Crystallography‎ > ‎Refinement‎ > ‎

Theory

Tronrud
Introduction to macromolecular refinement
Acta Crystallogr D Biol Crystallogr (2004) vol. 60 (Pt 12 Pt 1) pp. 2156-68

Constraints: Simplify the refinement problem by reducing the number of parameters
Restrains: Instead work by increasing the number of observations, the parameters, of the model

The 'B' factors and occupancy
The B factor provides an estimate of an atom's vibration about its central position. The usual form is either to define the B factor as ISOTROPIC, meaning that the atoms vibrates equally in all direction and can be visualized as lying within a sphere, or to define an ANISOTROPIC B factor, which describes vibration of the atom within an ellipsoid centered at the atomic coordinate.
The most significant inappropriate constraint applied generally to protein models is the ISOTROPIC B factor. It is quite certain that atoms in crystals that diffract to resolution lower than 2 A move anisotropically and yet they are routinely modeled as isotropic.
A parameterization that allows fewer parameters than an independent anisotropic B factor for each atom is called TLS (translation, libration and screw). In this system the motion of a group of atoms is described by three matrices, one for a purely translational vibration of the group, a second for libration (wobbling) of the group about a fixed point and a third for a translation and libration that occurs in concert. The explicit assumption of TLS B factors is that the group of atoms move as a rigid unit.

{Note: TLS parameterization allows to partly take into account anisotropic motions at modest resolution (> 3.5 A) and it might improve refinement statistics of several percent}

More about B (or Temperature) factors.

Guides to convergence
Iterative real- and reciprocal-space refinement is monitored by comparing the measured structure-factor amplitude |F obs| [which are proportional to (I obs)^1/2] with amplitudes |F calc| from the current model. In calculating the new phases at each stage, we learn what intensities our current model, if correct, would yield. As we converge to the correct structure, the measured Fs and the calculated Fs should also converge.

The most widely used measure of convergence is the residual index, or R-factor

R=[sum||F obs| - |F calc||]/sum |F obs|

In this expression, each |F obs| is derived from a measured reflection intensity and each |F calc| is the amplitude of the corresponding structure factor from the current model.

A more demanding and revealing criterion of model quality and of improvements during refinement is the free R-factor, (R free). R free is computed with a small set of randomly chosen intensities, the 'test set', which are set aside from the beginning and not used during refinement. The are used only in the cross-validation or quality control process of assessing the agreement between calculated (from the model) and observed data. At any stage in refinement, R free measures how well the current atomic model predicts a subset of the measured intensities that were not included in the refinement, whereas R measures how well the current model predicts the entire data set that produced the model.