Gordon M. Crippen

A novel approach to calculation of conformation: Distance geometry

This paper considers the calculation of the conformation of a molecule by the unusual means of proposing the matrix D of all interatomic distances subject to a priori energetic and geometric constraints, and then calculating the corresponding atomic coordinates. The necessary and sufficient conditions on D from distance geometry are cited. Results are given for trials of numerical methods for carrying out such conformational calculations on cyclohexane and trypsin inhibitor.

Stable calculation of coordinates from distance information

A new method is described for the calculation of Cartesian coordinates for n points given the n × n matrix of interpoint distances. The algorithm is faster than some earlier methods, and it is remarkably stable with respect to both numerical roundoff errors and errors in the given distance matrix. The resultant coordinates have their origin near the center of mass and axes approximately along the three principal rotational axes. The calculation is described of distances to the center of mass directly from the distance matrix. Results of computer trials of the algorithm are given.

The tree structural organization of proteins

Abstract We offer an objective definition of the domains of a protein, given its C α coordinates from high-resolution X-ray crystal studies. This is done by an algorithm which groups segments of the polypeptide chain together when there are a relatively large number of contacts between the two segments. The result is an organizational tree showing a hierarchy of segments grouping together, then clusters merging until all parts of the chain are included. In this view the highest level clusters correspond well to more subjective definitions of folding domains and the lowest level, the segments, roughly match the usual assignments of pieces of secondary structure. The intermediate level clusters suggest possible folding mechanisms, which are discussed.

Calculation of protein tertiary structure

Abstract We describe a method for calculating the tertiary structure of proteins given their amino acid sequence. The algorithm involves locally minimizing an energylike expression as a function of the Cartesian co-ordinates of the C β of all residues. Although the approximation to the true polypeptide geometry and conformational energies is extremely approximate, quite respectable results have been obtained for the small proteins rubredoxin and trypsin inhibitor, where the root mean square errors are as low as 4.0 A and 4.7 A, respectively.

Note rapid calculation of coordinates from distance matrices

Abstract In conclusion, we have devised and tested a practical algorithm for rapidly solving (1) in the difficult case of large n and numerous strong constraints. Computer time increases only quadratically and memory requirements increase only linearly with n , and there is little difficulty with multiple minima. We hope this will make the distance geometry approach to conformational calculation more feasible for large, highly constrained systems.

Global optimization and polypeptide conformation

Abstract The problem of calculating the conformation of a molecule by global minimization of its free energy is precisely formulated. Various bounds and estimates are derived for the number of energy evaluations necessary to perform the task, independent of the search algorithm used. The algorithm by Schubert is cited as optimal for the problem as formulated, and an improvement for starting it is presented. In light of the estimates for computer time and memory for the optimal method, ab initio global minimization is proven to be infeasible for calculating conformations of even oligopeptides.

Directional structural features of globular proteins.

We describe a novel presentation of the conformation of the backbone atoms for proteins of known structure. Given the Cα atom cartesian co-ordinates from X-ray crystallography, a matrix is calculated, where the ijth element of the matrix is the cosine of the angle between the direction of the chain at residue i and the direction of the chain at residue j. These “direction matrices” have distinctive patterns which correspond to α-helix, extended structure, straight or bent segments, “superhelix”, and many other important structural features. We discuss the direction matrices for a number of proteins, and make some generalizations on the basic principles of protein folding.