Environ. Sci. Technol. 2004, 38, 2397-2405
Metal Speciation Dynamics and Bioavailability: Bulk Depletion Effects J O S EÄ P . P I N H E I R O , * , † JOSEP GALCERAN,‡ AND HERMAN P. VAN LEEUWEN§ Centro Multidisciplinar de Quı´mica do Ambiente, Departamento de Quı´mica e Bioquı´mica, Faculdade de Cieˆncias e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal, Departament de Quı´mica, Universitat de Lleida, 191 Av. Rovira Roure, 25198 Lleida, Spain, and Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands
Under conditions of bulk depletion, the speciation and bioavailability of trace metals must be considered at two different time scales: (i) the time scale of the biouptake flux, as determined by diffusion of the bioactive free metal, dissociation of the bioinactive complex species, and the internalization rate; and (ii) the time scale of depletion of the bulk medium. The implications of these two time scales for the speciation dynamics are discussed in terms of experimental conditions. The geometry of the system is taken into account via a spherical cellular model. It considers a spherical organism depleting a spherical volume in a nonstirred medium and assumes linear adsorption of the metal at the biointerface and first-order internalization kinetics. In cases where the rate of biouptake is fully controlled by the internalization step, concentration gradients in the medium are insignificant. Then the biouptake becomes independent of the geometry of the system, and the model has a much simpler solution. Examples of trace metal uptake by microorganisms are analyzed: (i) cobalt uptake by Prochlorococcus in the presence of NTA, under conditions where bulk depletion is the controlling process due to the large number of organisms and high internalization rates, (ii) silver uptake by Chlamydomonas reinhardtii with significant effects of bulk depletion, due to the high internalization rate; (iii) lead uptake by Chlorella vulgaris with pratically negligible bulk depletion due to the low internalization rate of the metal; and (iv) lead uptake by intestinal Caco-2 cells, illustrating the simplification of the bulk depletion model for a system with different geometry where internalization is the rate-controlling step.
Introduction In many studies on the influence of speciation on the biouptake of trace metals, bulk concentrations remain essentially constant. This is the case for many natural systems where the medium is essentially infinite, relative to the scale * Corresponding author e-mail:
[email protected]; telephone: +351289-800905; fax: +351-289-819403. † Universidade do Algarve. ‡ Universitat de Lleida. § Wageningen University. 10.1021/es034579n CCC: $27.50 Published on Web 03/04/2004
2004 American Chemical Society
at which the biouptake occurs. However there are also situations where depletion occurs (e.g., in uptake of essential metals by plankton in the ocean). The depletion in dissolved metal is so large that in most areas of the ocean the surface concentrations are a small fraction of those in the deep waters (1-3). Bulk depletion may also be important in other types of systems, like bioreactors and biofilms (4, 5), “in vitro” intestinal uptake (6-8), and nutrient uptake by plants (9). In addition many authors, including Morel and MorelLaurens (10) and recently Gerringa et al. (11), recognized that adding chelators to maintain the free metal concentration constant is not without problems. Citing Morel “It is an often poorly appreciated paradox that to mimic the effects of metals at nanomolar and lower concentrations in oceanic water with very low organic content it is in fact necessary to use culture media containing micromolar concentration of metals and artificial organic ligands”. In this work, we intend to show that in principle it is not always necessary to take measures to avoid bulk depletion since it can be modeled with a relatively simple model, especially in the case of internalization control. We actually show that the course of depletion may be used to determine parameters of the uptake process. The main objective is to develop a basic understanding of the influence of bulk depletion on the biouptake rates and their dependence on dynamic trace metal speciation. A general mathematical framework is formulated in terms of a cellular model that considers a (spherical) organism depleting a (spherical) volume in a nonstirred medium. The model includes linear adsorption of the metal at the surface of the organism and first-order internalization kinetics. For more involved systems, such as bioreactors and biofilms (4, 5), refined consideration of the kinetics, transport, geometry, and adsorption isotherms may be required. Although such situations cannot be solved analytically, the basic arguments related to the effects of bulk depletion on the biouptake are still applicable. Therefore, we present this model as a first step toward exploration of the behavior of systems containing metal complexes under conditions of bulk depletion.
Formulation of the Model Biological System. Let us consider a finite volume (V) of a dispersion containing a certain number (z) of randomly distributed spherical microorganisms with volume (V0) and radius (r0). For purposes of modeling, we formally divide the volume (V) into z spherical units with volume (Vf) and radius (rf). Thus
4 V ) zVf ) πzrf3 3
(1)
implying that the volume of medium per cell (Vm) equals (4/3)π(rf3 - r03). The depletion volumes around an individual cell are considered independent since the behavior of small organisms is controlled by viscosity and not by inertia and they move coherently with the medium (12). For the spherical symmetry adopted, the number of moles of a given metal (M) present in the volume (Vm) at a time (t) is generally given by
Ω(t) ) 4π
∫rc rf 2
r0
M,T(r,t)
dr
(2)
which simply becomes cM,TVm for a homogeneous system. Consumption of metal by the organism depletes the volume (Vm), thus rendering the number of moles (Ω) and the VOL. 38, NO. 8, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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concentration (cM,T) time-dependent quantities. Furthermore, it is assumed that the microorganisms are constant in size and number for the duration of the experiment. To satisfy this assumption, the usual procedure is to conduct relatively short experiments (13, 14). If the experiments are performed over several days, then the model can only be used as a first approximation because other factors should be taken into account: (i) the growth kinetics and mortality of the organisms (15); (ii) the excretion (efflux) of the internalized metal species (16); (iii) the possibility that the organism exudes ligands to the medium, thus modifying speciation and, therefore, possibly the uptake pathways (17-20); (iv) variation in the concentration of transporter sites due to exposure conditions (e.g., changes in pH or hardness, synthesis of transporter sites). The extent of depletion is given by the decrease of the number of moles in the medium (Ω(0) - Ω(t)), which equals the time integral of the biouptake flux (Ju(t)) (i.e., the internalized amount) counted as positive for transport from the medium to the organism: 2
Ω(0) - Ω(t) ) 4πr0
∫ J (t′) dt′ t
0 u
(3)
where t′ is a dummy time integration variable. The biouptake flux (Ju) is usually described by a MichaelisMenten type of equation, representing a fast Langmuirian adsorption of the free metal, followed by a first-order internalization step. The adsorption process is generally fast as compared with the diffusion and internalization processes, and this ensures local equilibrium between adsorbed and free metal ions (1, 21, 22). To simplify the mathematical treatment, we use the linear portion of the Langmuir isotherm for low metal concentrations, also known as the Henry isotherm (ΓM(t) ) KHc0M(t)), and the biouptake flux becomes
Ju(t) ) kintΓM(t) ) KHkintc0M(t)
(4)
where KH is the Henry coefficient, kint is the internalization rate constant, ΓM is the amount of metal adsorbed, and c0M(t) is the metal concentration at the organism surface. Apart from the transporter sites, cell walls and plasma membranes may contain sites that adsorb a significant amount of metal ions (13). The key issue is that, as long as the metal adsorbed to the mere adsorption sites is in equilibrium with the internalization sites and with the solution, the internalization rates will not be affected by their existence. However the bulk concentration will be affected by the adsorption, which usually appears as a fast initial decrease in concentration since the adsorption process is generally fast. As long as adsorption at all the relevant sites (i.e., the transporter sites and the mere adsorption sites) remains in the linear regime, an overall Henry isotherm applies with an effective adsorption coefficient (23), which is a weighted average of the Henry coefficients of the adsorption and transporter sites. Under these conditions, the internalized amount is correctly described using eq 4, but discrimination between the Henry coefficients for the transporter and adsorption sites would require knowledge of their respective concentrations and affinities for the metal of concern. The metal concentration at the surface of the organism (c0M(t)) is related to the bulk concentration through the mass transport in solution. For small organisms, radius below 10-5 m, (24, 25) spherical diffusion is the predominant and very efficient mode of mass transport. Convective terms can then generally be neglected, which is similar to the mathematical treatment encountered with microelectrodes (26). Note that this is applicable even for the case of swimming organisms, where the motion could enhance mass transport. Studies on chemoreception showed that an increase in uptake is not 2398
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TABLE 1. Flux Time Constants, τSS (eq 5), and Depletion Time Constant, τ (Solution Diffusion Control, eq 14 with β ) 0), DM ) 1 × 10-9 m2 s-1, for Different Sizes of the Cell (Expressed by Radius, r0) and Cell Densities, n τ (s)
r0
(10-6 0.3 1.0 3.0 10.0
m)
τSS (s) 9 × 10-5 1 × 10-3 9 × 10-3 1 × 10-1
n)
1011
m-3
n ) 1013 m-3
2650 800 265 80
26.5 8 2.65 0.8
very significant even in the limiting situations of fast uptake where the surface of the organism behaves as a perfect metal sink (27). A very relevant aspect is that, in spherical diffusion, steady state is rapidly attained for small organisms (Table 1), the flux time constant τSS being given by (28)
O(τSS) ) O(r02/DM)
(5)
where the symbol O denotes order of magnitude and DM is the diffusion coefficient of the metal. The model assumes that the concentration profile in the diffusion layer is essentially at steady state while the bulk concentration is being depleted (on a time scale much longer than the time needed for the re-adjustment of the concentration profile). For this assumption to be valid, the radius of the spherical unit (rf) must be much larger than the organism radius (r0) (see Supporting Information). Metal diffusion in solution can be affected by the presence of ligands. To describe bulk depletion, two cases will be considered: (i) the medium contains the metal only in its free form and (ii) the medium contains free metal ions as well as complexes. The complexes only participate in the biouptake process via dissociation into free metal and ligand. Free Metal Only Case. The steady-state diffusive flux of metal toward a microsphere is given by (29)
Jdif(t) ) DM [c/M(t) - c0M(t)]/r0
(6)
and the resulting concentration profile for semi-infinite steady-state diffusion is
cM(r,t) ) c/M(t) - [c/M(t) - c0M(t)](r0/r)
(7)
where c0M(t) and c/M(t) denote the concentrations of M at the organism surface and in the bulk, respectively, for a certain time (t). Applying a profile this type to our finite cellular model implies that we equate the bulk concentration (c/M(t)) with the concentration at the outer edge of the spherical unit cell with radius (rf) or cM(rf,t) ) c/M(t). Due to the spatial constraints of the confined medium, the steady-state profile defined by eq 7 implies a finite gradient of metal at the edge of the cell unit (i.e., at r ) rf). For an alternative mathematical model based on the boundary condition at r ) rf, the resulting expressions are much more involved and require numerical solutions (see Supporting Information). Transient effects in the flux are neglected, and this implies that the gradual development of the profile represented by eq 7 is modeled as being generated instantaneously. Substituting the steady-state concentration profile (cM(r,t)) (eq 7) in depletion function (eq 2) and neglecting the terms containing V0 (see Supporting Information) yields the expression:
Ω(t) ) Vfc/M(t) where Ff is the ratio rf/r0.
3Vf / (c (t) - c0M(t)) 2Ff M
(8)
TABLE 2. Overview of the Expressions for Biouptake Flux, J(t)a, and Depletion Time Constant, τb, in the Relevant Limiting Cases for Biouptake Flux in the Presence and Absence of Ligands internalization control
complexesc
free metal/inert dynamic complexes
condition
J(t)
KHkint , DM/r0 KHkint , pDM/r0
KHkintc/M(t) KHkintc/M(t)
solution diffusion control τ
condition
J(t)
τ
Vf/(A0KHkint) Vf(1 + K′)/(A0KHkint)
KHkint . DM/r0 KHkint . pDM/r0
(DM/r0)c/M(t) (pDM/r0)c*M(t)
Vf/(A0(DM/r0)) Vf(1 + K′)/(A0(pDM/r0))
inert complexesd (ξ ) 0) nonlabile complexes (ξ , 1) labile complexes (ξ ) 1)
p)1 p ) 1 + κa1/2 p ) 1 + K′
a Given by eqs 12 and 19. b Given by eqs 14 and A8. c Inert/dynamic complexes on the depletion time scale as defined by eq 18. nonlabile/labile complexes on the flux time scale as defined by eq 22.
Steady state means the uptake flux (Ju) and the diffusive flux (Jdif) are equal, which can be exploited to compute the metal concentration at the surface:
c0M(t) ) [DM/r0/(DM/r0 + KHkint)]c/M(t) )
(1 + β)-1c/M(t) (9)
with β defined as
β ) KHkintr0/DM
(10)
The uptake rate parameter β is identical to the ratio b/a (b and a being the governing transport and affinity parameters in the Best equation) (21); for complex systems, this ratio is equal to what has been denoted as the “bioavailability number”, Bn (30). Combination of eq 8 with the dependence of the surface concentration on the internalization and transport parameters (eq 9) allows for the elimination of the surface concentration c0M(t):
Ω(t) )
Vfc/M(t)
3Vf β c/ (t) 2Ff 1 + β M
(
)
(11)
The steady-state flux (J) for a given time (t) is
J(t) )
(
DM / r0 1 [cM(t) - c0M(t)] ) + r0 KHkint DM
)
-1
c/M(t)
(12)
where DM/r0 represents the diffusive conductance and KHkint represents the internalization conductance (31). Using eqs 11 and 12 in eq 3 allows solving for the bulk metal concentration evolution:
c/M(t) ) c/M(0) exp(- t/τ)
(13)
in which τ is the time constant of the depletion process given by
τ)
Vf(1 + β) - (3/2)(Vf/Ff)β A0KHkint
(14)
where A0 is the surface area per organism. For first-order uptake kinetics, the depletion is thus represented by an exponential decay with time constant (τ), which is a function of (i) the radius of the organism (r0); (ii) the internalization rate parameter KHkint; and (iii) the radius of the depletion volume (rf) that according to eq 1 and for a constant total volume of solution (V) is determined by the cell density, n () z/V). Table 2 shows the limiting situations for biouptake (i.e., the cases where the biouptake flux is fully controlled either by the internalization process (β , 1) or by the diffusion in solution (β . 1). In the case of internalization control, no concentration gradient is created in solution; hence, τ (given
d
Inert/
by eq 14) reduces to Vf/A0KHkint. In the case of diffusion control, the internalization rate is so fast that the metal concentration at the surface tends to zero, leading to the highest rates of bulk depletion. From Table 1, it can be seen that in this limiting case the prerequisite that the time constant of reaching steady state must be much higher than the depletion time constant holds for organisms with radius up to 3 × 10-6 m, even for large cell densities. Metal in the Presence of Ligands Case. Let us consider the situation where the medium contains a ligand (L) with which the metal (M) forms the bioinactive complex (ML). The ligand is assumed to be in such excess over M that its concentration is essentially constant at any r:
cL(r,t) ) c/L(t)
(15)
In such a case, the equilibrium concentrations of metal ion and complex are linearly related by the effective complexation constant (K′):
K′ ) Kc/L(t) ) c/ML(t)/c/M(t)
(16)
The rates of interconversion between M and ML derive from the formation rate constant (ka) and the dissociation rate constant (kd). Complex systems are divided into inert and dynamic categories. This distinction is based on the values of the intrinsic lifetimes of M and ML associated with the complex formation/dissociation in solution 1/k′a (where k′a ) kac/L(t)) and 1/kd, respectively, relative to the effective time scale of the experiment (32, 33). Inert Complexes. Complexes are inert on the depletion time scale if
k′aτ , 1 and kdτ , 1
(17)
In this situation, conversion of M into ML and vice-versa is insignificant in the bulk of the medium. Hence, only the free metal is taken up, and the complexes do not interfere in any way. Therefore, even in the presence of a large excess of complexes, there is no buffering of the free metal on the depletion time scale, and the biouptake is identical to the free metal only case. Dynamic Complexes. The system is dynamic on the depletion time scale if the rates of association/dissociation of the complex are fast enough to maintain equilibrium in the bulk (k′aτ . 1 and kdτ . 1). In the transition from dynamic to inert, the complexes contribute less to the uptake, and the steady state flux is increasingly sustained by the free metal ion. Under usual experimental conditions, τ . τSS, and thus many complexes that are inert on the flux time scale may become dynamic on the depletion time scale. From Table 1, we see that an organism of radius 10-6 m has a τSS of 10-3 s; hence, complexes will be inert on the flux time scale (k′aτSS , 1 or kdτSS , 1) if k′a and kd , 103 s-1. On the depletion time scale, the potentially less dynamic situation VOL. 38, NO. 8, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 3. Parameters for Metal Uptake in the Presence of Complexing Agents by Various Organisms r0 (10-6 m)
n (1011 m-3)
Co-NTA Prochlorococcus Ag-Cl Chlamydomonas
0.31 2.0
300 0.1
Pb-NTA Chlorella Pb-Caco2
1.8
100
a
KHkint (10-4 m s-1) 1.6 1.5 0.56 0.00022 0.0027
Conditional K calculated for the experimental conditions (seawater).
(smaller τ) occurs for a system under full diffusion control, and from Table 1, we obtain for a cell density of 1013 cell m-3 a τ value of O(10) s. From eq 17, the system is considered inert if k′a or kd , 10-1 s, 4 orders of magnitude lower than for the flux time scale. Any complex system that is inert on the flux time scale and dynamic on the depletion time scale will show fluxes that reflect free metal only and depletion characteristics reflecting free metal plus bound metal. For a metal in the presence of ligands, Ω(t) is defined by the sum of free and complexed metal species in the spherical cellular volume and can be expressed in a manner similar to eq 2:
Ω(t) ) 4π
∫ r [c rf 2
r0
M(r,t)
+ cML(r,t)] dr
(18)
Using the reasoning for the free metal case, we obtain the same expression as given by eq 13 but with a different depletion time constant (τ) that now depends on the extent of lability of the complexes. Two different expressions for τ are found again by equating the internalization and pertaining diffusive fluxes (Ju and Jdif). The diffusive flux (Jdif) now depends on the behavior of the complex species (ML). Qualifications such as “labile”, which indicates the full contribution of complexes to the flux, and “nonlabile”, where the flux is kinetically controlled, are based on a comparison of fluxes that are controlled by finite rates of dissociation of ML to the limiting diffusional fluxes (32). The biouptake flux under conditions of excess ligand and spherical diffusion is now given by
J(t) )
(
)
r0 1 + KHkint pDM
-1
c/M(t)
(19)
in which the parameter (p) expresses the contribution from dissociation of ML to the flux (35):
p ) (1 + K′ξ)
(20)
where ξ is the degree of lability ranging from 0 for inert complexes (p ) 1) to 1 for fully labile complexes (p ) 1 + K′) and is defined as (34, 35):
ξ)
κ1/2 a (K′(1 + K′))1/2 + κ1/2 a
(21)
where is DML/DM and κa is the dimensionless kinetic constant defined as (35):
κa ) kac/Lr02/DM
(22)
which compares the characteristic time of diffusion (r20/DM) to the mean lifetime (1/kac/L) of the free metal. The analytical expression of τ for dynamic complexes (Appendix A) is quite involved since it now also depends on K′, , and ka. Therefore, we will consider the relevant limiting situations for both internalization control and diffusion control (Table 2). 2400
9
DM (10-9 m2 s-1)
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b
E
0.64 1.5
0.55 1.0
0.8 0.8
0.44
ka (ML) (105 mol-1 m3 s-1) 2 1.2 6200 6200
K (ML) (mol-1 m3) 5800a 2.0b 180c 2.9 × 108 10
pH 7.8 7 6 7
AgCl0. c AgCl2-.
If the biouptake flux is governed by the internalization, there is no diffusion limitation and only the bulk free metal concentration is relevant to the uptake (this is essentially the free ion activity model (1)). Comparing this situation with the metal only case, the difference lies in the thermodynamic complexation factor (1 + K′). Usually, K′ > 1, so τ will be larger in the complex case, and the physical explanation is that the complex behaves as a free metal buffer. In the case where the biouptake flux is governed by diffusion in the medium, the dissociation of complex species on the flux time scale may become relevant (21). We can distinguish the usual kinetically limiting situations of inert and dynamic complexes and, within the dynamic situation, the cases of nonlabile and labile complexes (32). If the complexes are inert on the flux time scale (i.e., if κa , 1 (ξ approaches 0, p ) 1), the diffusive flux is a function of the free metal alone. Since κa ) kac/L r02/DM ) k′aτSS, the condition κa f 0 is analogous to the condition for complex inertness. In Table 2, substituting the p value for inert complexes in the τ equation, we see that this result parallels the one obtained for the free metal case at the limit of diffusion control, the only difference being the presence of the thermodynamic complexation factor (1 + K′). Like in the case of control by internalization, the free metal concentration governs the biouptake flux. For diffusion control, the complexes are inert on the flux time scale and therefore cannot contribute to the flux. However, since the complexes are dynamic with respect to the depletion time scale, the amount of complex acts as a buffer of free metal in the bulk medium. If the complexation reaction is dynamic on the flux time scale (κa . 1), two limiting possibilities exist with respect to the nature of the flux: labile or nonlabile complexes. The complexes are nonlabile if κa , K′(1 + K′), implying that K′ . 1 and ξ approaches κ1/2 a /K′. The contribution of ML to the transport of M is then kinetically controlled with p f (1 + κ1/2 a ), and the biouptake flux (23) is the sum of the diffusive free metal flux and the kinetic flux due to the slowly dissociating complex. The depletion time constant (τ) decreases with increasing κa, indicating that due to the kinetic contribution to the flux it takes less time than in the inert case to deplete the medium for a given amount of total metal. In the case of fully labile complexes (i.e., κa . K′(1 + K′)), ξ approaches 1, p f (1 + K′), and the complex ML fully contributes to the flux, which for spherical diffusion becomes the sum of the purely diffusive fluxes of the free metal and the labile complex. Note that if the diffusion coefficient of the complex is equal to that of the metal ( ) 1), τ reduces to the expression for the free metal case. This means that the time required to deplete the bulk, in the presence of a fully labile complex with ) 1, is the same as if all the metal was free. This conclusion is easily drawn from the flux equation: when equals 1, then cM,T ) (1 + K′)cM and the flux becomes DMc/M,T/r0. If the diffusion coefficient of the complex is smaller than the diffusion coefficient of the metal, then there is a retardation effect in the flux that will be reflected by a longer depletion time scale.
/ / FIGURE 1. Variation of cM,T (t)/cM,T (0) with time displaying the effect of cell density n on bulk depletion (eq 13) for the free metal only case. Values of the parameters used are as follows: (9) Pb-Chlorella, DM ) 8 × 10-10 m2 s-1, r0 ) 1.8 × 10-6 m, KHkint ) 2.2 × 10-8 m s-1, n ) 1013 cell m-3; (O) Ag-Chlamydomonas, DM ) 1.5 × 10-9 m2 s-1, r0 ) 2.0 × 10-6 m, KHkint ) 1.0 × 10-4 m s-1, n ) 1010 cell m-3; and Co-Prochlorococcus DM ) 6.4 × 10-10 m2 s-1, r0 ) 3.1 × 10-7 m, n ) 3.1 × 1013 cell m-3, KHkint ([) 0.3 × 10-4 m s-1, (4) 3.3 × 10-4 m s-1.
FIGURE 2. Time course of cobalt uptake by Prochlorococcus in the presence of 1.0 × 10-2 mol m-3 NTA. Experimental data from Figure 3A of ref 36 for (9) 0.94 × 10-10 mol m-3 57Co, (O) 2.9 × 10-10 mol m-3 57Co, and ([) 9.4 × 10-10 mol m-3 57Co. Values of the parameters used in the model are (thick line) as follows: DM ) 6.4 × 10-10 m2 s-1, r0 ) 3.1 × 10-7 m, n ) 3.1 × 1013 cell m-3, ka ) 2 × 108 mol-1 m3 s-1, E ) 0.55, K ) 5.8 × 103 mol-1 m3, pH 7.8, and KHkint ) 1.6 × 10-4 m s-1.
Discussion The influence of bulk depletion on biouptake will be illustrated using four different metal uptake studies as examples: (i) cobalt uptake by Prochlorococcus (36), (ii) silver uptake by Chlamydomonas reinhardtii (14) (iii) lead uptake by Chlorella vulgaris (13), and (iv) lead uptake by Caco-2 cells (6). The cobalt uptake by Prochlorococcus is an example of growth limitation where the trace metal is a nutrient, the biouptake rate being relatively high. The lead uptake by Ch. vulgaris is a typical example of uptake of a toxic metal, and the silver uptake by C. reinhardtii is an example of uptake of a non-essential element. These experiments were performed using a cell washing procedure, meaning that reported values are the true internalized excluding the metal adsorbed in the mere adsorption sites of the cell walls and
membrane. The case of lead uptake by the Caco-2 illustrates the applicability of the bulk depletion model for systems with different geometries in the situation of internalization control. The relevant parameters used in the model are presented in Table 3. Co-Prochlorococcus. Marine cyanobacteria belonging to the Prochlorococcus genus are the most abundant photosynthetic organisms on earth (37) with typical concentrations between 1 and 3 × 1011 cell m-3 and a cell diameter between 0.5 and 0.7 µm. This diameter means that spherical diffusion is the dominant transport process, thus optimizing the uptake of nutrients such as N or P, the concentration of which is often below the detection limit in central parts of the oceans, dominated by Prochlorococcus. Saito et al (36) estimated that an average population of Prochlorococcus (1011 cell m-3), with VOL. 38, NO. 8, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Time course of silver uptake by Chlamydomonas reinhardtii at free Ag+ concentration (1.0 × 10-5 mol m-3). Values of the parameters used in the model are as follows: DM ) 1.5 × 10-9 m2 s-1, r0 ) 2.0 × 10-6 m, ka ) 8.9 × 1010 mol-1 m3 s-1, E ) 1.0, K ) 2.0 mol-1 m3, pH 7. (a) Comparison between experimental data from Figure 2 of ref 14 for n ) 1010 cell m-3 and the results obtained with the model. Experimental data: low chloride case (complexes negligible) (b) and 4.0 mol m-3 chloride, total metal 1.04 × 10-3 mol m-3 (O). Model: KHkint ) 5.6 × 10-5 m s-1, free metal case (thick line) in the presence of 4.0 mol m-3 chloride (thin line) and in the presence 4.0 mol m-3 chloride (dashed line) using a KHkint ) 1.5 × 10-4 m s-1. (b) Estimated bulk depletion for n ) 3 × 1011 cell m-3 and KHkint ) 1.5 × 10-4 m s-1 for the free metal case (thick line) and in the presence of 4.0 mol m-3 of chloride (dashed line). the calculated uptake rates and a growth rate of 0.4 d-1 and the measured particulated cobalt value of 1.5 × 10-3 Co/ mol-1 C (38), would incorporate 0.3 × 10-9 mol m-3 d-1 of cobalt in new cells and that incorporation at these levels would represent the use of a significant fraction of the total dissolved cobalt reservoir in the ocean. In the Prochlorococcus example, the bacteria were cultivated in a cobalt-depleted medium in the presence of excess of other nutrients to induce cobalt-limited growth (36). For the Co-Prochlorococcus system, the size of the bacteria is not reported, but we can estimate a spherical radius of 3.1 × 10-7 m from the cell volume reported for the same strain 2402
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of 1.3 × 10-19 m3 (39). Using the uptake rates and free cobalt concentration values reported in Table 1 of ref 36, we calculate a range of KHkint values between 0.3 and 3.3 × 10-4 m s-1, which together with the very high cell density used in the experiments of 3 × 1013 m-3 would give an extremely fast bulk depletion for free metal alone that for the higher value of the internalization rate parameter would be larger than 99% after 10 min (Figure 1). The estimated value for the uptake rate parameter β for this system is 0.08, which means that the biouptake is largely controlled by the internalization. This is due to the very small radius of the bacteria since maintaining the other parameters
FIGURE 4. Time course of lead uptake by Caco-2 cells at free Pb2+ concentration (2.6 × 10-3 mol m-3). Values of the parameters used in the model are as follows: DM ) 8 × 10-10 m2 s-1, Vm ) 2 × 10-6 m3, A0 ) 4.2 × 10-4 m2, K ) 10.0 mol-1 m3, pH 7, and KHkint ) 2 × 10-7 m s-1: free metal (thick line), 1.0 mol m-3 complexing groups (dashed line). and increasing the radius 10-fold, the system would be predominantly under diffusion control (β ) 0.80). From the values in Table 3 we obtain a flux time constant (τSS) of O(10-4) s and a depletion time constant (τ) of O(102) s. Applying the dynamic criteria for the values of k′a(O(103) s-1) and kdO(101) s-1, we see that the system is dynamic in the depletion time scale but is not dynamic in the flux time scale (kdτSS ) 10-3). Thus, the contribution of the complexed cobalt to the uptake can only be realized via dissociation in the bulk. Figure 2 shows the experimental results of an uptake experiment in the presence of NTA at different total concentrations of 57Co. The uptake rates decrease with time due to bulk depletion, which is relatively fast due to the high cell density and the high uptake rate. The model successfully fits the three data sets with one set of parameter values, using a KHkint value of 1.6 × 10-4 m s-1, which falls in the range of estimated values (0.3-3.3 × 10-4 m s-1) for uptake experiments in the presence of EDTA and DTPA and absence of bulk depletion (36). Ag-Chlamydomonas. For the Ag-Chlamydomonas system, the authors report a maximum depletion of 8% (14) for a 1-h exposure. Using eq 13 (in conjunction with eq 14), a range of KHkint between 0.5 and 1.5 × 10-4 m s-1 can be computed from the slopes of the curves presented in Figures 1 and 2A and curve A of Figure 5 in ref 14, which illustrates the difficulty of obtaining these values accurately even in laboratory experiments. Using an average value of 1.0 × 10-4 m s-1, 15% depletion is obtained for 1-h exposure whereas 99% depletion would require approximately 1 d. In this example, the uptake of silver in the presence of chloride is larger than expected, which was tentatively attributed to the contribution of silver chloride complexes to the uptake (14). We will try to verify such a hypothesis by taking into account the potential effect of bulk depletion on the uptake. The experimental data correspond to an average uptake rate parameter β of 0.15, which indicates that the diffusive rate is of the same order of magnitude as the internalization rate. Sufficiently labile complexes may thus contribute to the biouptake flux. Using the values of Table 3, τSS is found to be O(10-3) s and, for a chloride concentration of 4.0 mol m-3, we obtain a k′aτSS of O(103) and a kdτSS of O(102). This implies that the system is dynamic on the time
scale of the steady-state flux and certainly on the depletion time scale. The degree of lability (ξ) is equal to 0.8, indicating that the system is mostly labile; therefore, the complex silver species pratically contribute to the uptake flux in proportion to their concentrations. Figure 3a shows the effect of the contribution from the fully labile complexes to the biouptake in the absence of bulk depletion. This figure depicts results calculated from values obtained from Figure 2A of ref 14, in which the authors attribute the observed differences in silver uptake between the low chloride case (complexes are negligible) (b) and the high chloride case 4.0 mol m-3 (O) to the contribution of complexed species to the uptake flux. The free metal concentration is the same in both experiments (1.0 × 10-5 mol m-3); however, the total metal present in the high chloride medium is 1.04 × 10-3 mol m-3. We simulated both experiments using the calculated range of KHkint values (5.6 × 10-5 m s-1 and 1.5 × 10-4 m s-1) and found that the contribution of complex species for the high chloride case lies well below the experimental results for the smaller value (β ) 0.07) and approximately reproduce the results for the higher value (β ) 0.2) (dotted line). This supports the interpretation that there is a contribution of complex species to the uptake, only if the internalization rate constant is high enough to compel a significant contribution of the diffusion control (high β). Our model predicts that an increased cell density of organisms would provoke bulk depletion in the course of the experiment. Figure 3b shows that for an n of 3 × 1011 m-3, after 20 min, 90% of the free metal would be depleted. In a high chloride medium there is a significant contribution of the complexed metal to the biouptake in much greater proportion as compared to the case of no bulk depletion. It is important to emphasize that bulk depletion may be observed even in the presence of a relatively strong complexing agent for a short experiment if the uptake rate parameter β value and cell density are sufficiently high. Pb-Chlorella. For the Pb-Chlorella system, less than 5% depletion is expected for 1-h exposure (Figure 1) whereas 99% depletion would require approximately 6 d. This is due to the low values for the internalization rate parameter, VOL. 38, NO. 8, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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characteristic for uptake of toxic metals (KHkint ) 2.2 × 10-8 m s-1 calculated using the data from ref 13). Due to the internalization control, if complexes are present in solution they do not contribute to the uptake flux. From the values in Table 3, we obtain a depletion time constant (τ) of O(105) s; applying the dynamic criteria for the values of k′a(O(105) s-1) and kdO(1) s-1, we see that the system is dynamic in the depletion time scale. From Table 2 we can see that τ is directly proportional to (1 + K′), which is typical for FIAM, and in this case the presence of complexing agents renders bulk depletion negligible, which is probably a conclusion valid for the majority of toxic trace metals in aquatic system. Intestinal Lead Uptake. The intestinal system consists of a flat sheet of Caco-2 cells in contact with a simulated gastrointestinal track solution (chyme). For this geometry, the uptake rate parameter β takes the form of KHkintδ/DM where δ is the thickness of the steady-state linear diffusion layer. With a δ of O(10-4) m and an apparent permeability KHkint/(1 + K′) of O(10-8) ms-1, it is clear that β , 1 (i.e., that the internalization rate controls the uptake). Most of the lead is present in the form of lead-phosphate and lead-bile complexes with K values of O(10) mol-1 m3, which were found to be labile on the flux time scale (6). This system displays bulk depletion on the order of 30% after 1 d (6). Since the complexes are labile on the flux time scale, we can use the equation for the depletion time constant presented in Table 2, whereby replacing Vf by the volume of the solution and A0 by the surface area of the Caco-2 cells. From the measured 30% bulk depletion, we calculate a τ of 2.4 × 105 s and using the volume-to-area ratio of (2.1 × 102 m-1) to compute a permeability of 2 × 10-8 m s-1. This compares favorably with the reported value of 1.7 × 10-8 m s-1 and shows that bulk depletion studies may well be utilized in addition to accumulation studies. From the permeability, KHkint/(1 + K′), the reported K and the ligand concentration value of 10 and 1.0 mol m-3, respectively (obtained from the potentiometric complexing capacity), we estimate a KHkint value of 2 × 10-7 m s-1. Figure 4 compares the internalized lead in the presence and absence of labile complexes. For one and the same free lead concentration the slope of the two curves is the same until bulk depletion starts to affect the free metal line. The amount of metal eventually internalized is much larger in the presence of complexes, illustrating their role in buffering the free metal concentration in the bulk medium. The contribution of complexes to the uptake is opposite to the lead uptake by Chlorella, although both cases are described by the same equation (i.e., internalization control in the presence of dynamic complexes) (Table 2). This is due to the differences in binding strength of the chyme ligands and NTA with lead (7 orders of magnitude) and contact time (1 d instead of 1 h). Since in this case the medium cannot be changed in order to avoid bulk depletion by addition of a strong complexing agent, it illustrates the importance of the complex strength in bulk depletion effects. Bulk depletion allows the bioinactive complex species to contribute to the metal uptake in a system where this would seem to be impossible considering the internalization rate control (β , 1).
Appendix A
Acknowledgments
Supporting Information Available
J.P.P. acknowleges support from Fundac¸ a˜o para a Cieˆncia e Tecnologia (B-Sab 317). J.G. acknowledges support of his research by the Spanish Ministry of Education and Science (DGICYT: Project BQU2000-0642), by the European Community under Contract EVK1-CT2001-86, and from the “Comissionat d’Universitats i Recerca de la Generalitat de Catalunya”. The authors thank J. Buffle and K. Wilkinson for their useful comments and, in the case of J.P.P. and J.G., acceptance during a summer period at CABE, Sciences II, Universite´ de Gene`ve.
Various details on the approximation used in the modeling and checking of the model against the particular case of free metal with no-flux condition at r ) rf. This material is available free of charge via the Internet at http://pubs.acs.org.
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The integral in eq 18 can be computed using the exact solution for the profiles under steady-state diffusion using the methodology given in ref 35:
cM(r,t) )
{
c/M(t) 1 -
(
)(
)}
(A1)
)(
)}
(A2)
c0M(t) 1 + h + K′ exp(h(1 - F)) 1- / F(1 + K′ + h) cM(t)
cML(r,t) )
{
c/ML(t) 1 -
(
c0M(t) 1 + h - exp(h(1 - F)) 1- / F(1 + K′ + h) cM(t) /
being F ) r/r0. As said before, we use c i(t) instead of ci(rf,t) which is an approximation valid only if rf . r0 is obeyed. One finds a equation similar to eq 8:
Ω(t) ) Vf(1 + K′)c/M(t) - f1(c/M(t) - c0M(t))
(A3)
where h ) (κa(1 + 1/K′))1/2 and
f1 )
(
4πr03 (( - 1)K′)
(1 + h - (1 + hFf) exp(- h(Ff - 1)))
+ h2(1 + K′ + h) (1 + h)(Ff2 - 1) (A4) (1 + K′) 2(1 + K′ + h)
)
We now derive a relationship similar to eq 9:
[ (
c0M(t) ) 1 + β/ 1 +
)]
-1
K′κ1/2 a (K′(1 + K′))1/2 + κ1/2 a
c/M(t) )
c/M(t) (A5) 1 + β/p
For simplification we used p (eq 20). Substituting eq A5 into eq A3 yields
Ω(t) ) {Vf(1 + K′) - f1((β/p)/(1 + β/p))}c/M(t) (A6) Assuming that the time to re-adjust to steady state is negligible, we integrate the parallel of eq 3 (dΩ ) -4πr20J(t) dt) using eq A6 and the flux supply being now:
J(t) )
(
KHkintc*M(t) r0 1 + ) 1 + β/p KHkint pDM
)
-1
c/M(t)
(A7)
The result of the integration is eq 14 where the depletion time constant τ is
τ)
Vf(1 + K′)(1 + β/p) - f1β/p A0KHkint
(A8)
Symbols A0
area of one organism (m2)
c/i
concentration of species i in bulk solution (mol m-3)
/ ci,T
total concentration of species i in bulk solution (mol m-3)
ci(x,t)
volume concentration of the species i at distance x from surface of organism and time t (mol m-3)
DM
diffusion coefficient of the free metal in solution (m2 s-1)
DML
diffusion coefficient of the complex in solution (m2 s-1)
J(t)
steady-state flux at the surface of the organism at a given time (mol m-2 s-1)
Ju(t)
biouptake flux at a given time (mol m-2 s-1)
K
complexation constant (mol-1 m3)
K′
effective complexation constant () Kc/L)
ka
formation rate constant of the complex ML (mol-1 m3 s-1)
kd
dissociation rate constant of the complex ML (s-1)
KH
Henry adsorption constant (m)
kint
internalization rate constant (s-1)
n
cell density (m-3)
p
(1 + K′ξ), see eq 20
r0
radius of the organism (m)
rf
radius of the depleting volume for one organism (m)
V0
volume of one organism
Vf
depleting volume for one organism (m3)
Vm
medium volume for one organism (Vf -V0) (m3)
z
total number of organisms
Greek Symbols β
uptake rate parameter
ΓM
surface concentration (mol m-2)
) DML/DM, dimensionless diffusion coefficient
ξ
degree of lability
κa
dimensionless kinetic constant
F
nondimensional distance (r/r0)
τ
depletion time constant (s)
τSS (s)
time constant to the establish steady state or flux time constant
Ff
nondimensional border (rf/r0)
Ω(t)
metal in the “volume per capita” at a given time (mol)
Literature Cited (1) Morel, F. M. M.; Hering J. G. Principles and Applications of Aquatic Chemistry; John Wiley and Sons: New York, 1993; pp 319-420. (2) Sunda, W. G. In The Biogeochemistry of Iron in Seawater; Turner, D., Hunter, K. A. Eds.; John Wiley & Sons: New York, 2001; pp 41-84.
(3) Morel, F. M. M.; Price, N. M. Science 2003, 300, 944. (4) de Beer, D.; Stoodley, P. Water Sci. Technol. 1995, 32 (8), 11. (5) Picioreanu, C.; van Loosdrecht, M. C. M.; Heijnen, J. J. Water Sci. Technol. 1999, 39 (7), 115. (6) Oomen, A. G.; Tolls, J.; Sips, A. J. A. M.; Groten, J. P. Arch. Environ. Contam. Toxicol. 2003, 44, 116. (7) Ruby, M. V.; Schoof, R.; Brattin, W.; Goldade, M.; Post, G.; Harnois, M.; Mosby, D. E.; Casteel, S. W.; Berti, W.; Carpenter, M.; Edwards, D.; Cragin, D.; Chappell, W. Environ. Sci. Technol. 1999, 33, 3697. (8) Jumarie, C.; Fortin, C.; Houde, M.; Campbell, P. G. C.; Denizeau, F. Toxicol. Appl. Pharmacol. 2001, 170, 29. (9) Roose, T.; Fowler, A. C.; Darrah, P. R. J. Math. Biol. 2001, 42, 347. (10) Morel, F. M. M.; Morel-Laurens, N. M. L. In Trace Metals in Sea Water; Wong, C. S., Boyle, E., Bruland, K. W., Burton, J. D., Goldberg, E. D., Eds.; NATO Conference Series IV: Marine Chemistry; Plenum Press: New York, 1983; pp 841-869. (11) Gerringa, L. J. A.; de Baar, H. J. W.; Timmermans, K. R. Mar. Chem. 2000, 68, 335. (12) Karp Boss, L.; Boss, E.; Jumars, P. A. Oceanogr. Mar. Biol. 1996, 34, 71. (13) Slaveykova, V. I.; Wilkinson, K. J. Environ. Sci. Technol. 2002, 36, 969. (14) Fortin, C.; Campbell, P. G. C. Environ. Toxicol. Chem. 2000, 19, 2769. (15) Kovarova-Kovar, K.; Egli, T. Microbiol. Mol. Biol. Rev. 1998, 62, 646. (16) Mirimanoff, N.; Wilkinson, K. J. Environ. Sci. Technol. 2000, 34, 616. (17) Gonzales-Davila, M.; Santana-Casiano, J. M.; Perez-Pena, J.; Millero, F. J. Environ. Sci. Technol. 1995, 29, 289. (18) Volker, C.; Wolf-Gladrow, D. A. Mar. Chem. 1999, 65 (3-4), 227. (19) Wilhelm, S. W.; Trick, C. G. Limnol. Oceanogr. 1994, 39, 1979. (20) Vasconcelos, M. T. S. D.; Leal, M. F. C. Environ. Sci. Technol. 2001, 35, 508. (21) Van Leeuwen, H. P. Environ. Sci. Technol. 1999, 33, 3743. (22) Xue, H. B.; Stumm, W.; Sigg, L. Water Res. 1988, 22, 917. (23) Galceran, J.; Monne, J.; Puy, J.; van Leeuwen, H. P. Mar. Chem. 2004, 85, 89. (24) Stulik, K.; Amatore, C.; Holub, K.; Marecek, V.; Kutner, W. Pure Appl. Chem. 2000, 72, 1483. (25) Buffle, J.; Horvai, G. In In Situ Monitoring of Aquatic Systems: Chemical Analysis and Speciation; Buffle, J., Horvai, G., Eds.; John Wiley & Sons: New York, 2000; pp 11-12. (26) Oldham, K. B. In Microelectrodes: Theory and Applications; Montenegro, M. I., Queiros, M. A., Daschbach, J. L., Eds.; NATO ASI Series E 197; Kluwer: Dordrecht, The Netherlands, 1991; p 35. (27) Wiegel, F. W. Phys. Rep. 1983, 95 (5), 283. (28) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; John Wiley & Sons: New York, 2001; p 161. (29) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, 1964. (30) Bosma, T. N. P.; Middeldorp, P. J. M.; Schraa, G.; Zehnder, A. J. B. Environ. Sci. Technol. 1997, 31, 248. (31) Hudson, R. J. M. Sci. Total Environ. 1998, 219 (2-3), 95. (32) Van Leeuwen, H. P.; Buffle, J.; Cleven, R. Pure Appl. Chem. 1989, 61, 255. (33) Van Leeuwen, H. P.; Puy, J.; Galceran, J.; Cecilia, J. J. Electroanal. Chem. 2002, 526, 10. (34) Galceran, J.; Puy, J.; Salvador, J.; Cecilia, J.; Mas, F.; Garces, J. L. Phys. Chem.-Chem. Phys. 2003, 5, 5091. (35) Galceran, J.; Puy, J.; Salvador, J.; Cecı´lia, J.; Leeuwen, H. P. J. Electroanal. Chem. 2001, 505, 85. (36) Saito, M. A.; Moffett, J. W.; Chisholm, S. W.; Waterbury, J. B. Limnol. Oceanogr. 2002, 47, 1629. (37) Partensky, F.; Hess, W. R.; Vaulot, D. Microbiol. Mol. Biol. Rev. 1999, 63, 106. (38) Sherrel, R. M.; Boyle, E. A. Earth Planet. Sci. Lett. 1992, 111, 155. (39) Shaw, S. L.; Chisholm, S. W.; Prinn, R. G. Mar. Chem. 2003, 80, 227.
Received for review June 10, 2003. Revised manuscript received January 19, 2004. Accepted January 23, 2004. ES034579N
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