Team:ETH Zurich/Modeling/Lactate Module
- Project
- Modeling
- Lab
- Human
Practices - Parts
- About Us
Lactate Module
Single cell model
The initial idea was to measure the lactate production rate. We assumed that on our timeframe, the lactate production might not reach steady state. Because of that, we could not base our design on absolute values but rather on relative values between cancer and normal cells. That is why our sensor has the topology of a fold change sensor, so that he could be able to measure the fold change in the lactate production rates. However due to the topology of the natural detection system of lactate in E coli. Our system does not behave as a fold change sensor but rather amplifies the difference between the production of cancer and normal cells.
Description of the design
In the absence of lactate, LLdR, the regulatory protein binds to the promoter and represses transcription of LacI, but also represses transcription of GFP. When Lactate is present, Lactate binds to LldR. Thus both transcription of LacI and GFP are initiated. When LacI reaches a certain threshold, LacI represses transcription of GFP after a certain delay. Both LacI and Lldr should be absent in order to have transcription of GFP. In the correct set of parameter space, this can work as a fold change sensor. In our system, if repression of LacI is less strong it will repress transcription of GFP for low levels of lactate and allow GFP transcription for high levels of lactate.
Chemical species
| Name | Description |
|---|---|
| Lacout | Lactate produced by mammalian cells |
| Lacin | Lactate inside E. coli cells |
| L2 | Dimer of LldR, regulatory protein of lld operon, acts as a repressor |
| DLL | Dimer formed between Lactate and LLdr dimer |
| LacI | Lac 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 |
| LldP | Lactate permease, active transporter |
Reactions
\begin{align*} \text{Lac}_{\text{out}}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{m,p}},v_\mathrm{max,p}} \text{Lac}_{\text{in}}\\ 2 \cdot \text{Lac}_{\text{in}} + \text{L}_{2} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{DLL}}}^{k_{\mathrm{-DLL}}} \text{DLL}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{L}_2}} \text{LacI}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{L}_2}} \text{GFP}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\bot}^{\text{LacI}}} \text{GFP}\\ \text{IPTG} + \text{LacI} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{IL}}}^{k_{\mathrm{-IL}}} \text{IL}\\ \end{align*}Initial model and predictive modeling
Equations
Assumptions
- We used the quasi steady state approximation to model the fast dimerization of Lactate to L2 and of IPTG to LacI.
- We assumed that the Hill coefficient for Lldr was equal to two, since two Lactate molecules bind to one dimer of Lldr. The Hill coefficient for LacI was also set to two.
Equations
\begin{align*} [L_2] &=\frac{L_\mathrm{2tot}}{\frac{[Lac_\mathrm{in}]^2}{K_\mathrm{d,DLL}}+1}\\ \frac{d[LacI]}{dt}&=\frac{a_{\mathrm{LacI}}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^2}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^2}*\frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}})^2}-d_{\mathrm{GFP}}[GFP]\\ \end{align*}Non dimensionalized equations
For the initial model, we chose to model the input of lactate as a step input. We non-dimensionalized the system in order to simplify the system
\begin{align*} l_0&= [ \tilde{L_2}]=\frac{[L_2]}{K_{\mathrm{R,L}}}\\ l_1&=[\tilde{LacI}]=\frac{[LacI]}{K_{\mathrm{R,LacI}}}\\ l_2&=[GFP]\\ \tau &=d_{\mathrm{LacI}}\cdot t\\ B&=\frac{Lac_\mathrm{initial}^2}{K_\mathrm{d,DLL}}\\ l_0 &=\frac{\gamma_1}{F_c^2 \cdot \alpha^2 \cdot B+1}\\ \frac{dl_1}{d\tau}&=\frac{a_1}{1+l_0^2}-l_1\\ \frac{dl_2}{d\tau}&=\frac{b_1}{1+l_0^2}\cdot \frac{1}{1+(\frac{l1}{\gamma_2 +1})^2}-b_2l_2\\ a_1&=\frac{a_\mathrm{LacI}}{d_\mathrm{LacI}\cdot K_{RLacI}}\\ b_1 &= \frac{a_\mathrm{GFP}}{d_\mathrm{LacI}}\\ b_2&= \frac{d_\mathrm{GFP}}{d_\mathrm{LacI}}\\ K_\mathrm{d,DLL} &= \frac{k_\mathrm{-DLL}}{k_\mathrm{DLL}}\\ \gamma_1 &= \frac{L_\mathrm{2tot}}{K_\mathrm{R,L}}\\ \gamma_2&=\frac{IPTG_\mathrm{tot}}{K_\mathrm{d,IL}}\\ \end{align*}