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Appl Environ Microbiol, June 1998, p. 2295-2300, Vol. 64, No. 6
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
Computation of the Electrical Double Layer Properties of
Semipermeable Membranes in Multicomponent Electrolytes
Evgeny
Wasserman* and
Andrew R.
Felmy
Pacific Northwest National Laboratory,
W. R. Wiley Environmental Molecular Sciences Laboratory,
Richland, Washington 99352
Received 9 January 1998/Accepted 1 April 1998
 |
ABSTRACT |
A methodology is presented for calculating of the surface
potential, Donnan potential, and ion concentration profiles for semipermeable microbial membranes that is valid for an arbitrary electrolyte composition. This model for surface potential,
Donnan potential, and charge density was applied to recently reported experimental data for gram-positive bacteria, including Bacillus brevis, Rhodococcus opacus, Rhodococcus
erythropolis, and Corynebacterium species. These
calculations show that previously unconsidered trace amounts of
divalent and trivalent cations at very low concentrations (10
6 M) can have significant effects on the calculated
surface and Donnan potentials, at ionic strengths of I 
0.01 M,
and that these effects need to be considered in accurate modeling of
microbial surface. In addition, the calculated ion concentration
profiles show that owing to the relatively high surface charges that
can develop in microbial membranes, electrostatic effects can act to significantly concentrate divalent (factors of 5 × 103) and trivalent (factors of 2 × 104)
cations within the bacterial cell wall. Comparison of the calculated concentration factors with those derived from experiments shows that a
significant fraction of the uptake of metal by bacteria can be
explained by the proposed electrostatic model.
| |
TEXT |
Characterization of the electrical
double layer of bacterial surfaces has been the subject of a great deal
of recent research efforts owing to the fact that the bacterial surface
charge and electrical double layer properties affect many important
environmental properties of bacteria including attachment-detachment of
bacteria, metal binding, mobility, and zeta potential (1-8,
10-13, 16-21). As a result, models of the charge potential
relationships at bacterial surfaces and across bacterial cell walls
have been studied extensively (9, 10, 13, 14, 17, 18, 21),
usually by assuming Donnan equilibrium deep within the membrane and
then solving the Poisson-Boltzmann equation for a specific electrolyte
type and concentration with fixed boundary conditions to obtain the
surface potential distribution. All of the proposed numerical or
analytic solutions of these equations to obtain the relationships among the Donnan potential, surface potential, and charge density for microbial membranes are strictly valid only for pure 1-1 electrolytes (see, for example, reference 17) or at most mixtures
of 1-1 and 2-1 electrolytes (14). Unfortunately, natural
geochemical systems often contain a complex mixture of different
chemical components and ionic species, and the question as to the
applicability of equations derived for simple mixtures to complex
multicomponent natural systems arises.
In this paper, a solution procedure to the Poisson-Boltzmann equation
for a semipermeable membrane that is valid for any arbitrary mixture of
charged ions is presented. The importance of considering all ions in
the calculation of surface and Donnan potentials for microbial
membranes in dilute solutions is demonstrated by using surface charging
data for bacterial membranes reported recently in by van der Wal et al.
(17). The results show that the presence of trace amounts of
divalent or trivalent cations in relatively dilute solutions
(I, 0.01 M) can significantly alter both the calculated
surface and Donnan potential as well as the distribution of counter and
co-ions within the membrane.
This model does not account for metabolism in the bacteria. However,
the rate at which the thermodynamic equilibrium described by this model
is reached is likely to be significantly higher than the rate of
metabolism.
Abbreviations used.
For a complete list of abbreviations and
units used in notation of equations, see Table
1.
Computational procedure.
Equations for the Donnan potential,
surface potential, and the potential profiles across a planar
ion-penetrable membrane have been published in the literature for
several particular special cases. The most comprehensive treatment for
mixed electrolyte solutions was presented by Ohshima and Kondo
(14), who described 1-1 electrolytes, symmetric
electrolytes, and a mixed solution of 1-1 and 2-1 electrolytes. In this
paper, the treatment found is in that reference is extended to the case
of an arbitrary electrolytic solution of charged species, and the
validity of assumptions made to solve these equations is discussed. The
published analytic solutions follow from our general equations as
special cases.
Following earlier treatments (
14,
17), the solution for the
planar geometry with the only coordinate (
x) perpendicular
to the surface of the membrane is obtained. Use of a planar geometry
is
justified, since the radius of curvature of the cell (approximately
10
4 Å) is significantly greater than the thickness of the
area in
which a nonzero electrostatic field exists. An infinite
thickness
of the membrane and solution is also assumed. Once again,
after
solving the equations under this assumption we note that the
electric
field exists only in the area thinner than the membrane,
justifying
the infinite-thickness assumption.
The problem is formally described as follows. The membrane
(
x < 0) has fixed charge density (charge in a unit
volume)
m and dielectric constant
m. The solution has
N different
aqueous ions, and its dielectric constant is
s. Each
ion-labeled
i has the
formal charge
Zi and bulk concentration
in
solution (far away from the membrane)
Cib.
These quantities are the input for the calculation, and the
goal is to
solve for the equilibrium potential profile and profile
of
concentrations. The surface charge density can be deduced from
proton
titration experiments. Titration experiments do not provide
any
information about the distribution of charge throughout the
cell wall.
It would be unreasonable to assume that all of the
charge is located on
the surface of the cell wall like surface
charge density on a metallic
electrode. It must have spatial distribution
through the cell wall. Not
knowing the exact details of this distribution,
it is natural to assume
it to be uniform as we did.
Donnan potential.
Provided the cell wall is relatively thick,
the charge density (fixed plus induced) well within the cell wall
vanishes. The sufficient thickness depends on the values of the fixed
charge density m and dielectric constant
m. In our numerical examples below, this
sufficient thickness was of the order of 60 Å. We note here that for
all of the bacteria listed in Table 1 of reference
17, the cell wall thickness was greater than this
value. From the Poisson equation, the potential becomes constant within
the membrane at the Donnan potential, 
D. All
ions are assumed to freely penetrate the membrane. The energy of an ion
(i) in the field of the potential is
Zie, and according to the Boltzmann
distribution, the concentration well inside the membrane
i is given by the following
equation:
|
(1)
|
where
Cib is the concentration
in the bulk solution. The charge neutrality equation then becomes
|
(2)
|
This equation is then solved for the Donnan potential. At this
point, it is convenient to introduce the notation
and reduce the numerical problem to finding the positive root of
the following polynomial:
|
(3)
|
where
Zmax is the maximum charge of the ion
in the solution. The degree of this polynomial is
Zmax 
Zmin, where
Zmin is
the minimum ionic charge (the formal
charge of the most negatively
charged anion). Algorithms and computer
codes for finding the
roots of a polynomial are readily available
(
15). We determined
the Donnan potential by finding the
single positive root of equation
3 (which takes a fraction of a second
on a
desktop computer) and
then used
to obtain the Donnan potential itself.
This expression is general, and the expression for the 1-1 electrolyte
follows from equation 3 as a special case. In case
of a 1-1 electrolyte,
Zmax = 1 and
Zmin =
1, while
N = 2. Therefore,
in this particular case equation 3 is a quadratic equation
whose
solution yields the well-known expression for the Donnan
potential
for the membrane in contact with a 1-1 electrolytic solution
(
14):
|
(3`)
|
We never used equation 3', since we always considered solutions
with tracer species present. It is given here just to demonstrate
consistency with the results of reference
14 for the
special
case treated therein.
Potential profile across the membrane.
To obtain the potential
profile across the membrane, we solved the Poisson equation
2 = /(0) (in SI units),
where [C/m3] is the charge density and is the
dielectric constant of the medium (Table 1). The zero of the
x axis is located on the boundary between the membrane
(x < 0) and solution (x > 0). The
relative dielectric constant of the membrane is denoted
m, and that of the solution is denoted
s. Since we have only one spatial variable
for the planar geometry, 2 = 
. The charge
density () consists of the fixed-charge density
m (present only in the membrane) and the
induced-charge density. The latter is determined from the Boltzmann
distribution as in equation 1. Therefore, the Poisson equations for the
membrane and the solution, respectively, have the following forms:
|
(4)
|
|
(5)
|
The boundary conditions for the potential are
(
x =
0) =
(
x = +0),
(
x =
) =
D, and
(
x = +
) = 0. The other
boundary conditions
involve the electric field
E =
.
The only component
of the electric field present in planar geometry
is normal to the
surface. This yields the boundary condition
mE(
x =
0) =
sE(
x = +0). Naturally, we
expect the electric
field to vanish far away from the boundary,
E(
x =
) =
E(
x =
+
) = 0.
This boundary value problem can be solved numerically, but such
solutions are inefficient for infinite-boundary conditions.
As a
result, the problem is transformed into an initial value
problem by
integrating equations 4 and 5 once and then applying
the boundary
conditions for the electric field. This procedure
yields a single
transcendental equation for the surface potential.
In solving equations 4 and 5, it is convenient to introduce the
dimensionless (reduced) potential
y through
y =
.
Then, equations 4 and 5 are reduced to the following equations:
|
(6)
|
|
(7)
|
Since the right-hand side does not depend on
x,
equations 6 and 7 can be integrated by introducing new variables
u =
dy/dx,
then
Integration of this equation yields the following expression for
u2 with the constant of integration
U1:
|
(8)
|
where
b = 
and
ai = 
.
To determine
U1, we note that as
x
,
yyD and
u0. Substituting
this in equation 8 yields
|
(9)
|
and the expression for the membrane is
|
(10)
|
For the solution, as
x,
y0 and
u0. A similar procedure yields the
following equation for
x > 0:
|
(11)
|
Since
u is proportional to
E,
|
(12)
|
The transcendental equation for the reduced surface potential
y0 is obtained algebraically as follows:
![<FR><NU>ϵ<SUB>m</SUB></NU><DE>ϵ<SUB>s</SUB></DE></FR><FENCE>&rgr;<SUB>m</SUB>(y<SUB>D</SUB> − y<SUB>0</SUB>) − <LIM><OP>∑</OP><LL>i = 1</LL><UL>N</UL></LIM>C<SUP>b</SUP><SUB>i</SUB> F[<UP>exp</UP>(<UP>−</UP>Z<SUB>i</SUB>y<SUB>D</SUB>) − <UP>exp</UP>(<UP>−</UP>Z<SUB>i</SUB>y<SUB>0</SUB>)]</FENCE>](/content/vol64/issue6/fulltext/2295/img024.gif)
|
(13)
|
This equation is solved by bracketing the root around
yD and then using the Brent method
(
15).
Note that for the special case of a 1-1 electrolyte and with the
additional assumption
m =
s (which we
believe is
incorrect), the analytical expression
given in reference
14 follows from equations 13 and 3'. We
never used this expression, since it is inapplicable when the
trace
species are present; besides, in our case
m 
s. So we solved equation 13 numerically
instead, which
took approximately 1 s on a desktop computer.
To obtain the potential profile across the membrane, the initial value
problem for differential equations 6 for
x < 0 and
7 for
x > 0 is solved. The initial values for the
reduced potential
y(
x = 0) =
y0 are obtained by solving equation 13 for
y0 and
x = 0 is calculated from equations 10 and 11, respectively. The sign
of
u =
dy/dx is determined from the
following equation:
![<UP>sign</UP>[u(x = 0)] = <UP>−sign</UP>[y<SUB>0</SUB>]](/content/vol64/issue6/fulltext/2295/img028.gif)
|
(14)
|
This initial value problem is solved numerically by
the Runge-Kutta method with adaptive stepsize control as
implemented
by Press et al. (
15).
The Fortran code that performs these calculations is available from the
authors upon request.
Application to bacterial surfaces.
In this section, the
previously described model relating surface potential, Donnan
potential, and charge density for an arbitrary ion composition is
applied to the surface charging data for five gram-positive bacteria
reported by van der Wal et al. (17). Use of data from the
latter authors also allows a direct comparison of our results with
their similar calculations for simple 1-1 electrolytes.
In calculating the potentials for different electrolytes, the only
variables are the concentration and charge of each ion,
the surface
charge density, and the thickness of the cell wall.
In the calculations
that follow, the surface charge density at
each ionic strength and the
thickness of the cell wall were fixed
at the values given in Table 2 of
reference
17. These values
correspond to neutral pH
and are listed in that reference for
three values of ionic strength,
i.e., 0.1, 0.01, and 0.001 M.
As the first approximation, the bulk
charge density
m in the membrane was
calculated by dividing the surface charge
density by the cell wall
thickness. We assumed
m = 60
as suggested by
van der Wal et al. (
17) and
s = 78,
which is the experimental dielectric constant of water. Since
we
are interested only in solutions at low concentrations, the
difference
between the dielectric constant of pure water and the
solution is
relatively small and does not affect the results significantly.
The
effects on the surface and Donnan potentials of other ions
present in
minor concentrations were then calculated. The objective
was to
demonstrate the conditions (i.e., in terms of ionic strength
and minor
species charge and concentration) under which species
present at low
concentration need to be included in the calculations
in order to
obtain accurate representations of the surface and
Donnan potentials.
The calculations were performed at three different
1-1 electrolyte
concentrations (i.e., 0.001, 0.01, and 0.1 M)
with trace (i.e.,
10
6 M) concentrations of divalent or trivalent ions
added. The presence
of such low concentrations of divalent or trivalent
ions is very
likely in groundwater systems.
As a first example, the potentials as a function of distance were
calculated for
Bacillus brevis (selected since it is a
common
microorganism in soil and vadose sediments) as a function of
electrolyte
concentration, with a minor charged +3 cation (actually 3-1 electrolyte)
included in the solution at a fixed concentration of
10
6 M (Fig.
1). As can
readily be seen, at an electrolyte concentration
of 0.1 M, the presence
of such a minor concentration of trivalent
cation has little or no
effect on the surface and Donnan potentials,
which are essentially
identical to the values reported previously
by van der Wal et al.
(
17). However, at ionic strengths of as
low as 0.01 M, a
detectable difference begins to appear, and at
lower ionic strengths
(0.001 M) the differences become marked,
with the calculated
electrostatic potentials being considerably
lower than the calculated
values in the absence of the trace trivalent
cation. Divalent cations
also can effect the calculated potentials
at this lower ionic strength
(Fig.
2), although the effect is
not as
large. Interesting, however, are the ion concentration
plots for both
trivalent and divalent cations present in the bulk
solution at
10
6 M (Fig.
3), which show
extremely large concentration effects
relative to the concentrations in
bulk solution (i.e., ~2 × 10
4 for trivalents and
5 × 10
3 for divalents) at an ionic strength of 0.001 M. In fact, in these
solutions 65% of the total surface charge is
countered by trivalent
cations. Clearly, including trace species in the
calculation of
the surface and Donnan potentials is necessary for
accurate modeling
of these parameters at these low ionic
strengths. This model corresponds
to the case of a small amount
of bacteria in a significant volume
of solution (infinite dilution with
respect to the concentration
of bacteria). Clearly, if an insufficient
mass of the trace species
is available, the concentration factor will
be smaller. The calculations
performed here were based on the
values of the surface charge
density for neutral pH. Increasing the pH
of the solution makes
the concentrating effect for the tracer divalent
and trivalent
cations more pronounced, since the magnitude of the
charge density
increases (although this effect may be compensated for
by increased
metal ion hydrolysis and reduced species charge). This
effect
decreases in acidic solutions. In addition, given the
potentially
high surface charges that develop under these conditions at
bacterial
cell surfaces (i.e.,
0.56 C/m
2 for
B. brevis), any accurate model of microbial metal uptake
or
charge development would need to include these factors. Similar
calculations for all five bacteria examined by van der Wal et
al.
(
17) are given in Fig.
4.
These results show that although
there are certainly some
differences among the various bacteria,
the results for
B. brevis are likely typical for gram-positive
bacteria, and the
following results are reported only for
B. brevis.
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FIG. 1.
Calculated surface and Donnan potentials for
B. brevis with and without considering the effects
of trace trivalent cation concentration (106 M) as a
function of electrolyte composition. Note the significant effect at a
0.001 M concentration and the totally negligible effect at a 0.1 M
concentration.
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FIG. 2.
Calculated surface and Donnan potentials for
B. brevis with and without considering the effects
of a trace trivalent and divalent cation concentration
(106 M) in 0.001 M 1-1 electrolyte.
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FIG. 3.
Calculated ion concentration profiles for trace divalent
and trivalent cations (106 M bulk concentration) in 0.001 M 1-1 electrolyte.
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FIG. 4.
Calculated ion concentration profiles for trace
trivalent cations (106 M bulk concentration) in 0.001 M
1-1 electrolyte for five different bacteria. Surface charging data are
from reference 17.
|
|
With these factors in mind, it is also of interest to examine how the
trace species can affect the ion concentration profiles
of the bulk,
singly charged, counter-, and co-ions (Fig.
5 and
6).
These calculations show that the differences in potential
calculated
for the cell wall in the presence and absence of a
trace trivalent
cation in dilute electrolyte (Fig.
1) have marked
effects on the ion
concentration profiles for both the bulk cation
(Fig.
5) and the bulk
anion (Fig.
6). In the case of the bulk
cation, the incorporation of
significant concentrations of trivalent
species significantly reduces
the amount of sorbed bulk, singly
charged cation. Correspondingly, the
overall reduction in calculated
surface and Donnan potentials in the
presence of trace trivalent
cation results in significant increases in
adsorbed bulk co-ion
(Fig.
6) compared to the case in which no
trace cation is present.
These large cation concentration
effects make it of interest to
determine the thickness of the
area from which a membrane is able
to concentrate the divalent
and trivalent cations. Considering
the thickness of the cell wall
of
B. brevis (750 Å) (
17) and
the concentrating
effect of 2 × 10
4 found for trivalent cations, we
estimated an upper bound of 1.5
mm for this distance. The same
calculation for divalent cations
yielded 0.375 mm. In other words,
simple electrostatic considerations
dictate that these microorganisms
have the ability to scavenge
trace species at length scales 1,000 times
greater than the cell
size.
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FIG. 5.
Calculated ion concentration profiles for the bulk +1
counter ion and the trace trivalent cation (106 M bulk
concentration) in 0.001 M 1-1 electrolyte with and without considering
the effects of trace trivalent cation concentration for B. brevis.
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FIG. 6.
Calculated ion concentration profiles for the bulk 1
co-ion with and without considering the effects of trace trivalent
cation concentration (106 M bulk concentration) for
B. brevis in 0.001 M 1-1 electrolyte.
|
|
Although consideration of minor highly charged species is certainly
important, it needs to be pointed out that the very large
concentration
effects of minor divalent and trivalent species
predicted for the
microbial membranes is limited to more dilute
solutions (
I,
0.01 M). For example, comparing the predicted concentrations
within
the microbial membrane at the same concentration of trace
trivalent
cation (10
6 M) between ionic strengths of 0.001 and 0.1 M
(Fig.
7) shows
that despite an increase
in surface charge from
0.56 to
1.23
C/m
2 that occurs
between these two solutions, the trivalent cation
is concentrated by a
factor of ~10
3 less in the 0.1 M solution than in the
0.001 M solution. Clearly,
the electrostatic effects on trace metal ion
are much more effective
in dilute electrolytes than in concentrated
electrolytes.
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FIG. 7.
Calculated ion concentration profiles for trace
trivalent cations (106 M bulk concentration) in 0.001 and
0.1 M 1-1 electrolyte.
|
|
It is also of interest to compare the concentration factors predicted
based on this electrostatic model with experimental
data on bacterial
metal uptake. Unfortunately, we are aware of
no definitive study that
measures all of the parameters (i.e.,
ionic strength, cell wall charge,
pH, metal uptake, and cell wall
volume) necessary to perform definitive
calculations. Given these
facts, the best data set available to test at
least qualitatively
the importance of electrostatics on the uptake of a
wide range
of metal ions is the data described by Beveridge and Murray
(
4),
who measured the uptake of several divalent and
trivalent metal
ions on
B. subtilis at a total metal
concentration of 5 × 10
3 M. Although the high metal
concentrations used in these studies
are not expected to result in
large concentration factors, it
is still of interest to contrast the
experimental data with a
model predicting metal uptake based solely
on electrostatic concentration
effects. Unfortunately,
the ionic strength of the solution, pH,
and surface charge
density were not reported by Beveridge and
Murray (
4).
Therefore, in order to proceed, the following two
assumptions are
necessary: (i) the surface charge density is estimated
from the data
for
B. brevis (
17) and (ii) the solutions are
assumed to be neutral (pH = 7). Since the surface charge density
is critical, we present the calculated concentration factors for
three
surface charge densities at different ionic strengths (Table
2). As a reference point, the ionic
strengths of a 5 × 10
3 M MeX
2 or
MeX
3 solution would be 0.015 and 0.03 M, respectively,
assuming that no additional components are present and no metal
hydrolysis or precipitation occurred. The experimental data for
several
metals (
4), expressed as dimensionless ratios of
concentrations,
are included in Table
3. Note that the uptake values vary
dramatically
between different cations but that the electrostatic
calculations
alone (Table
2) appear to explain the uptake of several
metal
ions, including Sr
2+, Ni
2+,
Pb
2+, and, possibly Ca
2+. The concentration
factors of Mg
2+, Mn
2+, Cu
2+, and
Fe
3+ are far too large to be explained by electrostatic
concentration
alone. Clearly, specific binding to the cell wall
material is
indicated. We speculate that the zero value for Al is
related
to metal ion hydrolysis or precipitation processes. In any
event,
accurate models of the electrical double layer are of
significance
in determining the mechanisms of metal uptake. Mechanisms
for
determining the amounts of metal ions specifically bound to
bacterial
surfaces and the concentrations bound solely as a result of
electrostatic
considerations (
4,
6) are needed. It is hoped
that with
improved models of the electrical double layer, such as those
proposed here, as well as with improved experimental data on surface
charge and metal binding, better understanding of the specific
mechanisms of metal uptake can be obtained.
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TABLE 2.
Concentration factors for divalent and trivalent tracer
cations as a function of surface charge calculated
for B. brevisa
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TABLE 3.
Concentration factors for several divalent and trivalent
cations for B. subtilis as measured by Beveridge and
Murray (4)a
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| |
ACKNOWLEDGMENTS |
This work was supported by a Laboratory Directed Research and
Development program entitled Molecular Mechanism of Microbial Attachment and Surface Reaction at the Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated for the
U.S. Department of Energy by Battelle Memorial Institute under contract
no. DE-AC06-76RL0 1830.
We thank James K. Fredrickson and James R. Rustad of PNNL for helpful
comments on the manuscript. Lisa Onishi is thanked for her help in the
preparation of the manuscript.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Environmental
Molecular Sciences Laboratory, Pacific Northwest National Laboratory, MS K8-96, P.O. Box 999, Richland, WA 99352. Phone: (509) 376-4528. Fax: (509) 376-3650. E-mail: evgeny{at}emsl.pnl.gov.
| |
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Appl Environ Microbiol, June 1998, p. 2295-2300, Vol. 64, No. 6
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
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Abstract
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