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.

Geometry and boundary conditions

Figure 1 Millifluidic compartments used for the diffusion model

Three different compartments are considered :

• The wells and channels, which are filled with liquid medium and where extracellular AHL only can diffuse, with its diffusion coefficient D, and degrade with the degradation rate d_AHLext.

• The beads without the cells inside them, whose volumes are approximated to the volume of beads themselves, and where extracellular AHL only can degrade and diffuse, with an a priori different diffusion coefficient C_beads* D_AHLext.

• The cells, which are considered altogether as one compartment, and which contains all species except extracellular AHL.

Figure 2 Geometry used for simulation on Comsol Multiphysics

In order to fulfill these conditions in Comsol Multiphysics, the cells are not drawn but instead, beads are considered as compartments with a certain density of cells. Rates of diffusion through the membrane are added as reaction rates and intracellular species are contained in beads via a Neumann boundary condition applied to the surface of the beads, that is : $$ \nabla C \cdot \textbf{n} =0$$ $$\text{for all species except extracellular AHL}$$ $$\text{for every vector n normal to the surface of a bead}$$

Species

Reactions

These reactions happen in every bead of the pattern on the grid.

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

Parameter estimation

Initial conditions determined by bead preparation step

Non-induced cells, that is all cells except the one from the middle well of the first row have an initial LuxR concentration of LuxR due to constitutive production and degradation : $$ [LuxR]_i=\frac{a_{LuxR}}{d_{LuxR}} $$ The cells of the first row are induced by 10 μ M AHL just before they are encapsulated in alginate. Then, bead formation takes half-an-hour before these cells are added in the first well. During this half-an-hour AHL can diffuse out. Therefore initial conditions in the first cell completely depend on these previous steps. We have also simulated this bead preparation in order to get these initial conditions.

Figure 3 Beads stored in calcium chloride, 10mM

The geometry is a simple bead in a bigger compartment filled with calcium chloride. We look at the concentrations at the middle of the bead. Initial concentration of LuxR before bead diffusion starts is a classical steady state a_LuxR/d_LuxR

Figure 4 Signaling molecules

Figure 5 Regulators and GFP

Extracellular AHL diffuses very fast through the membrane, which makes intracellular AHL increase, until both reach the same value and diffusion through the membrane reaches an equilibrium. From this point on AHL starts to bind to LuxR to form RLux, so AHL decreases, until RLux triggers enough production of LuxI which catalyses again AHL production, and makes AHL concentration become stable.

After 30 minutes, all LuxR is bound to AHL and converted to RLux, so that we have $$ [LuxR]_i= 0nM$$ $$ [RLux]_i= 200 nM$$ $$ [AHLint]_i=[AHLint]_i=2 μ M$$

RLux has already started to trigger the production of LuxI and GFP, so that we have $$ [mRNAGFP]_i = 40nM$$ $$ [GFP]_i = 900nM$$ $$ [mRNALuxI]_i = 40nM$$ $$ [LuxI]_i = 16000nM$$

Results

Pattern developing

The developing pattern shows that every row senses the signal coming from above, amplifies it and is capable to send a signal to the next row, which is high enough to enable propagation until the bottom of the grid. We can see that the dynamics of the model are very well fitted to experimental conditions.

The background in our experimental setup is very high and this is high we see actual fluorescence appear only after 11 hours. In order to account for this, we also set up a background in the simulated pattern by adjusting the scale.

Dynamics by row

Here you can see the quorum sensing mechanism happening in several rows at different moments. The first colony has a high initial concentration of AHL so at the beginning, AHL diffuses quickly out of the first colony. However as the first cell is strongly induced because of this AHL, a positive feedback is produced which will enable AHL concentration to reach a steady state. The colonies in the row below receive AHL progressively as opposed to the first colony which had an initial concentration, so the feedback in these colonies below has the time to happen immediately when AHL is received, and this is why AHL increases so fast in these colonies.

Figure 6

For simulation of the pattern, in order to be able to simulate fluorescence of the beads which is influenced by cell growth, we had to plot the concentration of GFP overall in a bead as opposed to the concentration of GFP in cells. Here we compare concentration of GFP in cells (doesn't account for growth) and overall concentration of GFP in a bead (accounts for growth) :

Figure 7 GFP in beads

Figure 8 GFP in single cell