Computational protocol: A Mathematical Model of Quorum Sensing Induced Biofilm Detachment

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[…] For our computer simulations we restrict ourselves to the two-dimensional setting with a rectangular computational domain Ω = [0, L] × [0, H]. The substratum, on which biofilm colonies form is the bottom boundary, x 2 = 0, see also . The substratum is assumed to be impermeable to biomass and dissolved substrate, so we pose homogeneous Neumann boundary conditions there i.e. ∂n M = ∂n N = ∂n C = ∂n A = 0, for x 2 = 0. We consider our rectangular computational domain as part of a larger biofilm reactor. At the lateral boundaries, where x 1 = 0 or x 1 = L, we assume a symmetry boundary condition, which allows us to view the domain as a part of a continuously repeating larger domain. Therefore, we pose here as well homogeneous Neumann conditions for all dependent variables i.e. ∂n M = ∂n N = ∂n C = ∂n A = 0, for x 1 = 0 or x 1 = L.At the top boundary, x 2 = H, we pose homogeneous Dirichlet conditions for the biofilm biomass M. The degeneracy D(0) = 0 in leads to a finite speed of interface propagation in the sense that initial data with compact support imply solutions with compact support. Therefore, as long as biomass does not reach the boundary of the domain, the model satisfies simultaneously homogeneous Dirichlet and Neumann conditions, which are combined in the no-flux conditions D(M)∇M = 0. Since our simulations will be terminated before biomass reaches the top of the domain, the choice of boundary conditions there is not critical. For the nutrient C, we pose at the top boundary, x 2 = H, an inhomogeneous Dirichlet condition. C is set there to the bulk concentration value, which reflects that substrate is added to the system through this segment of the domain boundary. The dispersed cell density and the autoinducer concentration are set there to nil. This enforces a diffusion gradient from the biofilm in the interior of the domain to the boundary and mimics removal of quorum sensing molecules and dispersed cells into the surrounding bulk phase where they are negligible due to instantaneous dilution. Thus we have C = C ∞, M = N = A = 0 at x 2 = H.Initially biofilm biomass is placed in small pockets with M > 0 at the substratum only. The locations and initial sizes of these pockets will be chosen randomly or explicitly specified a priori. Thus ∂Ω2(0)∩{x 2 = 0} ≠ ∅, ∂Ω2(0)∩∂Ω \ {x 2 = 0} = ∅ and ∫Ω2(0) dx ≪ ∫Ω dx. Ω2(0) is typically not connected, i.e. several inoculation sites are usually considered and all have a boundary with x 2 = 0. We will assume that initially no dispersing cells and no autoinducers are in the system, and that the concentration of nutrients is initially at bulk levels, i.e. C = C ∞, N = A = 0 at t = 0.Eqs ()–() are discretized on a regular grid using a cell centered finite difference-based finite volume scheme for space and semi-implicit time-integration, adapted from [, , ] to account for the new dependent variable A, N, which are treated in the same manner as C. In every time step, four linear algebraic systems are solved, one for each dependent variable. These linear systems are sparse and at least weakly diagonally dominant. They are efficiently solved with the stabilized biconjugate gradient method []. The linear solver is prepared for parallel execution on multi-core and shared memory multiprocessor architectures using OpenMP, as described in []. Simulations will be terminated when the biofilm reaches a set target size or when a set maximum simulation time is reached. For the visualization of simulation results we use the Kitware Paraview visualization package (spatially resolved plots) and gnuplot (lumped results).For better interpretation of the computer simulations of the model, the following quantitative lumped measures will be used Biofilm size relative to the domain size ω(t):=∫Ω2(t)dx∫Ωdx(7) Average nutrient concentration in Ω2: Cavg(t):=∫Ω2(t)C(t,x)dx∫Ω2(t)dx(8) Total sessile biomass in the biofilm: Mtot(t):=∫ΩM(t,x)dx(9) The total amount of dispersed cells: Ntot(t):=∫ΩN(t,x)dx(10) Average concentration of the quorum sensing molecules in Ω2, non-dimensionalized with respect to τ: Aavg(t):=∫Ω2(t)A(t,x)dxτ∫Ω2(t)dx(11) Biomass loss K(T): This is the relative difference between the net biomass gain and the produced sessile biomass over a period of time T defined as follows K(T)=∫0T∫Ω[μCk1+C]Mdxdt-[Mtot(T)-M0]∫0T∫Ω[μCk1+C]Mdxdt.(12) where M tot(T) is the amount of biomass in the system at t = T and M 0 is the amount of biomass initially present in the system.The ratio of dispersed cells that are re-attached and the cells that are detached, at time t: Z(t)=η2η1[∫Ω(Mk5+M)Ndx∫Ω(An1+An)Mdx](13) A measure for the amount of dispersed cells (i.e. the diffusive flux) that left the domain over the time interval [0, T] P(T)=∫0T∫0L∂N∂n|y=Hdx1dt(14) Biofilm size relative to the domain size ω(t):=∫Ω2(t)dx∫Ωdx(7) Average nutrient concentration in Ω2: Cavg(t):=∫Ω2(t)C(t,x)dx∫Ω2(t)dx(8) Total sessile biomass in the biofilm: Mtot(t):=∫ΩM(t,x)dx(9) The total amount of dispersed cells: Ntot(t):=∫ΩN(t,x)dx(10) Average concentration of the quorum sensing molecules in Ω2, non-dimensionalized with respect to τ: Aavg(t):=∫Ω2(t)A(t,x)dxτ∫Ω2(t)dx(11) Biomass loss K(T): This is the relative difference between the net biomass gain and the produced sessile biomass over a period of time T defined as follows K(T)=∫0T∫Ω[μCk1+C]Mdxdt-[Mtot(T)-M0]∫0T∫Ω[μCk1+C]Mdxdt.(12) where M tot(T) is the amount of biomass in the system at t = T and M 0 is the amount of biomass initially present in the system.The ratio of dispersed cells that are re-attached and the cells that are detached, at time t: Z(t)=η2η1[∫Ω(Mk5+M)Ndx∫Ω(An1+An)Mdx](13) A measure for the amount of dispersed cells (i.e. the diffusive flux) that left the domain over the time interval [0, T] P(T)=∫0T∫0L∂N∂n|y=Hdx1dt(14) The default model parameters used in the simulations are summarized in . Parameter values that are varied in the simulations will be stated in the text where the simulation experiments are described. […]

Pipeline specifications

Software tools ParaView, Gnuplot
Application Miscellaneous
Organisms Escherichia coli