## 3 Methodology
### Covalent Modification Cycle with Direct Positive Autoregulation and a Novel Controller BC-DPAR  
#### 3.2.1 Covalent Modification Cycle with Direct Positive Autoregulation (CMC-D)  
A block diagram showing the idea of the covalent modification cycle with direct positive autoregulation (CMC-D) is illustrated in Figure 5. Compared with the CMC structure (described in Figure 2), the structure CMC-D has the following same mechanism: \(U^{*}\) can be combined with \(X_{a}\) to form \(U\), while \(U\) can react with \(X_{d}\) to form \(U^{*}\). In addition, the following structure exists in CMC-D: the inactive species \(U^{*}\) can be transformed into \(U\) under the action of species \(U\), i.e., the reaction \(U^{*}+U\to C_{3}\to U+U\). CRN representations of the structure CMC-D are given by  
\[ \begin{split}& X_{a}+U^{*}\xrightarrow{k_{b1}}C_{1}\\ & C_{1}\xrightarrow{k_{c1}}X_{a}+U\\ & X_{d}+U\xrightarrow{k_{b2}}C_{2}\\ & C_{2}\xrightarrow{k_{c2}}X_{d}+U^{*}\\ & U^{*}+U\xrightarrow{k_{b3}}C_{3}\\ & C_{3}\xrightarrow{k_{c3}}U+U\\ \end{split}  \tag{10}\]  
ODEs of abstract chemical reactions in (10) can be obtained by  
\[ \begin{split}&\frac{d[C_{1}]_{t}}{dt}=k_{b1}[X_{a}]_{t}[U^{*}]_{t }-k_{c1}[C_{1}]_{t}\\ &\frac{d[C_{2}]_{t}}{dt}=k_{b2}[X_{d}]_{t}[U]_{t}-k_{c2}[C_{2}]_{ t}\\ &\frac{d[C_{3}]_{t}}{dt}=k_{b3}[U]_{t}[U^{*}]_{t}-2k_{c3}[C_{3}]_{ t}\\ &\frac{d[U^{*}]_{t}}{dt}=-k_{b1}[X_{a}]_{t}[U^{*}]_{t}+k_{c2}[C_{2 }]_{t}-k_{b3}[U]_{t}[U^{*}]_{t}\\ &\frac{d[U]_{t}}{dt}=-k_{b2}[X_{d}]_{t}[U]_{t}+k_{c1}[C_{1}]_{t}- k_{b3}[U]_{t}[U^{*}]_{t}+2k_{c3}[C_{3}]_{t}\\ &\frac{d[X_{a}]_{t}}{dt}=-k_{b1}[X_{a}]_{t}[U^{*}]_{t}+k_{c1}[C_{ 1}]_{t}\\ &\frac{d[X_{d}]_{t}}{dt}=-k_{b2}[X_{d}]_{t}[U]_{t}+k_{c2}[C_{2}]_{ t}\\ \end{split}  \tag{11}\]  
Figure 5: Mathematical framework for the covalent modification cycle with direct positive autoregulation. The network diagram in Figure 6 is constructed to intuitively describe the connections between species and multiple reactions in Equation (10). Equation (12) can be obtained due to Equation (11). \[ \begin{split}\frac{d[C_{1}]_{t}}{dt}+\frac{d[X_{a}]_{t}}{dt}=0\\ \frac{d[C_{2}]_{t}}{dt}+\frac{d[X_{d}]_{t}}{dt}=0\\ \frac{d[C_{1}]_{t}}{dt}+\frac{d[C_{2}]_{t}}{dt}+\frac{d[C_{3}]_{t }}{dt}+\frac{d[U^{s}]_{t}}{dt}+\frac{d[U]_{t}}{dt}=0\end{split}  \tag{12}\]  
Note that the results \((d[C_{1}]_{t}/dt)+(d[X_{a}]_{t}/dt)=0\), \((d[C_{2}]_{t}/dt)+(d[X_{d}]_{t}/dt)=0\) and \((d[C_{1}]_{t}/dt)+(d[C_{2}]_{t}/dt)+(d[C_{3}]_{t}/dt)+(d[U^{s}]_{t}/dt)+(d[U]_{t }/dt)=0\) imply that the total mass \(C_{1}+X_{a}\), \(C_{2}+X_{d}\) and \(C_{1}+C_{2}+C_{3}+U^{s}+U\) is conserved through the lifetime of the process. 2.2 Proposed an Application of the Covalent Modification Cycle with Direct Positive Autoregulation--BC-DPAR  
In Section 3.2.1, the BC is constructed utilizing the covalent modification cycle, as shown in Figure 3. Based on this idea, a novel controller, BC-DPAR, is constructed using the covalent modification cycle with direct positive autoregulation (depicted in Figure 5). Furthermore, the schematic of the BC-DPAR in Equation (13) is shown in Figure 7.  
Figure 6: Graphical representation of Equation (10). Figure 7: Schematic diagram of the BC-DPAR scheme for a process. According to Figure 7, there are some differences in structure between BC and BC-DPAR. Compared with BC, species \(U^{*}\) in BC-DPAR can be converted into \(U\) under the action of species \(U\), i.e., the reaction \(U+U^{*}\xrightarrow{\mu_{c}}2U\). For the BC-DPAR, the corresponding CRNs can be given by  
\[ \begin{split}& R\xrightarrow{k_{c}}R+R_{r}\\ & Y\xrightarrow{\theta_{c}}Y+R_{y}\\ & R_{r}+R_{y}\xrightarrow{\gamma_{c}}R_{r}\cdot R_{y}\\ & R_{r}\xrightarrow{\phi_{c}}\varnothing\\ & R_{y}\xrightarrow{\phi_{c}}\varnothing\\ & R_{r}+U^{*}\xrightarrow{a_{c}}U+R_{r}\cdot R_{y}\\ & U+U^{*}\xrightarrow{\mu_{c}}2U\\ & R_{y}+U\xrightarrow{\beta_{c}}U^{*}\end{split}  \tag{13}\]  
Using the mass action kinetics, the ODEs of Equation (13) are given by  
\[