A finite-difference computer model of solute diffusion in bacterial films with simultaneous metabolism and chemical reaction

We present a mathematical model for growth and control of facultative anaerobic bacterial biofilms in nutrient rich environments. The growth of the microbial population is limited by protonated lactic acids and the local pH value, which in return are altered as the microbial population changes. The process is described by a non-linear parabolic system of three coupled equations for the dependent variables biomass density, acid concentration and pH. While the equations for the dissolved substrates are semi-linear, the equation for bacterial biomass shows two non-linear diffusion effects, a power law degeneracy as the dependent variable vanishes and a singularity in the diffusion coefficient as the dependent variable approaches its a priori known threshold. The interaction of both effects describes the spatial spreading of the biofilm. The interface between regions where the solution is positive and where it vanishes is the biofilm/bulk interface. We adapt a numerical method to explicitly track this interface in x -t space, based on the weak formulation of the biofilm model in a moving frame. We present numerical simulations of the spatio-temporal biofilm model, applied to a probiotic biofilm control scenario. It is shown that in the biofilm neighbouring regions co-exist in which pathogenic bacterial biomass is produced or killed, respectively. Furthermore, it is illustrated how the augmentation of the bulk with probiotic bacteria leads to an accelerated decay of the pathogenic biofilm.

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“…[9,48] and with simulation studies in Ref. [10] for the neutral pH range. Note that the model in Ref.…”

Section: General System Description and Model Behaviormentioning

confidence: 82%

Modeling and Simulation of a Bacterial Biofilm That Is Controlled by pH and Protonated Lactic Acids

Khassehkhan

Eberl

2007

Computational and Mathematical Methods in Medicine

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“…The computer simulation of the two-stage diffusion process allowed an improved fit to be achieved and, as is so often the case with such numer ical modelling methods, gave extra insight into the pro cesses involved. It models simple clearance experiments to better than 0.1% accuracy [Dibdin, 1992].…”

Section: Discussionmentioning

confidence: 99%

The Water in Human Dental Enamel and Its Diffusional Exchange Measured by Clearance of Tritiated Water from Enamel Slabs of Varying Thickness

Dibdin¹

1993

Caries Res

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Tracer clearance studies were used to measure the amount and mobility of the water in human dental enamel. Slabs of enamel whose thickness ranged from 0.4 to 0.8 mm were equilibrated with tritiated water in dilute neutral buffer at 35°C for 1 week, rinsed and allowed to exchange at the same temperature with stirred unlabelled buffer. This was sampled and counted periodically to allow calculation of diffusion coefficients from the clearance rate. Exchangeable water amounted to 2.0 ± 0.4% w/w (6.0 ± 1.2% v/v). Some samples were masked on one side to double the diffusion distance. Diffusion calculations assuming homogeneous diffusion gave anomalous coefficients varying with sample thickness, indicating a more complex system was involved. In contrast, analysis assuming a biphasic distribution of diffusion pathways was much more successful. This assumed slow exchange from an inner pore system ‘A’ (probably associated with the prism cores) followed by much faster diffusion along a minority of larger pores ‘B’, such as exist at the prism boundaries. The ‘B’ pore-volumes were based on past studies of porosity in enamel, and the ‘A’ volumes on this work. Analysis gave an overall effective diffusion coefficient (D_e) for thick samples of 4.3 × 10^––7 cm² s^––1. The results were consistent with a D_B of 0.9 × 10^––5 cm² s^––1 and a rate-constant (k_AB) for exchange between the two pore systems which, if operating over a half-prism width (say 2 μm) would correspond to an ‘inner’ (D_AB) diffusion coefficient in the region of 6 × 10^––12 cm² s^––1.

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“…This will not apply to other computer modelling studies, e.g. those of Dibdin [1990Dibdin [ , 1992, where glycolytic activity was measured in the presence of a 1% peptone supplement. As found in previous studies with several types of chewing gum [Dawes and Macpherson, 1992], peak salivary flow rates with both gums X and Z occurred during the first minute of gum chewing.…”

Section: Discussionmentioning

confidence: 99%

“…Dibdin [1990Dibdin [ , 1992). However, this gave modelled Stephan curve minima more than half a pH unit higher than observed experimentally.…”

Section: Statistical Analysesmentioning

confidence: 99%

“…The computer model itself was based on that of Dibdin [1990Dibdin [ , 1992 as later modified for the study of the effect on a Stephan curve of: (a) chewing sugar-containing and sugar-free gums [Dibdin et al, 1995] and (b) the breakdown of different concentrations of salivary urea by plaque [Dibdin and Dawes, 1998]. For the present study, these were combined to include modelling of time-dependent changes in the concentration of salivary urea due to the use of a urea-containing chewing gum, together with the effects of sucrose rinses.…”

Section: Statistical Analysesmentioning

confidence: 99%

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Salivary Concentrations of Urea Released from a Chewing Gum Containing Urea and how These Affect the Urea Content of Gel–Stabilized Plaques and Their pH after Exposure to Sucrose

Dawes

Dibdin

2001

Caries Res

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The objectives were to: (1) determine the salivary concentrations of urea during 20 min chewing of a sugar–free gum containing 30 mg of urea; (2) measure the degree to which this urea would diffuse into a gel–stabilized plaque; (3) study the effect of the urea on the fall and subsequent rise in pH (Stephan curve) on exposure to 10% sucrose for 1 min; (4) model the measurements 2 and 3 mathematically. In point 1, the salivary urea concentration of the 12 subjects peaked at 47 mmol/l in the first 2 min of gum chewing, falling within 15 min to the unstimulated salivary concentration of 3.4 mmol/l. Recovery of urea from the saliva averaged 81.5%. ‘Plaques’ of 1% agarose or 67% dead bacteria in agarose accumulated urea from the saliva roughly as expected, whereas those plaques containing 8% live and 59% dead Streptococcus vestibularis showed negligible accumulation. Computer modelling showed this difference to be due to urease of live bacteria breaking down the urea as rapidly as it entered the plaque. Simulation of the effect of gum chewing subsequent to initiation of a Stephan curve in the latter type of plaque showed a rapid rise in pH but then a fall again on return to unstimulated conditions. This fall had not been seen in previous studies, with Streptococcus oralis, nor was it predicted by the computer modelling. Neither experimental simulation nor computer modelling suggested that chewing urea–containing gum before exposure to sucrose would have any effect on a subsequent Stephan curve. Thus chewing gum is only likely to inhibit caries when it is chewed after consumption of fermentable carbohydrate, rather than before.

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A finite-difference computer model of solute diffusion in bacterial films with simultaneous metabolism and chemical reaction

Cited by 13 publications

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Modeling and Simulation of a Bacterial Biofilm That Is Controlled by pH and Protonated Lactic Acids

Modeling and Simulation of a Bacterial Biofilm That Is Controlled by pH and Protonated Lactic Acids

The Water in Human Dental Enamel and Its Diffusional Exchange Measured by Clearance of Tritiated Water from Enamel Slabs of Varying Thickness

Salivary Concentrations of Urea Released from a Chewing Gum Containing Urea and how These Affect the Urea Content of Gel–Stabilized Plaques and Their pH after Exposure to Sucrose

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