Team:SEU/Model
Model
In this part, we provide our computation model in chemical reactions and prove the validity via kinetic analysis.
In our system, numerical values are represented by concentrations of certain species. Chemical reactions are simplified as formal reactions such as \(A+B \xrightarrow{k} C\). A formal reaction consists of reactants, products (\(A,B and C\)) and rate constant (\(k\)). For each computation operation, given initial concentrations of certain species as inputs, the outputs are representd by the concentration of another species in the system.
According to [1], formal reactions 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.
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. The analysis in our model is totally based on classic mass action kinetics.
Addition:
To calculate the sum of several values and represent the value by concentration of another species in our system, we utilize:
(\(\sum_{i=1}^n[A_i](0)\)) \(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O,... \quad A_n \xrightarrow{k_n} O\). It calculates \([O](\infty)=\sum_{i=1}^n[A_i](0)\).
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.