# The Design of 2 DOF IMC-PID Controller in Biochemical Reaction Networks
## 3 Results
### Building the Division Gate Control System to Restrain Leak Reactions  
It has been verified that the design of the 2 DOF IMC-PID controller is implementable within the framework of CRNs, and, next, the controller will be used to construct a division gate control system to suppress leak reactions. The division gate is established by combining the three operations of addition, subtraction, and multiplication, which achieves the ratio operation of the two biomolecular signals \(A\) and \(B\)[22]. The division gate in Figure 12 is built based on CRNs by Equations (60)-(63). \[A^{\pm}\xrightarrow{n_{1}}A^{\pm}+E^{\pm},W^{\pm}\xrightarrow{n_{1}}W^{\pm}+E^ {\mp} \tag{60}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control System** & **Rise Time** & **Settling Time** & **Overshoot** \\ \hline
2 DOF IMC-PID Control System \(F_{1}\) & 32,188 s, 112,726 s, 192,486 s & 40,165 s, 123,250 s, 202,671 s & 0.51\%, 1.15\%, 0.48\% \\
1 DOF PID Control System \(F_{2}\)[29; 30] & 5601 s, 85,619 s, 165,459 s & 47,740 s, 137,540 s, 212,244 s & 26.8\%, 51.7\%, 27.9\% \\
2 DOF PID Control System \(F_{3}\)[30] & 19,153 s, 99,125 s, 180,331 s & 40,211 s, 125,989 s, 205,643 s & 4.59\%, 10.2\%, 4.72\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance of three control systems \(F_{i}(i=1,2,3)\) in DSD reaction networks. Figure 11: Output responses of three control systems \(F_{i}(i=1,2,3)\). \[E^{\pm}\overset{\mu}{\rightarrow}\varnothing,E^{+}+E^{-}\overset{k_{\text{exp}}}{ \rightarrow}\varnothing \tag{61}\]  
\[E^{\pm}\overset{i_{1}}{\rightarrow}E^{\pm}+C^{\pm},C^{\pm}\overset{i_{2}}{ \rightarrow}\varnothing,C^{+}+C^{-}\overset{k_{\text{exp}}}{\rightarrow}\varnothing \tag{62}\]  
\[C^{\pm}+B^{\pm}\overset{H_{2}}{\rightarrow}C^{\pm}+B^{\pm}+W^{\pm},W^{\pm} \overset{H_{3}}{\rightarrow}\varnothing,W^{+}+W^{-}\overset{k_{\text{exp}}}{ \rightarrow}\varnothing \tag{63}\]  
Equations (60) and (61) represent the error signal \(E\) and set \(\mu_{1}=0.1\) nM/s. Equation (62) produces the calculation result \(C\), and the reaction rates \(i_{1}=0.01\) nM/s and \(i_{2}=1\times 10^{-8}\) nM/s. Equation (63) offers the multiplication result of the output signal \(C\) with the input signal \(B\) and sets \(\mu_{2}=1\) nM/s. For example, the initial concentrations of the species are set to \(\left[A^{+}\right]_{0}=6\) nM and \(\left[B^{+}\right]_{0}=3\) nM. When the initial specie \(A\) is affected by the different extent of the leak reactions, it makes \(\left[A^{+}\right]_{0}<6\) nM and leads to the calculation of the division gate to a smaller \(C<\frac{A}{B}\). Similarly, when the initial specie \(B\) is affected by the different extent of the leak reactions, it makes \(\left[B^{+}\right]_{0}<3\) nM and leads to the calculation of the division gate to a bigger \(C>\frac{A}{B}\). The CRNs of the leak reactions are given as follows. \[A\overset{\gamma}{\rightarrow}\varnothing \tag{64}\]  
\[B\overset{\gamma^{\prime}}{\rightarrow}\varnothing \tag{65}\]  
In Equations (64) and (65), the leak parameters \(\gamma\) and \(\gamma^{\prime}\) reflect the extent to which leak reactions occur. Figure 13 presents the effect of various levels of leak reactions on the computations. As the leak reaction occurs more severely, the error of the calculation result of the divisor gate gets larger. The leak at the dividend leads to small calculations and the leak  
Figure 12: The structure of the division gate. Figure 13: Leak reactions lead to errors in calculation results: (**a**) leak in the initial specie \(A\); (**b**) leak in the initial specie \(B\). **Remark 6**: _Another case is discussed for when the initial species \(A\) and \(B\) leak simultaneously, and how the calculation result of the division gate changes. The calculation of the division gate is correct when the initial species \(A\) and \(B\) leak to the same level, but this is not the ideal result. Once the two species leak to different extents, the calculation result of the division gate will be wrong.