Lactate Module

Overview

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 it 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

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.

Current Model

A closer look at the mechanism of lldR

After looking at these(link) puzzling results, we realized that our first model of the mechanism of action of LldR was not realistic. In the literature, we found a compatible explanation, depicted here. In the paper from Aguilera, they suggest that LldR may be required for the transcription machinery. Hence, instead of having only repression by LldR, LldR might play a dual role as a repressor and an activator. It suggests that when Lactate is present, it destabilizes the DNA loop and induces a conformational change of LldR.This results in the transcription of the gene of interest (goi). This mechanism is consistent with our results. In the following, we will describe the mathematical equations corresponding to this mechanism.

Reactions

\begin{align*} \varnothing&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{LldR}}} \text{LldR}\\ 2 \cdot \text{LldR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{K_{\mathrm{d,1}}} \text{LldR}_2\\ 2 \cdot \text{Lact}+\text{LldR}_2 &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{K_{\mathrm{d,2}}}2 \cdot \text{LactLldR}\\ \text{Lact}+\text{LldR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}^{K_{\mathrm{d,3}}}\text{LactLldR}\\ \text{LldR} &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{lldR}}} \varnothing\\ \text{LldR}_2 &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{lldR_2}}} \varnothing\\ \text{LactLldR} &\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LactLldR}}} \varnothing\\ \end{align*}

Mathematical model

According to the previous description, the gene of interest is activated by LactLldR and repressed by LldR dimer. Hence, if gfp is the gene of interest, then we have the following equation for the ODE:

\begin{align*} \frac{d[GFP]}{dt}&=\frac{a_{GFP}}{1+(\frac{[LldR_2]}{K_R})^{n_r}}\cdot \frac{(\frac{[LactLldR]}{K_a})^{n_a}}{1+(\frac{[LactLldR]}{K_a})^{n_a}}-d_\mathrm{GFP}[GFP] \end{align*}

We can now apply the conservation of mass to LldR, we obtain the following:

\begin{align*} \text{LldR}_{tot}&=[LldR]+[LactLldR]+2 \cdot [LldR_2] \end{align*}

Simplification

The previous equations are anyway unidentifiable for us, that is why we are going to simplify the system by approximating the transcription of gfp by an Hill activation function:

\begin{align*} \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP} \cdot (\frac{[Lact]}{K_\mathrm{lact}})^{n}}{1+(\frac{[Lact]}{K_\mathrm{lact}})^{n}}-d_\mathrm{GFP}[GFP]\\ \end{align*}

Full module simplified model

Chemical species

Name	Description
Lactout	Lactate produced by mammalian cells
Lactin	Lactate inside E. coli cells
LacI	Lac operon 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{Lact}_{\text{out}}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M,LldP}},v_\mathrm{LldP}} \text{Lact}_{\text{in}}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}_{in}}} \text{LacI}\\ &\mathop{\xrightarrow{\hspace{4em}}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}_{in}}} \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*}

Equations for full module

Consistent with the simplification describes in the previous section, we derive these equations:

\begin{align*} \frac{d[Lact_{in}]}{dt}&= \frac{v_\mathrm{LldP} \cdot \frac{[Lact_\mathrm{out}]}{K_\mathrm{M,LldP}}}{1+\frac{[Lact_\mathrm{out}]}{K_\mathrm{M,LldP}}}- d_\mathrm{Lact_{in}}[Lact_\mathrm{in}]\\ \frac{d[LacI]}{dt}&=\frac{a_\mathrm{LacI}\cdot (\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}{1+(\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}\cdot (\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}}{1+(\frac{[Lact_{in}]}{K_{\mathrm{A,Lact}}})^{n_\mathrm{1}}} \cdot \frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}})^{n_\mathrm{2}}}-d_{\mathrm{GFP}}[GFP]\\ \end{align*}

Initial model and predictive modeling

Overview

In the following we describe our initial model. Thanks to that model, we were able to make decisions concerning the design of our system. We also derived precise functional specifications for our system.

How did we derive the model ?

In this system, the mechanism of action of LacI is well known, whereas the action of LldR and Lactate is not. Therefore, we derived the model for the mechanism of LldR by analogy to similar metabolic pathway. The paper from [Aguilera 2008], indicates that members of FadR family, including LldR, are highly similar. For example, GntR binds to two operator sites to negatively regulate the transcription of the gntT gene. Total repression of gntT was suggested to be achieved by DNA looping through interaction between the two GntR molecules. From this, we assumed that :

• LldR exists as a dimer in solution.
• 2 molecules of Lactate bind to one LldR dimer (L2).
• Lldr dimer bind to the two operator sites when no LldR is present.
• Lactate releases the binding of LldR dimer to the operators.

Chemical species

Name	Description
Lactout	Lactate produced by mammalian cells
Lactin	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 operon 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{Lact}_{\text{out}}&\mathop{\xrightarrow{\hspace{4em}}}^{K_{\mathrm{M,LldP}},v_\mathrm{LldP}} \text{Lact}_{\text{in}}\\ 2 \cdot \text{Lact}_{\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*}

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{[Lact_\mathrm{in}]^2}{K_\mathrm{d,DLL}}+1}\\ \frac{d[LacI]}{dt}&=\frac{a_{\mathrm{LacI}}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^{n_1}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[GFP]}{dt}&=\frac{a_\mathrm{GFP}}{1+(\frac{[L_2]}{K_{\mathrm{R,L}}})^{n_1}}*\frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}})^{n_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{Lact_\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^{n_1}}-l_1\\ \frac{dl_2}{d\tau}&=\frac{b_1}{1+l_0^{n_1}}\cdot \frac{1}{1+(\frac{l1}{\gamma_2 +1})^{n_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*}

Initial States

Every time, we set the initial states of our model to be the steady states when only some Lactate in the medium.

Characteristics of the system

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

In order to satisfy this condition, we need:

\begin{align*} n_2&=1 \end{align*}

If we apply the two necessary conditions in the MATLAB model, we obtain a perfect fold change sensor.

The Fold change represents the fold change between different production rates between mammalian cells. Hence, Fold Change =1 represents the production of a normal cell and the other curves represent potential production of cancer cells.

Observation: The fold change pulse will probably be too short, and the basal level of GFP is probably too high with this system.

Our system specifications

We want a system that amplifies the difference in production rates between cancer and normal cells. One example of such behavior is the following:

Here we can observe that for a fold change of 5 for the input, we obtain a 200 fold change at the output. We have amplified the response compared to the input, but also compared to the natural Lactate sensor (fold change in the response is about 15). In the next section, we will discuss the influence of the parameters on that ratio.

Parameter search

Using the literature and our own estimations, we estimated a reasonable range of parameters in which we think the set of parameters of the biological system is located. To see the range of parameters used: click here.

According to our specifications, we want to amplify the signal difference between cancer and normal cells' production of Lactate. That's why our objective function is to maximize the following ratio:

\begin{align*} \frac{\text{GFP}_\mathrm{\text{ss,Cancer}}}{\text{GFP}_\mathrm{\text{ss,Normal}}} \end{align*}

To obtain the following figure, we had first to compute the optimal parameters in the ranges chosen. The set of optimal parameters was obtained thanks to constrained non-linear optimization.We then computed the cost for every pair of parameters on 2D grid, fixing the other parameters to their optimal values.

What do the variables represent?

• \(\gamma_1\) represents the repression by LldR.
• \(\alpha \cdot \sqrt{B}\) represents the production of Lactate by a normal cell.
• \(\frac{a_1}{\gamma_2 +1}\) represents the repression by LacI.\(\gamma_2\) represents the equivalent amount of IPTG. So the more we increase \(\gamma_2\), the more we reduce the amount of active LacI in the cell.
• \(n_1\) is the Hill coefficient of LldR.
• \(n_2\) is the Hill coefficient of LacI.

Observations

From this figure, we can make the following observations:

• If we increase \(\gamma_1\) then we increase the range where our system show high amplification.
• If we increase \(\frac{a_1}{\gamma_2 +1}\) then we increase the range of possible values for \(\gamma_1\) .
• \(n_1\) has a strong influence on the output of the system. If the system is more cooperative, then the ratio can reach really high values. Whereas, \(n_2\) does not have a clear influence at least on the ranges of parameters chosen.