Josep Lluís Garcés

Effect of crowding by dextrans on the hydrolysis of N-Succinyl-L-phenyl-Ala-p-nitroanilide catalyzed by α-chymotrypsin

Traditionally, studies on the diffusion-controlled reaction of biological macromolecules have been carried out in dilute solutions (in vitro). However, in an intracellular environment (in vivo), there is a high concentration of macromolecules, which results in nonspecific interactions (macromolecular crowding). This affects the kinetics and thermodynamics of the reactions that occur in these systems. In this paper, we study the crowding effect of large macromolecules on the reaction rates of the hydrolysis of N-succinyl-L-phenyl-Ala-p-nitroanilide catalyzed by α-chymotrypsin, by adding dextrans of various molecular weights to the reaction solutions. The results indicate that the volume occupied by the crowding agent, but not its size, plays an important role in the rate of this reaction. A v(max) decay and a K(m) increase were obtained when the dextran concentration in the sample was increased. The increase in K(m) can be attributed to the slowing of protein diffusion, due to the presence of crowding. Whereas the decrease in v(max) could be explained by the effect of mixed inhibition by product, which is enhanced in crowded media. As far as we know, this is the first reported experiment on the crowding effect in an enzymatic reaction with a mixed inhibition by product.

Lability and mobility effects on mixtures of ligands under steady-state conditions

Analytical solutions for the steady-state flux arriving at an active surface from a mixture (in which one active species reacts with non-active ligands in the medium) can be helpful in a variety of problems: voltammetric techniques, heterogeneous processes in reactors, toxic or nutrient uptake, techniques of diffusive gradients in thin films (DGT), etc. Under any geometry that sustains steady-state, a convenient combination of the reaction–diffusion equations leads to a simpler formulation of the problem for arbitrary diffusivities of the species and arbitrary rate constants of the first-order conversion between the active species and the non-active species. The resulting problem can be characterised in terms of a list of dimensionless parameters involving the kinetic and mobility properties of each species. A lability degree for each 1∶1 complex in terms of the surface concentrations leads to: (i) a lability criterion specific for each complex in the mixture and (ii) the assessment of the relative contribution of each complex to the resulting flux. Semi-infinite spherical diffusion (as in the Gel Integrated MicroElectrode, GIME, biouptake modelling of micro-organisms, etc.) is specifically considered and some consequences of its full analytical solutions are discussed.

Diffusion of alpha-Chymotrypsin in solution-crowded media. A fluorescence recovery after photobleaching study

Fluorescence recovery after photobleaching (FRAP) is one of the most powerful and used techniques to study diffusion processes of macromolecules in membranes or in bulk. Here, we study the diffusion of alpha-chymotrypsin in different crowded (Dextran) in vitro solutions using a confocal laser scanning microscope. In the considered experimental conditions, confocal FRAP images could be analyzed applying the uniform circular disk approximation described for a nonscanning microscope generalized to take into account anomalous diffusion. Considering the slow diffusion of macromolecules in crowded media, we compare the fitting of confocal FRAP curves analyzed with the equations provided by the Gaussian and the uniform circular disk profile models for nonscanning microscopes. As the fitted parameter variation with the size and concentration of crowders is qualitatively similar for both models, the use of the uniform circular disk or the Gaussian model is justified for these experiments. Moreover, in our experimental conditions, alpha-chymotrypsin shows anomalous diffusion (alpha < 1), depending on the size and concentration of Dextran molecules, until a high concentration and high size of crowding agent are achieved. This result indicates a range of validity of the idealized fitting expressions used, beyond which other physical phenomena must be considered.

Use of activity coefficients for bound and free sites to describe metal–macromolecule complexation

A thermodynamic formalism to describe the small molecule–macromolecule complexation equilibrium, based on the concept of free and occupied sites (formal species) is presented. The formalism is particularly useful in systems with a large number of species and allows for the inclusion of either transport phenomena or adsorption on the boundary of the system. The homogeneous and independent complexation behaviour is denoted ideal complexation. All other behaviours of complexation are treated as deviations from such an ideal system by means of their activity coefficients, this allows the definition of a thermodynamical equilibrium constant, K, for any complexation process, written in terms of formal species. Irrespective of the model of complexation considered in the system, the concentration equilibrium relationship for formal species tends to K when the concentration of the small molecule tends to vanish (limit of ligand excess). The experimental recording of activity coefficients is straightforward and there is no need for numerical derivatives of experimental data. The analysis of the activity coefficients vs. the free metal concentration plots allows an easy and general characterization of the complexation process. Two particular cases of non-ideal complexation (interactions between bound sites and the presence of chelates) are selected to illustrate the general characteristics of the activity coefficients, and to relate them to the affinity spectrum. Expressions for the first two moments of the affinity distribution in terms of the characteristics of the activity coefficients are given.

Interpretation of speciation, measurements on labile metal-macromolecular systems by voltammetric techniques

Phenomena such as non-ligand excess, adsorption and heterogeneous or non-independent complexation models are discussed in order to improve the interpretation of the plots of the normalized limiting currents (Φ)versus the total ligand concentration in speciation studies. It is shown that under ligand excess conditions, all complexation models behave like the homogeneous and independent model. This result enhances the reliability of stability constants obtained under excess conditions using an equilibrium relationship between formal species. The convenience of Φversus total metal concentration plots in order to detect the mentioned phenomena and to obtain quantitative information about the complexation and adsorption processes from the limiting Φ value at low metal concentrations is emphasized. Finally, it is shown that non-ligand excess conditions with their corresponding interpretative framework are required for heterogeneity studies.

Model-independent link between the macroscopic and microscopic descriptions of multidentate macromolecular binding: Relationship between stepwise, intrinsic, and microscopic equilibrium constants

The binding of ions or other small molecules to macromolecules and surfaces can be macroscopically characterized by means of the stepwise (or stoichiometric) equilibrium constants, which can be obtained experimentally from coverage versus concentration data. The present work presents a novel, simple, and direct interpretation of the stepwise constants in terms of the microscopic, site-specific, stability constants. This formalism can be applied to the most general case, including the heterogeneity of the sites, interactions among them, multicomponent adsorption, and so forth, and, in particular, to chelate complexation. We show that the stepwise equilibrium constants can be expressed as a product of two factors, (i) the average number of free potential sites (per bound ion) of the microscopic species to be complexed (stoichiometric factor) and (ii) the average of the microscopic stability constants of their free potential sites. The latter factor generalizes the concept of the intrinsic equilibrium constant to systems with chelate complexation and reduces to the standard definition for monodentate binding. However, in the case of heterogeneous multidentate complexation, the stoichiometric factor cannot be known a priori, so that the finding of the intrinsic constants is not trivial. One option is to approximate the stoichiometric factor by the value that would correspond to identical active centers. We investigate the accuracy of this assumption by comparing the resulting approximate intrinsic constants to those obtained by Monte Carlo simulation of several binding models. For the cases investigated, it is found that the assumption is quite accurate when no correlated structures (typical of short-range interactions) are formed along the chain. For adsorption of particles attached to a large number of active centers, the formalism presented here leads to the Widom particle insertion method.

Influence of the adsorption phenomena on the NPP and RPP limiting currents for labile metal-macromolecule systems

Abstract The impact of adsorption effects on limiting currents obtained in reverse pulse (RPP) and normal pulse polarography (NPP) is analysed critically for induced reactant adsorption systems with any ligand to metal ratio. Qualitative explanations in terms of the concentration profiles are provided. We demonstrate that, for labile systems, no influence on the RPP limiting current can be found from the complex adsorption. However, ligand adsorption increases the RPP limiting current, this influence decreasing as the total metal concentration decreases. So, RPP normalised limiting current values ( φ ) are suggested to improve the fitting of the stability constant at very low total metal concentrations. If the total ligand concentration is chosen for maximum sensitivity, the remaining effect of the ligand adsorption leads to a bias in the stability constant of less than 14%, provided that the diffusion coefficient of the complex and ligand is 20-fold lower than the diffusion coefficient of the metal ion. Because the limiting NPP and RPP currents are equal for any ligand to metal ratio and without adsorption, approximate expressions derived for NPP facilitate the determination of the stability constant from the φ versus total metal concentration plot. This procedure is applied to some experimental results of Zn and Cd complexed with poly(methacrylic) acid.

Complexation isotherms in metal speciation studies at trace concentration levels. Voltammetric techniques in environmental samples

Is any complexation isotherm valid to analyse macromolecular binding of trace metal ions in environmental samples? To answer this question, a detailed study of the behaviour of some usual complexation isotherms with different underlying affinity spectra at intermediate and low coverages is performed. Metal-complexation is characterised through the behaviour of the average equilibrium function, defined as Kc ≡ [occupied sites]/([free metal][free sites]), which can be interpreted, at any metal concentration, as the average of the microscopic stability constants of the remaining free sites of the system. It is shown that, at intermediate coverages, the value of Kc depends basically on the average, μ, and standard deviation, σ, of the distribution function p(log k), the binding curve showing low sensitivity with respect to the concrete shape of the affinity spectrum. As a result, isotherms with the same values of μ and σ describe binding data of H+ to humic acid (HA) with similar accuracy. In contrast, at low coverages, Kc tends to the average of the microscopic stability constants of all the sites in the naked macromolecule, 〈k〉, which, for a given μ and σ, strongly depends on the shape of the isotherm. This result imposes an important constraint on the isotherms used and warns against extrapolation since isotherms with very similar behaviours at intermediate coverages, can predict very different ones for low coverages. This different behaviour is clearly evidenced by simulated voltammetric experiments, suggesting their suitability for speciation studies of trace metals in environmental samples. Normalised limiting currents of Cd/HA are used to validate isotherms suitable to describe both low and intermediate coverage.

Voltammetric currents for any ligand-to-metal concentration ratio in fully labile metal-macromolecular complexation. Easy computations, analytical properties of the currents and a graphical method to estimate the stability constant

In order to enable a wider use of voltammetric methods in speciation analysis, it is convenient not to be restricted by ligand excess conditions. This work assumes labile ideal complexation of a metal ion by a ligand, planar electrode, no electrodic adsorption and equal diffusion coefficients for the complex and the ligand, but very different from the metal ion. It is shown that the system of non-linear equations describing the diffusion of the species in a potentiostatic experiment for any ligand to metal ratio can be reduced to only one ordinary differential equation by means of a change of variable. Standard numerical methods can then be used in the computation of the solution with a great saving of computational time and resources in comparison with other existing methods. Some properties of the currents are also proved: (i) Cottrellian behaviour for any current in normal pulse polarography (NPP) and for limiting currents in reverse pulse polarography (RPP), (ii) the dependence of the normalised limiting current (φ) on just three parameters, and (iii) the equality of limiting NPP and RPP currents. The normalised current for high stability constant values depends on just two parameters, one of which is the ratio of total metal/total ligand concentrations, and can be found from an implicit algebraic equation. A new representation for the normalised limiting currents is suggested: the iso-φ diagram, which for each ratio of diffusion coefficients, e, describes the currents for any stability constant in a unique drawing. A new graphical procedure arising from this diagram is suggested and then applied to data corresponding to Zn/poly(methacrylic) acid at pH 6 and fixed ionic strength.

Affinity distribution functions in multicomponent heterogeneous adsorption. Analytical inversion of isotherms to obtain affinity spectra

An analytical approach for the interpretation of multicomponent heterogeneous adsorption or complexation isotherms in terms of multidimensional affinity spectra is presented. Fourier transform, applied to analyze the corresponding integral equation, leads to an inversion formula which allows the computation of the multicomponent affinity spectrum underlying a given competitive isotherm. Although a different mathematical methodology is used, this procedure can be seen as the extension to multicomponent systems of the classical Sipss work devoted to monocomponent systems. Furthermore, a methodology which yields analytical expressions for the main statistical properties (mean free energies of binding and covariance matrix) of multidimensional affinity spectra is reported. Thus, the level of binding correlation between the different components can be quantified. It has to be highlighted that the reported methodology does not require the knowledge of the affinity spectrum to calculate the means, variances, and covariance of the binding energies of the different components. Nonideal competitive consistent adsorption isotherm, widely used in metal/proton competitive complexation to environmental macromolecules, and Frumkin competitive isotherms are selected to illustrate the application of the reported results. Explicit analytical expressions for the affinity spectrum as well as for the matrix correlation are obtained for the NICCA case.