Petri nets and ODEs as complementary methods for comprehensive analysis on an example of the ATM–p53–NF-$$\kappa$$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-$$\kappa$$ κ 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.

Mathematical modeling of regulatory networks of intracellular processes – Aims and selected methods

“…In such models, variables usually represent concentrations, levels or the number of molecules of molecular species that are taken into account in a given system. Sometimes, such approaches are expanded by adding a component from other methodologies, like fuzzy logic propositions [35] , or combining dynamics analysis with a parallel one that is concerned with a model structure, not dynamics, as is the case with Petri nets [19] , [48] . Good reviews of these methods can be found in [31] , [50] .…”

“…Surprisingly, although various modeling approaches have been used to detect key elements in disease-related signaling pathways and have led to numerous promising molecular targets for drugs, recent work shows that the results obtained with different methods do not always overlap [48] . Comprehensive structural and parametric analysis of ATM/p53/NF-κB signaling pathway model based on ODEs and Petri net has been performed and resulted in finding different subsets of important processes that might indicate prospective targets for drugs.…”

“…Comprehensive structural and parametric analysis of ATM/p53/NF-κB signaling pathway model based on ODEs and Petri net has been performed and resulted in finding different subsets of important processes that might indicate prospective targets for drugs. While some of the processes appeared in each analysis method, others were found only by one of them and would be missed if only that method was applied [48] . This discovery emphasizes that each method has its own advantages and limitations, and none is better than the other.…”

Mathematical Modeling Support for Lung Cancer Therapy—A Short Review

“…Sometimes, such approaches are expanded by adding a component from other methodologies, e.g., fuzzy logic [95], Petri nets, [96] or others [97].…”

Holistic View on the Structure of Immune Response: Petri Net Model

“…PNs have a broad application in the modeling of biological systems [26][27][28]. PNs have been extended with fuzzy logic approaches in the context of gene regulatory network modeling [29], and have been combined with ODEs to model signaling networks [30]. In the field of biological PN models, the analysis of invariants has proven valuable in application such as gene regulation [31], signal transduction [32][33][34][35][36], and metabolic networks [37,38].…”