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

Figure 1: Topology of the Lactate sensor

The network depicted here is equivalent to an incoherent feed forward loop since Lactate activates production of LacI and GFP.

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. Because of the AND gate implemented, 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.

In order to allow fine tuning of the levels of active LacI inside the cells, we decided to add IPTG to the model.

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

The model displays perfect fold change behaviour when the steady state of GFP does not depend on the input Lactate. In order to do so, we need to supress all the saturation terms and then :

\begin{align*} [LldR]&\propto \frac{1}{[Lac]^2}\\ [LacI]&\propto \frac{1}{(\frac{1}{[Lac]^2})^{n_1}}\\ [GFP]&\propto \frac{1}{(\frac{1}{[Lac]^2})^{n_1}} \cdot \frac{1}{[LacI]^{n_2}}\\ [GFP]&\propto \frac{[Lac]^{2\cdot n_1}}{[Lac]^{2\cdot n_1 \cdot n_2}} \end{align*}

In order to satisfy this condition, we need:

\begin{align*} n_1&=1 \end{align*}