^{1}, Francesc Mas

^{1}and Jaume Puy

^{2,a)}

### Abstract

The concept of conditional stability constant is extended to the competitive binding of small molecules to heterogeneous surfaces or macromolecules*via* the introduction of the conditional affinity spectrum (CAS). The CAS describes the distribution of effective binding energies experienced by one complexing agent at a fixed concentration of the rest. We show that, when the multicomponent system can be described in terms of an underlying affinity spectrum[integral equation (IE) approach], the system can always be characterized by means of a CAS. The thermodynamic properties of the CAS and its dependence on the concentration of the rest of components are discussed. In the context of metal/proton competition, analytical expressions for the mean (conditional average affinity) and the variance (conditional heterogeneity) of the CAS as functions of are reported and their physical interpretation discussed. Furthermore, we show that the dependence of the CAS variance on allows for the analytical determination of the correlation coefficient between the binding energies of the metal and the proton. Nonideal competitive adsorption isotherm and Frumkin isotherms are used to illustrate the results of this work. Finally, the possibility of using CAS when the IE approach does not apply (for instance, when multidentate binding is present) is explored.

The authors gratefully acknowledge support of this research by the Spanish Ministry of Education and Science (DGICYT: Project No. BQU2003-9698), by the European Community under Contract No. EVK1-CT2001-86, and from the “Comissionat d’Universitats i Recerca de la Generalitat de Catalunya.”

I. INTRODUCTION

II. THE CONDITIONAL AFFINITY SPECTRUM

III. MEAN AND VARIANCE OF THE CAS: CONDITIONAL AFFINITY AND CONDITIONAL HETEROGENEITY

A. Mean of the CAS

B. Variance of the CAS

IV. QUANTIFYING THE BINDING CORRELATION BY USING THE CONDITIONAL AFFINITY SPECTRUM

A. The covariance matrix corresponding to an arbitrary multicomponent isotherm

B. Finding the covariance matrix from the CAS

C. Binding correlation in Frumkin isotherm

V. CONDITIONAL AFFINITY SPECTRA IN SOME CASES WHERE THE INTEGRAL EQUATION APPROACH IS NOT SUITABLE: SPECIFIC AND MULTIDENTATE BINDING

A. A system where not all the ions compete for all the sites

B. A system with multidentate complexation

VI. CONCLUDING REMARKS

### Key Topics

- Protons
- 55.0
- Equilibrium constants
- 20.0
- Adsorption
- 8.0
- Integral equations
- 8.0
- Macromolecules
- 7.0

## Figures

(a) Three-dimensional plot of the multidimensional affinity spectrum underlying NICA isotherm. Parameters: . (b) Affinity spectrum of (a) represented as a contour plot. Iso- curves for (a), (b), (c), and are also depicted in the figure.

(a) Three-dimensional plot of the multidimensional affinity spectrum underlying NICA isotherm. Parameters: . (b) Affinity spectrum of (a) represented as a contour plot. Iso- curves for (a), (b), (c), and are also depicted in the figure.

(a) Binding curve vs in a system where two cations (proton denoted as 1 and a metal ion denoted as 2) compete for the binding sites at different values: (a), (b), (c), (d), (e), and (f). The binding curves correspond to a NICA isotherm with , , , and . (b) Conditional affinity spectra underlying the binding curves depicted in (a) indicating the effective distribution of affinities seen by the metal ion at the corresponding proton concentration.

(a) Binding curve vs in a system where two cations (proton denoted as 1 and a metal ion denoted as 2) compete for the binding sites at different values: (a), (b), (c), (d), (e), and (f). The binding curves correspond to a NICA isotherm with , , , and . (b) Conditional affinity spectra underlying the binding curves depicted in (a) indicating the effective distribution of affinities seen by the metal ion at the corresponding proton concentration.

Mean of the CAS corresponding to a NICA isotherm as a function of . Parameters: and (a), (b), (c), (d).

Mean of the CAS corresponding to a NICA isotherm as a function of . Parameters: and (a), (b), (c), (d).

Variance of the CAS corresponding to a NICA isotherm as a function of . Parameters: , , and (a), (b), (c), (d), (e). Curves (f) and (g) correspond to the fully correlated and fully uncorrelated cases, respectively.

Variance of the CAS corresponding to a NICA isotherm as a function of . Parameters: , , and (a), (b), (c), (d), (e). Curves (f) and (g) correspond to the fully correlated and fully uncorrelated cases, respectively.

Variance of the CAS underlying the Frumkin isotherm as a function of . Parameters: , and (a), (b), (c), (d), (e), and (f). Inset: correlation coefficient, , for the same isotherm with as a function of .

Variance of the CAS underlying the Frumkin isotherm as a function of . Parameters: , and (a), (b), (c), (d), (e), and (f). Inset: correlation coefficient, , for the same isotherm with as a function of .

CAS of a system with selective (1) and nonselective sites for the metal at different values: (a) , (b) , (c) , (d) , and (e) . The peak (1) corresponds to the selective sites with weight described by a Sips distribution and parameters . The peak (2) corresponds to the nonselective sites of weight described by the CAS underlying the NICA isotherm with , , and .

CAS of a system with selective (1) and nonselective sites for the metal at different values: (a) , (b) , (c) , (d) , and (e) . The peak (1) corresponds to the selective sites with weight described by a Sips distribution and parameters . The peak (2) corresponds to the nonselective sites of weight described by the CAS underlying the NICA isotherm with , , and .

Linear chain able to bind protons (with stability constant ) and metals. The metals can be bound in two different forms: single bond (with stability constant ) and bidentate bond (with stability constant ).

Linear chain able to bind protons (with stability constant ) and metals. The metals can be bound in two different forms: single bond (with stability constant ) and bidentate bond (with stability constant ).

(a) vs curves as given in Eq. (53) and corresponding to the model depicted in Fig. 7 for different values: (a) , (b) , (c) , (d) , (e) , (f) , and (g) . Parameters: , , and . CAS corresponding to the binding curves depicted in (a).

(a) vs curves as given in Eq. (53) and corresponding to the model depicted in Fig. 7 for different values: (a) , (b) , (c) , (d) , (e) , (f) , and (g) . Parameters: , , and . CAS corresponding to the binding curves depicted in (a).

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