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 concentration over time is given by an ordinary differential equation (ODE), that comprises production and linear degradation or decay. However, the types of reactions that take place depend on the localization, i.e. extracellular or cytoplasmic; in the latter case, we have to further distinguish between the two type of cells encountered in our system, sender cells or so called mine cells and receiver cells. Mine cells synthesized the signalling molecule which depends on the product of luxI gene, while for the degradation we assumed a linear behaviour. For the receiver cells only linear degradation take place, and we assume that is under the same rate as mine cells. However, for the extracellular decay we consider that OOHL degrades at a different rate, because the intracellular process is driven by enzymatic degradation, whereas the extracellular decay is a non active process. Furthermore, we include a dilution factor, taking into account 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.