Tissue components hydrolyzing matrix metalloproteinases (MMPs) exhibit a high sequence similarity (56 -64% in catalytic domains) and yet a significant degree of functional specificity. The hexapeptide-binding sites of 24 known human MMPs were compared in terms of their force field interaction energies with five probes that are most frequently encountered in substrates and inhibitors. The probes moved along a grid enclosing partially flexible binding sites in rigid catalytic domains that were represented by published experimental structures and comparative models and new comparative models for nine most recently characterized MMPs. For individual MMPs, representative interaction energies were obtained as averages for all suitable experimental structures. Correlations of the representative energies for all MMP pairs were succinctly catalogued for individual probes, subsites, and correlation levels. Among the probes (neutral sp 3 carbon and sp 3 oxygen, positive sp 3 nitrogen and hydrogen, and negative carbonyl oxygen), the last probe is least distinctive. Similarities of subsites are decreasing as S1 > S2 > S3 > S1 ϳ S3 > S2 . Most interesting, occupancies of subsites in published structures of MMP-inhibitor complexes follow an almost parallel trend, alluding to overall low selectivity of known MMP inhibitors. Flexible subsite S1 that appears as the specificity pocket in rigid x-ray structures is actually very similar among individual MMPs. Several correlations indicated that MMPs 3, 8, and 12 have similar binding sites. Modeling results are corroborated with published experimental data on MMP inhibition and substrate specificities. The results provide numerous clues for development of specific inhibitors and substrates, as well as for selection of MMPs for testing that provides maximum information without redundant experiments.
Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n z= 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains only three (when n = 6) or four (when n = 7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n > 8 will also be discussed.Keywords: polygonal knots, space polygons, knot spaces, knot invariants. 245 J. Knot Theory Ramifications 2001.10:245-267. Downloaded from www.worldscientific.com by UNIVERSITY AT BUFFALO on 02/04/15. For personal use only. (V 2 ~ Vi) X (v 3 -Vi) • {V2 ~ Vi) X (v 4 -Vi) < 0, (V 4 -V 3 ) X (vi -V 3 ) • («4 -V3) X ( V 2 -V3) < 0. J. Knot Theory Ramifications 2001.10:245-267. Downloaded from www.worldscientific.com by UNIVERSITY AT BUFFALO on 02/04/15. For personal use only.
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