Complexation to macromolecules with a large number of sites

This paper presents an approach based on the saddle-point approximation to study the equilibrium interactions between small molecules and macromolecules with a large number of sites. For this case, the application of the Darwin–Fowler method results in very simple expressions for the stoichiometric equilibrium constants and their corresponding free energies in terms of integrals of the binding curve plus a correction term which depends on the first derivatives of the binding curve in the points corresponding to an integer value of the mean occupation number. These expressions are simplified when the number of sites tends to infinity, providing an interpretation of the binding curve in terms of the stoichiometric stability constants. The formalism presented is applied to some simple complexation models, obtaining good values for the free energies involved. When heterogeneous complexation is assumed, simple expressions are obtained to relate the macroscopic description of the binding, given by the stoichiom...


I. INTRODUCTION
The study of the equilibria and kinetics of the interactions between small moleceules and macromolecules provides important information about many biological and environmental processes; typically, the interactions between biological macromolecules ͑proteins, DNA͒ and small molecules are crucial for understanding many metabolic routes; 1,2 in the environment, the complexation of heavy metals with fulvic and humic compounds in soils or in natural waters determines to a large extent their bioavailability, toxicity, and mobility. 3,4 macromolecule usually contains several complexing sites, to which small molecules can be bound, ranging from two sites ͑as is the case in many proteins͒ 5 to a very large number ͑as in polymeric complexation͒. 1,6,7The presence of a large number of complexing sites can enormously complicate the interpretation of the experimental binding data because of the great number of chemical species and physicochemical phenomena involved ͑heterogeneity, positive and negative cooperativity, polyelectrolytic behavior, steric effects, conformational changes, linkage or competition effects, etc͒. 3,82][13] Several magnitudes, such as the Hill coefficient, 11 the binding capacity, introduced by di Cera, 2,5 or the activity coefficients of free and bound sites, 9 have been used to quantify such deviations.
The large number of sites present in many macromolecules complicates the fitting of the stoichiometric constants if no hypothesis on the complexation model is imposed.In some cases the value of the stability constants obtained by a nonlinear fitting of the Adair equation can be unstable and depend strongly on the experimental errors. 14Some general properties of different magnitudes related to coverage data can be extremely useful for improving this fitting, such as the symmetrical properties of the binding curve 2,5 as well as some properties of the activity coefficients of free and bound sites and some characteristics of the average equilibrium function. 9Even when the fitting of the Adair equation is successful, much microscopic information is lost in the global analysis of complexation.The free energy corresponding to a stoichiometric equilibrium is an average energy of all the microscopic species involved ͑with a fixed number of bound small molecules͒.The description of site-specific effects demands resolution of more parameters than those yielded by the global description.Local coverage data are required for a description of binding and linkage effects taking place at individual sites of a multisite macromolecule. 15ome approximate procedures to model the macromolecular complexation processes have also been developed.The most common start assuming a model of complexation.In many cases, if positive cooperativity is not detected, a model with heterogeneous and independent sites is assumed, which involves superposition of local Langmuirian isotherms.Some procedures to fit the affinity spectrum with no a priori assumption of the discrete or continuous set of affinities involved have been developed. 4,16,17Klotz demonstrated that all the complexation models can be expressed as a summation of Langmuirian isotherms, if imaginary stabil-a͒ Author to whom correspondence should be addressed.Electronic mail: fmas@qf.ub.es;Fax: 34-93-402 12 31.ity constants are allowed. 18The demonstration is based on the factorization of the macrocanonical partition function.Nevertheless, as Klotz himself recognizes, imaginary stability constants have no physical meaning.
Another important approximation, valid when the number of sites is large enough, supposes that thermodynamical limit conditions can be applied to a macromolecule. 1,9Darwin and Fowler demonstrated 19 that as the number of terms of a partition function increases, its value can be approximated by taking the maximum term.The assumption of thermodynamical limit conditions, although restrictive, allows to obtain analytical expressions which can be easily used to fit the binding curve.
This work describes a treatment for systems with high number of sites, obtaining the thermodynamical limit value for the stoichiometric stability constants and a first corrective term suited for cases with a number of sites large enough to apply the hypothesis on which the Darwin-Fowler method ͑saddlepoint approximation͒ is based.Section II introduces the nomenclature and some definitions used in the work.Section III describes the application of the Darwin-Fowler method to macromolecular complexation.The discussion of the limiting case of very large number of sites and some consequences of such approximation on the values of the stability constants and on the physical interpretation of the binding curve are included in Sec.IV.The results obtained in Secs.III and IV are applied in Sec.V to simple complexations models; the homogeneous model with interactions between bound sites and the heterogeneous ideal model.The mathematical details are described in the Appendix.

II. THERMODYNAMICAL TREATMENT OF THE COMPLEXATION OF A SMALL MOLECULE TO A MACROMOLECULE
In order to avoid ambiguities, let us assume that:

͑i͒
A complexing site S is a set of coordinating groups of a macromolecule, P, to which a small molecule ͑M͒ ͑known in the biochemical literature as ligand͒ can bind; ͑ii͒ M j P labels the chemical species with the same number j of bound ligands, regardless of the specific sites complexed; ͑iii͒ All full covered macromolecules have the same number, s, of bound ligands; ͑iv͒ M, MS, and S label the so called formal species defined by their concentrations c M ͑concentration of free ligand͒, c MS ϵ ͚ jϭ0 s jc M j P ͑concentration of bound ligand͒ and c S ϵ⌺ jϭ0 s (sϪ j)c M j P ͑concentration of free sites of the macromolecule͒; ͑v͒ Ideal dilute solution behavior for the real species M j P ( jϭ0,...,s) is assumed.
Let us represent the sequential complexation of the macromolecule as where the stability constants for these processes, K j ( j ϭ1,...,s), are known as stoichiometric constants.
The most common techniques used in the study of the macromolecular binding are based on measures of c M .Once this value and that of the total concentrations are known it is possible to calculate the mean occupation number, , the coverage, , or the average equilibrium function, K c , at a concentration c M , as 3,5 ͑c where c T,S is the total concentration of macromolecular sites.The equilibrium relationships corresponding to Eqs. ͑1͒ allow us to relate the mean occupation number to a polynomial in terms of c M , the Adair equation, 2,14 which can be written as

͑5͒
where the coefficients of the polynomial, b j ϵK 1 K 2 ...K j ( j ϭ1,...,s; b 0 ϵ1), are the Adair coefficients, and ⌶(c M ) ϵ ͚ jϭ0 s b j c M j is the macrocanonical partition function.The Adair coefficients lead directly to the stoichiometric constants as K j ϭ(b j /b jϪ1 ).
At very low concentrations of the complexing agent, all complexation systems behave like the ideal one. 9,20The concentration of the formal species do not differ among different complexation models, and the average equilibrium function tends to a constant given by

͑6͒
Because of this limiting behavior, a formalism in which K plays the role of a thermodynamical constant for the process M ϩS⇔M S has been developed, the activity coefficients for formal species relating the concentration of formal species to the concentration that would have been present if the complexation had been ideal.Moreover, at very high concentrations of the complexing agent, the average equilibrium function tends to

͑7͒
which allows us to compute the last stoichiometric stability constant K s , if s is known.
It has also been reported 9 that the macrocanonical partition function, ⌶(c M ), is in fact a polynomial of the product Kc M .Therefore, K is a normalization constant for the coefficients of the macrocanonical partition function, and a new set of coefficients ͕a j ͖ can be defined as b j ϵa j K j ͑8͒ so that ͕a j ͖ are independent of K. Thus, Eq. ͑5͒ can always be rewritten in terms of ͕a j ͖ if a suitable change of the concentration scale is chosen.Therefore, systems with different K can be compared and reduced to the case Kϭ1 with a suitable election of the concentration units.In the following we assume that Kϭ1 without loss of generality.For ideal complexation, the expressions for K j and b j reduce to 5,9 b j id ϭͩ s j ͪK j ; a j id ϭͩ s j ͪ; where the superscript ͑id͒ refers to the ideal complexation case.Any complexation model can be tackled by defining an excess equilibrium constant K j E as and, labeling ⌬ j G 0 the standard Gibbs energy for the complexation equilibrium of the j-ligand, an excess Gibbs energy can defined as indicating the deviation from the ideal complexation case.
The Gibbs energy involved in the binding of j ligands to the naked macromolecule is given by ͑13͒ Finally, we will refer to the binding capacity, B(c M ), a function of c M introduced by di Cera 2,5 and defined as Di Cera has proved 5 that the function B is always positive, and an increase or decrease of c M leads to an increase or decrease of B, respectively.In fact, the binding capacity plays the role of a physical response function such as heat capacity or compressibility, and its positive character is related to the stability principles embodied by the second law of thermodynamics. 21

III. APPLICATION OF THE SADDLE POINT APPROXIMATION TO THE DETERMINATION OF EQUILIBRIUM CONSTANTS
We are interested in obtaining simple approximate expressions for b j which can be useful to avoid unstabilities in the fitting of the Adair equation ͑5͒. 14For high s values, a usual way to deal with problems in statistical mechanics is to apply the classical approximation of Darwin and Fowler, 19 which allows a drastical simplification of the expressions involved.
Let us integrate expression ͑5͒ for the mean number of occupation.Thus, where Equations ͑15͒ and ͑16͒ relate the macrocanonical partition function ⌶ at a concentration c M to the area covered by the curve (c M )/c M vs c M .The Wyman integral equation 2,5 can be easily derived from Eqs. ͑15͒ and ͑16͒.The integration, by parts, of Eq. ͑16͒ leads to Equation ͑18͒, known as the Wyman integral equation indicates that b s can be obtained from the area of the curve ln c M vs .
We are now trying to extend the Wyman integral Eq. ͑18͒ for b s to any b j .Since ⌶(c M ) is a polynomial in c M , the coefficients ͕b j ͖ can be expressed using the Laurent development 22 as where z is a complex variable and the integration takes place over a path containing zϭ0.Let us define j (z) as The Appendix shows that if z takes values on a complex plane, the function j (z) has a local minimum in the real direction and a local maximum in the imaginary direction at zϭc jϩ1 ͑where c jϩ1 indicates the value of c M corresponding to ϭ jϩ1) whenever jϽsϪ1; i.e., zϭc jϩ1 is a saddle point.This is an interesting mathematical property because the integration ͑19͒ can be easily performed on a circle with center zϭ0 and radius rϭc jϩ1 ; since zϭc jϩ1 is the highest maximum on such a circle ͑see Appendix͒, the major contribution to the value of b j , is located close to zϭc jϩ1 , in the imaginary direction.
Performing the integral ͑19͒ as described in the Appendix Eq. ͑A14͒ leads to the following value for b j : which, using the definition of b j in terms of K j , becomes Expression ͑22͒ provides a method to obtain an estimation of the first sϪ2 stoichiometric stability constants from experimental data of ln c M vs .The first term in the rhs of Eq. ͑22͒ corresponds to the area under the curve ln c M vs the mean occupation number, , between ϭ j and ϭ jϩ1.Therefore, for a large number of binding sites, the calculation of the stoichiometric constants only needs a small interval of points around ϭ j.The constants K sϪ1 and K s cannot be calculated by this method because the function j (z) does not have a saddle point for jϭsϪ1 and jϭs as described in the Appendix.Obviously, both K s and b s can be calculated as indicated in Eqs.͑7͒ and ͑18͒, respectively.When K s and b s are known, it is possible to evaluate the values of b sϪ1 , and also K sϪ1 , the remaining constant.

"s˜ؕ…
For many systems, the number of sites per macromolecule is high enough to obtain a good description taking the limit s˜ϱ, an assumption which is known as the thermodynamical limit.We will particularize the results ͑21͒ and ͑22͒ to these conditions in order to deduce simple expressions for the set of stoichiometric constants of complexation and to provide a physical interpretation of the binding curve.
A. An expression for the set of stability constants ˆKj ‰ in the limit s˜ؕ In the limit s˜ϱ, the second term of the rhs of Eq. ͑22͒ disappears.This is easily seen if Eq. ͑22͒ is written in terms of and it is divided by s.For large values of s, it becomes and the standard Gibbs energy, ⌬ j G 0 , can be calculated by computing the area under the curve ln c M vs or between the points (ln c j , ϭj or j ) and (ln c jϩ1 , ϭjϩ1 or jϩ1 ).Furthermore, for very large values of s, expression ͑23͒ can be simplified applying the mean value theorem to the integral

͑24͒
Since the interval of integration in Eq. ͑24͒ decreases as s increases, we can assume Ϸ0 for large values of s; therefore, Eq. ͑23͒ becomes Hence, under thermodynamical limiting conditions, the value of the standard Gibbs energy associated with K j is the opposite of the logarithm of c j .Expression ͑25͒ can also be rewritten as providing a very simple method for obtaining the set of stoichiometric constants and their corresponding Gibbs energies in a system with a large number of sites.Some remarks of the result ͑26͒ should be emphasized: ͑i͒ This result provides a simple physical interpretation to the inverse of the binding curve (ln c M vs v); the value of ln c M at ϭ j gives directly the standard Gibbs energy involved in the complexation of the jth ligand, ⌬ j G 0 ͑see Fig. 1͒.Likewise, a plot of 1/c M vs leads directly, for integer values of , to the stoichiometric stability constants.͑ii͒ If thermodynamical limiting conditions are fulfilled, K j must decrease with j for any complexation model, reaching the values provided by Eq. ͑26͒.Now, we remark the generality of this result.Actually, if K j increases with j for s˜ϱ, (K jϩ1 /K j )у1, Eq. ͑26͒ yields c jϩ1 рc j .This means that the free ligand concentration corresponding to ϭ jϩ1 is lower than the value of c M corresponding to ϭ j and leads immediately to a negative value of the slope vs ln c M (B Ͻ0).Therefore, Eq. ͑26͒ is directly related to the thermodynamical stability condition (BϾ0) for s˜ϱ.͑iii͒ According to Eq. ͑25͒, we can write Thus, for s˜ϱ, the inverse of the B value can be interpreted as a measure of the difference between the affinity of the ligand jϩ1 and the ligand j, i.e., as the standard Gibbs energy needed to complex a new ligand, relative to the standard Gibbs energy needed by the last complexed one.

B. Relationship between the average equilibrium function, K c , and the stoichiometric constants K j in the limit s˜ؕ
According to Eq. ͑10͒, For s large enough the term 1/s can be neglected and using Eq.͑26͒ for the value of K j and Eq.͑4͒ for the average equilibrium function, In terms of ⌬ j G E , Eq. ͑31͒ is rewritten as Thus, for a high enough s value, that of K c /K when the mean number of occupation is an integer is merely the value of the excess stability constant K j E , and its logarithm is proportional to the excess Gibbs energy.

V. APPLICATION TO SOME PARTICULAR COMPLEXATION MODELS
In this section, we apply the results above described to some simple models of complexation; an homogeneous complexation with interaction between sites, and an heterogeneous case of complexation without interaction between sites.Both cases are widely used in the literature.

A. Complexation with interaction between neighboring sites
A simple model to describe systems with interaction between bound species considers a fixed interaction energy ␦ between the nearest neighboring occupied sites ͑1D Ising model͒.The excess stability constant ͑12͒, within the 1Dmean-field approximation, is given by 9,23-25

͑33͒
where ␤ϭ(1/k B T), thus showing that K j depends exponentially on j.
FIG. 2. Exact values ͓in dotted lines with marker ᮀ ͑empty square͔͒ of the Gibbs energies G j 0 ͑a͒, increments of Gibbs energies (⌬ j G 0 ) ͑b͒, and excess Gibbs energies (⌬ j G E ) ͑c͒, for the interaction model of complexation with interaction parameter ␤␦ϭ2 ͑negative cooperativity͒, number of sites s ϭ20 and Kϭ1 M Ϫ1 , compared with those obtained by using the different approximations proposed in this work; marker ᭹ ͑filled circle͒ for approximation given by Eqs.͑21͒ ͑a͒ and ͑22͒ ͑b͒; marker ᭝ ͑empty triangle͒ for approximations given by Eqs.͑25͒ ͑b͒ and ͑32͒ ͑c͒.
As described in Sec.II, results obtained for systems with any value of K can be compared with those present here if a suitable change of the concentration scale is performed.As K is merely a concentration scale factor we have used Kϭ1 in the present calculations.
As usual, two contributions ͑an ''enthalpic'' or ''energetic'' effect and an ''entropic'' effect͒ can be considered in G j 0 (ϭH j 0 ϪTS j 0 ) and ⌬ j G 0 (ϭ⌬ j H 0 ϪT⌬ j S 0 ).Taking K ϭ1, the enthalpic contribution only includes the effect of the lateral interactions ͑positive for repulsive interactions and negative for attractive ones͒.Concerning the entropic contribution, while jϽs/2, S j 0 increases as j increases and then S j 0 decreases for jϾs/2 because the number of accessible microstates for the binding of j ligands to the naked macromolecule decreases.Accordingly, ⌬ j S 0 is a monotonically decreasing function taking positive values for jϽs/2 and negative ones for jϾs/2 regardless the sign of ␦.
Figures 2͑a͒, 2͑b͒, and 2͑c͒ show the exact G j 0 , ⌬ j G 0 , and ⌬ j G E ( jϭ1,...,sϪ2) values obtained with Eq. ͑33͒ for a case with repulsive interactions (␤␦ϭ2) and sϭ20 ͑as an estimation of the thermodynamic limit conditions͒ together with the results obtained from the approximate expressions ͑21͒, ͑22͒ proposed in this work and their corresponding limits when s˜ϱ, Eqs.͑25͒ and ͑32͒.The exact G j 0 values are in good agreement to those obtained with the approximate expression ͑21͒, as Fig. 2͑a͒ shows.A good estimation is also obtained for ⌬ j G 0 ͑related to the stoichiometric equilibrium constant, K j ) using the approximate expressions ͑22͒ and ͑25͒ in Fig. 2͑b͒ and a comparison between the exact ⌬ j G E values and those obtained from the average equilibrium function, K c , by using the approximate expression ͑32͒, is shown in Fig. 2͑c͒.Decreasing s ͑see Fig. 3, where the same calculations as Fig. 2 have been performed for a smaller number of sites, sϭ10), the thermodynamical limit approximate expressions begin to deviate, especially Eq. ͑25͒ for ⌬ j G 0 ͓Fig.3͑b͔͒, but the corrective term given by Eq. ͑21͒ for G j 0 ͓Fig.3͑a͔͒ and by Eq. ͑22͒ for ⌬ j G 0 ͓Fig.3͑b͔͒, yield still accurate results.
As Figs. 2 and 3 correspond to a repulsive case, for low values of j, G j 0 decreases with increasing j because the entropic contribution predominates over the enthalpic one, whereas for intermediate and large values of j, the repulsive interaction becomes more important and G j 0 increases even to positive values.Accordingly, ⌬ j G 0 and ⌬ j G E always increase with j and, therefore, K j and K c always decrease with j, as expected.
When positive cooperativity is considered ͑see Fig. 4, where ␤␦ϭϪ1) only the entropic contribution for jϾs/2 contributes to increase G j 0 .So, G j 0 decreases with increasing j until a j value larger than that found in the repulsive case.For this attractive case, the agreement between the real and approximate values of G j 0 is not so good ͓Fig.4͑a͔͒ as it is in the repulsive case ͑Fig.2͒, especially for the lowest and the highest values of j.This can be explained from the relationship between the coverage and the ligand concentration c M , which under thermodynamical limiting conditions is Kc M ϭ͑1Ϫ ͒exp͑ Ϫ2␤␦ ͒. ͑34͒ The condition BϾ0 ͓the binding capacity defined in Eq. ͑14͔͒, applied to Eq. ͑34͒, leads to

␤␦ϾϪ2 ͑35͒
which indicates that the system is thermodynamically stable for s˜ϱ only when ␤␦ϾϪ2.This is a consequence of the use of the mean-field approximation ͑which implies a phase transition at ␤␦ϭϪ2 for the 1D-Ising model͒. 19,21Since approximations ͑21͒-͑22͒, ͑25͒, and ͑32͒ have been deduced using the thermodynamic limit as the starting point, their values will diverge from the real ones as we approach ␤␦ ϭϪ2, since the model used leads to a non realistic description of the system if ␤␦ϽϪ2 as s increases.This difficulty can be avoided taking into account the correlations between the interactions of different ligands in the macromolecule, which we neglected with the mean-field approximation.

B. Heterogeneous complexation: Relationship between the stoichiometric and the intrinsic stability constants
Due to its direct microscopic interpretation, the heterogeneous model is widely used especially when there is some previous physical information suggesting the consideration of sites with different affinities.
Let us consider a system with m kinds of sites, and let s i be the number of sites of type i with an intrinsic stability FIG. 3. As in Fig. 2͑a͒ and Fig. 2͑b͒ but with the number of sites sϭ10.
constant k i (iϭ1,...,m).Assuming a Langmuirian local isotherm, the mean occupation number can be related to the ligand concentration, without assuming thermodynamical limiting conditions, by The relationship between the stoichiometric constants and the intrinsic stability constants described by Klotz 10 is cumbersome for large s values.Nevertheless, expression ͑25͒, valid for the stoichiometric stability constants in the limit s˜ϱ, together with Eq. ͑36͒, provide a very simple expression for such relationship, valid for large s values, Equation ͑37͒ indicates that the stoichiometric stability constants K j can be considered as an average of the intrinsic stability constants, where w i, j , the weighting factors, corresponds to the mean number of free sites of the ith type at ϭ j and i, j ϵ(k i c M )/(1ϩk i c M ) c M ϭc i and i, j ϵs i i, j are the coverage and the mean occupation number of the sites of type i at the free ligand concentration c j .Thus, K j are calculated as an average of the intrinsic stability constants k i with a weighting factor given by the mean number of free sites of type i at the concentrations c j , respectively.This indicates that the stoichiometric stability constant of the binding of an additional ligand to a macromolecule P is an average of the intrinsic stability constants not yet occupied.
The relationship between K c and the intrinsic stability constants is easily obtained.From Eqs. ͑4͒ and ͑36͒ it is obtained 26

͑39͒
where x i ϵ(s i Ϫ i )/(sϪ) provided that ͚ iϭ1 m x i ϭ1, and i is the mean occupation number of the sites of ith type at the concentration c M .
This expression holds independently of the s-value and indicates that the average equilibrium constant is an average of the intrinsic stability constants with a weighting factor given by the fraction of mean number of free sites of type i with respect to the total of free sites at the concentration c M .
Figures 5͑a͒ and 5͑b͒ consider a heterogeneous system with two types of sites with the same probability (s 1 ϭs 2 ), but with a different total number of sites s, 16 and 30, respectively.The approximate expressions ͑22͒ and ͑25͒ are used to reproduce the value of the free energies involved.In general, a good accordance between the exact results and those given by ͑22͒ and ͑25͒ is obtained, especially for the highest number of sites (sϭ30).As expressions ͑22͒ and ͑25͒ work out the results for the stoichiometric constants from the plot 1/c M vs ͓Fig.1͑b͔͒, care must be taken to minimize the experimental error in such plots, because they have direct influence on the stoichiometric constants.
Regarding the results described above: ͑i͒ Equation ͑37͒ relates K j to the intrinsic stability constants k i .Replacing c j in Eq. ͑37͒ in terms of K j , we have Expression ͑40͒ relates the Adair approach, a macroscopic description of the binding, to a microscopic or local formalism widely used in complexation problems; the affinity spectrum method ͓usually, in this last formalism, the summation involved in Eq. ͑40͒ is replaced by an integral when a continuous distribution of intrinsic stability constants is assumed͔.͑ii͒ In some cases, it is reasonable to replace the local Langmuirian isotherm in Eq. ͑36͒ by other local isotherms ͑for instance, the Fowler-Guggenheim or Frunkim isotherm͒ of the kind f (k i ,c M ).Then Eq. ͑36͒ could be generalized as FIG. 4. As in Fig. 3 but with ␤␦ϭϪ1 ͑positive cooperativity͒ and sϭ20.
In such a case, a relationship between microscopic and macroscopic parameters like Eq. ͑40͒ can also be written as which generalizes the relationship between the Adair approach with the affinity spectrum method given by Eq. ͑40͒ for a wide variety of local isotherms used.

VI. CONCLUDING REMARKS
The Darwin-Fowler method, a classical result of statistical thermodynamics, has been reviewed in order to obtain approximate expressions for the stoichiometric constants of the complexation of small ligands to macromolecules with a number of sites so high that thermodynamical limiting conditions are used.The Darwin-Fowler method is based on the so-called saddle point approximation.The goodness of such approximation to analise complexation problems is studied in this work.
Very simple expressions for the value of the first sϪ2 Gibbs energies of the sequential complexation processes have been deduced.These Gibbs energies are calculated directly from the binding curve without making any hypothesis on the model of complexation involved, and contain two contributions; the first one is an integral term corresponding to the area under the curve ln c M vs the mean occupation number between j and jϩ1.The second contribution, which can be considered as a corrective term, depends on the binding curve derivatives on the points with an integer value for the mean occupation number.This last term can be neglected for high s values.
The value of the stoichiometric stability constants K j tends to the inverse of the free ligand concentration at the point of mean occupation number equal to j as the number of sites s increases.This provides a simple interpretation of the average equilibrium function, K c , which takes the value of the excess stability constant, K j E , at integer values of the mean occupation number (ϭ j).
The expressions derived are applied to a simple heterogeneous complexation case and to an homogeneous complexation with interaction between bound sites.The results obtained reproduce quite well the real values of the stability constants, and the accuracy increases as s increases.
For the heterogeneous case, in the limit s˜ϱ, we have derived very simple expressions to relate the stoichiometric stability constants ͑a macroscopic description of the com-plexation͒ with the intrinsic or microscopic stability constants represented in the affinity spectrum.

APPENDIX: APPLICATION OF THE SADDLE POINT APPROXIMATION TO THE MACROMOLECULAR BINDING
Coefficients b j of the macrocanonical partition function are obtained as the result of the integral ͑19͒ where z takes values in the complex plane and C is a closed path on this plane around zϭ0.
We have defined an auxiliar function j (z) in Eq. ͑20͒ as Equation ͑A2͒ indicates that the function j (z), the integrand of Eq. ͑A1͒, is an analytical function except for z ϭ0.Therefore, j (z), is an harmonic function 22 5. Exact increments of Gibbs energies (⌬ j G 0 ) for a heterogeneous system with two kinds of independent sites (k 1 ϭ10 M Ϫ1 , k 2 ϭ10 4 M Ϫ1 ), compared with those obtained by using the different approximations proposed in this work with markers as in Fig. 2͑b͒ ͑a͒ s 1 ϭs 2 ϭ8; ͑b͒ s 1 ϭs 2 ϭ15.
where x and y indicate, respectively, the real and imaginary components of the complex variable z.
Restricting z to real values, whereas jϽsϪ1, j (z) has a minimum along this x-axis at some point zϭz j because exp(sF(z)) is a monotonically increasing function ͑a polynomial of degree s with all the coefficients b i being positive͒ and 1/z jϩ1 is a monotonically decreasing function along the real axis.Hence Eq. ͑A3͒ for zϭz j leads to where the first equality is derived from the condition of extreme at zϭz j , the second indicates the presence of a minimum in the real direction and the last condition comes from Eq. ͑A3͒.j (z) is an analytical function and satisfies the Cauchy-Riemann equations, 22 which ensures that the condition of extreme for j (z j ) yields to ‫ץ(‬ j /‫ץ‬x) zϭz j ϭϪi(‫ץ‬ j /‫ץ‬y) zϭz j ϭ0, which together with the last inequality of Eq. ͑A4͒ indicates the simultaneous presence of a maximum for j (z) along the imaginary direction at z ϭz j .Therefore, z j is a saddle point on the complex plane.For jϭsϪ1 or jϭs there is no minimum for j (z) along the real axis, since function j (z) decreases monotonically.
It is easy to compute the real value z j corresponding to the minimum of j (z).The condition of extreme, j Ј(z ϭz j )ϭ0, using Eq.͑A2͒ becomes which, taking into account that the lhs ͑ϵleft hand side͒ of Eq. ͑A5͒ corresponds to the mean occupation number, (5), indicates that at zϭz j , ϭ jϩ1.Thus, z j corresponds to the concentration c M for which ϭ jϩ1, labeled as c jϩ1 , z j ϭc jϩ1 ; ͑c jϩ1 ͒ϭs͑ c jϩ1 ͒ϭ jϩ1.͑A6͒ To calculate the second derivative of j (z) at zϭz j we use the auxiliar function where it has been used the condition of extreme j Ј(c jϩ1 ) ϭg j Ј(c jϩ1 )ϭ0.Equation ͑A8͒ gives the concavity of j (z) at zϭz j .It indicates that the concavity increases with s leading to a sharper maximum on the imaginary direction and a deeper minimum on the real direction appear.If we choose the contour of integration of Eq. ͑A1͒ to be a circle about zϭ0 with radius c jϩ1 , the contour will pass through z j in the imaginary direction, in which point and direction the integrand has an extremely sharp maximum.This is the highest maximum on the circle.The absolute value of j (z) over the circle with ͉z͉ϭc jϩ1 is

͑A9͒
which reaches its maximum value when all the terms are real.As all the b i coefficients are real and positive, all the terms are also real and positive if and only if zϭz j .Hence, j (z) reaches the highest maximum along the circle ͉z͉ ϭc jϩ1 over the real axis.
As there is no maximum comparable in height along the contour, the main contribution to the integral ͑A1͒ comes from the neighborhood of c jϩ1 .Expanding g j (z) around z ϭc jϩ1 , g͑z ͒ϭg͑ c jϩ1 ͒ϩ 1 2 gЉ͑c jϩ1 ͒͑ zϪc jϩ1 ͒ 2 ϩ¯͑A10͒ replacing Eq. ͑A10͒ in Eq. ͑A1͒, and using the polar form of z and dz, providing that the circle of integration has the radius ͉z͉ϭc jϩ1 (zϭc jϩ1 e i , dzϭic jϩ1 e i d), the integral ͑A1͒ becomes where it has been taken as the real part since b j is real.
From the definition of g(z) and Eq.͑A2͒, gЉ(c jϩ1 ) can be evaluated as If the integral ͑A11͒ is performed only in a small interval around c jϩ1 ,e i can be replaced by 1ϩi, and taking into account that the only relevant contribution to the integral is located in this small interval, the limits of integration can be extended to ͑Ϫϱ, ϱ͒.
Then, substituting Eq. ͑A12͒ into the integral ͑A11͒, the evaluation of the resulting quadratic integral yields  mean number of the free sites of the ith type x real part of the complex variable z x i fraction of free sites of the ith type with respect to the total number of free sites y imaginary part of the complex variable z z complex variable zϭxϩyi Greek letters ␤ϭ1/k B T inverse of the thermic energy ␦ interaction energy between adjacent bound ligands j (z) auxiliar complex function related to the complex macrocanonical partition function phase of a complex variable ͑dimensionless real variable͒ mean occupation number i mean occupation number of sites of the ith type i, j mean occupation number of sites of the ith type when ϭ j coverage or fraction of occupied sites in a macromolecule i, j coverage of sites of the ith type when vϭ j ⌶ macrocanonical partition function of the system ͕M j P; jϭ0,...,s͖ Superindices id ideal 0 standard state

FIG. 1 .
FIG.1.Schematic plot of the inverse of the binding curve, ln c M vs .The value of ln c M at the integer values of ϭ j yields directly, in the limit s˜ϱ, Ϫln K j .
ln b j ϭϪ ͵ Ϫϱ ln c jϩ1 ͑c M ͒d ln c M Ϫ͑ jϩ1 ͒ln c jϩ1 ln c M ͪ c M ϭc jϩ1 ͮ ͑A13͒and integrating the first term of the rhs of Eq. ͑A13͒ by parts, this equation can be written as ln b j ϭϪ ͵