Team:Heidelberg/Modelling/aptakinetics
Table 1:Model equations describing target binding of switchable AptaBodies
\begin{align}
\frac{d[D]}{dt}&=-k_{L}[L][D]+k_{-L}[D_{L}]-k_{H,D}[H][D]+k_{-H,D}[D_{H}]\\
\frac{d[D_{L}]}{dt}&=k_{L}[L][D]-k_{-L}[D_{L}]-k_{H,D_{L}}[H][D_{L}]+k_{-H,D_{L}}[D_{LH}]\\
\frac{d[D_{H}]}{dt}&=-k_{L}[L][D_{H}]+k_{-L}[D_{LH}]+k_{H,D}[H][D]-k_{-H,D}[D_{H}]\\
\frac{d[D_{LH}]}{dt}&=k_{L}[L][D_{H}]-k_{-L}[D_{LH}]+k_{H,D_{L}}[H][D_{L}]-k_{-H,D_{L}}[D_{LH}]\\
\frac{d[L]}{dt}&=-k_{L}[L][D]+k_{-L}[D_{L}]-k_{L}[L][D_{H}]+k_{-L}[D_{LH}]\\
\frac{d[H]}{dt}&=-k_{H}[H][D]+k_{-H}[D_{H}]-k_{H,D_{L}}[H][D_{L}]+k_{-H,D_{L}}[D_{LH}]
\end{align}
Table 3:Stepwise changes from the basic model to the optimal variant 4 and to simplifications of variant 4 to variants 4a to 4c
| Model variant | Subsequent modifications relative to basic model or previous variant | Changes in fitting quality |
|---|---|---|
| 1 | Michaelis-Menten instead of linear kinetics for active template | no improvement |
| 2 | Individual $k_{syn}$ and $n_A$ values for different polymerase concentrations | improvement |
| 3 | $n_A$ depends on function of $T_{act}$ and $A$ $n_A=n_{A,0} A^{k} /T_{act}^{l}$ |
improvement, $k\approx0$ |
| 4, best model | Setting $k=0$ | improvement |
| 4a | No degradation of P in variant 4 | decrease |
| 4b | No degradation of A in variant 4 | decrease |
| 4c | Binding of $P$ to $T$ in steady state in variant 4 | decrease |
Table 4:Model equations for the basic model and Variants 1 to 4c
| Model species | Variant | Equation |
|---|---|---|
| $P$ | Basic model Variants 1 to 4, 4c |
$ \frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]-k_{deg,P}[P] $ |
| Variant 4a | $ [P](t)=[P](t_{0})\exp\left(-k_{deg,P}t\right) $ | |
| Variant 4b | $ \frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}] $ | |
| $T$ | Basic model Variants 1 to 4, 4b, 4c |
$ \frac{d[T]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}] $ |
| Variant 4a | $ [T]=[T_{tot}]-[T_{act}] $ | |
| $T_{act}$ | Basic model Variants 1 to 4, 4b, 4c |
$ \frac{d[T_{act}]}{dt}=k_{on}[T][P]-k_{off}[T_{act}] $ |
| Variant 4a | $ [T_{act}]=\frac{[T_{tot}][P]}{K_{d,P}} $ | |
| $A$ | Basic model Variants 2 to 4, 4a, 4b |
$ \frac{d[A]}{dt}=-k_{syn}[A][T_{act}]-k_{deg,A}[A] $ |
| Variant 1 | $ \frac{d[A]}{dt}=-k_{syn}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]}-k_{deg,A}[A] $ | |
| Variant 4c | $ \frac{d[A]}{dt}=-k_{syn}[A][T_{act}] $ | |
| $M$ | Basic model Variant 2 |
$ \frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}[A][T_{act}] $ |
| Variant 1 | $ \frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]} $ | |
| Variant 3 | $ \frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]^{k}}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[A]^{1-k}[T_{act}]^{1+j} $ | |
| Variants 4, 4a, 4b, 4c | $ \frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[T_{act}]^{1+j} $ |