Difference between revisions of "Team:Heidelberg/Modelling/rtsms"

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RTSMS
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<html>
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<head>
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<title></title>
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</head>
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<body>
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<div class="container">
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<div class="content">
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<p>
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<strong>Studying determinants of polymerase efficiency based on an aptamer sensor</strong></p>
 +
<p>
 +
Table 3 shows iterative steps, which lead to the refined model version &nbsp;4. Further modifying this variant to the models 4a, 4b or 4c did not improve fit quality.</p>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
Table 3. Stepwise changes from the basic model to the optimal variant 4 and to simplifications of variant 4 to variants 4a to 4c</p>
 +
<table border="1" cellpadding="0" cellspacing="0">
 +
<tbody>
 +
<tr>
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<td style="width:92px;height:60px;">
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<p>
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Model variant</p>
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</td>
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<td style="width:350px;height:60px;">
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<p>
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Subsequent modifications relative to basic model or previous variant</p>
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</td>
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<td style="width:200px;height:60px;">
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<p>
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Changes in fitting quality</p>
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</td>
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</tr>
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<tr>
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<td style="width:92px;">
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<p>
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1</p>
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</td>
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<td style="width:350px;">
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<p>
 +
Michaelis-Menten instead of linear kinetics for active template</p>
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</td>
 +
<td style="width:200px;">
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<p>
 +
no improvement</p>
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</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
2</p>
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</td>
 +
<td style="width:350px;">
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<p>
 +
Individual $k_{syn}$ and $n_A$ values for different polymerase concentrations</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
improvement</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
3</p>
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</td>
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<td style="width:350px;">
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<p>
 +
$n_A$ depends on function of $T_{act}$ and $A$</p>
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<p>
 +
$n_A=n_{A,0} A^{k} /T_{act}^{l}$</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
improvement, $k\approx0$</p>
 +
<p>
 +
&nbsp;</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
4, best model</p>
 +
</td>
 +
<td style="width:350px;">
 +
<p>
 +
Setting $k=0$</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
improvement</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
4a</p>
 +
</td>
 +
<td style="width:350px;">
 +
<p>
 +
No degradation of P in variant 4</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
decrease</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
4b</p>
 +
</td>
 +
<td style="width:350px;">
 +
<p>
 +
No degradation of A in variant 4</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
decrease</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:92px;">
 +
<p>
 +
4c</p>
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</td>
 +
<td style="width:350px;">
 +
<p>
 +
Binding of $P$ to $T$ in steady state in variant 4</p>
 +
</td>
 +
<td style="width:200px;">
 +
<p>
 +
decrease</p>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<p>
 +
&nbsp;</p>
 +
<br clear="all" />
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
Table 4. Model equations for the basic model and Variants 1 to 4c</p>
 +
<table border="1" cellpadding="0" cellspacing="0">
 +
<tbody>
 +
<tr>
 +
<td style="width:55px;">
 +
<p>
 +
Model species</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Variant</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
Equation</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td rowspan="3" style="width:55px;">
 +
<p>
 +
$P$</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Basic model</p>
 +
<p>
 +
Variants 1 to 4, 4c</p>
 +
</td>
 +
<td style="width:436px;">
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<p>
 +
\[</p>
 +
<p>
 +
\frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]-k_{deg,P}[P]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 4a</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
[P](t)=[P](t_{0})\exp\left(-k_{deg,P}t\right)</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 4b</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td rowspan="2" style="width:55px;">
 +
<p>
 +
$T$</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Basic model</p>
 +
<p>
 +
Variants 1 to 4, 4b, 4c</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[T]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 4a</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
[T]=[T_{tot}]-[T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td rowspan="2" style="width:55px;">
 +
<p>
 +
$T_{act}$</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Basic model</p>
 +
<p>
 +
Variants 1 to 4, 4b, 4c</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[T_{act}]}{dt}=k_{on}[T][P]-k_{off}[T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 4a</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
[T_{act}]=\frac{[T_{tot}][P]}{K_{d,P}}</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td rowspan="3" style="width:55px;">
 +
<p>
 +
$A$</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Basic model</p>
 +
<p>
 +
Variants 2 to 4, 4a, 4b</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[A]}{dt}=-k_{syn}[A][T_{act}]-k_{deg,A}[A]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 1</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\frac{d[A]}{dt}=-k_{syn}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]}-k_{deg,A}[A]</p>
 +
<p>
 +
&nbsp;</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 4c</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[A]}{dt}=-k_{syn}[A][T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td rowspan="4" style="width:55px;">
 +
<p>
 +
$M$</p>
 +
</td>
 +
<td style="width:151px;">
 +
<p>
 +
Basic model,</p>
 +
<p>
 +
Variant 2</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}[A][T_{act}]</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variants 1</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]}</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variant 3</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]^{k}}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[A]^{1-k}[T_{act}]^{1+j}</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:151px;">
 +
<p>
 +
Variants 4, 4a, 4b, 4c</p>
 +
</td>
 +
<td style="width:436px;">
 +
<p>
 +
\[</p>
 +
<p>
 +
\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[T_{act}]^{1+j}</p>
 +
<p>
 +
\]</p>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<p>
 +
&nbsp;</p>
 +
<br clear="all" />
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
Table 5. Parameter estimates for switchable AptaBody candidates</p>
 +
<table align="left" border="1" cellpadding="0" cellspacing="0">
 +
<tbody>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
parameter</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
best fit</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
lower CI</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
upper CI</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
%</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{deg,A}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,00001055</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,000009087</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,00001157</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
23,5</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{deg,P}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,000307</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,0002983</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,0003017</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
1,1</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{on,P}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,002639</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,0007579</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,0007629</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,2</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{off,P}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,002742</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,003122</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,003196</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
2,7</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
l</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,09085</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,07886</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,07999</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
1,2</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{syn,M,P=1c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,06863</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,2054</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,2060</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,9</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
n_{A_0,P=1c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
126,67</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
126,4511826</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
127,8631277</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
3,2</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{syn,M,P=0.5c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,02570</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,25683523</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,257400898</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,7</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
n_{A_0,P=0.5c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
425,8</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
421,0343254</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
428,7790799</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
5,8</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
k_{syn,M,P=0.1c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,5591</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
1,930027572</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
1,932774173</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
0,5</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="width:143px;">
 +
<p>
 +
n_{A_0,P=0.1c_0}</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
293,5</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p align="right">
 +
159,0489132</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
Infinity</p>
 +
</td>
 +
<td style="width:98px;">
 +
<p>
 +
Infinity</p>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
&nbsp;</p>
 +
<p>
 +
&nbsp;</p>
 +
</div>
 +
</div>
 +
</body>
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Revision as of 03:55, 19 September 2015


Studying determinants of polymerase efficiency based on an aptamer sensor

Table 3 shows iterative steps, which lead to the refined model version  4. Further modifying this variant to the models 4a, 4b or 4c did not improve fit quality.

 

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}]

\]

Variants 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}

\]

 


 

Table 5. Parameter estimates for switchable AptaBody candidates

parameter

best fit

lower CI

upper CI

%

k_{deg,A}

0,00001055

0,000009087

0,00001157

23,5

k_{deg,P}

0,000307

0,0002983

0,0003017

1,1

k_{on,P}

0,002639

0,0007579

0,0007629

0,2

k_{off,P}

0,002742

0,003122

0,003196

2,7

l

0,09085

0,07886

0,07999

1,2

k_{syn,M,P=1c_0}

0,06863

0,2054

0,2060

0,9

n_{A_0,P=1c_0}

126,67

126,4511826

127,8631277

3,2

k_{syn,M,P=0.5c_0}

0,02570

0,25683523

0,257400898

0,7

n_{A_0,P=0.5c_0}

425,8

421,0343254

428,7790799

5,8

k_{syn,M,P=0.1c_0}

0,5591

1,930027572

1,932774173

0,5

n_{A_0,P=0.1c_0}

293,5

159,0489132

Infinity

Infinity