International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 8.3, pp. 698701
Section 8.3.2.1. Stereochemical constraints as observational equations^{a}NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,^{b}Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 200151305, USA, and ^{c}Laboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 203755000, USA 
As described in Section 8.1.2 , given a set of observations, , that can be described by model functions, M_{i}(x), where x is the vector of model parameters, we seek to find x for which the sum is minimum. For restrained refinement, S is composed of several classes of observational equations, including, in addition to the ones for structure factors, equations for interatomic distances, planar groups and displacement factors.
Structure factors yield terms in the sum of the form The distances between bonded atoms and between nextnearestneighbour atoms may be used to require bonded distances and angles to fall within acceptable ranges. This gives terms of the form where σ_{d} is the standard deviation of an empirically determined distribution of values for distances of that type. Groups of atoms may be restrained to be near a common plane by terms of the form (Schomaker, Waser, Marsh & Bergman, 1959) where and are parameters of the plane, σ_{p} is again an empirically determined standard deviation, and · indicates the scalar product.
If a molecule undergoes thermal oscillation, the displacement parameters of individual atoms that are stereochemically related must be correlated. These parameters may be required to be consistent with the known stereochemistry by assuming a model that gives a distribution function for the interatomic distances in terms of the individual atom parameters and then restraining the variance of that distribution function to a suitably small value. The variation with time of the distances between covalently bonded atoms can be no greater than a few hundredths of an ångström. Therefore, the thermal displacements of bonded atoms should be very similar along the bond direction, but they may be more dissimilar perpendicular to the bond. If we make the assumption that the atom with a broader distribution in a given direction is `riding' on the atom with the narrower distribution, the variance of the interatomic distance parallel to a vector v making an angle with the direction of bond j is (Konnert & Hendrickson, 1980) where is the normal distance for that type of bond, = , and and are the mean square displacements parallel to v of atom a and atom b, respectively. The restraint terms then have the form . For isotropic displacement factors, these terms take the particularly simple form , but with the disadvantage that, when isotropic displacement parameters are used, the displacements cannot be suitably restrained along the bonds and perpendicular to the bonds simultaneously.
Several additional types of restraint term have proved useful in restraining the coordinates for the mean positions of atoms in macromolecules. Among these are terms representing nonbonded contacts, torsion angles, handedness around chiral centres, and noncrystallographic symmetry (Hendrickson & Konnert, 1980; Jack & Levitt, 1978; Hendrickson, 1985). Contacts between nonbonded atoms are important for determining the conformations of folded chain molecules. They may be described by a potential function that is strongly repulsive when the interatomic distance is less than some minimum value, but only weakly attractive, so that it can be neglected in practice, when the distance is greater than that value. This leads to terms of the form which are included only when . Macromolecules usually gain flexibility by relatively unrestricted rotation about single bonds. There are, nevertheless, significant restrictions on these torsion angles, which may, therefore, be restrained by terms of the form where and are dihedral angles between planar groups at opposite ends of the bond.
Interatomic distances are independent of the handedness of an enantiomorphous group. If r_{c} is the position vector of a central atom and , , and are the positions of three atoms bonded to it, such that the four atoms are not coplanar, the chiral volume is defined by where × indicates the vector product. The chiral volume may be either positive or negative, depending on the handedness of the group. It may be restrained by including terms of the form
Table 8.3.2.1 gives ideal coordinates, in an orthonormal coordinate system measured in Å, of various groups that are commonly found in proteins. The ideal conformations of pairs of amino acid residues, from which the ideal values to be used in restraint terms of various types may be determined, are constructed by combining the coordinates of the individual groups. For example, consider a dipeptide composed of glycine and alanine joined by a trans peptide link, giving the molecule The origin is placed at each of the Cα positions in turn, and interatomic distances to nearest and nextnearest neighbours are computed. Planar groups and possible nonbonded contacts are identified, and torsion angles and chiral volumes for chiral centres are computed. Table 8.3.2.2 is a summary of the restraint information for this simple molecule. In order to incorporate this information in the refinement, these ideal values are combined with suitable weights. Table 8.3.2.3 gives values of the standard deviations of the various types of constraint relation that have been found (Hendrickson, 1985) to give good results in practice.



Even for a small protein, the normalequations matrix may contain several million elements. When stereochemical restraint relations are used, however, the matrix elements are not equally important, and many may be neglected. Convergence and stability properties can be preserved when only those elements that are different from zero for the stereochemical restraint information are retained. The number of these elements increases linearly with the number of atoms, and is typically less than 1% of the total in the matrix, so that sparsematrix methods (Section 8.1.5 ) can be used. The method of conjugate gradients (Hestenes & Stiefel, 1952; Konnert, 1976; Rae, 1978) is particularly suitable for the efficient use of restrainedparameter least squares.
References
Hendrickson, W. A. (1985). Stereochemically restrained refinement of macromolecular structures. Methods in enzymology, Vol. 115. Diffraction methods for biological macromolecules, Part B, edited by H. W. Wyckoff, C. H. W. Hirs & S. N. Timasheff, pp. 252–270. New York: Academic Press.Hendrickson, W. A. & Konnert, J. H. (1980). Incorporation of stereochemical information into crystallographic refinement. Computing in crystallography, edited by R. Diamond, S. Ramaseshan & D. Venkatesan, pp. 13.01–13.26. Bangalore: Indian Academy of Sciences.
Hestenes, M. & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. J. Res. Natl Bur. Stand. 49, 409–436.
Jack, A. & Levitt, M. (1978). Refinement of large structures by simultaneous minimization of energy and R factor. Acta Cryst. A34, 931–935.
Konnert, J. H. (1976). A restrainedparameter structurefactor leastsquares refinement procedure for large asymmetric units. Acta Cryst. A32, 614–617.
Konnert, J. H. & Hendrickson, W. A. (1980). A restrained parameter thermalfactor refinement procedure. Acta Cryst. A36, 344–350.
Rae, A. D. (1978). An optimized conjugate gradient solution for leastsquares equations. Acta Cryst. A34, 578–582.
Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959). To fit a plane or a line to a set of points by least squares. Acta Cryst. 12, 600–604.