Team:ETH Zurich/Modeling/AHL Module
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AHL Module
Introduction and Goals
The present design is meant to respond to the colocalization of cells on the mammalian cell surface therefore, the single model is provided only as a sanity check. First, we checked the validity of our equations with this model. Then, we study here the influence of the degradation and the influence of adding a riboswitch controlling the expression of LuxI. Indeed, due to the dimensions of the chip in which the bacteria and the mammalian cells are located, We expect the signaling molecule to spread really quickly in the well.
Description of the design
When AHL concentration is high enough due to high density of cells, the lux promoter is triggered, which produces more AHL. When the threshold concentration of activated LuxR (LuxRAHL) is reached, GFP production starts. We have here, an irreversible switch.
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 |
Single Cell Model
Reactions
\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}}}_{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}\\ \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
- 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
- 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)
- 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}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-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}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-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.
Observations
We can observe that AiiA degradation has no influence on GFP output. It is because of LuxI accumulation in the cell. Indeed, AiiA acts on AHL level, but because of the leakiness of the promoter, AiiA is unable to counteract production of AHL level by LuxI.
Single cell model with riboregulated LuxR responsive promoter
Adding a riboregulator
We wanted to check the influence of a riboregulator on GFP output. Looking at the characterization from ETHZ 2014, we saw that riboswitches were able to reduce the leakiness by 65 fold. However, as well as reducing the leakiness, riboswitches reduce the final expression levels by 10 fold. As we have two different promoters controlling the expression of LuxI and GFP. We decided to use one riboswitch only for LuxI, to prevent LuxI accumulation in the cell.
Simplified equations
\begin{align*} [LuxRAHL]&= \frac{[AHL]\cdot LuxR_\mathrm{tot}}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=L_{Lux,ribo}+\frac{a_{\mathrm{LuxI,ribo}}(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL,ribo}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{LuxRAHL,ribo}}})^{n_{lux}}}-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}}})^{n_{lux}}}{1+(\frac{[LuxRAHL]}{K_{\mathrm{a,LuxRAHL}}})^{n_{lux}}}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \end{align*}Simulation
Expected limitations and Conclusion
Our system should respond to the colocalization of the e-coli cells on the mammalian cell surface. The different concentrations of e-coli in the bulk and around the mammalian cells will determine the response of the sensor. However, we anticipate the following problems:
- If the concentration of the protein LuxR is too high, the leakiness of the promoter might self activate the sensor too quickly.
- If AHL diffuses in the nano well plate too fast, the fluctuations in the concentration of e coli cells might not be detected by the quorum sensing module.
The reaction-diffusion model is meant to answer these questions.
Click here for reaction diffusion models