Shape superposition-based

Methods for molecular shape similarity can be roughly divided into two categories those that require finding the optimal superposition of the molecules being compared (superposition-based) and those that, by contrast, are independent of molecular orientation and position (superposition-free). Here we are restricting our focused review to those techniques that have demonstrated to perform shape similarity and its suitability to bioisosteric replacement in small molecules. 

In the superposition-free category, techniques are typically based on exploiting interatomic distances in a way that is independent of molecular orientation and position. One group of superposition-free methods measuring molecular shape was based on atom triplet distances. Bemis and Kuntz [16] devised a method that considered each molecule as the set of its atom triplets. Molecular shape histograms were calculated with the perimeters of the triangle formed by each atom triplet and... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

In the solid, dynamics occurring within the kHz frequency scale can be examined by line-shape analysis of 2H or 13C (or 15N) NMR spectra by respective quadrupolar and CSA interactions, isotropic peaks16,59-62 or dipolar couplings based on dipolar chemical shift correlation experiments.63-65 In the former, tyrosine or phenylalanine dynamics of Leu-enkephalin are examined at frequencies of 103-104 Hz by 2H NMR of deuterated samples and at 1.3 x 102 Hz by 13C CPMAS, respectively.60-62 In the latter, dipolar interactions between the 1H-1H and 1H-13C (or 3H-15N) pairs are determined by a 2D-MAS SLF technique such as wide-line separation (WISE)63 and dipolar chemical shift separation (DIP-SHIFT)64,65 or Lee-Goldburg CP (LGCP) NMR,66 respectively. In the WISE experiment, the XH wide-line spectrum of the blend polymers consists of a rather featureless superposition of components with different dipolar widths which can be separated in the second frequency dimension and related to structural units according to their 13C chemical shifts.63... 

From the beginnings, attempts to model the line shapes of collision-induced absorption spectra were based on the assumption that the various rotational lines of induced spectra, Figs. 3.10 through 3.14, are superpositions of scaled and shifted line profiles, g(C)(v), °f a small number of different, e.g., overlap- and quadrupole-induced, types [313, 404],... 

H-bonded systems may require additional diffuse or polarization functions. For example, the 6-311++G(d,p) basis set had been found to be suitable for H-bonded systems [78-81], It may be necessary to include Basis Set Superposition Errors (BSSE) [82] and Zero-Point-Energy (ZPE) corrections in evaluating the relative stabilities. Such corrections are often of the same magnitude as the energy differences among the dominant conformers. Moreover, the relative conformer energies may also differ noticeably with the basis sets used. All these factors will affect the Boltzmann factors predicted for different conformers and therefore the appearance of the population weighted VA and VCD spectra. Thus, an appropriate selection of DFT functionals and basis sets is very important for VCD simulations. A scale factor of 0.97-0.98 is usually applied to the calculated harmonic frequencies to account for the fact that the observed frequencies arise from an anharmonic force field instead of a harmonic one. A Lorentzian line shape is typically used in simulations of VA and VCD spectra. The full-width at half maximum (FWHM) used in the spectral simulation is usually based on the experimental VA line widths. 

ROCS is using a shape-based superposition for identifying compounds that have similar shape. Grant and Pickup (118) showed that using atomic-centered Gaussians instead of a spherical function can dramatically reduce the time required for a shape alignment of two molecules. This improved routine allows the program to perform shape-based database searches at an acceptable speed (300-400 conformers/s). 

The measurement model method for distinguishing between bias and stochastic errors is based on using a generalized model as a filter for nonreplicacy of impedance data. The measurement model is composed of a superposition of line-shapes that can be arbitrarily chosen subject to the constraint that the model satisfies the Kramers-Kronig relations. The model presented in Figure 21.8, composed of Voigt elements in series with a solution resistance, i.e.. 

To overcome this problem, we have designed a novel combined scheme that efficiently accounts for bilayer deformations together with the electrostatic PB solution self-consistently [27]. Our strategy is based on representing the membrane interface shape (contour) as a linear superposition of N Gaussian functions (used here as a basis set) centered at different locations on the surface of the membrane. In this manner, we can approximate the local membrane height h(x,y) at any point (x. y) by the following sum [27] ... 

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