Compartment Model

Introduction and Goals

Figure 1. Full system genetic design. Lactate induces the lactate sensor. The output of the lactate sensor is LuxR. LuxR through initiation of the leakiness triggers the quorum sensing module. From this point, the E. coli cells are able to sense each other via AHL signaling

The combined compartment model studies the connections between the lactate module and the AHL module. The lactate sensor's output is LuxR. LuxR is the input of the AHL module. The system seems sequential but the actual input of the AHL module is the signaling molecule AHL. LuxR only induces leaky expression of the signaling molecule. However to activate the AHL module, a significant density of E. coli has to be reached. That is why the system should work as an AND gate.

AND gate

In our system we want to reduce the amount of false positives . That’s why cells displaying intermediary characteristics should not be detected by our system. We consider that cells showing increased lactate production rate but do not expose phosphatidylserine, or cells exposing phosphatidylserine but not an increased lactate production rate should not be recognized by our system. We implemented the system to obtain an AND gate. The system works as two sequential filtering step. The sequential design was used in order to limit the self-activation of the quorum sensing module. Indeed as we have seen in the AHL module, the difference between the two modules strongly depends on the amount of LuxR in the E. coli . This design has a disadvantage though, it requires fine-tuning in order to avoid that one signal prevails on the second one. In the scheme shown below, we describe in which situation, the E. coli should fluoresce.

One particularity of our system is that even healthy cells will produce lactate. That is why we implemented a lactate module that works as a fold-change sensor. The fold-change sensor will produce a pulse of LuxR. We will study here how the pulsed response influence the output of the system.

Figure 2. Description of the AND gate

Combined Compartment Model

Overview

With this model wanted to investigate whether our system works as an AND gate. We compared the output from the simple lactate detection system with the one from the system including the fold-change sensor.

The equations are the integration of both modules in one compartment model.

Assumptions

We assume:

1. Instant diffusion of AHL in the compartments.
2. In all the following we assume the nanowell volume to be 1 nL.
3. All the E. coli receive the same amount of lactate. This is the worst-case scenario since probably the E. coli on the doughnut will sense a higher lactate production rate than the ones not bound, i.e., in the bulk.

Equations

The equations are a combination of the compartment model, the AHL module and the lactate module.

The internal states, lactate, LuxR, LuxRAHL, LuxI have the same ODEs than the single cell model. Equivalent to the compartment model, we added convection between the different compartments.

Simulation when the lactate concentration reach the same steady states

Below, a comparison is shown between the response of a system including the fold change sensor and the one harboring only the natural detection system for the four different cases described earlier. The lactate inputs give two different delayed pulses of LuxR. The different pulses of LuxR lead to different delay of self-activation of the GFP output. The first GFP output to activate is the one with the cancer cell, the second and third one represent cells with intermediary characteristics. The latest activation time is for the one with no colocalization and no lactate production. With the fold-change sensor, we obtain a significant time difference between the different inputs. With only the natural detection system we do not obtain a clear difference between the different cases. The lactate signal prevails on the quorum sensing signal.

Figure 3. Simulation of the full system with the fold change sensor. Case 1) Lactate steady states are identical. Response of the system in four different situations. Lact+/QS+ corresponds to an E. coli next to a cancer cell, i.e., high lactate production and colocalization on the cancer cell's surface. Lact-/QS- correspond to the healthy cell's case, i.e. low lactate production rate and no colocalization. The other two situations displayed represent intermediary cell characteristics, i.e. high lactate production rate or colocalization on the mammalian cell's surface.

Figure 4. Simulation of the full system with the natural lactate detection system. Case 1) Lactate steady states are identical.

Simulation when the lactate concentration reach different steady states

Here we will compare again the response of the system when including the fold change sensor and the natural lactate detection system.

We can see that here again the fold change gives a better response for the same parameters.

Figure 5. Simulation of the full system with the fold change sensor . Case 2) Lactate steady states are separated.

Figure 6. Simulation of the full system with the natural lactate detection system . Case 2) Lactate steady states are separated

Choosing the best design

The fold-change sensor solved the problem of leakiness and the problems about the size of the nanowell plate we had earlier. The pulse of LuxR should be large enough in order to activate the quorum sensing module. A broader peak can be obtained by introducing a delay in the response of LacI. This can be easily implemented biologically (if needed) by introducing an intermediary protein between lactate induction and LacI production.

In silico GFP measurement experiment

As we have seen earlier, lactate production signals still prevail on the quorum sensing signal. In the broader context of our system, the experimenter will take the measurement at a specific time point. The heatmap represents the ratio of GFP output with and without colocalization on the doughnut for different rates of lactate production and different number of E. coli colocalized on the doughnut. In both cases, the E. coli receive the same amount of lactate.

We can see that we obtain an AND gate, however it is not an optimal AND-gate since we obtain some GFP fluorescence when lactate production is high.

Figure 7. In silico GFP measurement experiment with the fold change sensor . The measurement was taken at a specific time point. We obtain an AND gate.

Single cell model

Overview

The single cell model is provided here to simulate the combined model.

Chemical species

Reactions

\begin{align*} &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LacI},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LacI}\\ \text{IPTG} + \text{LacI} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{IL}}}^{k_{\mathrm{-IL}}} \text{IL}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{A,appLact}}^{\displaystyle\mathop{\downarrow}^{\text{Lact}}} \text{LuxR}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_{LuxR},K_{R,LacI}}^{\displaystyle\mathop{\bot}^{\text{LacI}}} \text{LuxR}\\ \text{AHL} + \text{LuxR} &\mathop{\mathop{\xrightarrow{\hspace{4em}}}^{\xleftarrow{\hspace{4em}}}}_{k_{\mathrm{LuxRAHL}}}^{k_{\mathrm{-LuxRAHL}}} \text{LuxRAHL}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{LuxI},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{LuxI}\\ &\mathop{\xrightarrow{\hspace{4em}}}_{a_\mathrm{GFP},K_{\mathrm{a,LuxRAHL}}}^{\displaystyle\mathop{\downarrow}^{\text{LuxRAHL}}} \text{GFP}\\ \end{align*}

\begin{align*} \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{a_{\mathrm{AHL}}}\text{AHL}+\text{LuxI}\\ \text{LuxR}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxR}}}\varnothing\\ \text{AHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{AHL}}}\varnothing\\ \text{LuxRAHL}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxRAHL}}}\varnothing\\ \text{LuxI}&\mathop{\xrightarrow{\hspace{4em}}}^{d_{\mathrm{LuxI}}}\varnothing\\ \end{align*}

Equations including the fold change sensor

Combining all of the equations from the two different modules, it yields the following system:

\begin{align*} \frac{d[LacI]}{dt}&=\frac{a_\mathrm{LacI} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}-d_{\mathrm{LacI}}[LacI]\\ \frac{d[LuxR]}{dt}&=\frac{a_\mathrm{LuxR} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}} \cdot \frac{1}{1+(\frac{[LacI]}{K_{\mathrm{R,LacI}}\cdot (\gamma_2+1)})^{n_\mathrm{2}}}-d_{\mathrm{LuxR}}[LuxR]\\ [LuxRAHL]&= \frac{[AHL]\cdot [LuxR]}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \gamma_2 &= \frac{IPTG_{tot}}{K_{IL}} \end{align*}

Equations including the natural detection system

\begin{align*} \frac{d[LuxR]}{dt}&=\frac{a_\mathrm{LuxR} \cdot (\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}{1+(\frac{[Lact]}{K_\mathrm{A,appLact}})^{n_1}}-d_{\mathrm{LuxR}}[LuxR]\\ [LuxRAHL]&= \frac{[AHL]\cdot [LuxR]}{K_{\mathrm{d,LuxRAHL}}+[AHL]}\\ \frac{d[LuxI]}{dt}&=a_{\mathrm{LuxI}}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_{\mathrm{LuxI}}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{LuxI}}[LuxI]\\ \frac{d[AHL]}{dt}&=a_{\mathrm{AHL}}[LuxI]-d_{\mathrm{AHL}}[AHL]\\ \frac{d[GFP]}{dt}&=a_\mathrm{GFP}k_{\mathrm{leaky}}([LuxR]-[LuxRAHL])+\frac{a_\mathrm{GFP}(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}{1+(\frac{[LuxRAHL]}{K_{\mathrm{A,LuxRAHL}}})^2}-d_{\mathrm{GFP}}[GFP]\\ K_\mathrm{d,LuxRAHL} &= \frac{k_\mathrm{-LuxRAHL}}{k_\mathrm{LuxRAHL}}\\ \gamma_2 &= \frac{IPTG_{tot}}{K_{IL}} \end{align*}

Summary

The best design appears to be the fold-change sensor compared to the simple lactate detection system, no matter what is the behavior of the lactate inputs . The pulse of LuxR appears to solve the leakiness of the quorum sensing module that was considered earlier. By simulating an in silico GFP measurement, we can prove that the system can function as expected and provide a measurable GFP fluorescence. In theory, the measurement could be taken 3 hours after loading the cells into the chip. We then obtain a fast and robust method to detect circulating tumor cells.

However, several beneficial effects were not considered in this model:

1. The lactate capture effect is not considered here. This effect should increase the sensed lactate only for E. coli localized on the doughnut..
2. The microscopy effect is also not included in this model: id the E. coli are localized on the surface, the experimenter will see apparent higher fluorescence just thanks to the E. coli colocalized on the cancer cell.
3. Will flow in the microfluidic chip increase the quorum sensing effect?.

These effects are considered in the reaction diffusion model.