Conditional equilibrium constants in multicomponent heterogeneous adsorption : The conditional affinity spectrum

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 pH are reported and their physical interpretation discussed. Furthermore, we show that the dependence of the CAS variance on pH 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.


I. INTRODUCTION
Binding of small molecules to heterogeneous surfaces and macromolecules is a matter of fundamental interest in several fields of theoretical and applied chemistry.][10] In several other disciplines, such as biochemical substract-ligand interactions, 11 chromatography, 12 or polymer science, 13,14 the heterogeneous binding also plays a relevant role.
Although several theoretical approaches to describe the heterogeneous adsorption have been proposed, 15 we focus here in the so-called affinity spectrum or integral equation ͑IE͒ approach.The IE approach is based on characterizing the ligand ͑a surface or a macromolecule͒ by a distribution of binding constants called affinity spectrum, which provides the fraction of sites with binding equilibrium constant lying in between the values k and k + dk.Hence, the coverage, i.e., the total fraction of occupied sites, is given by a weighted superposition of Langmuirian isotherms, 8,9,16 Usually, as indicated in Eq. ͑1͒, the affinity spectrum is given in terms of log͑k͒, which is proportional to the binding free energy.
Although such an approach has been widely used, in many systems of interest two or more components compete for the same macromolecular/surface sites.For instance, in natural waters the competition between the metal ion and the proton is in most cases unavoidable.A straightforward generalization of the affinity spectrum methodology to multicomponent systems relies on the definition of a multidimensional affinity spectrum. 9,17,18The binding properties of the system are then described by a superposition of multicomponent Langmuirian isotherms, where k i and c i represent, respectively, the concentration and the binding stability constant for component i, and x i = log͑k i ͒.Once a particular functional form for the affinity spectrum is suggested, the integrals appearing in ͑1͒ and ͑2͒ can, in principle, be performed and a binding isotherm can be obtained. 9,15,17n the other hand, sometimes one could be interested in the inverse problem: the finding of the affinity spectrum embedded in a suitable binding isotherm.For monocomponent systems, the inverse problem was analytically solved by Sips in his classical analysis of binding to heterogeneous surfaces. 16Numerical regularization techniques have also been successfully applied to solve the inverse problem from experimental binding data. 19,20For multicomponent systems, the n-dimensional nature of the integrals appearing in ͑2͒ complicates the interpretation of the binding isotherms in terms of an underlying affinity spectrum.However, the use of a multidimensional affinity spectrum is especially interesting, since it allows characterizing the binding in terms of the moments of this distribution.In addition to the mean values, which characterize the mean binding affinity, the variance provides information on the heterogeneity of the binding, i.e., on the dispersion of the binding affinities.Finally, the covariance measures the correlation between the affinity of the different components for a given site.For instance, in the context of metal-macromolecule complexation, when large values of the affinity of the proton occur together with large values for the affinity of the metal, the correlation is positive.When a large affinity value for the proton does not imply a large or low affinity value for the metal there is no correlation and we say that the two binding energies are independent. 17,21n analytical solution of the integral equation ͑2͒ has been reported in a recent paper. 22Thus, an analysis of some multicomponent isotherms, such as nonideal consistent competitive adsorption 17,23 ͑NICA͒ and Frumkin isotherms, 24 in terms of an underlying affinity spectrum has been reported.In the same paper, a way to obtain the main statistics ͑mean, variances, and covariances͒ of the multidimensional affinity spectra avoiding the detailed knowledge of the multidimensional spectrum was described.
Here we present an alternative ͑and complementary͒ approach to the study of multicomponent adsorption.A classical methodology in analytical chemistry to deal with multicomponent binding lies in the definition of the so-called conditional equilibrium constants. 25Conditional equilibrium constants allow considering a multicomponent system as a monocomponent one whenever the concentrations of the rest of components are kept constant.This is particularly useful for those systems in which only the concentrations of one ͑or more͒ of the species is, in equilibrium, present in a large range of values.For instance, in some natural waters, calcium and proton concentrations can approximately be taken as constants, while the concentrations of contaminants and heavy metal ions vary strongly depending on the presence or not of a pollution source. 1,2Conditional binding properties have indeed been used in a number of works.For instance, stability constants restricted to some fixed pH at a given ionic strength value have been reported in the literature for the binding of metal ions to synthetic or natural macromolecules. 7,26However, the theoretical extension of this procedure to deal with heterogeneous ligands is still lacking.
The aim of the present work is to explore the possibility of generalizing the concept of conditional equilibrium constant to heterogeneous systems.This generalization leads to the definition of the conditional affinity spectrum ͑CAS͒.Although the ideas here presented can be easily extended to other experimental situations, we will use the environmental terminology and restrict ourselves to metal/proton competition.In Sec.II, the CAS is defined as a natural generalization of a conditional equilibrium constant and the physical interpretation of such distribution is provided.The results are further applied to obtain the CAS associated to the NICA isotherm.In Sec.III, we analyze the main statistical properties of the CAS, i.e., its mean and variance, and their dependence on the pH value.In Sec.IV, we present a new methodology to obtain the main statistics of the multidimensional affinity spectra from extrapolation of the statistics of the CAS.In this way, the computing of multiple integrals can be avoided, and we are able to analytically obtain the statistics corresponding to NICA and Frumkin competitive isotherms.Finally, Sec.V shows that, although the existence of the multidimensional affinity spectrum guarantees the existence of the CAS, the CAS can also be used in cases where there is no multidimensional affinity spectrum, as defined in Eq. ͑2͒.Two of such situations are analyzed: ͑i͒ some sites bind selectively to only one component, and ͑ii͒ mono-and multidentate metal bindings coexist.

II. THE CONDITIONAL AFFINITY SPECTRUM
Let us first consider the simplest situation in which an arbitrary site, S, of the surface or macromolecule can be occupied by either the proton, M 1 , or a metal ion, M 2 present in the solution.When the sites are identical and independent, two equilibrium constants, k 1 and k 2 , can be associated with these binding processes,

͑3͒
The metal coverage, i.e., the fraction of sites occupied by the metal, is given by the well-known competitive Langmuirian isotherm, where c 1 and c 2 are the free proton and free metal concentrations, respectively.Dividing by 1 + k 1 c 1 both the numerator and the denominator, Eq. ͑4͒ can be rewritten as where kЈ͑c 1 ͒ = k 2 / ͑1+k 1 c 1 ͒ is the well-known conditional stability constant commonly used in analytical chemistry.Notice that kЈ͑c 1 ͒ evidences that the "effective" affinity of a given site for a metal ion at some fixed pH depends on the pH value.Equation ͑5͒ indicates that, by using kЈ, the multicomponent system has been transformed, from a practical point of view, into an equivalent effective monocomponent one.
As usual in the literature, it is convenient to rewrite the expressions in terms of logarithms of the stability constants, since the resulting terms can be easily related to free energy by simply multiplying it by a factor −RT ln 10.Accordingly, log kЈ͑c 1 ͒ can be rewritten as Two terms can be clearly identified in expression ͑6͒.The first one, x 2 , is associated to the "intrinsic binding free energy," that is, the free energy of the binding of a metal ion in absence of protons.͑Notice that the second term vanishes when c 1 = 0, i.e., at high enough pH values.͒On the other hand, the term −log͑1+k 1 c 1 ͒ can be related to the average work needed to perform the necessary exchange of the proton by the metal in order to complex the macromolecular site.This exchange work ͓given by RT͑ln 10͒log͑1+k 1 c 1 ͔͒ increases as the proton concentration increases, and, as a consequence, the conditional equilibrium constant is a decreasing function of c 1 .
Let us obtain the distribution pЈ͑xЈ ; c 1 ͒ of the effective stability constants, seen by a metal ion in the presence of a heterogeneous ligand, i.e., the density of sites with effective affinity xЈ = log kЈ at a given pH.From now onwards, we will refer to this distribution as the CAS.
The probability of finding a macromolecular site having affinity for the proton lying in between x 1 and x 1 + dx 1 and affinity for the metal ion in between x 2 and x 2 + dx 2 is, according to ͑2͒, given by p͑x 1 ,x 2 ͒dx 1 dx 2 .

͑7͒
Writing x 2 in terms of the conditional affinity xЈ, as given by Eq. ͑6͒, Since we are interested in the metal affinity, regardless of the proton affinity, we integrate ͑8͒ for all the x 1 values.This operation produces the conditional affinity spectrum which can then be written as ͑9͒ Equation ͑9͒ relates the CAS to the multidimensional affinity spectrum underlying multicomponent complexation.Notice that, according to ͑9͒, the existence of multidimensional affinity spectrum ensures the existence of the CAS.Therefore, the existence of the latter is a necessary condition for the existence of the former.However, the converse assumption might not be true: conditional affinity spectra could be realistic for conditions under which a multidimensional affinity spectrum, as defined in Eq. ͑2͒, cannot be assumed.This possibility will be investigated in Sec.V.
In terms of pЈ͑xЈ ; c 1 ͒, the metal coverage, using Eqs.͑2͒ and ͑9͒, can be rewritten as i.e., the metal coverage of the multicomponent system has been expressed by an integral equation corresponding to a monocomponent one.As a consequence, all the methods, both numerical and analytical, suggested in the literature to solve the integral equation ͑1͒ can then be applied to find the conditional affinity spectrum, bearing in mind that the resulting spectrum depends on the proton concentration, c 1 .This indicates, for instance, that the CAS can be obtained directly from the experimental binding data without the knowledge of the underlying multidimensional affinity spectrum.Figure 1 illustrates the computation of the conditional affinity spectrum from the multidimensional one.The starting point is the multidimensional affinity spectrum depicted as a three-dimensional ͑3D͒ plot in Fig. 1͑a͒ and as a contour plot in Fig. 1͑b͒.As pЈ͑xЈ ; c 1 ͒ is the density of sites of a given fixed effective affinity xЈ, we have to integrate the multidimensional spectrum over the curve x 2 − log͑1 + e x 1 + 1 ͒ = xЈ, where xЈ is kept constant, to obtain pЈ͑xЈ ; c 1 ͒, as Eq.͑9͒ indicates.For a fixed c 1 , lines for different constant xЈ values are plotted in Fig. 1͑b͒.Notice that for low enough x 1 values, all the lines are parallel with ordinate at x 1 → −ϱ given by the constant xЈ.Increasing x 1 , the curves move towards higher x 2 , this indicating that stronger complexing sites are required in order to compensate for the increased work needed to extract the proton.
Let us apply the procedure above developed to find the CAS associated to the NICA.This isotherm is widely used to describe the competitive binding of metal ions to natural organic matter in environmental chemistry.The metal coverage is then expressed as 23,27,28

͑11͒
We will restrict ourselves to the case n 1 = n 2 = n, a condition required for the existence of a multidimensional affinity spectrum underlying NICA isotherm ͑see, for instance, Refs.17 and 20͒.So proceeding, we attach to the interpretation of the NICA isotherm where both n i and p are heterogeneity parameters related to the particular ion and to the ligand, respectively. 21,28In order to obtain the CAS corresponding to the metal we have to consider the proton concentration as a fixed parameter of the system, and further solve the integral equation ͑1͒ for p͑x͒ with ͑c͒ given by ͑11͒.By using complex variable analysis, the inversion of a monocomponent isotherm is straightforward.Using the inversion formula proposed in Refs.16 and 29, bearing in mind that proton concentration must be taken as a constant, the underlying affinity spectrum is given by 16 pЈ͑xЈ;c 1 ͒ = ͑ln 10͒kЈͯIm 2 ͑c 1 ,z͒ where xЈ = log kЈ , z is a variable in the complex plane and "Im" represents the operation "take the imaginary part."Af-ter some algebra, Eq. ͑12͒ applied to the NICA isotherm leads to A pictorial representation of the procedure is given in Fig. 2: the initial information consist in a set of metal binding curves for different pH values ͓see Fig. 2͑a͔͒.In the illustrated case we have used the NICA isotherm with parameters n 1 = n 2 = n = 0.4.Each of such binding curves can now be regarded as a monocomponent binding isotherm for the metal, to which the usual inversion procedures can be applied.Using Eq. ͑13͒, a set of affinity spectra, one for each pH value ͓see Fig. 2͑b͔͒, have been computed.It can be observed that, in lowering the pH value, the CAS is shifted towards lower affinities indicating that the work required to exchange protons is increasing.Finally, for high enough pH values, the resulting spectra tend to collapse converging to a Sips distribution, since only one component ͑the metal ion͒ is being complexed.Actually, taking c 1 = 0 in expression ͑13͒, with

a Sips distribution function with average affinity k ¯2
and heterogeneity parameter m = n 2 p is recovered,

͑16͒
in agreement with the fact that NICA reduces to a Langmuir-Freundlich isotherm when only one component is present. 17

III. MEAN AND VARIANCE OF THE CAS: CONDITIONAL AFFINITY AND CONDITIONAL HETEROGENEITY
In the previous section we have seen that a multicomponent system can be treated as an equivalent monocomponent one by keeping constants the concentrations of the rest of components.This has led to the definition of the conditional affinity spectrum of this ion.On the other hand, as it has been pointed out previously, 30 the mean and the variance of a one-dimensional affinity spectrum are the parameters that almost determine the coverage at a wide range of concentrations of the complexing ion, i.e., very different affinity spectra sharing a common mean and variance lead to the same FIG.2. ͑a͒ Binding curve 2 vs log͑c 2 ͒ in a system where two cations ͑proton denoted as 1 and a metal ion denoted as 2͒ compete for the binding sites at different pH values: pH=2 ͑a͒, pH=4 ͑b͒, pH=6 ͑c͒, pH=8 ͑d͒, pH=9 ͑e͒, and pH=12 ͑f͒.The binding curves correspond to a NICA isotherm with k ¯1 =10 5 , k ¯2 =10 7 , n 1 = n 2 = 0.4, and p = 0.5.͑b͒ Conditional affinity spectra underlying the binding curves depicted in ͑a͒ indicating the effective distribution of affinities seen by the metal ion at the corresponding proton concentration.coverage at a very wide range of concentrations.In the same way, for a fixed pH and concentration of the rest of competing binding ions, the CAS distribution exhibits some mean affinity and a certain heterogeneity.The variance of the CAS distribution is a measure of this heterogeneity.Let us analyze the dependence of the mean and the variance of the CAS on the pH ͑the same procedure can be used to obtain the dependence of these statistics on the concentration of the other competing ions͒ and how this dependence can be interpreted in terms of the underlying binding events.

A. Mean of the CAS
The mean of the CAS can be calculated as where here and in the rest of this work, the symbol ͗͘ C is used to denote the average over the conditional affinity spectrum while ͗͘ will indicate the average over the twodimensional affinity spectrum.
Replacing the expression for the CAS, ͑9͒, in ͑17͒, By using the change of variables x 2 = xЈ + log͑1+k 1 c 1 ͒, and further integrating over x 2 , Eq. ͑18͒ becomes where ͗x 2 ͘ is the mean affinity of the metal, x 2 ͒dx 2 is the affinity spectrum of the proton in absence of metal ions.On the other hand, Eq. ͑19͒ can be rewritten as

͑20͒
Finally, noting that the term between brackets in Eq. ͑20͒ is the proton coverage in absence of metal ions, 1 ͑c 1 ,0͒, we obtain the sought expression for the conditional average affinity,

͑21͒
The resulting expression has a general validity, restricted only to the suitability of the IE approach to describe the thermodynamic properties of the system.A clear physical interpretation of the conditional average affinity emerges from Eq. ͑21͒.It indicates that the mean affinity seen by the metal ion at some pH is lower than the mean affinity that would experience in absence of protons.The substracting term, the second term of the right-hand side ͑rhs͒ of ͑21͒, increases as the proton concentration increases and just reflects the increasing work that the metal has to do to be bound, i.e., the interchanging energy.Furthermore, expression ͑21͒ has a remarkable property.Notice that in order to know the average affinity for the metal at a given pH, we only need a constant, ͗x 2 ͘, and the proton titration curve of the ligand in absence of metal.As a consequence, the mean conditional affinity of any metal is, up to an additive constant, independent of the nature of the metal, provided that Eq. ͑2͒ is accepted to describe the thermodynamic properties of the system.This fact is illustrated in Fig. 3, where we have plotted the conditional average affinity as a function of the pH, for systems in which proton binding is described by a Langmuir-Freundlich isotherm.The different curves correspond to different ions, whose binding properties can be, in principle, completely different, and thus be characterized by very different multicomponent isotherms.However, in agreement with the result ͑21͒, the FIG. 3. Mean of the CAS corresponding to a NICA isotherm as a function of pH. Parameters: curves only differ in a constant value, characteristic of the metal ion under study.Finally, it is worth noting that, since ͗xЈ͘ C and the ͐ −ϱ log c 1 1 ͑c 1 ,0͒d log c 1 can be experimentally determined in an independent way, Eq. ͑21͒ can be regarded as a test of the validity of the integral representation ͑2͒, by simply noting that the addition ͗xЈ͘ C + ͐ −ϱ log c 1 1 ͑c 1 ,0͒d log c 1 should be independent of the pH and equal to ͗x 2 ͘.

B. Variance of the CAS
A similar approach can be applied to obtain the variance of the conditional affinity spectrum, which is defined as and measures the spread of the conditional affinity values.As it happened with the conditional average affinity, ͗xЈ͘ C , the quantity Ј 2 ͑c 1 ͒ can be obtained directly from experimental data or from the multicomponent isotherm under study.In a previous paper, we showed that the variance of any affinity spectrum can be directly obtained from the isotherm, without the detailed knowledge of the underlying affinity spectrum. 22 By trivially adapting this result to the present situation, Ј 2 can be expressed as 3 .

͑24͒
Although useful from a practical point of view, expression ͑24͒ does not provide a physical interpretation of the apparent heterogeneity reflected by Ј 2 ͑c 1 ͒.With this aim, let us recall the definition of pЈ͑xЈ , c 1 ͒ to calculate Ј 2 ͑c 1 ͒.By using Eq.͑9͒, the computation of the first term in ͑22͒ leads to Inserting Eqs.͑19͒ and ͑25͒ in ͑22͒, the CAS variance can be expressed as and first component 2

͑29͒
Each one of the three terms appearing in ͑26͒ can be regarded as coming from a different source of apparent or conditional heterogeneity.
is the variance of the affinity spectrum for the metal in absence of protons, i.e., it is a measure of the intrinsic heterogeneity of the ligand for the metal ion and hence it is independent from pH. ͑ii͒ first component 2 ͑c 1 ͒ concerns the variance of the exchange work needed to bind metals at a given pH.As the exchange work requires the release of protons from the complexing sites, the variance of the exchange work will depend on the heterogeneity of the proton binding energy.As a result, first component 2 ͑c 1 ͒ increases the variance of the conditional affinity spectrum and it can be obtained from data of the system containing only protons.͑iii͒ Finally, corr 2 ͑c 1 ͒ is a very interesting term which depends on the correlation between the binding energy of proton and metal and which tends to decrease the variance of the conditional affinity spectrum.Correlation is a measure of the linear association between the affinity of the metal and that of the proton for the sites.When a large affinity value for the proton does not imply a large or low affinity value for the metal there is no correlation, and we say that the two binding energies are independent, i.e., p͑x 1 , x 2 ͒ = p 1 ͑x 1 ͒p 2 ͑x 2 ͒.In this case, corr 2 is zero, and Ј 2 ͑c 1 ͒ is the result of adding two independent contributions coming from the two components, i.e., On the other hand, when the metal and proton binding energies are fully correlated, i.e., p͑x 1 , which only depends on information concerning the proton binding.So we conclude that, when the binding is fully correlated or fully uncorrelated, the apparent heterogeneity only depends, up to an additive constant, on information that can be obtained using only the proton titration curve.Actually, this means that Eqs.͑26͒, ͑30͒, and ͑31͒ can be used to experimentally assess whether the competitive binding is correlated and to quantify the correlation of the binding energies of metal and proton as we will see in more detail in the next section.
In Fig. 4 we have plotted Ј 2 ͑c 1 ͒ in terms of the pH for bicomponent systems whose binding properties are described by means of a NICA isotherm with n 1 = n 2 = n.The product np has been kept constant while different values of the parameter p have been used in the different curves.Besides, we have plotted two limiting cases: the fully correlated and the fully uncorrelated systems.We can see that for noncorrelated systems, Ј 2 ͑c 1 ͒ is an increasing function of the proton concentration.In effect, according to ͑30͒, there are only two independent, and thus additive, sources of heterogeneity, 2 2 and first component 2 ͑c 1 ͒, the latter being related to the fluctuations of the exchange work.Conversely, in presence of some correlation, the variance of the conditional spectrum decreases.We could say that correlation introduces some additional ordering in the metal binding, resulting in a lower heterogeneity, which increases with c 1 .As a consequence, increasing the n value, while the product np is kept constant, Ј 2 ͑c 1 ͒ progressively tends to the conditional variance corresponding to the fully correlated system, in agreement with the role of parameter p as a common source of heterogeneity for all the ions. 17

A. The covariance matrix corresponding to an arbitrary multicomponent isotherm
We have shown that the analysis presented for the variance of the CAS leads to the conclusion that apparent heterogeneity and binding correlation are closely related.Let us provide some insight in this general assertion in the framework of the IE approach.The quantity of interest is the covariance of the affinity spectrum, C 12 , which is defined as the multiple integral, where

͑33͒
It is useful to express the binding correlation in terms of the correlation coefficient, a dimensionless quantity defined as According to ͑32͒, the computation of R implies the computation of the multidimensional affinity spectrum, by inverting Eq. ͑2͒ and further performance of the multiple integrals appearing in ͑32͒.The main difficulty of this pro-cedure is that the affinity spectrum for most of isotherms is not known.An alternative strategy was provided in Ref. 22. There, we showed that R can be directly computed from the expression of the multicomponent isotherm without the knowledge of the underlying affinity spectrum.The following expression for C 12 was reported: with = 1 + 2 .Following this way, an analytical expression for the correlation of the NICA isotherm with n 1 = n 2 = n has been obtained 22 which indicates that NICA isotherm always considers some degree of positive correlation between proton and metal binding.Notice that for n = 1, we have R = 1, which implies full correlation, in agreement with the analysis performed in Sec.III.

B. Finding the covariance matrix from the CAS
Nevertheless, the use of Eqs.͑32͒-͑34͒ to obtain analytical expressions for the binding correlation is limited by the fact that, for most multicomponent isotherms, the double integrals appearing in ͑34͒ cannot be performed analytically.This problem is especially relevant in the cases where the coverages cannot be explicitly expressed in terms of the concentrations, as it happens in most multicomponent isotherms ͑Frumkin, Temkin, etc.͒.Here we propose a new method which allows the computing of the binding correlation by profiting from the properties of the CAS variance.
Taking the limit c 1 → ϱ in expressions ͑28͒ and ͑29͒ and noting that, in this limit, k 1 c 1 ӷ 1, we have FIG.4. Variance of the CAS corresponding to a NICA isotherm as a function of pH.Parameters: k ¯1 =10 5 , k ¯2 =10 7 , np = 0.5 and p = 0.6 ͑a͒, p =0.7 ͑b͒, p = 0.8 ͑c͒, p = 0.9 ͑d͒, p =1 ͑e͒.Curves ͑f͒ and ͑g͒ correspond to the fully correlated and fully uncorrelated cases, respectively.lim where 1 2 represents the intrinsic heterogeneity for the proton and C 12 is the covariance between the binding energies of both components.
Thus, in the limit c 1 → ϱ, the variance of the conditional affinity spectrum becomes lim The covariance C 12 can hence be expressed as

͑39͒
Equation ͑39͒ is the sought expression for the binding correlation.All the terms in the right-hand side of ͑39͒ can be directly computed by using just the properties of monocomponent systems: the calculation of the variance of the affinity spectrum for monocomponent systems implies the calculation of single integrals, as indicated by Eq. ͑23͒, and 1 2 ͑and 2 2 ͒ can be directly obtained by taking c 2 → 0 ͑and c 1 → 0͒ in the multicomponent isotherm, lim Moreover, expressions ͑38͒ and ͑40͒ indicate that the difference between the limiting Ј 2 values at low and high enough pH values gives lim c 1 →ϱ Ј 2 − lim c 1 →0 Ј 2 = 1 2 −2C 12 .Thus, the computation of Ј from experimental metal binding data in the limit of very low and very high pH range together with the proton titration curve can provide a way of determining the correlation.
Let us apply the result ͑39͒ to provide analytical expressions for the binding correlation in the case of the multicomponent Frumkin isotherm.

C. Binding correlation in Frumkin isotherm
The Frumkin isotherm was originally derived to take into account the interaction between the binding agents by means of the introduction of some interaction parameters ij .For a bicomponent system, it reads The calculation of the correlation coefficient in a bicomponent Frumkin isotherm will clarify the advantages of the new approach based on the properties of the CAS and outlined in Sec.IV A. First, it is convenient to rewrite ͑41͒ as Clearly, in the limit c 1 → ϱ the right-hand side of ͑43͒ tends to zero.Hence, ͑44͒ On the other hand, dividing Eq. ͑41͒ by ͑42͒ and replacing ͑43͒ in the resulting equation we have Rearranging terms in ͑45͒, the macromolecular coverage for the metal becomes which can be recognized as a monocomponent Frumkin isotherm with affinity parameter b ˜= ͑b 2 / b 1 c 1 ͒e ͑ 11 − 12 ͒ and interaction parameter The variance of the spectrum underlying the monocomponent Frumkin was reported in Ref. 30.Applied to ͑47͒ it yields 2 = ͑ ˜2 /12+ ˜͒ / ͑ln 10͒ 2 .Likewise, 1 2 and 2 2 are given by i With these results at hand, the limit in ͑38͒ becomes lim and replacing ͑48͒ in Eq. ͑39͒, the correlation coefficient corresponding to the affinity spectrum for the multicomponent Frumkin isotherm is given by It is worth noticing from ͑48͒ that, since the conditional variance is a positive quantity, i.e., Ј 2 Ͼ 0, there is a restriction on the values of the Frumkin parameters for the existence of the CAS.This condition is i.e., the interaction energy between different components has to be smaller than the average of the intrinsic monocomponent interaction energies.Furthermore, since the existence of the CAS is necessary for the existence of the affinity spectrum, ͑50͒ can also be seen as a necessary condition for the existence of a multidimensional affinity spectrum underlying a Frumkin isotherm.Figure 5 plots the variance of the CAS corresponding to a Frumkin isotherm as a function of the pH for different 12 values.Inside the figure, we have depicted the correlation coefficient as a function of 12 .Notice that the correlation coefficient increases from R =0 to R =1 as 12 increases from 0 to 12 = 11 + 22 / 2. A further increase of 12 yields negatives values of Ј 2 .These values do not have physical meaning ͓see curve ͑f͒ in the figure͔, this indicating the nonexistence of the CAS ͑and, thus, of affinity spectrum͒ for this set of parameters ͑ 11 = 22 = 2.0 in the figure͒.Notice that there are three parameters determining the Frumkin isotherm, just the number of independent elements of the covariance matrix.Thus, there is no restriction on the values of these elements and all the range of R values can be covered.Figure 5 also shows that all 12 values lead to a common Ј 2 value at high pH range.This Ј 2 value is just 2 2 = ͑ 22 2 /12 + 22 ͒ / ͑ln 10͒ 2 , the intrinsic variance of the binding energy of the metal according to ͑40͒.The Ј 2 value at low pH range is given by ͑48͒ and decreases as the correlation increases in agreement with ͑38͒.
The procedure outlined in this section can be applied to any multicomponent isotherm and, among them, to the NICA isotherm.As expected, expression ͑35͒ for the correlation coefficient, previously obtained by performing the multidimensional integrals involved in Eq. ͑34͒, is recovered.

V. CONDITIONAL AFFINITY SPECTRA IN SOME CASES WHERE THE INTEGRAL EQUATION APPROACH IS NOT SUITABLE: SPECIFIC AND MULTIDENTATE BINDING
Up to now, we have dealt with systems which can be described by an underlying multidimensional affinity spectrum ͑IE approach͒.We have shown that, if Eq. ͑2͒ is accepted as a good thermodynamic description of the system, a CAS can always be defined.However, the fulfillment of this condition is too restrictive for the existence of the CAS, so that a conditional spectrum could in some cases be defined even when Eq. ͑2͒ does not apply.
Although a general CAS formulation is out of the scope of the present work, there are at least two kinds of systems of interest for which the IE approach does not apply.We will use these cases to generalize now the CAS.Let us first con-sider the case of a ligand which has some selective sites for one of the competing cations, i.e., sites to which only one of the cations, for instance, the cation i, is able to be bound.Another case of special interest is the treatment of competitive systems where the competing cations bind with different stoichiometries to the functional groups of the ligand.Actually, Eq. ͑2͒ requires that both components bind to the sites in a one-to-one stoichiometry, so that Eq. ͑2͒ does not account for the possible formation of chelates.However, in metal binding to macromolecules, multidentate complexation has been widely reported 26,31,32 and exchange ratios higher than 1 have been recorded. 23Let us analyze these two situations in the framework of the conditional affinity spectrum.

A. A system where not all the ions compete for all the sites
When some sites are able to selectively bind a metal cation, while protons cannot be bound to these sites, the metal coverage can be written as where q ns and q s =1−q ns represent the fraction of sites which have nonselective and selective bindings for the metal ions, respectively.The first term of the rhs of ͑51͒ specifies the occupation of the nonselective sites.The structure of this term indicates that the nonselective sites fulfill all the properties given in Secs.II and III for the CAS underlying a multidimensional affinity spectrum.Thus the global CAS of this system consists in the CAS distribution associated to the nonselective binding sites plus the affinity distribution of the selective sites.The resulting distribution is depicted in Fig. 6 for different pH values.The parameters of the figure have been selected so that the distribution is bimodal in order to evidence the effect of pH.Notice that the distribution asso- ciated to the competitive sites ͑those corresponding to the metal binding sites able to be protonated͒ is shifted when changing the pH value as described in Sec.III, while the affinity of the sites able to bind only metals is not affected by a pH change.Finally, we remark that Eq. ͑51͒ assumes that there is no interaction between metals bound to selective and nonselective sites.If interactions were present, some changes in the selective binding distribution should be observed as a response to a pH change.The mean conditional affinity can then be written as which differs from the average conditional affinity provided in Sec.III ͓see Eq. ͑21͔͒.This is a consequence from the fact that Eq. ͑21͒ was deduced under the hypothesis of the suitability of the IE approach which now does not apply.

B. A system with multidentate complexation
Let us address the question of multidentate complexation considering a linear chain able to bind protons and metal ions either in monodentate or bidentate bonding as depicted in Fig. 7.This system will be used to exemplify the application of the CAS in describing multidentism phenomena.Interactions between occupied sites are assumed to be negligible.In a recent paper, Koper et al. 33 provided a general method to deal with chelate complexation of metals to linear chains.The method is based on generalizing the classical transfer-matrix techniques used in Ising-type models.By using these techniques, the resulting isotherm for the present binding system is given by where k 1 is the equilibrium constant for the proton, k 2 and k 22 the equilibrium constants for the metal in the monodentate and bidentate forms, respectively, and is the highest eigenvalue of the transference matrix, Under a constant proton concentration, Eq. ͑53͒ becomes a monocomponent isotherm which is plotted in Fig. 8͑a͒.Notice that while at low metal concentration the bidentate binding predominates, at high metal concentration the metal binding becomes monodentate independently of the values of the binding parameters k 2 and k 22 .
Applying the inversion formula ͑12͒ to Eq. ͑53͒, the CAS corresponding to a given pH value can be obtained.However, we will proceed in a different way.Simple algebra shows that isotherm ͑53͒ can be rearranged in the alternative form, The parameters k eff and u eff , which are functions of c 1 , are defined as FIG. 6. CAS of a system with selective ͑1͒ and nonselective sites for the metal at different pH values: ͑a͒ pH = 12, ͑b͒ pH=7, ͑c͒ pH=6, ͑d͒ pH=5, and ͑e͒ pH = 3.The peak ͑1͒ corresponds to the selective sites with weight q s = 0.5 described by a Sips distribution and parameters m = 0.5, k ¯=10 12 .The peak ͑2͒ corresponds to the nonselective sites of weight q ns = 0.5 described by the CAS underlying the NICA isotherm with k 1 =10 5 , k 2 =10 5 , and n = p = 0.5.FIG. 7. Linear chain able to bind protons ͑with stability constant k 1 ͒ and metals.The metals can be bound in two different forms: single bond ͑with stability constant k 2 ͒ and bidentate bond ͑with stability constant k 22 ͒.
For a fixed pH, the isotherm ͑53͒ so expressed, is equivalent to that of a system which merely consists in a linear chain to which a metal can be bound by only monodentate binding but showing an effective interaction energy between occupied sites given by eff =−kT ln u eff .The affinity spectrum of this system, previously reported in Ref. 29 reads pЈ͑xЈ;c 1 ͒ = ln 10 2 Equation ͑59͒ indicates that this affinity spectrum is symmetric with respect to log k eff + log u eff , being its existence always warranted by the fact that u eff Ͻ 1.Using Eq. ͑59͒ for the cases depicted in Fig. 8͑a͒, we obtain the CAS spectra depicted in Fig. 8͑b͒.One can see that two peaks appear, the distance between them being dependent on the value of u eff : the smaller u eff , the wider the spectrum.Since the parameters k eff and u eff are pH dependent, Eqs.͑57͒ and ͑58͒ explain the dependence of the CAS on the pH.Notice in Fig. 8͑b͒ that the apparent heterogeneity increases as the ratio k 22 / k 2 increases.For a better understanding of this behavior, it is useful to analyze the situation that appears in absence of protons.The interaction parameter u eff reduces to indicating that even in this case ͑c 1 =0͒, there is some heterogeneity that can be explained in terms of entropy changes in the binding process when multidentate complexation is present.It is known 30 that the decrease of the number of microstates ͑and thus the increasing of entropy͒ when a new metal is added to the macromolecule is lower in the case of chelate complexation than in monodentate binding, so that, when multidentism is present, the equilibrium constants decrease faster with metal loading.This is reflected in the spectrum through a wider distribution of affinities.On the other hand, when the proton concentration increases the spectrum shifts towards smaller free energies ͑as expected͒, and the apparent heterogeneity becomes smaller.A simple explanation can again be found in Eqs.͑57͒ and ͑58͒: the system behaves with an effective equilibrium con-FIG.8. ͑a͒ 2 vs log͑c 2 ͒ curves as given in Eq. ͑53͒ and corresponding to the model depicted in Fig. 7 for different pH values: ͑a͒ pH=2, ͑b͒ pH=3, ͑c͒ pH=4, ͑d͒ pH=5, ͑e͒ pH=6, ͑f͒ pH=8, and ͑g͒ pH = 12.Parameters: k 1 =10 15 , k 2 =10 8 , and k 22 =10 12 .CAS corresponding to the binding curves depicted in ͑a͒.
stant which is the sum of a term due to bidentate binding ͓k 22 / ͑1+k 1 c 1 ͒ 2 ͔ and a term corresponding to monodentate binding ͓k 2 / ͑1+k 1 c 1 ͔͒.Clearly, the first term is much more affected by a change in the proton activity than the second, since the binding of a bidentate metal can imply, in principle, the removing of two protons.In other words, the chelate contribution, responsible for the heterogeneity of the system, decreases as the pH decreases.In the limiting case of very acidic media ͑c 1 → ϱ͒, the system becomes fully monodentate and homogeneous ͑u eff =1͒.
Finally, it is worth to explore the limiting case in which monodentate binding is absent ͑k 2 → 0͒.Then, the low affinity peak simply disappears from the spectrum.Now it can be shown that the spectrum normalizes to 1 / 2 instead of 1, due to the fact that only half of the sites can be occupied.The resulting spectrum becomes pЈ͑xЈ;c 1 ͒ = ln 10 2͑͑4k eff /kЈ͒ − 1͒ 1/2 .͑61͒ Furthermore, it is easy to show that the average affinity is given by which, as happened in the case discussed in Sec.V A, does not correspond to the expression reported in Sec.III ͓Eq.͑21͔͒ as a consequence of the nonsuitability of IE approach in the case of multidentate complexation.

VI. CONCLUDING REMARKS
The concept of conditional equilibrium constant, a fundamental tool in the study of multicomponent binding, has been extended to describe the competition in systems where the complexing agents can be bound to the ligand according to a distribution of affinities.In this case, the extension of the conditional affinity constant has led to the conditional affinity spectrum ͑CAS͒.The CAS can be understood as an "effective" or "apparent" affinity spectrum experienced by a particular component when the concentrations of the rest of components are kept constant.It has been shown that the CAS always exists if the competitive binding can be described by means of a multicomponent affinity spectrum ͑IE approach͒.However, CAS can also exist even in some cases for which the IE approach does not apply.
Since, via the CAS, a multicomponent system is contracted into an equivalent monocomponent one, all the properties of the monodimensional affinity spectra can be applied.For instance, for a system with two binding cations ͑metal and proton͒, one can obtain the conditional affinity spectrum of the NICA or Frumkin isotherms for the metal ions by simply applying the Sips inversion formula to the multicomponent isotherm, keeping constant the proton concentration.The CAS obtained will now be a function of the pH of the system.
As a result, the mean and the variance of the CAS are functions of the proton concentration.When the IE applies, the mean of the CAS, which we call "mean conditional affinity," can be expressed in terms of the mean free binding energy characteristic of the metal ion ͑corresponding to the system in absence of protons͒, and a term only depending on the proton titration curve.This fact implies that if one plots the apparent affinity as a function of the pH, the curves obtained for different metal ions must be parallel, i.e., they differ just in an additive constant.The verification of this property could be a first test of validity of the affinity spectrum approach for multicomponent systems.
On the other hand, the CAS variance represents a measure of the "conditional heterogeneity" of the reduced system.In contrast with the apparent affinity, the apparent heterogeneity can be expressed as a sum of a constant plus a term coming from the proton titration curve only if there is no correlation between the binding of the metal and protons.If the multicomponent system presents binding correlation, an additional term appears.This dependence of the CAS variance on the binding correlation provides a procedure to compute the correlation matrices corresponding to multicomponent affinity spectra.By using this procedure, analytical expressions for the binding correlation of NICA and Frumkin isotherms were obtained.In this way, the binding parameters of such isotherms can be interpreted in terms of the heterogeneity properties of the system.
Finally, the possibility of extension of CAS to cases for which IE approach fails in describing the system is highlighted.Two cases are considered: selective binding ͑some sites bind only one of the components͒ and multidentism.In both cases the CAS are obtained.Proper dependences of the conditional average affinity on the pH are reported for these cases highlighting that the properties of these CAS spectra are different from those in which the IE approach applies.