Team:Heidelberg/Modelling/aptakinetics
Assisting the optimization of switchable AptaBodies by mathematical modeling
In our project, several parts are based on the prediction of optimal aptamers for binding target molecules and for designing nucleotide stems in switchable AptaBodies that can sense their targets by creating peroxidase activity based on molecular dynamics simulations. To test predictions of optimal aptamers and nucleotide stems, we had developed a high-throughput method for quickly screening the function and the dose-response of switchable AptaBodies. In 96-wells, powers of ten in concentrations of target ligands are tested if they can accelerate the catalysis of luminol, which is detected by a luminescence camera. Here, we use the decay curves of the luminol signal to assess the function of switchable AptaBodies. In its switched-off state, a functional AptaBody does accelerate the luminol catalysis. Whereas, in its on-state, binding of the target changes the conformation of the hemin-binding quadruplex. This facilitates the binding of hemin, increases the peroxidase activity and therefore accelerates the catalysis of luminol, which is visible by an accelerated decay of the signal.
Assuming an excess of $H_2O_3$ over luminol, the catalysis depends on the concentration of luminol reagent $R$ and the concentration of active enzymes. A low binding affinity of hemin $H$ to the unligated AptaBody $D$ leads to a certain concentration of hemin-bound DNA-AptaBodies $D_H$. In contrast, after binding to the ligand $L$, AptaBodies $D_{LH}$ are supposed to have a stronger affinity for hemin. Because both forms catalyse the luminol reaction, its time-dependent concentration change can be described by the equation
\begin{equation}
\frac{d[R]}{dt}=-k_{cat}[R](D_{H}+D_{LH}),
\end{equation}
in which $k_{cat}$ is the kinetic parameter for the catalysis. This equation can be solved to
\begin{equation}
[R](t)=[R](t_{0})\exp\left(-k_{cat}(D_{H}+D_{LH})t\right).
\end{equation}
Because we are interested in the enzyme activities, this facilitates an elegant way of parameter estimations. Instead of fitting to time series of experimental data, which can be affected by noise and background signal, we only need to extract the decay constant of the signals. For this purpose, we fitted exponential functions
\begin{equation}
f(t)=a+exp(-bt)
\end{equation}
to the fluorescence signals from the AptaBody screening experiments and calculated enzyme activities
\begin{equation}
b=k_{cat}(D_{H}+D_{LH})
\end{equation}
and half-lives of the signal
\begin{equation}
t_{\nicefrac{1}{2}= \frac{ln2}{b}=\frac{\ln2}{k_{cat}(D_{H}+D_{LH})}}.
\end{equation}
These enzyme activities were used for parameter estimations. In the following, the mathematical model shall be described.
Modeling target binding of switchable AptaBodies
To characterize the functionality of combinations of predicted aptamers and stems in switchable Aptabodies, our model describes binding of hemin $H$ and ligands $L$ to DNA-Aptabodies $D$, resulting in $D_L$, $D_H$ and $D_{LH}$. $D_H$ and $D_{LH} possess peroxidase activity and catalyze the luminol reaction (Figure 1). Table 1 shows the model equations. Therein, $k_L$ and $k_{-L}$ describe the binding and unbinding of the ligand $L$, $k_{H,D}$ and $k_{-H,D}$ binding and unbinding of hemin to the unligated AptaBody $D$, and $k_{H,D_L}$ and $k_{-H,D_L} of hemin to the ligand bound AptaBodies. Because a stronger enzymatic activity could be observed in the presence of the ligand, $k_{H,D_L}$ can be assumed to be larger than $k_{H,D}$.
Figure 1. The model for switchable AptaBodies describes ligand and hemin binding and enzyme catalysis. Reversible binding of the target ligand (L) and hemin (H) leads to $D_H$ and $D_{LH}$, which catalyze the reaction of the reporter (R) luminol, which creates the luminescence signal $h\nu$.
Table 1. Model equations describing target binding of switchable AptaBodies
|
\[ \frac{d[D]}{dt}=-k_{L}[L][D]+k_{-L}[D_{L}]-k_{H,D}[H][D]+k_{-H,D}[D_{H}] \] |
(1) |
|
\[ \frac{d[D_{L}]}{dt}=k_{L}[L][D]-k_{-L}[D_{L}]-k_{H,D_{L}}[H][D_{L}]+k_{-H,D_{L}}[D_{LH}] \] |
(2) |
|
\[ \frac{d[D_{H}]}{dt}=-k_{L}[L][D_{H}]+k_{-L}[D_{LH}]+k_{H,D}[H][D]-k_{-H,D}[D_{H}] \] |
(3) |
|
\[ \frac{d[D_{LH}]}{dt}=k_{L}[L][D_{H}]-k_{-L}[D_{LH}]+k_{H,D_{L}}[H][D_{L}]-k_{-H,D_{L}}[D_{LH}] \] |
(4) |
|
\[ \frac{d[L]}{dt}=-k_{L}[L][D]+k_{-L}[D_{L}]-k_{L}[L][D_{H}]+k_{-L}[D_{LH}] \] |
(5) |
|
\[ \frac{d[H]}{dt}=-k_{H}[H][D]+k_{-H}[D_{H}]-k_{H,D_{L}}[H][D_{L}]+k_{-H,D_{L}}[D_{LH}] \] |
(6) |
Because none of the interaction species is in excess, we have to include equations also for $L$ and $H$ and cannot assume constant concentrations. In experiments, we left a time interval of several minutes between the mixture of $H$, $D$ and $L$, and the addition of the luminol reagent $R$ to obtain a steady state before starting the luminol reaction. For parameter estimations, because we were only interested in the binding constants $K_L= k_{-L}/k_{L}$, $K_{H,D}= k_{-H,D}/k_{H,D}$ and $K_{H,D_L}= k_{-H,D_L}/k_{H,D_L}$, we fixed the unbinding parameters $k_{-L}$, $k_{-L}$ and $k_{-L}$ to one, together with preponing the integration start time for the ODE solver relative to the time of luminol addition. This procedure forces the model to a steady state before the addition of luminol. Then, sums of model variable values for $D_H$ and $D_{LH}$ can be fitted to values of experimentally determined enzyme activities. Therein, the parameter $k_{cat}$ is scaling factor between variables and measurements, which is estimated at the same time.
We estimated parameters for three experimentally tested kanamycin aptamers included in AptaBodies, one of them with two different stems, and a literature Aptamer variant. Figure 2A shows that the model can well explain the experimental data. When fitting $k_{H,D}$ and $k_{H,D_L}$ at the same time, we observed that only $k_L$ values were identifiable. Therefore, we decided to only estimate the ratios $K_{H,D}/K_{H,D_L}$. We were particularly interested in these ratios because they reflect the affinity change for hemin binding that is caused by ligand binding. When fitting the ratios together with $k_L$, all parameters were identifiable, which was assessed by profile likelihood estimation (Table 2). In Figure 2B, $K_L=1/k_L$ and the $K_{H,D}/K_{H,D_L}$ ratios are visualized for all tested candidates.
Table 2. Parameter estimates for switchable AptaBody candidates
|
candidate |
parameter |
best fit |
lower CI |
upper CI |
% |
|
CGGGGGT, stem III
|
$k_L $ in $1/(s \muM)$ |
0,0022 |
0,0021 |
0,0023 |
5,6 |
|
$K_{H,D}/K_{H,D_L}$ |
3,3977 |
3,3501 |
3,4524 |
3,0 |
|
|
GCTGTCG, stem II
|
$k_L $ in $1/(s \muM)$ |
0,0035 |
0,00212773 |
0,0034 |
0,0037 |
|
$K_{H,D}/K_{H,D_L}$ |
1,8702 |
3,35011103 |
1,8559 |
1,8772 |
|
|
GCTGTCG, stem III
|
$k_L $ in $1/(s \muM)$ |
0,011 |
0,010 |
0,0034 |
0,0037 |
|
$K_{H,D}/K_{H,D_L}$ |
1,548 |
1,536 |
1,561 |
1,6 |
|
|
CGGGGAT, stem V
|
$k_L $ in $1/(s \muM)$ |
0,0070 |
0,0088 |
0,0121 |
46,7 |
|
$K_{H,D}/K_{H,D_L}$ |
1,4680 |
1,4123 |
1,4482 |
2,4 |
|
|
KAN Aptamer Lit, stem IV |
$k_L $ in $1/(s \muM)$ |
0,0049 |
0,0049 |
0,0059 |
20,7 |
|
$K_{H,D}/K_{H,D_L}$ |
1,2790 |
1,2625 |
1,2818 |
1,5 |
| 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 |
| 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} $ |