Electrostatic similarity matrices

In the case of E2 ubiquitin conjugating enzymes, the interaction properties of about 200 protein structures were compared [49]. The pairwise similarity matrix was visualized as a dendrogram and a kinemage projection to three-dimensional space (see www.ubiquitin-resource.org). The analysis revealed relations between functional groupings and electrostatic properties at specific parts of the protein structure. 

The current version of this approach uses N x N matrices (N being the number of molecules included in the data set), in which the 3D similarity indices of each molecule as compared to every other molecule of the whole set are the non-diagonal elements of the similarity matrix [1064, 1065]. In an application to the affinities of 31 steroids to corticosteroid- and testosterone-binding globulins (same data as used in CoMFA studies, e.g. [36]), Carbo index similarity matrices were calculated for the shape and the electrostatic potentials of all molecules. Using the similarity... 

Each of the N molecules is compared with every other to generate an N X N similarity matrix, which is compared to the biological potency values using PLS60.63,198 or a neural network. Frequently, the cross-validation statistics are slightly better for models based on similarities than for those using the full matrix of electrostatic and steric potentials. 

A molecular dynamics force field is a convenient compilation of these data (see Chapter 2). The data may be used in a much simplified fonn (e.g., in the case of metric matrix distance geometry, all data are converted into lower and upper bounds on interatomic distances, which all have the same weight). Similar to the use of energy parameters in X-ray crystallography, the parameters need not reflect the dynamic behavior of the molecule. The force constants are chosen to avoid distortions of the molecule when experimental restraints are applied. Thus, the force constants on bond angle and planarity are a factor of 10-100 higher than in standard molecular dynamics force fields. Likewise, a detailed description of electrostatic and van der Waals interactions is not necessary and may not even be beneficial in calculating NMR strucmres. 

Belosludtsev and coworkers (2001) propose that the unmodified probes on the weakly cationic surface, although prevented from diffusing off the surface because of electrostatic interaction, nevertheless are available for hybrid nucleation. They suggest that such probe behavior could be viewed similarly to behavior observed in a liquid crystal matrix. This would be in sharp contrast to models describing the covalent attachment of short oligonucleotides as "oligo lawns" or monolayers of coiled probes (see Figure 3.6). 

We discussed in detail the properties of the matrix elements of the electrostatic energy operator for shell lN. The corresponding expressions for the remaining two-electron operators may be found in a similar way, therefore, here we shall present only final results. For the case of relativistic corrections H2, H and H s to the Coulomb energy (formulas (19.8), (19.11) and (19.12), respectively) we have... 

The matrix element of the operator of the energy of magnetic interactions between two subshells may be found in a similar way to the case of electrostatic interaction. The final result is as follows for direct dH 2 and exchange eH 2 parts, respectively ... 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

It includes the interactions of distributed multipole moments Q (up to a quadrupole) labeled t and u. The T matrix provides the Coulomb energy appropriate for particular multipoles and includes the distance between sites a and b and their relative orientations. The short range (penetration) component of the electrostatic energy, in a manner similar to the Ar-CC>2 case, can be absorbed into the exchange repulsion term. 

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