How fast do solutes diffuse into or out of a biofilm?

We have tools to visualize diffusion in biofilms

Image analysis of data from these types of experiments afforded an estimate of the effective diffusion coefficient of Rhodamine B in the biofilm. The average value of De, the effective diffusion coefficient in the biofilm, was 11 percent of its value in water. For details, see Abdul Rani et al. (2005) Antimicrob. Agents Chemother., in press.We can also calculate diffusion times

Suppose a stain is added to the medium bathing a biofilm. How long will it take this dye to permeate, by diffusion, to the interior of a cell cluster or to the bottom of the biofilm? To get started, we will define a single, simple measure of diffusive penetration time. There are two versions of this measure, depending on the geometry of the system. The time required for a solute added to the fluid bathing a biofilm to attain 90% of the bulk fluid concentration at the base of flat slab (uniformly thick) biofilm is given very simply by:

Here, L is the biofilm thickness, and D_e is the effective diffusion coefficient in the biofilm.

The time required for a solute to attain 90% of the bulk fluid concentration at the center of a spherical biofilm cell cluster is given by:

where R is the cluster radius and D_e is the effective diffusion coefficient in the biofilm. The first step in performing these calculations is to estimate the effective diffusion coefficient in biofilm.

Biofilms are mostly water, and the appropriate starting point for estimating a diffusion coefficient in a biofilm is to determine the value of the diffusion coefficient in pure water (D_aq). Some aqueous diffusion coefficients have been experimentally measured and can be found in the literature. Others can be estimated from a predictive correlation such as the Wilke-Chang correlation. The single number relevant to biofilms is 2.6, the diffusion coefficient for water.

Aqueous diffusion coefficients of selected solutes of interest in microbial systems are tabulated in Table 1. The values of diffusion coefficients in water of various biocides and antibiotics have been summarized elsewhere (32, 34).

The diffusion coefficient in water depends on temperature, both directly and through the effect of temperature (T) on the solution viscosity (µ). This temperature dependence of aqueous diffusion coefficients can be calculated through the relationship D_aqµ/T = constant.

The value of the effective diffusion coefficient in the biofilm will be reduced compared to the diffusion coefficient in water due to the presence of microbial cells, extracellular polymers, and abiotic particles or gas bubbles that are trapped in the biofilm. This reduction is described by the ratio D_e/D_aq. Experimental measurements of this ratio, termed the relative effective diffusivity, have been reviewed elsewhere (33) and that article presents guidelines and formulae for estimating De/Daq in biofilms. There are also sophisticated approaches for calculating this ratio (40). The relative effective diffusivity depends on the biomass density in the biofilm and the physiochemical properties of the solute. D_e/D_aq in biofilm ranges from ca. 0.2 to 0.8 with a mean value of ca. 0.4. The figure below presents consensus values of D_e/D_aq for selected solutes. A value for D_e/D_aq of 0.6 is suggested for small inorganic ions and light gases (e.g., oxygen, nitrous oxide, carbon dioxide, or methane), and a value for D_e/D_aq of 0.25 is suggested for most organic solutes.

The values shown in Figure 2 derive from a reanalysis of the data compiled by Stewart et al. (33). The error bars indicate the standard deviation.

Four example calculations

Armed with an effective diffusion coefficient and an estimate of the biofilm thickness or cluster radius, the calculation of diffusive penetration times is straightforward. The best way to illustrate this is with some example calculations.

Example calculation 1 is presented below in both text and as a narrated slide show. Prepared and narrated by Dr. Philip Stewart, Professor of Chemical and Biological Engineering at Montana State University, Bozeman, Montana

Script from the slide show (the slide show is currently unavailble): Slide 1: In this example calculation, we'll estimate the time it takes for an antimicrobial mouthwash to penetrate a patch of dental plaque. Slide 2: More specifically we want to address the question of how long must one rinse with mouthwash containing chlorhexidine, a common antimicrobial agent, to have this agent penetrate right down to the tooth surface. Slide 3: Our starting point is this equation, which gives us the time required to attain, at the deepest part of this hemispherical patch of dental plaque-the center point of the sphere, 90% of the applied bulk fluid concentration. Slide 4: There are just two parameters required in this calculation: R, the radius of this patch of biofilm, and De, the effective diffusion coefficient of chlorhexidine in the biofilm. The radius, we're going to take as given as 280 microns. Slide 5: Our starting point for estimating the diffusion coefficient of chlorhexidine in the biofilm is the value of the diffusion coefficient of chlorhexidine in water, Daq, here estimated using the Wilke-Chang correlation at 30 degrees Celsius. Slide 6: Now, in the dental plaque, the diffusion coefficient is reduced due to the presence of cells and polymers and precipitates. From looking at the literature on measurements of diffusion coefficients in dental plaque, we can estimate that a solute the size of chlorhexidine will only diffuse at about 20% of its rate in water, and so, to calculate De, we just take one-fifth of Daq, and here is that number. Slide 7: When these numbers are plugged into the equation, the only trick is that we need to make sure that the distance units will cancel-so we've put the radius in centimeters. Now the centimeters squared in the numerator, and the centimeters squared in the denominator will cancel and give us an answer in units of seconds. Slide 8: The answer is 289 seconds. Try rinsing with mouthwash for 5 minutes for a new appreciation of just how long 300 seconds is! This calculation is telling us that, in a pretty healthy patch of dental plaque two- or three-hundred microns thick, a typical rinse of 20 or 30 seconds probably is not adequate to deliver the full concentration of the antimicrobial in the mouthwash to the interior of the plaque.

Example calculation 1: Text version

Consider the penetration of a mouthwash into a roughly hemispherical patch of dental plaque with a radius of 280 µm. How long must one rinse with a mouthwash containing chlorhexidine to have this antimicrobial agent penetrate down to the tooth surface underneath the center of the plaque? The diffusion coefficient of chlorhexidine in water, as estimated by the Wilke-Chang correlation, is 3.7 x 10^-6 cm²/ s (34). Holding the mouthwash in the mouth will raise its temperature; assume that the mean temperature is 30°C. The aqueous diffusion coefficient at 30°C is then 4.2 x 10^-6 cm²/s. From Stewart (33) the value of D_e/D_aq in dental plaque for a solute such as chlorhexidine is predicted to be ca. 0.2. This gives a value of D_e of 0.84 x 10^-6 cm²/s. Using the formula for a sphere (0.31 R²/D_e), which also applies to a hemisphere on an impermeable plane, the diffusion time is calculated to be 289 s. Try swishing mouthwash for this period to develop a new appreciation for just how long five minutes is.

Now you try it. . .

A researcher plans to test the role of a homoserine lactone (HSL) quorum sensing signal on biofilm phenotypes by adding exogenous HSL to the bulk fluid. If the biofilm is treated as a flat slab that is 80 microns thick, estimate the time needed for N-(3-oxododecanoyl)-L-homoserine lactone to reach 90 percent of its bulk fluid concentration at the base of a biofilm. The temperature is 25°C.

Resources:

Equation 1: Calculating diffusion time in a flat slab biofilm

View Table 1: Diffusion coefficients in water at 25°C

View Table 2: Temperature dependence of aqueous diffusion coefficients

Calculator for Equation 1

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Example calculation 2

A biofilm growing in a flow cell at room temperature (22°C) is to be stained with a fluorescein-tagged probe. The biofilm consists of tightly packed mushroom-shaped clusters that are 100 µm tall. How long must the sample be incubated with this reagent to ensure that the stain reaches to the full depth of the biofilm? Suppose that the diffusion coefficient of the fluorescent probe in water is half that of fluorescein, or 2.7 x 10^-6 cm²/s at 25°C. At 22°C this value becomes 2.5 x 10^-6 cm²/s. Lacking specific information about the biofilm, we can take D_e/D_aq to be 0.25. This gives a value of D_e of 0.63 x 10^-6 cm²/s. Though the biofilm is not uniformly thick, you can treat it as if it were. The result will be a conservative estimate of the required staining time. The diffusion time is 1.03 L²/D_e, which equals 165 s. Staining for a few minutes should suffice.

Example calculation 3

A mixed-species biofilm grown for 6 days is 1,000 µm thick and locally uniform in thickness. Sodium chloride is added to the bulk fluid, and the appearance of the chloride ion is detected by using a microelectrode positioned near the substratum at the base of the biofilm. Estimate how long it takes for the chloride concentration at the substratum to reach 90% of the bulk fluid concentration. The system temperature is 24°C. The diffusion coefficient of chloride ion in water at 24°C is 20 x 10^-6 cm²/s. Taking D_e/D_aq to be 0.70, D_e is 14 x 10^-6 cm²/s. The required penetration time is 714 s or almost 12 min. This calculation is based on an experiment reported by Stewart et al. (35), and the calculated time scale is in rough agreement with the experimental result (1,000 s).

Example calculation 4

An experimenter proposes to test the role of a certain gene in biofilm development by overexpressing the gene at a particular stage in the maturation of the biofilm. The gene is placed under the control of a promoter that can be induced by adding IPTG (isopropyl-ß-D-thiogalactopyranoside). This inducer will be added when young microcolonies are forming. These microcolonies can be approximated as hemispherical patches of radius 5 µm. It is desired to calculate any delay in gene expression that might result from the time needed for the IPTG to diffuse throughout the nascent colony. The system temperature is 37°C. IPTG is a modified monosaccharide, so we can estimate that its diffusion coefficient in water at 25°C will be ca. 6.5 x 10^-6 cm²/s. Scaling to 37°C and taking D_e/D_aq to be 0.25, D_e is found to be 2.2 x 10^-6 cm²/s. The required penetration time is a mere 0.12 s. The delay anticipated for diffusive ingress is negligible.

Four assumptions implicit in these calculations

These two issues are discussed at greater length in the following paragraphs. The derivation of Equations 1 and 2 also assumes an impermeable substratum and a uniform effective diffusivity within the biofilm.

The term external mass transfer resistance refers to the barrier to diffusion posed by slow-moving fluid adjacent to a biofilm. The fluid may be moving, but its direction of flow is generally parallel to the biofilm surface when very close to the biofilm. This means that although there is convective transport in the direction of flow, there is little or no convection in a direction that would carry a solute into or out of a cell cluster. The extent of external mass transfer resistance depends on the hydrodynamics of the system. There is probably little external mass transfer resistance in high-velocity, turbulent flows. External mass transfer resistance is likely important in many laminar flow systems and is assuredly an issue in systems in which the fluid is static. There are quantitative approaches to calculating the effects of external resistance, but these are beyond the scope of this chapter. Suffice it to say that all of the diffusion-related phenomena discussed in this article will only be exacerbated when external mass transfer resistance is present.

Penetration times calculated by using Equation 1 or 2 will be reasonable estimates as long as there is no significant reaction or sorption of the solute in the biofilm. It may be easiest to appreciate this caveat by considering a couple of examples in which these equations break down. The diffusive penetration of IPTG calculated in the last example above depends on there being no metabolism of this molecule. While sugars have diffusion coefficients in water that are similar to that of IPTG, it would be incorrect to apply the IPTG result to the transport of a sugar if the microorganisms are capable of metabolizing that saccharide. Microbial utilization of the sugar will reduce its concentration as it diffuses into the biofilm. It would also be incorrect, for example, to equate the penetration time for the chloride ion (Cl-) calculated in the third example above with the penetration time for the hypochlorite ion (OCl-), even though these two ions have essentially the same diffusion coefficient in water. Whereas the chloride ion is inert, the hypochlorite ion is highly reactive and will be neutralized by reactions with organic matter in the biofilm. These reactions profoundly retard penetration of this species (6, 35). When a solute reacts in the biofilm it may never fully penetrate the biofilm. Reaction and diffusion achieve a steady balance that leads to persistent concentration profiles within the biofilm. This interaction is the subject of the next section.