Team:ETH Zurich/Modeling/Lactate Module

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

Lactate Module

Single cell model

Figure 1: Lactate sensor design

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 absc

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

  1. We used the quasi steady state approximation to model the fast dimerization of Lactate to L2 and of IPTG to LacI.
  2. 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*}

Characteristics of the model

Fold change behaviour