a MN fate (i.e. to achieve a greater than 7-fold increase in Gli1 concentration) at various constant extracellular Shh concentrations. (E-H) The spatiotemporal evolution of various Shh network constituents in a wild-type embryo is shown: (E) Shh extracellular concentration; (F) Ptc intracellular concentration; (G) Ptc-Shh complex intracellular concentration; and (H) Gli1 intracellular concentration. Bolded lines in each figure correspond to concentration profiles at the end of the V3/MN developmental time window (t=83 hours). Simulation initial conditions were: [Shh]=0; [PtcShh_in]=0; [PtcShh_out]=0; [Ptc_out]=2.0 nM; [Ptc_in]=0.33 nM; [Gli1]=1.63 nM; [Gli3]=5.81 nM; and [Gli3R]=61.2 nM. Parameters for core pathway: D_Shh=1.0x10^-7 cm²/s; k_off=0.10 min^-1; k_on=120,000,000 M^-1 min^-1_; k_Cin=0.2 min^-1; k_Cout=0.00181 min^-1; k_Cdeg=0.00198 min^-1; k_Pmax=2.25x10^-9 M min^-1; k_Pbas=1.73x10^-11 M min^-1; k_Pin=0.03 min^-1; k_Pout=0.00036 min^-1; k_Pdeg=0.09 min^-1; k_Gmax=2.74x10^-10 M min^-1; k_Gbas=2.11x10^-12 M min^-1; k_deg=0.009 min^-1; r_g3b=3.1x10^-19 M² min^-1; k_g3r=0.0117 min^-1; K_g3rc=0.12; K_ptc=3.32x10^-11 M; and K_Gli3=8.3x10^-10 M (see Table S1 in the supplementary material for parameter descriptions and sources). Boundary conditions for all species were impermeable at the source (/x=0 at x=0 µm) and zero at large distances (concentration=0 at x=300 µm).

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