Reduced Atom Representation

One mode of simplification concerns the representation of the polymer itself. The backbone of the protein polymer is illustrated below. 

Another approach to simplify the protein chain is to reduce the conformational space allowed to the protein. The argument is simply that one of the major forces in protein structure is the formation of a core, a hydrophobic core, in which the side chains are tightly packed with no free volume. It is indeed postulated to be one of the first steps in protein folding. If this is the case, the core of the protein can be simplified to the points on a lattice [43-47], 

One recent study [44], used a protein of known structure. They examined every possible conformation and whether it lead to the correct protein fold, in essence the kinetics of the folding process. Surprisingly, they found no correlation between the presence of secondary structural elements and finding the correct fold Instead the important feature that lead to the correct fold was the presence of the native state as a well-defined energy minimum. The correct energy landscape of configuration is the determinant feature for the correct folding. 

The author would like to thank Dr. Maria Pellegrini and Eduaro Mercurio for fruitful discussions and reading of the manuscript. 

Pauling, L., Corey, R. B. The structure of synthetic polypeptides Proc. Natl. Acad. Sci. USA 1951 37, 241-250. 

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Levitt Warshel [17, 18] were the first to show that reduced representations may work they used Ca atoms and virtual atoms at side chain centroids. OOBATAKE Crippen [24] simplified further by only considering the Ca atoms. This is snfficient since there are reasonably reliable methods (Holm Sander [11, 12]) that compute a full atom geometry from the geometry of the Ca atoms. (All atom representations are used as well, but limited to the prediction of tiny systems such as enkephalin.)... 

Figure 7-14. All-atom and united-atom representation of the amino acid isoleucine. In this example, 13 atoms, which are able to form explicit non-bonding interactions, are reduced to only four pseudo-atoms,...

Use the set of atomic orbitals as the basis for a representation of the group, and reduce this representation to its irreducible components. 

To determine how to form a set of trigonally directed hybrid orbitals, we begin in exactly the same way as we did in the MO treatment. We use the three a bonds as a basis for a representation, reduce this representation and obtain the results on page 219. However, we now employ these results differently. We conclude that the s orbital may be combined with two of the p orbitals to form three equivalent lobes projecting from the central atom A toward the B atoms. We find the algebraic expressions for those combinations by the following procedure. 

Representation of the Macromolecu-lar Receptor. The most straightforward approach for representing the macromolecular structure in a docking application would be by atomic coordinates of the entire protein. A full atomic representation, however, is generally impractical because of the size and complexity of protein structures. The structural information therefore needs to be reduced to a manageable yet representative size and form. 

Reduce each representation from Step 3 to its irreducible representations. This is equivalent to finding the symmetry of the group orbitals or the symmetry-adapted linear combinations (SALCs) of the orbitals. The group orbitals are then the combinations of atomic orbitals that match the symmetry of the irreducible representations. 

Another solution involves reduced representations of the macromolecules. For this purpose, many simplified models of proteins were developed first for folding processes [19-22] and subsequently for docking problems [17, 23]. The various proposed models differ in their degree of simplification. A residue may be described by only one point per side chain [21, 24, 25] or more [20, 22, 26]. Other researchers prefer to use discrete positions with a grid representation of the molecules [4, 27]. A small interval enables obtaining a nearly atomic representation [18] a larger one yields a sharply reduced model [17, 23]. Many other techniques have been proposed to simplify the systems, e.g. spherical harmonics [28, 29], Connolly surfaces [8, 30, 31] etc. 

For example, larger side chains could be represented by two united atoms [138]. Alternatively, an all-atom representation of the main chain could be employed with a reduced representation of the side chains [28]. Using this kind of representation and more elaborate statistical potentials, the structure of short peptides such as melittin, pancreatic polypeptide inhibitor, apamin [28], PPT, and PTHrP [138] have been predicted with an accuracy ranging from 1.7-A root-mean-square deviation, (RMSD) (measured for the a-carbon positions) for the small single helix of melittin to... 

Within the context of the above, let us try to formulate some necessary requirements for the design of a moderate resolution reduced model (based on united-atom representation and knowledge-based potentials) of real proteins and its force field ... 

As for a cluster orbitals, the transformation of basis to give 7t cluster orbitals from TT-type basis atomic orbitals is not unique. To define a transformation, we must first form and reduce the representation, spanned by the... 

Using the x, y, and z coordinates for each atom in Xep4, determine the reducible representation for aU molecular motions reduce this representation to its irreducible components and classify these representations into translational, rotational, and vibrational modes. 

A common method of proceeding with the valence problem is to construct, and then reduce, the representation of the valence urbilals in C- v It is assumed that both o--type and r-type bonds will be formed in the molecule, and at first these may be treated independently. The (r-valence orbitals are symmetric to rotations about the bond axes and may be represented by lines between the atoms. The transformation properties of these lines are then identical to the transformation of the valence orbitals. The reducible representation r([Pg.105]

In an FEP study, Rao and Singh used AMBER 3.3 to determine the relative binding free enei of pepstatin and its derivatives to Rhizopus pepsin. A united-atom and an all-atom representation were used for the residues of the enzyme and for the inhibitors, respectively. The pepstatin-Rhizopus pepsin complex was modeled starting from the high resolution crystal structure, where the inhibitor was in the reduced form. All pepstatin derivatives studied had modifications made on statine. The AAGlfia mutation of... 

In constructing this type, it is first clear that the different sets of equivalent atoms can be treated separately, except that all degenerate sots of S3 mmetry coordinates belonging to one species must be identically oriented, i.e., must have the same transformation coefficients. In many cases a further breakdown is possible which separates the z coordinates, say, from the x and y coordinates. This will be the case for ail noncubic point groups. By properly orienting the x and y axes separately for each atom, it is sometimes convenient to separate the x from the y coordinates initially. All these remarks represent simple methods of partially reducing the representation formed by the cartesian coordinates. 

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