Modelling the spatial dynamics of plasmid transfer and persistence

Abstract

Bacterial plasmids are extra-chromosomal genetic elements that code for a wide variety of phenotypes in their bacterial hosts and are maintained in bacterial communities through both vertical and horizontal transfer. Current mathematical models of plasmid-bacteria dynamics, based almost exclusively on mass-action differential equations that describe these interactions in completely mixed environments, fail to adequately explain phenomena such as the long-term persistence of plasmids in natural and clinical bacterial communities. This failure is, at least in part, due to the absence of any spatial structure in these models, whereas most bacterial populations are spatially structured in microcolonies and biofilms. To help bridge the gap between theoretical predictions and observed patterns of plasmid spread and persistence, an individual-based lattice model (interacting particle system) that provides a predictive framework for understanding the dynamics of plasmid-bacteria interactions in spatially structured populations is presented here. To assess the accuracy and flexibility of the model, a series of experiments that monitored plasmid loss and horizontal transfer of the IncP-1beta plasmid pB10 : : rfp in Escherichia coli K12 and other bacterial populations grown on agar surfaces were performed. The model-based visual patterns of plasmid loss and spread, as well as quantitative predictions of the effects of different initial parental strain densities and incubation time on densities of transconjugants formed on a 2D grid, were in agreement with this and previously published empirical data. These results include features of spatially structured populations that are not predicted by mass-action differential equation models.

Fig. 1

Lattice structure of IPS model. Coloured circles represent different cell types, the 9 sites in the light green square indicate the local growth neighbourhood for the central site, and the 49 sites combined in the light green and dark green regions constitute the nutrient neighbourhood for the central site. The model employs periodic boundary conditions.

Fig. 2

Simulations (left) and photographs (right) of patterns of plasmid loss in two different bacteria bearing plasmid pB10 ∷ rfp. In (a–c), two examples of simulation results are given to illustrate variability between runs. In experiments: white, segregants; pink, plasmid-bearing cells. In simulations: green, nutrients; light blue, 2 recipients; dark blue, 1 recipient; pink, 2 donors or transconjugants; red, 1 donor or transconjugant. The conjugation rate was set to 1 for all four simulations, recipient maximal growth rate was 1, and θ₁=1, θ₂=2. Spatial scales: for simulations and for experiments, the region shown corresponds to approximately 1 mm and 5 cm per side, respectively.

Fig. 3

Experimental data and model simulations for transfer of plasmid R1drd19 between E. coli cells on agar slides after 30 h of incubation, as a function of initial densities of donor and recipient populations. Dashed lines with open symbols, data from Simonsen (1990); solid lines with filled symbols, data from simulations. □ and ■, recipient; ○and ●, donor; △ and ▲, transconjugants; grid size for simulation, 1000×1000. Maximum growth rates are ψ_R=1, ψ_D=ψ_T=0.9, minimum growth rates are all 0; τ=0.005, θ₁=0, θ₂=4, γ_D^max=γ_T^max=3,γ_D^min =γ_D^min=0.03; subscripts are D, donor; (derepressed) transconjugant; R, recipient.

Fig. 4

Simulations showing microscopic development of plasmid-bearing (T) and plasmid-free (R) microcolonies. Colours: red, T; white, Ts that arose through conjugation; blue, R; green, nutrient. Pictures are taken from two time points of the same run: left side is just after the start of the simulation, right side is at final (stationary) time. Simulations were initialized at low cell density (top) with a fraction 0.005 of T and 0.005 of R, and at higher cell density (bottom) with fraction 0.05 of T and 0.05 of R. Grid size, 500×500. Maximum growth rates: ψ_R=1, ψ_D=ψ_T=0.73; minimum growth rates: 0; τ=0.005; conjugation rate: 2; θ₁=1, θ₂=2. The final densities yielded about 44 % T, 56 % R (top) and 57 % T, 43 % R (bottom), with roughly four times as many conjugation events when the initial density of cells was higher (bottom).

Fig. 5

Time series of (a) donor, (b) recipient, and (c) transconjugant densities during transfer of plasmid pB10 ∷ rfp between E. coli strains on filters. Solid lines, simulations; dotted lines, experimental data (means from duplicate mating experiments). Grid, 1000 × 1000. Grids were initialized with a fraction 0.00068 of T, 0.0202 of R and 0.0202 of D. Maximum growth rates, ψ_R=1, ψ_D=1.01, ψ_T=0.98; minimum growth rates: 0; τ=0.00001; γ_D^max=γ_T^max=1; γ_D^max=γ_T^max=0.1 θ₁=0, θ₂=4. Subscripts are as in Fig. 3.