# What is a B-factor?

The following is heavily based on Trueblood et al., 1996 and references therein.

Displacement of atoms from their mean position in a crystal structure diminishes the scattered X-ray intensity. The displacement may be the result of temperature-dependent atomic vibrations or static disorder in a crystal lattice. If the atomic displacements are assumed to be harmonic and anisotropic, the attenuation of atomic scattering factors can be represented by the atomic anisotropic Gaussian Debye-Waller factor

\(T(\textbf{h}) = \exp{[-8\pi^2 \langle u_\textbf{h}^2 \rangle \frac{\sin^2 \theta}{\lambda^2}]}\). (1)

(equation 1.4.12 from Trueblood et al., 1996). Here \(u_\textbf{h}\) is the projection of the atomic displacement \(\textbf{u}\) on the direction of diffraction vector \(\textbf{h}\), \(\theta\) is the scattering angle and \(\lambda\) is the X-ray wavelength. The average in the exponent denotes an average over space and time. The mean-square displacements are termed anisotropic displacement parameters (ADPs) as the average depends on the direction of \(\textbf{h}\). Referred to a Cartesian basis, an equivalent representation of eq. (1) is

\(T(\textbf{h}) = \exp{[-2\pi^2 \textbf{h}^\textbf{T}\textbf{Uh}]}\). (2)

The ADPs define the symmetric atomic mean-square displacement tensor

\(\textbf{U} = \begin{pmatrix} U_{11} & U_{12} & U_{13} \\ U_{12} & U_{22} & U_{23} \\ U_{13} & U_{23} & U_{33} \end{pmatrix}\). (3)

An element \(U_{ij}\) of \(\textbf{U}\) has dimension (length)^{2}. If the displacements are (assumed to be) isotropic, the average in eq. (1) is constant and the Debye-Waller factor only depends on the magnitude of \(\textbf{h}\):

\(T(|\textbf{h}|)=\exp{[-8\pi^2 \langle u^2 \rangle \frac{\sin^2 \theta}{\lambda^2}]}\). (4)

(equation 1.4.13 from Trueblood et al., 1996). The B-factor is directly related to the mean square isotropic displacement of the atom:

\(B = 8\pi^2 \langle u^2 \rangle\). (5)

A macromolecular crystal structure is typically represented by three coordinates (X,Y,Z), an occupancy parameter, and a "temperature factor" in the coordinate section of PDB files (Figure 1). These parameters are estimated during a computational process crystallographers refer to as *refinement*. The goal of refinement is to improve the agreement between the modeled structure and the reflections measured in the diffraction experiment. The (an)isotropic displacement parameters are variables during a typical refinement and may therefore also contain contributions of apparent displacements resulting from the use of an inadequate model or from overlooked errors in the X-ray data. The "temperature factor" field in the ATOM records of PDB files normally contains the isotropic B-factor ( Figure 1) or \(B_{eq}\), the value that represents isotropic displacement of atoms that were described by anisotropic ADPs during refinement.

```
ATOM 1 N THR A 1 17.047 14.099 3.625 1.00 13.79 N
ATOM 2 CA THR A 1 16.967 12.784 4.338 1.00 10.80 C
ATOM 3 C THR A 1 15.685 12.755 5.133 1.00 9.19 C
ATOM 4 O THR A 1 15.268 13.825 5.594 1.00 9.85 O
ATOM 5 CB THR A 1 18.170 12.703 5.337 1.00 13.02 C
ATOM 6 OG1 THR A 1 19.334 12.829 4.463 1.00 15.06 O
ATOM 7 CG2 THR A 1 18.150 11.546 6.304 1.00 14.23 C
```

**Figure 1** Thr 1 from 1crn. After the atom's identifiers the XYZ coordinates, occupancy, and B-factor are listed in these ATOM records.

The term "displacement parameter" is preferred over "temperature factor" because the atomic displacements are not only a consequence of thermal vibration but also of different atomic positions in different unit cells. The equivalent B-factor can be derived from the equivalent isotropic displacement parameter \(U_{eq}\):

\(B_{eq} = 8\pi^2 U_{eq} = \frac{8}{3} \pi^2 (U_{11} + U_{22} + U_{33})\). (6)

\(U_{eq}\) is equivalent to the sum of the eigenvalues of \(\textbf{U}\) and represents the mean-square displacement averaged over all directions. The ANISOU records of PDB entries normally contain the six independent elements of the symmetric tensor scaled by a factor 10^{4} (Figure 2).

```
ATOM 3293 N GLY B 635 15.522 -6.753 35.480 1.00 67.46 N
ANISOU 3293 N GLY B 635 7637 10155 7840 125 844 10 N
ATOM 3294 CA GLY B 635 16.002 -5.527 34.860 1.00 66.06 C
ANISOU 3294 CA GLY B 635 7519 9866 7715 35 723 -268 C
ATOM 3295 C GLY B 635 14.981 -4.446 34.608 1.00 69.48 C
ANISOU 3295 C GLY B 635 7997 10158 8244 -4 724 -496 C
ATOM 3296 O GLY B 635 15.107 -3.358 35.185 1.00 71.22 O
ANISOU 3296 O GLY B 635 8207 10502 8350 -17 684 -734 O
```

**Figure 2** Gly 635 from 3zzw. U_{11}, U_{12}, U_{13}, U_{12}, U_{13}, and U_{23} are stored in the ANISOU records after the identifier columns. The last numeric column of the ATOM records is the equivalent B-factor, B_{eq}, calculated from the six U_{ij} elements (eq. 3) in the ANISOU records (eq. 6).

### Occupancy

B-factors in PDB files commonly are seen as a measure of (local) mobility in the (macro)molecule. As mentioned above, this is only partly true.

When 3D structures are solved by crystallography, then, as the name suggests, crystals are needed. And in crystals by necessity crystal artefacts are observed. The three most important artefacts are 1) crystal packing artefacts at those locations where the (macro)molecules interact with other (macro)molecules in the crystal; 2) parts of the crystal have a different conformation of the molecule (static disorder); 3) alternate locations when parts of the macromolecule are happy in multiple conformations (dynamic disorder). Crystal packing artefacts are a whole story in themselves and are not of relevance for the BDB. You find information about these artefacts in other PDB-facilities. Alternate locations are most often seen for amino acid side chains when these can have multiple rotamers that are energetically similarly favourable. If this is observed in the crystal, then you will find multiple side chain atoms with alternate location indicators A, B, etc. The occupancies normally add up to unity (Figure 3). If no alternates have been observed, the occupancy simply equals 1.00 (e.g. Figure 1, Figure 2)

```
ATOM 435 N SER A 69 78.391 18.901 31.786 1.00 8.05 N
ATOM 436 CA ASER A 69 77.622 17.702 31.446 0.70 8.36 C
ATOM 437 CA BSER A 69 77.646 17.698 31.413 0.30 8.16 C
ATOM 438 C SER A 69 76.425 18.043 30.558 1.00 8.31 C
ATOM 439 O SER A 69 76.220 17.431 29.497 1.00 8.35 O
ATOM 440 CB ASER A 69 77.152 17.000 32.717 0.70 9.01 C
ATOM 441 CB BSER A 69 77.274 16.853 32.636 0.30 8.36 C
ATOM 442 OG ASER A 69 76.383 15.862 32.393 0.70 10.15 O
ATOM 443 OG BSER A 69 76.364 17.530 33.477 0.30 8.59 O
```

**Figure 3** Ser 69 from 4jf1. Two alternate locations have been observed for this side-chain (A and B) with occupancies 0.7 and 0.3.

It is often impossible to accurately model occupancy and ADPs separately because they are highly correlated. Only at high resolution it may be possible to resolve close alternate positions instead of modeling the atoms with large B-factors.