Lactate Module

Single cell model

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

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