Analysis of a 2D model of quorum sensing characterized by a transition from bistability in spatially coupled signalling compartments

Intercellular signalling and communication are used by bacteria to regulate a variety of behaviours. In a type of communication called quorum sensing (QS), bacteria adjust their behaviour at a colony-wide level in response to reaching a critical cell density. Mathematically, these changes can include the emergence of chemical oscillations as well as transitions between bistable steady-states, leading to hysteresis-like phenomena. We formulate and analyse a 2D mathematical model of the prototypical LuxI/LuxR QS system in a bacterial colony. We use a coupled cell-bulk PDE-ODE framework in which the bacteria are modelled as small `holes' of a common small radius coupled through a bulk diffusion field. The method of matched asymptotic expansions is used to construct steady-state solutions and determine linear stability properties thereof. In a colony of identical cells, we show that there are bistable equilibria and use a hybrid asymptotic-numeric approach to construct bifurcation diagrams. Linear stability is determined through the analysis of a type of nonlinear eigenvalue problem called a globally coupled eigenvalue problem (GCEP). The result of coupling the cells to a bulk diffusion field induces an 'effective' parameter that can be viewed as function of the colony population. We demonstrate QS, as well as the more general concept of efficiency sensing, when this parameter passes through a saddle-node bifurcation. In the limit of large but finite bulk diffusion, the cell-bulk system can be well-approximated by a simpler system of ODEs. This reduced system is used to study the effect of cell location on QS behaviour. Moreover, in this regime the critical population required for a 'quorum' can be computed from a simple analytical formula which offers much physical insight. We verify the predictions from asymptotic theory with full numerical solutions of the cell-bulk system and find good agreement between the two.Science, Faculty ofMathematics, Department ofGraduat