Team:ETH Zurich/Modeling/Single-cell Model
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Compartment Model
Introduction
Logic of an AND-gate
In our system we want to reduce the amount the amount of false positives . That’s why cells displaying intermediary characteristics should not be detected by our system. We consider that cells showing increased lactate production rate but do not expose phosphatidylserine, or cells exposing phosphatidylserine but not an increased lactate production rate should not be recognized by our system. We implemented the system to obtain an AND GATE . The system works as two sequential filtering step. The sequential design was used in order to limit the self-activation of the quorum sensing module . Indeed as we have seen in the AHL module, the difference between the two modules strongly depends on the amount of LuxR in the E. coli . This design has a disadvantage though, it requires fine-tuning in order to avoid that one signal prevails on the second one. In the scheme displayed below, we describe in which situation, the E. coli should display fluorescence.
One particularity of our system is that even healthy cells will produce lactate. That is why we implemented a lactate module that works as a fold-change sensor. The fold change sensor will produce a pulse of LuxR. We will study here how the pulsed response influence the output of the system.
Description of the AND-GATE
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. We will compare the output if we use the simple lactate detection system or the fold-change sensor.
The equations are the integration of both modules in one compartment model.
Assumptions
We assume:
- Instant diffusion in the compartments.
- In all the following we assume the same nanowell volume to be 1 nL.
- All the E. coli receive the amount of lactate. This is the worst case scenario since probably the E. coli on the doughnut will sense a higher lactate production.
Equations
The equations are the combination of the compartment model of the AHL module and the lactate module.
The internal states: Lactate, LuxR, LuxRAHL, LuxI have the same ODEs than the single cell model. As already done in the compartment model, we just add convection between the different compartments.
Simulation when the lactate concentration reach the same steady states
Below you will find one example of a comparison between the response of a system including the fold change sensor and the one harboring only the natural detection system for the four different cases described earlier. The lactate inputs give two different delayed pulses of LuxR. The different pulses of LuxR lead to different delay of self-activation of the GFP output. The first GFP output to activate is the one with the cancer cell, the second and third one represent cells with intermediary characteristics. The latest activation time is for the one with no colocalization and no lactate production. With the fold-change sensor, we obtain a significant time difference between the different inputs. With only the natural detection system we do not obtain a clear difference between the different cases. The lactate signal prevails on the quorum sensing signal.
Simulation of the full system with the fold change sensor
Simulation of the full system with the natural lactate detection system
Simulation when the lactate concentration reach different steady states
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 including the fold change sensor
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*}Equations including the natural detection system
\begin{align*} \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}}-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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