Restrained least-squares

The method of Konnert and Hendrickson [116-118] is the most widely used. The known stereochemical features of the amino acids are included as restraints and these 

For N parameters the least-squares equations lead to an V x AT normal matrix. Because the restraints involve only near neighbour atoms, the matrix is sparse, with the majority of non-diagonal terms being zero and less than 1% of the elements nonzero [117]. For n atoms and m distance restraints the number of elements to be stored is 6n-l-9m. For example, with a small protein of 812 atoms and 2030 restraints (approximately 3 x the number of atoms), the number of elements is 23 142. For phosphorylase b with 6640 atoms there are 26 561 parameters and some 229451 nonzero elements on the normal matrix, which is still only 0.03% of the total matrix elements. In the restrained least-squares refinement (and many of the other refinement methods) the normal equations are solved by the conjugate-gradient algorithm [129]. 

The relative weights for the X-ray observations and the restraints may be adjusted. In the early stages of refinement, the weights for the restraints might be relatively high in order to achieve a stereochemically sensible model. Gross errors in the structure can be detected by difference Fourier syntheses (section 2(h)). As the refinement progresses, the restraints may be relaxed. The final R value depends upon resolution and the restraints, and it is important that both values are quoted so that the stereochemical reasonableness of the structure can be assessed. 

The Konnert-Hendrickson method is relatively expensive in terms of computing power [131], but recent developments that combine the use of fast Fourier transform methods (section iii below) have provided dramatic increases in speed. 

As an example of the application of the method we quote values for the refinement of protease A from Streptomyces griseus [97]. The R value for some 12 662 reflections in the resolution range 8.0-1.8 A was 0.139 for some 5912 variable parameters for 1250 protein atoms and 175 water molecules. The final structure differed from ideal bond lengths by an overall root mean square (rms) deviation of 0.02 A and the probable error in atomic co-ordinates was of the order of 0.15 A. 

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Data analysis procedures have developed substantially over the last few years. In particular, use of least square refinement methods have been developed. Recent progress with theoretical development for the treatment of multiple scattering has resulted in Ugand group refinement such as an imidazole. We can expect further development in this area which ought to lead us to restrained least square refinement procedures for EXAFS data analysis. This type of restrained refinement is commonly used for macromolecular crystallographic structure determination where a similar problem of imderdeterminancy exists... 

In addition to the studies of various crystalline forms of domestic hen egg-white lysozyme, the structures of human and tortoise egg-white lysozymes have been determined (for crystal data see Table III). Artymiuk and Blake (1981) refined the structure of the human enzyme to 1.5 A resolution. The main objectives of this study were to determine the extent of differences in structure from that of the hen egg-white protein, to discover the location of water molecules, and to test the validity of the method of restrained refinement. The particular restrained least-squares approach to refinement described in their paper appears to have been validated. The two proteins were found to be closely homologous, but there were small differences (e.g., in a helices), details of which can be obtained from consulting their paper. 

Finzel, B. C.( 1987). Incorporation of fast Fourier transforms to speed restrained least-squares refinement of protein structures. J. Appl. Crystallogr. 20,53-55. 

Konnert, J. H. A restrained-parameter structure-factor least-squares refinement procedure for large asymmetric units. Acta Cryst. A32, 614-617 (1976). Sussman, J. L., and Podjarny, A. D. The use of a constrained-restrained least-squares procedure for the low-resolution refinement of a macromolecule, yeast tRNAfActa Cryst. B39, 495-505 (1983). 

Herzberg, O., and Sussman J. L. Protein model building by the use of a constrained-restrained least-squares procedure. J. Appl. Cryst. 16, 144-150 (1983). 

Figure 3. The segment consisting of residues Cys-192 and His-193 of the 2.8 A resolution structure of a single site mutant of aspartate aminotransferase [23]. Superimposed are the initial structure (dotted lines) obtained by fitting the atomic model to a multiple isomorphous replacement map, the structure obtained after several cycles of rebuilding and restrained least-squares refinement (thick lines), the structure obtained after simulated annealing refinement (thin lines), and the structure obtained after conjugate gradient minimization (dashed lines).

Guss, J.M., Bartunik, H.D., Freeman, H.C. Accuracy and precision in protein structure analysis Restrained least-squares refinement of the structure of poplar plastocyanin at 1.33 A resolution. Acta Crystallogr. B 1992, 48, 790-811. 

Thioredoxin from E. coli has been studied extensively using biochemical, spectroscopic and X-ray diffraction techniques. The protein consists of a single polypeptide chain of 108 amino acid residues of known sequence. The protein has been cloned and expressed. Thioredoxin of E. coli is a compact molecule with 90% of its residues in hehces, beta-strands or reverse turns. This protein transports electrons via an oxidation-reduction active disulfide". The oxidized form thioredoxin-(S2) is reduced to thioredoxin-(SH)2. In particular, this protein was found to participate in the reduction of ribonucleotides to deoxyribonucleotides. In Fig. 1, the optimized stracture is shown with a carbon backbone for clarity only. The molecule consists of two conformational domains, connected by two helices. The beta-sheet forms the core of the molecule packed on either side by clusters of hydrophobic residues. Helices form the external surface. We used a crystal stracture of the oxidized form of thioredoxin from Escherichia coli that has been refined by the stereochemically restrained least-squares procedure at 1.68 A resolution". ... 

The structure was refined using a version of the restrained-least-squares refinement program PROLSQ (Hendrickson 1985). The starting point for the refinement was the model for deoxy Hb (including 89 water molecules). The course of the refinement is charted in figure 10.2(a) showing the fluctuation of the f -factor during the refinement. The final f -factor was 19.6% for all data between 10 A and 1.5 A. The quality of... 

Refinement of Triclinic Lysozyme. II. The Method of Stereochemically Restrained Least Squares. 

Previously we reported the refinement of the structure of the RC from the R-26 strain using the Hendrickson and Konnert restrained least-squares refinement (9). Refinement cycles were alternated with model building using an interactive graphics terminal and the program FRODO (10). This yielded a R factor of 24.0% (1,3). 

E. A. Padlan and W. E. Love,/. BioL Chem., 260,8272 (1985). Refined Crystal Structure of Deoxyhemoglobin S. I. Restrained Least-Squares Refinement at 3.0-A Resolution. 

CORELS computer program for COnstrained-REstrained Least-Squares X-ray... 

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