Environ. Sci. Technol. 2001, 35, 894-900
Metal Speciation Dynamics and Bioavailability. 2. Radial Diffusion Effects in the Microorganism Range J O S EÄ P . P I N H E I R O † A N D HERMAN P. VAN LEEUWEN* Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands
The free ion activity model for the biouptake of metals from complex media is limited to cases where mass transfer is not flux determining. This paper follows a previous paper (Van Leeuwen, H. P. Environ. Sci. Technol. 1999, 33, 3743) where speciation dynamics and bioavailability of metals are analyzed in terms of bioconversion kinetics and simultaneous metal transport in the medium coupled with dissociation kinetics. Such analysis shows under what conditions labile complex species contribute to the biouptake process or, equivalently, under what conditions the free ion activity model is not obeyed. The present work addresses the theoretical extension of the expressions for the metal flux in the medium by a radial diffusion term so that these are also applicable in the microorganism size range. The transition from macroscopic to microscopic surfaces affects not only the nature of the flux but also the extent of lability of complex species (Van Leeuwen, H. P.; Pinheiro, J. P. J. Electroanal. Chem. 1999, 471, 55), and this can have a dramatic influence on the rate of biouptake of metal ions. Labilities of metal complexes and the ensuing limiting metal fluxes are therefore systematically analyzed for various dimensions of the uptaking surface. Different conditions of bioaffinities and bioconversion capacities are considered, and a number of examples of metal complexes with specified kinetic features are discussed.
Introduction In part 1 (1), the speciation and bioavailability of metals in complex media were analyzed on the basis of their conversion kinetics at the biointerface and their transport and formation kinetics in the medium. For the two kinetically limiting situations of inert and fully labile systems, the bioavailabilities of metal complexes were analyzed under conditions where the actual biouptake is described by a Michaelis-Menten type of steady-state flux, and the supply of free metal is governed by diffusion of free metal or coupled diffusion of the different labile metal species. The resulting steady-state fluxes were given in terms of two basic quantities, i.e., the relative bioaffinity parameter (a) and the ratio between the limiting uptake flux and the limiting transport flux (b). The analysis precisely reveals under what conditions labile complex species contribute to the biouptake process. * Corresponding author e-mail:
[email protected]; telephone: +31-317-482269; fax: +31-317-483777. † On leave from Centro Multidisciplinar de Quimica do Ambiente, UCEH-A. D. Quimica, Algarve University, Campus de Gambelas, 8000-810 Faro, Portugal. 894
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If the dimension of the receiving biosurface is of the same order of magnitude or smaller than that of the operational diffusion layer, the mass transport is convergent and therefore enhanced as compared to the linear macroscopic case. The expressions for the metal flux in the medium (Jm) then have to be extended with a radial diffusion term. Here we shall consider spherical microorganisms, but the treatment applies equally well to cases of larger organisms where the uptake takes place at individual, well-separated microregions. As in part 1, the analysis is focused on the cases of (i) free metal and/or inert complexes and (ii) labile complexes. In the transition from macroscopic to microscopic surfaces, not only the nature of the flux but also the extent of lability of a complex depends on the dimensions (2). Changes in lability can have a dramatic influence on the rate of biouptake of metal ions; therefore, it is important to always check lability features under the appropriate geometrical conditions, i.e., for the applicable mix of linear and radial transport. The literature contains only a few theoretical studies linking metal speciation to biouptake under conditions of nonnegligible radial transport. Jackson and Morgan (3) wrote down the correct set of starting equations, but mixed up diffusion layers and reaction layers and incorrectly accounted for effects of the dissociation and association reactions. In any case, their theory was only used to analyze the Fe-EDTA system, under conditions where this is inert and hence not subject to possible contribution from labile complexes. Berg and Purcell (4) and others (5-9) pioneered the work on the mechanism and kinetics of biouptake with radial transport of the metal ion only. Currently, the interest in generating knowledge on the uptake of metals from complex media by various types of microorganisms seems to be large (10-14). To develop a comprehensive theoretical basis for speciation dynamics and biouptake in the microorganism range, we will systematically analyze the labilities of metal complexes and the ensuing metal fluxes for various dimensions of the uptaking surface. By way of illustration, a number of examples of metal complexes with specified kinetic features will be considered in detail under different conditions of bioaffinities and bioconversion capacities.
Theory Starting points for a theory that relates metal speciation dynamics and bioavailability can be found in part 1 (1). The biouptake process is modeled by a single Best equation, which expresses the normalized steady-state flux Q () J/J/u) in terms of the bioaffinity parameter a () KM/c/M) and the bioconversion flux capacity parameter b () J/u/J/m). In dimensionless form, the Best equation reads
Q)
{ [
]}
(1 + a + b) 4b 12b (1 + a + b)2
1/2
(1)
with J, J/u, and J/m representing the flux, the limiting biouptake flux, and the limiting transport flux in the medium, respectively. A strong bioaffinity is characterized by a low KM and hence by a relatively low a. The parameter b compares the limiting biouptake flux J/u to the limiting transport flux in the medium J/m. Static Systems (Inert Complexes). First, we consider the case where the medium contains only free metal (M) and/or complexes (ML) with rates of dissociation and association so slow that they are inert on the effective timescale (δ2/D) (15):
kdδ2/DML, ka′δ2/DM , 1 10.1021/es000042n CCC: $20.00
(2)
2001 American Chemical Society Published on Web 02/02/2001
TABLE 1. Lability Criteria (L . 1) for Linear and Radial Diffusion r0 > δ > µ
δ > r0 > µ
linear
radial
bioconversion capacity
(kd/DML)µr0
(bs
diffusion
L
TABLE 2. Limiting Cases of Q for Free Metal or Inert Complexes
(kd/DML)µδML
bioaffinity
high
For diffusion to a sphere, the steady-state radial flux of metal can be obtained starting from Fick’s second law and corresponds with (16)
( )( )
Jsm ) -DM c/M 1 -
c0M
c/M
1 1 + δM r 0
(3)
where δM is the thickness of the diffusion layer and r0 is the radius of the spherical surface. Whether diffusion is predominantly linear or radial basically depends on the dimensions of the biosurface as compared to the thickness of the diffusion layer. The limiting value of the spherical difusion flux Js/ m is / Js/ m ) -DM cM
(
1 1 + δ M r0
)
(4)
where in the limit of r0 . δM, the flux approaches that of the linear steady-state case. In the modification of the best equation, the change from linear to spherical transport will only affect the b parameter, which we will denote as bs: s
b )
J/u Js/ m
)
J/u DM c/M
(
1 1 + δ M r0
)
kdδ2/DML, ka′δ2/DM . 1
µ ) (DM/kacL)1/2
1
decreases correspondingly. Thus in studying metal uptake by microorganisms from media with complexes, not only the type of diffusion (linear or radial) is important but also the lability because this depends on r0 as well. Nonlabile Complexes: Kinetic Flux. If the system is dynamic but nonlabile (L , 1), the flux of metal is governed by the kinetic flux, which, for the linear case, is given by (19)
Jkin ) kdc/ML µ
(7)
With decreasing micoorganism size (r0), there is a decrease of lability of the metal complex because the transport in the diffusion layer becomes more and more radial. The diffusional flux Jdif increases; hence, the ratio Jkin/Jdif decreases and lability
(8)
with µ given by eq 7. These fluxes must be compared with J/u. If Jkin . J/u, then Q ) 1. If Jkin , J/u, then Q ) Jkin/J/u, defined by eq 8. For intermediate cases, one has to go back to the Michaelis-Menten equation and couple this to the kinetic flux via the surface concentration of M (c0M) (20). Labile Complexes: Speciation and Fluxes. For the case of labile complexes and linear steady-state diffusion, the flux Jm was defined as (1)
(5)
Rigorous lability criteria are based on a further condition concerning the interfacial fluxes and were obtained for the steady-state linear transport situation as described in part 1 (1). When considering radial diffusion, the operational lability of complexes also depends on the dimension of the organism, and changes in lability can have a drastic influence on the rate of uptake of metal ions. To evaluate lability in case of radial diffusion, a series of cases was analyzed for microelectrodes (2), and the relevant conclusions are developed below. Lability Criteria under Conditions of Radial Diffusion. The lability criterion parameter, L, for the cases of linear and radial diffusion is presented in Table 1. These expressions are valid for sufficiently strong complexes with DMLK ′/DM . 1 (18). Some coefficients of order unity have been removed since these are insignificant on the level of the inequalities given. The parameter L compares the rate of supply of M by a purely diffusion-controlled flux of ML (Jdif) to the rate that results from the kinetically limited dissociation of ML (Jkin). L is usually expressed in terms of the reaction layer thickness, µ, the fundamental kinetic quantity defined by (19)
a/bs . 1 f 1/a a/bs , 1 f 1/bs 1/a
low (bs , 1)
-1
(6)
low (a . 1)
1/bs
. 1)
Jm ) D h c/M
Dynamic Systems. The dynamic situation is characterized by rates of association and dissociation of ML that are so fast that on the effective timescale (17):
high (a , 1)
()
( )
c0M 1 (1 + K ′) 1 - / δ h c
(9)
M
where D h is the mean diffusion coefficient of M and ML and δ h is the corresponding (mean) diffusion layer thickness. Depending on the hydrodynamic conditions of the biouptake situation, δ h varies with D h according to some power function D h R with R usually between 1/3 and 1/2 (21). Quite generally, we can express δ h as γD h R where γ is the constant coefficient in δ h that does not depend on D h . The flux ratio parameter b can then be written as
b)
J/u J/m
J/u
) D h c/M
1 (1 + K ′) δ h
()
)
γJ/u / D h (1-R)cM,t
(10)
For radial diffusion, the flux is extended according to eq 3
Jsm ) D h c/M
( ) ( ) c0M 1 1 + (1 + K′) 1 - / δ h r0 c
(11)
M
so that the bs for radial conditions follows as
bs )
J/u
J/u
Jm
/ D h cM,t
) s/
(
1 1 + γD h R r0
)
-1
(12)
Again for r0 . δ h , this reduces to the linear diffusion case (10). For r0 , δ h (purely radial diffusion), we get bs ) J/ur0/D h c/M,t, where the dependence of the flux parameter on the effective diffusion coefficient D h (power -1) is different from that in eq 10 (power between -2/3 and -1/2).
Results and Discussion Static Systems. As compared to the linear transport situation outlined in part 1 (1), the differences caused by radial diffusion arise when the fluxes depend on the transport parameter b. Table 2 collects the limiting situations with respect to the VOL. 35, NO. 5, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Log J vs log r0. Free metal and/or inert complexes. Semiinfinite linear diffusion (bold curve), radial diffusion (thin curve), and mixed radial and linear diffusion (O) δ ) 1.0 × 10-5 and (b) δ ) 5.0 × 10-5 m. Other parameters are bulk concentration c* ) 0.5 mol m-3, surface concentration c0 ) 0, DM ) 8 × 10-10 m2 s-1. values of bs and a. It can be seen that the normalized flux Q depends on b only for bs . 1 with a , 1 or a . 1 with a/bs , 1. In these cases, we have Q ) 1/bs (J ) Js/ m), which means that the biouptake flux is given by the maximum radial transport flux in the medium. For a given type of biological system, with a fixed value of J/u, the difference between the linear and the radial cases is given by the dependence of J on r0. In Figure 1 (log J vs log r0), we can observe that the radial flux is strongly dependent on r0. The value of J (eq 3) tends to the linear limit when r0 . δM and tends to the purely radial case when r0 , δM. The effect of sphericity on the flux can be very large, up to several orders of magnitude for low values of r0. We can also see that a change in δ (e.g., stirring conditions) has a strong effect in the linear steady state, but a much smaller effect in the mixed case, and as expected, none at all in the purely radial transport. Since Js/ m increases with decreasing r0, bs decreases proportionally with r0. High Bioaffinity Case (a , 1). Since the limiting flux ratio bs decreases with decreasing r0, it may happen that, for a given J/u, bs is larger than unity for large r0 but of order unity or less for small r0. Thus, the applicability of the simple expressions for the uptake rates in the limit of high bioconversion capacity (bs . 1) has to be carefully checked when going from linear to radial transport. Anyway, it is clear that a change in transport conditions from linear to radial implies an increase in the rate of biouptake per unit surface area. For high bioaffinities, the flux Q can therefore vary from 1/bs to 1 or, in other terms, the increased transport flux may approach the biouptake limiting flux J/u. On the other hand, if the linear steady state provides enough metal such that already J/u , J/m (b , 1), then increasing the transport efficiency will not affect the uptake flux since it is already equal to the maximum uptake flux J/u. Then the enhancement of the transport in solution due to radial diffusion effects is not relevant. Low Bioaffinity Case (a . 1). For high bioconversion capacities, there are two limiting cases: a/bs . 1 or a/bs , 1. For a given bioaffinity a, the change from linear to radial transport means that a/bs is increasing. In this case Q ) 1/a (a/bs . 1) becomes more probable than the other extreme Q ) 1/bs (a/bs , 1). The case of low bioconversion capacity is trivial. If linear steady-state transport provides enough metal so that already J/u , J/m, then increasing the transport rate will not affect the uptake flux. Dynamic Systems: Labile Complexes. In the case of labile complexes, it is important to understand how the fluxes depend on the radius r0 and how other parameters influence 896
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FIGURE 2. Log Js/ m vs log r0. Labile complexes. Semi-infinite linear diffusion (bold curve), radial diffusion (thin curve), and mixed radial and linear diffusion for different r values (b) 0, (0) 1/3, ([) 1/2, and (O) 1. Other parameters: c/M,t ) 10-3 mol m-3, c/L,t ) 0.1 mol m-3, c0 ) 0, E ) DML/DM ) 0.1, DM ) 1 × 10-9 m2 s-1, DML ) 1 × 10-10 m2 s-1, K ′ ) 103.
FIGURE 3. Log Js/ m vs log r0. Labile complexes. Semi-infinite linear diffusion (bold curve), radial diffusion (thin curve), and mixed radial and linear diffusion for different E (DML) values: (b) 0.01 (1 × 10-11 m2 s-1), (0) 0.1 (1 × 10-10 m2 s-1), (O) 1 (1 × 10-9 m2 s-1). Other parameters: c/M,t ) 10-3 mol m-3, c/L,t ) 0.1 mol m-3, c0 ) 0, r ) 1/3, DM ) 1 × 10-9 m2 s-1, K ′ ) 103. this dependency. Equations 10 and 12 define b and bs, and the key parameters governing these equations are D h , R, and r0. The dependency on R is important for the linear and mixed cases (eqs 10 and 12), but it does not affect the purely radial case, as can be seen in Figure 2. For different values of a (0, 1/3, 1/2, and 1), the radial term remains constant and the linear term decreases with decreasing a. The speciation also appears through the mean diffusion coefficient D h , which depends on () DML/DM) and K ′. The effect of decreasing (or DML) is shown in Figure 3. This decrease leads to a decrease in flux because D h becomes smaller. The decrease in the linear term is proportional to D h (1-a) whereas in the radial term it decreases with D h . The radial term becomes dominant for smaller r0 values, which can be explained by the increase of the term 1/(γD h R) with decreasing D h . A similar development is obtained with strong complexes (increasing K ′) since this generates a concomitant decrease of D h. Effect of Radial Diffusion on Normalized Flux Q for Labile Complexes. We are interested in tracing the influence of going from linear to radial steady-state diffusion on the contribution of labile complexes to uptake rates. Therefore, we will compare the linear, mixed, and radial transport fluxes for different values of the parameters a and bs, assuming that the Best equation is applicable. Table 3 shows the limiting cases of Q and outlines whether the contribution from the labile complexes is essential.
TABLE 3. Conditions for Bioavailability of Bioinactive Labile Metal Complexes and Corresponding Fluxes for General Case of Mixed Linear/Radial Diffusion bioavailability of complex bioconversion capacity
low
(bs
, 1)
bioaffinity
limiting value of Q
low (a . 1) high (a , 1)
1/a 1
low (a . 1) high (a , 1)
1/a 1
1/a
low (a . 1) a /b s , 1
1/a high
(bs
. 1) low (a . 1) a /b s . 1
1/bs
high (a , 1)
1/bs
complex contributes
ifa
J/u , (c/MDM + c/MDML)(1-R)(c/M,t)R J/u c/M
, (DM)(1-R)
1 1 + (c/MDM + c/MDML) γ r0
yes no
1 1 + DM γ r0
no yes
J/uc/M 1 1 , (c/MDM + c/MLDML)(1-R)(c/M,t)R + (c/MDM + c/MLDML) KM γ r0 J/u KM
, (DM)(1-R)
1 1 + DM γ r0
yes no
J/uc/M 1 1 . (c/MDM + c/MLDML)(1-R)(c/M,t)R + (c/MDM + c/MLDML) KM γ r0 1 1 J/u . (c/MDM + c/MDML)(1-R)(c/M,t)R + (c/MDM + c/MDML) γ r0
yes yes
a In all expressions, the first term corresponds to the linear diffusion term, and the second one corresponds to the radial diffusion term. Limiting conditions for the linear and radial cases are found by neglecting either the second or the first term, respectively.
FIGURE 4. Log Q vs log r0 for labile complexes calculated using eq 13 for different K′ values: (thin curve) 10, (- - -) 102, (s s) 103, (bold curve) 104. Other parameters: a e 0.1, bs ) 0.05, K ) 103, c/M,t ) 10-5 mol m-3, c0 ) 0, r ) 1/3, e ) 0.5, DM ) 1 × 10-9 m2 s-1. Low Bioconversion Capacity (bs , 1). The condition bs , 1 means that the transport of free plus complexed metal is fast as compared to the bioconversion of M or / J/u, D h cM,t
(
)
1 1 + γD h R r0
(13)
Using the definition of D h , this can be transformed into
1 1 / J/u , (c/MDM + c/MLDML)(1-R)(cM,t ) + (c/MDM + c/MLDML) γ r0 (14) where the first term in the right-hand side is the linear contribution and the second term is the radial contribution. It is now obvious that there are two possible situations: (i) the free metal alone can fulfilll the uptake of the organism:
J/u , c/M(DM)(1-R)
1 1 + c/MDM γ r0
(15)
or (ii) the contribution of the complex is necessary, and then eq 14 applies. In Figure 4, we plot log Q vs r0 using eq 15 for different values of K ′, i.e., different stabilities of the labile complexes. It can be seen that for low K ′, eq 15 tends to the limit of Q
) 1 (eq 15) but with increasing K ′ it becomes more and more difficult to achieve that limit. Increasing K ′ increases the value of a but in this case all the values presented in Figure 4 still obey the condition a , 1. The increased transport due to the radial term is also obvious since for lower r0 it is possible to reach the limit Q ) 1 for a large range of complex stabilities K ′ whereas the linear transport does not allow this. For r0 < δ h , the diffusion flux is predominantly radial, and a smaller concentration of free metal is sufficient to completely sustain the uptake of the organism. This situation applies for both high and low bioaffinities since in both situations the uptake flux is much lower than Js/ m. For high bioaffinity Q ) 1 (J ) J/u), and for low bioaffinity Q ) 1/a (J ) J/u c/M/KM). High Bioconversion Capacity (bs . 1). For high bioconversion rates so that the maximum uptake rate J/u is much larger than the maximum supply rate Js/ m, we have / h cM,t J/u . D
(
)
1 1 + R r γD h 0
(16)
High Bioaffinity (a , 1). A high value of bs means that the maximum uptake flux of the metal by the organism cannot be matched by the transport capacity of the medium. Then the labile complexes may fully contribute to the uptake, up to the maximum transport flux in the medium. For complex systems with a high degree of complexation, the condition a ) KM/c/M , 1 is a very restrictive one since c/M is already much smaller than c/M,t. If an increasingly radial flux leads to a bs no longer . 1, then the value Q will increase from 1/bs to ultimately 1, i.e., the increased transport flux may approach the biouptake limiting flux J/u. Low Bioaffinity (a . 1). As before in the low bioaffinity range, several cases can be distinguished. This is so because both a and bs are much larger than 1; hence, the derivation of the limits of the Best equation has to take into consideration the value of the bioavailability parameter a/bs. (i) a/bs , 1. This is a very limiting restriction since if a . 1, then bs . a . 1, which means that the bioconversion capacity must be extremely high. Then J/u c/M/KM . Js/ m, which means that eq 14 becomes VOL. 35, NO. 5, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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J/u c/M 1 / . (c/MDM + c/MLDML)(1-R)(cM,t )R + KM γ 1 (c/MDM + c/MLDML) (17) r0 This is even more restrictive in the case of radial diffusion since bs < b. The limit is the same as for the high bioaffinity case above, and in both cases full contribution of the labile complex ML is necessary to sustain the metal uptake. The difference is that if, in the high bioaffinity case, bs is no longer . 1, Q is changing from 1/bs to 1 when going from linear to radial. In the corresponding low bioaffinity case, Q is going from the diffusion-controlled 1/bs to the value 1/a prescribed by the linear part of the Michaelis-Menten equation. (ii) a/bs . 1. This case is similar to the one with bs , 1 and a . 1 but now requires a/bs . 1 or J/u c/M/KM , Js/ m instead of bs , 1. This means that eq 14 for the contribution of labile and free metal becomes
J/uc/M 1 / , (c/MDM + c/MLDML)(1-R)(cM,t )R + KM γ 1 (c/MDM + c/MLDML) (18) r0 which reduces to
[
]
Examples
FIGURE 5. Q and L vs log r0 for Pb2+ and bs ) 0.1 ( δ to r0 < δ (2).
From Figure 5a, we can see that for the chosen values of the parameters the lability criterion parameter L equals 50 for r0 . δ and the system is labile as long as r0 > 1.0 × 10-5m (L > 10) and becomes nonlabile for r0 < 1.0 × 10-7m (L < 0.1). Q ) 1 or 1/a. First we choose KM ) 10-9 mol m-3 and J/u ) 5 × 10-10 mol m-2 s-1. For the parameter set of Figure 5a, c/M is 9.9 × 10-6 mol m-3; hence, bs ) 0.1 and a ) 10-4. For this case, condition 15 is satisfied and Q equals 1 (see Table 3). Figure 5a confirms that Q is not affected by loss of lability of the complex if the free metal by itself is capable of sustaining the uptake. However, if we choose different conditions with a larger K ′, the lability decreases and eq 15 is no longer satisfied. Then the complex is bound to contribute to the flux in order to maintain Q ) 1. Figure 5b, based on eqs 14 and 15, shows the modifications for the same J/u and KM when cL,t is increased to 0.1 mol m-3 (K ′ ) 103). It illustrates that, with decreasing r0 and concomitant loss of lability, the complex contributes less, resulting in a decrease of the flux. However, that decrease in the flux is counter-balanced by the increase due to the growing radial nature of the diffusion; for r0 smaller than 10-6 m, the free metal by itself is again capable of supplying the uptake. Q ) 1/bs. We now choose KM ) 10-8 mol m-3 and J/u ) 4 × 10-3 mol m-2 s-1, corresponding to a very high uptake capacity. For a c/M of 9.9 × 10-6 mol m-3, we have bs () 106) . 1 and a () 10-3) , 1. From Table 3, it can be seen that then Q ) 1/bs. Figure 6 shows that for r0 ) 10-4 m the system is labile and Q is relatively low (1/bs). With decreasing r0, Q increases due to the enhancement of transport by radial diffusion until the complex starts loosing lability for r0 < 10-5
J/u , KM (DM)(1-R)
1 1 + DM γ r0
(19)
as the condition for the situation where the free metal only is able to satisfy the uptake of the organism. Radial diffusion increases the flux, which favors this situation, in contrast with the case a/bs , 1.
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FIGURE 6. Q and L vs log r0 for Pb2+ and bs ) 106 (.1), a ) 10-3 (,1), KM ) 10-8, J/u ) 4 × 10-3 mol m-2 s-1. L is given by the full curve, and its macroscopic limit is given by the horizontal bar. The biouptake flux Q calculated using the Best equation is given by (9), and the kinetic flux is given by (2). Other parameters: DM ) 10-9 m2 s-1 (µ ) 10-7 m), E ) 0.1, c/M,t ) 10-3 mol m-3, c/L,t ) 10-2 mol m-3 , K ) 104 mol-1 m3, K ′ ) 102, r ) 1/3, δM ) 5 × 10-5 m. m. In the region 10-7m < r0 < 10-5 m, Q will be between 5 × 10-6 and 2.5 × 10-5 with both diffusional and kinetic contributions to the flux. For r0 < 10-7 m, the complexes become completely nonlabile, and the contributions from the complex are purely kinetically controlled. The loss of lability does not necessarily mean that the actual uptake flux will be reduced. This is so because the kinetic flux at small r0 might still be larger than the diffusional flux at large r0. For a single microorganism with a given r0, different metals may have remarkably different behavior due to the metalspecific association and dissociation kinetics of the complexes involved. Take for example, an organism of r0 ) 10-5 m under the conditions of case b, which is simultaneously taking up Pb2+ and Ni2+ (a metal with relatively slow complex association/dissociation kinetics (22)) and which has a similar affinity a toward both metals. The Pb(II) species are labile, and their flux is controlled by coupled diffusion, whereas the Ni(II) complexes are inert, and the flux is determined by diffusion of free Ni(II). However for similar organisms with decreasing r0, the Pb complexes are loosing lability, resulting in a strong dimensional dependence of the relative uptake characteristics for the two metals. Again the main conclusion is that radial transport influences the biouptake in two different ways: it not only generates changes in the magnitude of the diffusional fluxes toward the surface but it also affects the labilities and, hence, the bioavailabilities of complex species.
List of Symbols R
power of D h in expressions for δ
δ h
mean diffusion layer thickness
δ
diffusion layer thickness
δM
diffusion layer thickness for free metal
()DML/DM)
γ
constant coefficient in δ h that does not depend on D h
µ
) (DM/kacL)1/2, reaction layer thickness
a
) KM/c/M, normalized bioaffinity parameter
b
) J/u/J/m, limiting flux ratio
c*
concentration in the bulk
c0
concentration at the surface
cL
free ligand concentration
cL,t
total ligand concentration
cM
free metal concentration
cM,t
total metal concentration
D h
mean diffusion coefficient
Di
diffusion coefficient of species i
J
biouptake flux
Jdif
metal transport flux due to diffusion
Jkin
metal transport flux due to dissociation of ML
Jm
metal transport flux in the medium
J/m
limiting metal transport flux in the medium
Jsm
metal transport flux considering spherical diffusion
Js/ m
limiting value for the spherical diffusion flux
J/u
limiting biouptake flux
K
) cML/(cM cL), stability constant
K′
) KcL,t stability constant times total ligand concentration
ka
association rate constant
ka′
) kacL,t, association rate constant times total ligand concentration
kd
dissociation rate constant
KM
bioaffinity parameter
L
ligand
L
lability criterion: labile L . 1, nonlabile L , 1
M
free metal
ML
metal complex
Q
) J/J/u, normalized flux
r0
radius of spherical organism
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Received for review February 28, 2000. Revised manuscript received November 28, 2000. Accepted December 4, 2000. ES000042N
