Two forms of atomic displacement parameters (ADPs) are commonly used in crystallographic models. Models refined against atomic-resolution data usually include a

symmetric tensor

**U** for each atom that describes anisotropic displacement. Models refined at lower resolution usually include instead an isotropic term

*B*
_{iso} for each atom. Both the isotropic and anisotropic ADPs describe a three-dimensional Gaussian probability density function centered on the mean position of the corresponding atom. Because crystals of macromolecules typically diffract only to modest resolution, the use of

*B*
_{iso} has been particularly common in structural models for macromolecules. Indeed, the standard ATOM record for structures archived in the Protein Data Bank (PDB) contains a field for a single ‘Temperature factor’ (Berman

*et al.*, 2007

), and many software tools that manipulate PDB files expect an isotropic ADP to be provided here.

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, 1999

*b*
) 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.

This distinction becomes important when an anisotropic model, for example a specific set of groups to be described by TLS, is chosen based on an existing model that was refined isotropically. In this case one wants to select from among many possible alternative anisotropic models the one model that makes the observed set of *B*
_{iso} values most likely. Thus a better estimator for *B*
_{iso} may lead to improved model selection.