Thin-film modelling of biofilm growth and quorum sensing

Biofilms are slimy films of bacteria that typically grow on solid surfaces with a fluid interface. Two mathematical models for nutrient-dependent, early-stage biofilm growth and quorum sensing (QS) are proposed and studied. The aim of this study is to use thin-film methods to derive a dimensionally reduced model on a stronger theoretical basis to that of Ward et al. (J Math Biol 47:23-55, 2003). The biofilm structure is described as a viscous fluid, and the two models apply different boundary conditions between the biofilm and the solid surface: namely (1) shear-stress-free; and (2) no-slip conditions. The complexity and dimensionality of each of the models are reduced by the application of asymptotic methods in biologically relevant thin-film limits. The first model reduces to a system of first-order PDEs, whilst the second results in a high-order, degenerate Fisher-type equation. We present a hodograph solution for model (1) and analyse model (2) in the travelling-wave limit. A key difference between the models is that the long-time solutions have contrasting properties: for model (1) the large-time solutions are dependent on the initial conditions, whilst in model (2) the long-time attractor is a travelling wave, the speed of which is approximated in a certain limit that depends on conditions at the biofilm edge and substratum interface. The QS aspects of the model derived using the systematic asymptotic approach are shown to be relatively unchanged from those of Ward et al. (2003), and the results therein are thus applicable to the current models.