RMS distance

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

The conformational distance does not have to be defined in Cartesian coordinates. Eor comparing polypeptide chains it is likely that similarity in dihedral angle space is more important than similarity in Cartesian space. Two conformations of a linear molecule separated by a single low barrier dihedral torsion in the middle of the molecule would still be considered similar on the basis of dihedral space distance but will probably be considered very different on the basis of their distance in Cartesian space. The RMS distance is dihedral angle space differs from Eq. (12) because it has to take into account the 2n periodicity of the torsion angle. 

It is up to the researcher to decide whether to use a Cartesian similarity measure or a dihedral measure and what elements to include in the summation [29]. It should be stressed that while the RMS distances perfonn well and are often used, there are no restrictions against other similarity measures. Eor example, similarity measures that emphasize chemical interactions, hydrophobicity, or the relative orientation of large molecular domains rather than local geometry may serve well if appropriately used. 

Here r is the center-to-center distance, and m is a unit vector along the line joining the centers of the two fragments, so that r = rm. Distances are made dimensionless with respect to the radius of a fragment (a), shear rates and time with respect to the characteristic shear rate y = V2D D, and Fe = FC with respect to H a. These equations are identical in form to... 

Further on, the measure RMS distance that is to be optimized is a valuable point of information in itself. It is used, for example, to compare predictions with crystal structures and invaluable for clustering similar placements. However, caution must be taken to avoid problems with symmetry in the molecules. Again, the problem of correspondence must be treated carefuUy, since, for example, a rotation of 180° of a phenyl ring should not affect the result of such a quality assessment. 

Because there are many possible ends in branched polymers, it is customary to use the radius of gyration S) instead of r for such polymers. The radius of gyration is actually the RMS distance of a chain end from the polymer s center of gravity. S is less than the end-to-end distance (r), and for linear polymers = 6S. ... 

A polymer stretched out to its full contour length is only one of the myriad conformations possible for a polymer at temperatures above Tg, or if the polymer is entirely crystalline. The chain length is expressed statistically as the RMS distance, which is only a fraction of the contour length. 

As you move the cursor pointing to amino acid residues of the aligned sequences in the Align window, the corresponding residues of the superimposed molecules in the Display window blink from blue to yellow, allowing an easy visualization of the overlap. In addition, the rms distance of the superimposed residues is displayed. 

To improve the superimposition (lowering rms distance), select the Iterative Magic Fit tool from the Fit menu. 

In addition to the RMS end-to-end distance, the dimensions of linear chains are often characterized in terms of the RMS distance of a chain segment from the center of mass of the molecule. It is defined as the square root of the average squared distance of all the repeating units of the molecule from its center of mass and is known simply as the RMS radius of gyration, S It is thus given by [14]... 

Scheme 3.1.2 Some structures determined by laboratory X-ray powder diffraction. RMS distance between atomic coordinates for structures solved by powder diffraction and single crystal diffraction were A 0.026 B 0.026, C 0.165, D 0.204 A.

Fig. 16J. The relationship between the percentage sequence differences and the rms distance differences between topologically equivalent C positions in optimally superposed pairs of homologous protein three-dimensional structures (taken from [2]). The are for all equivalent amino acids and the + are for those whose side chains contribute to the solvent inaccessible core. The lines in a are the best unweighted least-squares fits of quadratic equations for the following sets of points TA is for all points TI is for all + points BA and BI are for P sheet proteins while AA and AI are for those with a helices, b shows data for P sheet proteins and c for a helical proteins. Lines linking and + points for individual proteins are plotted in b and c...

Fig. 16.4. Phylogenetic trees or cladograms based on the sequence differences (SEQ) and three-dimensional structural differences (STR) of immunoglobulin fragments. The structural distance metric is a function of both the rms distance difference between superposed structures and the number of topologically equivalent positions in each pairwise comparison (taken from [11]). The lower part of the diagram shows a multidimensional scaling analysis based on structural differences. The constant (C) and variable (V) domains cluster together for the light (L) and heavy (H) chains of the immunoglobulin fragments...

For a given hexapeptide, similar peptides are generated by allowing for all possible substitutions of amino acids within a similarity distance threshold of 0.3 according to Figure 17.1 a. To obtain a measure for the similarity between two hexapeptides, we define the rms difference of the six distances between amino acids in respective positions. This rms distance ranges between 0.0 for identical peptides and the threshold distance if all amino acids are replaced by congeners at the threshold limit. For practical purposes we define a peptide similarity index which is 1.0 for identical or nearly identical peptides (rms distances <0.1) and decreases linearly to 0.0 as the rms distance increases to the threshold limit. 

The type of information that can be retrieved from this database is best illustrated by two characteristic examples shown in Ikble 17.2. The first example compiles hexapeptides related to the C-terminal part of the first and main a helix in the bacterial protein barnase (Ikble 17.2a), the structure of which is known from X-ray analysis [67], but is not contained in our database. Of over 30,000 related peptides generated, nine could be found in the database. They are moderately similar to the query peptide as reflected by rms distances greater than 0.1 and similarity indices below 0.8. Seven of the nine related peptides adopt full or partial helical folds in their respective protein structures, whereas only two less related peptides are found in partially extended conformations. This highly consistent picture is also borne out by the cumu-... 

The fully extended length of the chain is equal to nl so that the maximum extensibility of a random coil with r = Zrms is from n l to nl, i.e. a factor of rfi. The value of Znns also gives a measure of the spatial extent of a chain. A second useful measure of this is the radius of gyration, r, which is the RMS distance of the atoms of the chain from the centre of gravity of the chain. Debye showed that, provided that n is very large, Zg = r /s/6. 

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