Figure 4-3. This animated sequence
(left panel) shows the diffusion
of Rhodamine B (red) into a biofilm cluster over
a period of 8 minutes.
First, a Staphylococcus epidermidis biofilm was grown in a
flow cell. Working on the confocal scanning laser microscope, a cell
cluster was located using the transmission mode (panel B, right). After the microscope was changed from transmission
mode to detect red fluorescence, the fluid feeding the flow cell was
spiked with Rhodamine B. The dye first stained the periphery of the
cell cluster, then progressively moved inward toward the center of
the cluster. At 360 seconds, the flow was switched back to buffer
lacking Rhodamine B, allowing the outward diffusion of the dye to be
visualized.
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.
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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:
|
(1) |
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Here, L
is the biofilm thickness, and De 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: |
(2) |
where
R
is the cluster radius and
De 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 (Daq).
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.
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Aqueous diffusion coefficients of selected solutes of
interest in microbial systems are tabulated in
Table 4-1.
The values of diffusion coefficients in water of various biocides
and antibiotics have been summarized elsewhere (32,
34).
Use the
scrollbar to read this table, or
View Table 4-1
in
a new window. |
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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
Daqµ/T = constant.
Use the
scrollbar to read this table, or
View Table 4-2
in a new window. |
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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
De/Daq. 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.
De/Daq in biofilm ranges from ca. 0.2 to 0.8 with a
mean value of ca. 0.4.
Figure 4-4,
below, presents consensus
values of
De/Daq for selected solutes. A value for
De/Daq 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
De/Daq of 0.25 is suggested for most organic solutes.
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Figure 4-4.
Relative effective diffusion coefficients of selected solutes
in biofilms. These values derive from a reanalysis of the data
compiled by Stewart et al. (33).
The error bars indicate the standard deviation.
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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.
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Example
calculation 4-1 is presented below in both text and as a
narrated slide show. |
Example calculation 4-1: Slide show
version
Prepared and narrated by Dr.
Philip Stewart, Professor of Chemical and Biological
Engineering at Montana State University, Bozeman, Montana
|
Click to view
show
(Time: 2:21)
Slide
show script
|
Example calculation 4-1: 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-1 (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-1.
From Stewart (33)
the value of De/Daq
in dental plaque for a solute such as chlorhexidine is
predicted to be ca. 0.2. This gives a value of
De
of 0.84 x 10-6 cm2
s-1. Using the formula for a sphere
(0.31 R2/De),
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.
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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 25oC.
Resources:
Equation
4-1: Calculating diffusion time in a flat slab biofilm
View Table 4-1: Diffusion coefficients in water at
25°C
View Table 4-2: Temperature dependence of aqueous
diffusion coefficients
Calculator for
Equation 4-1
I'd like some
HELP.
Is
the correct answer:
14 s
54 s
80 s
327 s
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|
Example calculation 4-2
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-1 at 25°C.
At 22°C this value becomes 2.5 x 10-6 cm2 s-1.
Lacking specific information about the biofilm, we can
take De/Daq to be 0.25. This gives a value of
De of 0.63 x 10-6 cm2 s-1. 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/De,
which equals 165 s. Staining for a few minutes should
suffice.
Example calculation 4-3
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-1. Taking De/Daq to be 0.70,
De
is 14 x 10-6 cm2 s-1. 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 4-4
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-1.
Scaling to 37°C and taking De/Daq
to be 0.25,
De
is found to be 2.2 x 10-6 cm2 s-1.
The required penetration time is a mere 0.12 s. The
delay anticipated for diffusive ingress is negligible.
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Four
assumptions implicit in these calculations
Having made these
calculations, it is now appropriate to disclose the
four assumptions implicit in them. Two of the
assumptions are particularly critical because they are
often invalid. The first of these is that resistance
to mass transfer outside of the biofilm can be
neglected. In plain language, this means that the
concentration of the diffusing solute right at the
biofilm surface is essentially the same as its
concentration in the well-mixed bulk fluid. The other
key assumption is that the solute whose penetration is
being analyzed does not
sorb or react in the biofilm.
Both assumptions, if violated, will cause the actual
penetration time to be longer than the calculated
time. The preceding calculations, therefore, are best
regarded as lower bounds on the actual diffusion time.
These two issues are
discussed at greater length in the following
paragraphs. The derivation of Equations 4-1 and 4-2 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 module. 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 4-1 or 4-2 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.
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Take the Section 2 quiz
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