Bioaccumulation Dynamics of Metal Ions and Complexes by Aquatic Organisms: An Extended Best and Michaelis–Menten Framework for the Interplay between Chemodynamics, Bioavailability, and Biouptake Kinetics

Abstract

This study elaborates a theoretical framework to understand metal bioaccumulation beyond the limitations of the equilibrium-based Biotic Ligand Model (BLM). By integrating the dynamics of metal speciation in aquatic media with the kinetics of metal biouptake, the formalism predicts metal bioavailability under conditions where the BLM fails, such as high degrees of metal complexation, diffusion-limited metal uptake, and/or internalization of intact metal complexes. The theory accounts for the reactive transport of free metal ions and complexes, incorporating different uptake mechanisms by facilitated and passive diffusion. Extended Best expressions are derived for the flux of metal biouptake by coupling extracellular metal chemodynamics (intertwined diffusion and complexation) with Michaelis–Menten uptake kinetics. The approach provides a unified rationale for various bioaccumulation situations, including the uptake of lipophilic complexes, and the concomitant uptake of free and complexed metals through distinct or shared diffusion-facilitated pathways with competitive, noncompetitive, and/or uncompetitive inhibitions. Computational examples are detailed to illustrate the intricate interplay between metal species transport dynamics and biouptake kinetics for all bioaccumulation cases, particularly addressing how the metal internalization flux is impacted by the (bioavai)­lability of metal complexes, whether or not they are internalized intact. The benefits of the formalism are further illustrated through an analysis of well-characterized experimental data on neodymium (Nd) uptake by Chlamydomonas reinhardtii in the absence/presence of well-defined organic ligands. This analysis not only derives key metal biouptake and bioaffinity parameters but also provides solid evidence of the BLM’s failure to describe the data. The quantitative interpretation of Nd bioaccumulation data leads to the identification of two potential uptake mechanisms, and a methodology to distinguish between them is discussed. Overall, this work establishes a more accurate and comprehensive theoretical foundation for predicting metal bioaccumulation in aquatic systems, thereby fundamentally challenging the common equilibrium-based perception of metal bioavailability.

1. Introduction

The bioavailability of metal ions depends on their chemical speciation, i.e., their distribution over different physicochemical forms such as the free (hydrated) metal ion and the complexes it forms with various inorganic and organic ligands (L). Generally, the free metal ion (M) is the bioreactive (i.e., internalizable) metal form with few exceptions, e.g., lipid soluble metal complexes are proposed to cross biological membranes intact by passive diffusion. , Biouptake of metal ions involves (i) their diffusion toward the biological interface (biointerface in short), (ii) their association with a biological ion transporter, followed by (iii) their internalization. Depending on the relative magnitudes of the diffusive supply flux of free metal ions from the medium toward the biointerface and the metal internalization flux, a gradient in the free metal ion concentration may be established on the external medium side of the biointerface. The presence of a gradient in free metal ion concentration provides a driving force for the dissociation of metal complexes (ML), which buffers the diffusive supply flux of free M to an extent determined by the so-called lability of ML on the time scale of biouptake. , Thus, the bioavailability and bioaccumulation kinetics of metal ions are governed by the interplay between chemical speciation dynamics in the exposure medium and the time scale of processes occurring at the biological interface and within the organism. Various metal bioaccumulation scenarios are shown schematically in Figure , and all of them underpin a possible contribution of metal complexes to metal bioavailability, as further detailed in this work.

1.

Schematic illustrations of the bioaccumulation scenarios considered here for the uptake of free metal (M) and/or complexed metal (ML), and representation of the interplay between the kinetics of the complexation reaction M + L ↔ ML and the diffusion of M and ML from the bulk medium to the surface of the metal-consuming organism. (A) Uptake of M via a facilitated diffusion pathway. (B) Uptake of M and ML via two separate facilitated diffusion pathways. (C) Uptake of M and ML via a facilitated diffusion pathway and a passive diffusion route, respectively. (D) Uptake of M and ML via a single facilitated diffusion pathway and occurrence of inhibition processes between M and ML along the lines detailed in Figure . S_u,M and S_u,ML represent the internalization sites for M and ML, respectively, while S_u corresponds to the internalization sites operational for both M and ML species. More specifically, S_u,M and S_u,ML are the active sites of the protein ensuring the translocation of M and ML across the biological barrier (cell membrane), respectively. See text for further details.

Despite the inherent dependence of metal bioavailability and bioaccumulation on the kinetics of metal complexation and time scales of diffusion and biouptake, there is a large body of literature which entirely disregards the dynamic nature of the involved processes. For instance, the widely used biotic ligand model (BLM) , implicitly assumes that the diffusional supply flux of free metal ion is not rate-limiting, and the only relevant concentration of chemical species for biouptake is then the equilibrium concentration of free M in the bulk exposure medium. The free M concentration is typically computed by various equilibrium speciation models, each with their own limitations. Furthermore, there is typically a paucity of information on the physicochemical properties of the involved organic ligands, which may include natural and engineered molecular and nanoparticulate complexants. − It has been shown that the assumption that only the free M is involved in biouptake is valid only if the diffusive flux of free M (unsupported by coupled transport of labile complex species) is much greater than the biouptake flux. ,− This dynamic condition may be rather severe under typical conditions in aquatic ecosystems, e.g., those with a high degree of metal complexation, and particularly so in the presence of nanoparticulate complexes. − In making the connection between biouptake and toxicity, the BLM assumes that toxicity occurs once the extent of metal ion binding to so-called “biotic ligands” exceeds a certain threshold level, irrespective of the time scale. ,

Within BLM-type frameworks, the influence of various factors such as pH and dissolved organic carbon on metal ion bioavailability and toxicity is considered in terms of their effects on the equilibrium concentration of the free metal ion in the bulk exposure medium and equilibrium competition for association with the biotic ligands. This disregard of dynamic factors likely underpins the need for inclusion of many empirical factors and recalibration of the BLM to fit experimental observations for each organism and for each exposure condition considered. The observation that fully empirical multiple linear regression models perform as well asor better thanBLM for predictions of metal bioavailability and potential toxicity underscores the shortcomings of the BLM approach. ,

Herein we elaborate a comprehensive theoretical framework to describe a range of metal bioaccumulation scenarios where the bioavailability of M may not conform to the standard BLM equilibrium predictions. In detail, the processes considered include the interplay between the reactive transfer of metal ions and metal complexestermed hereafter as metal chemodynamics, ruled by metal diffusion transport and complexation reaction kineticstoward the surface of a metal-consuming organism (biointerface), and the bioaccumulation of M and/or ML via different uptake pathways, i.e., facilitated and/or passive diffusion routes with or without competitive inhibition between M and ML (Figure ). In all cases involving facilitated diffusion, the relevant kinetics of metal translocation process across the biological membrane is derived from the Michaelis–Menten mechanism. , Our findings highlight the (limited) range of kinetic conditions under which BLM-type models are applicable. They further provide a solid theoretical rationale of a large spectrum of metal bioaccumulation cases identified in literature as exceptions to the standard BLM model, and never previously formulated with the proper account of intertwined metal chemodynamics and kinetics of M and/or ML biouptake. The benefits of the theory are illustrated by our detailed analysis of selected experimental data taken from well-documented metal bioaccumulation literature, beyond the restrictive biouptake BLM framework and without a priori equilibrium-based considerations. While the theoretical framework developed herein focuses on the coupling between metal chemodynamics in solution and interfacial biouptake kinetics, it is important to recognize that biological uptake can be governed in some situations by specific membrane transport protein mechanisms. For instance, metal internalization can be influenced by ion exchange processes (symport or antiport) or the presence of required cosubstrates. Notable examples include the role of chloride as an exchange ion for copper, , or the requirement of bicarbonate for the transport of divalent metals like Zn, Mn, and Cd via ZIP8 transporters. , Although these specific biological transport pathways are not explicitly considered in the current theory, they may represent critical biological layers that - depending on the system - may complement the metal chemodynamic and bioaccumulation processes addressed in this work.

2. Materials and Methods

Selection of Experimental Data to be Analyzed with the Here-Reported Theory. While there are numerous publications on the biouptake of metal ions by various organisms, most are not sufficiently documented for our purposes. We identified studies which provided sufficient details on the exposure medium composition and biouptake data as a function of time. These studies refer to the biouptake of neodymium by the unicellular green algae Chlamydomonas reinhardtii in the absence or presence of organic ligands (malic acid and citric acid). For this system, the authors evidenced the failure of the standard BLM formulation for predicting metal biouptake and they invoked the bioavailability of formed complexes as a possible source of deviation between experiments and BLM computations.

3. Theory

3.1. Setting the Stage

In the following, we consider the reactive transfer of free metal ions (M) and metal complexes (ML) from a medium containing a metal-complexing ligand (L) to the surface of a spherical metal-consuming organism with curvature radius a. We elaborate below the so-far missing expressions that define the M and ML biouptake fluxes, denoted as $J_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{ML}}$ (mol m² s^–1), respectively, for the different bioaccumulation scenarios schemed in Figure . The approach accounts explicitly for the coupling between M/ML chemodynamics in the medium and the kinetics of the relevant M/ML interfacial biouptake processes.

In detail, we differentiate cases where the bioreactive (internalizable) metal forms are only the free metal species M (Figure A) from those where both M and ML can be internalized (Figures B–D). In the latter situation, we further distinguish scenarios where M and ML are bioaccumulated according to (i) two distinct facilitated diffusion pathways involving 2 different transporters, one for M and the other for ML (Figure B), (ii) a facilitated diffusion of M and a passive diffusion of ML (Figure C), and (iii) a single facilitated diffusion pathway mediated by one transporter type operating both for M and ML (Figure D). The bioaccumulation scenarios pictured in Figure cover the situations denoted as Type 1 to Type 5 by Zhao et al. (cf. Figure 2 therein) and qualified by those authors as the major exceptions to classical equilibrium models when predicting the bioavailability of metal complexes. Namely:

• Type 1-situation is that where there is passive diffusion of lipophilic metal complexes. It corresponds to the case sketched in Figure C in the limit where there is no uptake of free M, i.e., $J_{\mathrm{u},\mathrm{M}}=0$ . Examples include the uptake by fish and algae of complexes formed between Cd²⁺, Ni²⁺, and Pb²⁺ and ligands such as dithiocarbamates, xanthates, dialkyldithiophosphates and pyridinethiones.

• Type 2-situation is that where there is uptake of hydrophilic metal complexes through a ligand transporter system. It corresponds to the case sketched in Figure B in the limit $J_{\mathrm{u},\mathrm{M}}=0$ . Examples include the uptake of Cd-citrate complexes by the unicellular algae Pseudokirchneriella subcapitata, or the uptake of Ag through the S₂O₃ ^2– transport system by Chlamydomonas reinhardtii.

• Type 3-situation is that where there is reaction of a complex ML with a biotic ligand at the cell surface and the ensuing formation of a so-called ternary complex. It corresponds to the case sketched in Figure D in the limit $J_{\mathrm{u},\mathrm{M}}=0$ , with the ternary complex being formed between the active site of the protein transporter, S_u, and the complex ML. Examples include the uptake of Sc³⁺ by C. reinhardtii above pH 6.5.

• Type 4-situation represents an example where labile metal complexes can contribute to metal uptake (in the case of mass-transport limitation). It corresponds to the dynamic limit in Figure A where the rates of formation/dissociation of (fully labile) ML are fast on the effective time scale of their (slow) diffusion from the bulk medium to the surface of the organism. Examples include the uptake of Cu²⁺ by periphyton in the presence of NTA (nitrilotriacetic acid).

• Type 5-situation represents the situation where metal exchange between the medium and biotic ligand is rate-limiting. It refers to the case where equilibrium between free M and the biotic ligand occurs on a comparable, or slower, time scale as that of the internalization step. It thus corresponds to the kinetic limit in Figure A, where free M features a slow formation/dissociation kinetics with the biotic ligand on the effective time scale of their (fast) diffusion from the bulk medium to the surface of the organism. Examples are rare due to the difficulty to distinguish experimentally between the situation where metal uptake is under kinetic control and the one where the uptake is thermodynamically controlled (BLM). A famous illustration is Fe uptake by two species of marine phytoplankton.

It is stressed that the above classification found in literature is generally based on evaluating the ratio between the (theoretical) diffusional flux of free M and the (measured) M biouptake flux, and the use of the Michaelis–Menten equation involving or not competitive M-ML equilibrium surface adsorption. In turn, the interpretation of biouptake flux data never fully integrates the interplay between chemodynamics and biouptake kinetics of M and/or ML with the proper level of rigor. Here we present a theoretical framework with explicit expressions of the associated fluxes, which enables rigorous interpretation of the involved processes.

In Figure , steady-state metal transport at the organism/medium interface is rapidly achieved (within ∼ ms) for organisms whose size a is of the order of 1 to 10 μm, e.g., bacteria, microalgae or consuming organs of invertebrates such as gills in fish. ,,, In the bulk medium, far from the biological metal-consuming surface, equilibrium between M, ML and L is maintained, and their respective bulk concentrations, denoted as $c_{i=\mathrm{M},\mathrm{L},\mathrm{ML}}^{*}$ , satisfy $K_{\mathrm{ML}}=c_{\mathrm{ML}}^{*}/(c_{\mathrm{L}}^{*}{c}_{\mathrm{M}}^{*})$ where $K_{\mathrm{ML}}$ (m³ mol^–1) is the ML stability constant. In practice, there is usually an excess of L over (trace) M so that the concentration of L remains constant regardless of the advancement of the complexation reaction M + L ⇄ ML. This legitimates the introduction of the dimensionless ML stability constant, ${\stackrel{-}{K}}_{\mathrm{ML}}$ , defined by ${\stackrel{-}{K}}_{\mathrm{ML}}=K_{\mathrm{ML}}{c}_{\mathrm{L}}^{*}$ . For the purpose of demonstration and mathematical simplification, we discard below the possible depletion of metal species from the bulk solution during bioaccumulation. The reader is referred to previous reports − , for the modeling of such scenarios when M is the only bioreactive metal form. In addition, we assume that metal-induced toxicity effects on the biouptake processes are negligible, and we do not account for metal excretion. The former assumption is generally valid for short-term bioaccumulation and/or sufficiently low bulk metal concentrations. The latter approximation is particularly applicable to organisms such as microalgae, for which metal bioaccumulation data are the most extensively documented and can be analyzed by mechanistic theory. The overall process of M transfer involves the diffusion of M and ML, with diffusion coefficients $D_{i=\mathrm{M},\mathrm{ML}}$ ­(m² s^–1), and the interconversion kinetics between M and ML in the external aqueous medium, with association and dissociation kinetic constants $k_{\mathrm{a}}$ (m³ mol^–1 s^–1) and $k_{\mathrm{d}}$ (s^–1), respectively, connected to $K_{\mathrm{ML}}$ by $K_{\mathrm{ML}}=k_{\mathrm{a}}/k_{\mathrm{d}}$ . Meanings of symbols in this work are defined in a Glossary to ease reading. Below, we derive the expressions of the biouptake fluxes $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ for the metal bioaccumulation scenarios shown in Figure . This derivation (Section ) follows the evaluation in Section of the M and ML concentration profiles in the medium, and their corresponding diffusional supply fluxes to the metal-consuming biointerface.

3.2. Expressions of M/ML Concentration Profiles and Supply Fluxes to the Organism’s Metal-Consuming Surface

Herein we adopt the radial coordinate $r$ with the origin $r=0$ positioned at the center of the metal-consuming organism. Under the conditions outlined in Section , the spatial distributions of M and ML concentrations, denoted as $c_{\mathrm{M}}(r)$ and $c_{\mathrm{ML}}(r)$ , respectively, are governed by the coupled steady-state Nernst–Planck equations corrected for the chemical terms pertaining to ML formation and dissociation kinetics

$$\{\begin{array}{lr}{D}_{\mathrm{M}}{\nabla }_r^2{c}_{\mathrm{M}}(r)+k_{\mathrm{d}}{c}_{\mathrm{ML}}(r)-k_{\mathrm{a}}{c}_{\mathrm{ML}}(r)c_{\mathrm{L}}^{*}=0& (\mathrm{a})\\ D_{\mathrm{ML}}{\nabla }_r^2{c}_{\mathrm{ML}}(r)-k_{\mathrm{d}}{c}_{\mathrm{ML}}(r)+k_{\mathrm{a}}{c}_{\mathrm{ML}}(r)c_{\mathrm{L}}^{*}=0& (\mathrm{b})\end{array}$$ 1

where ${\nabla }_r^{2}fr^{-2}\mathrm{d}(r^2\mathrm{d}f/\mathrm{d}r)/\mathrm{d}r$ is the Laplacian operator in spherical geometry with $f$ a dummy function of the radial position r. The boundaries associated with eq specify M and ML surface concentrations that hold far from- and at- the organism’s surface, i.e.,

$$\{\begin{array}{lr}{c}_{i=\mathrm{M},\mathrm{L},\mathrm{ML}}(r\to \infty )=c_{i=\mathrm{M},\mathrm{L},\mathrm{ML}}^{*}& (\mathrm{a})\\ c_{i=\mathrm{M},\mathrm{ML}}(r=a)=c_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{a})}& (\mathrm{b})\end{array}$$ 2

where $c_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{a})}$ stand for the M and ML surface concentrations. For mathematical convenience, we introduce the dimensionless M and ML concentration profiles defined by ${\stackrel{-}{c}}_{i=\mathrm{M},\mathrm{ML}}(\stackrel{-}{r})=c_{i=\mathrm{M},\mathrm{ML}}(\stackrel{-}{r})/c_{i=\mathrm{M},\mathrm{ML}}^{*}$ with $\stackrel{-}{r}=r/a$ the scaled radial position, and $\epsilon =D_{\mathrm{ML}}/D_{\mathrm{M}}$ the ML and M diffusion coefficients ratio. Following the procedure outlined in Supporting Information (SI-A), the solution of eqs and can be written in the concise form

$$\{\begin{array}{lr}{\stackrel{-}{c}}_{\mathrm{M}}(\stackrel{-}{r})=1-\frac{1}{\stackrel{-}{r}}[1-\frac{{\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}{\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}}{1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}& \\ +\frac{\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}{1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}({\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}-{\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}){\mathrm{e}}^{-{\stackrel{-}{h}}_{\mathrm{a}}(\stackrel{-}{r}-1)}]& (\mathrm{a})\\ {\stackrel{-}{c}}_{\mathrm{ML}}(\stackrel{-}{r})=1-\frac{1}{\stackrel{-}{r}}[1-\frac{{\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}{\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}}{1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}& \\ -\frac{1}{1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}({\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}-{\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}){\mathrm{e}}^{-{\stackrel{-}{h}}_{\mathrm{a}}(\stackrel{-}{r}-1)}]& (\mathrm{b})\end{array}$$ 3

where ${\stackrel{-}{c}}_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{a})}=c_{i=\mathrm{M},\mathrm{ML}}(\stackrel{-}{r}=1)/c_{i=\mathrm{M},\mathrm{ML}}^{*}$ refers to the dimensionless M and ML surface concentrations, and the dimensionless ${\stackrel{-}{h}}_{\mathrm{a}}$ in eq is defined by

$${\stackrel{-}{h}}_{\mathrm{a}}=\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})/(\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})}$$ 4

In eq , ${\stackrel{-}{\kappa }}_{\mathrm{a}}=k_{\mathrm{a}}{a}^2{c}_{\mathrm{L}}^{*}/D_{\mathrm{M}}$ is the dimensionless Damkhöler number that compares the time scale of M diffusion, $a^2/D_{\mathrm{M}}$ , and the ML formation time scale, $1/(k_{\mathrm{a}}{c}_{\mathrm{L}}^{*})$ . Equivalently, $\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}$ compares the organism radius $a$ with the thickness $\sqrt{D_{\mathrm{M}}/(k_{\mathrm{a}}{c}_{\mathrm{L}}^{*})}$ of the reaction layer (Figure ) which represents the distance traveled by a free M before it reassociates with L to form ML.

The diffusional supply fluxes of M and ML from the bulk solution to the surface of the metal-consuming organism, denoted as $J_{i=\mathrm{M},\mathrm{ML}}$ , are defined by

$$J_{i=\mathrm{M},\mathrm{ML}}=D_{i=\mathrm{M},\mathrm{ML}}(\mathrm{d}{c}_{i=\mathrm{M},\mathrm{ML}}/\mathrm{d}r){|}_{r=a}$$ 5

Combining eqs and , we obtain the following expressions that define $J_{i=\mathrm{M},\mathrm{ML}}$

$$\{\begin{array}{lr}{J}_{\mathrm{M}}/J_{\mathrm{Diff},\mathrm{M}}^{\max }=1+{\gamma }_{\mathrm{M}}y-(1+{\gamma }_{\mathrm{M}})x& (\mathrm{a})\\ J_{\mathrm{ML}}/J_{\mathrm{Diff},\mathrm{ML}}^{\max }=1+{\gamma }_{\mathrm{ML}}x-(1+{\gamma }_{\mathrm{ML}})y& (\mathrm{b})\end{array}$$ 6

where we have written the dimensionless M and ML surface concentrations in the concise forms $x={\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}$ and $y={\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}$ , and we have further introduced $J_{\mathrm{Diff},i=\mathrm{M},\mathrm{ML}}^{\max }$ the maximum M and ML diffusion fluxes from bulk solution to the organism’s surface, defined by $J_{\mathrm{Diff},i=\mathrm{M},\mathrm{ML}}^{\max }=D_{i=\mathrm{M},\mathrm{ML}}{c}_{i=\mathrm{M},\mathrm{ML}}^{*}/a$ , which is achieved when ${\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}=0$ . Equation involves the dimensionless parameters ${\gamma }_{i=\mathrm{M},\mathrm{ML}}$ given by γ_ML = ${\gamma }_{\mathrm{M}}/(\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})$ = ${\stackrel{-}{h}}_{\mathrm{a}}/(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})$ , where ${\mathrm{h̅}}_{\mathrm{a}}$ is defined by eq . These parameters describe how the coupling between M complexation kinetics and M/ML diffusions (Figure ) impact the supply fluxes $J_{i=\mathrm{M},\mathrm{ML}}$ at given M and ML surface concentrations. Using eq , the parameters ${\gamma }_{i=\mathrm{M},\mathrm{ML}}$ can be rewritten in the convenient forms

$$\{\begin{array}{lr}{\gamma }_{\mathrm{M}}=\sqrt{\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}{\stackrel{-}{\kappa }}_{\mathrm{a}}/(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})}& (\mathrm{a})\\ {\gamma }_{\mathrm{ML}}=\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}/[\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})]}& (\mathrm{b})\end{array}$$ 7

3.3. Expressions of the M and ML Biouptake Fluxes

For all bioaccumulation scenarios shown in Figure , steady-state metal transport imposes that the M and ML biouptake fluxes, $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ , must fulfill the following condition

$$J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}=J_{i=\mathrm{M},\mathrm{ML}}$$ 8

where the M and ML diffusional supply fluxes, $J_{i=\mathrm{M},\mathrm{ML}}$ , are defined by eqs and . Consistent with literature, we assume that facilitated M and/or ML diffusionwhen operationalproceeds according to the Michaelis–Menten mechanism. , This involves a rapid adsorption of the bioreactive M and/or ML at the corresponding M and/or ML internalization sites (denoted as S_u,M and S_u,ML in Figure , representing the active sites of the M and ML protein transporters, respectively). This fast adsorption is then followed by a rate-limiting translocation of M and/or ML across the biological barrier, with kinetic constant $k_{\mathrm{int},\mathrm{M}}$ ­(s^–1) and $k_{\mathrm{int},\mathrm{ML}}$ ­(s^–1), respectively. Using an analogy to the well-established inhibition principles described by Michaelis–Menten enzyme kinetics theory, , we can further categorize the bioaccumulation scenario depicted in Figure D. Specifically, we distinguish between cases involving M-ML competitive inhibition, noncompetitive inhibition, uncompetitive inhibition and mixed inhibition. Considering the inhibition of M uptake by ML for the sake of demonstration, Figure illustrates each of these inhibition processes, clarifying how M and ML bind to the active (internalization) and/or nonactive (inhibitory) sites on the protein transporter that facilitates M translocation across the biological barrier.

2.

Expressions of the normalized M uptake flux, $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ , for the bioaccumulation scenario sketched in Figure D in the case of (A) M/ML competitive inhibition, (B) M/ML noncompetitive inhibition, and (C) M/ML uncompetitive inhibition. The expression of $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ for a mixed inhibition process is provided in (D) (blue font). The characteristic half-saturation constants $K_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ and $K_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{I})}$ (mol m^–3) refer to the reciprocals of the stability constants (m³ mol^–1) that describe the M and ML binding with their respective active and inhibitory transporter sites, respectively. The M and ML concentrations at the surface of the metal-consuming organism are denoted as $c_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{a})}$ . The competitive, noncompetitive, and uncompetitive inhibitions by ML are limits of the mixed inhibition process for $K_{\mathrm{u},\mathrm{ML}}\ll K_{\mathrm{ML}}^{(\mathrm{I})}$ , $K_{\mathrm{u},\mathrm{ML}}=K_{\mathrm{ML}}^{(\mathrm{I})}$ , and $K_{\mathrm{u},\mathrm{ML}}\gg K_{\mathrm{ML}}^{(\mathrm{I})}$ , respectively. For the sake of conciseness, only the expressions of $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ are given here. Formulations of $J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }$ for the various inhibition scenarios of interest derive from those of $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ after inverting the ‘M’ and ‘ML’ notations in all of the involved symbols. The dimensionless expressions defining the metal biouptake fluxes relevant for the various scenarios pictured in this figure are defined in Section . See text for further details.

The expressions for M and ML biouptake fluxes are derived below for each bioaccumulation scenario shown in Figures and , starting with the case where M is the only bioreactive metal form (Figure A).

3.3.1. Case Where M Is the Only Bioreactive Metal Form (Figure A)

In this situation where ML is not internalized, the only relevant biouptake flux is that of the free metal species M, $J_{\mathrm{u},\mathrm{M}}$ . This flux depends on the (dimensionless) surface concentration of free M, $x={\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}$ , according to the Michaelis–Menten expression written in the dimensionless format ,,

$$J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/(a_{\mathrm{M}}+x)$$ 9

where we recall that $J_{\mathrm{u},\mathrm{M}}^{\max }$ is the maximum M biouptake flux. In eq , $a_{\mathrm{M}}$ corresponds to the dimensionless metal bioaffinity parameter given by $a_{\mathrm{M}}=K_{\mathrm{u},\mathrm{M}}/c_{\mathrm{M}}^{*}$ , where $K_{\mathrm{u},\mathrm{M}}$ (mol m^–3) is the characteristic M half-saturation constant. The latter corresponds to the reciprocal of the stability constant (in m³ mol^–1) that describes the binding of M with the active M-transporter sites at the cell membrane. Accordingly, a very high bioaffinity of M for the M-internalization sites corresponds to very low values of $a_{\mathrm{M}}$ or $K_{\mathrm{u},\mathrm{M}}$ . In the extreme $a_{\mathrm{M}}\ll 1$ , all M internalization sites are occupied by M (full coverage), and the limit $J_{\mathrm{u},\mathrm{M}}\to J_{\mathrm{u},\mathrm{M}}^{\max }$ is reached. It can be verified that $J_{\mathrm{u},\mathrm{M}}^{\max }$ depends on $K_{\mathrm{u},\mathrm{M}}$ via $J_{\mathrm{u},\mathrm{M}}^{\max }=K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}{K}_{\mathrm{u},\mathrm{M}}$ , where $K_{\mathrm{H},\mathrm{M}}$ (m) is the Henry constant for the fast M adsorption on the M internalization sites. Substituting in eq the expressions of $J_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{M}}$ defined by eqs and , respectively, and further applying the condition $J_{\mathrm{ML}}=0$ which reflects the absence of ML internalization with $J_{\mathrm{ML}}$ defined by eq , we obtain after some algebra the following expressions for the M and ML surface concentrations $x={\stackrel{-}{c}}_{\mathrm{M}}^{(\mathrm{a})}$ and $y={\stackrel{-}{c}}_{\mathrm{ML}}^{(\mathrm{a})}$

$$\{\begin{array}{lr}x=\frac{1}{2}(1-a_{\mathrm{M}}-b_{\mathrm{M}}+(1+a_{\mathrm{M}}+b_{\mathrm{M}})\times & \\ \sqrt{1-\frac{4b_{\mathrm{M}}}{(1+a_{\mathrm{M}}+b_{\mathrm{M}}{)}^2}})& (\mathrm{a})\\ y=(1+{\gamma }_{\mathrm{ML}}x)/(1+{\gamma }_{\mathrm{ML}})& (\mathrm{b})\end{array}$$ 10

Substituting eq in eq , we derive the following dimensionless expression for the M biouptake flux

$$Q_{\mathrm{M}}=\frac{1}{2}\left(\frac{1+a_{\mathrm{M}}+b_{\mathrm{M}}}{b_{\mathrm{M}}}\right)\left(1-\sqrt{1-\frac{4b_{\mathrm{M}}}{(1+a_{\mathrm{M}}+b_{\mathrm{M}}{)}^2}}\right)$$ 11

where $Q_{\mathrm{M}}=J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ is the M biouptake flux $J_{\mathrm{u},\mathrm{M}}$ normalized by the maximum biouptake flux, $J_{\mathrm{u},\mathrm{M}}^{\max }$ . Equation is similar in form to the original equation developed by Best and is corrected here to account for metal speciation and ML lability (cf. eqs and below). ,, This corrected equation aligns with van Leeuwen’s findings. In eqs and , the term $b_{\mathrm{M}}$ represents the dimensionless M bioconversion capacity. It is defined by $b_{\mathrm{M}}=J_{\mathrm{u},\mathrm{M}}^{\max }/({pJ}_{\mathrm{Diff},\mathrm{M}}^{\max })$ and it thus compares the maximum M biouptake flux, $J_{\mathrm{u},\mathrm{M}}^{\max }$ , with the maximum diffusional supply flux of M at the organism surface, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ . It also includes a dimensionless correction factor, $p(\ge 1)$ , that specifies the contribution of ML to the overall M diffusion flux. This parameter $p$ is defined according to ,

$$p=1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}\xi$$ 12

where ξ is the so-called ML lability index with values between 0 and unity, defined by ,,

$$\xi =\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}/(\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}+\sqrt{\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}}\sqrt{1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}})$$ 13

The limit $\xi \to 1$ corresponds to fully labile ML complexes, i.e., complexes whose formation/dissociation kinetics are sufficiently fast on the time scale of their diffusion toward the surface of the M-consuming organism for equilibrium to be maintained between M and ML at every spatial scale. The case of nonlabile ML complexes applies when M bioaccumulation is dictated by the diffusion of free M plus the kinetics of ML dissociation at the organism surface. In this latter situation, ξ and $p$ reduce to $\xi \to \sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}/(\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})$ and $p\to 1+\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}$ , respectively. , These extremes of fully labile and nonlabile ML complexes, with diffusion- and kinetic-controlled ML contributions to the bioupake flux, respectively, refer to so-called dynamic complexes, i.e., systems for which the characteristic lifetimes of M and ML (given by $1/(k_{\mathrm{a}}{c}_{\mathrm{L}}^{*})$ and $1/k_{\mathrm{d}}$ , respectively) are fast compared to the time scale ${\tau }_{\mathrm{SS}}=a^2/D_{\mathrm{M}}$ for establishment of steady-state diffusion of M from the medium to the M-consuming biological surface, i.e., $k_{\mathrm{a}}{c}_{\mathrm{L}}^{*}{\tau }_{\mathrm{SS}}\gg 1$ and $k_{\mathrm{d}}{\tau }_{\mathrm{SS}}\gg 1$ . ML systems in line with the conditions $k_{\mathrm{a}}{c}_{\mathrm{L}}^{*}{\tau }_{\mathrm{SS}}\ll 1$ and $k_{\mathrm{d}}{\tau }_{\mathrm{SS}}\ll 1$ are qualified as inert as they do not contribute, in any manner, to the M biouptake flux. For these systems, we have $p=1$ or, equivalently, $\xi =0$ . The reader is referred to previous reports for further details on the differentiations between inert, dynamic, labile and nonlabile ML complexes. , In particular, it can be shown that the conditions of nonlability and full lability of ML are met for $1\ll {\stackrel{-}{\kappa }}_{\mathrm{a}}\ll \epsilon {\stackrel{-}{K}}_{\mathrm{ML}}(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})$ and ${\stackrel{-}{\kappa }}_{\mathrm{a}}\gg \epsilon {\stackrel{-}{K}}_{\mathrm{ML}}(1+\epsilon {\stackrel{-}{K}}_{\mathrm{ML}})$ , respectively. ,, It is emphasized that ML complexes contributing to M biouptake (to an extent that depends on their lability) can be termed as bioavailable even though ML is not a bioreactive (internalizable) form in the bioaccumulation scenario of Figure A. As mentioned earlier, eq corresponds to the standard form of the Best equation, , corrected for the contribution of ML complexes to the flux. , It is easily verified that the normalized flux $Q_{\mathrm{M}}=J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ increases with increasing the bioaffinity of M for the internalization sites (i.e., decreasing $a_{\mathrm{M}}$ ) and/or with decreasing the bioconversion capacity of the organism (decreasing $b_{\mathrm{M}}$ ), with the limit $Q_{\mathrm{M}}\to 1$ reached for $a_{\mathrm{M}}\ll 1$ and $b_{\mathrm{M}}\ll 1$ . The ratio $a_{\mathrm{M}}/({\mathit{p}\mathit{b}}_{\mathrm{M}})$ identifies with the so-called Bosma number, denoted as ${\mathit{B}\mathit{n}}_{\mathrm{M}}$ , which compares the M diffusion conductance $D_{\mathrm{M}}/a$ (m s^–1) and the M internalization conductance $K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}$ (m s^–1), where we recall that $k_{\mathrm{int},\mathrm{M}}$ (s^–1) is the M internalization kinetic constant and $K_{\mathrm{H},\mathrm{M}}$ (m) is the Henry constant for the adsorption of M on the M internalization sites, ,,

$${\mathit{Bn}}_{\mathrm{M}}=a_{\mathrm{M}}/({\mathit{pb}}_{\mathrm{M}})=a^{-1}{D}_{\mathrm{M}}/(K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}})$$ 14

where the second equality is derived using the relationship $J_{\mathrm{u},\mathrm{M}}^{\max }=K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}{K}_{\mathrm{u},\mathrm{M}}$ . In the limit $1/{\mathit{B}\mathit{n}}_{\mathrm{M}}\ll 1$ , M bioaccumulation is kinetically limited by the internalization of M via translocation across the organism surface. ,,, This specific situation corresponds to that tackled by the commonly accepted and applied Biotic Ligand Model (BLM). In the other extreme $1/{\mathit{Bn}}_{\mathrm{M}}\gg 1$ , M internalization kinetics is so fast that the rate of M bioaccumulation becomes solely determined by metal diffusion transport in the external medium. For intermediate scenarios satisfying ${\mathit{Bn}}_{\mathrm{M}}\sim 1$ , both internalization kinetics and M diffusionsupported or not by ML diffusion depending on ML labilitydefine the overall kinetics of M accumulation by the organism.

In the Henry bioaccumulation regime achieved for sufficiently low bioaffinity of M, i.e., $a_{\mathrm{M}}=K_{\mathrm{u},\mathrm{M}}/c_{\mathrm{M}}^{*}\gg 1$ , the coverage of the internalization sites by M is well below unity and linearization of eq then leads to $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/a_{\mathrm{M}}$ or, equivalently, $J_{\mathrm{u},\mathrm{M}}=K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}{c}_{\mathrm{M}}^{(\mathrm{a})}$ . Under such a condition, it is easily verified that application of eq leads to the following expressions of M and ML surface concentrations

$$\{\begin{array}{lr}x=1/(1+({\mathit{pBn}}_{\mathrm{M}}{)}^{-1})& (\mathrm{a})\\ y=1+\xi (x-1)& (\mathrm{b})\end{array}$$ 15

which correctly compares with the results given elsewhere and with the expression of the ML lability, $\xi =1-y$ , given in refs , under the perfect sink boundary condition defined by $x=0$ . Within the Henry bioaccumulation regime, the normalized M bioaccumulation flux $Q_{\mathrm{M}}$ defined by eq then simplifies into

$$Q_{\mathrm{M}}=a_{\mathrm{M}}^{-1}/(1+({\mathit{pBn}}_{\mathrm{M}}{)}^{-1})$$ 16

which also agrees with previous findings.

3.3.2. Cases Where Both M and ML Are Bioreactive Metal Forms

3.3.2.1. Scenario Where M and ML Are Internalized by Different Facilitated Diffusion Pathways (Figure B)

Here, the M biouptake flux remains defined by eq , and the ML biouptake flux can be expressed by a Michaelis–Menten expression equivalent to that adopted for M, i.e.,

$$J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }=y/(a_{\mathrm{ML}}+y)$$ 17

where $J_{\mathrm{u},\mathrm{ML}}^{\max }$ is the maximum ML biouptake flux. In eq , $a_{\mathrm{ML}}$ is the dimensionless ML bioaffinity parameter defined by $a_{\mathrm{ML}}=K_{\mathrm{u},\mathrm{ML}}/c_{\mathrm{ML}}^{*}$ , where $K_{\mathrm{u},\mathrm{ML}}$ (mol m^–3) is the ML half-saturation constant corresponding to the reciprocal of the ML binding constant with ML-internalization sites at the cell surface. The bioaffinity parameter $a_{\mathrm{ML}}$ is analogous to $a_{\mathrm{M}}$ defined for free metal M in eq , Section . Applying the steady-state condition $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}=J_{i=\mathrm{M},\mathrm{ML}}$ (eq ) where $J_{i=\mathrm{M},\mathrm{ML}}$ , $J_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{ML}}$ are defined by eqs , , and , respectively, we show that M and ML surface concentrations x and y, respectively, are now defined by the following coupled quadratic equations in x and y

$$\{\begin{array}{lr}{A}_2{x}^2+A_1(y)x+A_0(y)=0& (\mathrm{a})\\ B_2{y}^2+B_1(x)y+B_0(x)=0& (\mathrm{b})\end{array}$$ 18

where $A_2$ and $B_2$ are constants, and $A_{0,1}(y)$ and $B_{0,1}(x)$ are functions of $y$ and $x$ , respectively, given by

$$\{\begin{array}{lr}\left[\begin{array}{c}{A}_0(y)\\ A_1(y)\\ A_2\end{array}\right]=\left[\begin{array}{c}-(1+{\gamma }_{\mathrm{M}}y)\\ 1+{\gamma }_{\mathrm{M}}+{\mathit{B}\mathit{n}}_{\mathrm{M}}^{-1}-a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}}y)\\ a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}})\end{array}\right]& (\mathrm{a})\\ \left[\begin{array}{c}{B}_0(x)\\ B_1(x)\\ B_2\end{array}\right]=\left[\begin{array}{c}-(1+{\gamma }_{\mathrm{ML}}x)\\ 1+{\gamma }_{\mathrm{ML}}+{\mathit{Bn}}_{\mathrm{ML}}^{-1}-a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}}x)\\ a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}})\end{array}\right]& (\mathrm{b})\end{array}$$ 19

In eq , we have introduced the Bosma number ${\mathit{B}\mathit{n}}_{\mathrm{ML}}$ which refers to the ML biouptake process and is the analogue of ${\mathit{B}\mathit{n}}_{\mathrm{M}}$ defined by eq for M. Similarly, ${\mathit{B}\mathit{n}}_{\mathrm{ML}}$ is given by the expression

$${\mathit{Bn}}_{\mathrm{ML}}=a_{\mathrm{ML}}/({\mathit{pb}}_{\mathrm{ML}})=a^{-1}{D}_{\mathrm{ML}}/(K_{\mathrm{H},\mathrm{ML}}{k}_{\mathrm{int},\mathrm{ML}})$$ 20

where $b_{\mathrm{ML}}=J_{\mathrm{u},\mathrm{ML}}^{\max }/({\mathit{pJ}}_{\mathrm{Diff},\mathrm{ML}}^{\max })$ is the ML bioconversion capacity of the organism and the counterpart of $b_{\mathrm{M}}$ for M, $k_{\mathrm{int},\mathrm{ML}}$ (s^–1) is the ML internalization kinetic constant, and $K_{\mathrm{H},\mathrm{ML}}$ (m) is the Henry constant for the ML adsorption at the ML-internalization sites. Equations and in x and y can be solved numerically (cf. Section ) and the bioaccumulation fluxes $J_{\mathrm{u},\mathrm{i}=\mathrm{M},\mathrm{ML}}$ are then subsequently evaluated according to eqs and .

Within the linear Henry regimes of M and ML bioaccumulation, applicable for $a_{i=\mathrm{M},\mathrm{ML}}=K_{\mathrm{u},i}/c_i^{*}\gg 1$ , the biouptake flux formulated according to eqs and can be linearized with respect to x and y, leading to $J_{\mathrm{u},\mathrm{M}}=K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}{c}_{\mathrm{M}}^{(\mathrm{a})}$ and $J_{\mathrm{u},\mathrm{ML}}=K_{\mathrm{H},\mathrm{ML}}{k}_{\mathrm{int},\mathrm{ML}}{c}_{\mathrm{ML}}^{(\mathrm{a})}$ . In these regimes, eqs and can be solved analytically, and after some developments we show that M and ML surface concentrations, $x$ and $y$ , are provided by the explicit expressions

$$\{\begin{array}{lr}x=1/\left\{1+\frac{1+[1+(\eta {\mathit{Bn}}_{\mathrm{M}}-1)\xi ]{\mathit{Bn}}_{\mathrm{ML}}^{-1}}{1+(1-\xi )({\mathit{pBn}}_{\mathrm{ML}}{)}^{-1}}({\mathit{pBn}}_{\mathrm{M}}{)}^{-1}\right\}& (\mathrm{a})\\ y=[1+\xi (x-1)]/[1+(1-\xi ){\mathit{Bn}}_{\mathrm{ML}}^{-1}]& (\mathrm{b})\end{array}$$ 21

where we have introduced the dimensionless quantity $\eta =J_{\mathrm{Diff},\mathrm{ML}}^{\max }/J_{\mathrm{Diff},\mathrm{M}}^{\max }=\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}$ and we have given the results as a function of the ML lability index ξ (eq ) using the relationships interconnecting ${\gamma }_{\mathrm{i}=\mathrm{M},\mathrm{ML}}$ and ξ

$$\{\begin{array}{lr}{\gamma }_{\mathrm{M}}=\eta \xi /(1-\xi )& (\mathrm{a})\\ {\gamma }_{\mathrm{ML}}=\xi /(1-\xi )& (\mathrm{b})\end{array}$$ 22

which are easily obtained by combining eqs and . In the linear Henry M and ML bioaccumulation regimes, the normalized steady-state M and ML biouptake fluxes, $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ , can be derived by substituting eq in eq . After algebraic arrangements, the results read as

$$\{\begin{array}{lr}{Q}_{\mathrm{M}}=a_{\mathrm{M}}^{-1}{\mathit{Bn}}_{\mathrm{M}}\{1+& \\ \frac{({\mathit{pBn}}_{\mathrm{M}}{)}^{-1}[\eta \xi +(\xi -p)\psi ]-1}{1+({\mathit{pBn}}_{\mathrm{M}}{)}^{-1}\{1+\psi [(1-\xi )(1+{\mathit{Bn}}_{\mathrm{M}}^{-1})+\eta \xi ]\}}\}& (\mathrm{a})\\ Q_{\mathrm{ML}}=a_{\mathrm{ML}}^{-1}{\mathit{Bn}}_{\mathrm{ML}}\{1+& \\ \frac{({\mathit{pBn}}_{\mathrm{M}}{)}^{-1}(\xi \psi -1)-1}{1+({\mathit{pBn}}_{\mathrm{M}}{)}^{-1}\{1+\psi [(1-\xi )(1+{\mathit{Bn}}_{\mathrm{M}}^{-1})+\eta \xi ]\}}\}& (\mathrm{b})\end{array}$$ 23

where we have defined the dimensionless ratio ψ which compares the relative significance of the ML and M biouptake pathways according to

$$\psi =({\mathit{Bn}}_{\mathrm{ML}}/{\mathit{Bn}}_{\mathrm{M}}{)}^{-1}=a_{\mathrm{M}}{b}_{\mathrm{ML}}/(a_{\mathrm{ML}}{b}_{\mathrm{M}})$$ 24

The situation where ML is not internalized is retrieved for $k_{\mathrm{int},\mathrm{ML}}\to 0$ , i.e., $1/{\mathit{Bn}}_{\mathrm{ML}}\to 0$ or, equivalently, $\psi \to 0$ . Under Henry bioaccumulation conditions, it is verified that eq satisfactorily reduces to $Q_{\mathrm{ML}}\to 0$ in the limit $\psi \to 0$ , while eqs and correctly simplify into eqs and , respectively. Most generally, it is verified that the solution of eqs and taken in the limit $1/{\mathit{Bn}}_{\mathrm{ML}}\to 0$ correctly reduces to eq . When expressed as a function of ${\gamma }_{\mathrm{M},\mathrm{ML}}$ , the defining expressions of $Q_{\mathrm{M},\mathrm{ML}}$ remain interchangeable if inverting the ‘M’ and ‘ML’ notations in all the symbols involved, regardless of the values of $a_{\mathrm{M},\mathrm{ML}}$ and thus of the applicability of the Henry bioaccumulation conditions. This property is due to the inherent symmetry of eqs , , , , and with respect to x and y, and ‘M’ and ML’ symbols. The result valid in the linear Henry regime reads according to eq S5 in SI–B and it is fully equivalent to eq .

We provide in Figures and the limits of the expressions defining $x$ , $y$ (eq ) and $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ (eq ), both valid in the Henry M and ML bioaccumulation regimes, across the entire spectrum of metal biouptake dynamics, i.e., when metal biouptake is controlled by M and/or ML internalization kinetics ( ${\mathit{Bn}}_{\mathrm{M}}^{-1}\ll 1$ and/or ${\mathit{Bn}}_{\mathrm{ML}}^{-1}\ll 1$ , respectively) or by M diffusion $({\mathit{Bn}}_{\mathrm{M}}^{-1}\gg 1)$ . In the latter situation, the limits of eqs and are derived for inert ML complexes, and for dynamic nonlabile and labile ML complexes with either internalization- or diffusion-controlled ML biouptake corresponding to ${\mathit{Bn}}_{\mathrm{ML}}^{-1}\ll 1$ and ${\mathit{Bn}}_{\mathrm{ML}}^{-1}\gg 1$ , respectively. We stress that ${\mathit{Bn}}_{\mathrm{M},\mathrm{ML}}^{-1}$ (eqs and ) reflect not only the coupling between M/ML diffusion and complexation reaction kinetics in the medium, but also some of the key properties of the metal-consuming biological surface via the involved metal bioaffinity and bioconversion parameters ( $a_{i=\mathrm{M},\mathrm{ML}}$ and $b_{i=\mathrm{M},\mathrm{ML}}$ , respectively).

3.

Limits of eq defining the M and ML concentrations at the surface of the metal-consuming organism in the Henry biouptake regime ( $x$ and $y$ , respectively) over the entire spectrum of M and ML uptake dynamic cases. Results apply here to the scenario where M and ML uptakes proceed according to different facilitated diffusion pathways (Figure B). The conditions for application of the standard BLM limit are specified. See text for details.

4.

Limits of eq defining the normalized M and ML biouptake fluxes at the surface of the metal consuming organism ( $Q_{\mathrm{M}}=J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }$ and $Q_{\mathrm{ML}}=J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }$ , respectively) in the Henry biouptake regime over the entire spectrum of M and ML uptake dynamic cases. Results apply here to the scenario where M and ML uptakes proceed according to different facilitated diffusion pathways (Figure B). The conditions for application of the standard BLM limit are specified. See text for details.

3.3.2.2. Case Where M and ML Are Internalized by Facilitated and Passive Diffusions, Respectively (Figure C)

In this scenario schemed in Figure C, the M uptake flux is still defined by eq and the ML uptake flux can be expressed in the following operational form

$$J_{\mathrm{u},\mathrm{ML}}=D_{\mathrm{o},\mathrm{ML}}{c}_{\mathrm{ML}}^{*}(y-y_{\mathrm{o}})/{\delta }_{\mathrm{o}}$$ 25

where $D_{\mathrm{o},\mathrm{ML}}$ stands for the ML diffusion coefficient across the biological barrier of thickness ${\delta }_{\mathrm{o}}$ , and $(y-y_{\mathrm{o}})/{\delta }_{\mathrm{o}}$ represents the ML concentration gradient across that barrier. For the sake of simplicity, we adopt here a sink boundary at the internal side of this barrier, i.e., $y_{\mathrm{o}}=0$ , which avoids the necessity to invoke the way ML species, once internalized, are handled by the organism via e.g., detoxification or storage strategies. Applying the steady-state condition $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}=J_{i=\mathrm{M},\mathrm{ML}}$ (eq ) where $J_{i=\mathrm{M},\mathrm{ML}}$ are defined by eq , $J_{\mathrm{u},\mathrm{M}}$ by eq and $J_{\mathrm{u},\mathrm{ML}}$ by eq , we show that M and ML surface concentrations x and y, respectively, are now defined by the algebraic equations

$$\{\begin{array}{lr}{A}_2{x}^2+A_{1}x+A_0=0& (\mathrm{a})\\ y=\frac{1+{\gamma }_{\mathrm{ML}}x}{1+{\beta }_{\mathrm{ML}}+{\gamma }_{\mathrm{ML}}}& (\mathrm{b})\end{array}$$ 26

where $A_{0,1,2}$ are all constants given here by

$$\left[\begin{array}{c}{A}_0\\ A_1\\ A_2\end{array}\right]=\left[\begin{array}{c}-(1+{\gamma }_{\mathrm{M}}+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})\\ (1-a_{\mathrm{M}}^{-1}+{\mathit{Bn}}_{\mathrm{M}}^{-1})(1+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})+{\gamma }_{\mathrm{M}}(1-a_{\mathrm{M}}^{-1}+{\beta }_{\mathrm{ML}})\\ a_{\mathrm{M}}^{-1}[{\gamma }_{\mathrm{ML}}+(1+{\gamma }_{\mathrm{M}})(1+{\beta }_{\mathrm{ML}})]\end{array}\right]$$ 27

The dimensionless parameter ${\beta }_{\mathrm{ML}}$ in eq is defined by the ratio between the ML diffusion conductance across the biological barrier and the ML diffusion conductance from bulk solution to the outer side of the biological barrier, that is

$${\beta }_{\mathrm{ML}}={\delta }_{\mathrm{o}}^{-1}{D}_{\mathrm{o},\mathrm{ML}}/(a^{-1}{D}_{\mathrm{ML}})$$ 28

After some algebra, the solutions $x$ and $y$ to eqs and can be written in the closed form expressions

$$\{\begin{array}{lr}x=\frac{(\lambda +{\beta }_{\mathrm{ML}})(1-a_{\mathrm{M}}^{-1})+{\mathit{Bn}}_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})+{\gamma }_{\mathrm{M}}{\beta }_{\mathrm{ML}}}{2a_{\mathrm{M}}^{-1}[\lambda +(1+{\gamma }_{\mathrm{M}}){\beta }_{\mathrm{ML}}]}\{-1+\frac{|(\lambda +{\beta }_{\mathrm{ML}})(1-a_{\mathrm{M}}^{-1})+{\mathit{Bn}}_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})+{\gamma }_M{\beta }_{\mathrm{ML}}|}{(\lambda +{\beta }_{\mathrm{ML}})(1-a_{\mathrm{M}}^{-1})+{\mathit{Bn}}_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})+{\gamma }_{\mathrm{M}}{\beta }_{\mathrm{ML}}}\times & \\ \sqrt{1+4a_{\mathrm{M}}^{-1}\frac{(\lambda +{\beta }_{\mathrm{ML}})[\lambda +{\beta }_{\mathrm{ML}}(1+{\gamma }_{\mathrm{M}})]}{[(\lambda +{\beta }_{\mathrm{ML}})(1-a_{\mathrm{M}}^{-1})+{\mathit{Bn}}_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{ML}}+{\beta }_{\mathrm{ML}})+{\gamma }_{\mathrm{M}}{\beta }_{\mathrm{ML}}{]}^2}}\}& (\mathrm{a})\\ y=\frac{1+{\gamma }_{\mathrm{ML}}x}{1+{\beta }_{\mathrm{ML}}+{\gamma }_{\mathrm{ML}}}& (\mathrm{b})\end{array}$$ 29

where λ is defined by $\lambda =1+{\gamma }_{\mathrm{M}}+{\gamma }_{\mathrm{ML}}$ . The normalized fluxes $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ then directly follow upon substitution of x and y given by eq into eq (or eq ) and eq . In the linear Henry regime for M bioaccumulation applicable for $a_{\mathrm{M}}=K_{\mathrm{u},\mathrm{M}}/c_{\mathrm{M}}^{*}\gg 1$ , it is straightforward to verify that the corresponding expressions of $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ ( $x$ and $y$ ) are exactly those given by eqs (eqs , respectively) after replacing therein the reciprocal ML Bosma number, ${\mathit{Bn}}_{\mathrm{ML}}^{-1}$ , by ${\beta }_{\mathrm{ML}}$ . In turn, under Henry bioaccumulation conditions, the biouptake scenarios in Figures B become formally equivalent, with the ML internalization conductance term K _H,ML k _int,ML (m s^–1) in Figure B playing the role of the ML barrier diffusion conductance D _o,MLδ_o ^–1 (m s^–1) in Figure C.

3.3.2.3. Scenario Where M and ML Are Internalized by a Single Facilitated Diffusion Pathway (Figure D)

3.3.2.3.1. Competitive Inhibition. In this scenario schemed in Figures D and A, M and ML species are bioaccumulated via a shared facilitated diffusion pathway where they compete for the same type of active sites of the transport protein in the cell plasma membrane. In turn, the fluxes $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ are defined by Michaelis–Menten expressions corrected for such a M/ML competitive inhibition according to

$$\{\begin{array}{lr}{J}_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/[a_{\mathrm{M}}(1+y/a_{\mathrm{ML}})+x]& (\mathrm{a})\\ J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }=y/[a_{\mathrm{ML}}(1+x/a_{\mathrm{M}})+y]& (\mathrm{b})\end{array}$$ 30

Following the strategy adopted for the bioaccumulation scenarios considered previously, the M and ML surface concentrations $x$ and $y$ , respectively, are derived from the steady-state condition $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}=J_{i=\mathrm{M},\mathrm{ML}}$ (eq ), where $J_{i=\mathrm{M},\mathrm{ML}}$ and $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ are given by eqs and , respectively. After some algebra, we find that $x$ and $y$ are governed by the following new coupled quadratic equations

$$\{\begin{array}{lr}{A}_2{x}^2+A_1(y)x+A_0(y^2,y)=0& (\mathrm{a})\\ B_2{y}^2+B_1(x)y+B_0(x^2,x)=0& (\mathrm{b})\end{array}$$ 31

where $A_0(y^2,y)$ , $B_0(x^2,x)$ , $A_1(y)$ and $B_1(x)$ are now functions of $y^2$ , $y$ , $x^2$ , and/or $x$ , and $A_2$ and $B_2$ are constants defined by

$$\{\begin{array}{lr}\left[\begin{array}{c}{A}_0(y^2,y)\\ A_1(y)\\ A_2\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{ML}}^{-1}y)(1+{\gamma }_{\mathrm{M}}y)\\ (1+a_{\mathrm{ML}}^{-1}y)(1+{\gamma }_{\mathrm{M}})+{\mathit{Bn}}_{\mathrm{M}}^{-1}-a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}}y)\\ a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}})\end{array}\right]& (\mathrm{a})\\ \left[\begin{array}{c}{B}_0(x^2,x)\\ B_1(x)\\ B_2\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{M}}^{-1}x)(1+{\gamma }_{\mathrm{ML}}x)\\ (1+a_{\mathrm{M}}^{-1}x)(1+{\gamma }_{\mathrm{ML}})+{\mathit{Bn}}_{\mathrm{ML}}^{-1}-a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}}x)\\ a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}})\end{array}\right]& (\mathrm{b})\end{array}$$ 32

where the parameters $a_{\mathrm{M},\mathrm{ML}}$ , ${\gamma }_{\mathrm{M},\mathrm{ML}}$ and ${\mathit{Bn}}_{\mathrm{M},\mathrm{ML}}$ have all been defined in the preceding developments. Equations and can be solved numerically (Section ), and the fluxes $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ are subsequently obtained by substituting the solutions found for $x$ and $y$ in eq (or, equivalently, in eq ). In the linear M and ML Henry bioaccumulation regimes reached for sufficiently low values of M and ML bioaffinities (i.e., $a_{i=\mathrm{M},\mathrm{ML}}\gg 1$ ), eq simplifies into the linearized expressions $J_{\mathrm{u},\mathrm{M}}=K_{\mathrm{H},\mathrm{M}}{k}_{\mathrm{int},\mathrm{M}}{c}_{\mathrm{M}}^{(\mathrm{a})}$ and $J_{\mathrm{u},\mathrm{ML}}=K_{\mathrm{H},\mathrm{ML}}{k}_{\mathrm{int},\mathrm{ML}}{c}_{\mathrm{ML}}^{(\mathrm{a})}$ which do not involve any term related to M/ML competitive inhibition. Accordingly, M/ML competitive inhibition becomes insignificant in the M and ML Henry bioaccumulation regimes, and the scenario in Figures D and A then becomes strictly and formally identical to the one sketched in Figure B (cf. Section ). In turn, in the limits $a_{i=\mathrm{M},\mathrm{ML}}\gg 1$ , the defining expressions for the M and ML surface concentrations and for the normalized M and ML biouptake fluxes for the competitive inhibition scenario reduce to eqs and , respectively.

3.3.2.3.2. Mixed Competitive Inhibition. In this situation illustrated by Figures D and D, M and ML are translocated across the biological barrier via a single facilitated diffusion pathway, and both species compete not only for the active sites of the transport protein but also for nonactive (inhibitory) sites. More specifically, while the binding of M and ML to the active transporter sites refers to the competitive inhibition case treated in the previous section, the mixed inhibition scenario also involves the binding of M and ML to the ML-transporter and M-transporter complexes, respectively, via inhibitory sites. Accordingly, in addition to the M and ML bioaffinity parameters $a_{i=\mathrm{M},\mathrm{ML}}=K_{\mathrm{u},i}/c_i^{*}$ , we introduce here the dimensionless M and ML bioaffinity parameters for the nonactive sites of the ML-transporter and M-transporter complexes, respectively, denoted as $a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}=K_i^{(\mathrm{I})}/c_{\mathrm{u},i}^{*}$ . The constants $K_{i=\mathrm{M},\mathrm{ML}}^{(\mathrm{I})}$ (in mol m^–3) correspond to the reciprocals of the stability constants describing the binding of M and ML with the nonactive (inhibitory) sites of the protein transporters. The biouptake fluxes $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ can be then written as a function of the surface concentrations $x$ and $y$ according to the following forms

$$\{\begin{array}{lr}{J}_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/[a_{\mathrm{M}}(1+y/a_{\mathrm{ML}})+x(1+y/a_{(\mathrm{I}),\mathrm{ML}})]& (\mathrm{a})\\ J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }=y/[a_{\mathrm{ML}}(1+x/a_{\mathrm{M}})+y(1+x/a_{(\mathrm{I}),\mathrm{M}})]& (\mathrm{b})\end{array}$$ 33

Application of the steady-state condition $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}=J_{i=\mathrm{M},\mathrm{ML}}$ (eq ) where $J_{i=\mathrm{M},\mathrm{ML}}$ and $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ are given by eqs and , respectively, leads to the following set of coupled polynomial equations in $x$ and $y$

$$\{\begin{array}{lr}{A}_2(y)x^2+A_1(y^2,y)x+A_0(y^2,y)=0& (\mathrm{a})\\ B_2(x)y^2+B_1(x^2,x)y+B_0(x^2,x)=0& (\mathrm{b})\end{array}$$ 34

where the functions $A_0(y^2,y)$ , $B_0(x^2,x)$ , $A_1(y^2,y)$ and $B_1(x^2,x)$ , $A_2(y)$ , and $B_2(x)$ are now defined by

$$\{\begin{array}{lr}\left[\begin{array}{c}{A}_0(y^2,y)\\ A_1(y^2,y)\\ A_2(y)\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{ML}}^{-1}y)(1+{\gamma }_{\mathrm{M}}y)\\ (1+a_{\mathrm{ML}}^{-1}y)(1+{\gamma }_{\mathrm{M}})+{\mathit{Bn}}_{\mathrm{M}}^{-1}-a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}}y)(1+a_{(\mathrm{I}),\mathrm{ML}}^{-1}y)\\ a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}})(1+a_{(\mathrm{I}),\mathrm{ML}}^{-1}y)\end{array}\right]& (\mathrm{a})\\ \left[\begin{array}{c}{B}_0(x^2,x)\\ B_1(x^2,x)\\ B_2(x)\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{M}}^{-1}x)(1+{\gamma }_{\mathrm{ML}}x)\\ (1+a_{\mathrm{M}}^{-1}x)(1+{\gamma }_{\mathrm{ML}})+{\mathit{Bn}}_{\mathrm{ML}}^{-1}-a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}}x)(1+a_{(\mathrm{I}),\mathrm{M}}^{-1}x)\\ a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}})(1+a_{(\mathrm{I}),\mathrm{M}}^{-1}x)\end{array}\right]& (\mathrm{b})\end{array}$$ 35

Equations and must be solved numerically to obtain $x$ and $y$ and subsequently evaluate the M and ML uptake fluxes $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ via use of eq (or eq ). In the limit where M and ML bioaffinities to the inhibitory sites are so low that $a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}^{-1}\to 0$ , it is easily verified that eqs – for the mixed inhibition scenario correctly reduce to eqs – applicable to the competitive inhibition case. Lastly, in the extreme where $a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}^{-1}\to 0$ and $a_{i=\mathrm{M},\mathrm{ML}}\gg 1$ , it is also straightforward to verify from eq that the biouptake scenario considered here becomes formally similar to that where M and ML are internalized by different facilitated diffusion pathways within the linear Henry bioaccumulation regimes (Figure B, Section ): expressions of x, y and $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ then reduce to eqs and in these regimes, respectively.

3.3.2.3.3. Noncompetitive Inhibition. The noncompetitive inhibition situation (Figures D and B) may be viewed as a particular case of the mixed competitive inhibition scenario where M and ML bioaffinity parameters for the active and inhibitory sites are identical, i.e., $a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}=a_{i=\mathrm{M},\mathrm{ML}}$ . In this limit, eq is still valid, and eqs and simplify into

$$\{\begin{array}{lr}{J}_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/[(x+a_{\mathrm{M}})(1+y/a_{\mathrm{ML}})]& (\mathrm{a})\\ J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }=y/[(y+a_{\mathrm{ML}})(1+x/a_{\mathrm{M}})]& (\mathrm{b})\end{array}$$ 36

and

$$\{\begin{array}{lr}\left[\begin{array}{c}{A}_0(y^2,y)\\ A_1(y^2,y)\\ A_2(y)\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{ML}}^{-1}y)(1+{\gamma }_{\mathrm{M}}y)\\ (1+a_{\mathrm{ML}}^{-1}y)[1+{\gamma }_{\mathrm{M}}-a_M^{-1}(1+{\gamma }_{\mathrm{M}}y)]+{\mathit{Bn}}_{\mathrm{M}}^{-1}\\ a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}})(1+a_{\mathrm{ML}}^{-1}y)\end{array}\right]& (\mathrm{a})\\ \left[\begin{array}{c}{B}_0(x^2,x)\\ B_1(x^2,x)\\ B_2(x)\end{array}\right]=\left[\begin{array}{c}-(1+a_{\mathrm{M}}^{-1}x)(1+{\gamma }_{\mathrm{ML}}x)\\ (1+a_{\mathrm{M}}^{-1}x)[1+{\gamma }_{\mathrm{ML}}-a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}}x)]+{\mathit{Bn}}_{\mathrm{ML}}^{-1}\\ a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}})(1+a_{\mathrm{M}}^{-1}x)\end{array}\right]& (\mathrm{b})\end{array}$$ 37

respectively. The M and ML uptake fluxes $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ are then obtained from eq where $x$ and $y$ are solutions of eq with $A_0(y^2,y)$ , $B_0(x^2,x)$ , $A_1(y^2,y)$ , $B_1(x^2,x)$ , $A_2(y)$ , and $B_2(x)$ being defined by eq .

3.3.2.3.4. Uncompetitive Inhibition. The uncompetitive inhibition situation (Figures D and C) corresponds to a mixed competitive inhibition (eqs –, Figures D and D) in the limit where M and ML bioaffinities for the nonactive inhibitory sites are much larger than those for their respective active sites, i.e., $a_{(\mathrm{I}),\mathrm{i}=\mathrm{M},\mathrm{ML}}^{-1}\gg a_{\mathrm{i}=\mathrm{M},\mathrm{ML}}^{-1}$ . In such a limit, it is straightforward to verify that eq defining M and ML biouptake fluxes become

$$\{\begin{array}{lr}{J}_{\mathrm{u},\mathrm{M}}/J_{\mathrm{u},\mathrm{M}}^{\max }=x/[a_{\mathrm{M}}+x(1+y/a_{(\mathrm{I}),\mathrm{ML}})]& (\mathrm{a})\\ J_{\mathrm{u},\mathrm{ML}}/J_{\mathrm{u},\mathrm{ML}}^{\max }=y/[a_{\mathrm{ML}}+y(1+x/a_{(\mathrm{I}),\mathrm{M}})]& (\mathrm{b})\end{array}$$ 38

and that the M and ML surface concentrations x and y are now determined by the coupled polynomial equations

$$\{\begin{array}{lr}{A}_2(y)x^2+A_1(y^2,y)x+A_0(y)=0& (\mathrm{a})\\ B_2(x)y^2+B_1(x^2,x)y+B_0(x)=0& (\mathrm{b})\end{array}$$ 39

where the functions $A_0(y)$ , $B_0(x)$ , $A_1(y^2,y)$ , $B_1(x^2,x)$ , $A_2(y)$ , and $B_2(x)$ are here provided by

$$\{\begin{array}{lr}\left[\begin{array}{c}{A}_0(y)\\ A_1(y^2,y)\\ A_2(y)\end{array}\right]=\left[\begin{array}{c}-(1+{\gamma }_{\mathrm{M}}y)\\ 1+{\gamma }_{\mathrm{M}}+{\mathit{Bn}}_{\mathrm{M}}^{-1}-a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}}y)(1+a_{(\mathrm{I}),\mathrm{ML}}^{-1}y)\\ a_{\mathrm{M}}^{-1}(1+{\gamma }_{\mathrm{M}})(1+a_{(\mathrm{I}),\mathrm{ML}}^{-1}y)\end{array}\right]& (\mathrm{a})\\ \left[\begin{array}{c}{B}_0(x)\\ B_1(x^2,x)\\ B_2(x)\end{array}\right]=\left[\begin{array}{c}-(1+{\gamma }_{\mathrm{ML}}x)\\ 1+{\gamma }_{\mathrm{ML}}+{\mathit{Bn}}_{\mathrm{ML}}^{-1}-a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}}x)(1+a_{(\mathrm{I}),\mathrm{M}}^{-1}x)\\ a_{\mathrm{ML}}^{-1}(1+{\gamma }_{\mathrm{ML}})(1+a_{(\mathrm{I}),\mathrm{M}}^{-1}x)\end{array}\right]& (\mathrm{b})\end{array}$$ 40

Within the framework of the bioaccumulation scenario considered here, the dimensionless M and ML uptake fluxes $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ are thus obtained from eq where $x$ and $y$ are solutions of eq and the functions $A_0(y)$ , $B_0(x)$ , $A_1(y^2,y)$ , $B_1(x^2,x)$ , $A_2(y)$ , and $B_2(x)$ are defined by eq .

3.4. Summary, Validation of the Numerical and Analytical Surface Concentration and Flux Computations

As a summary, Table reports the expressions we derived for the M and ML surface concentrations, $x$ and $y$ , and the normalized M and ML biouptake fluxes, $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ , applicable to each metal bioaccumulation scenario shown in Figures and . In all cases, evaluating these fluxes requires solving a set of two polynomial equations in $x$ and $y$ that involve the quantities $A_{0,1,2}$ and $B_{0,1,2}$ we defined earlier for each bioaccumulation situation. Closed-form solutions for $x$ , $y$ , and $Q_{i=\mathrm{M},\mathrm{ML}}$ can be rigorously obtained only for the bioaccumulation scenarios depicted in Figure A,C. For all other cases, explicit approximate solutions are obtainable under the Henry bioaccumulation conditions, where Q _i=M,ML depend linearly on $x$ and $y$ . For all situations addressed in the Results and Discussion section, the numerical evaluation of $x$ and $y$ at fixed values of the model parameters specifying M/ML speciation, diffusion and biouptake (cf. details in Section ), was performed by solving the relevant set of coupled polynomial equations within the Mathcad Prime 8.0 calculus environment and use of the Levenberg–Marquardt solving algorithm. The M and ML biouptake fluxes were then computed by substituting the obtained $x$ and $y$ values into the relevant expressions of Q _i=M,ML. We systematically verified that the numerical outcomes properly compared to predictions from rigorous or approximate analytical solutions within the range of applicable values of model parameters. Finally, we stress that the validity of all analytical developments detailed in Section was systematically verified using the symbolic calculus capabilities of Mathcad Prime 8.0. All codes are available upon request.

1. Theoretical Expressions Defining the Dimensionless M and ML Surface Concentrations (x and y, Respectively) and Dimensionless Biouptake Fluxes (Q _i=M,ML) for the Metal Bioaccumulation Scenarios Pictured in Figures and (Specified) .

^aThe derivations of the results collated in this table are detailed in Section and the limits reached within the linear Henry bioaccumulation regimes $\left(a_{i=\mathrm{M},\mathrm{ML}}\gg 1\right)$ and in the extreme where M/ML binding to inhibitory sites is insignificant $(a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}\gg 1)$ are further specified.

4. Results and Discussion

We present and discuss here illustrative numerical simulations that detail the dependence of metal bioaccumulation fluxes $Q_{i=\mathrm{M},\mathrm{ML}}$ on the relevant biophysiochemical parameters involved in the theory developed in Section . Specifically, these parameters describe: (i) the interplay between metal speciation and diffusive transport ( ${\stackrel{-}{K}}_{\mathrm{ML}}$ , ξ, η), i.e., the chemodynamics and lability of ML complex, (ii) the bioaffinities of M and ML to the internalization and inhibitory sites ( $a_{i=\mathrm{M},\mathrm{ML}}^{-1}$ , $a_{(\mathrm{I}),i=\mathrm{M},\mathrm{ML}}^{-1}$ ), and (iii) the kinetics of M/ML bioaccumulation, as subsumed in the Bosma numbers ${\mathit{Bn}}_{i=\mathrm{M},\mathrm{ML}}$ and their ratio $\psi ={({\mathit{Bn}}_{\mathrm{ML}}/{\mathit{Bn}}_{\mathrm{M}})}^{-1}$ . It is verified that specification of this set of dimensionless parameters (i)-(iii) is sufficient to fully describe the biouptake scenarios considered in Section and schematically presented in Figures and . Given that experimental reports classically represent metal bioaccumulation data as a function of free metal concentration, we provide below the evolution of the metal bioputake flux as a function of the dimensionless free metal concentration in the form of $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}$ , which is the reciprocal of the metal bioaffinity parameter $1/a_{\mathrm{M}}$ . Furthermore, unless e.g., specific isotopic labels are employed, it is virtually impossible to differentiate between internalized free and complexed metal forms from bioaccumulation experiments. Accordingly, we report below the normalized total metal biouptake flux, $Q_{\mathrm{T}}$ , defined by

$$Q_{\mathrm{T}}=(J_{\mathrm{u},\mathrm{M}}+J_{\mathrm{u},\mathrm{ML}})/J_{\mathrm{u},\mathrm{M}}^{\max }=Q_{\mathrm{M}}+\eta (b_{\mathrm{ML}}/b_{\mathrm{M}})Q_{\mathrm{ML}}=Q_{\mathrm{M}}+\eta \psi (a_{\mathrm{ML}}/a_{\mathrm{M}})Q_{\mathrm{ML}}$$ 41

where the second and third equalities are established using eq and the expressions $Q_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}/J_{\mathrm{u},i}^{\max }$ , $\eta =\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}$ , and $b_{i=\mathrm{M},\mathrm{ML}}=J_{\mathrm{u},i}^{\max }/({\mathit{pJ}}_{\mathrm{Diff},i}^{\max })$ (cf. Sections and ).

4.1. Illustrative Computational Results

4.1.1. Bioaccumulation Scenario Where M and ML Are Each Internalized by Their Own Distinct Facilitated Diffusion Pathway and There Is No Competition (Figure B)

Figure shows the total metal biouptake flux, $Q_{\mathrm{T}}$ , as a function of M affinity for the active sites ( $1/a_{\mathrm{M}}$ ) for various Bosma number ratios, $\psi ={({\mathit{Bn}}_{\mathrm{ML}}/{\mathit{Bn}}_{\mathrm{M}})}^{-1}$ . As detailed in the theory section, for $\psi \to 0$ (bold black curve featuring the plateau value at $1/a_{\mathrm{M}}\gg 1$ ), the intact ML complex is not taken up and $Q_{\mathrm{T}}$ corresponds solely to $J_{\mathrm{u},\mathrm{M}}$ (which includes any M dissociated from ML as determined by the degree of ML lability, ξ). For $\psi \to 0$ , the plateau in $Q_{\mathrm{T}}$ obtained at large values of $1/a_{\mathrm{M}}$ corresponds to saturation of the active biouptake sites for M, at which point the normalized flux approaches unity (as prescribed by the Michaelis–Menten equation). At the other extreme, for sufficiently large values of ψ, uptake of ML is diffusion-limited $\left(1/{\mathit{Bn}}_{\mathrm{ML}}\gg 1\right)$ and $Q_{\mathrm{T}}$ then approaches the maximum diffusive flux for ML, $J_{\mathrm{Diff},\mathrm{ML}}^{\max }$ (blue dashed lines) under the conditions of Figure where $\eta \gg 1$ . Figure A–D illustrates the interplay between M bioaccumulation kinetics, subsumed in its Bosma number ( ${\mathit{Bn}}_{\mathrm{M}}$ ), and the lability of ML (quantified by ξ). In Figure A, ML is inert $\left(\xi \to 0\right)$ and thus ML does not contribute to M biouptake (and vice versa). In addition, we have ${\mathit{Bn}}_{\mathrm{M}}^{-1}\ll 1$ , thus there is no diffusion limitation for M biouptake, i.e., the internalization step is rate-limiting and there is no concentration gradient in M at the biointerface ( $x=c_{\mathrm{M}}^{(\mathrm{a})}/{\mathrm{c}}_{\mathrm{M}}^{*}=1$ , independent of ψ, red line in Figure A). Accordingly, when ML is not taken up $\left(\psi \to 0\right)$ , $Q_{\mathrm{T}}$ remains well below the diffusive flux limit for M, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ (red dashed line in Figure A). As ψ increases, at a given $1/a_{\mathrm{M}}$ the contribution to $Q_{\mathrm{T}}$ from uptake of ML increases until the diffusive limit for ML is attained (blue dashed line in Figure A). At high values of $1/a_{\mathrm{M}}$ , the increase in $Q_{\mathrm{T}}$ above the saturation plateau with increasing ψ corresponds solely to only ML uptake. The surface concentration of ML $\left(y=c_{\mathrm{ML}}^{\left(\mathrm{a}\right)}/{\mathrm{c}}_{\mathrm{ML}}^{*}\right)$ evolves accordingly: as ψ increases, the uptake of ML increases and the surface concentration of ML decreases and ultimately approaches zero for ψ → ∞ (black lines in Figure A).

5.

Dependence of the (dimensionless) total metal biouptake flux $Q_{\mathrm{T}}$ ­(eq ) on the normalized free metal concentration for the situation where free M and the ML complex are internalized via distinct facilitated diffusion pathways (eqs , –, Figure B). The normalization factor is the reciprocal M affinity constant for the M-internalization sites, K _u,M. The resulting ratio $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}$ identifies with the reciprocal of the M bioaffinity parameter, 1/a _M. Results are further given for various values of the reciprocal M and ML Bosma numbers, Bn _M ^–1 and Bn _ML ^–1 or, equivalently, Bn _M ^–1 and ψ = (Bn _ML/Bn _M)⁻¹ (eq ), as specified. (A) M biouptake is rate-limited by the internalization step. (B–D) M biouptake is rate-limited by diffusion, and ML is (B) inert, (C) nonlabile, and (D) fully labile. Model parameters other than those provided in the figure: 1/a _ML = 1, η = 15. The red and blue dotted lines represent the maximum diffusion flux of free M and the maximum diffusion flux of free M plus ML complex, respectively, both normalized by J _u,M .

6.

Dependence of the (dimensionless) M and ML surface concentrations $x=c_{\mathrm{M}}^{(\mathrm{a})}/c_{\mathrm{M}}^{*}$ (red) and $y=c_{\mathrm{ML}}^{(\mathrm{a})}/c_{\mathrm{ML}}^{*}$ ­(black), respectively, on the normalized free metal concentration $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}=1/a_{\mathrm{M}}$ (which is the reciprocal of the M bioaffinity parameter) for the situation where free M and the ML complex are internalized via distinct facilitated diffusion pathways (eqs , –, Figure B). Results are presented under the conditions of Figure for various values of the reciprocal M and ML Bosma numbers, Bn _M ^–1 and Bn _ML ^–1 or, equivalently, Bn _M ^–1 and ψ = (Bn _ML/Bn _M)⁻¹ (eq ), as specified. (A) M biouptake is rate-limited by the internalization step. (B–D) M biouptake is rate-limited by diffusion, and ML is (B) inert, (C) nonlabile, and (D) fully labile. Model parameters other than those provided in the figure: as in Figure . In (D), the red and black curves (x and y, respectively) are superimposed.

In Figure B, ML is still inert $\left(\xi \to 0\right)$ and thus does not contribute to M biouptake. Also, we have ${\mathit{Bn}}_{\mathrm{M}}^{-1}=10$ , thus M biouptake is now limited by M diffusion from the bulk solution and a concentration gradient in M develops at the biointerface ( $x=c_{\mathrm{M}}^{(a)}/{\mathrm{c}}_{\mathrm{M}}^{*}$ is independent of ψ due to inert ML, cf. red line in Figure B). Accordingly, when ML is not taken up $\left(\psi \to 0\right)$ , $Q_{\mathrm{T}}$ corresponds here to the maximum diffusive flux for M, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ (red dashed line) until saturation of the active sites is reached (at sufficiently high $1/a_{\mathrm{M}}$ ) (Figure B). In the Henry regime, the surface concentration of M is $x\approx {\mathit{Bn}}_{\mathrm{M}}$ (Figure ), it then increases as $1/a_{\mathrm{M}}$ increases, eventually reaching a plateau of unity corresponding to saturation of the active M sites, at which point internalization has become rate-limiting (Figure B). The evolutions of $Q_{\mathrm{T}}$ and $y=c_{\mathrm{ML}}^{(a)}/{\mathrm{c}}_{\mathrm{ML}}^{*}$ as a function of ψ (Figure B and B) follow the same trend as described for Figures A and A.

In Figure C, ML is nonlabile $\left(\xi =0.1\right)$ and thus dissociation kinetics determine the extent to which ML can contribute to M biouptake, and ${\mathit{Bn}}_{\mathrm{M}}^{-1}=10$ , thus M biouptake is diffusion-limited by M diffusion from the bulk solution. Now, when ML is not taken up $\left(\psi \to 0\right)$ , so long as the M active sites are not saturated (i.e., in the Henry regime at sufficiently low $1/a_{\mathrm{M}}$ ), $Q_{\mathrm{T}}$ exceeds the maximum diffusive flux for M, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ (red dashed line), which reflects the contribution from M dissociated from ML. Since ML is nonlabile, there is some degree of coupling between the surface concentrations of M and ML (Figure C). For example, when $\psi \to 0$ there is no uptake of ML, but as the bioaffinity for M increases (i.e., as $1/a_{\mathrm{M}}$ increases), the surface concentration of M increases (red lines in Figure C) and thus so does the surface concentration of ML (black lines in Figure C). For these conditions ( ${\mathit{Bn}}_{\mathrm{M}}^{-1}\gg 1$ , nonlabile ML), in the Henry regime, the ψ dependence of $x$ is described by $x\approx [1+\sqrt{{\stackrel{-}{\kappa }}_{\mathrm{a}}}{\mathit{Bn}}_{\mathrm{ML}}]{\mathit{Bn}}_{\mathrm{M}}$ when ${\mathit{Bn}}_{\mathrm{ML}}^{-1}\gg 1$ (Figure ). At the other extreme, when there is diffusion-limited uptake of ML (ψ → ∞) and the surface concentration of ML approaches zero, the plateau in the surface concentration of M at large $1/a_{\mathrm{M}}$ is less than unity because the local equilibrium is shifted toward the formation of ML (Figure C).

In Figure D, ML is now labile $\left(\xi \to 1\right)$ and thus fully contributes to M biouptake via coupled M-ML diffusion, and the condition ${\mathit{Bn}}_{\mathrm{M}}^{-1}=10$ is here maintained so that M biouptake is still diffusion-limited. As compared to the nonlabile case (Figure C), below saturation of the M active sites, $Q_{\mathrm{T}}$ for $\psi \to 0$ (no ML uptake) exhibits greater exceedance of the maximum diffusive flux for M, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ (red dashed line) which is a consequence of the complete buffering of the M concentration by labile ML at all spatial scales. This is reflected in the surface concentration profiles for M and ML which are now fully coupled in accordance with the lability of the system (Figures and D). At sufficiently large $1/a_{\mathrm{M}}$ when saturation of M active sites is achieved, the M and ML concentration gradients disappear, i.e., internalization has become rate limiting. Surface depletion of M at low $1/a_{\mathrm{M}}$ in the presence of labile ML (Figure D) is less than that for nonlabile ML (Figure C) because M is fully buffered by the dissociation of ML. As ψ increases, the contribution of ML increases and as ψ → ∞, ML uptake is dominant (independent of $1/a_{\mathrm{M}}$ ) and the surface concentrations of both M and ML approach zero.

Further insights are obtained by considering the dependence of $Q_{\mathrm{T}}$ and the surface concentrations of M and ML on the ratio $\eta =J_{\mathrm{Diff},\mathrm{ML}}^{\max }/J_{\mathrm{Diff},\mathrm{M}}^{\max }=\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}$ at various degrees of ML lability (Figure ). As ψ increases, the contribution of ML increases, and $Q_{\mathrm{T}}$ becomes increasingly sensitive to ML lability, ξ, as highlighted in Figure A for $\psi ={10}^4$ . In this ψ-regime, as ξ increases, M is increasingly able to buffer the surface concentration of ML and thus $Q_{\mathrm{T}}$ increases and the surface concentrations of M and ML evolve accordingly (Figure B, and also Figures C,D). At sufficiently large values of η and ψ, the total flux $Q_{\mathrm{T}}$ becomes predominantly limited solely by ML diffusion from bulk solution (blue dashed line in Figure A), it becomes accordingly independent of ML lability and it well exceeds the maximum flux of M diffusion from solution (red dashed line). Interestingly, unlike common practice in the BLM literature, Figure A shows that metal complex (ML) lability can be a determining biouptake factor even when the metal internalization flux is well below the maximum metal diffusion supply flux. This is because the ML complex itself can be taken upa hypothesis that the BLM framework rejects from the outset and which is rarely validated or justified a posteriori.

7.

Dependence of the (dimensionless) (A) total metal biouptake flux $Q_{\mathrm{T}}$ (eq ) and (B) M/ML surface concentrations $x=c_{\mathrm{M}}^{(\mathrm{a})}/c_{\mathrm{M}}^{*}$ (red) and $y=c_{\mathrm{ML}}^{(\mathrm{a})}/c_{\mathrm{ML}}^{*}$ ­(black), on the ratio $\eta =J_{\mathrm{Diff},\mathrm{ML}}^{\max }/J_{\mathrm{Diff},\mathrm{M}}^{\max }=\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}$ . The considered situation is that where free M and ML complex are internalized via distinct facilitated diffusion pathways (eqs , –, Figure B). Results are presented for ${\mathit{Bn}}_{\mathrm{M}}^{-1}={10}^{-3}$ and various values of $\psi ={({\mathit{Bn}}_{\mathrm{ML}}/{\mathit{Bn}}_{\mathrm{M}})}^{-1}$ (eq ), as specified. For the specific case $\psi ={10}^4$ , i.e., ${\mathit{Bn}}_{\mathrm{ML}}^{-1}=10$ , results are given for various values of the ML lability, ξ, as specified. For situations where ${\mathit{Bn}}_{\mathrm{M},\mathrm{ML}}^{-1}\ll 1$ , results do not depend on ξ (i.e., ML is inert). Model parameters other than those provided in the figure: $1/a_{\mathrm{M}}=2\times {10}^{-4}$ , $1/a_{\mathrm{ML}}=0.1$ . See text for details. In (A), the red and blue dotted lines represent the maximum diffusion flux of free M and the maximum diffusion flux of free M plus ML complex, respectively, both normalized by $J_{\mathrm{u},\mathrm{M}}^{\max }$ .

4.1.2. Bioaccumulation Scenario Where M and ML Are Internalized Via a Single Shared Facilitated Diffusion Pathway

4.1.2.1. M and ML Competitive Inhibition

In this case, the biouptake fluxes $J_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{ML}}$ are defined by the Michaelis–Menten expressions corrected for competitive inhibition, eq . Making the analogy with first-principles of enzymatic kinetic theory, M and ML competitive inhibition results in an apparent increase of the M and ML half-saturation constants, $K_{\mathrm{u},\mathrm{M}}$ and $K_{\mathrm{u},\mathrm{ML}}$ (or, equivalently, $a_{\mathrm{M},\mathrm{ML}}$ ) while leaving unaffected the maximum uptake fluxes, $J_{\mathrm{u},\mathrm{M}}^{\max }$ and $J_{\mathrm{u},\mathrm{ML}}^{\max }$ , respectively. , Figure presents the total (dimensionless) uptake flux, $Q_{\mathrm{T}}$ , the ML uptake flux, $Q_{\mathrm{ML}}$ , and the surface concentrations of M and ML as a function of the reciprocal of M bioaffinity parameter for the active sites $(1/a_{\mathrm{M}})$ , and for various values of the ML bioaffinity parameter, $a_{\mathrm{ML}}$ . Figure A–C shows the results for the case of inert ML $\left(\xi =0\right)$ , and no diffusion limitation of M uptake $({\mathit{Bn}}_{\mathrm{M}}^{-1}\ll 1)$ . At $\psi \to 0$ there is no uptake of ML and $Q_{\mathrm{T}}$ thus corresponds solely to M uptake (black dashed line, Figure A). As $a_{\mathrm{ML}}$ increases at fixed value of ψ (solid black curves, Figure A), the competitive effect of ML on M uptake decreases and thus, at a given $1/a_{\mathrm{M}}$ , $Q_{\mathrm{T}}$ increases with increasing $a_{\mathrm{ML}}$ and attains a maximal plateau value at lower $1/a_{\mathrm{M}}$ , i.e., less M affinity is needed to overcome ML competition (Figure A). Consistent with these effects, as $a_{\mathrm{ML}}$ increases (i.e., decreasing ML affinity for the internalization sites), $Q_{\mathrm{ML}}$ decreases and approaches zero at progressively higher $1/a_{\mathrm{M}}$ due to the decreasing ability of ML to compete with M uptake (Figure B). In the given example, both ${\mathit{Bn}}_{\mathrm{M}}^{-1}$ and ${\mathit{Bn}}_{\mathrm{ML}}^{-1}$ are ≪1, thus there is no diffusion limitation of M or ML and no concentration gradient at the biointerface, i.e., $x\approx y=1$ (Figure C). At large $1/a_{\mathrm{M}}$ , $Q_{\mathrm{ML}}$ then reaches a limiting value equal to ∼ $1/(a_{\mathrm{ML}}/a_{\mathrm{M}})$ (eq ) and the resulting total uptake flux $Q_{\mathrm{T}}=Q_{\mathrm{M}}+\eta \psi (a_{\mathrm{ML}}/a_{\mathrm{M}})Q_{\mathrm{ML}}$ ­(cf. eq ) then reduces to $Q_{\mathrm{T}}=Q_{\mathrm{M}}+\eta \psi$ . For the conditions selected in Figure A, we have $\eta =15$ and $\psi =0.316$ (solid black curves), which yields $Q_{\mathrm{T}}=5.74$ at large $1/a_{\mathrm{M}}$ .

8.

Dependence of the (dimensionless) (A, D, G) total metal biouptake flux $Q_{\mathrm{T}}$ (eq ); (B,E,H) corresponding ML biouptake flux $Q_{\mathrm{ML}}$ and (C, F, I) associated M/ML surface concentrations $x=c_{\mathrm{M}}^{(a)}/c_{\mathrm{M}}^{*}$ (red) and $y=c_{\mathrm{ML}}^{(a)}/c_{\mathrm{ML}}^{*}$ (black), on the normalized free metal concentration $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}=1/a_{\mathrm{M}}$ , for the situation where free M and ML complex are internalized via a single facilitated diffusion pathway with competitive inhibition (eqs –, Figures D and A). Results are presented for various values of the ML bioaffinity parameter $a_{\mathrm{ML}}$ (specified), with (A–C) ${\mathit{Bn}}_{\mathrm{M}}^{-1}={10}^{-2}$ and $\xi =0$ ; (D–F) ${\mathit{Bn}}_{\mathrm{M}}^{-1}=10$ and $\xi =0$ ; and (G, H, I) ${\mathit{Bn}}_{\mathrm{M}}^{-1}=10$ and $\xi \to 1$ . In each panel, the computations pertaining to the limit where there is no competitive inhibition (eqs , –, Figure B) are indicated in green color (A, B, D, E, G, H) and in dotted lines (C, F, I), as specified. This limit formally corresponds to the one where M and ML biouptake proceeds according to 2 distinct pathways. In (A), the dotted black and green lines are predictions for $\psi \to 0$ or, equivalently, ${\mathit{Bn}}_{\mathrm{ML}}^{-1}\to 0$ . These predictions coincide in (D, G) with results given for $a_{\mathrm{ML}}={10}^{-2}$ . In (A, D, G), the red and blue dotted lines represent the maximum diffusion flux of free M and the maximum diffusion flux of free M plus ML complex, respectively, both normalized by $J_{\mathrm{u},\mathrm{M}}^{\max }$ . In (C, F, I), x and y do not depend on $a_{\mathrm{ML}}$ in the limit of 2 bioaccumulation pathways. Model parameters other than those provided in the figure: $\eta =15$ and ${\mathit{Bn}}_{\mathrm{ML}}^{-1}={10}^{-2.5}$ .

Figure D–F reports the results for the case of inert ML $\left(\xi =0\right)$ and diffusion limitation of M uptake $\left({\mathit{Bn}}_{\mathrm{M}}^{-1}=10\right)$ . Figure D shows that when ML competition is low (i.e., $a_{\mathrm{ML}}\ge 1$ ), $Q_{\mathrm{T}}$ corresponds to the maximum diffusive flux of M, $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ (below saturation of the active sites, black curves coincident with the red dashed line). As ML competition increases (decreasing $a_{\mathrm{ML}}$ ) at a given $1/a_{\mathrm{M}}$ value, $Q_{\mathrm{T}}$ decreases. As compared to Figure B, $Q_{\mathrm{ML}}$ is here basically greater at a given $a_{\mathrm{ML}}$ due to the diffusion limitation of M uptake (Figure E). This is a consequence of the lower surface concentration of M (Figure F), and thus the more effective competition by ML. As ML competition increases (i.e., as $a_{\mathrm{ML}}$ decreases), the surface concentration of M increases (Figure F) and M uptake becomes less impacted by the diffusion limitation. The limiting value of $Q_{\mathrm{T}}$ at large $1/a_{\mathrm{M}}$ is close to unity ( $\eta =15$ and $\psi =3.16\times {10}^{-4}$ ). Since ${\mathit{Bn}}_{\mathrm{ML}}^{-1}$ ≪ 1, there is no diffusion limitation of ML and thus no concentration gradient at the biointerface ( $y=1$ , Figure F).

Figure G–I details the results for the case of labile ML $\left(\xi \to 1\right)$ and diffusion limitation of M uptake $\left({\mathit{Bn}}_{\mathrm{M}}^{-1}=10\right)$ . The general trends are the same as the preceding case except that now when ML competition is low $\left(a_{\mathrm{ML}}\ge 1\right)$ , $Q_{\mathrm{T}}$ exceeds the maximum diffusive flux of M, due to buffering of the M concentration by labile ML (Figure G). As ML competition increases (i.e., as $a_{\mathrm{ML}}$ decreases), $Q_{\mathrm{T}}$ decreases here again at a given $1/a_{\mathrm{M}}$ value, i.e., progressively greater M bioaffinity is needed to overcome the increasing ML competition. As ψ → ∞, $Q_{\mathrm{T}}$ approaches the diffusion limit for ML, $J_{\mathrm{Diff},\mathrm{ML}}^{\max }$ (blue dashed line). The evolution of $Q_{\mathrm{ML}}$ as a function of $a_{\mathrm{ML}}$ and $a_{\mathrm{M}}$ lies intermediate between those presented in Figure B,E, and it reflects the coupled surface concentration profiles of M and ML (Figure I), i.e., ML lability inherently results in mutual buffering of M and ML at all spatial scales, thereby moderating the effect of ML competition as compared to the inert case (for which there is no concentration gradient in ML; Figure F).

In the absence of competitive inhibition (green lines in Figure ), the situation becomes formally similar to that when ML and M are taken up by 2 uncoupled internalization pathways (Section , Figures –). Under such conditions, there is no plateau in $Q_{\mathrm{T}}$ at large $1/a_{\mathrm{M}}$ (Figure A,D,G) because $Q_{\mathrm{ML}}$ is constant (Figure B,E) or, in the case of labile ML, increases as $1/a_{\mathrm{M}}$ increases (Figure H). In the latter case, the increase in $Q_{\mathrm{ML}}$ is due to buffering of the ML surface concentration by labile ML.

4.1.2.2. M and ML Uncompetitive Inhibition

In this case, the biouptake fluxes $J_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{ML}}$ are defined by the Michaelis–Menten expressions corrected for uncompetitive inhibition, eq . Under such conditions, drawing the analogy with enzyme kinetic theory, , uncompetitive inhibition results in an apparent and similar decrease in both $K_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{M}}^{\max }$ . We consider below the internalization of M and ML via a single shared facilitated diffusion pathway including uncompetitive inhibition of ML on M uptake. That is, for illustrative purposes, it is assumed that M has no uncompetitive effect on ML uptake and thus $Q_{\mathrm{ML}}$ is given by the standard expression, eq (i.e., $1/a_{(\mathrm{I}),\mathrm{M}}\to 0$ in eq ). Figure presents the total uptake flux, $Q_{\mathrm{T}}$ , and the surface concentrations of M and ML as a function of the affinity of M for the active sites $(1/a_{\mathrm{M}})$ , for various values of the ML bioaffinity parameter for the nonactive inhibitory sites, $a_{(\mathrm{I}),\mathrm{ML}}$ (and with considering a fixed ML bioaffinity parameter for the active sites, $a_{\mathrm{ML}}$ ). At $\psi \to 0$ , there is no ML contribution to the uptake and the plateau in $Q_{\mathrm{T}}$ at large $1/a_{\mathrm{M}}$ corresponds to saturation of the M internalization sites (with accounting for the inhibitory effect of ML). As $a_{(\mathrm{I}),\mathrm{ML}}$ increases, the inhibitory effect of ML on M uptake decreases, and thus the $Q_{\mathrm{T}}$ plateau increases and approaches unity, as illustrated by the black dashed curves (a) → (d) in Figure A,C,E. As ψ increases, the contribution of ML increases, as evidenced by the increases in $Q_{\mathrm{T}}$ beyond the plateau at sufficiently large $1/a_{\mathrm{M}}$ . When ML is inert and uptake of M is diffusion-limited (Figure C), the point at which $Q_{\mathrm{T}}$ increases beyond the plateau is shifted to greater $1/a_{\mathrm{M}}$ as compared to the nondiffusion-limited case (Figure A). This result is a consequence of the lower surface concentration of M in the diffusion-limited case (Figure D) and thus the more effective is uncompetitive inhibition by ML at a given bulk M concentration. When ML is labile, and uptake of M is diffusion-limited (Figure E), the lability of ML buffers the M and ML surface concentrations (Figure F) and thus moderates the impact of the uncompetitive inhibition. As detailed for the preceding cases, $Q_{\mathrm{T}}$ in the Henry regime (low $1/a_{\mathrm{M}}$ ) lies below $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ when M uptake is not diffusion-limited (Figure A), is equal to $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ when M uptake is diffusion-limited and ML is inert (Figure C), and exceeds $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ when M uptake is diffusion-limited and ML is labile (and approaches $J_{\mathrm{Diff},\mathrm{ML}}^{\max }$ as ψ → ∞; Figure E).

9.

Dependence of the (dimensionless) (A, C, E) total metal biouptake flux $Q_{\mathrm{T}}$ (eq ) and (B, D, F) corresponding M/ML surface concentrations $x=c_{\mathrm{M}}^{(a)}/c_{\mathrm{M}}^{*}$ and $y=c_{\mathrm{ML}}^{(a)}/c_{\mathrm{ML}}^{*}$ , on the normalized free metal concentration $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}=1/a_{\mathrm{M}}$ , for the situation where free M and ML complex are internalized via a single facilitated diffusion pathway with uncompetitive inhibition (eqs –, Figures D and C). Results are presented for various values of the ML bioaffinity parameter a _(I),ML, with a _(I),ML = 10^–2 (a), 10^–1 (b), 1 (c), 10³ (d), for (A, B) Bn _M ^–1 = 10^–1 and ξ = 0; (C, D) Bn _M ^–1 = 10 and ξ = 0; (E, F) Bn _M ^–1 = 10 and ξ → 1. In each panel, the computations are further provided for ψ→0 (black dotted lines) and three other specified values of ψ (black, red, and blue solid lines). In (A, C, E), the red and blue dotted lines represent the maximum diffusion flux of free M and the maximum diffusion flux of free M plus ML complex, respectively, both normalized by J _u,M . In (B, D, F), x and y do not depend on ψ. Model parameters other than those provided in the figure: 1/a _(I),M → 0, 1/a _ML = 0.3, η = 15. Under the conditions examined in (A, B) and (C, D), we have Q _ML ≈ 0.23 regardless of a _(I),ML and ψ. Under the conditions examined in (E, F), the variations of Q _ML with changing a _(I),ML are shown in Figure S1, SI–C.

4.1.2.3. M and ML Mixed Competitive Inhibition, and the Noncompetitive Inhibition Limit

For illustrative purposes herein, we assume that ML acts as a mixed competitive inhibitor for the uptake of M (eq ), and that M affects the uptake of ML via only competitive inhibition, i.e., $1/a_{(\mathrm{I}),\mathrm{M}}\to 0$ in eq . We further recall that the noncompetitive case (eq ) refers to a limit of the mixed competitive situation (eq ) where the affinity of e.g., ML for the active and nonactive uptake sites is the same, i.e., $a_{(\mathrm{I}),\mathrm{ML}}=a_{\mathrm{ML}}$ . Figure presents the total uptake flux, $Q_{\mathrm{T}}$ , the ML uptake flux, $Q_{\mathrm{ML}}$ , and the surface concentrations of M and ML as a function of the affinity of M for the active sites $\left(1/a_{\mathrm{M}}\right)$ and for various values of the affinity of ML for the nonactive inhibitory sites, $a_{(\mathrm{I}),\mathrm{ML}}$ . Under the conditions selected in Figure , the ML activity for the active sites, $a_{\mathrm{ML}}$ , is fixed, and the quantities $Q_{\mathrm{ML}}$ , $x$ , and $y$ are all independent of ψ essentially because $\psi \ll 1$ .

10.

Dependence of the (dimensionless) (A, D, G) total metal biouptake flux $Q_{\mathrm{T}}$ (eq ); (B, E, H) corresponding ML biouptake flux $Q_{\mathrm{ML}}$ and (C, F, I) associated M/ML surface concentrations $x=c_{\mathrm{M}}^{(\mathrm{a})}/c_{\mathrm{M}}^{*}$ and $y=c_{\mathrm{ML}}^{(\mathrm{a})}/c_{\mathrm{ML}}^{*}$ , on the normalized free metal concentration $c_{\mathrm{M}}^{*}/K_{\mathrm{u},\mathrm{M}}=1/a_{\mathrm{M}}$ , for the situation where free M and ML complex are internalized via a single facilitated diffusion pathway with mixed competitive inhibition (eqs –, Figures D and D). Results are presented for various values of the ML bioaffinity parameter a _(I),ML, with a _(I),ML = 10^–3 (a), 10^–2 (b), 10^–1 (c), 10² (d), for (A–C) Bn _M ^–1 = 10^–1 and ξ = 0; (D–F) Bn _M ^–1 = 10 and ξ = 0; (G–I) Bn _M ^–1 = 10 and ξ → 1. In each panel, the computations are further provided for ψ → 0 (black dotted lines) and two other specified values of ψ (blue and red solid lines). In (A, D, G), the red and blue dotted lines represent the maximum diffusion flux of free M and the maximum diffusion flux of free M plus ML complex, respectively, both normalized by J _u,M . In (B, E, H) and (C, F, I), the quantities Q _ML, x, and y do not depend on ψ. Model parameters other than those provided in the figure: 1/a _(I),M → 0, 1/a _ML = 0.3, η = 5. The noncompetitive inhibition limit reached for a _ML = a _(I),ML is intermediate between cases (c) and (d) under the here examined conditions.

Figure A–C shows the results for the case of inert ML, and no diffusion limitation of M uptake. Figure A shows that at $\psi \to 0$ (black dashed curve) there is no uptake of ML, and $Q_{\mathrm{T}}$ thus corresponds solely to M uptake. Under these conditions $\left(\psi \to 0\right)$ , the magnitude of the plateau in $Q_{\mathrm{T}}$ at large $1/a_{\mathrm{M}}$ increases toward unity as the uncompetitive inhibition by ML decreases (curves (a) → (d), referring to increasing values of $a_{(\mathrm{I}),\mathrm{ML}}$ ). As ψ increases, the contribution from ML to the overall metal uptake increases and thus at large $1/a_{\mathrm{M}}$ , $Q_{\mathrm{T}}$ becomes greater than the plateau level attained for $\psi \to 0$ . As discussed for the competitive competition case (Figure ), at large $1/a_{\mathrm{M}}$ , $Q_{\mathrm{ML}}$ reaches a limiting value equal to $y/(a_{\mathrm{ML}}/a_{\mathrm{M}})$ (Figure B), where $y$ approaches unity at sufficiently high $1/a_{\mathrm{M}}$ (Figure C). Thus, at large $1/a_{\mathrm{M}}$ , $Q_{\mathrm{ML}}$ approaches $1/(a_{\mathrm{ML}}/a_{\mathrm{M}})$ , and $Q_{\mathrm{T}}=Q_{\mathrm{M}}+\eta \psi$ , cf. eq . For the conditions in Figure A, $\eta =5$ and ψ = 10^–3 or 10^–2, which yields $Q_{\mathrm{T}}=Q_{\mathrm{M}}+5\times {10}^{-3}$ (red curves) or $Q_{\mathrm{M}}+5\times {10}^{-2}$ (blue curves), respectively, at large $1/a_{\mathrm{M}}$ . In this example, ${\mathit{Bn}}_{\mathrm{ML}}^{-1}$ is rather low (being 10^–3 for ψ = 10^–2 and 10^–4 for ψ = 10^–3), thus the contribution from ML becomes negligible relative to that from M as the uncompetitive component of the ML mixed inhibition process decreases (curves (a) → (d)). As $1/a_{(\mathrm{I}),\mathrm{ML}}\to 0$ (curve (d)), this ML uncompetitive component becomes negligible, and the curves for all ψ values converge to that for $\psi \to 0$ (Figure A).

A similar trend in $Q_{\mathrm{T}}$ is observed when M uptake is diffusion-limited and ML is inert (Figure D). Due to the low ψ values used, the plateau in $Q_{\mathrm{T}}$ is practically the same as that for M only, and the contribution from ML is only evident at the greatest level of uncompetitive inhibition shown (curve (a)) (ηψ is 5 × 10^–4 or 5 × 10^–5, in blue and red curves, respectively). In the Henry regime (low $1/a_{\mathrm{M}}$ ) $Q_{\mathrm{T}}$ is equal to $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ and the curves are shifted to greater $1/a_{\mathrm{M}}$ as compared to the nondiffusion-limited case (Figure A) due to the concentration gradient in M (Figure F), i.e., ML can compete more effectively at lower $1/a_{\mathrm{M}}$ . This finding is reflected in the evolution of the concentration gradient of M: as the uncompetitive component of the ML mixed inhibition decreases (curves (a) → (d)), the concentration gradient in M shifts to greater $1/a_{\mathrm{M}}$ (Figure F). The trend in $Q_{\mathrm{ML}}$ (Figure E) reflects the coupling between the dependence of $Q_{\mathrm{ML}}$ on $x/a_{\mathrm{M}}$ $\left(y=1\right)$ and the dependence of $x$ on $a_{(\mathrm{I}),\mathrm{ML}}$ (Figure F).

For the case where M uptake is diffusion-limited and ML is labile (Figure G), $Q_{\mathrm{T}}$ in the Henry regime (low $1/a_{\mathrm{M}}$ ) exceeds $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ due to the contribution from labile ML to the M flux. Again, the contribution from ML is barely discernible due to the low ηψ values considered. The position of the $Q_{\mathrm{T}}$ curves on the $1/a_{\mathrm{M}}$ axis is intermediate between those in Figure A,D because there are now concentration gradients in both M and ML (Figure I). Accordingly, the trend in $Q_{\mathrm{ML}}$ (Figure H) now reflects the intricate coupling between the dependence of $Q_{\mathrm{ML}}$ on $x/a_{\mathrm{M}}$ and $y$ , together with the dependence of $x$ on $a_{(\mathrm{I}),\mathrm{ML}}$ (Figure I).

5. Application to the Interpretation of Measured Metal Bioaccumulation Data

The literature data from Yang et al. selected for confrontation with theory (cf. Section ) refer to the bioaccumulation of neodymium (M = Nd) by the unicellular green algae Chlamydomonas reinhardtii in the absence or presence of organic ligands (namely, L = malic acid or citric acid), Figure (symbols), as a function of free Nd concentration. This specific data set was selected because it offers a very high level of experimental detail and consistency, which is essential for a rigorous confrontation with our theory. Critically, the original authors of this study demonstrated that the traditional Biotic Ligand Model (BLM) is not applicable for this system, as they observed an increase in metal biouptake flux with increasing ligand concentration. This makes the data set an ideal benchmark to test our framework, which specifically aims to elucidate “non-BLM” bioaccumulation scenarios where metal chemodynamics play a pivotal role. Furthermore, the data set provides experimental metal biouptake fluxes, effectively integrating the time dimension. These featurescombined with well-characterized physicochemical medium composition and a broad range of tested metal concentrations in the presence and absence of well-defined organic ligandsprovide the robust input parameters required to validate the interplay between metal chemodynamics and metal biouptake kinetics detailed by our theoretical framework. We sought to interpret these data sets by means of theory according to the two following different bioaccumulation scenarios: (i) Nd and Nd-L are internalized by two distinct facilitated diffusion pathways without competition (Figure A, eqs , –), and (ii) Nd and Nd-L are internalized by a single facilitated diffusion pathway and there is competitive inhibition (Figure B, eqs –). For both scenarios, we attempted to fit the data (in log–log format) using the relevant theoretical equations by iteratively adjusting the involved bioaffinity and maximum uptake flux parameters, $K_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}$ and $J_{\mathrm{u},i=\mathrm{M},\mathrm{ML}}^{\max }$ , respectively. The required values of $K_{\mathrm{u},\mathrm{M}}$ and $J_{\mathrm{u},\mathrm{M}}^{\max }$ were obtained by first fitting the bioaccumulation data measured without ligand (i.e., by considering eq in the limits $J_{\mathrm{u},\mathrm{ML}}\to 0$ and $1/a_{\mathrm{ML}}\to 0$ ). These values were then fixed when fitting the bioaccumulation data collected in the presence of L. Each data point referring in Figure A,B to measurement with a ligand corresponds to specific log–log combinations of concentrations of ligand and free Nd concentrations, which are detailed in Figure C (denoted as ‘Malate 1,2’ and ‘Citrate 1,2’). Derivation of the model parameters for each envisaged bioaccumulation scenario was performed by fitting the two relevant data sets 1 and 2 concurrently. Theoretical predictions were then evaluated with adopting the so-obtained model parameters and by varying free and complexed Nd concentrations according to the log–log regressions shown in Figure C (dashed blue and pink lines in Figure A,B). Data analysis was performed using a custom MATLAB code that we developed for that purpose, which is available upon request. The results are summarized in Table , along with the adopted literature values for the diffusion coefficients of Nd and Nd-L, $D_{\mathrm{M},\mathrm{ML}}$ , for moderately stirred solution, ,− and the Nd-L complexation constants, $K_{\mathrm{ML}}$ (cf. caption Table ). Data analysis leads to the following conclusions:

• the experimental data sets are fully consistent with the two types of bioaccumulation scenarios considered, as a similar goodness-of-fit is achieved for both cases (cf. R ² values in Table );

• in both situations, diffusion-limited uptake of M and ML can be excluded. This is supported by our evaluation of the reciprocal Bosma numbers, ${\mathit{Bn}}_{\mathrm{M},\mathrm{ML}}^{-1}$ , which are both much lower than unity (cf. Table ), and it is further confirmed by the independence of the theoretical fitting on the lability ξ of the Nd-L complex (which is inert);

• We note that the R ² values for the citrate system are lower than those obtained for the malate system. This discrepancy is primarily due to the significantly higher experimental data scattering observed in the citrate data set, rather than a limitation of the theoretical framework itself. Specifically, a close inspection of the citrate data reveals that measurements taken at nearly identical free metal ion concentrations (representing experiments performed at different citric acid concentrations)differing by only 0.01–0.02 log units in concentration (where concentration is in mol m^–3)exhibit biouptake fluxes (in mol m^–2 s^–1) that differ by as much as 0.2 log units (cf. Figure A,B). Such intrinsic experimental variance necessarily limits the maximum achievable R ² during the fitting process. Despite this scattering, the theory successfully captures the overall kinetic trends and the “nonstandard BLM” behavior of the system.

11.

(A, B) Measured Nd (=M) bioaccumulation flux, $J_{\mathrm{u}}$ , by Chlamydomonas reinhardtii (symbols) as a function of the free Nd concentration, $c_{\mathrm{M}=\mathrm{Nd}}^{*}$ (log–log representation). Experiments were conducted in the presence and absence of the ligand (L), specifically malic acid or citric acid (as specified in the legend boxes). Measurements were quantitatively interpreted using theory, represented by dashed and dotted lines (cf. legend boxes). For cases where a ligand was present, theoretical fitting of the measured data was attempted by considering two scenarios: (A) two separate and uncoupled bioaccumulation pathways for Nd and Nd-L complexes (eqs , –) and (B) a single, shared bioaccumulation pathway with competitive inhibition (eqs –). (C) The combinations of concentrations of the ligand and free Nd concentrations used in the experiments are specified by the symbols, while dotted and dashed lines refer to log–log regressions of these data. The model parameters obtained from fitting the data to theory are compiled in Table . The maximum diffusion flux of M (dashed black lines in (A) and (B)) is given by $J_{\mathrm{Diff},\mathrm{M}}^{\max }=D_{\mathrm{M}}{c}_{\mathrm{M}}^{*}/a$ , where $D_{\mathrm{M}}$ is corrected for the diffusion layer thickness in moderately stirred solution, as detailed in the caption of Table . See main text for details.

2. Fitted and Derived Parameters from the Analysis of Nd Bioaccumulation Data as a Function of Free Nd Concentration in the Absence or Presence of the Ligand (Malic Acid or Citric Acid) for Chlamydomonas reinhardtii (Log-log Representation, Figure ) ^,

^aData analysis was performed according to metal bioaccumulation flux expressions relevant for the scenarios where Nd and Nd-L complexes are internalized via (i) two separate, uncoupled bioaccumulation pathways (eqs , –, left column) or (ii) a single, shared bioaccumulation pathway with competitive inhibition (eqs –, right column). Nd bioaccumulation data in the absence of the ligand were fitted according to eq in the limits $J_{\mathrm{u},\mathrm{ML}}\to 0$ and $1/a_{\mathrm{ML}}\to 0$ . Values in brackets correspond to the lower and upper bounds of the 95% confidence intervals. Other model parameters required for the fitting of data to theory were $a=3.5$ μm (cell radius); $K_{\mathrm{ML}}={10}^{2.52}$ m³ mol^–1 (Nd-malate) and $K_{\mathrm{ML}}={10}^{5.7}$ m³ mol^–1 (Nd-citrate) being the stability constants applicable at the experimental ionic strength of 10 mM; $D_{\mathrm{M},\mathrm{ML}}=D_{\mathrm{M},\mathrm{ML}}^{(0)}(a+\delta )/\delta$ (apparent diffusion coefficient including correction for gentle hydrodynamic stirring conditions) with $\delta =8$ μm ,,− and $D_{\mathrm{M}}^{(0)}\approx D_{\mathrm{ML}}^{(0)}=6\times {10}^{-10}$ m² s^–1 at 25 °C (diffusion coefficient uncorrected for convection). The diffusion supply fluxes of M and ML are proportional to $D_{\mathrm{M},\mathrm{ML}}/a$ , i.e., $D_{\mathrm{M},\mathrm{ML}}^{(0)}(a^{-1}+{\delta }^{-1})$ , which correctly compares with the diffusion conductance term for moderately stirred solution. ,−

Yang et al. interpreted their data sets by proposing that Nd and Nd-L bioaccumulation takes place via a shared diffusion facilitated pathway. However, our rigorous theoretical analysis demonstrates that it is impossible to distinguish between this scenario and one where Nd and Nd-L are taken up by two distinct, uncoupled pathways, particularly within the linear Henry biouptake regime. In this regime, theoretical expressions of the total metal flux are indeed identical for both models, as discussed in Section “Competitive Inhibition”. A clear distinction between these two bioaccumulation scenarios can only be made if data are measured at free metal concentrations high enough to saturate the active M-sites (cf. Figure A vs Figure B). In the existing literature, a single, shared biouptake pathway is often invoked to interpret bioaccumulation data on metals, particularly trivalent ions such as rare earth metals and Al­(III), ,,− when the standard BLM model fails due to its nonaccount of ML bioavailability. Beyond the nonuniqueness of this interpretation, authors often exclude potential diffusion limitation (and ML lability contribution) by comparing $J_{\mathrm{u},\mathrm{M}}$ with $J_{\mathrm{Diff},\mathrm{M}}^{\max }$ and arguing $J_{\mathrm{u},\mathrm{M}}/J_{\mathrm{Diff},\mathrm{M}}^{\max }\ll 1$ . − This reasoning, however, holds strictly only for the case where the intact ML complex is not bioaccumulated. If ML is taken up, diffusion limitation may be operational depending on the magnitude of the specific Bosma number that is operative for the very ML biouptake process. Furthermore, in scenarios where M and ML are thought to share a single uptake pathway, maximum M and ML uptake fluxes, $J_{\mathrm{u},\mathrm{i}=\mathrm{M},\mathrm{ML}}^{\max }$ , are commonly assumed to be identical. ,, We show here that while this approximation is practically valid for Nd and Nd-Malate complexes $(J_{\mathrm{u},\mathrm{ML}}^{\max }/J_{\mathrm{u},\mathrm{M}}^{\max }=0.86)$ , it is not for Nd and Nd-Citrate $(J_{\mathrm{u},\mathrm{ML}}^{\max }/J_{\mathrm{u},\mathrm{M}}^{\max }=0.46)$ , cf. Table . Ultimately, this work illustrates the significant benefits of using a mechanism-based Best-Michaelis–Menten formalism to interpret bioaccumulation data, moving beyond the common application of simple competitive adsorption isotherms.

In view of the above elements, the developed framework identifies and resolves several fundamental limitations of existing models by replacing empirical assumptions with a rigorous biophysical derivation. Most notably, the theory provides a formal mechanism to address the failure of the standard Biotic Ligand Model (BLM) in “nonstandard BLM” scenariosspecifically those where biouptake flux increases despite an increasing ligand concentration in solution. By rigorously excluding the often-unjustified assumption of complex inertness (via evaluation of the required Bosma numbers) and by quantifying the specific contribution of each bioreactive (internalizable) metal form (free and complexed) via derivation here of the parameter ψ, the theory offers a precise accounting of the overall metal biouptake. Furthermore, in case of citric acid data sets the derivation demonstrates the inadequacy of the common approximation done in the literature which consists of equating the maximum uptake fluxes for each bioreactive metal form, proving instead that these parameters are different for Nd metal and Nd-citrate complexes. Perhaps most significantly, the theory provides a mathematical justification for the fact that it is fundamentally impossible to discriminate between a single common uptake pathway and two distinct pathways for free and complexed metals based on bioaccumulation data collected within the Henry regime, as is the case for the herein analyzed data sets. This finding directly nuances conclusions currently found in the literature based on basic Michaelis–Menten arguments.

In turn, the framework developed herein addresses a long-standing gap in the literature regarding the connections between metal chemodynamics and biouptake kinetics of free metal and/or complexed metal species. While the current validation focuses on neodymium due to the paucity of data sets providing a sufficiently broad range of metal concentrations and well-defined ligands, the resulting theoretical insightssuch as the mathematical indistinguishability of uptake pathways in the Henry metal bioaccumulation regimepossess a universal character. This approach is intended to serve as a foundational tool for interpreting non-BLM scenarios, including systems such as silver-thiosulfate which share mechanistic features with the Nd-malate/citrate systems discussed here. Furthermore, this theory is intended to guide the design of future experiments, providing the necessary analytical framework to extract previously inaccessible information regarding the complex interplay between chemical speciation, bioavailability, and bioaccumulation.

6. Conclusions

This study presents a comprehensive theoretical framework to advance the understanding of kinetics of metal accumulation by aquatic organisms. Our model explicitly accounts for the dynamic interplay between metal speciation in the aquatic medium and the kinetics of biological uptake, thereby challenging the prevalent, equilibrium-based assumptions of the Biotic Ligand Model (BLM). This framework is particularly required in cases where the BLM fails, such as under high-metal complexation conditions, when the rate of metal biouptake is limited by metal diffusion from the bulk solution to the biosurface, or when metal complexes are bioreactive (i.e., internalizable) and/or feature some degree of lability. Our framework integrates the reactive transport of free and complexed metals within the biouptake scenarios identified in the literature as exceptions to the standard BLM. These scenarios include the biouptake of lipophilic complexes, the internalization of free and complexed metals via distinct diffusion-facilitated uptake routes, or the formation of ternary metal–ligand complexes with active internalization sites shared by free and complexed metal species. By combining Michaelis–Menten biouptake kinetics, metal diffusion from the bulk solution and metal complexation kinetics in solution, we derive extended Best bioaccumulation flux expressions. These expressions define how the metal internalization flux depends on the relevant metal bioaffinity and complexation constants, as well as on the diffusion- and internalization-defining properties of both free metal ions and complexes. These expressions further account for the competitive, noncompetitive or uncompetitive inhibition operating between free and complexed metals when they share the same active sites on a membrane translocation protein. For all the biouptake cases considered, analytical expressions are elaborated where possible, especially for metal concentrations that align with the linear Henry bioaccumulation regime.

A detailed analysis of computated examples, which cover the biouptake situations mentioned previously, shows that a low metal uptake flux -relative to the maximum diffusion supply flux from solution- is not enough to conclude that only free metal is taken up and that ML lability is irrelevant. Instead, we demonstrate how the lability of metal complexes can be impacting the overall metal uptake flux, which depends on whether the complex itself is taken up through its own internalization pathway. The theory also shows how competition from metal complexes can significantly modify the uptake flux of free metals. The impact of this competition is determined not only by its nature (competitive, uncompetitive, noncompetitive, or mixed competitive inhibition), but also by the lability of the complexed metal species (when operative), and the specific uptake characteristics of both free and complexed metals, as determined by their respective Bosma numbers (diffusion- or absorption-limited biouptake). The comprehensive theory developed herein offers an alternative to the traditional and often restrictive BLM, for interpreting bioaccumulation data and testing the applicability of different internalization mechanisms. Unlike BLM, which is merely a specific situation covered by the model, our framework can quantitatively access the contribution, if any, of metal complexes to biouptake. These benefits are illustrated by analyzing existing literature data on the biouptake of neodymium by Chlamydomonas reinhardtii.

Overall, this work presents a rigorous and thorough theoretical framework for predicting metal bioaccumulation in aquatic environments, moving beyond the popular equilibrium-based view of metal bioavailability. Refinements of this framework include accounting for metal depletion from the bulk solution during bioaccumulation, analyzing speciation situations that involve multidentate species and ligand mixtures, − and integrating the lability of metal complexes when the ligand is a colloid or a nanoparticle. − ,− We also anticipate that the biouptake flux expressions derived herein will be valuable for interpreting the bioaccumulation and/or bioremoval of emerging metallic contaminants, such as rare earth metals, or for more accurately defining metal toxicity thresholds in risk assessment. − For the sake of example, our model, unlike standard BLM, can quantitatively predict an increase in metal uptake in the presence of ligands. This is a phenomenon that has been documented in the scientific literature, ,− ,− and our model offers several scenarios that are consistent with this non-BLM behavior.

Glossary

GLOSSARY (main symbols)

Latin

A _0,1,2 and B_0,1,2

Dimensionless coefficients involved in the polynomial expressions defining the dimensionless M and ML surface concentrations (cf. Table 1 and Section 3.3.2).

a

Cell radius (m)

a _i=M,ML

Dimensionless M and ML bioaffinity parameters, defined by the ratios K _u,i=M,ML/c _i=M,ML .

b _i=M,ML

Dimensionless M and ML bioconversion capacities, defined by the ratios J _u,i=M,ML /(pJ _Diff,i=M,ML ).

Bn _i=M,ML

M and ML Bosma numbers (dimensionless) defined by eqs 14 and 20, respectively.

c _i=M,L,ML

Concentrations of M, L and ML in bulk solution (mol m^–3).

D _i=M,ML

Diffusion coefficient of M and ML (m² s^–1).

J _Diff,i=M,ML

Maximum M and ML diffusion flux from solution to the metal-bioaccumulating surface (mol m² s^–1), defined by J _Diff,i=M,ML = D _i=M,ML c _i=M,ML /a (uncorrected for stirring in solution, cf. caption Figure 11).

J _i=M,ML

Diffusion supply flux of M and ML at the metal-bioaccumulating surface (mol m² s^–1).

J _u,i=M,ML

Internalization (or uptake) flux of M and ML (mol m² s^–1).

J _u,i=M,ML

Maximum internalization flux of M and ML defined by J _u,i=M,ML = K _H,i=M,ML k _int,i=M,ML K _u,i=M,ML (mol m² s^–1).

k _a,d

Kinetic constants for ML formation (k _a, m³ mol^–1 s^–1) and dissociation (k _d, s^–1).

k _int,i=M,ML

Kinetic constants for M and ML internalization (s^–1).

K _ML = c _ML /(c _L c _M )

Metal complexation constant or, equivalently, ML stability constant (m³ mol^–1).

K̅ _ML = K _ML c _L

Dimensionless metal complexation constant.

K _u,i=M,ML

M and ML half-saturation constants (mol m^–3), corresponding to the reciprocal of the stability constants (m³ mol^–1) that describe the M and ML binding with their respective active transporter sites.

K _i=M,ML

Reciprocals of the stability constants that describe the binding of M and ML with the nonactive (inhibitory) sites of the protein transporters (mol m^–3).

p

Defined by p = 1 + εK̅ _MLξ (dimensionless).

Q _i=M,ML

Dimensionless uptake flux of M and ML, defined by Q _i=M,ML = J _u,i=M,ML/J _u,i=M,ML .

Q _T

Total dimensionless metal uptake flux defined by eq 41.

x = c̅ _M

Dimensionless M concentration at the metal bioaccumulating surface defined by x =c _M(r̅ = 1)/c _M , where c _M(r̅ = 1) correspond to the surface concentration of M.

y = c̅ _ML

Dimensionless ML concentration at the metal bioaccumulating surface defined by y = c _ML(r̅ = 1)/c _ML , where c _ML(r̅ = 1) correspond to the surface concentration of ML.

Greek

${\beta }_{\mathrm{ML}}$

Diffusion conductance ratio defined by eq 28 (dimensionless).

${\gamma }_{i=\mathrm{M},\mathrm{ML}}$

Dimensionless constants defined by eq 7.

${\delta }_{\mathrm{o}}$

Membrane thickness involved in eq 25 (m).

δ

Diffusion layer thickness (m) in moderately stirred solution (cf. caption of Table 2).

$\epsilon =D_{\mathrm{ML}}/D_{\mathrm{M}}$

ML to M diffusion coefficients ratio (dimensionless).

η

Dimensionless scalar defined by η = $J_{\mathrm{Diff},\mathrm{ML}}^{\max }/J_{\mathrm{Diff},\mathrm{M}}^{\max }$ = $\epsilon {\stackrel{-}{K}}_{\mathrm{ML}}$ .

${\stackrel{-}{\kappa }}_{\mathrm{a}}=k_{\mathrm{a}}{a}^2{c}_{\mathrm{L}}^{*}/D_{\mathrm{M}}$

Dimensionless Damkhöler number comparing the time scale of M diffusion, $a^2/D_{\mathrm{M}}$ , and the ML formation time scale, $1/(k_{\mathrm{a}}{c}_{\mathrm{L}}^{*})$ .

ξ

ML lability, defined by eq 13 (dimensionless).

ψ

Defined by eq 24.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsenvironau.5c00293.

• Details on the derivation of eq (SI-A), the expression of the dimensionless uptake flux $Q_{i=\mathrm{M},\mathrm{ML}}$ in the linear Henry regime when M and ML are taken up by distinct diffusion-facilitated pathways (SI-B), and Figure S1 shows the variations of $Q_{\mathrm{ML}}$ under the conditions of Figure E,F (SI-C) (PDF)

CRediT: Jérôme F.L. Duval conceptualization, data curation, formal analysis, investigation, methodology, resources, software, validation, visualization, writing - original draft, writing - review & editing; Herman P. van Leeuwen investigation, writing - review & editing; Raewyn M. Town data curation, formal analysis, funding acquisition, investigation, validation, writing - original draft, writing - review & editing.

The authors declare no competing financial interest.