The choice to use isotropic ADPs in macromolecular crystallography is a concession to the limited number of Bragg reflections available at lower resolution; only four parameters per atom [ are required for a model with isotropic ADPs as opposed to nine parameters per atom for a model with anisotropic ADPs. It is not driven by an expectation that macromolecular crystals exhibit less anisotropy. On the contrary, the inherent flexibility of macromolecules combined with the high solvent content and relatively loose lattice packing they exhibit when crystallized leads to substantial atomic anisotropy (Hinsen, 2008 ). This is borne out experimentally both by refinement of anisotropic ADPs for the small fraction of protein structures that diffract to true atomic resolution (Schneider, 1996 ; Merritt, 1999b ) and by the improved R-factors obtained even for low-resolution structures when relatively simple descriptions of bulk anisotropy are added to the model (Merritt, 2011 ). Thus it is becoming standard practice in protein crystallography to include an explicit model for bulk anisotropic displacements (Zucker et al., 2010 ). The most common approach is to treat segments of the protein as approximately rigid groups exhibiting concerted displacements described by the translation/libration/screw (TLS) formalism (Trueblood, 1978 ; Howlin et al., 1989 ; Winn et al., 2001 ; Painter & Merritt, 2006 ). A second approach is to model concerted atomic displacements as arising from normal mode vibrations identified by an elastic network model (Poon et al., 2007 ). In both approaches these bulk models are applied to generate conventional anisotropic ADP descriptions for each atom, which are in turn used to calculate the gradients that drive crystallographic refinement (Winn et al., 2001 ; Chen et al., 2007 ). The individual anisotropic ADPs derived in this way are Gaussian approximations to the non-Gaussian distributions described by the TLS or normal mode displacements, an approximation that is strictly valid only in the limiting case of infinitesimal displacement amplitude.

The output from refining this sort of model thus consists of direct estimates for the atomic positional coordinates and for the bulk displacement parameters. From these can be derived estimates of per-atom anisotropic Gaussian displacements. As in the case of refining a model at atomic resolution with anisotropic Gaussian ADPs for each atom, there is no direct refinement of a quantity equivalent to B _iso. Nevertheless, as noted above, some software expects to have available a value representing the isotropic displacement of each atom. The quantity recommended for this purpose by the International Union of Crystallography (IUCr) (IUCr Commission on Journals, 1986 ) is where U is the conventional symmetric tensor describing the displacement of that atom as an anisotropic three-dimensional Gaussian in Cartesian coordinates (Hamilton, 1959 ; Trueblood et al., 1996 ). The physical interpretation of B _eq is the mean-square displacement averaged over all directions. This is appropriate for the generation of figures that use a mean-square displacement isosurface as a visual cue for atomic displacement, e.g. ORTEP (Burnett & Johnson, 1996 ). B _eq is also adequate for qualitative evaluation of the relative vibrational motion of various parts of a structure. But it is not necessarily the best choice for other quantitative purposes. In particular, it is not the best estimate of the isotropic ADP B _iso that would be obtained through direct refinement, as will be shown in this paper.

Since V is isotropic we do not care about the orientation of U and can therefore rotate it to yield the convenient form where , , are the eigenvalues of the original symmetric tensor. An equivalently convenient tensor form describing V in the same units as U is given by where the scalar value = .

Coordinates and observed structure factors for each structure in the test set were obtained from the Protein Data Bank. Structure factors were converted from intensities to amplitudes if necessary. Each structure was then subjected to 15 cycles of isotropic refinement using REFMAC version 5.6.0095, starting from the original positional coordinates and the isotropic B factor ( as given in the ATOM records of the corresponding PDB file. To minimize shifts in the positional coordinates during refinement, interatomic distances were strongly restrained to their initial values via the command ‘RIDG DIST SIGM 0.00001’. H atoms were added in riding positions. All other refinement parameters were left at default values.

A modified version of the program coruij (Merritt, 1999a ) was used to compare the ATOM and ANISOU records from the original anisotropic model with the ATOM records in the refined isotropic model. The program verifies that the positional shifts are negligible and outputs the original ADP eigenvalues, anisotropy, B _eq, B _est and B _iso for each atom. Analysis showed that B _est calculated from the original anisotropic ADPs was a better predictor than B _eq in estimating the refined value B _iso, particularly for atoms with strong anisotropy (Fig. 2 ).