OOHL: REACTION-DIFFUSION EQUATION

The change of OOHL concentration over time is influenced by two processes: (i) local chemical reactions and (ii) diffusion; which causes the molecule to spread over the agar plate (Eq. 1).

Equation 1: General partial differential equation for OOHL reaction-diffusion. D(OOHL(r,t),r) is the diffusive term, R(OOHL(r,t)) is the reaction term

For diffusion, we have a partial differential equation (Eq. 2) which describes density fluctuations over time and space. We do not model OOHL diffusion in and out cells explicitly; the underlying assumption is that this process is fast or that the molecule freely diffuses. From equation 2, D_OOHL(OOHL(r,t),r) denotes the collective diffusion coefficient for OOHL at location r. However, we assume that the diffusion coefficient does not depend on the density, i.e., D_OOHL is a constant. The value reported in the literature for the diffusion constant corresponds to measurements performed in water at 25^oC. Since diffusion in our system happens in agar, we scaled the diffusion constant by a factor C_agar (Fatin-Rouge et al., 2004).

Equation 2: Diffusive term for OOHL.

For the reaction component, the change of OOHL concentrations over time is given by an ordinary differential equation (ODE), that comprises production and linear degradation. The synthesis of the signalling molecule depends on the product of luxI gene. Now for the degradation, we consider that OOHL degrades at different rates depending on the localization, i.e. cytoplasmic or extracellular. Given that the intracellular degradation is driven by enzymatic degradation, whereas the extracellular decay is a non active process. Our model also includes a dilution factor due to the cell growth (Eq. 3).

Equation 3: Reaction term for OOHL. DF is the dimensionless dilution factor, where N₀ is the initial concentration and N_m is the carrying capacity (scaling factor, N_m = 1).

Finally, we need to specify the initial conditions (at time t = 0) and boundary conditions. At the starting point there is no OOHL in the agar plate, thus the initial concentration is zero ([OOHL(r,t=0)] = 0 M). For the boundary condition, we take into account that there is not flux out of the agar plate.

Equation 4: Neummann Boundary Condition.