Dynamics of Metal Partitioning at the CellâSolution Interface

Article pubs.acs.org/est

Dynamics of Metal Partitioning at the Cell−Solution Interface: Implications for Toxicity Assessment under Growth-Inhibiting Conditions Jérôme F. L. Duval,*,†,‡ Nathalie Paquet,§,∥ Michel Lavoie,§,⊥ and Claude Fortin§ †

Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), CNRS, UMR7360, Vandoeuvre-lès-Nancy, F-54501, France ‡ Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), Université de Lorraine, UMR7360, Vandoeuvre-lès-Nancy, F-54501, France § Centre Eau Terre Environnement (INRS-ETE), Institut National de la Recherche Scientifique, 490 de la Couronne, Québec G1K 9A9, Canada S Supporting Information *

ABSTRACT: Metal toxicity toward microorganisms is usually evaluated by determining growth inhibition. To achieve a mechanistic interpretation of such toxic effects, the intricate coupling between cell growth kinetics and metal partitioning dynamics at the cell−solution interface over time must be considered on a quantitative level. A formalism is elaborated to evaluate cell-surface-bound, internalized, and extracellular metal fractions in the limit where metal uptake kinetics is controlled by internalization under noncomplexing medium conditions. Cell growth kinetics is tackled using the continuous logistic equation modified to include growth inhibition by metal accumulation to intracellular or cell surface sites. The theory further includes metal− proton competition for adsorption at cell-surface binding sites, as well as possible variation of cell size during exposure to metal ions. The formalism elucidates the dramatic impacts of initial cell concentration on metal bioavailability and toxicity over time, in agreement with reported algae bioassays. It further highlights that appropriate definition of toxicity endpoints requires careful inspection of the ratio between exposure time scale and time scale of metal depletion from bulk solution. The latter depends on metal internalization−excretion rate constants, microorganism growth, and the extent of metal adsorption on nonspecific, transporter, and growth inhibitory sites. As an application of the theory, Cd toxicity in the algae Pseudokirchneriella subcapitata is interpreted from constrained modeling of cell growth kinetics and of interfacial Cd-partitioning dynamics measured under various exposure conditions.

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INTRODUCTION Algal and microbial toxicity assays based on cell growth measurements are routinely employed to evaluate the cidal or static nature of contaminants like metal ions.1−8 Results from growth inhibition bioassays are classically collated with measured concentrations of metals adsorbed at the cell surface, internalized, and dissolved in the extracellular volume.1−8 The analysis is sometimes refined by the determination of metal speciation in the exposure solution in order to discriminate between free and complexed metal forms.6,7 In fine, phenomenological relationships between cell division rate and internalized, adsorbed, or extracellular metal concentrations are established.1−6,8 Such relationships are then statistically analyzed (i) to identify the nature of the biotic ligands where metal accumulation leads to toxicity effects and (ii) to provide empirical toxicity descriptors like total solution concentration of metals leading to 50% growth inhibition. For numerous freshwater microalgae and metals, there is a large body of data © 2015 American Chemical Society

that convincingly evidence the statistical correlation between cell growth inhibition and concentrations of surface-bound and/or internalized metals.1,2,5,6,8 However, unambiguous identif ication of the toxicity-mediating sites is still lacking, since the differentiation between cell surface internalization sites and nonspecific cell-surface binding sites remains difficult,1,6 as argued by De Schamphelaere et al.1 Clearly then, evaluating the relevance of a given cell toxicity scenario requires the development of a theoretical framework to interpret, on a mechanistic and quantitative levels, the necessarily coupled cell growth inhibition kinetics and metal partitioning dynamics at metal-assimilating biointerphases. Received: Revised: Accepted: Published: 6625

February 2, 2015 April 17, 2015 May 6, 2015 May 6, 2015 DOI: 10.1021/acs.est.5b00594 Environ. Sci. Technol. 2015, 49, 6625−6636

Article

Environmental Science & Technology

The function Ψ(t) in eq 1 is the dimensionless growth inhibition factor at t defined from the ratio between growth rates observed in exposure and control media, i.e., Ψ(t) = 1 − {d[ln(X(t))]/ dt|c*M(t)}/{d[ln(X(t))]/dt|c*M0=0} with cM * (t) = cM(r=rc(t),t) being the metal concentration in bulk solution. The latter depends on t because the possible depletion of M from bulk medium is here accounted for.9,10,20,21 The limits Ψ(t) = 0 and Ψ(t) = 1 correspond to zero-toxicity and 100% growth inhibition situations, respectively. The variable μ in eq 1 is the cell growth rate in the exponential phase reached for X(t)/Xc ≪ 1 under conditions where Ψ(t) = 0. To further define the dependence of Ψ on time, one must specify the toxicity scenario envisaged for the metal−microorganism system considered. Numerous studies on the characterization of aqueous algistatic and bacteriostatic metal fractions1,2,5,6,8 suggest that Ψ depends on the internalized or cell-surface metal concentrations according to a sigmoid-like function, reminiscent of equilibrium metal adsorption processes. For the sake of demonstration, we thus tackle here cases where cell toxicity is caused by cell-surface metal adsorption or by metal internalization. Assuming a fast Langmuirian adsorption of M on the corresponding cell-surface and intracellular inhibitory ligands as compared to internalization and growth,20,22 the relative cell division rate 1 − Ψ(t) may be expressed as (SI, section I)

In this study, we provide the basis for such a formalism under conditions where metal complexation is low in the medium and overall metal uptake is rate-limited by the internalization step. The interplay between cell growth inhibition and metal partitioning is then fully solved in the extremes where cell toxicity is driven by metal binding either to noninternalization surface sites or to intracellular growth-inhibitory ligands. For each situation, the dependence of the growth inhibition factor on the initial concentrations of metals and cells and on the internalization−excretion kinetics is discussed on the basis of the corresponding time-dependent metal partitioning across the active biointerphase. The theory rationalizes the impacts of bulk metal depletion kinetics on bioavailability and toxicity assessments over time. It further demonstrates that measures often recommended to avoid bulk metal depletion in cell toxicity bioassays are not always necessary. Finally, it offers a solid biophysicochemical framework to identify the predominant mechanism of metal toxicity to microorganisms. This is illustrated by the successful analysis of the Cd toxicity mechanisms in the algae Pseudokirchneriella subcapitata from the modeling of Cd internalization/adsorption/depletion kinetics under various metal-exposure conditions. For the first time, this work elaborates a recent theory on metal biouptake dynamics9,10 within an ecotoxicological context.

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Ns,i

FORMULATION OF THE THEORY Microorganism Growth Inhibition. Let us consider a finite volume, VT, of a dispersion initially containing a single type of N0 spherical microorganisms (e.g., algae or bacteria) of radius a0 whose membrane possibly supports a soft surface layer of thickness d representing, for example, cell wall or extracellular polymeric substances (see the Glossary for the symbols with their associated units used in this paper).11−14 At any time t, the number N(t) of cells in the dispersion is determined by the level of growth inhibition by metal ions M of valence zM present at the initial concentration c*M0. We introduce the radial coordinate system r with r = a0 referring to the surface of the plasma membrane at t = 0. The concentration of metals at t and position r ≥ a(t) is hereafter denoted as cM(r,t), with a(t) being the cell radius that possibly depends on time t and is accessible by the experiments.6,15 For the sake of generality, metals M may adsorb at the cell surface and be internalized and excreted by the microorganisms. For modeling purposes, each microorganism is enveloped at any t by a virtual Kuwabara cell unit of radius rc(t) such that the microorganism/volume ratio therein identifies with the microorganism volume fraction ϕ(t) over the entire dispersion.9,10,16 In turn, rc(t) is defined by rc(t) = r0(t) ϕ(t)−1/3 with ϕ(t) = 4π[r0(t)]3N(t)/(3VT) and r0(t) = a(t) + d, the time-dependent particle size. For the sake of simplicity, the thickness and volume of the peripheral soft surface layer are taken to be constant over time t. Relaxation of this assumption leads to more complex mathematics, as detailed in the Supporting Information (SI). The quotient N(t)/VT, denoted as cp(t), represents the cell number density at t. The differential equation describing the rate of change in population size may be formulated according to the following modified continuous logistic growth model8,17−19 dX(t )/dt = μX(t )[1 − Ψ(t )][1 − X(t )/Xc]

1 − Ψ(t ) =

∑ j=1

(j) K s,i (j) + c M(r =a(t ),t ) K s,i

Nint,i

1 − Ψ(t ) =

∑ j=1

or

(j) K int ,i (j) K int ,i + ϕu(t )

(2a,b)

where Ns,i and Nint,i are the number of different surface and intracellular inhibitory sites, respectively, ϕu(t) is the concentration of internalized metals at t expressed in moles per unit −3 (j) microorganism surface area,9,10 and K(j) s,i (mol m ) and Kint,i (mol m−2) are the metal affinity constant for the jth type of surface and intracellular inhibitory sites, respectively. Without loss of generality, the developments below remain valid for more complex toxicity schemes involving both surface-adsorption- and internalization-mediated inhibition mechanisms or the adsorption of M to inhibitory sites other than those captured by the Langmuirian eqs 2a,b. Evaluation of cell growth kinetics from eqs 1 and 2a,b requires the determination of ϕu(t), cM * (t), and cM(r=a(t),t). This is done in the next section where partitioning dynamics of M between internalized, adsorbed, and extracellular fractions is formally detailed. Metal Partition Dynamics. The time-derivative of the amount of metals internalized by the N(t) microorganisms in the dispersion identifies with the difference between uptake flux and excretion flux at the microorganism surface corrected for growth biodilution and for variation of cell radius with time. Accordingly, it becomes (SI, section II) d[ϕu(t ) X(t )]/dt = [Ju (t ) − keϕu(t )]X(t ) − 2[ϕu(t ) X(t )/a(t )] da(t )/dt

(3)

−2 −1

Ju(t) is the uptake flux (mol m s ) at the membrane surface defined here by a Michaelis−Menten process.23−25 It corresponds to a fast Langmuirian adsorption of M at the transporter sites followed by a first-order internalization kinetic step.9,10,20 In N turn, Ju(t) = ∑j=1s,uJ*u (j)[cM(r=a(t),t)]/[K(j) s,u + cM(r=a(t),t)], where Ns,u is the number of different internalization sites considered, K(j) s,u is the affinity of M for the jth type of internalization site, and

(1)

with X(t) = N(t)/N0 being the dimensionless number of microorganisms at t and Xc = Nc/N0 being the dimensionless carrying capacity of the medium, where Nc is the maximum population size it can sustain under given nutrient conditions. 6626

DOI: 10.1021/acs.est.5b00594 Environ. Sci. Technol. 2015, 49, 6625−6636

Article


Ns,k max,(j) (j) written as Γs,k=i,u,p(t) = ∑j=1 Γs,k [cM(r=a(t),t)]/[Ks,k + (j) cM(r=a(t),t)], where Ks,p is the affinity constant of M to the jth nonspecific cell-surface binding site, Ns,p is the number of such sites, and Γmax,(j) s,k=i,u,p is the surface concentration of the jth type of Si, Su, and Sp sites. For purposes of simplicity, eq 4 ignores the term stemming from the possible adsorption of M across the soft surface layer of thickness d surrounding the membrane. The reader is referred to ref 9, where a procedure is detailed to include this adsorption contribution. For the sake of conciseness and according to one often respected assumption of the biotic ligand model, we hereafter analyze the situation where the overall metal uptake is kinetically limited by the sole internalization step,26,27 which holds for sufficiently fast transport of M from solution to cell surface as compared to the typical time for transfer of M across the membrane.9,10,28 The general situation where the uptake rate is determined by internalization kinetics and dynamics of conductive diffusion transport of M from/to the membrane deserves a separate study, which we shall detail in a forthcoming report. The metal concentration profile cM(r,t) involved in eq 4 is then rigorously defined here by the thermodynamic Boltzmann relationship cM(r,t) = βrc*M(t) where the Boltzmann factor βr at the position r is given by exp[−zMy(r)], with y(r) = Fψ(r)/RT being the dimensionless electrostatic potential at r, F the Faraday constant, T the temperature, R the gas constant, and ψ(r) the potential.9,10 For medium containing background electrolyte in excess over M, a situation commonly met in toxicity bioassays, βr does not depend on time, as justified in ref 10. The sign and magnitude of y(r) are governed by the electrolyte concentration in the medium and by the structure of the considered biointerphase, which includes the thickness and the hydrophobic/hydrophilic balance of the peripheral soft surface layer, the volume density of charges it carries, the membrane surface charge density, and the microorganism size.9,10 The reader is referred to ref 10, where a numerical evaluation of y(r) from the nonlinear Poisson−Boltzmann equation is reported together with explicit analytical solutions derived for relevant practical cases. Substituting cM(r,t) = βrc*M(t) in eq 4 and differentiating with respect to time, we then obtain the differential equation in terms of the searched cM * (t) (SI, section II)

(j) (j) J*u (j) = K(j) H,ukintKs,u is the maximum uptake flux pertaining to the (j) internalization of M via site j, with K(j) H,u and kint being the associated Henry adsorption affinity and internalization kinetic constant, respectively.9,10 ke is the excretion kinetic constant and keϕu(t) in eq 3 represents the excretion flux Je(t) that pertains to the easily exchangeable metal species only, with their activity in effect at the interface.9,10,20−22 For many metals and microorganisms, there are minimum concentrations of internalized M required to keep the cell alive. This prerequisite for cell viability can be easily implemented here according to the following conditional formulation for the excretion flux Je(t), where ϕ0u is the initial concentration of internalized metal required for cell viability: if ϕu(t) ≤ ϕ0u, then Je(t) = 0; otherwise, Je(t) = keϕu(t). At any t, the extent of M depletion from the extracellular solution is determined by the amount of metals adsorbed on the total biosurface area in the dispersion, Sa(t) × N(t), where Sa(t) = 4π[a(t)]2 is the cell surface area at time t, and by the amount of internalized metals. Within the framework of the Kuwabara representation, this mass balance condition becomes

X (t )

rc(t )

∫a(t)

=−

{∫0

ξ 2c M(ξ ,t ) dξ − t

∫a

rc0

ξ 2c M(ξ ,0) dξ

0

[Ju (ξ) − keϕu(ξ)]a(ξ)2 X(ξ) dξ + a(t )2 X(t )

Γs(t ) − (a0)2 Γs(0)

}

(4)

where ξ is a dummy integration variable and = rc(t=0) is the initial Kuwabara cell radius defined by r0c = [4πN0/(3VT)]−1/3. Γs(t) corresponds to the surface concentration of adsorbed metals per microorganism. Discriminating the adsorption of M to surface-inhibitory sites Si (relevant for the toxicity scenario given by eq 2a), to internalization sites Su, and to nonspecific cellsurface binding sites Sp, we write Γs(t) = Γs,i(t) + Γs,u(t) + Γs,p(t) where Γs,i(t), Γs,u(t), and Γs,p(t) pertain to the time-dependent surface concentrations of metals adsorbed at the Si, Su, and Sp surface sites, respectively. Assuming that all relevant adsorption processes are fast compared to growth and internalization,9,10,20,22 equilibrium relationships between adsorbed and free metal forms systematically apply. Then, Γs,k=i,u,p(t) may be r0c

j) ⎧ ⎡ A s,(jk)Γ s,max,( Ns, k k ∑ ∑ ⎪ ⎢ ( ) j 2 k = i,u,p j = 1 ⎪ (A s, k + c M ̅ *(t )) 2 dcM ̅ *(t )/dt = −⎨X(t )⎢⎢1 − V̅ + a ̅ (t ) ( ) j N J* ⎪ τL0 ∑ j =s,u1 u (j) ⎪ ⎢ A s,u ⎣ ⎩

⎧ ⎪ ⎪ a (t )2 X(t ) ×⎨ ̅ (j) ⎪ τ 0 ∑Ns,u Ju* ⎪ L j = 1 As,u(j) ⎩

⎛ ⎞ c * (t ) d X (t ) − ϕu̅ (t )⎟⎟ + ∑ Ju*(j)⎜⎜ (j) M̅ * dt ( ) + A c t ⎝ s,u ⎠ j=1 M ̅ Ns,u

N

+ 2 a ̅ (t )

⎫−1 ⎤ ⎪ ⎥ ⎪ ⎥ + V̅ ⎬ ⎥ ⎪ ⎪ ⎥⎦ ⎭

d a ̅ (t ) X (t ) dt

∑k = i,u,p ∑ j =s,k1 N

τL0 ∑ j =s,u1

max ,(j) cM ̅ *(t )Γ s, k

A s,(jk) + c M ̅ *(t ) Ju*(j) (j) A s,u

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

where we introduce the dimensionless internalized metal concentration ϕ̅ u(t) = keϕu(t)/∑Nj=1s,u Ju*(j); the metal concen*̅ (t) = cM * (t)/cM *0; the dimensionless affinities tration ratio cM (j) (j) 0 As,k=i,u,p = Ks,k=i,u,p/(βacM * ) of M for the jth type of Si, Su, and Sp

max ,(j) ⎞ ⎛ N c *(t )Γ a ̅ (t )2 ∑k = i,u,p ∑ j =s,k1 M̅ (j) s,k* ⎟ ⎜ A s, k + c M ̅ (t ) ⎟ * (t ) + ⎜(1 − V̅ ) c M ̅ (j) * N J ⎟ ⎜ τL0 ∑ j =s,u1 u (j) ⎟ ⎜ A s,u ⎠ ⎝

(5)

surface sites; and the dimensionless cell radius a(t) ̅ = a(t)/a0. The quantity V̅ is defined by the volume ratio V̅ = V0f /(βaṼ soft + V0f ), where βa is the Boltzmann factor at the membrane position, V0f = 4π(r0c )3/3 is the volume of the Kuwabara unit cell at t = 0, 6627


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Figure 1. (A) Cell growth curves (cell abundance relative to that at t = 0) under conditions where toxicity is determined by M adsorption to surface inhibitory sites, for different values of the dimensionless M affinity parameters: A(1) s,k=i,u = 1 (a), 0.17 (b), 0.09 (c), 0.07 (d), and 0.03 (e). Solid lines: rigorous solution of eqs 1, 3, and 5. Dotted lines: evaluation when M depletion is neglected (i.e., c*M(t) is arbitrarily fixed to c*M0 in eqs 1, 3, and 5). (B) Corresponding time-dependent concentration of M in bulk solution. Solid lines: rigorous solution. Dotted lines: evaluation when cell growth is (1) (1) −4 −1 (1) −5 neglected, i.e., μ → 0. Model parameters in parts A and B: βr = 1, a = 0.5 μm, d = 0, Xc = 14, Ns,k=u,i,p = 1, K(1) H kint = 2 × 10 m s , Ks,u = Ks,i = 10 mM, td max,(1) 0 −3 −1 0 −6 −7 −2 max,(1) = 1 h, ke ∼ 10 s , τL ∼ 100s [ϕ(t=0) = 8 × 10 ], Γs,k=i,u = 10 mol m , Γs,p = 0, a(t) ̅ = 1, and ϕu = 0. Time on the x axis is scaled to doubling time td in the absence of M.

and Ṽ soft = [3Vsoft(0)∫

rD 2

a0

ξ βξdξ]/{βaro(0)3(1 − α3)} is the initial

a(t) ̅ (accessible by the experiments) and of the initial boundary conditions ϕu(t=0) = ϕ0u, c*M ̅ (t=0) = 1, and X(t=0) = 1. The equations account for metal−microorganism electrostatic * (t), and interactions and bulk depletion effects. Once ϕu(t), cM X(t) are evaluated following the numerical procedure29 outlined in SI (section III therein) and validated from analytical solutions derived in the limits μ → 0 and a(t) ̅ = 1 (Table S1 in section IV of SI), the relevant surface concentrations of adsorbed metals can be computed from the expressions of Γs,k=i,u,p(t). The measurable amounts of internalized and cell-surface-bound metals over the entire microorganism dispersion may be further estimated from Qu(t) = N0X(t) ϕu(t) Sa(t) and Qs,k=i,u,p(t) = N0X(t) Γs,k=i,u,p(t) Sa(t), respectively, where the index k differentiates metals bound to growth inhibitory surface sites Si, to internalization sites Su, and to nonspecific surface sites Sp. Proton interactions with the transporter sites at the plasma membrane may be further important for proper evaluation of metal toxicity.30,31 Indeed, inhibition of metal uptake by protons is well-documented, although the nature (competitive or noncompetitive) of this inhibition is not always clear.32 The above theory can be straightforwardly extended to include metal−proton competitions for each of the introduced type of binding sites33 pending appropriate definition of the dimensionless metal affinity terms A(j) s,k=i,u,p. In the presence of such (j) competitions, the latter terms are written as A(j) s,k=i,u,p = (Ks,k / 0 H (j)‑H (j)‑H * )(1 + βa cH+/Ks,k ), where Ks,k=i,u,p refers to the affinity of βacM protons for the jth type of surface inhibitory (k = i), uptake (k = u), or nonspecific adsorption (k = p) sites (see the demonstration in SI, section II). βHa cH+ refers to the concentration of protons at the cell surface accounting for the required Boltzmann enhancement factor βHa = exp(−y(r)), and cH+ pertains to the bulk concentration of protons. The limit where there are no metal−proton competition terms is simply given by βHa cH+/ (j) (j) *0). In K(j)‑H s,k=i,u,p ≪ 1, which leads to As,k=i,u,p = Ks,k=i,u,p/(βacM practice, it is often sufficient to consider the binding sites with the highest affinities to protons and/or metals.34 From the above formulation, it is expected that A(j) s,k=i,u,p increases with decreasing pH; i.e., the effective affinity of metal ions for the cell surface binding sites is lowered. In turn, a decreased metal toxicity is

geometric volume {Vsoft(0) = 4π[r0(0) − a0 ]/3} of the surrounding soft surface layer formally corrected for electrostatics with α = a0/r0(0) (see SI for details). The radial position rD in the integral defining Ṽ soft satisfies r0(0) + κ−1 ≪ rD ≪ rc(t), where κ−1 is the Debye layer thickness that reflects the extension of the electric double layer from the outer cell surface.9,10 The formulation holds for cases where there is no electrostatic interaction between microorganisms over time, which is generally verified for practical values of medium salinity and microorganism volume fractions in standard toxicity tests. The time constant τ0L in eq 5 is the membrane transfer time scale of M at t = 0 defined by (section II in SI) 3

Ns,u ⎡ ⎤ ̃ + VT/N0)/⎢Sa(0) ∑ (1/R s(j))⎥ τL0 = (βaVsoft ⎢⎣ ⎥⎦ j=1

3

(6)

(j) (j) where R(j) s = 1/(KH,ukintβa) is the membrane transfer resistance associated with the uptake of M via the jth type of internalization site.9,10 τ0L may be viewed as the reciprocal of an uptake rate constant pertaining to the passage of metals from extracellular to intracellular volumes under conditions where the extracellular metal concentration profile is equilibrated. The dimensionless product keτ0L can be interpreted as the ratio between uptake and excretion time constants or, for that matter, as the thermodynamic partitioning constant of the easily exchangeable metal ions across the cell−solution interface. The formulation incorporates possible kinetic asymmetry in the membrane transfer mechanism with unequal routes inward and outward via the introduction of distinct internalization sites and of an overall metal excretion flux term [keϕu(t)]. In the limit X(t≥0) = 1, reached for μ → 0 and/or Ψ(t) = 1 (eq 1), and Ns,k=i,u,p = 1, eqs 3 and 4 correctly reduce to our formulation of metal uptake dynamics in microorganisms population of constant size.9 The set of nonlinear and coupled differential eqs 1, 3, and 5 fully defines the intertwined kinetics of cell growth and dynamics of metal partitioning between cell surface and intra- and extracellular volumes upon specification of

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affinities corresponding to A(1) s,i ≪ 1, Ψ(t) becomes unity and cell growth is completely inhibited, i.e., X(t) → 1. For decreasing values of A(1) s,i between these two extremes, X(t) decreases due to the increase in the amount of M bound to the inhibitory surface sites Si (Figure S1, section V in the SI). At fixed A(1) s,i and for sufficiently short exposure times where X(t) ≪ Xc, the linear dependence of ln(X) on t obtained in the absence of toxicity effects is lost due to the dynamic partitioning of M between internalized, adsorbed (Figure S1, SI), and free forms (Figure 1B). The resulting depletion of M from solution over time (Figure 1B) leads indeed to a time-dependent inhibition factor Ψ (Figure S2, section V in SI) that affects cell growth kinetics according to eq 1. The linear dependence of ln(X) on t (and the independence of Ψ with respect to t) is maintained for cells growing in a medium where M depletion is insignificant (Figure S2, SI). With increasing t, the ongoing uptake of metals (Figure S1, SI) leads to a falling supply of M to the microorganisms. Depletion of M from the solution then becomes pronounced, and the larger so for decreasing c*M0 or, equivalently, increasing A(1) s,i (Figure 1B). In turn, growth inhibition becomes weaker over time, which is in line with the decrease of Ψ with t illustrated in Figure S2 in the SI. Ultimately, for sufficiently large t, the cell growth rate drops to zero once the stationary phase [where X(t) → Xc] is reached. The lower the value of A(1) s,i , the stronger the metal toxicity and the longer the delay required to recover this stationary growth phase. This is in agreement with the reported action of, for example, Cd(II) and Ni(II) on Pseudomonas brassicacearum.3 Except at t → 0, where M depletion remains marginal, cell population size [based on X(t)] is significantly underestimated when ignoring M depletion, i.e., with arbitrarily fixing c*M(t) to c*M0 in eqs 1, 3, and 5 (dotted lines in Figure 1A). This is so because the concentration of toxic metals in solution * (t) of becomes necessarily larger than the true concentration cM bioavailable metals in the exposure medium (Figure 1B). Stated differently, if assuming a priori that the solid lines in Figure 1A correspond to cell growth curves measured in the absence of metal depletion from solution, then the half-maximal effective concentration EC50 at a given exposure time texp, rigorously defined as the quantity c*M(t=texp) leading to Ψ(t=texp) = 1/2, would be overestimated. This would come to underestimate metal toxicity to an extent that depends on the difference cM *0 − * (t=texp). It is stressed that the aforementioned definition of cM EC50 leads to the time-independent solution EC50 = K(1) s,i /βa, which is simply derived from eq 2a with Ns,i = 1. Figure 1B further * (t) obtained displays the time-dependent metal concentration cM from eqs 1, 3, and 5 together with that evaluated in the limit μ → 0, i.e., when ignoring cell growth kinetics (see Figure S1 of the SI for the corresponding internalized and adsorbed metal amounts). The latter limit corresponds to the case we detailed elsewhere:9 cM * (t) decreases with t before reaching a constant nonzero plateau value at t → ∞, the presence of which is symptomatic of the metal excretion process (i.e., ke ≠ 0 under the conditions of Figure 1). The time scale of M depletion from bulk solution and the magnitude of cM * (t→∞) intrinsically depend on the kinetic and thermodynamic parameters pertaining to the internalization−excretion−adsorption processes, as detailed in Table S1 of the SI and in refs 9 and 10. In the situation where cell growth is accounted for, Figure 1B highlights the existence of two successive kinetic steps whose origin may be deciphered from the comparison with data obtained in the limit μ → 0: (i) a short time window where M depletion is solely determined by the dynamics of the interfacial processes controlling metal internalization and bioaccumulation (dotted lines in Figure 1B) and (ii) a longer

anticipated at low pH, as confirmed by the results displayed in Figure 1 (and detailed below), where cell growth is reported for various values of A(j) s,i with j = 1. This finding is further in agreement with the experimental observation by, for example, Parent and Campbell30 for a decreased Al3+ toxicity on Chlorella pyrenoidosa in acidified water. Limitations and Potential Extensions of the Theory. The main objective of this work is to develop a basic understanding of the influence of metal partitioning dynamics on metal toxicity evaluation. Given the current state of the art, the above model should be viewed as a first step toward a better integration of physicochemical hurdles formed by (i) the intricate nature of the organism/medium interface, which generally features charge separation and Boltzmann modification of metal activity in the interfacial electric double layer, and (ii) the depletion processes influencing the partitioning of metals over time across the cell−solution interface. The above physical framework is formulated under conditions where all interfacial processes are at equilibrium; in particular, metal uptake is considered in the limit where conductive−diffusion transport of metals from solution to the active biomembrane is not limiting the rate of metal uptake. For more involved systems, a refined account of metal uptake kinetics, extracellular metal transport limitations, cell geometry, and isotherm of metal adsorptions at the biosurface may be mandatory. In addition, it is anticipated that due attention should be paid in the near future to intracellular metal speciation in either the thermodynamic, partial equilibrium, or metal-transport-limiting regimes in connection with the dynamics of metal partitioning at the very cell−solution interface. For example, it may be envisaged that in some situations the transport of metal ions inside the cell body is a lot slower than the transport in aqueous medium, therefore modifying interfacial metal partitioning over time. The accompanying theoretical advances will surely need supporting experimental developments for probing in situ the dynamic status (lability) of metal complexes formed in the cell volume. In that respect, studies combining proteomic and transcriptomic tools with interfacial metal partitioning analyses will surely be helpful for appreciating the connection between intracellular− extracellular metal speciation and cell metabolism.

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RESULTS AND DISCUSSION Dependence of Cell Growth Kinetics on Metal Partitioning Dynamics. Effect of Surface Metal Adsorption. Figure 1A shows the typical time evolution of cell population size normalized to its initial size, X(t), for various values of initial metal concentration, cM *0, under conditions where there is no variation of cell radius over time (a(t) = 1) and toxicity is ̅ determined by M adsorption at a single type of surface inhibitory site, i.e. eq 2a applies with Ns,i = Ns,u = 1. The time t is scaled with respect to the doubling time td = μ−1 ln[(Xc − 1)/(Xc/2 − 1)] of the cell population in the absence of M. The selected c*M0 values are further converted into the relevant (1) dimensionless metal affinity A(1) *0) for the surface s,i = Ks,i /(βacM inhibitory sites. Proton−metal competition terms are discarded in the given computational example, even though their impact on cell toxicity can be effectively apprehended along the lines set forth at the end of the “Metal Partition Dynamics” section. In the regime of weak metal affinities, i.e., A(1) s,i ≫ 1, metal toxicity is insignificant [Ψ(t) → 0] and the solution of eq 1 simply reads X(t) = 1/[(1−Xc−1)e−μt + Xc−1]: the growth curve then displays the expected exponential phase preceding the stationary phase reached at t/td ≫ 1, where X → Xc. For sufficiently large metal 6629


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(1) Figure 2. (A) Cell growth curves under conditions where toxicity is determined by M adsorption to surface inhibitory sites, for different values of K(1) H kint given in terms of the dimensionless product keτ0L (indicated). The dotted line in part A refers to situations where there is no toxicity. (B) Corresponding time-dependent concentration of M in bulk solution. Solid lines: rigorous solution of eqs 1, 3, and 5. Dotted lines: evaluation with μ → 0. Model −3 −1 s , ϕ(t=0) = 8 × 10−6, and other parameters as in Figure 1. parameters in parts A and B: A(1) s,k=i,u = 0.17, ke ∼ 10

amount of metals adsorbed at the inhibitory surface sites (Figure S6B, SI). Qualitatively, the dependences of X(t) and c*M(t) on time with increasing keτ0L are similar to those previously discussed 0 for decreasing values of A(1) s,i (Figure 1), except in the limit keτL ≫1. Indeed, contrary to the case A(1) ≪1 in Figure 1, this limit s,i does not lead to a complete suppression of cell growth. Under the conditions of Figure 2, cM * (t) reached in the extreme keτ0L ≫1 does not significantly exceed the affinity constant K(1) s,i , so Ψ(t) remains strictly lower than unity (eq 2a), and consequently, the growth rate d[ln(X)]/dt (eq 1) is strictly positive. Figure 3

time scale where M depletion is driven by the increase over time of the total biosurface area Sa(t) × N(t) in the dispersion. The lower the value of A(1) s,i , the more significant the cell growth inhibition and the broader the exposure time range where M depletion kinetics becomes independent of cell growth. It is noteworthy that Jansen et al.35 observed the aforementioned two-stage transient behavior with metal complex systems for which the interfacial processes listed above in item i were coupled by depletion of the bioreactive species in the medium and by slow dissociation of their complexes. The representation given in Figure 1 in terms of the dimensionless affinity constant A(1) s,i further captures the basic effects of cell−metal electrostatic interactions on cell growth inhibition, essentially because As,i(1) involves the surface Boltzmann factor βa. The larger (lower) the latter with respect to unity, the more significant the attraction (repulsion) between metal ions and cells surface.10 In turn, increasing βa (or, equivalently, decreasing A(1) s,i ) leads to a decrease in X(t) because the cell surface behaves like an electrostatic trap for metal ions, resulting in an increase of M adsorption to surface inhibitory sites and to an increase in cell growth inhibition. More detailed information is given in section V of the SI about the impact of electrostatics on metal toxicity assessment (Figures S3−S5 therein). Impact of Internalization/Excretion Kinetics. In Figure 2A, cell growth curves are provided for various values of the (1) dimensionless product keτ0L (varied with changing K(1) H,ukint ; see eq 0 6), where τL is the metal membrane transfer time scale defined by eq 6 and 1/ke is the time scale for metal excretion. Similarly to Figure 1, the scenario where M adsorption at the cell surface induces toxicity effects is considered, and unless otherwise specified, it will remain so in the following. The corresponding cM * (t) is given in Figure 2B together with results obtained in the limit μ → 0. As detailed elsewhere for a cell population of constant size (μ → 0),9 an increase in keτ0L basically reflects a faster metal excretion compared to internalization, which leads to a decrease in the amount of internalized metals (Figure S6, section V in the SI) and to a suppression of metal depletion in solution (Figure 2B). Figure 2A shows that cell growth inhibition is enhanced for larger values of keτ0L due to an increase in the concentration of bioavailable metals in bulk solution (Figure 2B) and to that of the

Figure 3. Growth inhibition factor Ψ evaluated at t = td under conditions of Figure 2.

displays the growth inhibition factor Ψ at t = td for the set of keτ0L values tested in Figure 2. As seen in eq 2a, all Ψ(t=td) data evaluated across a wide range of excretion−internalization conditions fall onto one master curve if represented as a function of the corresponding metal concentration c*M(t=td) in solution: the EC50 obtained from this plot identifies with K(1) s,i /βa. If reported as a function of cM *0, a representation found in literature,6 Ψ(t=td) decreases with decreasing keτ0L, in agreement with the dependence of X(t) on keτ0L detailed in Figure 2. This latter representation, however, does not reflect the occurrence over time of metal depletion in the exposure solution (Figure 2B). As a result, its use would lead to keτ0L-dependent “EC50 values” that 6630


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Environmental Science & Technology are overestimated at t = td as compared to the expected K(1) s,i /βa outcome (eq 2a), the discrepancy being as large as ca. 1 order of magnitude at keτ0L = 3 × 10−2 (Figure 3). Figure 3 thus identifies a common pitfall in evaluating so-called “toxicity descriptors” (like EC50) with no cautious examination of depletion effects.6 The resulting metal toxicity becomes necessarily underestimated when discarding the depletion process, in line with results given in the preceding section. Impact of Initial Cell Density. Figure 3 further sheds light on the effect of initial cell density on the outcome of microorganisms toxicity assays. Indeed, eq 6 shows that the initial time scale τ0L for transfer of M across the active membrane is inversely proportional to the initial cell density N0/VT: the larger the latter, the faster the overall metal flux from the solution into the organisms.9,10 In turn, decreasing N0/VT leads to an increase in τ0L or, for that matter, keτ0L. From Figures 2 and 3, a decrease in N0/ VT thus causes a suppression of M depletion from the exposure solution and an increase in growth inhibition following an increase in the concentration of bioavailable toxic metals in solution. This trend is indeed confirmed by the evaluation of X(t) for different values of the initial cell volume fraction ϕ(t=0) (Figure S7, section V in the SI), and for the sake of completeness, the corresponding partitioning dynamics of M across the cellsolution interphase is further provided in Figure S8, section V in the SI. If arbitrarily defining EC50 at a given exposure time as the initial metal concentration c*M0 leading to Ψ = 1/2, then the EC50 value would necessarily increase upon increasing the initial cell density N0/VT or, equivalently, upon decreasing keτ0L (see Figure 3). The above predictions on the relationships between initial cell *0 density, cell growth inhibition, EC50 values (identified with cM leading to Ψ = 1/2), and metal depletion in solution all agree with the experimental findings by Franklin et al.,6 who conducted bioassays to estimate toxicity of copper on various microalgal species. To the best of our knowledge, it is the first time that a theoretical formalism quantitatively elucidates how initial cell density may affect metal toxicity assessment. On the basis of the above results, we comment on the recommendations formulated by several authors6,36,37 to conduct bioassays with prescribed values of initial cell inoculum to avoid bulk metal depletion and underestimation of metal toxicity. These recommendations are actually not mandatory provided that (i) metal partitioning at the cell−solution interphase is adequately addressed under the tested conditions and (ii) toxicity descriptors such as EC50s are properly defined in terms of the true surface or intracellular metal concentrations reached at the considered exposure time. Beyond the legitimate worry to perform bioassays under conditions that realistically mimic natural waters composition,6 if proceeding according to conditions i and ii above, EC50 remains independent of N0/VT (Figure 3) and time, since it reduces to K(1) s,i /βa pending eq 2a applies with Ns,i = 1. Surface-Adsorption- versus Internalization-Mediated Toxicity Scenarios. For the sake of conciseness, a discussion is given in the SI for the case where toxicity is mediated by the internalized metal fraction; i.e., eq 2b applies (Figure S9, section (1) V in the SI). Briefly, with increasing keτ0L (via decreasing K(1) H,ukint ), the overall internalized metal amount, Qs,u(t), necessarily decreases due to the larger propensity of the microorganisms to expel metals from their intracellular compartment. As a result, growth inhibition now becomes weaker with increasing keτ0L and Ψ(t) decreases, which constitutes a difference with the situation treated in Figure 3. Such differences between the surfaceadsorption- and the internalization-mediated toxicity scenarios should allow deciphering of the importance of both processes

when these theoretical models are applied to analyze experimentally measured adsorbed or internalized metal in microorganisms. In addition, the half-maximal effective concentration EC50 at an exposure time texp is now rigorously defined by the quantity ϕu(t=texp), leading to Ψ(t=texp) = 1/2, with the result EC50 = K(1) int,i, as seen in eq 2b. The conclusions regarding the impact of the initial cell density N0/VT on metal toxicity assessment are identical to those formulated for the situation where cell growth inhibition is determined by metal surface adsorption (Figure S7, SI): the lower the value of N0/VT, the less marked the bulk metal depletion and the larger the amount of bioavailable metals for internalization, so cell growth inhibition becomes enhanced. Application to Cd(II) Accumulation and Toxicity in Pseudokirchneriella subcapitata. We now consider the situation where P. subcapitata, a temperate microalgae species, is exposed over time (texp = 12, 24, 48, 72, and 96 h) to different initial concentrations c*M0 of cadmium (0−300 nM). Following the methodology detailed elsewhere,38 the cell number density [cp(t), Figure 4A], the metal concentration in bulk solution * (t), Figure 4B], the amounts of internalized and cell-surface[cM bound metals [Qu(t) and ∑k=i,u,pQs,k(t), respectively] were measured over time under the various exposure conditions tested. From these data, the relevant ϕu(t) (Figure 4C) and Γs(t) (Figure 4D), both defined in the preceding sections, were evaluated with the proper account of the algae size [a ∼ 2.5−3 μm for t ≤ 96 h, ⟨Sa(t)/dt⟩ ∼ 0 or a(t) ̅ = 1, and the absence of significantly protuding surface structures from the membrane; i.e., d ≪ a]. The composition of the Cd exposure medium is detailed by Paquet et al.38 The free initial Cd2+ concentrations measured in the culture media at total measured cadmium concentrations lower than 62 nM were found to be close (although data variability was high) to the modeled free Cd2+ concentrations, suggesting that no significant Cd complexation occurred initially in the culture medium before algal inoculation at these low Cd2+ levels. Only at higher Cd2+ concentrations (>62 nM) did Paquet et al.38 measure significant initial complexation of Cd in solution, but they did not, however, provide an explanation for this apparent initial Cd complexation at high Cd2+ concentration. Despite this uncertainty, the set of data published by Paquet et al.38 constitutes a good start for confrontation with theory. To the best of our knowledge, there is no such available complete set of data where cp(t), c*M(t), ϕu(t), and Γs(t) are systematically measured under different exposure conditions, which includes different initial metal concentrations and distinct exposure times. In addition, very few studies pay attention to the measurement of the free metal concentration in exposure medium, most of the studies focusing on the experimental evaluation of total metal content and on the use of thermodynamic speciation software for retrieving, indirectly, the content of free metal ions in solution. The difference between simulated (with MINEQL+) and experimentally measured free Cd2+ concentration reported by Paquet al.,38 though not negligible, is further necessarily impacted by ill-defined experimental errors on both the metal titration and calibration method. For the lack of better, the data of Paquet et al.38 are used below to challenge the theory for differentiating the nature of the algistatic metal fraction over time. Cell growth is obviously inhibited at sufficiently large t and/or large c*M0 (Figure 4A). This is reflected by the decrease in cp(t) *0 at a fixed exposure time t ≥ 24 h and by the with increasing cM *0 (not shown). decrease of d[ln cp(t)]/dt with increasing cM 6631


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(for obvious practical reasons) indeed prevents one from drawing firm conclusions.1,6 As an alternative, we attempted to quantitatively interpret the * (t), ϕu(t), and Γs(t) under the tested exposure coupled cp(t), cM and c*M0 conditions using eqs 1, 3, and 5 applied in the limit where either eq 2a or eq 2b is a priori hypothesized. To do so, a straightforward analysis of cell growth in the absence of Cd provides the required value μ ∼ 0.07 ± 10% h−1 (or equivalently td ∼ 10 h) and further evidence an exponential growth phase under all conditions examined (i.e., Xc → ∞ in eq 1). In addition, cadmium is thought to be a nonessential element in freshwater algae and one cannot expect that P. subcapitata thus maintains a nonzero intracellular level of Cd2+ at t = 0. Accordingly, we set ϕu0 = 0 as initial boundary conditions for eq 3. Wolterbeek et al.39 modeled the uptake of Zn in P. subcapitata exposed to various free Zn concentrations buffered with NTA under long-term exposure (≈80 h). They found that Zn efflux at Zn2+ concentrations lower than 10−6 M was statistically illdefined and was a negligible parameter in their Zn uptake model. Since current evidence suggests that, at low Cd2+ concentrations, cadmium uptake occurs via Zn transporters in algae,34,40,41 the results by Wolterbeek et al.39 point out that excretion of Cd or Zn should be very low in P. subcapitata. In addition, at a free Cd2+ exposure concentration of 7 nM, Lavoie et al.40 showed that longterm (60 h) Cd uptake in the alga Chlamydomonas reinhardtii could be well-predicted with a BLM-type model without considering any efflux of Cd, thus suggesting that Cd efflux remained very low,42 even over long-term exposure. For all these reasons, Cd efflux in P. subcapitata is discarded in the following, which is done upon setting ke → 0 in eq 3. Finally, ionic strength in the exposure media (7.51 mM) corresponds to typical salinity where electrostatics cannot be a priori neglected.9,10,21 Accordingly, the various (effective) metal affinity constants accessible from constrained data fitting necessarily involve an electrostatic contribution (the surface Boltzmann factor βa), as detailed below. The pH of the medium was further constant at 7.0, and at such low concentrations of protons, no competition between protons and metals for the transporter sites is expected.32,43 Under the conditions of our study, the concentration Γs(t) = Γs,i(t) + Γs,u(t) + Γs,p(t) of cellsurface bound metals at time t is then given by Γs(t) = [βacM * (t)]/[K(j) * (t)]. Inspection of the ∑k=i,u,p∑Nj=1s,kΓmax,(j) s,k s,k + βacM measured adsorption data reveals that within experimental error, Γs(t) depends linearly on c*M(t). This basically means that for the system of interest here, it is legitimate to linearize Γs(t) with respect to βacM * (t); i.e., the adsorption falls within the Henry 9,22 regime, where the inequalities K(j) Then, it s,k ≫ βac* M(t) applies. Ns,k max,(j) (j) comes Γs(t) ≈ ∑k=i,u,p∑j = 1(Γs,k )/(Ks,k )βac*M(t). Writing Γs(t) * (t)/K̅ M, we then in the concise and general form Γs (t) = βaΓmax s cM Ns,k max,(j) /K(j) obtain after identification Γmax s /K̅ M = ∑k=i,u,p∑j=1 Γs,k s,k . Stated differently, the isotherm for metal adsorption at the algal surface (Figure 4D) can be adequately interpreted upon adjustment of two parameters only, an effective maximum surface concentration of metal adsorption sites (Γmax s ) and an effective metal affinity constant for the surface binding sites (K̅ M/ βa), which includes the electrostatic correction βa. There is thus no necessity to distinguish between the three types of sites for interpreting Γs(t) data. The comparison between experiments and theory is displayed in Figure 4 for the two toxicity scenarios envisaged. As a first attempt, we used a single type of inhibitory intracellular site (Nint,i = 1) or inhibitory surface site (Ns,i = 1) and we verified that the introduction of additional inhibitory sites did not improve data

Figure 4. Dependence of (A) P. subcapitata cell density, (B) Cd(II) * ), (C) internalized metal concenconcentration in bulk solution (cM tration (ϕu), and (D) cell-surface-bound metal concentration (Γs) on * (t=0) for different Cd(II)-exposure times initial Cd(II) concentration cM (indicated). Symbols: experimental data. Dotted and plain lines: theoretical analysis according to internalization-mediated and surfaceadsorption-mediated toxicity scenarios, respectively (eq 2b and 2a, respectively). For each toxicity scenario tested, respectively, values of the corresponding model parameters are specified in Table S2 of the SI (section VI).

Figure 4B further reveals that Cd depletion from bulk solution occurs. Last, the internalized and adsorbed metal fractions basically increase with increasing c*M0 and exposure time, as intuitively expected (Figure 4C,D). On the basis of these data, the purpose of the developments below is to rationalize the partitioning of Cd(II) at the P. subcapitata/solution interphase and identify the mechanisms governing Cd toxicity, i.e., the nature(s) of the algistatic Cd fraction(s). As a first attempt, the inhibition growth factor Ψ was evaluated from cell growth data as a function of exposure time for the various cM *0 conditions tested. The obtained Ψ(t) values were then plotted as a function of the corresponding measured c*M(t) and ϕu(t) (data not shown) in order to check the applicability of the toxicity scenarios captured by eqs 2a,b, respectively (see discussion of Figures 3 and S9B, SI). This classical way of proceeding, however, turns out to be inefficient for differentiating between internalization- and surface-adsorption-driven toxicity scenarios. The inherent scattering of the data collected over relatively large time intervals 6632


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Environmental Science & Technology fitting. In view of the inherent chemical heterogeneities describing the surface and intracellular compartments of microorganisms, the thermodynamic constants evaluated here and pertaining to the binding of metals at surface or intracellular inhibitory sites should probably be considered as effective averages of the ensemble of binding constants defining the interactions between metals and sites driving growth inhibition, or they correspond to the inhibitory sites with the highest metal affinity. Accordingly, the interpretation of the data was attempted upon adjustment of K̅ M/βa, Γmax (see above arguments), K(1) s s,i /βa, (1) or Kint,i (depending on the toxicity scenario a priori assumed) and (1) the kinetic and thermodynamic uptake parameters βaK(1) H,ukint and (1) /βa; i.e., five parameters were adjusted to attempt Ks,u reconstructing the 20 sets of time-dependent data plotted in Figure 4 versus cM * (t=0) {5 tested exposure times × 4 variables * (t), ϕu(t), and Γs(t)]}. The corresponding magnitudes [cp(t), cM of the only five adjustable parameters required to recover the coupled {cp,c*M,ϕu,Γs} data versus c*M0 and exposure times are reported in Table S2 (section VI in the SI). The treatment of cases where the Henry adsorption regime does not apply over the entire range of tested metal concentrations would further require * (t) vs cM * (t) a careful inspection of the plot dΓs(t)/dcM constructed from experimental data. The latter may indeed reveal the presence of several energetic binding domains (marked by maxima) that correspond to the different involved metal binding sites and to wavy modulations of Γs(t) upon variation of c*M(t).22 This would at least guide the modeling by defining the number of sites that should be considered for rationalizing metal adsorption isotherms. It is emphasized that this methodology requires, however, accurate data measured upon refined variation of c*M(t) for the derivative dΓs(t)/dc*M(t) to be properly evaluated. This derivation method is widely used and solidly established for modeling the adsorption of molecular probes (protons, gas) at solid surfaces.44 The conclusions of the modeling are summarized as follows: (I) It is difficult a priori to evaluate the nature of the toxicity mechanism after 12 h exposure because both tested toxicity scenarios lead to convincing reconstruction of the corresponding {cp,c*M,ϕu,Γs} data. However, when adopting the metal internalization-mediated toxicity scenario (eq 2b), it is found that the (1) (1) uptake flux Ju*(1) = K(1) H,ukint Ks,u obtained at texp = 12 h is ca. 5 times lower than that found at longer exposures (Table S2, SI). This result is in line with conclusions by Paquet et al.38 and Errécalde et al.,42 who showed the ability of P. subcapitata algal cells to reduce net Cd uptake rates on a short-time scale. These elements suggest that toxicity is caused by internalized metals, as confirmed below from data analysis at texp = 24 and 48 h. (II) At texp = 24 and 48 h, the internalization-driven toxicity scenario is appropriate to recover the dependences of * ,ϕu,Γs} on time and on cM *0. The model parameters {cp,cM required for that purpose remain constant within experimental uncertainties (Table S2, SI). In contrast, a successful application of the surface-adsorption-driven toxicity scenario at texp = 24 and 48 h is achieved only at the cost of ad hoc adjustment of K(1) s,i /βa within a range of values differing by an order of magnitude (Table S2, SI). Regardless, the quality of the fit obtained with such meaningless variations in K(1) s,i /βa remains worse than that achieved when eq 2b is considered (especially at texp = 48 h). (III) At texp = 72 and 96 h, parts A and C of Figure 4 evidence the shortcomings of the internalization-driven toxicity scenario to adequately interpret experimental data. In particular, the ad hoc introduction of a second type of internalization site is

required to minimize (but not suppress) the significant difference between theoretically predicted and experimentally measured ϕu at low cM * (t=0) (Figure 4C), and it is not sufficient to correctly capture the evolution of cp and ϕu over the whole range of c*M0 (Figure 4A,C). On the contrary, the application of eqs 1, 3, 5 with eq 2a (surface-adsorption-driven toxicity scenario) very convincingly reproduces the evolution of {cp,cM * ,ϕu,Γs} with time and cM *0 using a consistent set of (1) (1) parameters (Table S2, SI). The magnitudes of βaK(1) H,ukint , Ks,u /βa, , and K /β obtained at t = 72 and 96 h are further Γmax ̅ s M a exp comparable to those derived at texp ≤ 48 h, where toxicity is mediated by the internalized metal fraction (see points I and II above). This validates the realistic possibility of a transition from internalization- to surface-adsorption-driven toxicity mechanisms upon increasing texp from 48 to 72 h. Above a critical amount of metals bound to the cell surface, a situation reached at sufficiently large texp, adsorbed metals may indeed lead to severe membrane injury and may significantly disrupt cellular homeostasis,45 cause oxidative damage,46 and/or compete with uptake of essential elements,47 thereby significantly impacting cell growth. Our study suggests that the amount of metals adsorbed at the cell surface corresponds to the algistatic metal fraction at texp ≥72 h. For some practical cases, both toxicity mechanisms, envisaged here separately, could take place concomitantly. Analysis of such complex cases is possible with our theory pending replacement of the growth inhibition function Ψ in eq 1 by a linear or even nonlinear combination of eqs 2a,b (the algorithm used for solving the differential eqs 1, 3, and 5 can handle both types of coupling). The issue is then to have at one’s disposal additional data that could indicate the nature of such a coupling. In that respect, it is anticipated that “omic” experiments (transcriptomic and genomic analyses of cells before and after exposure to metals) would be very informative, as well as data defining the dynamic status of metal complexes at the membrane and in the intracellular volume. (IV) Though the modeling appropriately reproduces the significant depletion of Cd(II) from solution at texp ≤48 h (Figure 4B), the extent of that depletion for long-term exposure times remains systematically misevaluated, especially at t = 96 h. If some metal complex is contained in the initial solution, even at a very low level (which is the case for solutions with Cd2+ concentration lower than 62 nM, where 2% metal complex content is expected38), then its proportion will increase in the course of the experiment due to the decreasing metal to ligand ratio in the medium. This could explain the above discrepancy between theory and experiments. However, this discrepancy is observed even at very low metal concentration, where an excess ligand condition most likely applies. Under such excess ligand conditions, the proportion of free Cd2+ does not change upon varying the total metal concentration, and as the Cd 2+ concentration decreases with time (Figure 4B), metal complexes will dissociate, and a constant fraction of free Cd2+ should be maintained over time. Accordingly, and in agreement with previous findings,38 an increased release of algae exudates under long exposure conditions (t = 96 h) where the cell concentration is largest (Figure 4A) could better explain a significant metal complexation in the extracellular media and thus a misevaluation of metal depletion when using the mass balance condition given by eqs 4 and 5. This excretion of metal-binding exudates could correspond to a strategy developed by the algae to counteract metal toxic effects at long exposure times (Figure 4B) when cell density is largest. A refined mass balance formulation would be necessary to capture these effects upon including therein the rate 6633


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la Lutte contre les changements climatiques (MDDELCC), 2700 Rue Einstein, G1P 3W8, Québec, Canada. ⊥ Québec-Océan and Takuvik Joint Université Laval/CNRS Research Units, Département de Biologie, Université Laval, G1V 0A6, Québec, Canada.

of excretion of metal-binding exudates and the dynamics of their association with free metal ions. (V) For the sake of completeness, we evaluate a posteriori the * (t) underlying the validity of the condition cM(r,t) = βrcM applicability of our formalism. This condition is subjected to the requirement that the metal membrane transfer time scale τL isat any time tcomparable to the time scale τE for transfer of M from bulk solution to intracellular volume.9,10 As a first approximation, τL and τE are evaluated here from the expressions derived elsewhere10 in the limit μ → 0 after replacing therein the cell number term by N(t). Under the conditions of interest here, we obtain τE/τL ∼ 1 + Bn−1(1 − 3ϕ(t)1/3/2) (section VII of the (1) SI), where the Bosma number Bn = (DMa−1)/(βaK(1) H,ukint ) compares cell assimilation properties to the dynamic metalsupplying potential of the medium and DM is the diffusion coefficient of Cd(II) in solution. With DM ∼ 10−9 m2 s−1 and −7 −1 (1) βaK(1) H,ukint ∼ 10 m s (Table S2, SI), it comes τE/τL ∼ 1, which justifies a posteriori the use of eqs 1, 3, and 5 for the system Cd(II)−P. subcapitata. To conclude, the theoretical formalism detailed in this work clearly highlights the intimate relationships among the metalpartitioning dynamics at the cell−solution interphase, the biophysicochemical properties of that biointerphase, and the cell growth inhibition kinetics. It opens a new route for determining “a better mechanistic link (rather than a statistical one) between metal binding to surface sites, internal metals, and toxic effects”.1 The new theoretical framework developed in the present study, when applied to model experimentally measured toxicity and uptake data under chronic metal exposure conditions (without metal buffer), will help to identify whether internalized or cell-surface metal fractions predominantly lead to toxicity effects. In addition, the model offers possibilities to study the uptake, toxicity, and essentiality of different oxyanions, such as arsenate, selenate, or molybdate, for which there is no welldefined synthetic complexing agent to buffer the concentration of oxyanions. Extension of the formalism may be further envisaged for more complex cases where cell death affects M partitioning or inactivation of uptake occurs over time to meet metabolic cell demands.

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.F.L.D. is grateful to three reviewers for their valuable and constructive comments on this work.

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GLOSSARY a(t) time-dependent cell radius (without the protuding surface layer) (m) a0 cell radius (without the protuding surface layer) at t = 0 (m) a(t) dimensionless, time-dependent cell radius defined by ̅ a(t)/a0 0 A(j) K(j) s,k=i,u,p s,k=i,u,p/(βac* M ); dimensionless affinity of M for the jth type of Si, Su, and Sp surface sites in the absence of metal−proton competition at cell surface binding sites cM(r,t) concentration of free metals at the position r and time t (mol m−3) 0 cM * initial bulk concentration of free metals (mol m−3) 0 c*M (t) c * M(t)/c* M ; dimensionless concentration of free metal ̅ in bulk solution at time t cp(t) cell number density (m−3) d thickness of the peripheral cell surface layer (m) EC50 half-maximal effective concentration (mol m−3) at a given exposure time texp, rigorously defined as the quantity cM * (t=texp) leading to Ψ(t=texp) = 1/2 (see eq 2a) Ju(t) metal internalization flux at time t (mol m−2 s−1) (j) (j) (j) (j) J*u KH,u kintKs,u ; limiting (maximum) metal internalization flux (mol m−2 s−1) ke kinetic constant for metal excretion (s−1) k(j) kinetic constant for internalization of M via the jth int type of internalization sites (j = 1, ..., Ns,u) (s−1) (j) Ks,i metal affinity constant for the jth type of surface inhibitory sites (j = 1, ..., Ns,i) (mol m−3) K(j) metal affinity constant for the jth type of intracellular int,i inhibitory sites (j = 1, ..., Nint,i) (mol m−2) K(j) metal affinity constant for the jth type of nonspecific s,p adsorption sites (j = 1, ..., Ns,p) (mol m−3) K(j) metal affinity constant for the jth type of internals,u ization sites (j = 1, ..., Ns,u) (mol m−3) K(j) Henry coefficient for M adsorption at the jth type of H,u internalization sites (j = 1, ..., Ns,u) (m) K(j)‑H affinity of protons for the jth type of surface s,k=i,u,p inhibitory (k = i), uptake (k = u), or nonspecific adsorption (k = p) sites (mol m−3) M metal N(t) cell number at t Nc maximum population size the medium can sustain under given nutrient conditions N0 number of cells at t = 0 Ns,k=u,p number of different internalization sites (k = u) at the membrane surface and number of different nonspecific surface adsorption sites for metals (k = p) Ns,i number of different surface inhibitory sites

ASSOCIATED CONTENT

S Supporting Information *

Details on the derivation of eqs 2a,b (section I), eqs 3, 5 and an account of the proton−metal adsorption competition at cell surface sites (section II); details on the numerical procedure for solving eqs 1, 3, and 5 (section III) and on the solution in the limits μ → 0 and a(t) ̅ = 1 (Table S1, section IV); supplementary Figures S1−S9 (section V); magnitudes of the adjustable parameters required to fit the experimental data of Figure 4 (Table S2, section VI); and evaluation of τE/τL under the conditions of Figure 4 (section VII). The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.5b00594.

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AUTHOR INFORMATION

Corresponding Author

*Phone: + 33 3 83 59 62 63; fax: + 33 3 83 59 62 55; e-mail: [email protected]. Present Addresses ∥

Centre d’expertise en analyse environnementale du Québec, Ministère du Développement durable, de l′Environnement et de 6634


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Environmental Science & Technology ϕ0u

Nint,i

number of different intracellular inhibitory sites N0X(t) ϕu(t) Sa(t); measurable amount of internalized metals over the entire microorganism dispersion (mol) Qu(t) Qs,k=i,u,p(t) N0X(t) Γs,k=i,u,p(t) Sa(t); measurable amounts of cellsurface bound metals over the entire microorganism dispersion and the index k differentiates metals bound to growth inhibitory Si surface sites, to internalization sites Su and to nonspecific Sp surface sites (mol) r radial coordinate with the origin at the center of the spherical cells (m) rc(t) radius of the Kuwabara unit cell at time t (m) r0c rc(t=0); initial radius of the Kuwabara unit cell (m) rD radial position satisfying r0(t) + κ−1 ≪ rD ≪ rc(t) (m) a(t) + d; overall cell radius with peripheral surface r0(t) layer included at time t (m) (j) R(j) 1/(K(j) s H,ukintβa); membrane transfer resistance associated with the uptake of M via the jth type of internalization site (m−1 s) Sa(t) active cell surface area at time t (m2) Sk=i,u,p denote the inhibitory surface sites (k = i), the internalization sites (k = u) and the nonspecific adsorption surface sites (k = p) t time (s) texp exposure time (s) td μ−1 ln[(Xc − 1)/(Xc/2−1)]; doubling time of the cell population in the absence of M (s) VT volume of the dispersion containing metal-assimilating microorganisms (m3) V̅ dimensionless ratio defined by V0f /(βaṼ soft + V0f ) 0 Vf 4π(r0c )3/3; initial volume of the Kuwabara unit cell (m3) r Ṽ soft Ṽ soft = [3Vsoft(0)∫ D ξ2βξdξ]/{βaro(0)3(1 − α3)} is

concentration of internalized metals at time t = 0 (mol m−2) N s,u ϕ̅ u(t) keϕu(t)/∑j=1 Ju*(j); dimensionless concentration of internalized metals at time t Γs(t) metal surface concentration at time t [Γs(t) = Γs,i(t) + Γs,u(t) + Γs,p(t)] (mol m−2) Γs,i(t), Γs,u(t), and Γs,p(t) time-dependent surface concentrations of metals adsorbed at the inhibitory surface sites Si, at the internalization sites Su, and at the nonspecific cell-surface binding sites Sp, respectively (mol m−2) max,(j) Γs,k=i,u,p surface concentration of the jth type of Si, Su, and Sp sites (mol m−2) κ−1 Debye length defining the typical extension of the electric double layer (m) μ cell growth rate in the exponential phase in the absence of M (s−1) 0 τL metal membrane transfer time scale at t = 0 defined by eq 6 (s) τE time scale for transfer of M from bulk solution to intracellular volume at time t (s) τL metal membrane transfer time scale at time t (s) ψ(r) electrostatic potential at the radial position r (V) Ψ(t) dimensionless growth inhibition factor at time t

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a0

X(t)

y(r) zM

the initial geometric volume {Vsoft(0) = 4π[r0(0)3 − a03]/3}; geometric volume Vsoft of the surrounding soft surface layer formally corrected for electrostatics with α = a0/r0(0) (m3) N(t)/N0; dimensionless cell number or, equivalently, dimensionless cell density number (since VT is constant over time) in the dispersion of volume VT at time t Fψ(r)/RT; dimensionless electrostatic potential with F the Faraday, T the temperature, R the gas constant valence of metal ions

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GreekTerms

βa βHa

βD βr ϕ(t) ϕu(t)

REFERENCES

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