Reaction-diffusion model: genetic circuit with GFP as reporter gene

In our first spatio-temporal model, we wanted to find out if (i) a suitable OHHL gradient is formed and (ii) validate the model with experimental data, as a proof of concept. Essentially we model the receiver cells (E. coli DH5α strain) being transformed with a plasmid containing GFP under the control of pLux promoter. Subsequently, we simulate a 2D spatio-temporal reaction-diffussion system with COMSOL Multiphysics.

Video: GFP expression levels over 11 hours, concentration in mol/m³. Distance between colonies: 1,5 cm.

OHHL is the only molecule in the model that is allowed to diffuse in and out cells, however species concentrations are time- and space-dependent. Thus the change of concentrations over time is given by system of non-linear partial differential equations (PDEs). Each equation represents the rate of change of species' continuos concentration as a sum of terms representing biological processes, such as production, degradation and regulation; the later is captured with Michaelis-Menten functions. An important assumption is that specific cellular processes like transcription and translation are not modelled explicitly, but lumped together. Our system consists of two type of genetically engineered colonies: (i) Mine Cells and (ii) Receiver Cells, that can be treated as modules that communicate only by a single molecule, OHHL. Additionally, we include the decay of OHHL while diffusing in the agar plate.

Mine cells

Two proteins of interest for our design are produced by mine cells: (i) LuxI and (ii) NagZ; both genes are constitutively expressed. Aditionally, the synthesis of OHHL is carried on by this cells; the product of luxI gene is directly involved in the synthesis of the signalling molecule, using as substrates S-adenosylmethionine (SAM) and an acylated acyl carrier protein (ACP) from the fatty acid biosynthesis pathway (Schaefer et al., 1996). We assume precursor molecules are provided by the cell in non-limiting conditions.

The system of equations for the states involved in the sender module are given below:

Equation system 5: System of Differential equations for mine cells.

Agar plate

In the agar plate no reactions take place; hence only two processes of interested are considered: (i) diffusion and (ii)decay of the diffusible signal, OHHL (Eq 6).

Equation 6: Processes taking place on the agar plate: diffusion and decay of OHHL

Receiver cells

Receiver cells are responsible for processing the signal sent by the mine cells. OHHL acts as the input signal and the output is green fluorescence whose intensity correlates with the sensed OHHL concentration by receiver cells; for this purpose GFP expression is under the control of pLux promoter. Once OHHL molecules reach a non-mine colony, they can bind to LuxR protein, and the complex LuxR/OHHL acts as a transcriptional activator. Therefore, this module works as a high pass amplitude filter for GFP, since it is produced when the OHHL levels are sufficiently high.

The system of equations for the states involved in the sender module are given below:

Equation system 7: System of Differential equations for receiver cells

Results: Simulation

The goal of the simulation was to compare it to experimental results, thus it was conducted for 11 hours and with 3 mine cells to mimic the experimental setup (Figures 1-4). It can be seen that the predicted time scale for the expression of GFP from the simulation and the experimental results are congruent. In addition, it is very important to point out that:

1. The intensity of the GFP correlates with the number of adjacent mines, thus a suitable OHHL gradient is formed.

2. The contribution from the basal expression cannot be neglected.

Figure 1: Parallel between experimental, showed on the left, and simulation results, showed on the right, after 5.5 h. The experiments were performed in triplicates; Location of the mine cells: row 2 - third colony from left to right, row 3 - fifth colony from left to right, row 4 - fourth colony from left to right.

Figure 2: Parallel between experimental, showed on the left, and simulation results, showed on the right, after 6.5 h. The experiments were performed in triplicates; Location of the mine cells: row 2 - third colony from left to right, row 3 - fifth colony from left to right, row 4 - fourth colony from left to right.

Figure 3: Parallel between experimental, showed on the left, and simulation results, showed on the right, after 9.5 h. The experiments were performed in triplicates; Location of the mine cells: row 2 - third colony from left to right, row 3 - fifth colony from left to right, row 4 - fourth colony from left to right.

Figure 4: Parallel between experimental, showed on the left, and simulation results, showed on the right, after 11 h. The experiments were performed in triplicates; Location of the mine cells: row 2 - third colony from left to right, row 3 - fifth colony from left to right, row 4 - fourth colony from left to right.

It is important to point out that after 11 hours, it is possible to distinguish between 0, 1 and 2 adjacent mines. However, the difference in intensity between 2 and 3 adjacent mines, it is not significant. Hence, we decide to move to a three neighbours setup; removing the internal colony in the hexagonal grid.

The parameters used for the simulation correspond to values reported in literature and some of them are estimated from our experiments, like the growth rate and the affinity constant of the dimer LuxR/OHHL.

Regarding the OHHL concentration gradient after 11 hours, the experimental results are consistent with the simulation (Fig 5).

Figure 5: OHHL concentration gradient after 11 hours.

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