## 5 Dependent Shrink Algorithm and Case Study
### Dependent Shrink Algorithm
The following are available online at [[http://www.mdpi.com/1999-4893/10/1/4/s1](http://www.mdpi.com/1999-4893/10/1/4/s1)]([http://www.mdpi.com/1999-4893/10/1/4/s1](http://www.mdpi.com/1999-4893/10/1/4/s1)), Figure S1: Phenomenon model of the endocytosis signaling pathway, which is obtained from Petri net pathways ([[http://genome.ib.sci.yamaguchi-u.ac.jp/pmp/](http://genome.ib.sci.yamaguchi-u.ac.jp/pmp/)]([http://genome.ib.sci.yamaguchi-u.ac.jp/pmp/](http://genome.ib.sci.yamaguchi-u.ac.jp/pmp/))), Figure S2: Original endocytosis Petri net model, to which Algorithm 1 is to be applied, Figure S3: An intermediate net is obtained by shrinking all of the self-loops related to transitions, \(t_{2}\), \(t_{7}\), \(t_{8}\), \(t_{11}\), \(t_{14}\), \(t_{21}\), \(t_{22}\) and \(t_{26}\), and all of the conflict structures related to transition sets, \(\{t_{2}\), \(t_{3}\}\), \(\{t_{7}\), \(t_{8}\), \(t_{11}\), \(t_{14}\), \(t_{8}\), \(t_{10}\)], [\(t_{11}\), \(t_{13}\)], [\(t_{16}\), \(t_{18}\)], [\(t_{11}\), \(t_{22}\), \(t_{23}\)], [\(t_{21}\), \(t_{22}\)], [\(t_{26}\), \(t_{28}\)] and [\(t_{26}\), \(t_{30}\)]. As a result, \(t_{3}\), \(t_{7}\), \(t_{8}\), \(t_{10}\), \(t_{13}\), \(t_{14}\), \(t_{18}\), \(t_{21}\), \(t_{22}\), \(t_{23}\), \(t_{28}\) and \(t_{30}\) are shrunk, Figure S4: An intermediate net is obtained by processing single-input single-output structure related to places, Rab5a/GTP, ErbB1, active-SPRY2, Grb2-2, Grb2-c-Cbl, active-SPRY2/c-Cbl, EGFR-2, SHP1 and SHP2, and as a result, \(t_{1}\), \(t_{5}\), \(t_{16}\), \(t_{19}\), \(t_{25}\), \(t_{27}\) and \(t_{29}\) are shrunk, Figure S5: The resultant nets of the final and one step before, XML S1: IL-3, XML S2: endocytosis, PNML S1: IL-3, PNML S2: endocytosis. This work was partially supported by the Grant-in-Aid for Scientific Research (B) (16H02896) from the Japan Society for the Promotion of Science. [PERSON] and [PERSON] wrote the proofs and designed the algorithm. [PERSON] and [PERSON] conceived of the principle of the Petri net-based modeling of a signaling pathway. Writing of the paper was shared with these three authors. All authors have read and approved the final manuscript. The authors declare no conflicts of interest. ## References

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