## 1 Introduction
In this manuscript, the focus is to design novel biomolecular controllers based on enhanced covalent modification cycles. The covalent modification cycle with PAR is utilized to design two novel controllers, i.e., BC-DPAR (involving the covalent modification cycle with direct positive autoregulation) and BC-IPAR (involving the covalent modification cycle with indirect positive autoregulation). Considering the application of the covalent modification cycle in the BC controller which differs from DCRN-based controllers and circumvents the limitation of the dual rail representation due to the absence of subtraction modules, it is feasible to build novel controllers using the enhanced covalent modification cycle with PAR. On the one hand, the covalent modification cycle (CMC) is combined with the direct PAR to form a modified CMC structure (namely CMC-D), a new controller BC-DPAR; on the other hand, another enhanced CMC structure (namely CMC-I) is constructed integrating the CMC and indirect PAR, creating a novel controller BC-IPAR. Then, taking the enzymatic proteolysis reaction process as the research object, a corresponding biological model is constructed with simple chemical reactions to reflect the performance of the proposed biomolecular controllers. Considering the transformation relationship between CRNs and DNA strand displacement reactions, the abstract chemical reactions involving in the enzymatic proteolysis process will be transformed into the corresponding DNA strand displacement reactions in the following work. Finally, combined with the DNA strand displacement mechanism, CRN-based BC, BC-DPAR, BC-IPAR and the enzymatic protein hydrolysis model are converted, and the corresponding control schemes are designed using the proposed enzymatic protein hydrolysis process model, thereby analysing the corresponding regulatory properties. The results show that both the proposed novel controllers, BC-DPAR and BC-IPAR, can rapidly and stably achieve the desired concentration levels of protein hydrolysates compared to the existing BC. In addition, the DNA-based BC-DPAR control scheme with direct positive autoregulation is more effective in regulation than the BC-IPAR control with indirect positive autoregulation, requires less regulation time, and can achieve steady-state output more quickly. Based on prior research on biomolecular controllers, it is proposed to examine potential biological implementations of the proposed controllers via different alternative implementation routes based on RNA molecules or proteins, thereby confirming the experimental findings. The main contributions of the manuscript are as follows: (1) This paper proposes two novel biomolecular controllers within the context of chemical reaction network theory, i.e., BC-DPAR and BC-IPAR, which are based on the enhanced covalent modification cycle (involving the covalent modification cycle with direct or indirect positive autoregulation). In addition, these two controllers circumvent the limitation of dual rail representation in the CRNs designs owing to the absence of subtraction operations, thus reducing the number of abstract chemical reactions required for implementation. (2) The enzymatic protein hydrolysis reaction is modelled using CRNs, using the simplest form of chemical reactions to describe this typical enzymatic reaction process. The abstract chemical reactions are converted into DNA strand displacement reactions, and then the DNA-based enzymatic protein hydrolysis reaction model is constructed. (3) Combined with the proposed enzymatic protein hydrolysis process model and DNA-based controllers (including BC, BC-DPAR and BC-IPAR), the various control schemes are realized, and the corresponding regulation properties are analysed. Results show that the proposed two novel controllers, BC-DPAR and BC-IPAR, both can improve the output of enzymatic protein hydrolysis processes to a certain degree compared to the existing BC, especially the BC-DPAR control scheme involving the covalent modification cycle with direct positive autoregulation.