A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion.

We develop and analyse a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The formal continuum limit of the model is a nonlinear diffusion equation with a diffusivity which can become negative if the adhesion coefficient is sufficiently large. This appears to be related to the presence of spatial oscillations and the development of plateaus (pattern formation) in numerical solutions of the discrete model. A combination of stability analysis of the discrete equations and steady-state analysis of the limiting PDE (and a higher-order correction thereof) can be used to shed light on these and other qualitative predictions of the model.

A multi-phase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm.

It is well known that sessile bacteria have a strong tendency to exist in a biofilm phenotype, whereby bacterial cells aggregate and produce a gel-like extracellular matrix, which, in an infection scenario, offers a significant barrier to attack by conventional antibiotics and the immune system. In this paper we develop a multi-phase model of a maturing Pseudomonas aeruginosa biofilm, allowing for the production and secretion of exopolysaccharide (EPS). The primary quorum-sensing system of P. aeruginosa (namely the lasR system) is believed to be required for full biofilm development, and we thus take the synthesis of EPS to be regulated by the cognate signal molecule, 3-oxo-C12-HSL. We also take EPS and signal production, along with bacterial growth, to be limited by oxygen availability, thus factoring in the nutrient poor conditions deep inside the biofilm. We use simulations to examine the role played by quorum sensing in the biofilm maturation process, and to investigate the effect of anti-quorum sensing and antibiotic treatments on EPS concentration, signal level, bacterial numbers and biofilm growth rate. In addition, we undertake analysis of the associated travelling-wave behaviour.

Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured Pseudomonas aeruginosa population.

The bacterial cell to cell signalling system known as quorum sensing (QS) is essential for the regulation of virulence in many pathogens and offers a specific biochemical target for novel antibacterial therapies. Expanding on earlier work, in which consideration was given to the primary QS system (lasR system) in a homogeneous population of the common human pathogen Pseudomonas aeruginosa, we build a simple spatial model of an early-stage P. aeruginosa biofilm subject to treatment with topically applied anti-QS drugs (of two specific kinds) and conventional antibiotics. In the case of a slowly growing biofilm we show that both kinds of anti-quorum sensing drug are effective in reducing the level of the relevant signal molecule (3-oxo-C12-homoserine lactone; henceforth AHL), in each case obtaining an explicit bound on the steady-state AHL profile in terms of a prescribed surface drug concentration. Using numerical methods, we are also able to reproduce the hysteretic phenomena exhibited by the homogeneous model, in particular showing that for each kind of anti-QS drug there is a parameter regime in which a catastrophic collapse occurs in the steady-state AHL concentration as the surface drug concentration passes some critical value; an alternative way of interpreting this result is to say that, for a prescribed surface drug concentration, there is a critical biofilm depth such that treatment is successful until this depth is reached, but fails thereafter. In the thick-biofilm limit we show that the critical concentration of each drug increases exponentially with the biofilm thickness (or, conversely, that the critical depth increases logarithmically with surface drug concentration); this is dramatically different to the behaviour observed in the corresponding homogeneous model, where the critical concentrations grow linearly with bacterial carrying capacity, and thus highlights the relative difficulty of treating a large, spatially-structured population with diffusing antibacterials.

Mathematical modelling of therapies targeted at bacterial quorum sensing.

Bacteria commonly use diffusible signal molecules to synchronise their behaviour by facilitating population dependent co-ordination. This cell-to-cell signalling mechanism is known as quorum sensing (QS) and provides a way of ensuring that certain genes are 'switched on' only when a certain signal concentration (typically corresponding to a large population density) has been reached. In this paper we focus on the QS system of the human pathogen Pseudomonas aeruginosa, which employs a complex hierarchy of QS signalling systems, which regulate the formation of multiple exoproducts, swarming and biofilm differentiation. In P. aeruginosa, the signal molecules are N-acylated homoserine lactones (AHLs; e.g., N-(3-oxododecanoyl)-homoserine lactone [3-oxo-C12-HSL]), which bind to transcriptional regulator proteins (LasR in the case of 3-oxo-C12-HSL) to activate the expression of target genes including lasI, which codes for the 3-oxo-C12-HSL synthase. Since the virulence of P. aeruginosa is controlled by QS, agents (QSBs) designed to block this cell-to-cell communication have potential as novel antibacterials. By drawing on existing models for the reaction kinetics of this system, we model a growing population subject to treatment with two kinds of QSB, together with a conventional antibiotic. The first kind of QSB is assumed to act by diffusing through the cell membrane and then destabilising/sequestering LasR, while the second kind remains outside the cell and degrades the AHL signal molecule itself. Numerical and mathematical analysis of the resulting systems of ordinary differential equations reveals in particular that, while a sufficiently high dose of QSB is, in all cases considered, able to reduce the AHL concentration (and hence virulence) to a negligible level, the qualitative response to treatment is sensitive to parameter values.