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In this paper we study the statistical geometry of ensembles of poly (L-alanine) conformations computed by several different distance geometry algorithms. Since basic theory only permits us to predict the statistical properties of such ensembles a priori when the distance constraints have a very simple form, the only constraints used for these calculations are those necessary to obtain reasonable bond lengths and angles, together with a lack of short- and long-range atomic overlaps. The geometric properties studied include the squared end-to-end distance and radius of gyration of the computed conformations, in addition to the usual rms coordinate and phi/psi angle deviations among these conformations. The distance geometry algorithms evaluated include several variations of the well-known embed algorithm, together with optimizations of the torsion angles using the ellipsoid and variable target function algorithms. The conclusions may be summarized as follows: First, the distribution with which the trial distances are chosen in most implementations of the embed algorithm is not appropriate when no long-range upper bounds on the distances are present, because it leads to unjustifiably expanded conformations. Second, chosing the trial distances independently of one another leads to a lack of variation in the degree of expansion, which in turn produces a relatively low rms square coordinate difference among the members of the ensemble. Third, when short-range steric constraints are present, torsion angle optimizations that start from conformations obtained by choosing their phi/psi angles randomly with a uniform distribution between -180 degrees and +180 degrees do not converge to conformations whose angles are uniformly distributed over the sterically allowed regions of the phi/psi plane. Finally, in an appendix we show how the sampling obtained with the embed algorithm can be substantially improved upon by the proper application of existing methodology.

By generating classes of random structures for trypsin inhibitor and carp myogen, each consistent with a given set of experimental or theoretical information, we have assessed the relative utility of various experiments and theories in deducing the conformation of macromolecules. We compare the calculated structures with known x-ray coordinates and compute for each class an average error. Small errors mean that the experimental or theoretical constraints limit the structures to the vicinity of the crystal structure, whereas large errors show that the constraints permit a wide variety of tertiary conformations. We find the following points to hold true: (1) Qualitative information on all the distances, as might be obtained from the correct prediction of interresidue contacts, effectively determines the structure (error approximately 1 A). (2) Quantitative information on a limited number of distances, as might be obtained from nmr or crosslinking experiments, significantly restricts the range of possible structures only when the number of distances given is comparable to the number of residues (error approximately 3 A). (3) Quantitative information on the distances of each residue to the center of mass of the molecule, as might in part be obtained from solvent accessibility and solution x-ray studies, is not particularly restrictive by itself (error approximately 5 A). (4) Complete qualitative local distance information, as might be obtained from secondary prediction and CD/ORD studies, is clearly consistent with a wide variety of tertiary structures (error approximately 7 A).

A recently developed method of constrained optimization, known as the ellipsoid algorithm, is explored as a tool for determining sterically acceptable interactions between two molecules. These interactions are described by constraints on intermolecular distances. Upper distance bounds between specific pairs of atoms, one in the ligand and one in the enzyme, were used in two different types of docking problems. In the first type, knowledge of a small set of well-defined upper distance constraints was assumed, e.g., specific information about hydrogen bonds or experimentally determined atom–atom distances. The second approach assumes only knowledge of the location of the binding site of one of the molecules, but nothing about how the second, typically small, molecule is packed into this site. In this case, a set of upper distance bounds was used one at a time to explore systematically all possible ways of packing the ligand into the site. In all applications, van der Waals repulsions were used to define a lower distance bound for each atom pair. To specify the relative orientation of the two molecules, a new set of variables had to be introduced. They enable us to represent the set of all possible rotations in a vector space as required by the ellipsoid algorithm. Consequently, in contrast to the usual Euler angles, they assure additivity, so that the result of multiple changes of the variables is not sensitive to the order in which they are performed. In addition to the six degrees of freedom involved in docking rigid objects, conformational flexibility can be explicitly included in the individual molecules. Applications discussed include the docking of two macromolecules and the formation of an enzyme–inhibitor complex. For each constraint set tested, ten randomly chosen starting structures were optimized. The time for each of the runs was between 3 min and 1.3 h on a VAX 750 or a Microvax II; in larger problems, the time is spent primarily in van der Waals checking. The advantages of the ellipsoid algorithm compared to other methods are its robustness with respect to local minima, and the relatively small amounts of computer time and memory that it needs. A final discussion compares this method to some other docking methods, including distance geometry and the matching of molecular surfaces.