Mathematical modelling of quorum sensing in bacteria.

The regulation of density-dependent behaviour by means of quorum sensing is widespread in bacteria, the relevant phenomena including bioluminescence and population expansion by swarming, as well as virulence. The process of quorum sensing is regulated by the production and monitoring of certain molecules (referred to as QSMs); on reaching an apparent threshold concentration of QSMs (reflecting high bacterial density) the bacterial colony in concert 'switches on' the density-dependent trait. In this paper a mathematical model which describes bacterial population growth and quorum sensing in a well mixed system is proposed and studied. We view the population of bacteria as consisting of down-regulated and up-regulated sub-populations, with QSMs being produced at a much faster rate by the up-regulated cells. Using curve fitting techniques for parameter estimation, solutions of the resulting system of ordinary differential equations are shown to agree well with experimental data. Asymptotic analysis in a biologically relevant limit is used to investigate the timescales for up-regulation of an exponentially growing population of bacteria, revealing the existence of bifurcation between limited and near-total up-regulation. For a fixed population of cells steady-state analysis reveals that in general one physical steady-state solution exists and is linearly stable; we believe this solution to be a global attractor. A bifurcation between limited and near-total up-regulation is also discussed in the steady-state limit.