Model

The main experiment for investigating diffusion is a propagation of the whole pattern through the chip via the quorum sensing module only. We have used beads containing cells which are able to sense luxAHL, produce GFP when they sense it, and amplify the signal for the next row. The combination of the quorum sensing module with diffusion enables to check that cells will amplify the signal enough from one row to the next one, and to check what would be the time scale of the pattern formation.

We used a reaction-diffusion model to combine quorum sensing reactions and diffusion. Here are the different compartments considered :

Millifluidic compartments used for the diffusion model

Species

The beads are the compartments containing the cells. The number of cells per bead is called N. The fraction of volume occupied by cells to total bead volume is $$\frac{V_{\text{internal to cells}}}{V_{\text{bead}}}=N \ \ V_{E. coli}$$

Reactions

$$ \begin{align} \emptyset&\rightarrow LuxR \\ LuxAHLext & \leftrightarrow LuxAHLint\\ LuxAHLint+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{GFP}\\ mRNA_{GFP}&\rightarrow GFP \end{align}$$

Equations

Deriving diffusion rates

We are using a reaction-diffusion model, which means that for internal and external AHL species which diffuse, we have to include a diffusion rate in addition to reaction rates :

$$\frac{d[AHL]}{dt}=\mathcal{Diff}(AHL)+\mathcal{R}(AHL)$$

According to Fick's law of diffusion, the flow of AHL Φ(AHL_int) (number of molecules per second) from the bead into the cells and the flow of AHL Φ (AHL_ext) from cells into the bead into the bead are $$\Phi(AHL_{bead \rightarrow cells}) = N\sigma \mathcal{A} ([AHL_{ext}]-[AHL_{int}]) \\ \Phi(AHL_{cells \rightarrow bead }) = N \sigma \mathcal{A} ([AHL_{int}]-[AHL_{ext}])$$ $$\text{where }\sigma \text {is the membrane permeability, A is the area of the membrane and N is the number of cells per bead.}$$

Thus the diffusion rate of internal AHL (concentration per second) is : $$Diff(AHL_{int})=\frac{N \sigma \mathcal{A}}{V_{int}} ([AHL_{ext}]-[AHL_{int}])=D_m ([AHL_{ext}-[AHL_{int}])$$ $$\text{where} D_m=\frac{\sigma \mathcal{A}}{V_{E coli}} \text{is a lumped coefficient for diffusion through the membrane}$$

and the diffusion rate of external AHL is $$Diff(AHL_{ext})=\frac{N \sigma \mathcal{A}}{V_{bead}} ([AHL_{ext}]-[AHL_{int}])= \frac{N V_{E.coli}}{V_{bead}}D_m([AHL_{ext}]-[AHL_{int}])=\rho V_{E.coli}D_m([AHL_{ext}]-[AHL_{int}])$$ where ρ is the cell density (number of cells per bead).

Set of equations

$$ \begin{align*} \frac{d[AHLext]}{dt} &= \rho \ V_{E. coli}\ D_m (LuxAHL_{int}-AHL_{ext}) -d_{AHLext}[AHL_{ext}]\\ \frac{d[AHLint]}{dt} &= Dm (LuxAHL_{ext}-AHL_{int}) + k_{-RLux}[R_{Lux}]-k_{RLux}[LuxAHL_{int}][LuxR]-d_{AHLint}[AHL_{int}]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[AHL_{int}][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[AHL_{int}][LuxR] - k_{-RLux}[RLux] - d_{RLux} [RLux] \\ \frac{d[mRNA_{GFP}]}{dt} &= \frac{k_{mRNA_{GFP}}[RLux]^2}{K_{mLux}^2 + [RLux]^2}- d_{mRNA_{GFP}} [mRNA_{GFP}]\\ \frac{d[GFP]}{dt} &= k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]\\ \end{align*} $$

Results