Modeling cooperating micro-organisms in antibiotic environment

Abstract

Recent experiments with the bacteria Paenibacillus vortex reveal a remarkable strategy enabling it to cope with antibiotics by cooperating with a different bacterium-Escherichia coli. While P. vortex is a highly effective swarmer, it is sensitive to the antibiotic ampicillin. On the other hand, E. coli can degrade ampicillin but is non-motile when grown on high agar percentages. The two bacterial species form a shared colony in which E. coli is transported by P. vortex and E. coli detoxifies the ampicillin. The paper presents a simplified model, consisting of coupled reaction-diffusion equations, describing the development of ring patterns in the shared colony. Our results demonstrate some of the possible cooperative movement strategies bacteria utilize in order to survive harsh conditions. In addition, we explore the behavior of mixed colonies under new conditions such as antibiotic gradients, synchronization between colonies and possible dynamics of a 3-species system including P. vortex, E. coli and a carbon producing algae that provides nutrients under illuminated, nutrient poor conditions. The derived model was able to simulate an asymmetric relationship between two or three micro-organisms where cooperation is required for survival. Computationally, in order to avoid numerical artifacts due to symmetries within the discretizing grid, the model was solved using a second order Vectorizable Random Lattices method, which is developed as a finite volume scheme on a random grid.

Fig 1. A typical ringed pattern of a mixed P. vortex and E. coli bacterial colony on a 14 cm agar plate containing the antibiotic ampicillin.

The rings represent different bacterial densities in alternating behaviors of building and expansion. Reproduced from [7].

Fig 2. Grid effects.

Numerical solutions for the linear (k = 0, top) and non-linear (k = 1, bottom) diffusion equation ∂b/∂t = ∇∙(b^k∇b). With linear diffusion, a rectangular lattice (left) and a random lattice (right) yield similar results. However, with non-linear diffusion, the solution is compact and different lattices yield observably different numerical solutions. In particular, the 4-fold symmetry of the rectangular grid is apparent in figure (C).

Fig 3. Single-species simulations.

Individual species cannot grow. (A) On its own, P. vortex dies due to antibiotics and the colony does not expand. (B) On its own, E. coli does not expand because the bacteria are unable to move independently towards a nutrient rich area. Simulation parameters are detailed in Table 1. Simulation time is equivalent to about 50 hours.

Fig 4. Two-species simulations.

The joint P. vortex and E. coli bacterial colony in an antibiotic environment develops ring-like patterns. All simulation parameters are the same as in Fig 3.

Fig 5. Comparing experiments and simulations.

The colony radius as a function of time. Left: experiments (reproduced from [7] showing P. vortex alone (full blue squares), E. coli alone (empty red squares) and the combined colony (black diamonds). Right: Simulations. Both figures show the non-continuous increase in the radius of the joint colony but only small, marginal expansion of each species on its own. Simulation units were converted to experimental ones as explained in the text.

Fig 6. A snapshot of the simulated cross-sections of the different colonies: P. vortex and E. coli (solid black line), only P. vortex (dotted red) and only E. coli (dashed blue).

Fig 7. Simulation results with two colonies (P. vortex+E. coli).

Left: Initial colonies. Right: After 30 hrs. Upon contact, the builder/explorer phases of the two colonies synchronize.

Fig 8. Simulation results for a plate in which only the left half of the domain initially has antibiotics.

(A) The concentration of antibiotics, (B) P. vortex and (C) E. coli after 13hrs. The colony only grows to the right (no antibiotic) although in principle, the mixed colony can also grow to the left.

Fig 9. Simulation results for a 3-species system.

Fig 10. The stages of creating the VRL.

(a) A uniform reference lattice with a single node chosen uniformly in each cell. (b) The Voronoi diagram for the random nodes. (c) The Delaunay triangulation yields (d), the final VRL.