# The Role of Macrophage Dynamics in Atherosclerosis Analyzed Using a Petri Net-Based Model
## 3 Results and Discussion
### _t-Invariant-Based Knockout Analysis_  
To evaluate the importance of every functional biological unit in our model, we needed to perform a thorough analysis. To achieve this, we initially knocked out every MCT set, including the trivial ones (single transitions). This step was crucial in determining the number of transitions affected by each knockout.  
To understand the concept of an affected transition, it is crucial to first explore the definition of affected t-invariants. Affected t-invariants are characterized by having transitions that have been knocked out within their support. A transition is labeled as affected if it is part of the support of an affected t-invariant. For example, a transition \(t_{x}\) is not considered affected by \(t_{y}\) if it is not included in the support of any t-invariant that has \(t_{y}\) in its support. This criterion was also applied to non-trivial MCT sets. The results, which are elaborated on in Table 4, indicate only single transitions or non-trivial MCT sets had an impact higher than 2% of affected transitions.  
It should be noted that Table 4 presents two distinct percentage values for each MCT set, which quantify the effect of disabled transitions on t-invariants. The first value specifies the percentage of all disabled transitions, including those within the MCT set. The subsequent value, enclosed in parentheses, denotes the percentage of transitions outside the MCT set that become disabled as a result.  
Based on the obtained results, it is evident that the transitions comprising \(m_{1}\) have the potential to significantly impact a substantial portion of the network when they are knocked out. Nevertheless, it is important to note that transitions within an MCT set inherently occur simultaneously in t-invariants. Therefore, removing any transition from an MCT set will inevitably affect all other transitions within the same set. It should be emphasized that \(m_{1}\) is composed of 23 transitions, which make up 55% of the total transitions in the model. The transitions belonging to this MCT set represent the signaling pathways triggered by disruptions in cholesterol balance, resulting in the activation of pro-inflammatory cytokines like IL-23 and IL-17. This leads to inflammation, the mobilization of HSPCs due to neutrophil activity, and an increase in the production and differentiation of monocytes. Ultimately, these processes contribute to the formation of atherosclerotic plaque.  
Another MCT set that has a substantial inhibitory influence on the network is \(m_{5}\). If any of the transitions of \(m_{5}\) are knocked out, it has the ability to disable a portion of the network, specifically around 11.9% of it. This pertains to the mechanisms in which the release of the cytokine G-CSF is triggered by IL-17, resulting in the mobilization of HSPCs within the bone marrow. On the contrary, it can be observed that the elimination of sets \(m_{2}\), \(m_{3}\), \(m_{4}\), and \(m_{6}\) only impacts transitions that are part of these sets.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**MCT Set** & **Contained Transitions** & **Biological Interpretation** \\ \hline \(m_{3}\) & \(t_{12}\), \(t_{13}\), \(t_{14}\) & Reduced macrophages and osteoblasts in bone marrow, along with lower expression of CXCL12, which triggers the mobilization and expansion of hematopoietic stem and progenitor cells (HSPCs). \\ \hline \(m_{4}\) & \(t_{24}\), \(t_{25}\), \(t_{34}\) & The secretion of macrophage retention factors, chemokines, and cytokines by foam cells leads to an intensification of the local inflammatory process, which is further intensified by inflammatory mediators. \\ \hline \(m_{5}\) & \(t_{16}\), \(t_{17}\) & The release of G-CSF by peripheral cells, influenced by IL-17, results in its migration to bone marrow and subsequent neutrophil production. \\ \hline \(m_{6}\) & \(t_{36}\), \(t_{41}\) & Early phase of plaque formation and promotion of fibrous cap thinning. \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._