## 5 Conclusions  
This article proposes two novel biomolecular controllers, i.e., BC-DPAR and BC-IPAR, and demonstrates how to construct the two controllers using abstract chemical reactions. A novel controller BC-DPAR is proposed with the application of the covalent modification cycle with direct positive autoregulation; another novel controller BC-IPAR is constructed integrating the covalent modification cycle and indirect positive autoregulation. Then, an enzymatic protein hydrolysis process model based on CRNs is established, and the whole process is constructed in the simplest form of chemical reaction. In addition, the CRN-based process model obtains the corresponding DNA implementations under the action of DNA strand displacement mechanism. Finally, three different control schemes for DNA-based BC, BC-DPAR, and BC-IPAR are designed and analysed utilizing the proposed enzymatic proteolysis reaction model. Results show that the proposed novel BC-DPAR and  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controllers** & **BC** & **BC-IPAR** & **BC-DPAR** \\ \hline Setting time t (\(U=0\,\mathrm{nM}\)) & \(8.5\times 10^{2}\,\mathrm{s}\) & \(8.5\times 10^{2}\,\mathrm{s}\) & \(6.5\times 10^{2}\,\mathrm{s}\) \\ Setting time t (\(U=1\,\mathrm{nM}\)) & \(1.2\times 10^{3}\,\mathrm{s}\) & \(8.0\times 10^{2}\,\mathrm{s}\) & \(5.2\times 10^{2}\,\mathrm{s}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameterization of \(X_{A}\) actual output level results for the BC, BC-IPAR and BC-DPAR in the enzymatic protein hydrolysis model.  
BC-IPAR control schemes both shorten the setting time required to achieve the expected output levels of proteolytic products compared to the existing BC scheme, and the better regulation effect of the BC-DPAR scheme tends to achieve the ideal expected output. In addition, this paper provides alternative implementation routes for examining potential biological implementations of the proposed controllers to validate our experimental results. An interesting aspect, which has not been considered in this paper, is the combination of the proposed controller in RNA design and protein engineering to synthesize regulators with adjustable thresholds and gains. This is left as future work.  
Conceptualization, Y.X.; resources, X.W.; writing--original draft preparation, Y.X.; writing--review and editing, H.L.; funding acquisition, H.L. All authors have read and agreed to the published version of the manuscript.  
This work is supported by 111 Project (No. D23006), the National Natural Science Foundation of China (Nos. 62272079, 61972266), the Liaoning Revitalization Talents Program (No. XLYC2008017), the Natural Science Foundation of Liaoning Province (Nos. 2020-KF-14-05, 2021-MS-344, 2021- KF-11-03, 2022-KF-12-14), the Scientific Research Fund of Liaoning Provincial Education Department (No. IJKZZ20220147), the State Key Laboratory of Synthetical Automation for Process Industries, the State Key Laboratory of Light Alloy Casting Technology for High-end Equipment (No. LACT-006), the Postgraduate Education Reform Project of Liaoning province (No. LNYG2022493), the Dalian Outstanding Young Science and Technology Talent Support Program (No. 2022 RJ08).  
Not applicable.  
The data supporting the results reported here are available from the corresponding author upon request.  
The authors declare no conflict to interest.  
The following abbreviations are used in this manuscript: