Evaluation of the Koutecký–Koryta approximation for voltammetric currents generated by metal complex systems with various labilities

Impedance Spectroscopy for Bacterial Cell Monitoring, Analysis, and Antibiotic Susceptibility Testing

Conventional approaches for bacterial cell analysis are hindered by lengthy processing times and tedious protocols that rely on gene amplification and cell culture. Impedance spectroscopy has emerged as a promising tool for efficient real-time bacterial monitoring, owing to its simple, label-free nature and cost-effectiveness. However, its limited practical applications in real-world scenarios pose a significant challenge. In this review, we provide a comprehensive study of impedance spectroscopy and its practical utilization in bacterial system measurements. We begin by outlining the fundamentals of impedance theory and modeling, specific to bacterial systems. We then offer insights into various strategies for bacterial cell detection and discuss the role of impedance spectroscopy in antimicrobial susceptibility testing (AST) and single-cell analysis. Additionally, we explore key aspects of impedance system design, including the influence of electrodes, media, and cell enrichment techniques on the sensitivity, specificity, detection speed, concentration accuracy, and cost-effectiveness of current impedance biosensors. By combining different biosensor design parameters, impedance theory, and detection principles, we propose that impedance applications can be expanded to point-of-care diagnostics, enhancing their practical utility. This Perspective focuses exclusively on ideally polarizable (fully capacitive) electrodes, excluding any consideration of charge transfer resulting from Faradaic reactions.

Back Accumulation of Diffusive Gradients in Thin-Films Devices with a Stack of Resin Discs To Assess Availability of Metal Cations to Biota in Natural Waters

Determining species, concentrations, and physicochemical parameters in natural waters is key to improve our understanding of the functioning of these ecosystems. Diffusive Gradients in Thin-films (DGT) devices with different thicknesses of the resin or of the diffusive disc can be used to collect independent information on relevant parameters. In particular, DGT devices with a stack of two resin discs offer a simple way to determine dissociation rate constants of metal complexes from the accumulation of the target metal in the back resin disc. In this work, simple approximate expressions for the determination of the dissociation rate constant are reported and applied to a model Ni nitrilotriacetic complex as well as to Zn complexes in the Mediterranean Osor stream. Once the physicochemical parameters are known, one can plot the labile fraction of the metal complexes in terms of the thickness of the diffusion domain. These plots reveal a strong dependence on the nature of complexes as well as on the characteristics of the diffusion domain, and they are of high interest as predictors of availability to biota whose uptake is limited by diffusion.

Comparison between numeric and approximate analytic solutions for the prediction of soil metal uptake by roots. Example of cadmium

The dissociation of metal complexes in the soil solution can increase the availability of metals for root uptake. When it is accounted for in models of bioavailability of soil metals, the number of partial differential equations (PDEs) increases and the computation time to numerically solve these equations may be problematic when a large number of simulations are required, for example for sensitivity analyses or when considering root architecture. This work presents analytical solutions for the set of PDEs describing the bioavailability of soil metals including the kinetics of complexation for three scenarios where the metal complex in solution was fully inert, fully labile, or partially labile. The analytical solutions are only valid i) at steady-state when the PDEs become ordinary differential equations, the transient phase being not covered, ii) when diffusion is the major mechanism of transport and therefore, when convection is negligible, iii) when there is no between-root competition. The formulation of the analytical solutions is for cylindrical geometry but the solutions rely on the spread of the depletion profile around the root, which was modelled assuming a planar geometry. The analytical solutions were evaluated by comparison with the corresponding PDEs for cadmium in the case of the French agricultural soils. Provided that convection was much lower than diffusion (Péclet's number<0.02), the cumulative uptakes calculated from the analytic solutions were in very good agreement with those calculated from the PDEs, even in the case of a partially labile complex. The analytic solutions can be used instead of the PDEs to predict root uptake of metals. The analytic solutions were also used to build an indicator of the contribution of a complex to the uptake of the metal by roots, which can be helpful to predict the effect of soluble organic matter on the bioavailability of soil metals.

Chemodynamics of Soft Nanoparticulate Metal Complexes: From the Local Particle/Medium Interface to a Macroscopic Sensor Surface

The lability of a complex species between a metal ion M and a binding site S, MS, is conventionally defined with respect to an ongoing process at a reactive interface, for example, the conversion or accumulation of the free metal ion M by a sensor. In the case of soft charged multisite nanoparticulate complexes, the chemodynamic features that are operative within the micro environment of the particle body generally differ substantially from those for dissolved similar single-site complexes in the same medium. Here we develop a conceptual framework for the chemodynamics and the ensuing lability of soft (3D) nanoparticulate metal complexes. The approach considers the dynamic features of MS at the intraparticulate level and their impact on the overall reactivity of free metal ions at the surface of a macroscopic sensing interface. Chemodynamics at the intraparticulate level is shown to involve a local reaction layer at the particle/medium interface, while at the macroscopic sensor level an operational reaction layer is invoked. Under a certain window of conditions, volume exclusion of the nanoparticle body near the medium/sensor interface is substantial and affects the properties of the reaction layer and the overall lability of the nanoparticulate MS complex toward the reactive surface.

The biotic ligand model can successfully predict the uptake of a trivalent ion by a unicellular alga below pH 6.50 but not above: possible role of hydroxo-species

In many reported cases, the biotic ligand model (BLM) has been shown to predict the bioavailability of divalent metals toward aquatic biota successfully. However, studies on the bioavailability of nonessential trivalent metals, including aluminum (Al), are relatively scarce. In the present study, short-term scandium (Sc) internalization fluxes (Jint) were measured in Chlamydomonas reinhardtii in order to explore the applicability of the BLM to trivalent metals. Scandium was selected for its chemical similarities with Al and for its convenient radio-isotope (Sc-46). Apparent affinity constants of Sc(3+) with membrane transport sites (KSc-Rcell app) were surprisingly high, ranging from 10(8.08) M(-1) to 10(13.95) M(-1) over the pH range from 4.50 to 7.90. The competition of H(+) for binding with Sc(3+) transport sites explained this trend within the pH range of 4.50 to 6.00, but not from pH 6.50-7.90. In this latter pH range, predicted fluxes were smaller than observed fluxes and this divergence increased with pH, from a factor of 4 to approximately 1000. Above pH 6.50, the calculated supply of Sc(3+) to the biointerface by physical diffusion of the free Sc(3+) ion and by the dissociation of its hydroxo-complexes (ScOH(2+), Sc(OH)2(+) and Sc(OH)3) was insufficient to support the high observed internalization fluxes. We speculate that this failure of the BLM could be due to the transmembrane transport of undissociated Sc hydroxo-complexes. Scandium uptake could be modeled reasonably well using a simple semiempirical equation considering equal contributions from Sc(3+), ScOH(2+), Sc(OH)2(+), and Sc(OH)3 and no H(+) competition. Our work highlights the importance of studying the possible role of hydroxo-species in trace metal uptake.

Diffusion limitations in root uptake of cadmium and zinc, but not nickel, and resulting bias in the Michaelis constant

It has long been recognized that diffusive boundary layers affect the determination of active transport parameters, but this has been largely overlooked in plant physiological research. We studied the short-term uptake of cadmium (Cd), zinc (Zn), and nickel (Ni) by spinach (Spinacia oleracea) and tomato (Lycopersicon esculentum) in solutions with or without metal complexes. At same free ion concentration, the presence of complexes, which enhance the diffusion flux, increased the uptake of Cd and Zn, whereas Ni uptake was unaffected. Competition effects of protons on Cd and Zn uptake were observed only at a very large degree of buffering, while competition of magnesium ions on Ni uptake was observed even in unbuffered solutions. These results strongly suggest that uptake of Cd and Zn is limited by diffusion of the free ion to the roots, except at very high degree of solution buffering, whereas Ni uptake is generally internalization limited. All results could be well described by a model that combined a diffusion equation with a competitive Michaelis-Menten equation. Direct uptake of the complex was estimated to be a major contribution only at millimolar concentrations of the complex or at very large ratios of complex to free ion concentration. The true K(m) for uptake of Cd(2+) and Zn(2+) was estimated at <5 nm, three orders of magnitude smaller than the K(m) measured in unbuffered solutions. Published Michaelis constants for plant uptake of Cd and Zn likely strongly overestimate physiological ones and should not be interpreted as an indicator of transporter affinity.

Use of an exchange method to estimate the association and dissociation rate constants of cadmium complexes formed with low-molecular-weight organic acids commonly exuded by plant roots

Organic acids released from plant roots can form complexes with cadmium (Cd) in the soil solution and influence metal bioavailability not only due to the nature and concentration of the complexes but also due to their lability. The lability of a complex influences its ability to buffer changes in the concentration of free ions (Cd); it depends on the association (, m mol s) and dissociation (, s) rate constants. A resin exchange method was used to estimate and (m mol s), which is the conditional estimate of depending on the calcium (Ca) concentration in solution. The constants were estimated for oxalate, citrate, and malate, three low-molecular-weight organic acids commonly exuded by plant roots and expected to strongly influence Cd uptake by plants. For all three organic acids, the and estimates were around 2.5 10 m mol s and 1.3 × 10 s, respectively. Based on the literature, these values indicate that the Cd- low-molecular-weight organic acids complexes formed between Cd and low-molecular-weight organic acids may be less labile than complexes formed with soil soluble organic matter but more labile than those formed with aminopolycarboxylic chelates.

Modeling the measurements of cellular fluxes in microbioreactor devices using thin enzyme electrodes

An analytic approach to the modeling of stop-flow amperometric measurements of cellular metabolism with thin glucose oxidase and lactate oxidase electrodes would provide a mechanistic understanding of the various factors that affect the measured signals. We divide the problem into two parts: (1) analytic formulas that provide the boundary conditions for the substrate and the hydrogen peroxide at the outer surface of the enzyme electrode layers and the electrode current expressed through these boundary conditions, and (2) a simple diffusion problem in the liquid compartment with the provided boundary conditions, which can be solved analytically or numerically, depending on the geometry of the compartment. The current in an amperometric stop-flow measurement of cellular glucose or lactate consumption/excretion is obtained analytically for two geometries, corresponding to devices developed at the Vanderbilt Institute for Integrative Biosystems Research and Education: a multianalyte nanophysiometer with effective one-dimensional diffusion and a multianalyte microphysiometer, for which plentiful data for metabolic changes in cells are available. The data are calibrated and fitted with the obtained time dependences to extract several cellular fluxes. We conclude that the analytical approach is applicable to a wide variety of measurement geometries and flow protocols.

Protonation effects on dynamic flux properties of aqueous metal complexes

The degree of (de)protonation of aqueous metal species has significant consequences for the kinetics of complex formation/dissociation. All protonated forms of both the ligand and the hydrated central metal ion contribute to the rate of complex formation to an extent weighted by the pertaining outer-sphere stabilities. Likewise, the lifetime of the uncomplexed metal is determined by all the various protonated ligand species. Therefore, the interfacial reaction layer thickness, μ, and the ensuing kinetic flux, Jkin, are more involved than in the conventional case. All inner-sphere complexes contribute to the overall rate of dissociation, as weighted by their respective rate constants for dissociation, kd. The presence of inner-sphere deprotonated H2O, or of outer-sphere protonated ligand, generally has a great impact on kd of the inner-sphere complex. Consequently, the overall flux can be dominated by a species that is a minor component of the bulk speciation. The concepts are shown to provide a good description of experimental stripping chronopotentiometric data for several protonated metal–ligand systems.

Metal binding by heterogeneous ligands: kinetic master curves from SSCP waves

Stripping chronopotentiometry at scanned deposition potential, SSCP, is shown to be a powerful tool for determination of the distribution of metal dissociation rate constants, kd, for heterogeneous ligands. SSCP effectively scans a range of metal-to-ligand ratios from the foot to the plateau of the wave, thus allowing a portion of the distribution of k(d) values to be determined for a given bulk solution composition. In particular, the nature of the measurement renders accessible very low metal-to-ligand ratios which are otherwise difficult to attain by a bulk solution methodology, such as a potentiometric titration. In the presence of kinetic currents, the shape of the SSCP wave is modified as compared to the labile case. A data interpretation framework is developed which well-describes the SSCP wave for heterogeneous complexes in the kinetic current regime. The analysis utilizes the Freundlich binding isotherm together with the Koutecký-Koryta approximation, i.e., assuming a spatially discontinuous transition from labile to nonlabile behavior. For the case of Pb(II) and Cu(II) complexation by a peat fulvic acid, the existing body of data is drastically expanded to the low to very low metal-to-ligand ratio domain, thereby greatly improving the quality of the derived distribution parameters.

An exchange method to investigate the kinetics of Cd complexation in soil solutions

Knowledge of the dynamic speciation of metals in soil solution is necessary for soil-plant transfer models. The lability of a metal in a given context partly depends on the kinetics of interconversion between the free ion (M) and one complex (ML), governed by the association (c(k)a) and dissociation (c(k)d) rate constants of the reaction of formation of ML If only one single complex is assumed to be present in solution, a single kinetic experiment will enable estimation of ckd and cka', that is, the product of cka by the concentration of free ligand in solution. The method consists of the measurements over time of the radioactive metal remaining in solution after the labeling of a Ca resin--solution system in balanceforthe nonlabeled metal initially present. The method is described for Cd and was applied on four soil solution extracts. The order of magnitude of the estimated c(k)a'- and c(k)d-values was about 10(-3) s(-1). The advantages and drawbacks of the method are discussed.

Computing steady-state metal flux at microorganism and bioanalogical sensor interfaces in multiligand systems. A reaction layer approximation and its comparison with the rigorous solution

In complicated environmental or biological systems, the fluxes of chemical species at a consuming interface, like an organism or an analytical sensor, involve many coupled chemical and diffusion processes. Computation of such fluxes thus becomes difficult. The present paper describes an approximate approach, based on the so-called reaction layer concept, which enables one to obtain a simple analytical solution for the steady-state flux of a metal ion at a consuming interface, in the presence of many ligands, which are in excess with respect to the test metal ion. This model can be used for an unlimited number of ligands and complexes, without limit for the values of the association/dissociation rate constants or diffusion coefficients. This approximate solution is compared with a rigorous approach for the computation of the fluxes based on an extension of a previously published method (J. Galceran, J. Puy, J. Salvador, J. Cecília, F. Mas and J. L. Garcés, Phys. Chem. Chem. Phys., 2003, 5, 5091-5100). The comparison is performed for a very wide range of the key parameters: rate constants and diffusion coefficients, equilibrium constants and ligand concentrations. Their combined influence is studied in the whole domain of fully labile to non-labile complexes, via two combination parameters: the lability index, L, and the reaction layer thickness, mu. The results show that the approximate solution provides accurate results in most cases. However, for particular combinations of metal complexes with specific values of L or mu, significant differences between the approximate and rigorous solutions may occur. They are evaluated and discussed. These results are important for three reasons: (i) they enable the use of the approximate solution in a fully reliable manner, (ii) when present, the differences between approximate and rigorous solution are largely due to the coupling of chemical reactions, whose importance can thus be estimated, (iii) due to its simple mathematical expression, the individual contribution of each metal species to the overall flux can be computed.

Lability of a Mixture of Metal Complexes under Steady-State Planar Diffusion in a Finite Domain

The rigorous analytical solution for the fluxes from a mixture of 1:1 metal complexes toward an active surface under steady-state planar diffusion in a finite domain and excess ligand conditions allows for the computation of the global degree of lability of the system as well as particular degrees of lability of each complex in the mixture. This kind of system is found in a variety of fields ranging from electrochemical techniques (such as stripping chronopotentiometry at scanned deposition potential, SSCP) to analytical devices (such as diffusion gradients in thin-film gels, DGT). Among the specific effects arising from the presence of a mixture of ligands competing for the metal we highlight the following: (i) The degree of lability of a complex in the mixture differs from its degree of lability in an unmixed system with the same ligand concentration, and (ii) the degree of lability of one complex depends on (i.e., can be modified with) the concentrations of the ligands in the mixture. The impact of these characteristics on the metal flux crossing the active surface reaches the highest value when both complexes are partially labile. The complex contribution to the metal flux goes through a maximum when the thickness of the diffusion domain is varied. Thus, the thickness of the diffusion domain can be chosen to enhance the contribution of one particular complex. Lability criteria for each complex of the mixture within the reaction layer approximation are also reported. In particular, the reaction layer formulation for a complex is discussed in detail for two limiting cases: the rest of complexes are all nonlabile or the rest of complexes are all labile.

Labile Cd complexes increase Cd availability to plants

Dissolved trace metals are present in the environment as free ions and as complexes. Commonly used models to predict metal bioavailability consider the free ion as the major bioavailable species. However, increases in metal availability in the presence of metal complexes have repeatedly been found. We measured the uptake of cadmium (Cd) by spinach (Spinacia oleracea) from solution in absence or presence of synthetic ligands. At the same free ion concentration, the uptake of Cd ranged over almost 3 orders of magnitude and was largest in treatments with fast dissociating (i.e. labile) complexes. Similar results were found for the diffusional fluxes in these solutions, as measured with the DGT technique. The observed effect of Cd complexes on the plant uptake was in agreement with model calculations in which plant uptake was assumed to be governed by the diffusional flux. These results strongly suggest that Cd uptake is rate-limited by diffusion of the free ion to the root surface, even in stirred solutions. As a result, dissolved Cd complexes can increase Cd uptake, resulting in apparent exceptions from the free ion activity model. The magnitude of this increase depends both on the concentration and on the lability of the complexes. The free ion concept should therefore be reconsidered when transport limitations of the metal ion to the uptake site prevail.

Two grid refinement methods in the lattice Boltzmann framework for reaction–diffusion processes in complex systems

This paper studies the optimisation of a numerical model and a computer code to solve numerically reaction-diffusion processes in environmental or biological systems with complicated geometries and mixtures of reactions including time and spatial scales extending over several order of magnitude. In particular, we consider different grid refinement techniques in the framework of a lattice Boltzmann solver for reaction-diffusion systems. Two new grid refinement methods are proposed, which are both quantitatively good. The first method is based on the matching of the concentration profiles and fluxes across two adjacent sub-domains, while the second method is based on nested subgrids. The focus of our study is the trade off between accuracy and CPU time. We show how the different parameters of the method, such as the refinement factors, the location of the boundary between different grids or coupling methods at the interface affect the quality of the numerical solution and the efficiency of the method.

Lability Criteria for Successive Metal Complexes in Steady-State Planar Diffusion

The lability of sequential metal complexes, ML, ML2, ML3, ... , up to a general 1:n metal/ligand stoichiometric ratio is considered for the case of metal ions (M) being accumulated at a surface (analytical sensor or organism). The analytical solution for the steady-state diffusion of M within a sequential complexation scheme allows quantification of the contribution from the dissociation of all of the complex species to the metal flux through the so-called lability degree, xi. A lability degree for each sequential complexation step is also defined which, due to the sequential character of the complexation scheme, depends not only on the proper kinetic constants of the given complexation step but also on the kinetics of the previous ones. When all contributions from the complexes are diffusion limited, the system is fully labile and xi=1. To provide simple lability criteria, the reaction layer approximation is extended to specifically deal with this sequential complexation scheme, so that a reaction layer thickness is defined when the existence of one particular rate-limiting step is assumed. Expressions for the classical lability parameter, L, are formulated using the reaction layer approximation. The change of the lability of the system as the diffusion layer thickness is modified is analyzed in detail. The contribution of the complex flux reflects the evolution of the system from labile to inert as the thickness of the sensor is appropriately decreased.

Dynamic speciation analysis and bioavailability of metals in aquatic systems

Dynamic metal speciation analysis in aquatic ecosystems is emerging as a powerful basis for development of predictions of bioavailability and reliable risk assessment strategies. A given speciation sensor is characterized by an effective time scale or kinetic window that defines the measurable metal species via their labilities. Here we review the current state of the art for the theory and application of dynamic speciation sensors. We show that a common dynamic interpretation framework, based on rigorous flux expressions incorporating the relevant diffusion and reaction steps, is applicable for a suite of sensors that span a range of time scales. Interpolation from a kinetic spectrum of speciation data is proposed as a practical strategy for addressing questions of bioavailability. Case studies illustrate the practical significance of knowledge on the dynamic features of metal complex species in relation to biouptake, and highlight the limitations of equilibrium-based models.

LBGK method coupled to time splitting technique for solving reaction-diffusion processes in complex systems

A new approach to numerically solve a reaction-diffusion system is given, specifically developed for complex systems including many reacting/diffusing species with broad ranges of rate constants and diffusion coefficients, as well as complicated geometry of reacting interfaces. The approach combines a Lattice Boltzmann (LB) method with a splitting time technique. In the present work, the proposed approach is tested by focusing on the typical reaction process between a metal ion M and a ligand L, to form a complex ML with M being consumed at an electrode. The aim of the paper is to systematically study the convergence conditions of the associated numerical scheme. We find that the combination of LB with the time splitting method allows us to solve the problem for any value of association and dissociation rate constant of the reaction process. Also, the method can be extended to a mixture of ligands. We stress two main points: (1) the LB approach is particularly convenient for the flux computation of M and (2) the splitting time procedure is very well suited for reaction processes involving association-dissociation rate constants varying on many orders of magnitude.

Metal speciation dynamics and bioavailability: bulk depletion effects

Under conditions of bulk depletion, the speciation and bioavailability of trace metals must be considered at two different time scales: (i) the time scale of the biouptake flux, as determined by diffusion of the bioactive free metal, dissociation of the bioinactive complex species, and the internalization rate; and (ii) the time scale of depletion of the bulk medium. The implications of these two time scales for the speciation dynamics are discussed in terms of experimental conditions. The geometry of the system is taken into accountvia a spherical cellular model. It considers a spherical organism depleting a spherical volume in a nonstirred medium and assumes linear adsorption of the metal atthe biointerface and first-order internalization kinetics. In cases where the rate of biouptake is fully controlled by the internalization step, concentration gradients in the medium are insignificant. Then the biouptake becomes independent of the geometry of the system, and the model has a much simpler solution. Examples of trace metal uptake by microorganisms are analyzed: (i) cobalt uptake by Prochlorococcus in the presence of NTA, under conditions where bulk depletion is the controlling process due to the large number of organisms and high internalization rates, (ii) silver uptake by Chlamydomonas reinhardtii with significant effects of bulk depletion, due to the high internalization rate; (iii) lead uptake by Chlorella vulgaris with pratically negligible bulk depletion due to the low internalization rate of the metal; and (iv) lead uptake by intestinal Caco-2 cells, illustrating the simplification of the bulk depletion model for a system with different geometry where internalization is the rate-controlling step.

Dynamic Speciation Analysis of Heterogeneous Metal Complexes with Natural Ligands by Stripping Chronopotentiometry at Scanned Deposition Potential (SSCP)

Stripping chronopotentiometry at scanned deposition potential (SSCP) allows chemical heterogeneity in metal speciation to be unambiguously identified. In the labile regime, use of the Freundlich binding isotherm allows straightforward determination of parameters to describe the apparent stability and heterogeneity of metal complexes with humic substances. The extent of heterogeneity of metal binding by several humic substances follows the order Cu(ii) >> Pb(ii) > Cd(ii). The lability of metal complexes decreases from the foot to the top of the wave, and the greater the degree of heterogeneity, the more readily lability is lost. In the kinetic current regime, the Koutecký–Koryta approximation allows an expression to be obtained for the SSCP wave that provides a good estimate of the experimental data for metal complexes with moderate degrees of heterogeneity.