## 4 Implementations with DSD Reaction Networks
### Results  
Based on the above design ideas, the regulatory scheme of the enzymatic protein hydrolysis model with controllers (including BC, BC-IPAR and BC-DPAR) can be realized using DSD mechanism. All reaction rates involved in the proposed CRN-based controllers and the enzymatic process model are provided in Tables 1 and 2, respectively. For the controllers involved in this work, the initial values of the signals \(R_{r}\), \(R_{y}\) and \(R_{r}\cdot R_{y}\) are set to zero, i.e., \([R_{r}]_{0}=[R_{y}]_{0}=[R_{r}\cdot R_{y}]_{0}=0\,\mathrm{nM}\). According to Remark 2, there is a common point among the three controllers BC, BC-DPAR and BC-IPAR, i.e., the reaction \(Ut_{\mathrm{Total}}=Ut^{*}+U\). For the driving signal \(U\) of the enzymatic protein hydrolysis process, consider two cases \(U=0\,\mathrm{nM}\) and \(U\
eq 0\,\mathrm{nM}\). Integrating the enzymatic protein hydrolysis model of Equation (1) with the BC of Equation (7), BC-DPAR of Equation (13) and BC-IPAR of Equation (18), the following results can be obtained, as shown in Figure 14.  
**Remark 5**. : _It is worth noting that the parameter \(\mu_{c}\) in Table 1 is related to the reaction \(U+U^{*}\xrightarrow{\mu_{c}}2U\) of the BC-DPAR, whereas the parameter \(\delta_{c}\) are involved in the reaction \(U+U^{*}+R_{r}\xrightarrow{\delta_{c}}2U\) of the BC-IPAR._  
In addition, the output curves of protein hydrolysate production \(X_{A}\) (green curve in Figure 14) are quantified using the setting time as a metric, and the data are integrated as shown in Table 3.  
The following results can be obtained by combining Table 3 with Figure 14. On the one hand, the setting times of the BC and BC-IPAR at \(U=0\,\mathrm{nM}\) are \(1.2\times 10^{3}\) s and \(8.0\times 10^{2}\) s, respectively, while at \(U=1\,\mathrm{nM}\), the setting times of the BC and BC-IPAR are nearly identical, with a settling time of approximately \(8.5\times 10^{2}\) s. From a purely numerical standpoint, it can be seen that compared with the BC scheme, the proposed BC-IPAR can achieve the expected output level in a shorter time. On the other hand, the setting times of the BC and BC-DPAR at \(U=0\,\mathrm{nM}\) are \(8.5\times 10^{2}\) s and \(6.5\times 10^{2}\) s, respectively, while \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Descriptions** & **Nominal Values** \\ \hline \(k_{c}\) & Catalytic reaction rate & \(3.9\times 10^{-4}\) s\({}^{-1}\) \\ \(\theta_{c}\) & Catalytic reaction rate & \(3.9\times 10^{-4}\) s\({}^{-1}\) \\ \(\gamma_{c}\) & Binding rate & \(2.0\times 10^{4}\,\mathrm{M}^{-1}\) s\({}^{-1}\) \\ \(\phi_{c}\) & Degradation reaction rate & \(3.0\times 10^{-4}\) s\({}^{-1}\) \\ \(\alpha_{c}\) & Catalytic reaction rate & \(1.2\times 10^{4}\,\mathrm{M}^{-1}\) s\({}^{-1}\) \\ \(\beta_{c}\) & Binding rate & \(1.2\times 10^{4}\,\mathrm{M}^{-1}\) s\({}^{-1}\) \\ \(\mu_{c}\) (BC-DPAR) & Catalytic reaction rate & \(6.0\times 10^{-3}\) s\({}^{-1}\) \\ \(\delta_{c}\) (BC-IPAR) & Catalytic reaction rate & \(6.0\times 10^{-3}\) s\({}^{-1}\) \\ \(Ut_{\mathrm{Total}}\) & Total amount of \(U+Ut^{*}\) & 6.0\(\,\mathrm{nM}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parametric representation for controllers including the BC, BC-DPAR and BC-IPAR. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Descriptions** & **Nominal Values** \\ \hline k\({}_{1}\) & Binding rate & \(2.8\times 10^{-2}\) s\({}^{-1}\) \\ k\({}_{2}\) & Unbinding rate & \(1.4\times 10^{-2}\) s\({}^{-1}\) \\ k\({}_{3}\) & Degradation reaction rate & \(3.0\times 10^{-5}\) s\({}^{-1}\) \\ \(X_{\mathrm{Total}}\) & Total amount of \(X_{\mathrm{E}}+X_{\mathrm{P:E}}\) & 4.0\(\,\mathrm{nM}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parametric representation for the enzymatic protein hydrolysis model. at \(U=1\,\)nM, the setting times of the BC and BC-DPAR are \(1.2\times 10^{3}\,\)s and \(5.2\times 10^{2}\,\)s. Compared with the BC scheme, the proposed BC-DPAR scheme can realize the output of protein hydrolysate rapidly and stably.