## 1 Introduction  
A Petri net [1; 2] is a solid mathematical representation used in concurrent systems modeling, enabling a formal and clear representation of biological pathways at different time and scale levels [3]. Structurally, a Petri net is a directed bipartite graph, including two types of nodes, places and transitions and tokens contained in places. A state of a Petri net is expressed by a token distribution over the places and can be changed by the firing of transitions. The first Petri net model of a biological pathway [4] is a metabolic pathway, describing the conversion of metabolites through enzyme-catalyzed chemical reactions, which has been extensively studied regarding the structural [5; 6; 7; 8] and dynamic [9; 10; 11] properties. A signaling pathway is another major biological pathway, consisting of cascades of activated/deactivated proteins or protein complexes, through which signals are propagated from the cell surface to the nucleus. A Petri net has been used as a structural modeling framework for a signal pathway in a knowledge representation [12] and for data implementation in a database [13]. Modularization of a Petri net model into biologically-functional parts in a signaling pathway has been demonstrated using the fundamental behavioral property, T-invariant [14], which was formalized in metabolic pathways before [5; 6; 7; 8], and its extended property, the feasible T-invariant [15]. Here, a T-invariant is a firing counting vector of transitions in a periodic firing sequence that leads back to the original token distribution where it starts from. A technique to automatically classify T-invariants into functional modules in a biological pathway was proposed in [16] based on cluster analysis. Anotherclassification of T-invariants in terms of biological network structure was demonstrated in [17], where hierarchically-structured Petri net representations are derived automatically. T-invariant was further applied for the representation of the enzymatic activation process in a signaling pathway [18]. The dynamic behaviors of a signaling pathway were examined in Petri net models for a token amount, reflecting the concentration levels of molecules obtained based on the order of the transition firings specified by the topological motifs in a signaling pathway [19; 20], and were calculated after reducing the number of parameters through a simplification process of a Petri net model [21]. The experimentally uncovered biological facts are usually summarized in a picture of a network structured by figures of various shapes and several type of arrows reflecting biological images. A Petri net allows us to model a biological pathway with maintaining its structural information, owing to the characteristics of the graphical representation of Petri net. Accordingly, Petri net-based modeling does not produce a negative effect of the abstraction in a modeling process compared to other modeling methods, such as differential equation-based methods. The timed Petri net is a type of Petri net, in which time delay is associated with state transition, and is a promising candidate for modeling the dynamic property of a signaling pathway because the delay time of a transition is able to reflect the rate of a corresponding reaction in a signaling pathway. However, in the works above [18; 19; 20; 21], a timed Petri net has not been employed. The first attempt to use a timed Petri net for the modeling of a signaling pathway was in [22] (the \"time\" Petri net, which has a different extension in terms of time to the \"timed\" Petri net, was applied to model metabolic pathways in [11]), where the delay times were calculated based on a simple rule in which the sum of the consumption is equal to the production, so as to maintain the concentration equilibrium for each substance in a signaling pathway. Such Petri nets satisfying this simple rule are called retention-free Petri nets later and introduced as a new concept enabling a more flexible determination of the delay times using a concrete algorithm [23].