Distance geometry three dimensional coordinates

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

At this point, we have shown only that we can convert three-dimensional coordinates to the metric matrix G and then regenerate them. The key to distance geometry is that G also can be derived directly from the distance matrix, D, in which each component djj is the distance between atoms i and /. D is first converted into D< where each component is the distance between each atom i and the center of mass o. 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

DISCO considers three-dimensional conformations of compounds not as coordinates but as sets of interpoint distances, an approach similar to a distance geometry conformational search. Points are calculated between the coordinates of heavy atoms labeled with interaction functions such as HBD, HBA or hydrophobes. One atom can carry more than one label. The atom types are considered as far as they determine which interaction type the respective atom would be engaged in. The points of the hypothetical locations of the interaction counterparts in the receptor macromolecule also participate in the distance matrix. These are calculated from the idealized projections of the lone pairs of participating heavy atoms or H-bond forming hydrogens. The hydrophobic points are handled in a way that the hydrophobic matches are limited to, e.g., only one atom in a hydrophobic chain and there is a differentiation between aliphatic and aromatic hydrophobes. A minimum constraint on pharmacophore point of a certain type can be set, e.g. if a certain feature is known to be required for activity [53, 54]. 

Other simple geometrical descriptors are interatomic distances between pairs of atoms s and t. Interatomic distances are devided into intramolecular interatomic distances, i.e. distances between any pair of atoms (s, t) within the molecule, and intermo-lecular interatomic distances, i.e. distances between atoms of a molecule and atoms of a receptor structure, a reference compound or another molecule. While classical computational chemistry describes molecular geometry in terms of three-dimensional Cartesian coordinates or internal coordinates, the -> distance geometry (DG) method takes the interatomic distances as the fundamental coordinates of molecules, exploiting their close relationship to experimental quantities and molecular energies. 

Having understood the concepts of Ew ald summation techniques for three-dimensional bulk systems, we now turn to systems that are finite in at least one spatial dimension. We focus on a slab-like geometry where the fluid is confined by two plane parallel and structureless solid surfaces separated by a distance s along the 2-axis of the coordinate system and of infinite extent in the x-y plane (see also Section 1.3.2). Hence, for the time being, we shall be... 

All possible conformers lie between these upper and lower distance bounds—the task of distance geometry is to convert or embed this usually uncertain distance information into accurate three-dimensional Cartesian coordinates. Crippen and HavePi solved the problem for the case of an exact distance matrix, where all distances are known. Much additional effort has gone into finding efficient and practical methods for solving the general problem of a distance bounds matrix, where only a subset of the distances is known exactly. This remains a very difficult problem for large molecules with more than 1000 atoms. 

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