Difference between revisions of "Team:ETH Zurich/Modeling/AHL Module"

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<tr> <td>Aiia </td> <td>AHL-lactonase, N-Acyl Homoserine Lactone Lactonase </td> </tr>
 
<tr> <td>Aiia </td> <td>AHL-lactonase, N-Acyl Homoserine Lactone Lactonase </td> </tr>
 
</table>
 
</table>
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<h2>Reactions</h2>
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\begin{align*}
 
\begin{align*}
 
\varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LuxR}}} \text{LuxR}\\
 
\varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LuxR}}} \text{LuxR}\\
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     <li>As LuxR is constitutively produced in this model, we considered that LuxR was constant. We used then the conservation of mass to derive the equations</li>
 
     <li>As LuxR is constitutively produced in this model, we considered that LuxR was constant. We used then the conservation of mass to derive the equations</li>
 
     <li>On the other hand, the binding and unbinding of LuxR to AHL is fast compared to the synthesis of LuxI. We used the quasi steady state approximation (QSSA)</li>
 
     <li>On the other hand, the binding and unbinding of LuxR to AHL is fast compared to the synthesis of LuxI. We used the quasi steady state approximation (QSSA)</li>
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    <li>In this model we considered that no AHL diffuses out of the cell. </li>
 
</ol>
 
</ol>
 
<h3>Simplified equations</h3>
 
<h3>Simplified equations</h3>
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K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\
 
K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\
 
\end{align*}
 
\end{align*}
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<h2>Simulations</h2>
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<p> We decided to simulate the single cell model in order to check the influence of AHL degradation by AiiA, and to confirm the sequential design.</p>
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Revision as of 06:56, 25 August 2015

"What I cannot create I do not understand."
- Richard Feynmann

AHL Module

Introduction

Chemical species

Name Description
AHL Signaling protein, Acyl homoserine lactone (30C6-HSL)
LuxR Regulator protein, that can bind to AHL to form a complex
LuxRAHL Complex of LuxR and AHL, activates transcription of LuxI
LuxI Autoinducer synthase
Aiia AHL-lactonase, N-Acyl Homoserine Lactone Lactonase
\begin{align*} \varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LuxR}}} \text{LuxR}\\ \text{AHL} + \text{LuxR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{LuxRAHL}}}^{k_{\mathrm{-LuxRAHL}}} \text{LuxRAHL}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\ \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{AHL}}}\text{AHL}+\text{LuxI}\\ \text{LuxR}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxR}}}\varnothing\\ \text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{AHL}}}\varnothing\\ \text{LuxRAHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxRAHL}}}\varnothing\\ \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxI}}}\varnothing\\ \text{Aiia}+\text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M}},v_{\mathrm{Aiia}}}\text{Aiia}\\ \end{align*}

Equations

The equations are based on mass action kinetics and basic rate laws such as Hill equation.

Assumptions

  1. As LuxR is constitutively produced in this model, we considered that LuxR was constant. We used then the conservation of mass to derive the equations
  2. On the other hand, the binding and unbinding of LuxR to AHL is fast compared to the synthesis of LuxI. We used the quasi steady state approximation (QSSA)
  3. In this model we considered that no AHL diffuses out of the cell.

Simplified equations

\begin{align*} [LuxRAHL]&= \frac{[AHL]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^2}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]-\frac{v_\mathrm{Aiia}\cdot [AHL]}{K_{\mathrm{M,AiiA}}+[AHL]}\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}(LuxR_\mathrm{tot}-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^2}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \end{align*}

Simulations

We decided to simulate the single cell model in order to check the influence of AHL degradation by AiiA, and to confirm the sequential design.

CAPTION TEXT