Team:SEU/Model
Model
In this part, we provide our computation model in chemical reactions and prove the validity via kinetic analysis.
According to [1], formal reactions such as \(A+B \xrightarrow{k} C\) can be mapped to DNA strand displacement (DSD) reactions [2] without losing the kinetic features of the reaction. We borrow such a model in our project to implement calculation operations.
The whole proof is based on mass action kinetics. The corresponding DSD reaction implementation is given as well. As only addition, subtraction and multiplication are required in our project, we only provide the implementation of such computation. For each calculation operation, we firstly provide formal reactions, then provide kinetic analysis and finally propose DSD reaction implementation.
Addition:
\(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O,... \quad A_n \xrightarrow{k_n} O\)
Proof:
\(\dfrac{d [A_i](t)}{d t}=-k_i[A_i](t) (i=1,2...n)\) \(\Rightarrow [A_i](t)=[A_i](0)e^{-k_it} (i=1,2...n)\), \(\dfrac{d [O](t)}{d t}=\sum_{i=1}^n k_i[A_i](t)\) \(\Rightarrow [O](t)=-(\sum_{i=1}^n[A_i](0)e^{-k_it})+\sum_{i=1}^n[A_i](0)\) \(\Rightarrow [O](\infty)=\sum_{i=1}^n[A_i](0)\). Thus addition is successfully implemented.
The DSD implementation:
Subtraction:
\(A+B \xrightarrow{k_1} \phi\)
Proof:
Apparently, \([A](t)=[B](t)+\Delta \).
If \(\Delta \neq 0\), \(\dfrac{d [A](t)}{d t}=-[A](t)([A](t)-\Delta)\) \(\Rightarrow [A](t)=\dfrac{[A](0)\Delta}{-[A](0)+[A](0)e^{\Delta t}+\Delta e^{\Delta t}} (\Delta \neq 0).\) If \(\Delta > 0\), \([A](\infty)=\Delta\). Otherwise \([A](\infty)=0\).
If \(\Delta =0\), \([A](t)=\dfrac{[A](0)}{1+[A](0)t}\). \([A](\infty)=0\). Hence substraction is implemented.
The DSD implementation:
Multiplication:
\(\alpha \xrightarrow{k_1} \phi, A+B+\alpha \xrightarrow{k_2} A+B+\alpha+C\)
Proof:
\(\dfrac{d [\alpha](t)}{d t}=-k_1[\alpha](t)\) \(\Rightarrow [\alpha](t)=[\alpha](0)e^{-k_1t},\) \(\dfrac{d [A](t)}{d t}=\dfrac{d [B](t)}{d t}=0, \dfrac{d [C](t)}{d t}=k_2[A](t)[B](t)[\alpha](t)\) \(\Rightarrow [C](\infty)=\int_0^\infty [A](0)[B](0)[\alpha](t)=k_2/k_1[\alpha](0)[A](0)[B](0)\). Hence multiplication is implemented.
The DSD implementation:
References
[1] D. Soloveichik, G. Seelig, E. Winfree, "DNA as a universal substrate for chemical kinetics," Proceedings of the National Academy of Sciences, vol. 107, no. 12, pp. 5393–5398, 2010.