Fig 1.
The colony front advances with a constant speed after a transient period.
Numerical solution of the model equations (Eqs (3–5)) over a period of 250 hours with the following parameters: g = 0.173 h−1, D0 = 2 μm2min−1, Dp = 220 μm2min−1, DN = 104 μm2 min−1, m = 4, α/e0 = 19 h−1, β = 16 h−1. (A) The advance and shape of the colony front (solid lines) between equidistant time points becomes constant at longer times, which indicate that our model equations exhibit a traveling wave solution. The colony front has a sharp edge, which is a consequence of large decrease in diffusion rate below a threshold EPS level, which depends on the population density. (B) The EPS levels (dashed lines) are proportional to population density and display similar travelling wave behavior. (C) The nutrient level decreases with time as the population expands and consumes more nutrients. The nutrient level is measured in arbitrary units ensuring 1:1 nutrient to cell density conversion in Eqs (3–5). (D,E) The long-term colony expansion rate scales similar to Fisher waves (as ) with the diffusion rate Dp and growth rate gmax for non-diffusing nutrients (DN = 0) (solid line and closed circles). The circles (closed/open) represent the colony expansion rate computed from numerical solution of model equations and the lines (solid/dashed) represent the expansion rate calculated from a phase-space analytical method (see Methods section). The open circles in 1(D,E) correspond to the expansion rate in the presence of diffusing nutrients and deviate from the square root scaling in Dp and is captured in the expansion rates determined from phase-space analysis (dashed line). (F) The steady state expansion rate increases from a minimum value
to a maximum value
as the initial nutrients level increases. Here c' (dotted line) is the expansion rate with the EPS level set to its half saturation level e0, i.e., the minimum level of EPS at which S-motility becomes active.
Fig 2.
The model fits the density-dependent expansion data from social-motility assay experiments.
(A) Colony radius of M. xanthus strain DZ2 (A+S+) on 0.5% agar CYE plates after 24 h at different initial cell densities is represented by the stars. The solid line represents the numerical prediction from our model with parameters as in Fig 1. (B) The expansion rate at different initial cell densities for M. xanthus A−S+ strains (an average of three S-motile strains: DK1217, DK1218, DK1219) on 1.5% agar CTT plates at 8 hr is represented by the stars. The solid line represents the numerical prediction from our model in which most of the parameters are the same as in Fig 1, except g = 0.154 h−1,Dp = 16 μm2min−1, m = 4, α/e0 = 150 h−1.
Fig 3.
The expansion rate saturates after a transient lag phase that depends on the initial cell density.
The model predicts a transient lag phase in the expansion rate with density dependence followed by a constant steady-state expansion rate for all densities. At low density the EPS level is low and slow expansion occurs due to the basal diffusion rate D0. Over time the EPS level increases leading to a steady state phase with a constant expansion rate determined by Dp.
Fig 4.
Quantitative analysis of long-term expansion experiments.
(A) Cell density-dependent social motility of M. xanthus strain DK1218 (A−S+) on 0.5% agar. We observe that the transition from the slow-expansion phase to the constant-expansion phase occurs earlier for cells initiated with high densities as compared to low-density conditions. The bar represents 1700 μm. (B) The net radial expansion of the colony over time shows that colonies with different initial cell densities (points with standard error) expand with equal rates after an initial transition period, as predicted by the numerical results (lines) of our model.