To alleviate the problems in the receptor-based design of metalloprotein ligands due to inadequacies in the force-field description of coordination bonds, a four-tier approach was devised. Representative ligand-metalloprotein interaction energies are obtained by subsequent application of (1) docking with metal-binding-guided selection of modes; (2) optimization of the ligand-metalloprotein complex geometry by combined quantum mechanics and molecular mechanics (QM/MM) methods; (3) conformational sampling of the complex with constrained metal bonds by force-field based molecular dynamics (MD); and (4) a single point QM/MM energy calculation for the time-averaged structures. The QM/MM interaction energies are, in a linear combination with the desolvation-characterizing changes in the solvent-accessible surface areas, correlated with experimental data. The approach was applied to structural correlation of published binding free energies of a diverse set of 28 hydroxamate inhibitors to zinc-dependent matrix metalloproteinase 9 (MMP-9). Inclusion of step 3 and step 4 significantly improved both correlation and prediction. The two descriptors explained 90% of variance in inhibition constants of all 28 inhibitors, ranging from 0.08 to 349 nM, with the average unassigned error of 0.318 log units. The structural and energetic information obtained from the timeaveraged MD simulation results helped understand the differences in binding modes of related compounds.

MMPs and TACE (ADAM-17) assume independent, parallel or opposite pathological roles in cancer, arthritis, and several other diseases. For therapeutic purposes, selective inhibition of individual MMPs and TACE is required in most cases due to distinct roles in diseases and the need to preserve activities in normal states. Toward this goal, we compared force-field interaction energies of five ubiquitous inhibitor atoms with flexible binding sites of 24 known human MMPs and TACE. The results indicate that MMPs 1−3, 10, 11, 13, 16, and 17 have at least one subsite very similar to TACE. S3 subsite is the best target for development of specific TACE inhibitors. Specific binding to TACE compared to most MMPs is promoted by placing a negatively charged ligand part at the bottom of S2 subsite, at the entrance of S1' subsite, or the part of S3' subsite that is close to catalytic zinc. Numerous other clues, consistent with available experimental data, are provided for design of selective inhibitors.

Inhibitory potencies of 28 hydroxamate derivatives to matrix metalloproteinase 9 (MMP‐9) have been modeled using the Linear Response method that describes free energy of binding as a linear combination of ensemble averages of van der Waals, electrostatic, and cavity‐related terms. Individual terms are differences in the respective quantities between the bound and free ligands, which were determined using two methods: a hybrid Monte Carlo method within the frame of Surface‐generalized Born Continuum Solvation Model and Molecular Dynamics simulation. The MD simulations of ligands in the hydrated MMP‐9 active site, as well as of free ligands in water were carried out for 200 ps. The time‐averaged structures of the ligands bound with the protein and of the free ligands were collected at 5, 10, 25, 50, 100, and 200 ps. The best correlations between experimental and calculated binding energies, obtained using the training set containing 21 structurally diverse compounds, explained about 90% of experimental variance. The predicted binding affinities for the test set of 7 inhibitors were in good agreement with the experimental data (RMSE=0.944). Similar RMSE values were observed with 200 random selections of the 7‐member test set. The loss of polar and nonpolar solvent accessible surface areas upon binding was identified as the main phenomenon contributing to affinity, accompanied by the enhancement of van der Waals interactions upon binding.

The Linear Response (LR) approximation and similar approaches belong to practical methods for estimation of ligand-receptor binding affinities. The approaches correlate experimental binding affinities with the changes upon binding of the ligand electrostatic and van der Waals energies, and of solvation characteristics. These attributes are expressed as ensemble averages that are obtained by conformational sampling of the protein-ligand complex and of the free ligand by molecular dynamics or Monte Carlo simulations. We observed that outliers in the LR correlations occasionally exhibit major conformational changes of the complex during sampling. We treated the situation as a multi-mode binding case, for which the observed association constant is the sum of the partial association constants of individual states/modes. The resulting nonlinear expression for the binding affinities contains all the LR variables for individual modes that are scaled by the same 2-4 adjustable parameters as in the one-mode LR equation. The multi-mode method was applied to inhibitors of a matrix metalloproteinase, where this treatment improved the explained variance in experimental activity from 75% for the uni-mode case to about 85%. The predictive ability scaled accordingly, as verified by extensive cross-validations.

Abstract. It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any L 1 -contraction with mean ergodic (ME) modulus, and for any positive contraction of Lp with 1 < p < ∞. We extend the return times theorem by proving that if S is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g bounded measurable {S n g(ω)} is a universally good weight for a.e. ω. We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any L 1 -contraction with mean ergodic modulus converge in L 1 -norm. In order to produce weights, good for weighted ergodic theorems for L 1 -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of L 1 -contractions is the product of their moduli, and that the tensor product of positive quasi-ME L 1 -contractions is quasi-ME.

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