Petri nets and ODEs as complementary methods for comprehensive analysis on an example of the ATM-p53-NF-[Formula: see text]B signaling pathways

Abstract

Intracellular processes are cascades of biochemical reactions, triggered in response to various types of stimuli. Mathematical models describing their dynamics have become increasingly popular in recent years, as tools supporting experimental work in analysis of pathways and regulatory networks. Not only do they provide insights into general properties of these systems, but also help in specific tasks, such as search for drug molecular targets or treatment protocols. Different tools and methods are used to model complex biological systems. In this work, we focus on ordinary differential equations (ODEs) and Petri nets. We consider specific methods of analysis of such models, i.e., sensitivity analysis (SA) and significance analysis. So far, they have been applied separately, with different goals. In this paper, we show that they can complement each other, combining the sensitivity of ODE models and the significance analysis of Petri nets. The former is used to find parameters, whose change results in the greatest quantitative and qualitative changes in the model response, while the latter is a structural analysis and allows indicating the most important subprocesses in terms of information flow in Petri net. Ultimately, both methods facilitate finding the essential processes in a given signaling pathway or regulatory network and may be used to support medical therapy development. In the paper, the use of dual modeling is illustrated with an example of ATM/p53/NF-[Formula: see text]B pathway. Each method was applied to analyze this system, resulting in finding different subsets of important processes that might be prospective targets for changing this system behavior. While some of the processes were indicated in each of the approaches, others were found by one method only and would be missed if only that method was applied. This leads to the conclusion about the complementarity of the methods under investigation. The dual modeling approach of comprehensive structural and parametric analysis yields results that would not be possible if these two modeling approaches were applied separately. The combined approach, proposed in this paper, facilitates finding not only key processes, with which significant parameters are associated, but also significant modules, corresponding to subsystems of regulatory networks. The results provide broader insight into therapy targets in diseases in which the natural control of intracellular processes is disturbed, leading to the development of more effective therapies in medicine.

Figure 1

ODE-based wiring diagram vs. Petri net model.

Figure 2

Time courses of selected variables in the the ATM/p53/NF-$\kappa$B signaling pathway model described in Jonak et al.. Simulation was performed with two input signals: (TOP PANELS) 10 ng/ml TNF and 4 Gy IR; (BOTTOM PANELS) 10 ng/ml TNF and 10 Gy IR. TNF stimulation is constant during the simulation, IR radiation is turned on after 24 h simulation for 1 h.

Figure 3

The proposed model of the ATM–p53–NF-$\kappa$B pathways with two excitations (TNF and IR) (541 t-inv). The presented Petri net is divided into several modules: ATM (green), p53 (gray), NF-$\kappa$B (red), WIP1 (blue), CREB (purple) and inputs of the system (yellow). The places which are marked with the same color and name correspond to the same particle (logic place), they are used only for transparency in the model.

Figure 4

Parameters ranking for the ATM/p53/NF-$\kappa$B signaling pathway model described in Jonak et al.. Simulation was performed with two input signals: (TOP PANEL) 10 ng/ml TNF and 4 Gy IR; (BOTTOM PANEL) 10 ng/ml TNF and 10 Gy IR. Parameter names corresponding to the numbers on the ranking are given in Supplementary Table S4.

Figure 5

Impact of increasing/decreasing the value of the parameter $m{a}_3$ (par. No. 75) on the model’s response. Simulation was performed with two input signals: (TOP PANELS) 10 ng/ml TNF and 4 Gy IR; (BOTTOM PANELS) 10 ng/ml TNF and 10 Gy IR. TNF stimulation is constant during the simulation, IR radiation is turned on after 24 h simulation for 1 h.

Figure 6

Impact of increasing/decreasing the value of the parameter $m{c}_1$ (par. No. 80) on the model’s response. Simulation was performed with two input signals: (TOP PANELS) 10 ng/ml TNF and 4 Gy IR; (BOTTOM PANELS) 10 ng/ml TNF and 10 Gy IR. TNF stimulation is constant during the simulation, IR radiation is turned on after 24 h simulation for 1 h.