Geometry distance

One way to describe the conformation of a molecule other than by Cartesian or internal coordinates is in terms of the distances between all pairs of atoms. There are N(N - l)/2 interatomic distances in a molecule, which are most conveniently represented using an N X N S5munetric matrix. In such a matrix, the elements (i, j) and ( , i) contain the distance between atoms i and j and the diagonal elements are all zero. Distance geometry explores conformational space by randomly generating many distance matrices, which are then converted into conformations in Cartesian space. The crucial feature about distance geometry (and the reason why it works) is that it is not possible to arbitrarily assign values to the 

Distance geometry uses a four-stage process to derive a conformation of a molecule [Crippen 1981 Crippen and Havel 1988]. First, a matrix of upper and lower interatomic distance bounds is calculated. This matrix contains the maximum and minimum values periiutted to each interatomic distance in the molecule. Values are then randomly assigned to each interatomic distance between its upper and lower bounds. In the third step, the distance matrix is converted into a trial set of Cartesian coordinates, which in the fourth step are then refined. 

We can now proceed to the generation of conformations. First, random values are assigned to all the interatomic distances between the upper and lower bounds to give a trial distance matrix. This distance matrix is now subjected to a process called embedding, in which the distance space representation of the conformation is converted to a set of atomic Cartesian coordinates by performing a series of matrix operations. We calculate the metric matrix, G, each of whose elements (i, j) is equal to the scalar product of the vectors from the origin to atoms i and  

The elements Gij can be calculated from the distance matrix using the cosine rule  

It is usual to take the centre of the molecule as the origin of the coordinate system The distance of each atom from the centre can be calculated directly from the interatomic distances using the following expression  

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Crippen G M 1977 A novel approach to calculation of conformation distance geometry J. Comput. Rhys. 24 96-107... 

Metzler W J, Hare D R and Pardi A 1989 Limited sampling of conformational space by the distance geometry algorithm implications for structures generated from NMR data Bioohemistry 2S 7045-52... 

J.J. More and Z. Wu, Global continuation for distance geometry problems, SIAM J. Optimization 7 (1997), 814-836. 

Soc. 1974, 96, 4S34--i842. pi3] D. Weininger, SMILES - a language for molecules and reactions, in Handbook of Chemoinformatics, J. Gasteiger (Ed.) Wiley-VCH, Weinheim, 2003, Chapter 11, Section 3. pi4] G. M. Crippen, T.F. Havel, Distance geometry and molecular conformations, in Chemometrics Research Studies Series 15 D. Bawden (Ed.), Research Studies Press (Wiley), New York, 1988. 

The greatest value of molecular dynamic simulations is that they complement and help to explain existing data for designing new experim en ts. Th e sun ulation s are in creasin gly n sefn I for stnictural relinemcnt of models generated from XMR, distance geometry, an d X-ray data. 

Five-carbon fragment to illustrate distance geometry algorithm. 

Dmparison of various methods for searching conformational space has been performed cycloheptadecane (C17H34) [Saunders et al. 1990]. The methods compared were the ematic search, random search (both Cartesian and torsional), distance geometry and ecular dynamics. The number of unique minimum energy conformations found with 1 method within 3 kcal/mol of the global minimum after 30 days of computer processing e determined (the study was performed in 1990 on what would now be considered a / slow computer). The results are shown in Table 9.1. 

Blaney J M and J. S Dixon 1994. Distance Geometry in Molecular Modeling, In Lipkowitz K B and D E Boyd (Editors) Reviews in Computational Chemistry Volume 5. New York, VCH Publishers, pp. 299-335. 

Crippen G M and T F Havel 1988. Distance Geometry and Mdecidar Conformation. Chemorridtrics Research Studies Series 15. New York, Jolin Wiley Sons. 

Ensemble Distance Geometry, Ensemble Molecular Dynamics and Genetic Algorithms... 

A variant of distance geometry called ensemble distance geometry [Sheridan et al. 1986] can be used to simultaneously derive a set of conformations with a previously defined set of pharmacophoric groups overlaid. Ensemble distance geometry uses the same steps as... 

Distance matrix used in ensemble distance geometry. There are Ni atoms in the first molecule, N2 in the and so on. 

Fig. 12.12 Four molecules used to derive the nicotinic pharmacophore by distance geometry and the pharmacophore obtained.

Sheridan R P, R Ndakantan, J S Dixon and R Venkataraghavan 1986. The Ensemble Approach to Distanc Geometry Application to the Nicotinic Pharmacophore. Journal of Medicinal Chemistry 29 899-906. 

Dirac equation one-electron relativistic quantum mechanics formulation direct integral evaluation algorithm that recomputes integrals when needed distance geometry an optimization algorithm in which some distances are held fixed... 

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