## 2 Basic Definitions and Modeling Rules  
In this section, we briefly provide the necessary definitions of a Petri net. For detailed descriptions, the reader is suggested to refer to [2]. **Definition 1**. : _A Petri net (or a net for short), denoted by \(PN=(T,P,E,\alpha,\beta)\), is a bipartite graph consisting of elements shown in Figure 1, where:_  
* \(T\) _is a set of transitions_ \(\{t_{1},t_{2},\cdots,t_{|T|}\}\)_;_
* \(P\) _is a set of places_ \(\{p_{1},p_{2},\cdots,p_{|p|}\}\)_;_
* \(E=E^{+}\cup E^{-}\)_, and_ \(E^{+}\) _and_ \(E^{-}\) _are respectively the sets of arcs from transitions to places_ \(\epsilon=(t,p)\) _and from places to transitions_ \(e=(p,t)\)_;_
* \(\alpha_{e}\) _is the weight of the arc expressed by positive integer_ \(e=(p,t)\)_; and_
* \(\beta_{e}\) _is the weight of the arc expressed by positive integer_ \(e=(t,p)\)_._  
**Definition 2**. : _Let \(PN=(T,P,E,\alpha,\beta)\) be a Petri net._  
1. \({}^{\bullet}t\) _(_\(t^{\bullet}\)_) is the set of input (output) places of_ \(t{\in}T\)_, and_ \({}^{\bullet}p\) _(_\(p^{\bullet}\)_) is the set of input (output) transitions of_ \(p{\in}P\)_._
2. _A transition without an input arc is called the source transition, and the set of source transitions is denoted by_ \(T_{src}=\{t_{1}^{src},\cdots,t_{a}^{src}\}(a\geq 1)\)_._
3. _A transition without an output arc is called the sink transition, and the set of sink transitions is denoted by_ \(T_{sink}=\{t_{1}^{sink},\cdots,t_{b}^{sink}\}(b\geq 1)\)_._
4. _A transition_ \(t\) _is called the synchronous transition if there exists a set of input places_ \(P_{s}\) _that for any_ \(p\in P_{s}\)_,_ \(p^{\bullet}=\{t\}\) _holds, and is defined by_ \(T_{sync}=\{t_{1}^{sync},\cdots,t_{c}^{sync}\}(c\geq 1)\)_._
5. _A place can hold a positive integer that represents a number of tokens. An assignment of tokens in places expressed in the form of a vector_ \(M\) _is called a marking, which varies during the execution of a Petri net. Given an initial marking_ \(M_{0}\)_, a Petri net is called a marked Petri net and is denoted by_ \(MPN\ =\ (PN,M_{0})\)_._  
Figure 2 shows the source (i) and sink (ii) transitions, and Figure 3 shows a synchronous transition. Note that we use discrete Petri nets in the present study. Figure 1: Petri net elements. Figure 3: Synchronous transition. Figure 2: Source and sink transitions. **Definition 3**. : _A transition \(t\) is fireable if each input place \(p_{1}\) of \(PN\) has at least \(\alpha_{e}\) tokens, where \(\alpha_{e}\) denotes the weight of an arc \(e=(p_{1},t)\). The firing of a transition \(t\) removes \(\alpha_{e}\) tokens from each input place \(p_{1}\) of \(t\) and deposits \(\beta_{e}(e=(t,p_{O}))\) tokens to each output place \(p_{O}\) of \(t\), where \(\beta_{e}\) denotes the weight of an arc \(e=(t,p_{O})\). A source transition is always fireable._  
[PERSON] et al. [18] gave the following modeling rules for signaling pathways based on a Petri net representation. 1. Places denote static elements, including chemical compounds, conditions, states, substances and cellular organelles participating in the biological pathways. Tokens indicate the presence of such elements. The number of tokens can be regarded as a representation of the amount of chemical substances. The current assignment of tokens to the different places is expressed in the form of a vector, namely a marking, as defined above. 2. Transitions denote active elements, including chemical reactions, events, actions, conversions and catalyzed reactions. A transition fires by taking off tokens from their individual input places and creating new tokens that are distributed to the output places if the input places have at least as many tokens in them as the arc weight from the place to the transition. 3. Directed arcs connecting places and transitions represent the relations between corresponding static elements and active elements. Arc weights \(\alpha\) and \(\beta\) (defined in Definition 1) describe the quantities of substances required before and after a reaction, respectively.