## 5 Dependent Shrink Algorithm and Case Study
### Dependent Shrink Algorithm
Another way is to establish a series of linear equations for all of the places according to retention-freeness, Equation (8), and to solve these equations to finally get the solutions with the form \(f_{nonvalid}=r_{nonvalid,valid}f_{valid}\). Here, \(f_{nonvalid},f_{valid}\) are the firing frequencies corresponding to non-valid and valid reaction speed data. Suppose \(t_{i}\) is only one transition whose corresponding speed data are valid; we can derive the firing frequencies for all of the other transitions of the IL-3 Petri net model in any one of two ways as follows:

\[ \begin{split} f_{i}=&\quad f_{1}=f_{2}=\cdots=f_{2 8}=1\\ f_{29}=&\quad f_{30}=\cdots=f_{50}=f_{o}=\frac{1}{2}.\end{split}  \tag{12}\]

In addition to IL-3, we also apply our algorithm to the endocytosis Petri net model and give the shrunk result as can be found in the Supplementary Materials, in which, IL-3 and endocytosis respectively consist of 43 places, 54 transitions and 109 arcs and 24 places, 34 transitions and 70 arcs. Moreover, the System Biology Markup Language (SBML) [31] and Petri Net Markup Language (PNML) [32] files of those Petri net models can also be obtained from the Supplementary Materials. ### Discussion on the Resultant Structure from the Algorithm

As mentioned, when the Algorithm 1 terminates, there are two possible cases: (1) the resultant net is a single transition; (2) the resultant net is such that all places possess a single output transition and multiple input transitions. Case (1) indicates that all transitions of the given net \(PN\) satisfy a dependency relation, and they comprise an equivalence class, that is, if the firing frequency of any one of the transitions is decided, then the frequencies of all of the others are automatically decided as well. Case (2) may be further divided into: (i) the remaining net consisting of only such transitions that do not satisfy dependency relations with one another; and (ii) some of the transitions satisfying a dependency relation, but not having been shrunk into a single transition. Case (2) (i) indicates that all transitions in the remaining net are independent, and each is a representative of an equivalence class, as can be illustrated in Figure 16. In this Figure 16a, each transition set is rounded by a line, and the transitions included in each set satisfy a dependency relation with one another. Applying dependent shrink, the transformed net is obtained as shown in Figure 16b. Further, the equivalence classes of \(PN\) represented by \(t_{1},t_{2},t_{3},t_{7},t_{8},t_{9},t_{10}\) are shown in Figure 16c. Case (2) (ii) implies that the remaining net includes a subnet that consists of transitions that all have a dependent relation, but cannot be shrunk into a single transition by shrinking the single-input and single-output structure (Figure 7) or the conflict structure (Figure 8). \(PN^{\prime}_{t_{2}}\), shown in Figure 17, is such a subnet, where each transition has at least one input place with one output, which means \(|P^{\prime}_{t_{2}}|\geq|T^{\prime}_{t_{2}}|\) in \(PN^{\prime}_{t_{2}}\). To find such a subnet, we should establish a series of linear equations for all places according to the retention-freeness in which the input and output token flows are equal and solve these equations. Because \(|P^{\prime}_{t_{2}}|\geq|T^{\prime}_{t_{2}}|\) in \(PN^{\prime}_{t_{2}}\), the number of equations is more than that of the variables (firing frequencies of the transitions). Thus, if there are exactly \(|T^{\prime}_{t_{2}}|-1\) linear independent equations,then all of the transitions satisfy a dependency relation. A detailed discussion of this remains as a future issue. ## 6 Conclusions

Despite the recent rapid progress in high throughput measurements of biological data, it is still difficult to gather all reaction speed data in the biological pathways.