Difference between revisions of "Team:ETH Zurich/Modeling/Single-cell Model"

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<p>In this section, we describe the behaviour of the combined model. </p>
 
<p>In this section, we describe the behaviour of the combined model. </p>
 
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<h2> Combined Compartment Model </h2>
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<h3> Overview</h3>
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<p> In this model we plan to simulate whether our system can work as an AND-GATE. Therefore we simulated the system using compartment to model the density of the <i> E. coli </i> as already explained in the <a href="https://2015.igem.org/Team:ETH_Zurich/Modeling/AHL_Module#Compartment_Model">AHL module</a>. </p>
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<p> First, we will simulate the model with no amplification of the lactate input to match our experimental results. In a second part, we will simulate the full model.
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As <a href="https://2015.igem.org/Team:ETH_Zurich/Modeling/AHL_Module#Simulation">already done</a> in the AHL module, we will compare three conditions:</p>
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<li>No degradation by AiiA, no riboregulator.</li>
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<li>Degradation of AHL by AiiA.</li>
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<li>Riboregulator controlling LuxI expression.</li>
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<h3> Results</h3>
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<p> These equations are the integration of both modules in one compartment model. <p>
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<h4> Assumptions</h4>
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<p> Here we will assume that .</p>
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<h4> Equations</h4>
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<h2> Single cell model </h2>  
 
<h2> Single cell model </h2>  
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</a>
 
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<h3> Simulation</h3>
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<p> Here, we expect to find the features from the two models: <p>
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Revision as of 15:13, 14 September 2015

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

Combined Model

Introduction

Genetic design

In this section, we describe the behaviour of the combined model.

Combined Compartment Model

Overview

In this model we plan to simulate whether our system can work as an AND-GATE. Therefore we simulated the system using compartment to model the density of the E. coli as already explained in the AHL module.

First, we will simulate the model with no amplification of the lactate input to match our experimental results. In a second part, we will simulate the full model. As already done in the AHL module, we will compare three conditions:

  1. No degradation by AiiA, no riboregulator.
  2. Degradation of AHL by AiiA.
  3. Riboregulator controlling LuxI expression.

Results

These equations are the integration of both modules in one compartment model.

Assumptions

Here we will assume that .

Equations

Single cell model

Overview

The single cell model is provided here to simulate the combined model.

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
Lact Lactate
LacI Lac operon repressor, DNA-binding protein, acts as a protein
IPTG Isopropyl β-D-1-thiogalactopyranoside, prevents LacI from repressing the gene of interest
IL Dimer formed between LacI and IPTG

Reactions

\begin{align*} &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LacI},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LacI}\\ \text{IPTG} + \text{LacI} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{IL}}}^{k_{\mathrm{-IL}}} \text{IL}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LuxR}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{R,LacI}}^{\displaystyle\mathop{\bot}^{\text{LacI}}} \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}}}_{a_\mathrm{LuxI},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{GFP},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{GFP}\\ \end{align*} \begin{align*} \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

Combining all of the equations from the two different modules, it yields the following system:

\begin{align*} \frac{d[LacI]}{dt}&=\frac{a_\mathrm{LacI} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[LuxR]}{dt}&=\frac{a_\mathrm{LuxR} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}} \cdot \frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}\cdot (\gamma_2+1)})^{n_\mathrm{2}}}-d_{\mathrm{LuxR}}[LuxR]\\ [LuxRAHL]&= \frac{[AHL]\cdot [LuxR]}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}([LuxR]-[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]-[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}}\\ \gamma_2 &= \frac{IPTG_{tot}}{K_{IL}} \end{align*}

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