## 3 Methodology
### Covalent Modification Cycle with Indirect Positive Autoregulation and a Novel Controller BC-IPAR  
#### 3.3.1 Covalent Modification Cycle with Indirect Positive Autoregulation (CMC-I)  
Figure 8 shows the mathematical framework of the covalent modification cycle with indirect positive autoregulation (CMC-I). Compared with the structure CMC (illustrated in Figure 2) and CMC-D (described in Figure 5), CMC-I has a special mechanism: the inactive species \(U^{*}\) can be transformed into \(U\), i.e., under the action of species \(U\) and an activator \(X_{a}\), i.e., the reaction \(U+X_{a}\to C_{3}+U^{*}\to C_{4}\to U+U\). Figure 8: Mathematical framework for the covalent modification cycle with indirect positive autoregulation. Its abstract chemical reactions can be described by  
\[ \begin{split}& X_{a}+U^{*}\xrightarrow{k_{b4}}C_{1}\\ & C_{1}\xrightarrow{k_{c1}}X_{a}+U\\ & X_{d}+U\xrightarrow{k_{b2}}C_{2}\\ & C_{2}\xrightarrow{k_{c2}}X_{d}+U^{*}\\ & UI+X_{a}\xrightarrow{k_{b3}}C_{3}\\ & U^{*}+C_{3}\xrightarrow{k_{b4}}C_{4}\\ & C_{4}\xrightarrow{k_{c4}}U+C_{3}\end{split}  \tag{15}\]  
The network diagram in Figure 9 is constructed to intuitively describe the connections between species and multiple reactions in Equation (15). ODEs of abstract chemical reactions in (15) can be obtained by  
\[ \begin{split}&\frac{d[C_{1}]_{t}}{dt}=k_{b1}[X_{a}]_{t}[U^{*}]_{t}- k_{c1}[C_{1}]_{t}\\ &\frac{d[C_{2}]_{t}}{dt}=k_{b2}[X_{d}]_{t}[U]_{t}-k_{c2}[C_{2}]_{t }\\ &\frac{d[C_{3}]_{t}}{dt}=k_{b3}[U]_{t}[X_{a}]_{t}\\ &\frac{d[C_{4}]_{t}}{dt}=k_{b4}[U^{*}]_{t}[C_{3}]_{t}-k_{c4}[C_{4} ]_{t}\\ &\frac{d[U^{*}]_{t}}{dt}=-k_{b1}[X_{a}]_{t}[U^{*}]_{t}+k_{c2}[C_{2 }]_{t}-k_{b4}[U^{*}]_{t}[C_{3}]_{t}\\ &\frac{d[U]_{t}}{dt}=-k_{b2}[X_{d}]_{t}[U]_{t}+k_{c1}[C_{1}]_{t}- k_{b3}[U]_{t}[X_{a}]_{t}+k_{c4}[C_{4}]_{t}\\ &\frac{d[X_{a}]_{t}}{dt}=-k_{b1}[X_{a}]_{t}[U^{*}]_{t}+k_{c1}[C_{1 }]_{t}-k_{b3}[U]_{t}[X_{a}]_{t}\\ &\frac{d[X_{d}]_{t}}{dt}=-k_{b2}[X_{d}]_{t}[U]_{t}+k_{c2}[C_{2}]_{ t}\\ \end{split}  \tag{16}\]  
Figure 9: Graphical representation of Equation (15). Equation (17) can be obtained due to Equation (16). \[ \begin{split}&\frac{d[C_{1}]_{t}}{dt}+\frac{d[C_{3}]_{t}}{dt}+\frac{d[ X_{a}]_{t}}{dt}=0\\ &\frac{d[C_{2}]_{t}}{dt}+\frac{d[X_{d}]_{t}}{dt}=0\\ &\frac{d[C_{1}]_{t}}{dt}+\frac{d[C_{2}]_{t}}{dt}+\frac{d[C_{3}]_{ t}}{dt}+\frac{d[C_{4}]_{t}}{dt}+\frac{d[U^{*}]_{t}}{dt}+\frac{d[U]_{t}}{dt}=0 \end{split}  \tag{17}\]  
Note that the results \((d[C_{1}]_{t}/dt)+(d[C_{3}]_{t}/dt)+(d[X_{a}]_{t}/dt)=0\), \((d[C_{2}]_{t}/dt)+(d[X_{d}]_{t}/dt)=0\) and \((d[C_{1}]_{t}/dt)+(d[C_{2}]_{t}/dt)+(d[C_{3}]_{t}/dt)+(d[C_{4}]_{t}/dt)+(d[U^{* }]_{t}/dt)+(d[U]_{t}/dt)=0\) imply that the total mass \(C_{1}+C_{3}+X_{a}\), \(C_{2}+X_{d}\) and \(C_{1}+C_{2}+C_{3}+C_{4}+U^{*}+U\) is conserved through the lifetime of the process. **Remark 1**. : _The transition in structures from the CMC to CMC-D, then to CMC-I, makes some changes in the results derived from the corresponding abstract chemical reaction networks. The most obvious change is from the reaction \(C_{1}+C_{2}+U^{*}+U\) to \(C_{1}+C_{2}+C_{3}+U^{*}+U\), then to \(C_{1}+C_{2}+C_{3}+C_{4}+U^{*}+U\)._  
3.2 Proposed an Application of the Covalent Modification Cycle with Indirect Positive Autoregulation--BC-IPAR  
The controller BC is constructed utilizing the covalent modification cycle, while a novel controller BC-DPAR can be proposed using the covalent modification cycle with direct positive autoregulation. In this part, combined with the covalent modification cycle with indirect positive autoregulation (depicted in Figure 8), a novel controller BC-IPAR is proposed in the following work. Figure 10 shows the schematic of the BC-IPAR. Compared with the BC-DPAR, species \(U^{*}\) and \(R_{r}\) can be converted to species \(U\) under the action of species \(U\) in the controller BC-IPAR, i.e., the reaction \(U+U^{*}+R_{r}\xrightarrow{\delta_{c}}2U\). CRNs representations of the BC-IPAR can be obtained by  
\[