Parameter identifiability and sensitivity analysis predict targets for enhancement of STAT1 activity in pancreatic cancer and stellate cells

Abstract

The present work exemplifies how parameter identifiability analysis can be used to gain insights into differences in experimental systems and how uncertainty in parameter estimates can be handled. The case study, presented here, investigates interferon-gamma (IFNγ) induced STAT1 signalling in two cell types that play a key role in pancreatic cancer development: pancreatic stellate and cancer cells. IFNγ inhibits the growth for both types of cells and may be prototypic of agents that simultaneously hit cancer and stroma cells. We combined time-course experiments with mathematical modelling to focus on the common situation in which variations between profiles of experimental time series, from different cell types, are observed. To understand how biochemical reactions are causing the observed variations, we performed a parameter identifiability analysis. We successfully identified reactions that differ in pancreatic stellate cells and cancer cells, by comparing confidence intervals of parameter value estimates and the variability of model trajectories. Our analysis shows that useful information can also be obtained from nonidentifiable parameters. For the prediction of potential therapeutic targets we studied the consequences of uncertainty in the values of identifiable and nonidentifiable parameters. Interestingly, the sensitivity of model variables is robust against parameter variations and against differences between IFNγ induced STAT1 signalling in pancreatic stellate and cancer cells. This provides the basis for a prediction of therapeutic targets that are valid for both cell types.

Figure 1. Reaction network of the IFNγ stimulated STAT1 signalling pathway.

The network shows key reactions of the pathway. IFNγ activates the type II IFN receptor. To keep the model simple, Janus kinases (JAK) are not considered separately but as part of the active receptor complex only. The receptor-associated JAKs phosphorylate cytosolic STAT1 (STAT1Uc), followed by rapid and high affinity formation of homodimers (STAT1Dc). STAT1Dc translocates into the nucleus (STAT1Dn). Nuclear STAT1D can be dephosphorylated, leading to nuclear export of the resulting STAT1Un into the cytoplasm. STAT1Uc can also shuttle into the nucleus. As a transcription factor, STAT1 induces the transcription of specific target genes. The network considers STAT1 itself and SOCS1 as target genes of IFNγ-activated signalling. SOCS1 is a potential negative feedback regulator; inhibiting the phosphorylation of STAT1Uc. The annotation delay refers to temporal differences between IFNγ action at the receptor level and consecutive steps. We could omit the binding of nuclear phosphorylated STAT1 to the DNA in this work comparing to , , because the slightly simplified model leads to indistinguishable fits, as shown in Figures S1 and S2 in Text S1.

Figure 2. IFNγ-induced STAT1 pathway in PSC and PC: Comparison between experimental time series and model simulations.

Experimental time series and model simulations that differ between the two cell types are shown. The left column in Figure 2 contains two subfigures of Figure S1 in Text S1 and the right column in Figure 2 contains two subfigures of Figure S2 in Text S1 for IFNγ = 100 ng/ml. The observation time is given on the x-axis of each subfigure. Experimentally determined expression levels of phospho-STAT1 protein are given in arbitrary units (a.u.). Immunofluorescence analysis data obtained by confocal microscopy were processed by calculating the ratio of nuclear versus cytoplasmic STAT1 concentration. Measured data are presented as blue circles with error bars. The simulated time courses resulting from the mathematical model with optimized parameter values are presented by red solid lines. The quality of the fit in the upper right subfigure is commented in the captions of Figure S2 in Text S1. Experimental time series are replotted from , .

Figure 3. Profile likelihood estimates for the calibrated PSC model.

Model parameters or initial conditions of variables are given on the x-axis of each subfigure. The PLE (black line) together with the point wise (red dashed lower horizontal line) and simultaneous confidence levels (red dashed upper horizontal line) are shown on the y-axis. The values of the x-axis where the PLE crosses the confidence levels yield the lower and upper boundary of the point wise and simultaneous confidence intervals, respectively. A parameter is identifiable if both confidence intervals are finite. We used the simultaneous confidence levels.

Figure 4. Profile likelihood estimates for the calibrated PC model.

Model parameters or initial conditions of variables are given on the x-axis of each subfigure. The PLE (black line) together with the point wise (red dashed lower horizontal line) and simultaneous confidence interval (red dashed upper horizontal line) are shown on the y-axis. For further details see Figure 3.

Figure 5. Amount of overfitting of the calibrated PSC and PC model.

Upper figures: Comparison of the normalized frequency distribution with the probability distribution . Lower figures: Respective cumulative distributions. The are chosen to represent the theoretically expected degrees of freedom (solid line), the degrees of freedom most compatible with the frequency distribution (dashed line) and the calculated degrees of freedom (dotted line). Left column: results for the calibrated PSC model, right column: results for the calibrated PC model.

Figure 6. Trajectories for parameter sets along profile likelihoods show differences in phosphorylated STAT1 increase.

The upper row shows the PLEs of the parameter from Figures 3 and 4. The lower rows show trajectories generated from -parameter sets within the PLE confidence intervals of the variables active receptor concentration (IIr) and phosphorylated STAT1 (STAT1D) for both cell types stimulated with IFNγ = 100 ng/ml. The observation time is given on the x-axis of each subfigure. The trajectories for -parameter sets are shown on the y-axis. For good visibility approximately 11 trajectories are plotted in equal distance according to the PLE in each subfigure. A “−” indicates the location of the trajectory for the smallest parameter value. A “+” indicates the location of the trajectory for the largest parameter value.

Figure 7. Correlations between model parameters after calculation of the PLEs.

The parameter for phosphorylation () is inversely correlated to the total receptor concentration () for PSC and PC as presented in the upper row. The lower row, left column shows the correlation between the parameters nuclear dephosphorylation () and nuclear import of STAT1Dc () and for PSC. A respective correlation equation is inserted in each subfigure. The lower row, right column shows for PC: One finds min⁻¹ which is located inside the CI for min⁻¹. However, one finds min⁻¹ which is located outside the upper boundary of its CI for min⁻¹.

Figure 8. Trajectories for parameter sets along profile likelihoods show nuclear accumulation of phosphorylated STAT1 in PSC.

The upper row shows the PLEs of the parameters from Figure 3. The lower rows show trajectories generated from parameter sets within the PLE confidence intervals of the variables cytoplasmic and nuclear phosphorylated STAT1 (STAT1Dc,STAT1Dn). Stimulation was done with IFNγ = 100 ng/ml. For higher values of and the plateau of the PLE trajectories further decrease, such that for min⁻¹ the plateau is located at a.u. For further details see captions of Figure 6.

Figure 9. Trajectories for parameter sets along profile likelihoods show missing nuclear accumulation of phosphorylated STAT1 in PC.

The upper row shows the PLEs of the parameters from Figure 4. The lower rows show trajectories generated from parameter sets within the PLE confidence intervals of the variables cytoplasmic and nuclear phosphorylated STAT1 (STAT1Dc, STAT1Dn). Stimulation was done with IFNγ = 100 ng/ml. A “*” indicates trajectories for min⁻¹. In this region one finds min⁻¹ (see Figure 7), which is located outside the upper boundary of its CI. For further details see captions of Figure 6.

Figure 10. Time dependent metabolic control coefficients for nuclear phosphorylated STAT1 of PSC model.

Stimulation with IFNγ = 100 ng/ml. The observation time is given on the x-axis of each subfigure. The metabolic control coefficient is given on the y-axis. The symbol “∼” in the y-axis label is a placeholder for the respective parameter or initial condition name. Each parameter or initial condition is independently perturbed by −1%. The metabolic control coefficients for the perturbed default parameter set (See Tables S1, S2, S3 in Text S1) are shown as red triangles. The black lines show the lower and upper CI boundaries of the MCCs. For few parameters one or both black lines are behind the red line.

Figure 11. Time dependent metabolic control coefficients for nuclear phosphorylated STAT1 of PC model.

Stimulation with IFNγ = 100 ng/ml. The observation time is given on the x-axis of each subfigure. The metabolic control coefficient is given on the y-axis. The symbol “∼” in the y-axis label is a placeholder for the respective parameter or initial condition name. Each parameter or initial condition is independently perturbed by −1%. The metabolic control coefficients for the perturbed default parameter set (See Tables S1, S2, S3 in Text S1) are shown as red triangles.The black lines show the lower and upper CI boundaries of the MCCs. For few parameters one or both black lines are behind the red line.

Figure 12. Nuclear accumulation of STAT1 in primary PSC and PC cells.

Isolated primary PSC from rat pancreas and PC cell lines of rat (DSL-6A/C1) and human (BxPC-3, MIA PaCa-2) origin were stimulated with 100 ng/ml species-specific IFNγ for the indicated times. STAT1 nuclear translocation was detected by immunofluorescence analysis. Data obtained by confocal microscopy were processed by calculating the ratio of nuclear versus cytoplasmic STAT1 concentration. Measured data are presented as circles with error bars, mean (n≥10) ± SEM.