Triplet correlation

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

The term cooperativity will be used almost synonymously with correlation, except for restricting its usage to a particular type of event, namely, site i is occupied and site j is occupied. In Eq. (1.1.21), we defined the pair correlation between two such events. In Eq. (1.1.22), we defined the triplet correlation among three such events. 

In this chapter we discuss three-site systems. We extend the three models treated in Chapter 4 direct correlation, indirect correlation mediated through the adsorbent molecule, and indirect correlation mediated by a chain of communicating subimits. Here, we discuss separately two possible structures of the system, a linear and a triangle arrangement of the sites (Fig. 5.1). Two fundamentally new features are discussed in considerable detail the nonadditivity of the triplet correlation and the possibility of long-range correlations. 

In all these cases the triplet correlation is expressible in terms of the pair correlations and the temperature dependence of the correlation is predictable, knowing the ligand-ligand interactions. This is, in general, not the case for systems with indirect correlations, discussed in the following sections. 

THREE STRICTLY IDENTICAL SITES NONADDITIVITY OF THE TRIPLET CORRELATION... 

We have seen in Section 4.5 that the conditions for having indirect pair and triplet correlations are the same as those for conformational changes induced by the binding process. As in Sections 3.4 and 4.5, the change in the mole fraction of the L form is the same also in this model, i.e.. 

Figure 5.4. The triplet correlation as a function of the pair correlation for the approximate additive casey(l, 1, l) = y(l, 1) and for the exact case given by Eq. (5.4.15) (with the minus sign and AT= 10" ...

A second warning signal, which applies to any three (or more) site systems, is the nonadditivity of the triplet correlations. Again, based on the normal behavior of the direct correlations in Eq. (5.4.1), one expects similar behavior from the total correlation, and writes... 

We next examine the triplet correlation in this system. This is defined, similarly to Eq. (5.4.7), by... 

The minus sign is correct for 1 < iT < 1. Thus, for any given K there exists a relationship between the triplet correlation and all the pair correlations, and it is not simple like Eq. (5.5.8) or (5.5.9). For = 1, this reduces to the simple relationship... 

The general expressions for the indirect correlations are fairly complicated. Nevertheless, we can find conditions under which there exist pair and triplet correlations by factoring out the factors that determined the sign of the indirect correlations. In terms of the parameters h, h, h, and T), T, the pair correlations may be written as... 

The triplet correlation in this system is quite complicated, even when abc is assumed to be independent of the conformation of the subunits. Nevertheless, we can make the following statements regarding the indirect triplet correlations ... 

When one of is unity, it makes two of the indirect pair correlations unity, but the indirect triplet correlation becomes equal to the third pair correlation. For example, when = 1, we have... 

If the long-range interaction 17(1, 3) can be neglected compared with 1/(1, 2), the (direct) triplet correlations can be written as ... 

Table 5.1 presents some values for the intrinsic binding constants, and also pair and triplet correlations for benzoic acid, benzene dicarboxylic acids, and benzene tricarboxylic acids. 

The pair and triplet correlations and the corresponding free energies are... 

In writing these equations, it is implicitly assumed that (1) there is no long-range correlation, i.e., = 1 and (2) the triplet correlation can be expressed as the sum... 

We note that in spite of the large differences in the two sets of results reported in Table 5.3, the binding isotherms computed with these three sets of results were almost indistinguishable on the scale of Fig. 5.19. The most important differences between the calculated correlations and those reported by Senear et al. are, first, there is a large negative cooperativity between sites a and c, while Senear et al. assumed from the outset that no long-range cooperativity exists, and second, the triplet correlation is not additive, i.e., neither Sj nor 62 is zero, while Senear et al. assumed from the outset that 82 = 0. 

In the phenomenological approach there is no way of interpreting (to which we have referred as the triplet correlation function). However, if one treats as if it were an "interaction energy f then the natural consequence would be to write, for a linear arrangement of three sites. 

We stress, however, that these two properties of the free energy of interaction are valid only in very special cases. We have shown in Sections 5.4-5.7, that, in general, the interaction free energies do not conform to this behavior there could be long-range correlations, i.e., [or g(a, c)], that differ from unity, and triplet correlations that differ from the product g(a, b)g(b, c). [There is no known example where the triplet correlation may be written as a sum of pair correlations g a, b) + gib, c), as assumed by Ackers et ai]... 

It is of interest that the nonadditivities of the triplet correlations are quite different for the three isomers (see also Table 5.1). The nonadditivities, computed as... 

From now on we use only the GPF for the cyclic system and drop the subscript C. Since our system has m identical units, the sites will always be identical in the weak sense. There is always one intrinsic constant for the first site but, in general, we have more than one pair correlation, triplet correlation, etc. As in Section 7.1 we develop, for simplicity, the case of two states/= 2, but most of the results are quite general. 

By a straightforward generalization we can write the triplet and higher-order correlations. For instance, the triplet correlation 2 3( i = ot> P Y) obtained from the corresponding triplet probability... 

Figure F.l. Schematic illustration of the three equivalent systems a, b, and c corresponding to Eqs. (F.2), (F.3), and (F.4). In a and b all the sites are independent. In c the three sites on each molecule are correlated by pair and triplet correlations.

In Fe8oB2o, Jacobaeus et al,96 claim to be able to derive a triplet correlation of atomic positions from field ion images. Specifically, they find that the distribution of bond angles between neighbor atoms exhibits a fairly distinctive peak at 60°, a broad peak between 90 and 125°, and a... 

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