## 2 Materials and Methods
### Petri Nets and Stochastic Petri Nets
It is possible since there are many extensions of Petri nets, which allow for including quantitative properties of various types into the model [34; 35]. Among such properties are those related to time dependencies existing in the modeled system. They can be described using time Petri nets or stochastic Petri nets. In this paper, the latter will be used. A stochastic Petri net is an extension of a classical Petri net (i.e., the one described above), which possesses all the properties of the latter. In this extension, probabilistic firing rates are associated with each transition. Formally, a stochastic Petri net is \(6\)-tuple \(Q_{S}=(P,T,F,W,M_{0},\Lambda)\), where \(P,T,F,W,M_{0}\) have the same meanings as in a classical Petri net, while \(\Lambda\) is a vector \(\Lambda=(\lambda_{1},\lambda_{2},...,\lambda_{m})\) of marking dependent firing rates, which are associated with every transition [39]. Transition firing times are exponentially distributed, and the distribution of random variable \(X_{j}\) of transition \(t_{j}\) firing time is described by \(f_{X_{j}}(\tau)=1-e^{-\lambda/\tau}\), \(\tau\geq 0\), \(j=1,2,\ldots,m\)[40]. In Figure 3, a simple diagram is given, summarizing all the steps from model creation to the results. In a stochastic Petri net, the firing of transitions follows mass-action kinetics. The necessary transition stochastic rates were computed using the heuristic devised in [41]. Six various constant times approximating the actual values have been used (\"very, very short\"--seconds; \"very short\"--seconds to a minute; \"short\"--up to a minute; \"medium\"--approximately half as long as the synthesis and expression; \"long\"--minutes to hours; \"very long\"--days), similarly as in [42]. This comes from the fact that the exact dependencies  
Figure 3: The comprehensive operational framework outlines the necessary procedures for constructing a model based on stochastic Petri nets, conducting an analysis of the model, and obtaining the desired outcomes. of time values between the components of the processes are not known. To determine those values for our model, first we had to determine the processes with the longest and shortest durations. Next, a time scale has been developed with values ranging from 1 to 500, using knowledge about the processes from the literature and experts. The rate constants of stochastic transitions have been calculated as a reciprocal of those values (see [41]). The model includes various molecules, complexes, and receptors, and their exact concentrations are not precisely known. To address this issue, three different values were used to approximate their actual amounts, following a similar approach as in [41]. These values represent high concentration (100), medium concentration (10), and low concentration or absence (0). The distribution of these values over the places constitutes the initial marking of the system, which defines the physiologically normal state. The stochastic Petri net model was analyzed using the Gillespie stochastic simulation algorithm (SSA) [43] implemented in the Snoopy software [44]. By executing a step-by-step simulation of the potential transitions within the stochastic Petri net, the SSA algorithm ensures that the resulting output consistently represents a valid state of the underlying stochastic process throughout the entire simulation. A detailed description of the algorithm can be found in [44]. To ensure the accuracy of the simulation results, the number of steps for the algorithm was determined based on previous analyses [41; 45]. In order to obtain meaningful results, a considerable number of simulation runs were performed to explore different traces of state changes over time [44]. To improve the reliability of the results, the traces of multiple simulation runs were averaged.