The effect of pH, temperature and plaque thickness on the hydrolysis of monofluorophosphate in experimental dental plaque.

Monofluorophosphate (MFP), an anti-caries agent commonly used in toothpaste, is known to be degraded to fluoride and orthophosphate by bacterial phosphatases in dental plaque. We have examined the effect of pH, temperature, plaque thickness and some ions on this process. Both natural plaque and artificial microcosm plaque incubated with purified MFP at pH 4-10 showed an optimum pH of approximately 8 for hydrolysis. Diffusion and concomitant hydrolysis were examined in an apparatus in which artificial plaque was held between rigid membranes separating two chambers. When MFP diffused through a plaque of 0.51-mm thickness over 4 h it was almost completely hydrolysed at pH 8, but hydrolysis on diffusion decreased as the pH deviated from 8. MFP in toothpaste extract showed a similar pH susceptibility to hydrolysis, according to the inherent pH of the toothpaste. Hydrolysis of MFP in the toothpaste was reduced by no more than 10% when compared with a matched-pH control, suggesting that other toothpaste ingredients had no major influence on hydrolysis. Transport was slower and hydrolysis at pH 6 more complete the thicker the plaque, but hydrolysis was not significantly slower at 23 degrees C than at 37 degrees C. The addition of various potential activating or inhibiting ions at 0.1 and 1.0 mmol/l had small and non-significant effects on hydrolysis. The results suggest that MFP toothpaste should be formulated and used to maximise enzymic hydrolysis of this complex anion, and that plaque pH control is probably the most important factor.

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.

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.

An in vitro system for the analysis of changes in depth distribution of diffusates in bacterial films.

No suitable technique exists which allows simultaneous analysis of time- and depth-dependent concentrations of all components of interest in single samples of intercellular fluid from dental plaque biofilms. We have developed an in vitro model which allows detailed study of these interactions by analysis of intercellular fluid and pH measurement at the film base. Compact, defined-depth films of Streptococcus mutans were formed in a micrometer-controlled, variable-depth well and bathed in synthetic saliva. Films exposed to synthetic saliva containing glucose for 2 min followed by clearance with glucose-free synthetic saliva gave typical 'Stephan-type' pH profiles. Intercellular fluid isolated from successive 200-microm-thick sections of 600-microm-deep films was analysed by ion chromatography. A concentration gradient of lactate, falling with depth, was measured. The experimental system described here, the first of its type to be described, can easily be used to analyse pH changes and the depth-dependent distribution of diffusates in a model bacterial film. Although this bacterial film is far removed from a natural biofilm, the apparatus has potential for the study of grown biofilms and is an important advance towards position-dependence analysis of diffusates in biofilms.

A mathematical model of the influence of salivary urea on the pH of fasted dental plaque and on the changes occurring during a cariogenic challenge.

Urea diffusing from saliva into dental plaque is converted to ammonia and carbon dioxide by bacterial ureases. The influence of normal salivary urea levels on the pH of fasted plaque and on the depth and duration of a Stephan curve is uncertain. A numerical model which simulates a cariogenic challenge (a 10% sucrose rinse alone or one followed by use of chewing-gum with or without sugar) was modified to include salivary urea levels from 0 to 30 mmol/l. It incorporated: site-dependent exchange between bulk saliva and plaque surfaces via a salivary film; sugar and urea diffusion into plaque; pH-dependent rates of acid formation and urea breakdown; diffusion and dissociation of end-products and other buffers (acetate, lactate, phosphate, ammonia and carbonate); diffusion of protons and other ions; equilibration with fixed and mobile buffers; and charge-coupling between ionic flows. The Km (2.12 mmol/l) and Vmax (0.11 micromol urea/min/mg dry weight) values for urease activity and the pH dependence of Vmax were taken from the literature. From the results, it is predicted that urea concentrations normally present in saliva (3-5 mmol/l) will increase the pH at the base of a 0.5-mm-thick fasted plaque by up to 1 pH unit, and raise the pH minimum after a sucrose rinse or sugar-containing chewing-gum by at least half a pH unit. The results suggest that plaque cariogenicity may be inversely related to salivary urea concentrations, not only when the latter are elevated because of disease, but even when they are in the normal range.

Mathematical modeling of biofilms.

A set of mathematical equations constitutes a mathematical model if it aims to represent a real system and is based on some theory of that system's operation. On this definition, mathematical models, some very simple, are everywhere in science. A complex system like a biofilm requires modeling by numerical methods and, because of inevitable uncertainties in its theoretical basis, may not be able to make precise predictions. Nevertheless, such models almost always give new insight into the mechanisms involved, and stimulate further investigation. The way in which diffusion coefficients are measured for use in a model, particularly whether they include effects of reversible reaction, is a key element in the modeling. Reasons are given for separating diffusion from reversible reaction effects and dealing with them in a separate subroutine of the model.

Effect of pH and calcium concentration on calcium diffusion in streptococcal model-plaque biofilms.

Demineralization during a cariogenic episode is affected by storage and transport in dental plaque of ions released from enamel, and by the effect on both of plaque fluid pH and ion concentrations. To investigate this, 45Ca effusion from a condensed film of streptococci was measured at pH 7, 6 and 5, and 0-20 mmol/l calcium. Cells were loaded into effusion chambers and the appearance of 45Ca and [3H]-inulin in carrier-containing but initially tracer-free buffer was measured. Ratios of 45Ca and [3H]-inulin activity in the initial suspending solution and at equilibrium in the clearance solution, permitted calculation of extracellular volume and bound calcium. The rate of Ca appearance was proportional to the retarded diffusion coefficient (rDe), which was related to the effective diffusion coefficient (De) by: rDe = De/(1 + R) in which R is the ratio of bound to free Ca2+. The rate of Ca2+ effusion increased with calcium concentration, converging on a value of 2.8 x 10(-10) m2/sec. At low pH it reached convergence at a lower [Ca]. This demonstrates that calcium effusion is dependent on binding, so a high proportion of binding sites in plaque will reduce mineral loss in vivo. Loss of binding sites at low pH will increase mineral loss.

Mathematical model of beta-lactam penetration into a biofilm of Pseudomonas aeruginosa while undergoing simultaneous inactivation by released beta-lactamases.

We present a mathematical model that describes penetration of an antibacterial agent into a bacterial biofilm and, in particular, the penetration of a beta-lactam compound into a biofilm of Pseudomonas aeruginosa. As well as dealing with this penetration, and the consequent bacterial lysis, the model considered diffusion of the released beta-lactamases in the extracellular space and the consequent inactivation there of further incoming antibiotic; it also allowed for any chosen fraction of the total beta-lactamase to be permanently accessible to exogenous substrate. The modelling scheme was validated against analytical solutions under appropriately simplified conditions. Using published experimental data we show here that lysis of cells in the surface layers of a film could have an important protective effect on the viability of underlying bacteria, especially in thick biofilms.

Inhibition of acid production in Streptococcus mutans R9: inhibition constants and reversibility.

End-product inhibition of acid production in Streptococcus mutans R9 was investigated by measuring effects of varying concentrations of H+ and of undissociated lactic or acetic acids on acid production rates in the pH stat. H+ caused purely uncompetitive inhibition (inhibition constant Kiu 0.018 mmol 1(-1). Lactic acid caused mixed inhibition with inhibition constants of Kiu 4.24 mmol 1(-1) and Kic 4.55 mmol 1(-1). Reversibility of inhibition by H+ showed only a statistically significant reduction only at pH < 4.5. Reversibility of inhibition by lactic and acetic acids decreased with increasing inhibitor concentration. In all cases, reversibility correlated with the extent to which viability was retained, suggesting that loss of reversibility was due to cell death. These results suggest that, after a low-pH episode in dental plaque, caused by fermentation of dietary carbohydrate, the ability of plaque organisms to produce acid in subsequent exposures to carbohydrate may be reduced.

Computer modeling of the effects of chewing sugar-free and sucrose-containing gums on the pH changes in dental plaque associated with a cariogenic challenge at different intra-oral sites.

Variation in salivary access to different intra-oral sites is an important factor in the site-dependence of dental caries. This study explored, theoretically, how access is modified by chewing sugar-free and sugar-containing gums. A finite difference computer model, described elsewhere, was used. This allowed for diffusion and/or reaction of substrate, acid product, salivary buffers, and fixed-acid groups. Site-dependent saliva/plaque exchange was modeled in terms of a 100-microns-thick salivary film covering the plaque (a) flowing directly from the salivary ducts, (b) flowing from the intra-oral salivary pool, or (c) exchanging with the pool. Computed flow-velocities or rates of exchange were based on previous intra-oral measurements. The model was also tested against an in vitro study conducted by two of the authors. In addition, the three proposed models of saliva/plaque interaction were compared, and the effect of salivary film thickness investigate. Results suggested that: (1) although sugar-free gum chewed during a cariogenic challenge causes a rapid rise in plaque pH, sucrose-containing gums cause the pH, after a temporary rise resulting from increased salivary flow, to stay low for an extended period; (2) the computer model reproduced in vitro tests reasonably well; (3) although the three models of the plaque/saliva interaction start from different assumptions, two lead to closely related predictions; and (4) increasing the assumed salivary film thickness by a large amount (e.g., from 50 to 200 microns) caused no change in modeled Stephan curves, as long as these changes were accompanied by appropriate reductions in film velocity, in accord, theoretically, with the practical clearance data.

Calcium and water diffusion in single-species model bacterial plaques.

The diffusion of tritiated water and 45Ca through single-species model plaques was measured in a diaphragm cell. Using plaques of Streptococcus downei, the apparent diffusion coefficient for tritiated water showed a small but significant decrease between pH 7.0 and 5.0 with a smaller, non-significant decrease for calcium. These effects are attributed to possible plaque shrinkage at reduced pH, which overcomes effects due to charge-dependent changes in permeability. Effective diffusion coefficients for 45Ca obtained from lag-time measurements were much reduced at low carrier concentrations, demonstrating a binding which correlated with previous results from equilibrium dialysis. Strep. sanguis and Corynebacterium matruchotii showed few differences from Strep. downei, although the corynebacterium previously demonstrated stronger calcium binding. Permeability to water and calcium were slightly affected by pH, while effective diffusion of calcium depended on concentration. This may have important implications for the process of mineral loss in dental caries.

The diffusion and enzymic hydrolysis of monofluorophosphate in dental plaque.

Although the ability of dental plaque to hydrolyze sodium monofluorophosphate (MFP) has been known for some time, its effect on the F- concentration at the plaque-enamel interface is undefined. We have determined enzyme kinetic values for MFP hydrolysis and diffusion coefficients so that the penetration and degradation of MFP in plaque can be modeled by computer. The KM and Vmax values for natural human plaque were 1.77 mmol/L and 41.4 nmol/min/mg protein, respectively, at pH 8.0. At pH 6.0, the Vmax was lower, 15.6 nmol/min/mg, but KM was not significantly different. Competitive inhibition by orthophosphate gave a Ki of 4.55 mmol/L. The diffusion coefficient for MFP in artificial plaque was 1.91 x 10(-6) cm2/sec. When these data were used for mathematical modeling of the effects of rinsing with MFP and F- solutions, compared with an equivalent NaF application, the concentration of F- from MFP was lower at the inner surface of plaque, and the peak occurred later. Both pH and plaque thickness had a marked effect on the amount of MFP that could penetrate: At pH 8.0, almost none reached the inner surface of a 1-mm-thick plaque intact. At pH 6.0, however, more MFP was able to penetrate, due to lower MFPase activity. While MFP diffusion is inherently slower than that of F-, enzymic degradation increases the gradient for inward diffusion. If the conventional view that MFP in toothpaste acts as a source of F- is true, then MFP toothpaste should be formulated to optimize MFPase activity in dental plaque.

Effect of the bathing fluid on measurements of diffusion in dental plaque.

The suggestion that loss of labile components from dental plaque into contacting aqueous buffer during some diffusion measurements might change the plaque's diffusion behaviour, and therefore invalidate the results, has been tested using tracer clearance. (a) Diffusion of [14C]-acetate and 3H2O was determined before and after a 3-4 h equilibration with neutral buffer. No differences or trends were found. (b) Mean normalized [14C]-acetate clearance curves from live plaque for measurements before and after such equilibration were identical in shape and showed no change in diffusion coefficient during clearance, as evidenced by their fits to the theoretical clearance curve. No significant difference was found between coefficients for the two periods (paired sample test). In addition, [14C]-lactate and 3H2O were found to permeate plaque in a miniature diaphragm diffusion cell at rates independent of whether the plaque was bathed either in saliva supernatant, plaque fluid or neutral buffer. This lent some further support to the main findings. It is concluded that diffusion through dental plaque is little affected by the bathing fluid.

Short algorithm to compensate for sampling-volume errors in diffusion-cell studies.

This communication describes a short algorithm enabling arbitrarily large but equal samples to be taken from the two chambers of a diaphragm diffusion cell without introducing any systematic error. It creates a transformed time-scale, dependent on sampling volume, which linearizes the output, fits this to a straight line by simple linear regression, and computes the diffusion coefficient from the slope. The advantages of sampling equal fractions from both chambers are discussed in terms of the greater precision this allows. The procedure may have uses in other fields.

The water in human dental enamel and its diffusional exchange measured by clearance of tritiated water from enamel slabs of varying thickness.

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 degrees 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 (De) for thick samples of 4.3 x 10(-7) cm2 s-1. The results were consistent with a DB of 0.9 x 10(-5) cm2 s-1 and a rate-constant (kAB) for exchange between the two pore systems which, if operating over a half-prism width (say 2 microns) would correspond to an 'inner' (DAB) diffusion coefficient in the region of 6 x 10(-12) cm2 s-1.

A quantitative study of calcium binding and aggregation in selected oral bacteria.

By means of micro-equilibrium dialysis, calcium binding capacities and affinities were measured in three different oral bacteria, and the effects of extracellular polysaccharide, pH, and aggregation were investigated. Binding capacities of 31.0 +/- 2.1 (C. matruchotii), 34.7 +/- 3.7 (S. sanguis), and 41.5 +/- 5.4 (S. downei) mumol calcium/g wet weight of cells were found at pH 7.0, falling to 21.4 +/- 0.8 mumol calcium/g wet wt. cells at pH 5.0 for S. downei. Dissociation constants were found to vary between 0.78 +/- 0.24 and 1.77 +/- 0.30 mmol/L (at pH 7.0, depending on species), and between 0.62 +/- 0.04 and 1.77 +/- 0.30 (in the pH range 5.0 to 7.0, for S. downei only). Examination suggested that at pH 7.0 calcium-facilitated bacterial association occurs in the streptococci with calcium uptake curves analogous with those of positively cooperative systems. Desorption of calcium from aggregated S. downei suggested that the mechanism of desorption differed from that of uptake. This may be an important factor in plaque formation and in the binding of cells to the surface of formed plaque. Plaque calcium forms a reservoir, readily released by a pH drop, which may increase plaque fluid saturation and reduce demineralization.

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

This finite-difference computer model is designed to simulate complex diffusion/reaction events in bacterial films. It is modular, each module mirroring closely a particular physical, chemical or biochemical factor. It is capable of handling > 20 diffusing/reacting species, but can be easily expanded or simplified to match particular systems. It was originally designed for modelling the events in dental plaque leading to tooth decay, but should find application in other fields. It allows for ion-exchange interactions with, for example, fixed charges on bacterial surfaces, which can act as pH and cation buffer sites. pH-dependent utilization of substrate is modelled implicitly, combining Michaelis-Menten kinetics with diffusion in a single iterative procedure. Advantages are given for computing diffusion of all other species explicitly using single-species diffusion coefficients, with charge-coupling by means of the algorithm Q-COUPLE. Activity corrections and enzyme pH-dependence are included. Chemical equilibria and mineral deposition/dissolution are computed iteratively node by node. The program is tested against some problems having analytical solutions, and an example is given of its application to demineralization of teeth as a result of bacterial action in dental plaque.

Precise charge-coupling calculations for finite difference diffusion problems using a modification of the add-on algorithm Q-COUPLE.

This note describes a minor modification to the recently published algorithm 'Q-COUPLE' designed for adding charge-charge interactions between diffusing species to time-dependent one-dimensional finite difference diffusion calculations. The original proposal concerned a simple way of doing this for Crank-Nicolson central time-difference schemes, and gave useful, but only approximate, agreement with theory in tests where the charge-coupling could also be calculated analytically. The new, slightly modified algorithm, when used with explicit (forward time-difference) modelling, gave analytically exact results for the charge interaction part of a similar trial calculation.

Plaque fluid and diffusion: study of the cariogenic challenge by computer modeling.

Every cariogenic challenge involves a mixture of convective transport, diffusive transport, and biochemical reactions, plus physico-chemical reactions (including charge-coupling of diffusion), all of which together require numerical methods for their analysis. This presentation describes a one-dimensional finite-difference computer model of the cariogenic process, and some conclusions obtained from it. Sugar clearance from the mouth, together with site-dependent exchange between the bulk saliva and plaque surface via a salivary film, is combined with a finite-difference model of events occurring within the dental plaque. The latter includes: sugar diffusion and pH-dependent acid production; diffusion and dissociation equilibria for two acid end-products of fermentation and their anions (acetate and lactate); diffusion and dissociation equilibria of phosphate buffer; diffusion of potassium and chloride; diffusion of protons and simultaneous equilibration with fixed and mobile buffers. So that proper concentration distributions consistent with local charge neutrality can be ensured, an algorithm called Q-couple is used to impose charge-coupling between the fluxes of different ions including fixed charges. Mineral dissolution and precipitation are modeled as part of the same equilibrium calculations. The predictions of the model are compared with those of an earlier, much simpler one, in which fixed buffers were not included. It is shown that the known concentration of fixed buffer greatly extends the low pH of a Stephan curve. The isoelectric point of the plaque bacteria also appears to be of importance. The effects of various concentrations of mobile buffers, including acetate, are investigated. It is also shown that varying plaque/saliva contact over the known range derived from published studies has a profound effect on the modeled cariogenic challenge.

Effect on a cariogenic challenge of saliva/plaque exchange via a thin salivary film studied by mathematical modelling.

Computer models can be powerful tools for studying complex interacting processes. The computer model of events in dental plaque during a cariogenic challenge described here simulates diffusion and metabolism of substrate, plus coupled diffusion/reaction of fourteen other species, charged and uncharged, including acidic metabolic products and fixed buffers. Its extension to deal with the effects of poor contact with bulk saliva when the plaque is presumed covered by a thin salivary film is here considered. Site-specific mixing rates between film and salivary pool were modelled phenomenologically, using data from the literature. Fast mixing was assumed during an initial carbohydrate intake phase (2 min sugar rinse), followed by site-dependent mixing and logarithmic clearance. The analysis also suggested a possible way of estimating local salivary film thickness. Increasing the halving time for exchange between film and bulk saliva was shown to prolong the pH minimum greatly, and to increase mineral loss. The respective roles of fixed buffers as stores of protons and of mobile buffers (especially bicarbonate) as exporters of protons from the inner plaque were emphasised.

Estimation of the velocity of the salivary film at some different locations in the mouth.

Previously, we studied the clearance rates of KCl from agarose gels positioned at different locations in the mouth, and showed that the rates were much slower than when clearance was into a well-stirred solution. We designed the present in vitro study to test the effect on KCl clearance of the velocity of a 0.1-mm-thick film of water flowing over an agarose gel of the same diameter and composition as those used in vivo. The thickness of the salivary film overlying dental plaque has been estimated to be about 0.1 mm, and we assumed that when clearance rates in vitro matched those found in vivo, velocities of the fluid film (in vitro) and the salivary film (in vivo) must be equal. On this basis, it was calculated in the present experiments that when salivary flow was unstimulated, the velocity of the salivary film at the level of the teeth varied between about 0.8 mm/min (upper-anterior buccal region) and 8.0 mm/min (lower-anterior lingual region). When salivary flow was stimulated, this was estimated to increase the velocity of the salivary film from 2 to 40 times, depending on the location in the mouth. It is postulated that the slow movement of the salivary film when flow is unstimulated allows for accumulation of diffusants from dental plaque, which reduces the concentration gradient for diffusion from plaque and prolongs the clearance time of such metabolic products as acid.