Searching for significant reactions and subprocesses in models of biological systems based on Petri nets.

The primary aim of this research was to propose algorithms enabling the identification of significant reactions and subprocesses within models of biological systems constructed using classical Petri nets. These solutions allow to performance of two analysis methods: an importance analysis for identifying individual reactions critical to the functioning of the model and an occurrence analysis for finding essential subprocesses. To demonstrate the utility of these methods, analyses of an example model have been performed. In this case, it was a model related to the DNA damage response mechanism. It is worth noting that the proposed analyses can be applied to any biological phenomenon represented using the Petri net formalism. The presented analysis methods represent an extension of classical Petri net-based analyses. Their utility lies in their potential to enhance our comprehension of the biological phenomena under investigation. Furthermore, they can lead to the development of more effective medical therapies, as they can aid in the identification of potential molecular targets for drugs.

Control of Cholesterol Metabolism Using a Systems Approach.

Cholesterol is an essential component of mammalian cells and is involved in many fundamental physiological processes; hence, its homeostasis in the body is tightly controlled, and any disturbance has serious consequences. Disruption of the cellular metabolism of cholesterol, accompanied by inflammation and oxidative stress, promotes the formation of atherosclerotic plaques and, consequently, is one of the leading causes of death in the Western world. Therefore, new drugs to regulate disturbed cholesterol metabolism are used and developed, which help to control cholesterol homeostasis but still do not entirely cure atherosclerosis. In this study, a Petri net-based model of human cholesterol metabolism affected by a local inflammation and oxidative stress, has been created and analyzed. The use of knockout of selected pathways allowed us to observe and study the effect of various combinations of commonly used drugs on atherosclerosis. The analysis results led to the conclusion that combination therapy, targeting multiple pathways, may be a fundamental concept in the development of more effective strategies for the treatment and prevention of atherosclerosis.

Interrelations between Iron and Vitamin A-Studied Using Systems Approach.

A deficiency of vitamin A (VAD) and iron is the most common nutritional problem affecting people worldwide. Given the scale of the problem, the interactions between vitamin A and iron levels are widely studied. However, the exact mechanism of the impact of vitamin A on the regulation of iron metabolism remains unclear. An extremely significant issue becomes a better understanding of the nature of the studied biological phenomenon, which is possible by using a systems approach through developing and analyzing a mathematical model based on a Petri net. To study the considered system, the t-cluster analysis, the significance analysis, and the analysis of the average number of transition firings were performed. The used analyses have allowed distinguishing the most important mechanisms (both subprocesses and elementary processes) positively and negatively regulating an expression of hepcidin and allowed to distinguish elementary processes with a higher frequency of occurrence compared to others. The analysis also allowed to resolve doubts about the discrepancy in literature reports, where VAD leads to positive regulation of hepcidin expression or to negative regulation of hepcidin expression. The more detailed analyses have shown that VAD more frequently positively stimulates hepcidin expression and this mechanism is more significant than the mechanism inhibiting hepcidin expression indirectly by VAD.

A Stochastic Petri Net-Based Model of the Involvement of Interleukin 18 in Atherosclerosis.

Interleukin 18 (IL-18) is a proinflammatory and proatherogenic cytokine with pleiotropic properties, which is involved in T and NK cell maturation and the synthesis of other inflammatory cytokines and cell adhesion molecules. It plays a significant role in orchestrating the cytokine cascade, accelerates atherosclerosis and influences plaque vulnerability. To investigate the influence of IL-18 cytokine on atherosclerosis development, a stochastic Petri net model was built and then analyzed. First, MCT-sets and t-clusters were generated, then knockout and simulation-based analysis was conducted. The application of systems approach that was used in this research enabled an in-depth analysis of the studied phenomenon. Our results gave us better insight into the studied phenomenon and allow revealing that activation of macrophages by the classical pathway and IL-18-MyD88 signaling axis is crucial for the modeled process.

A Role of Inflammation and Immunity in Essential Hypertension-Modeled and Analyzed Using Petri Nets.

Recent studies have shown that the innate and adaptive immune system, together with low-grade inflammation, may play an important role in essential hypertension. In this work, to verify the importance of selected factors for the development of essential hypertension, we created a Petri net-based model and analyzed it. The analysis was based mainly on t-invariants, knockouts of selected fragments of the net and its simulations. The blockade of the renin-angiotensin (RAA) system revealed that the most significant effect on the emergence of essential hypertension has RAA activation. This blockade affects: (1) the formation of angiotensin II, (2) inflammatory process (by influencing C-reactive protein (CRP)), (3) the initiation of blood coagulation, (4) bradykinin generation via the kallikrein-kinin system, (5) activation of lymphocytes in hypertension, (6) the participation of TNF alpha in the activation of the acute phase response, and (7) activation of NADPH oxidase-a key enzyme of oxidative stress. On the other hand, we found that the blockade of the activation of the RAA system may not eliminate hypertension that can occur due to disturbances associated with the osmotically independent binding of Na in the interstitium. Moreover, we revealed that inflammation alone is not enough to trigger primary hypertension, but it can coexist with it. We believe that our research may contribute to a better understanding of the pathology of hypertension. It can help identify potential subprocesses, which blocking will allow better control of essential hypertension.

Systems Approach to Study Associations between OxLDL and Abdominal Aortic Aneurysms.

Although abdominal aortic aneurysm (AAA) is a common vascular disease and is associated with high mortality, the full pathogenesis of AAA remains unknown to researchers. Abdominal aortic aneurysms and atherosclerosis are strongly related. Currently, it is more often suggested that development of AAA is not a result of atherosclerosis, however, individual factors can act independently or synergistically with atherosclerosis. One of such factors is low-density lipoprotein (LDL) and its oxidized form (oxLDL). It is known that oxLDL plays an important role in the pathogenesis of atherosclerosis, thus, we decided to examine oxLDL impact on the development of AAA by creating two models using Petri-nets. The first, full model, contains subprocess of LDL oxidation and all subprocesses in which it participates, while the second, reduced model, does not contain them. The analysis of such models can be based on t-invariants. They correspond to subprocesses which do not change the state of the modeled system. Moreover, the knockout analysis has been used to estimate how crucial a selected transition (representing elementary subprocess) is, based on the number of excluded subprocesses as a result of its knockout. The results of the analysis of our models show that oxLDL affects 55.84% of subprocesses related to AAA development, but the analysis of the nets based on knockouts and simulation has shown that the influence of oxLDL on enlargement and rupture of AAA is negligible.

Selected Aspects of Tobacco-Induced Prothrombotic State, Inflammation and Oxidative Stress: Modeled and Analyzed Using Petri Nets.

Many factors, such as endothelial dysfunction, inflammation and hemostatic disturbances, affect formation and progression of atherosclerotic plaque. In our study, we have focused on hemostatic disturbances with particular emphasis on the extrinsic pathways of coagulation. Thrombin is a main enzyme of coagulation and it is engaged in many different subprocesses. It leads to activation of factors of the coagulation cascade, transition of fibrinogen to fibrin monomer, endothelial damage, inflammation, activation of platelets and proliferation. In our study, selected aspects of disorders in prothrombotic states influenced by cigarette smoke have been modeled and analyzed. Tobacco-induced increased tissue factor, which is associated with less plasminogen activator and increased plasminogen activator-1 inhibitor, has been included in the presented model. These disorders together with accompanying inflammatory state are closely related to thrombus formation and cardiovascular disease promotion. The proposed model has been built using Petri nets and the analysis has been based mainly on t-invariants. Using the Petri net theory to model and analyze the investigated phenomena allows to better understand them by revealing some interesting dependencies in the studied biological system. It explains how tobacco smoke affects the analyzed processes and how harmful these effects are.

Theoretical Studies on the Engagement of Interleukin 18 in the Immuno-Inflammatory Processes Underlying Atherosclerosis.

Interleukin 18 (IL-18) is one of the pro-inflammatory cytokines expressed by macrophages, suggesting that it plays important physiological and immunological functions, among the others: stimulation of natural killers (NKs) and T cells to interferon gamma (IFN- γ ) synthesis. IL-18 was originally identified as interferon gamma inducing factor and now it is recognized as multifunctional cytokine, which has a role in regulation of innate and adaptive immune responses. Therefore, in order to investigate IL-18 contribution to the immuno-inflammatory processes underlying atherosclerosis, a systems approach has been used in our studies. For this purpose, a model of the studied phenomenon, including selected pathways, based on the Petri-net theory, has been created and then analyzed. Two pathways of IL-18 synthesis have been distinguished: caspase 1-dependent pathway and caspase 1-independent pathway. The analysis based on t-invariants allowed for determining interesting dependencies between IL-18 and different types of macrophages: M1 are involved in positive regulation of IL-18, while M2 are involved in negative regulation of IL-18. Moreover, the obtained results showed that IL-18 is produced more often via caspase 1-independent pathway than caspase 1-dependent pathway. Furthermore, we found that this last pathway may be associated with caspase 8 action.

The role of Fenton reaction in ROS-induced toxicity underlying atherosclerosis - modeled and analyzed using a Petri net-based approach.

The superoxide-driven Fenton reaction plays an important role in the transformation of poorly reactive radicals into highly reactive ones. These highly reactive species (ROS), especially hydroxyl radicals can lead to many disturbances contributing to the endothelial dysfunction being a starting point for atherosclerosis. Although, iron has been identified as a possible culprit influencing formation of ROS, its significance in this process is still debatable. To better understand this phenomenon, the influence of blockade of Fenton reaction in a proposed Petri net-based model of the selected aspects of the iron ROS-induced toxicity in atherosclerosis has been evaluated. As a result of the blockade of iron ions formation in the model, even up to 70% of the paths leading to the progression of atherosclerosis in this model has been blocked. In addition, after adding to the model, the blockade of the lipids peroxidation paths, progression of atherosclerotic plaque has been not observed. This allowed to conclude that the superoxide-driven Fenton reaction plays a significant role in the atherosclerosis.

Hemojuvelin-hepcidin axis modeled and analyzed using Petri nets.

Systems biology approach to investigate biological phenomena seems to be very promising because it is capable to capture one of the fundamental properties of living organisms, i.e. their inherent complexity. It allows for analysis biological entities as complex systems of interacting objects. The first and necessary step of such an analysis is building a precise model of the studied biological system. This model is expressed in the language of some branch of mathematics, as for example, differential equations. During the last two decades the theory of Petri nets has appeared to be very well suited for building models of biological systems. The structure of these nets reflects the structure of interacting biological molecules and processes. Moreover, on one hand, Petri nets have intuitive graphical representation being very helpful in understanding the structure of the system and on the other hand, there is a lot of mathematical methods and software tools supporting an analysis of the properties of the nets. In this paper a Petri net based model of the hemojuvelin-hepcidin axis involved in the maintenance of the human body iron homeostasis is presented. The analysis based mainly on T-invariants of the model properties has been made and some biological conclusions have been drawn.