Simplified schematic representation of the ErbB model. Receptor interaction, internalization, recycling and activation of MAPK and PI3K/Akt pathways are shown. Phosphatase, Ras-GTP hydrolysis and feedback regulation are indicated by red arrows. Matrix shows properties of each receptor dimer relevant to the topology of the model; see in for further information. The prefix C on certain proteins denotes an abbreviated representation of multiple species in the model, e.g. C:Shc is composed of all receptor-bound Shc molecules, which may include ErbB1/1-Shc, ErbB1/2-Shc and so on. However, each of these Shc complexes is represented explicitly by one or more dynamic variables in the model.

Experimental and simulated dynamics for three cell lines, each treated with two concentrations of EGF and HRG. Twenty good fits are shown for each cell line. (A) A431, (B) H1666 and (C) H3255 were stimulated with 5 nM or 0.01 nM EGF or 5 nM or 1 nM HRG as indicated. Vertical axes denote normalized signals representing the phosphorylated (and nominally active) forms of ErbB1, ERK and Akt such that the highest measured and simulated value for one signal, across all treatment conditions, was set to 100. In each panel, the fit having the lowest value of the objective function is indicated by a bold line; slightly less good fits are indicated by faded lines. Experimental data are indicated by dots and standard error of the mean (biological triplicates) by the vertical bars. (D) Distributions of 8 parameter values across 50 independent parameter optimization runs; parameter identities are described in more detail in .

Correlation of sensitivities between five best A431 fits that have been optimized for parameters by simulated annealing. Panels give time-integrated sensitivities for (A) pErbB1, (B) pERK and (C) pAkt dynamics. Within each panel, small boxes in the lower triangle show the correlation plot for parameters sensitivities between two fits; upper triangle shows correlation value (R) associated with each correlation plot.

Regional sensitivity analysis. Regional sensitivities were calculated according to . (A) Changes in sensitivity rankings as a function of number of randomization iterations used in sampling for regional sensitivity. Changes in rankings were calculated relative to the rankings of the run with the next smallest number of iterations. (Inset) Power-law relationship between rank changes and increasing sample size on a log-linear scale with an exponent of 0.4. (B–F) Regional sensitivities of five fits to A431 data, showing the top 10 sensitive parameters for pErbB1 (B), pERK (C, E) and pAkt (D, F) dynamics integrated over time following stimulation with 5 nM EGF (B–D), or HRG (E, F). Shading of entries is as follows: black—parameters with a rank order in the top 10 across all five fits; dark grey—in top 10 across four fits; light grey—in top 10 across three fits.

Dependency of parametric sensitivity on model feature or condition. Sensitivities were calculated according to . (A–D) Correlation plots comparing two dynamic variables (model features) or ligands (conditions). Vertical and horizontal dotted lines divide the plots into four quadrants (as labeled in red on a yellow background): I—insensitive parameters; II–III—parameters exhibiting differential sensitivities; IV—parameters along the diagonal showing joint sensitivities. Colors depict the biological activity or part of the pathway to which a parameter belongs as follows: red—receptor; blue—phosphatase; green—MAP kinase cascade; purple—PI3K–Akt cascade; yellow—Ras and its regulators. (A) pERK sensitivities compared for stimulation with 5 nM EGF versus HRG stimulation and (B) pAkt sensitivities under the same conditions. pERK versus pAkt sensitivities compared under 5 nM EGF (C) or HRG (D).

Differential sensitivity of pERK and pAkt to gefitnib treatment (A) and lapatinib treatment (B). Experimentally measured pERK (red diamonds and lines) and pAkt (black squares and lines) 5 min after EGF stimulation in A431 cells treated with inhibitor 10 min prior to ligand addition. Error bars indicate standard error of the mean.

(A) Dose–response of pErbB1, pShc, pERK and pAkt (5 min after ligand addition) over a 10⁶ range of EGF concentrations in A431 cells. (B) Amplification Ψ of signal calculated with (5 min after ligand addition) over a 10⁶ range of EGF concentrations in A431 cells for pERK, pAkt and pShc. Points represent data and lines represent simulation; error bars denote standard error of the mean. Gray boxes show high error measurements at low EGF concentrations, which arise due to phospho-ErbB1 signals being at the limit of detection.

(A) Dose–response of pERK (red line) and pErbB1 (black line) computed with best IERMv1.0 fit to A431 data. (B) pERK (red line) and pMEK (blue line) responses in a MAPK module isolated from IERMv1.0. (C) Apparent Hill coefficient H_app (as defined using in text) of pERK during Monte Carlo sampling of parameters. (D) Parameters controlling apparent Hill coefficient: c44—Raf phosphatase abundance; k48—phosphatase binding to MEK; c41—Raf abundance; kd49—phosphatase catalysis of MEK; c53—MEK phosphatase abundance; c26––Ras abundance; k44––Raf binding to MEK. (E) Dose–response of pERK for an IERMv1.0 fit with high Ras (parameter c26) abundance (dotted line) or low Ras (solid line). (F) Dose–response of pMEK (blue line) and pERK (red line) in a MAPK module isolated from a IERMv1.0 fit having low values of parameter c26 (left panel) or high values (right panel). (G) Dose–response of pERK for an IERMv1.0 fit with high k44 (dotted line) or low k44 (solid line). (H) Dose–response of pMEK (blue line) and pERK (red line) in a MAPK module isolated from an IERMv1.0 fit having low values of k44 (left panel) or high values (right panel).

(A) Experimentally determined dose–response of pErbB1 (upper panel), pERK (central panel) and pAkt (lower panel) at t=5 min across a range of EGF concentrations in multiple cancer cell lines. Hill coefficients for each EC₅₀ as calculated by nonlinear regression are given in the legend. (B) EC₁₀—effective concentration at 10% of maximum—of three signals for each cell line. Boxes indicate the position of the EC10. Data points and fits to a Hill curve (lines) are colored as follows: black—AU565; red—HBL100; green—MCF7; blue—T47D. The levels of pErbB1 in MCF7 were below the level of detection, but activation of ERK and Akt signals in response to EGF treatment was easily measured. Hill coefficients as calculated by nonlinear regression for activation of pErbB1, pERK and pAkt are, respectively, in AU565 cells, 1.36, 0.82 and 0.58; HBL100 cells, 1.20, 0.89 and 0.88; T47D cells, 1.18, 1.61 and 1.02. For MCF7 cells, Hill coefficients for activation of pERK and pAkt were, respectively, 1.53 and 1.03. (B) EC₁₀—effective concentration for 10% maximal activation for three phospho-signals in each cell line (box). Colored lines above boxes indicate the width of the corresponding activation profile of the Hill curve, defined as the concentration range that spans a rise in activation from 10 to 90%.