Pairwise superposition

Fig. 24.1 Schematic drawing representing the FITIT procedure. The conformational space is reduced by pairwise superpositioning of energetically allowed conformations of the different chemical classes. A second superpositioning step for all selected FITIT pairs finally yields the common pharmacophoric conformations.

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

Tom Blundell has answered these questions by superposing the Ca atoms of the two motifs within a domain with each other and by superposing the Ca atoms of the two domains with each other. As a rule of thumb, when two structures superpose with a mean deviation of less than 2 A they are considered structurally equivalent. For each pair of motifs Blundell found that 40 Ca atoms superpose with a mean distance of 1.4 A. These 40 Ca atoms within each motif are therefore structurally equivalent. Since each motif comprises only 43 or 44 amino acid residues in total, these comparisons show that the structures of the complete motifs are very similar. Not only are the individual motifs similar in stmcture, but they are also pairwise arranged into the two domains in a similar way since superposition of the two domains showed that about 80 Ca atoms of each domain were structurally equivalent. 

The three-body spectra and their associated correlation functions may be considered to be a superposition of three components of different nature. One part arises from two-body dynamics where the third atom acts strictly as a perturbing field. The second part represents the contributions of the irreducible three-body dynamics to the pairwise-additive induction. The third part is due to the three-body induction mechanism and contains the irreducible dipole. These agents vary differently with temperature and could in principle be separated on that basis. 

Fig. 13. Superposition of a/ji barrel domains of MR (white/red), MLE I (blue/red), enolase (yellow/red). Red coloring represents most similar regions among each pairwise comparison as described in the text.

Figure 10 One-dimensional potential energy parallel to the surface (a) empty surface with a single particle bound with adsorption energy FjiK (b) superposition of the potential energy in (a) with a pairwise interaction potential of partitles on the surface (/>, ) which nia be either attractive or repulsive...

This way we obtain 10 Viterbi paths. In the next step these are now combined to pairwise Viterbi paths by simple superposition of the Viterbi paths which belong to a combination. This produces five pairwise Viterbi... 

By another superposition of the five clustered pairwise Viterbi paths a global Viterbi path is obtained. This path contains 1114 different states, due to the fact that the states of the clustered pairwise Viterbi paths can be combined in any way giving a theoretical maximum of 5 4 5 4 4 = 1600 possible global states (of which only 1114 actually occur). Setting up the transition matrix again yields a sparse stochastic matrix in which more than 99% of the entries are equal to zero. 

Table 1. Pairwise comparison of the topology and primary sequence of members of the short spacer family. The alpha carbon atoms defining the zinc protease fold (orange segment. Fig. 3) have been used in the topological superposition [56]. The distances refer to the root mean square deviations of this fold between pairs of structures. The corresponding pairwise primary sequence homology is also shown.

The history of the search for an integral equation for the pair correlation function is quite long. It probably started with Kirkwood (1935), followed by Yvon (1935, 1958), Born and Green (1946), and many others. For a summary of these efforts, see Hill (1956), Fisher (1964), Rushbrooke (1968), Munster (1969), and Hansen and McDonald (1976). Most of the earlier works used the superposition approximation to obtain an integral equation for the pair correlation function. It was in 1958 that Percus and Yevick developed an integral equation that did not include explicitly the assumption of superposition, i.e., pairwise additivity of the higher order potentials of mean force. The Percus-Yevick (PY) equation was found most useful in the study of both pure liquids as well as mixtures of liquids. 

May, A. C. and M. S. Johnson, Improved genetic algorithm-based protein structure comparisons pairwise and multiple superpositions. Protein Eng, 1995. 8(9) p. 873-82. 

Fig. 9. Pairwise comparisons of the mean NMR structure of the globular domain in bPrP(121-230) (green) with (A) hPrP(121-230) (red), (B) mPrP (121-231) (yellow), and (C) shPrP(121-231) (pink). Same presentation as in Figure 5 Aspline function was drawn through the C positions. The variable radius of the cylindrical rods is proportional to the mean global displacement per residue, as evaluated after superposition for best fit of the atoms N, C , and C of the residues 124—227 in the 20 energy-minimized conformers used to represent the NMR structure.

Figure 16. Core-core repulsion correction in Hgj for two-valence electron Hg ECPs. Errors per atom due to a superposition of pairwise eorrections in larger highly-symmetric Hg clusters (n=3 equilateral triangle n=4 tetrahedron n=6 octahedron n=13 icosahedron) are also shown. The underlying core-core repulsion corrections were determined from small-core (20-valence electron) PP frozen-core calculations on the Hg + core systems. The interatomic distances in Hg2 and the bulk are indicated by vertical lines.

In this approach it is assumed that the basis set superposition error in the many-body cluster can be approximated by the sum of the Boys-Bemardi function counterpoise corrections for pairs of bodies. Hence the total interactions for an N-body cluster using the pairwise additive function counterpoise correction is given by... 

This formal resemblance can be misleading. Equation (3.6.53) is the exact isotherm for a system with direct interactions only. The two independent parameters of the model are Ki and 5. On the other hand, Eq. (3.6.51) has been derived on the basis of the pairwise additivity (or superposition approximation) assumptions (3.6.46) and (3.6.47). We have already seen that this approximation is unjustified for the indirect correlations. Since we know that in hemoglobin direct interactions are negligible, we have concluded that all correlations are due to indirect interactions, therefore (3.5.51) is incorrect. If we insist on expressing the isotherm in terms of the pair correlation function y, 1), we must also include nonadditivity effects [see Eq. (3.6.58) below]. But this is not necessary. A simpler and exact expression can be written in terms of the fundamental parameters of the model. This is essentially Eq. (3.6.37), where the Ki are defined in (3.6.36). 

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