Root-mean-square-deviation approximation

In large subunit enzymes (PVC and HPII), a short segment of about 30 residues links the a-helical domain to the C-terminal domain (Fig. 8). The latter segment is a conspicuous addition to the small subunit containing about 150 residues folded into a structure that resembles flavodoxin. For example, there is a root mean square deviation of 3.0 A between flavodoxin and approximately 100 residues of the C-terminal domains of either HPII or PVC. This can be compared to the 1.8 A root mean square deviation for 134 centers between the C-terminal domains of HPII and PVC. Unlike the N-terminal end, the final C-terminal residue Ala753 is visible in the structure of HPII. The C-terminal domain contains extensive secondary structure in the form of four a-helices (al5-18) and eight fi-strands (fi9-16). Despite the obvious structural similarity to flavodoxin, there is no evidence of nucleotide binding in the domain and its function remains a mystery. 

Also in the staggered model approach the u-values for the halogen-halogen distances are composed of contribution both from framework vibration and torsional motion. The torsional motion part may be expressed by athe root-mean-square deviation from the minimum position. For the molecules so far described, the value of 00 is to a good approximation equal for the gauche and tram peaks. (This is of course not the case for molecules like 1,2-dihaloethanes). 

The crystal structures of the E. coli DHFR-methotrexate binary complex (Bolin et al., 1982), of the Lactobacillus casei (DHFR-NADPH-methotrexate ternary complex (Filman et al., 1982), of the human DHFR-folate binary complex (Oefner et al., 1988), and of the mouse (DHFR-NADPH-trimethoprim tertiary complex (Stammers et al., 1987) have been resolved at a resolution of 2 A or better. The crystal structures of the mouse DHFR-NADPH-methotrexate (Stammers et al., 1987) and the avian DHFR—phenyltriazine (Volz et al., 1982) complexes were determined at resolutions of 2.5 and 2.9 A, respectively. Recently, the crystal structure of the E. coli DHFR—NADP + binary and DHFR-NADP+-folate tertiary complexes were resolved at resolutions of 2.4 and 2.5 A, respectively (Bystroff et al., 1990). DHFR is therefore the first dehydrogenase system for which so many structures of different complexes have been resolved. Despite less than 30% homology between the amino acid sequences of the E. coli and the L. casei enzymes, the two backbone structures are similar. When the coordinates of 142 a-carbon atoms (out of 159) of E. coli DHFR are matched to equivalent carbons of the L. casei enzyme, the root-mean-square deviation is only 1.07 A (Bolin et al., 1982). Not only are the three-dimensional structures of DHFRs from different sources similar, but, as we shall see later, the overall kinetic schemes for E. coli (Fierke et al., 1987), L. casei (Andrews et al., 1989), and mouse (Thillet et al., 1990) DHFRs have been determined and are also similar. That the structural properties of DHFRs from different sources are very similar, in spite of the considerable differences in their sequences, suggests that in the absence, so far, of structural information for ADHFR it is possible to assume, at least as a first approximation, that the a-carbon chain of the halophilic enzyme will not deviate considerably from those of the nonhalophilic ones. 

The values of coefficients in equation (1) and the root-mean-square deviations o of experimental points from the approximate dependence are given in table 1. This equation has been used to calculate T,x and p,x diagrams of the liquid-liquid phase equilibrium for the mixture under investigation. 

Crystal structures have been determined of several ferrichromes. Most of this work was performed by van der Helm and coworkers. The crystal structures of the members of this siderophore family can be superposed and a root-mean-square deviation of the 49 atoms, which are common to all members, is obtained of approximately 0.30 A. A comparison of the conformational angles around the cyclic hexapeptide ring shows differences of not more than 25°. This indicates that the structures and conformations are not the same but that they are similar with some conformational freedom. In all structures, the iron coordination site is on one side of the molecule, the coordination of the metal is K-cis, and the conformation of the amino acids is L. A (II) bend and a (I) bend of the cyclic peptide skeleton is found. In addition, extensive conformational analyses of siderophores in solution were performed by Lhnas and coworkers employing H andNMR. ... 

Comparisons between optimized and X-ray structures were once again made by calculating root-mean-square (RMS) deviations. When comparing all heavy atoms in the protein, the total RMS deviation is approximately 1.7 A, irrespective of method for the model system or the ONIOM implementation (mechanical, ONIOM-ME, or electronic embedding, ONIOM-EE). The largest deviations occur for residues in the vicinity of the second monomer. Therefore, adding the second monomer to the model should improve the calculated geometries. 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

The comparison of the genuine theoretical results with those predicted by this approximation shows a root-mean-square (rms) deviation of 0.2kcal/mol with those obtained in the HF/6-31G(if) calculations reported in Table 9.1. This result is all the more remarkable as it includes polycychc molecules (15-21), boat-cyclohexane stmctures (15, 21), as well as a twist-boat structure (19, twistane = tricyclo [4.4.00 ]decane). The use of this approximation for ZPE + — Hq in problems... 

Ratio of standard deviation to root mean square of the data. b Values for PhCO approximated using those for MeCO. r CONHj approximated as C02R. 

The BC and K0 models, on the other hand, show much smaller root mean square errors, typically in the 1 % range, over an amazingly substantial range x of intensities fitted, Fig. 5.8, lower set of data points. Maximal deviations from the exact profiles amount to no more than twice the root mean square errors shown, that is well within the experimental uncertainties of the best measurements. The BC model is especially well suited to approximate quadrupole-induced profiles. The K0 model, on the other... 

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