While cooperating, it is important to know more about the genetic circuit designed, and many users want to
predict the expression rate with the help of our software. To meet this need, we construct an ODE system to
simulate the dynamic behaviors of the genetic system mathematically. We adopt the Hill functions as the
blueprints of our formula, and construct a general model. After complete their circuit in designer, our
software can create an ODE system automatically.
we use mathematical formula to present general interaction between chemical materials and circuits, like
promotion and inhibition.
If a circuit express $\alpha$ without other inhibition and promotion from other material, we set the basic
Concentration-Time ODEs form of a certain protein $\alpha$ as below:
$$\frac{d[\alpha]}{d_t}=k_\alpha[\alpha_{\textrm{gene}}]-d[\alpha]$$
$K_{\alpha}$ represents the production or translation rate of $\alpha _{\textrm{gene}}$ .
$d$ represents the degradation rate of $\alpha _{\textrm{protein}}$.
$[\alpha _{\textrm{gene}}]$, $[\alpha]$ mean the concentration of $\alpha _{\textrm{gene}}$ and $\alpha _{\textrm{protein}}$.
If a circuit express $\alpha$ with promotion from $\beta$ , we set the Concentration-Time ODEs form as below:
$$\frac{d[\alpha]}{d_t}=k_\alpha[\alpha_{\textrm{gene}}][\beta]-d[\alpha]$$
$k_\alpha$ represents the production or translation rate of $\alpha _{\textrm{gene}}$. $d$ represents the degradation
rate of $\alpha _{\textrm{protein}}$. $[\alpha _{\textrm{gene}}]$, $[\alpha]$, $[\beta]$ mean the concentration of $\alpha
_{\textrm{gene}}$ ,$\alpha _{\textrm{protein}}$ and $\beta_{\textrm{protein}}$
If a circuit express $\alpha$ with inhibition from $\gamma$, we set the concentration-Time ODEs form as blow:
$$\frac{d[\alpha]}{d_t}=k_\alpha[\alpha_{\textrm{gene}}][\frac{1}{1+[\gamma]^{n\gamma}}]-d[\alpha]$$
$K_{\alpha}$ represents the production or translation rate of $\alpha _{\textrm{gene}}$ .
$N_{\gamma}$ represents the $\gamma$ repression coefficient.
$d$ represents the degradation rate of $\alpha _{\textrm{protein}}$.
$[\alpha _{\textrm{gene}}]$, $[\alpha]$ ,$[\gamma]$ mean the concentration of $\alpha _{\textrm{gene}}$ ,$\alpha _{\textrm{protein}}$ and
$\gamma _{\textrm{protein}}$.
If a chemical is influenced by more than one substance, the multi-one relationship is demonstrated as the
product of terms of each factor. For example, if a circuit expression $\alpha$ with inhibition from $\beta$ and
$\gamma$, we set the Concentration-Time ODEs form as blow:
$$\frac{d[\alpha]}{d_t}=k_\alpha[\alpha_{\textrm{gene}}][\frac{1}{1+[\gamma]^{n\gamma}}][\frac{1}{1+[\beta]^{n\beta}}]-d[\alpha]$$
$K_{\alpha}$ represents the production or translation rate of $\alpha _{\textrm{gene}}$ .
$N_{\gamma}$ represents the $\gamma$ repression coefficient.
$N_{\beta}$ represents the $\beta$ repression coefficient.
$d$ represents the degradation rate of $\alpha _{\textrm{protein}}$.
$[\alpha _{\textrm{gene}}]$,$[\alpha]$ ,$[\beta]$ ,$[\gamma]$mean the concentration of $\alpha _{\textrm{gene}}$ ,$\alpha
_{\textrm{protein}}$ ,$\beta _{\textrm{protein}}$,$\gamma _{\textrm{protein}}$
