Team:SEU/Model

Model

Computation method

Addition:

\(A_1 \xrightarrow{k_1} O,\quad A_2 \xrightarrow{k_2} O\)

Proof: \(\dfrac{d [A_i](t)}{d t}=-k_i[A_i](t)\) \(\Rightarrow [A_i](t)=[A_i](0)e^{-k_it}, \) \(\dfrac{d [O](t)}{d t}=\sum_{i=1}^2 k_i[A_i](t)\) \(\Rightarrow [O](\infty)=\int_0^\infty \sum_{i=1}^2 k_i[A_i](t)dt = [A_1](0)/k_1+[A_2](0)/k_2.\) If \(k_1\approx k_2\), then addition is successfully implemented.

The DNA strand displacement implementation:

Subtraction:

\(A+B \xrightarrow{k_1} \phi\)

Proof: It is identical with [1]. 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.

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.

References

[1]C. Fang, Z. Shen, Z. Zhang, X. You and C. Zhang, "Synthesizing a Neuron Using Chemical Reactions," 2018 IEEE International Workshop on Signal Processing Systems (SiPS), Cape Town, 2018, pp. 187-192.

[2]M. Vasic, D. Soloveichik, S. Khurshid, "CRN++: Molecular Programming Language." arXiv preprint arXiv 1809.07430.