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Additivity of Energy Contributions in Multivalent Complexes

Hans‐Jörg Schneider

FR Organische Chemie, Universität des Saarlandes, 66123, Saarbrücken, Germany

1.1 Introduction

Additivity of individual binding contributions is the very basis of multivalency. In classical coordination chemistry such simultaneous actions are described as the chelate effect. They offer almost unlimited ways to enhance the affinity [1,2,3,4,5,6], and therefore within certain limitations also the selectivity [7] of synthetic and natural complexes. Although additivity is often implied in experimental and theoretical approaches it is subject to many limitations which will be also discussed in the present chapter.

1.2 Additivity of Single Interactions – Examples

If only one kind of interaction is present in a complex one can expect a simple linear correlation between the number n of the individual interaction free energies ΔΔGi and the total ΔGt (Equation 1.1), as illustrated in Figure 1.1 for salt bridges [8]. Even though the organic ion pair complexes are based on cations and anions of very different size and polarizability one observes essentially additive salt bridges; the slope of the correlation indicates an average of ΔΔG = (5 ± 1) kJ/mol per salt bridge. The value of (5 ± 1) kJ/mol is observed in usual buffer solution, but varies as expected from the Debye–Hückel equation with the ionic strength of the solution [9]. Scheme 1.1 shows a corresponding value of K ≈ 10 M−1 per salt bridge for typical complexes where the affinity depends as expected on the degree of protonation [7].

Figure 1.1 Additive ion pair contributions in a variety of complexes with a number nC of salt bridges. From slope: average (5 ± 1) kJ/mol per salt bridge. A,B and C,C' – complexes of a tetraphenolate cyclophane (4−) with Me4N+ and an azoniacyclophane (4+) with mono‐ and dianionic naphthalene derivatives; D – anionic (sulfonate or carboxylate) with cationic (ammonio) triphenylmethane derivatives; E – organic dianions with organic dications; F – cationic azamacrocycle (6+ charges) with aliphatic dicarboxylates; G – cationic azacrowns with adenosine mono‐, di‐ and triphosphates.

Source: Ref. [8]. Reproduced with permission of John Wiley and Sons.

Scheme 1.1 Complexation log K values of anions 1–5 with a macrocyclic amine as function of the degree of protonation of the amine; and ion pairing with some representative complexes; log K values in water; n is the estimated number of salt bridges.

The additivity depicted in Figure 1.1 and Scheme 1.1 for salt bridges is in line with the Bjerrum equation, which describes ion pair association as a function of the ion charges zA and zB; Figure 1.2 shows for over 200 ion pairs a linear dependence of log K vs. zAzB [3]. For inorganic salts one finds similar ΔΔG values of 5–6 kJ/mol per salt bridge and a similar dependence on charges [10]. At zero ionic strength the stability decreases in the order Ca2+ > Mg2+ > > Li+ > Na+ > K+ and can be described by Equation 1.2 [11]. Additivity is observed although ion pairing in water is determined entirely by entropic contributions[11], unless other contributions dominate [12].

Figure 1.2 Ion pair association constants at zero ionic strength as a function of charge product, calculated for 203 ion pairs.

Source: Ref. [8]. Reproduced with permission of John Wiley and Sons.

If there is more than one kind of interaction, Equation 1.3 applies. Often however, only one of the contributions is the same, like salt bridges in complexes of nucleotides with a positively charged host (Scheme 1.2) [13]. Additivity is then observed by the constant stability difference of 2 × ΔΔG ≈ 10 kJ/mol between complexes with charged nucleotides and neutral nucleosides. The 10 kJ/mol reflects the presence of two salt bridges between the phosphate dianion and the host ammonium center, which agrees with structural analyses by NMR spectroscopy.

Scheme 1.2 Complexation free energies ΔG of nucleotides and nucleosides with the cyclophane CP66.

The complexes shown in Scheme 1.2 exhibit constant single ΔΔGA values only for the salt bridges, whereas the second contribution ΔΔGB varies as a function of the different nucleobases. Figure 1.3 illustrates a case where both ΔΔGA and ΔΔGB remain constant, the latter reflecting cation–π interactions. In principle one could use Equation 1.3 to derive both ΔΔGA and ΔΔGB, but more reliable values are obtained if for one interaction a ΔΔG value is used which is known from independent analyses, such as ΔΔGA = 5 kJ/mol for each salt bridge (see above). Then one observes a rather linear correlation with the number of phenyl units which shows a contribution of ΔΔGB ≈ 1.5 kJ/mol for the single +N–π interaction [14].

Figure 1.3 Ion pairs exhibiting both salt bridges and cation–π interactions; if ΔΔGA = 5 kJ/mol for each salt bridge are subtracted from ΔGt of each complex. Outliers (open circles) are due to conformational mismatch.

Source: Ref. [14]. Reproduced with permission of American Chemical Society.

The effect of nitro substituents on dispersive interactions is another example of additive energy contributions (Figure 1.4) [15,16]. Additivity with respect to substituent effects is observed in Hammett‐type linear free energy relationship correlations; Figure 1.5 shows an example for hydrogen bonds with C─H bonds as donor and with hexamethylphosphoramide as acceptor [17].

Figure 1.4 Additive ΔΔGX increments in complexes of porphyrins bearing cationic or anionic substituents R in meso position (TPyP or TPS) in water, after deduction of 5 kJ/mol for ion pair contribution where applicable. ΔΔGX increments in TPyP complexes for nitro substituents as an example (deviation for ortho‐dinitro due to steric hindrance); correlation between measured complexation energies ΔGexp and ΔGcalc calculated on the basis of experimentally determined averaged single contributions ΔGS. Filled circles, complexes with TPyP; open circles, complexes with TPS.

Source: Ref. [15]. Reproduced with permission of John Wiley and Sons.

Figure 1.5 Hammett‐type correlation of equilibria of hydrogen bonds with hexamethylphosphoramide as acceptor and para‐substituted tetrafluorobenzenes or phenylacetonitriles as donor; log K versus Hammett substituent constants.

Source: Ref. [17]. Reproduced with permission of John Wiley and Sons.

1.3 Limitations of Additivity

1.3.1 Free Energy Values ΔG Instead of Enthalpic and Entropic Values ΔH, TΔS

The examples shown above as well as most others in the literature rely on free energy values ΔG, although consideration of the corresponding ΔH and TΔS parameters could shed more light on the underlying binding mechanisms. As pointed out earlier by Jencks, the empirical use of ΔG “avoids the difficult or insoluble problem of interpreting observed ΔH and TΔS values for aqueous solution” [18]. Furthermore, according to Jencks, there is often an additional “connection Gibbs energy, ΔGS” (Equation 1.4) which he ascribed largely to changes in translational and rotational entropy. These connection ΔGS can be either negative or positive and will be discussed as major liming factors for additivity below in the context of cooperativity and allostery.

The success of using free energy values instead of enthalpic and entropic values is in an essential part due to entropy–enthalpy compensation which has empirically been found to hold with many complexations, although it is theoretically not well‐founded [19,20,21]. Another factor is that in typical supramolecular complexes the loss of translatory freedom is already paid by a single association step. The loss of rotational freedom upon complex formation has been experimentally [9] found to be smaller than theoretically expected (see below).

Entropy contributions pose particular problems, not only for the precise experimental determination, which in the past often relied on the temperature dependence of equilibrium constants (the Van ‘tHoff method) instead...