A multi-phase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm
Introduction
In this paper we continue to develop our work on Pseudomonas aeruginosa (henceforth PA) infections, thus far having considered antibacterial therapies targeted at the primary quorum-sensing system of a well-mixed, planktonic population [1] and an early-stage (closely packed) biofilm [2]. In particular, we take the model of [2] and now allow for (QS-controlled) EPS production and the entrainment of water as the biofilm develops. We then proceed to model antibacterial treatment of the biofilm with diffusing, topically applied antibiotics and anti-quorum sensing drugs. The modelling is relevant to the treatment of a chronic burn-wound infection, for example.
We recall that quorum sensing (QS) is a cell-to-cell communication phenomenon whereby bacterial cells are able indirectly to monitor their own population density and regulate their behaviour in a multicellular fashion as the density changes [22], [23]; in an infection scenario QS allows pathogens such as PA to switch on virulence genes rapidly and in unison, thus increasing their chances of overcoming the host response [17]. The development of drugs specifically designed to interfere with QS (and thus reduce virulence) in infecting bacteria could therefore be of considerable medical value, given the continuing increase in resistance to conventional antibiotics exhibited by many pathogenic species (including PA) [10], [14], [15], [16]. This observation provided the motivation for our previous modelling work, as presented in [1], [2].
As described in [1], the QS system of PA involves several signal molecules, two of which are acylated homoserine lactones (AHLs), existing in a regulatory hierarchy [23]. The primary system consists of an AHL-synthase, LasI, responsible for production of N-(3-oxododecanoyl)-homoserine lactone (3-oxo-C12-HSL), and a transcriptional activator LasR. The AHL signal molecule acts by binding to, and so activating, the LasR protein. This in turn binds to a specific chromosomal DNA sequence (known as a lux-box) upstream of target genes such as lasI (the 3-oxo-C12-HSL synthase) and possibly also lasR, thus enhancing their transcription. It is believed by many that the lasI/lasR system is necessary for successful biofilm development, it having been shown, in some flow chamber experiments at least, that biofilms formed by wild-type strains are much thicker, more dispersed and less susceptible to treatment with antibiotics or surfactants than those formed by lasI mutants [6], [7], [11], [18]. We note, however, that some experimentalists observe no significant physiological difference between wild-type and lasI-negative biofilms, obtaining for each PA genotype a thin biofilm with a homogeneous, closely packed structure [12]. It is not at all clear how to account for this rather contradictory picture, although one might speculate that differences in flow chamber and nutrient conditions could have a role to play.
In [1] we formulated and analysed a mathematical model of the reaction kinetics of this system in a growing batch culture population subject to treatment with two kinds of anti-QS drug (specific examples of which are known to exist in nature), along with a conventional antibiotic. The first kind of anti-QS drug (modelled on a certain halogenated furanone produced by the macroalga Delisea pulchra) diffuses into bacterial cells and degrades/sequesters the LasR protein, while the second kind is an enzyme (such as homoserine lactonases produced by certain Bacillus species) which remains outside the cell and degrades the AHL signal molecule directly (see Fig. 1 of [1] for a schematic of the relevant reaction pathways). In [2] we factored these basic kinetics into a reaction–diffusion–advection model of an early-stage PA biofilm growing on a flat, impermeable surface, and in Section 2 of the present paper we develop the model by allowing for the production and secretion of EPS, which generates a watery extracellular matrix surrounding the bacteria.
It is worth noting that, although there is by now quite an extensive literature on both QS and biofilm modelling, some of it being devoted to modelling bacterial EPS production explicitly, very little attention has been paid to a possible role for QS in EPS production/biofilm maturation. For example, Ward et al. [20] considered the role of QS in an early-stage (closely packed) biofilm, with no EPS production, whereby they observed travelling waves of QS behaviour in a thin, flat PA population, spreading laterally as a consequence of binary fission. Dockery and Klapper [9] treated their biofilm as a single viscous incompressible fluid satisfying Darcy’s Law, taking bacterial growth to be dependent on oxygen diffusing in from the upper surface; the nutrient limitation was shown to give rise to a fingering instability, which allowed for the growth of mushroom-like structures in the biofilm profile. This model was developed further in [5], where the biofilm was considered as a biological gel, consisting of networked polymer (EPS), a fluid solvent (water) and a scattering of bacterial cells; in the modified model EPS-induced osmotic pressure is taken to be an additional factor contributing to the redistribution of biomass, and one sees the same kind of frontal instability as in [9]. Kreft and Wimpenny [13] employed an individual-based model in which the biofilm is taken to consist of spherical bacterial cells and blobs of EPS. Mitotic growth and EPS production are dependent on a diffusing nutrient, and biofilm motion is generated as neighbouring cells push each other apart. Simulations reveal that bacteria which produce EPS at a high rate form inhomogeneous biofilms in which live cells are concentrated near the (nutrient-rich) surface, while biofilms generated by bacteria producing very little EPS tend to be homogeneous, with a closely packed population throughout. The approach taken by Chopp et al. [4] was to incorporate a model of QS in a two-phase biofilm model, the first phase consisting of live bacterial cells and the second of dead cells, EPS and water; in their model there is a term accounting for EPS production, but neither the dependence of EPS production on QS nor the dependence of biofilm growth on EPS production is investigated.
In an attempt to address this situation, the model biofilm of this paper is taken to consist of four different phases, namely live cells, dead cells, EPS and watery extracellular fluid. Bacterial growth is driven by a nutrient (oxygen) diffusing into the biofilm from its upper surface, and bacterial cells are subject to attack by a conventional antibiotic, which is also assumed to diffuse in from the surface. Cell growth and death, along with EPS production/secretion, generate bulk motion of the biofilm constituents and, under a physically reasonable simplifying assumption on the void fraction of the EPS matrix, one obtains a closed system of equations for the volume fractions and velocities of the four phases. As far as QS is concerned, the AHL signal molecule is produced by live cells (at a rate dependent on the local intracellular AHL and oxygen concentrations) and can diffuse into and out of live and dead cells, as well as through the extracellular fluid; see [21] and references therein for alternative approaches to QS modelling. For simplicity the AHL concentration is taken to be zero on the biofilm surface, which could be approximated in practice by washing, by rapid natural AHL-degradation in the surroundings, or by applying a high concentration of anti-AHL drug on the surface. EPS production is assumed to be dependent on the level of LasR/AHL dimer and on the local nutrient level.
In Section 3 we use our model equations to simulate numerically bacterial growth and QS behaviour, in particular investigating the differences between biofilms produced by wild-type PA (with an intact QS system) and those produced by lasI-negative mutants (with a deficient QS system). The results turn out to be in good qualitative agreement with those of [7], for example.
In Section 4 we simulate anti-QS treatment of a growing biofilm. As one expects, diffusing anti-QS drugs reduce EPS production, resulting in much thinner, uniform and closely packed biofilms, which are presumably more susceptible to dispersal with surfactants. Section 4 also contains a brief numerical investigation of conventional antibiotic treatment. We show that a topically applied antibiotic slows biofilm growth, and that a sufficiently high surface antibiotic concentration will halt biofilm growth altogether and lead to the eradication of all bacterial cells. Section 5 contains an analysis of the travelling-wave behaviour observed in the simulations, and Section 6 contains our conclusions.
Finally, here we note that, as in [2], the choice of parameter values for the basic QS and anti-QS kinetics in our simulations (as listed in the figure captions) was guided by the choices which were made in [1] for our simulations of the corresponding homogeneous model equations. For the parameters determining bacterial growth rates and chemical diffusivities we used the estimated values which were employed in [2]. The effect of varying the EPS-production parameters on biofilm growth and antibacterial efficacy is investigated throughout the paper.
Section snippets
Mass conservation
We now derive our mathematical model for a maturing PA biofilm subject to treatment with antibiotic and anti-quorum sensing drugs; the modelling of biomass growth dynamics is in the spirit of [2], [4], [20] and that of quorum-sensing kinetics follows [1], [8]. Firstly, the bacteria are assumed to be growing in a densely packed fashion on a flat surface given by (Cartesian coordinate) x = 0. The population is also assumed to be infinite in extent and homogeneous in the y and z directions, whereas
Biofilm growth; dependence of EPS production on quorum sensing
The equations obtained in Section 2 are solved numerically, using an iterative method already successfully employed in [2], [20]; the hyperbolic equations are treated with an implicit upwind method, and the diffusion equations with NAG routine D03PCF, which is based on the method of lines. Also, the two advective velocities v and vw are obtained from the integral expressions (6), (7) by means of the trapezium method. Our first task is to check that the model equations reproduce behaviour which
Treatment strategies
From the observations of the previous section one expects anti-QS drugs to be effective in slowing biofilm growth, thus resulting in thinner, less robust biofilms. This is indeed the case, as we illustrate in this section.
Formulation
The numerical solutions of this study and those of [2] suggest that in the large-time limit the biofilm depth increases at a linear rate for a wide range of parameter values. In this section we explore such large-time behaviour in the slow-growth limit, and determine upper and lower bounds on the growth speed, in the formas t → ∞, where the associated speeds C0, C1 are positive constants.
Following the approach of Section 4 of [2], we consider the behaviour on a long timescale in
Conclusions
We have derived a multi-phase model of a developing one-dimensional P. aeruginosa biofilm, and have used numerical simulations to examine the role played by quorum-sensing controlled EPS production in the maturation process, and in the response to antibacterial- and anti-QS treatments.
For an untreated wild-type PA biofilm (with an intact LasRI system) our simulations produce results which agree qualitatively with many experimental observations (see [7], for example); most of the biomass deep
Acknowledgments
The authors gratefully acknowledge support by a Leverhulme Trust Special Research Fellowship (SRF/40048) (KA) and by the Medical Research Council (JRK).
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