Statistical mechanical approach to competitive binding of metal ions to multi-center receptors

A microscopic site binding model to treat binding of several metal ions to multi-center receptors is proposed.

The model introduces the appropriate parameterization in terms of microscopic complexation constants and metal–metal pair interaction energies.

The model is solved with statistical mechanical techniques, including direct enumeration or transfer matrices.

We obtain microscopic and macroscopic complexation constants, microstate probabilities, and binding isotherms for chain-like receptors, including the long-chain limit.

Various examples to illustrate the usefulness of the model are given.


Interactions of metal ions with ligands represent an important topic in coordination chemistry.

In the past, research mainly focussed on mono-center complexes containing a single metal ion and one or more ligands.

More recently, attention shifted toward supramolecular multi-center assemblies, which result from the interactions between several ligands and metal ions.1

Researchers have focussed on the preparation and structural characterization of sophisticated metallo-supramolecular architectures, which can be obtained in high-yield by the simple mixing of the various components (i.e., self-assembly).2

However, the thermodynamic basis of the formation mechanisms of such complexes is poorly understood, and has not been addressed in much detail so far.

Proper understanding of these processes is essential for stimulating molecular programming, and for switching from the usual deductive chemical approach toward induction.3

The latter bottom-up process necessitates an exploitation and modelling of the experimental kinetic and thermodynamic data on complexation reactions.

In this context, multi-metal supramolecular lanthanide complexes are particularly interesting, as the search for new receptors displaying selective binding along the 4f-block series provides a substantial database of macroscopic and microscopic stability constants.4

The recent design of heterometallic f–f′ complexes evidencing large deviations from statistical distributions in solution demonstrates the need for mathematical models of multi-center complexation processes.5

The partial rationalization of the formation of the heterobimetallic and trimetallic complexes represents a first step toward this goal.6–11

Examples of such complexes are shown in Fig. 1, and the structure of the ligands in Fig. 2.

The formation of metal–ligand complexes can be quantified in terms of chemical equilibria and the corresponding complexation constants.

However, as soon as multi-center complexes are being considered, the number of constants becomes very large, and this approach becomes impractical.

For this reason, a parameterization of these constants in terms of only a few chemically meaningful parameters is necessary.

Ercolani's work12 focusses exactly on this problem, as he attempts to express formation constants of multi-center complexes in terms of two intrinsic constants and symmetry numbers.

A classical topic, where such questions arise, are acid–base equilibria of polyelectrolytes (i.e., proton binding).

In the early fifties, Kirkwood,13 Steiner,14 and Marcus15 had shown how to formulate a site binding model for this situation, and this approach has been extended until recently.16,17

The free energy of a microscopic protonation state is expressed in terms of a small number of parameters.

Normally, these include microscopic binding constants for each site and site–site pair interaction energies.

For an infinite polyelectrolyte chain, the binding isotherms can be evaluated by standard techniques borrowed from statistical mechanics.

For finite oligomers, one recovers the classical description in terms of chemical equilibria, and the corresponding protonation constants can be expressed in terms of microscopic binding constants and pair interaction energies.17

This parameterization appears to be the appropriate one to obtain the necessary simplification of the description of acid–base equilibria of polyprotic acids or bases.18–20

One encounters a similar problem, when one attempts to interpret the binding of metal ions by polyelectrolytes.21–25

Such charged macromolecules have linear or branched structures, and offer a large number of coordination sites.

Polyelectrolytes may therefore bind a larger number of metal ions.

For example, dendrimers have been reported to complex several hundreds of copper(ii) ions.22

Again, such systems cannot be described by a small number of complexation constants, and an alternative approach is needed.

To approach such questions, microscopic site binding models have been proposed to address the competition between protons and multi-dentate metal binding to polyelectrolytes.21

In this publication, we develop this topic further and discuss a site binding model to describe binding of different metal ions to a multi-center receptor with well-defined binding sites, see Fig. 1.

The receptor may consist of several preassembled ligands.

This paper is structured as follows.

First, we introduce the appropriate parameterization in terms of microscopic complexation constants and metal–metal pair interaction energies, elucidate the differences between microscopic and macroscopic equilibria in these systems, and extend the description to macromolecular receptors.

The model will be first discussed with simple examples, including small and infinitely large receptors.

In the last section the applicability of the model to experimental binding constants is illustrated.

The experimentally inclined reader is welcome to proceed to this section directly.

Binding of metal ions to a multi-center ligand receptor

Consider a receptor with several binding sites, each of which can bind a single metal ion.

Once the site is occupied with a metal ion, it can no longer bind any others.

If a metal ion of type ω is bound to a site i, the corresponding element of the state vector s(ω)i is defined to be unity, and all the other elements are zero.

The state of all sites can be then described with a state vector si.

To a first approximation, the effective free energy of the metal-receptor complex can be written as a general linear and quadratic expression in the state variables, namely17,21,26 where the chemical potential term can be related to the activity of metal αω as µ(ω)i = kT ln αωK(ω)i, where K(ω)i is the microscopic association constant of the center i for the metal of type ω, and kT the thermal energy.

We have further denoted by E(ωη)ij the pair interaction energy between two metal ions ω and η occupying the sites i and j.

Note that the pair interaction energy is symmetric upon exchange of the indices ω and η, as well as i and j.

The free energy eqn. (1) is a straightforward generalization of the treatment of protonation equilibria.16,17

Given the free energy of the metal–receptor complex, we can express the semi-grand canonical partition function as26where the sum extends over all possible states {si}.

The binding isotherm for the metal ω and center i can be written as a thermal average over the state variable, namelywhere the thermal average of a function f({si}) is defined as.

The overall binding fraction of the metal ω can be expressed aswhere N is the total number of sites of the receptor.

These isotherms obey the thermodynamic consistency relation

Macroscopic equilibrium constants

Based on the present description, one can obtain expressions for equilibrium constants for the chemical equilibria between different species.

For the sake of simplicity, let us consider two types of metal ions M(1) and M(2) only.

The generalization of the following arguments to an arbitrary number of metal ions is straightforward.

Commonly, solution equilibria are described in terms of macrospecies, where only the total number and type of metal ions are specified, but it is not specified to which site they are bound.

The formation of such macrospecies can be described in terms of a chemical reaction L + nM(1) + mM(2) ⇌ LM(1)nM(2)mwhich can be characterized in terms of macroscopic cumulative equilibrium constants βnm.

The latter can be obtained by expanding the partition function in terms of the activities a1 and a2, and as the result one obtains the binding polynomial.

From this expression, the probability of a macrospecies, or in other words its relative concentration, can be expressed as.

From these macrospecies probabilities, the binding isotherms can be directly obtained as.

These expressions reflect the basic fact that the overall binding isotherms depend on the probabilities of the corresponding macrostates.

Microscopic equilibrium constants

Each macrospecies normally corresponds to a collection of several microspecies.

A microspecies is defined with a set of given state variable vectors {si}.

The probability of such a species (i.e., its relative concentration) can be written as.

Let us express the macrostate probability in terms of the microstate probabilities where δn,m is unity for n = m and zero otherwise.

From this expression we find that the probability of the microspecies can be split into two terms, namely, p({si}) = π({si}|nm)Pnm(α1,α2)the macrostate probability Pnm(α1,α2), and the conditional microstate probability where the Kronecker's δn,m is unity for n = m and zero otherwise.

This conditional probability expresses the relative probability to find a given microstate within a particular macrostate.

The important advantage of introducing this conditional probability is that this quantity is a constant and independent of the metal activities.

The entire concentration dependence is captured by the macrostate probability.

Equilibrium constants of microscopic equilibria can be discussed in this framework as well.

Recall that the binding state of the receptor is uniquely defined with a set of given state variable vectors {si}.

Consider now a chemical equilibrium between two species L({si}) ⇌ L({si})the state vectors of which are identical with the exception of the elements sj(λ)j = 0 and s′(λ)j = 1.

This reaction corresponds to a microscopic equilibrium, where the site j eliminates a metal of type λ, but the binding state of the rest of receptor remains unchanged.

The successive equilibrium constant for the reaction eqn. (16) can be obtained from the free energy difference of these states, and yields the result.

Note that the previously introduced binding constant K(λ)j corresponds to the inverse of the microconstant for the release of a metal of type λ from site j when all other sites of the receptor are empty.

Transfer matrix techniques

Different binding states of receptors with a small number of sites can be enumerated by hand.

Larger structures can be enumerated on a computer, even though the number of sites accessible with this technique remains limited to about 20–30.

Larger systems can be either treated by Monte Carlo simulation, or certain structures by transfer matrix techniques.27

In particular, the latter can be used to address systems in the limit of large number of sites in a straightforward fashion.

To introduce the transfer matrix technique, consider a linear structure and nearest neighbor pair range interactions.27

Thereby, one introduces a recursion relation between the partition function of a chain with n sites and with n + 1.

In order to define this recursion, let us consider a restricted partition function for a sub-chain of length n, such that the values of the state vector of the last site are restricted to given values s, namely where δs,s is unity when s = s′ and zero otherwise.

By arranging the elements of the restricted partition function Ξn(s) into a column vector Ξn, the transfer matrix T can be defined as Ξn + 1 = TΞn.

Adopting this notation, we assume that the sites are numbered from the left to the right.

As will be discussed in the following, transfer matrices have simple forms for nearest neighbor pair interactions.

The recursion relation eqn. (19) can be initiated with the partition vector for the first site Ξ1.

However, it is usually more practical to generate this restricted partition function by the application of the transfer matrix to a suitably chosen generating vector Vg such that Ξ1 = TVg.

As will be exemplified below, the advantage of introducing this vector is its simple structure.

The partition function of the entire chain can be then obtained by summing over all the states of the sub-chain.

Again, this operation can be formulated in a matrix notation, by defining a terminating vector Vt.

Usually this vector contains unit entries, and one can write eqn. (20) as a scalar product Ξ = ṼtΞNwhere Ṽt denotes the transposed (row) vector.

The partition function can be now written in a compact fashion as Ξ = ṼtTNVg.

Based on this formulation, many properties of finite and infinite chains can be obtained in a straightforward way.

For example, the long chain limit is typically dominated with the largest eigenvalue λ of the transfer matrix T.

The partition function is asymptotically given by ΞλN for N → ∞If the largest eigenvalue cannot be found analytically, it can be easily calculated numerically.

The binding isotherms are then obtained from eqn. (5).

Binding of a single metal ion

This situation is completely analogous to the situation of proton binding discussed in detail elsewhere.14–17

Nevertheless, it is important to understand this case for our discussion of the multicomponent case.

For this reason, a few major features will be summarized here.

In this situation, the state vector has only one element, and the index ω can be suppressed.

In the situation of identical sites and nearest neighbor interactions the free energy is given by where µ is a chemical potential term, and Eij is the pair interaction energy between the sites i and j.

Due to Coulombic forces between charged metal ions, the interactions are repulsive and Eij > 0.

The chemical potential term µ can be related to the metal ion activity a and the microscopic binding constant K of the metal to a binding center when all other centers are empty by µ = kT ln aK.

Let us now consider three examples.

The first two concern small receptors, and the last one the infinite linear chain.

The situation of a symmetric receptor with three equivalent sites is simple to discuss, see Fig. 3.

Since all the sites are equivalent, there is only one type of the metal–receptor complex, and the microstate is equivalent to the macrostate.

The microscopic binding constant to bind a metal ion to the site will be denoted as K, which is the inverse of the microconstant to remove it.

The macroscopic formation constant β10 involves a statistical factor of 3, since there are three sites on which the metal ion can bind to the receptor.

When two metal ions bind to the receptor, there is again only one type of the metal–receptor complex, and the microstate is also equivalent to the macrostate.

The macroscopic formation constant β20 involves the square of the microconstant, as two metal ions are being bound, the statistical factor of 3, and the pair interaction parameter u, which is related to the pair interaction energy E by E = −kT ln u.

The inverse microconstant for removing a metal ion involves the binding constant K and one pair interaction u.

The macroconstant β30 for the fully occupied receptor involves the cube of the microconstant, three pair interactions, and no statistical factor.

When we remove a metal ion of the fully occupied receptor, we overcome the binding constant and two pair interactions, obtaining the inverse of Ku2.

The second example involves a linear receptor, with two equivalent terminal sites, and one central site, see Fig. 4.

There are two possible microstates, when one binds one metal ion to the receptor, namely either to the terminal site or to the central site.

We assume that the microconstant of these sites is the same for all sites, and will be denoted by K. The inverse of this value corresponds to the microconstant for the removal of the metal ion.

Since these constants are the same for all three sites, the conditional probability to find the microspecies with the central site occupied is 1/3, while when the terminal site is occupied 2/3.

The latter factor of two arises since there are two equivalent terminal sites.

The macroconstant β10 is given by three times the microconstant K, since each species contributes in the same way.

One also obtains two microspecies when the receptor binds two metal ions.

In the first species, one terminal site and one central site are occupied.

This microspecies invokes a nearest neighbor pair interaction, denoted by the parameter u, and the square of the microconstants.

To remove a metal ion from this species, the inverse microconstant involves a factor K and we eliminate a pair interaction, which involves a factor u.

In the other microspecies, both terminal sites are occupied.

For this species, which has a symmetry number of 2, it may be necessary to consider the typically weaker next neighbor pair interaction.

However, this interaction will be neglected here.

The microconstant to remove a metal ion is now given by the inverse of K. Summing up the contributions of both species, we thus find the macroconstant β20 and the conditional probabilities of each microspecies.

The receptor fully occupied with metal ions invokes two nearest neighbor pair interactions, and thus invokes a factor u2.

From these considerations, we obtain the conditional microstate probabilities and the macroconstants shown in Fig. 4.

To remove a metal ion, there are two different microconstants as the terminal site involves only one pair interaction, and the central one two.

In the case of linear chain, one can use the transfer matrix technique to obtain these results.

In the case of a single metal ion, equivalent sites, and nearest neighbor interactions, the transfer matrix is given by16,21,27 where we have introduced the reduced activity z = αK.The two supplementary (transposed) partition vectors are Ṽg = (1,0) and Ṽt = (1, 1).

The partition functions can be evaluated by inserting these quantities into eqn. (22).

For example, for a chain with three sites (N = 3) one obtains Ξ = 1 + 3z + (1 + 2u)z2 + u2z3.

Inserting eqn. (26) in eqn. (27) and comparing term-by-term with eqn. (8) the macroscopic binding constants βn0 follow, in agreement with Fig. 4.

Let us now consider the long chain limit as the last example.

This model is equivalent to the so-called Ising model, which is amply discussed in the statistical mechanics literature.27

The largest eigenvalue of eqn. (25) can be evaluated analytically as.

With eqns. (23) and (5), an analytical expression of the binding isotherm of a linear chain follows θ(z) = [2 + (λ(z)/z)(1 − zu)/(1 − u + λ(z)u)]−1.

Note that this isotherm is identical to the case of proton binding to a polyelectrolyte.15,16,21

In the absence of interactions (E = 0 or u = 1), this isotherm reduces to the familiar Langmuir isotherm.

This behavior of this isotherm is shown in Fig. 5.

In the absence of interactions, the isotherm has the classical sigmoidal shape in the semi-logarithmic representation.

With increasing repulsive interactions, it develops a two-hump structure with an intermediate plateau at θ = 1/2.

The intermediate plateau originates from the stabilization of an intermediate structure, where a bound metal ion alternates with an empty site, see Fig. 6(a).

With attractive interactions, a sharp transition takes place.

Within this transitions, the metal ions cluster together, see Fig. 6(b).

For repulsive interactions, the first hump in the binding isotherm is located at log a = log K, while the second hump at log a = log K −2ε where ε = −log10u = E/(kTln10).

The location of these humps is simply understood.

Initially, the metal ions bind to the empty sites, and thus with the microconstant log K. Beyond the plateau, the metal ions must bind between two occupied centers.

They have to overcome two pair interactions, and thus bind with a microconstant of log K − 2ε.

Competitive binding of several metal ions

Let us now present the treatment of two types of metal ions using the three examples already discussed.

The first example is the symmetric receptor with three equivalent sites.

In the presence of two metal ions, three types of complexes form.

Complexes will form only with the first metal ion, provided the activity of the second metal ions is small, see Fig. 3.

The same complexes form with the second metal ion, when the activity of the first metal ion is small.

In the intermediate situation, one obtains mixed complexes shown in Fig. 7.

Obviously, there are only mixed complexes with two or three metal ions.

Due to symmetry, the microspecies are again equivalent to the macrospecies.

The complex with two different metal ions involves one pair interaction energy E(12), and this interaction will be abbreviated with the parameter u12.

The macroconstant β11 is obtained by realizing that the microspecies has a symmetry number of six.

The microconstants to remove each metal ion can be again evaluated by considering the affinities of the metal ions to the sites and by considering the pair interaction, and their inverse values are given in Fig. 7.

There is one complex involving three metal ions, with one from one kind and two from the other kind.

This complex involves three pair interactions, namely one between the same metal ions, and two between different ones.

The macroconstants β12 and β21 of both complexes follow by taking into account the symmetry number of three.

Let us now consider the second example of the linear receptor with three sites, which has two equivalent terminal sites and one central site.

For the matter of illustration, let us assume that the affinity of the terminal and central sites are the same.

Again, three types of complexes will form.

Complexes with the first metal ion for sufficiently low activity of the second metal ion, complexes with the second metal ion for sufficiently low activity of the first metal ion, and mixed complexes.

The complexes with one metal ion has been discussed above, see Fig. 4, while Fig. 8 summarizes the mixed complexes.

There are three nonequivalent microstates, each of which has a symmetry number of two.

Either one metal ion occupies the central site, and a nearest neighbor pair interaction has to be considered in this case.

This case leads to two microstates.

Alternatively, both metal ions occupy the terminal sites.

These three microspecies have to be considered, when the macroconstant β11 of the complex with two different metal ions is evaluated.

The inverse values of the various microconstants to remove the respective metal ions are also indicated.

When three metal ions are bound, two sites are occupied by one type of the metal ion, and the remaining site with the second type.

Two macrospecies are possible, depending whether there are two metal ions of type one or of type two.

In either situation, we have two microstates, depending on the fact whether the two terminal sites are occupied with the same metal ion or with two different ones.

The macroconstants β12 and β21 are again obtained by considering the pair interactions and the symmetry numbers.

The microconstants follow as above.

The transfer matrix formalism can be extended to competitive metal binding to linear receptors in a straightforward fashion.

Consider two metal ions and assume nearest neighbor interactions.

The transfer matrix reads where we have introduced the abbreviations zω = αωK(ω) and uωη = exp(−E(ωη)/kT).

The generating and terminating vectors are given by g = (1,0,0) and t = (1,1,1).

Again, the properties of short chains can be evaluated analytically.

For example, the trimer (N = 3) is now described with a partition function.

Again, by equating the coefficients of eqn. (32) with eqn. (8) one obtains the macroscopic binding constants βnm, as shown in Figs. 4 and 8.

In long chain limit, the largest eigenvalue cannot be evaluated analytically in the general case.

However, an analytical solution is possible if we specialize to the important case of equal pair interactions, namely u11 = u12 = u22 = u.

The solution can be expressed in terms of the solution of the linear chain for a single metal ion.

Some algebra leads to the closed form of the adsorption isotherm where θ(z) is the mono-component isotherm given by eqn. (29) evaluated at z = z1 + z2.

In the absence of interactions, this isotherm reduces to the competitive Langmuir isotherm.

In the general case, when u11u12u22 the isotherm can be found numerically by evaluating the largest eigenvalue of the transfer matrix.

Figs. 9 and 10 illustrate the main features of these isotherms.

On the top of both figures, the binding isotherm for the first metal ion is shown as a function of its activity for fixed activities of the second metal ion.

On the bottom, the isotherm for the second metal is shown in the same representation.

The results for the analytically soluble case with equal interaction parameters are shown in Fig. 9.

At low activities of the second metal ion, the receptor fills up through the alternating full-empty configuration discussed above, see Fig. 6(a).

At high activities of the second metal ion, the second metal exchanges with the first one.

This process follows the one-to-one exchange isotherm since there is no difference in the interactions between the two metal ions.

As a consequence, a disordered intermediate structure is formed (Fig. 6(c)).

An example of unequal interaction parameters is shown in Fig. 10.

At low activities of the second metal ion, the receptor again fills up through the alternating full–empty configuration (Fig. 6(a)).

At high activities of the second metal ion, on the other hand, the second metal exchanges with the first one following an effective repulsive interaction between both metal ions, and leading to an intermediate plateau.

The intermediate structure is again alternating (Fig. 6(d)).

The binding isotherm of the second metal shows an interesting transition region, as the isotherm becomes steep in the intermediate activity regime.

When both components are in excess, one can show that the model can be again reduced to the linear chain model with the effective pair interaction energy E = E(11) + E(22) − 2E(12)or equivalently.

The normalized activity now is z = z1u212/(z2u222) for the first metal ion.

Depending on the magnitude of the individual interactions, at intermediate loading one can obtain random arrangements for E = 0, alternating arrangements for E > 0, or clustering of the same metal ions for E < 0 (see Fig. 6).

For the situation shown in Fig. 9 a random structure forms (E = 0), while in the one shown in Fig. 10 the alternating structure forms (E = 15 kJ mol−1 > 0).


Multi-center complexes of lanthanides provide good examples for the mechanisms discussed here.

For this reason, we shall focus on those, but applications to other complexes should be envisioned as well.

Trimetallic inositol complexes

As a first example, consider trimetallic complexes of 1,3,5-triamino-1,3,5-trideoxy-cis-inositol (taci) L1H3 with Nd3+, Sm3+, and Eu3+ (see Fig. 1 and 2).11

These complexes form in water according to the equilibrium 2L1H3 + xNd3+ + ySm3+ + (3 − xy)Eu3+ ⇌ [NdxSmyEu3−xy(L1)2]3+ + 6H+where x, y = 0, 1, 2, 3 with x + y ≤ 3 and L13− corresponds to the deprotonated ligand.

The measured equilibrium constants11 are reported in Table 1.

In order to apply the site binding model, which treats the binding of metal ions to one receptor, we consider the conditional equilibrium 2L1tot + xNd3+ + ySm3+ + (3 − xy)Eu3+ ⇌ [NdxSmyEu3−xy(L1)2]3+where L1tot refers to all solution species of the ligand.

We choose pH 8 as reference.

From the known ionization constants of L1H3,28 one can evaluate the conditional constants of these equilibria.

These conditional constants are given in Table 1, and only those will be considered further.

The site binding model introduces three microscopic binding constants K(Nd), K(Sm) and K(Eu) and six pair interaction energies E(ωη).

We simplify the interaction energy matrix further by assuming that the diagonal elements E(ωω) and off-diagonal elements E(ωη) (ωη) are the same, respectively, leaving us with two pair interaction energies to be determined.

Since only the complexes with three lanthanide ions exist, one cannot estimate both interaction energies independently.

For this reason, we set the diagonal term E(ωω) = 18.7 kJ mol−1.

This value can be estimated from interaction of an effective charge near +2, which are obtained from ab-initio calculations,29 and calculated for the distance of 3.61 Å invoking the Coulomb law in water.

With this value, the model contains four adjustable parameters, which we extract by non-linear least squares fit from the constants of the nine complexes containing one or two different metal ions (see Table 1).

We obtain three microscopic constants log K(Nd) = 10.56, log K(Sm) = 12.20, and log K(Eu) = 12.64.

The off-diagonal pair interaction turns out to be E(ωη) = 19.4 kJ mol−1 (ωη) and recall that we set E(ωω) = 18.7 kJ mol−1.

The fit cannot be substantially improved by allowing for further variation in the interaction parameters.

To test the predictive capabilities of the model, we have calculated the constant of the ternary complex [NdSmEu(L1)2]3+ (see Table 1).

The experimental value of its common logarithm is 25.90 and the model predicts 25.98.

This result can be considered as very satisfactory, since the model was calibrated on binary systems only.

The model therefore captures the interactions between the different lanthanide ions properly.

From this model description, we can extract various properties of interest.

Naturally, once the macroscopic constants are known, the macrostate probabilities can be computed (see Fig. 11).

The macrostate probabilities are displayed as a function of the free concentration ratio [Sm3+]/[Nd3+], where the free concentrations have been set equal to the activities for simplicity.

The total concentration of the three metals was set to 0.1 M and the overall mole fraction xEu of the Eu3+-ion was fixed.

In the absence of Eu3+, one observes the classical evolution from the [Nd3(L1)2]3+ complex, through the mixed binary complexes [Nd2Sm(L1)2]3+ and [NdSm2(L1)2]3+, to the [Sm3(L1)2]3+ complex (see Fig. 11(a)).

At xEu = 0.01, various mixed complexes appear, in particular the mixed ternary complex [NdSmEu(L1)2]3+ (see Fig. 11(b)).

At xEu = 0.1, binary complexes not containing Eu3+ disappear, and the evolution is dominated by the complexes [Eu3(L1)2]3+, [Eu2Sm(L1)2]3+, [EuSm2(L1)2]3+ and [Sm3(L1)2]3+ (see Fig. 11(c)).

The microscopic picture is very simple in this case, as each macrostate consists of a single microstate (see Fig. 12).

However, as the pair interaction energies between different metal ions are somewhat larger than between the same metal ions, the microscopic constants increase with increasing number of different neighboring metal ions.

This effect leads to a destabilization of the metal ions in the mixed complexes (see Fig. 12).

Moreover, one observes that Nd3+ is bound the weakest in the complex, Sm3+ is intermediate, and Eu3+ is bound the strongest.

Linear bimetallic and trimetallic helicates

As a second example, consider the monometallic and bimetallic helicates of the general formula [LaxEuy(L2)3]3(x+y)+ (x, y = 0, 1, 2) and trimetallic helicates of the formula [LaxEu3−x(L3)3]9+ (x = 0, 1, 2, 3, see Fig. 1 and 2).7,8

To analyze the formation constants, we assume that the bimetallic and trimetallic complexes have equal interactions, and thus we use the whole series to extract the microscopic parameters.

The formation constants to be analyzed are summarized in Table 2.

We note that the complexes [La2(L3)3]6+ and [Eu2(L3)3]6+ have been also described, but their structure is probably rather different, and they are not considered here.7

In order to simplify the structure of the model, let us consider the equilibrium constant of the exchange reaction [LaLa(L2)3]6+ + [EuEu(L2)3]6+ ⇌ 2[LaEu(L2)3]6+.

This exchange constant has been shown to be 4 within experimental error,7 meaning that the fraction in eqn. (41) can be well approximated by unity.

In other words, the pair interaction energies obey the mixing rule.

The off-diagonal terms are now simply approximated by an arithmetic average of the corresponding diagonal terms.

Note that this mixing rule imposes no effective interactions in the saturated state (cf. eqn. (36)).

The site binding model now contains the following parameters.

These include microscopic binding constants for both metal ions to the terminal site K(La)t and K(Eu)t and to the central site K(La)c and K(Eu)c.

When we consider nearest neighbor interactions only, we now have two pair interaction energies, namely E(LaLa), E(EuEu).

The cross term E(LaEu) is obtained from eqn. (42).

These six parameters can be extracted from the nine constants given in Table 2.

The experimental equilibrium constants can be described with the site binding model extremely well (see Table 2).

The resulting parameters are the microscopic constants for the terminal sites log K(La)t = 16.70 and log K(Eu)t = 19.10.

For the central site we have log K(La)c = 17.91 and log K(EuEu)c = 21.08.

The nearest neighbor pair interaction energies are E(LaLa) = 47.5 kJ mol−1 and E(EuEu) = 69.5 kJ mol−1.

Independent fit of the off-diagonal terms gives a value of the off-diagonal terms, which is very close to the value predicted by the mixing rule eqn. (42), and leads to no improvement to the fit.

Consideration of a next nearest neighbor interaction energy gives equally no effect.

Based on these model parameters, various macroscopic and microscopic properties of these helicates can be predicted.

Fig. 13 shows the macroscopic properties of the trimetallic helicates with the formula [LaxEu3−x(L3)3]9+ (x = 0, 1, 2, 3).

The macrostate probabilities, which are normalized to the total concentration of the saturated complexes, are shown in Fig. 13a.

As the concentration ratio [Eu3+]/[La3+] is increased, one goes through the characteristic sequence of the macrospecies [La3(L3)3]9+, [La2Eu(L3)3]9+, [LaEu2(L3)3]9+ and [Eu3(L3)3]9+.

Fig. 13(b) shows the relative fraction of metal bound in the trimetallic helicates.

One observes that the exchange between Eu3+ and La3+ occurs in a relatively narrow concentration range.

For each mixed complex (macrospecies), there are two possible structures (microspecies), each of which has always a fixed relative fraction within the macrostate.

Fig. 14 summarizes their microscopic properties.

The macrospecies [La2Eu(L3)3]6+ consists of the two microspecies, the dominant asymmetric complex [EuLaLa(L3)3]9+ and the symmetric minority complex [LaEuLa(L3)3]9+.

On the other hand, in the macrospecies [LaEu2(L3)3]9+ the symmetric complex [EuLaEu(L3)3]9+ is dominant, while the asymmetric complex [LaEuEu(L3)3]9+ is in minority.

This observation is consistent with NMR results, since the two minority complexes could not be detected.8

The microconstants to remove the respective metal ions from each site are summarized in Fig. 14.

One observes that the terminal metal ions are much more tightly bound that the central ones.

This effect is mainly caused by the additional electrostatic repulsion between the sites.

One further observes that the binding of La3+ is weakened through the presence of neighboring Eu3+ ions.

The same trend can be also observed for Eu3+, as its binding is also weakened by the presence of neighboring La3+ ions.


In this paper, we have presented a microscopic site binding model treating binding of several metal ions to preassembled multi-center receptors with well-defined binding sites within a statistical mechanical framework.

We introduce the appropriate parameterization in terms of microscopic complexation constants and metal–metal pair interaction energies.

From this model, the distinction between microscopic and macroscopic equilibria in these systems enters in a natural way.

With the present approach one can not only treat receptors with a small number of binding sites, but also macromolecular receptors capable of binding a large number of metal ions.

This situation is illustrated with linear chains.

The applicability of the model has been illustrated with examples of lanthanide complexes, namely mixed trimetallic cis-inositol complexes and multi-metallic helicates.

In both situations, we find that consideration of pair interactions is important to obtain an acceptable description of the data.

However, the general structure of the pair interaction matrix is not obvious at this point.

While for multi-metallic helicates a simple mixing rule appears to represent a reasonable approximation of the off-diagonal elements of the matrix, for the inositol complexes deviations from this rule are observed.

Further experimental data sets are necessary to settle such questions in detail.

Predictive capabilities of the site binding model were illustrated with the inositol complexes.

Thus, one can predict binding constants of metal complexes from a limited set of experimental data.

The example of the multi-metallic helicates did show that from a homologous series one can obtain a sufficient number of experimental quantities in order to fix the model parameters.

In both cases, the model is able to predict all details of the microscopic equilibria, including microstate probabilities and microscopic equilibrium constants.

An analogous formulation can be applicable when the receptor undergoes conformational transitions.

However, the parameters considered must be interpreted as thermal averages over the different conformational states.

Moreover, more long-range interactions and eventually higher order interactions (e.g., triplets) must be considered.

The present formulation certainly represents an appropriate way to treat binding of several metal ions to one type of multi-center receptor.

In most situations, however, the receptor is composed of different ligands, which have to assemble in order to form the complex.

As long as all complexes are composed of the same number of ligands (e.g., helicates) the present approach applies.

On the other hand, when the complexes contain different numbers of sub-units, the present approach must be generalized in order to incorporate this situation.

In the protein literature,30 this problem is referred to as aggregation of ligands, but a useful parameterization of this problem is not known.

These aspects will be the subject of future studies.