Team:ETH Zurich/Modeling/Lactate Module
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Lactate Module
Introduction
The initial idea was to implement a fold change sensor in order to measure the lactate production rate. We assumed that on our timeframe, the lactate production might not reach steady state. That is why our sensor has the topology of a fold change sensor. However due to the topology of the natural detection system 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.
Chemical species
| Name | Description |
|---|---|
| Lacout | Lactate produced by mammalian cells |
| Lacin | Lactate inside E. coli cells |
| L2 | 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{DLL}}+1}\\ \frac{d[LacI]}{dt}&=\frac{a_{\mathrm{LacI}}}{1+(\frac{[L_2]}{K_{\mathrm{RL}}})^2}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}}{1+(\frac{[L_2]}{K_{\mathrm{RL}}})^2}*\frac{1}{1+(\frac{[LacI]}{K_{\mathrm{RLacI}}})^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{RL}}}\\ l_1&=[\tilde{LacI}]=\frac{[LacI]}{K_{\mathrm{RLaci}}}\\ l_2&=[GFP]\\ \tau &=d_{\mathrm{LacI}}\cdot t\\ B&=\frac{Lac_\mathrm{initial}^2}{K_\mathrm{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_{Laci}}{d_{Laci}\cdot K_{RLaci}}\\ b_1 &= \frac{a_{gfp}}{d_{Laci}}\\ b_2&= \frac{d_{gfp}}{d_{Laci}}\\ K_\mathrm{DLL} &= \frac{k_\mathrm{-DLL}}{k_\mathrm{DLL}}\\ \gamma_1 &= \frac{L_\mathrm{2tot}}{K_\mathrm{RL}}\\ \gamma_2&=\frac{IPTG_t}{K_\mathrm{IL}}\\ \end{align*}