No abstract

This paper describes a procedure which optimizes the fitting of a 'model' of a protein to an electron density map. The technique seeks to minimize ~ (0o-Om)Zdv where Oo is the observed electron density and 0,~ is a density associated with a model in terms of which the observed densities are interpreted. 0,~ consists of a Gaussian density centred on each atomic centre, and a floating background level. Interactions due to overlapping densities of neighbouring atoms are allowed for and the model is normally treated as a flexible chain so that bond lengths are conserved during movement. Alternatively, the atoms may be.aUowed to move independently. Site occupations and atomic radii are also refinable. The calculation is organized in terms of a 'molten zone' of up to ten residues, which moves along the chain one residue at a time, linear or non-linear constraints being applied to preserve chain continuity at each end of the zone. Provision is made for the zone to become active or inactive in predetermined regions of the molecule. A difference map (0o-Ore) is available at the end of the calculation, as is a molecular listing with revised coordinates and dihedral and inter-bond angles. Inter-bond angles may be treated either as constants or as variables, and if variable may be made elastically stiffer than dihedral angles.The procedure is well suited to maps of 2 to 3/~ resolution, but is not limited to this range. It has produced convergent shifts exceeding 1"5 ~ in a map of 2 A resolution, and, except for shifts exceeding 1 A, convergence is essentially complete in one pass. The procedure has, so far, been applied to four proteins. A, B a t all B C

Two modifications to the commonly used protocols for calculating NMR structures are developed, relating to the treatment of NOE constraints involving groups of equivalent protons or nonstereoassigned diastereotopic protons. Firstly, a modified method is investigated for correcting for multiplicity, which is applicable whenever all NOE intensities are calibrated as a single set and categorised in broad intensity ranges. Secondly, a new set of values for 'pseudoatom corrections' is proposed for use with calculations employing 'centre-averaging'. The effect of these protocols on structure calculations is demonstrated using two proteins, one of which is well defined by the NOE data, the other less so. It is shown that failure to correct for multiplicity when using 'r(-6) averaging' results in overly precise structures, higher NOE energies and deviations from geometric ideality, while failure to correct for multiplicity when using 'r(-6) summation' can cause an avoidable degradation of precision if the NOE data are sparse. Conversely, when multiplicities are treated correctly, r(-6) averaging, r(-6) summation and centre averaging all give closely comparable results when the structure is well defined by the data. When the NOE data contain less information, r(-6) averaging or r(-6) summation offer a significant advantage over centre averaging, both in terms of precision and in terms of the proportion of calculations that converge on a consisten result.

A method of optimally superimposing n coordinate sets on each other by rigid body transformations, which minimizes the sum of all n ( n -1)/2 pairwise residuals, is presented. In the solution phase the work load is approximately linear on n, is independent of the size of the structures, is independent of their initial orientations, and terminates in one cycle if n = 2 or if the coordinate sets are exactly superposable, and otherwise takes a number of cycles dependent only on genuine shape differences. Enantiomorphism, if present, is detected, in which case the option exists to reverse or not to reverse the chirality of relevant coordinate sets. The method also offers a rational approach to the problem of multiple minima and has successfully identified four distinct minima in such a case. Source code, which is arranged to enable the study of the disposition of domains in multidomain structures, is available from the author. Keywords: algorithms; comparisons; quarternions; rotations; superpositionsWith the development of NMR and molecular dynamics techniques it is becoming increasingly common to wish to compare a number of actual or putative structures by superimposing their coordinates using rigid-body transformations (translation and strain-free rotation) and to measure the root mean square (rms) coordinate differences that result.The superposition of one structure on one other is a problem to which many solutions have been offered, such as those of McLachlan (1972McLachlan ( , 1979McLachlan ( , 1982, Kabsch (1976Kabsch ( , 1978, Diamond (1976Diamond ( , 1988, Lesk (1986), andKearsley (1989), but the optimal superposition of ensembles of structures has only recently received similar attention. The situation is complicated by the fact that if structure A is superimposed on structure C, and structure B is superimposed on structure C, then, in general, structure A is not optimally superimposed on structure B. In these circumstances the superposition of A on B is only optimal if two of the three structures are identical in shape.If the sum of the squares of the coordinate differences between A and B is designated EaB, and if it is wished t o compare n structures by superposition, there are n ( n -1 )/2 interactions such as EAB, and the method described in this paper minimizes the sum of all of these in Reprint requests to: R. Diamond,