Given a light mask, the reaction-diffusion model calculates the time- and position-dependent expression level of the β-galactosidase (β-gal) output gene. The model consists of (i) a partial differential equation describing 3OC6
HSL production, degradation, and diffusion and (ii) two algebraic equations describing the steady-state concentrations of CI and β-gal in response to 3OC6
HSL and light. In dimensionless form, these equations are
, and u3
represent the concentrations of 3OC6
HSL, CI, and β-gal at a position on the plate whose polar coordinates are given by (r, θ). The flight
functions quantify the transcription rates of the light-dependent ompC
promoter and the CI-repressed, LuxR::3OC6
The constants κ1
quantify the maximum production rate and the degradation rate of 3OC6
HSL, respectively. The production rate of 3OC6
HSL is estimated so that the maximum AHL concentration on the plate is 2.5 nM while the degradation rate of 3OC6
HSL is slow; it has a half-life of about 2.5 days at pH 6.6 (Flagan et al., 2003
). The conversion factor between the ompC
transcription rate, characterized by flight
, and CI concentration is κ3
= 0.8 nM/Miller. The constant κ4
is the maximum β-gal concentration, which is 289 Miller units. This value was determined in batch culture experiments as described above at 500nM (maximum) exogenous AHL in the absence of any CI protein (plasmid pJT105).
When solving these equations, the space and time coordinates are dedimensionalized so that r* = r / R and t* = tD/R2
where r is the radial position from the center of the plate, R is the radius of the plate, t is time and D = 1.67×10−7
/sec is the diffusivity of 3OC6
HSL (Basu et al., 2005
). The system is an agarose plate with radius R = 4.25 cm (3.55 mm operating depth), homogeneously filled with stationary bacteria. Because the bacterial photographs are crisp in our system we assume that there is no appreciable bacterial movement in the agarose plates. There is a no-flux boundary condition (Neumann type) at r* = 1 and a uniformly zero initial 3OC6
The differential equations in Equations (8
) are solved using the finite difference method. We substitute 2nd
order central differences for all spatial derivatives to create a sparse system of ordinary differential equations. The ordinary differential equations are solved using the Matlab (Mathworks, Natick, MA) ode23s stiff numerical integrator with a final time of 24 hours (t* = 0.0027). A sufficient number of radial and axial elements are used to accurately resolve each light mask. The solution yields the dynamics of edge formation in response to a given light mask.
Quantifying the Effect of Angle of Intersection on Edge Intensity
The effect of changing the angle of intersection between light and dark boundaries on the edge intensity is examined, comparing the model predictions to the experimentally observed behaviors. We create a series of unit circle in silico masks where θ degrees of the circle are in the light with 360-θ degrees in the dark and where θ is varied from 50 to 345 degrees. For each mask, the solution of the reaction-diffusion model is computed, which predicts the maximum edge intensity. The maximum edge intensity is the β-galactosidase concentration at the edge location. The model predictions compare favorably with the experimentally observed edge intensities of the asymmetrical silhouette mask at the selected angle intersections (). The image analysis procedure to obtain the experimental data is described above.
Calculating the Radial β-gal Profile
The radial edge intensity profile of the circle images are compared to the in silico radial β-galactosidase profile from the model solution (). We compute the in silico radial β-galactosidase profile by first inputting the circle light mask into the model and determining the solution. Then, the β-galactosidase concentration in terms of Miller units (u3) is outputted along the radial coordinate (r = 0 to 1.8 cm) and divided by the value of u3 at r = 0 to obtain the normalized intensity in .