From deep TLS validation to ensembles of atomic models built from elemental motions

Abstract

The translation-libration-screw model first introduced by Cruickshank, Schomaker and Trueblood describes the concerted motions of atomic groups. Using TLS models can improve the agreement between calculated and experimental diffraction data. Because the T, L and S matrices describe a combination of atomic vibrations and librations, TLS models can also potentially shed light on molecular mechanisms involving correlated motions. However, this use of TLS models in mechanistic studies is hampered by the difficulties in translating the results of refinement into molecular movement or a structural ensemble. To convert the matrices into a constituent molecular movement, the matrix elements must satisfy several conditions. Refining the T, L and S matrix elements as independent parameters without taking these conditions into account may result in matrices that do not represent concerted molecular movements. Here, a mathematical framework and the computational tools to analyze TLS matrices, resulting in either explicit decomposition into descriptions of the underlying motions or a report of broken conditions, are described. The description of valid underlying motions can then be output as a structural ensemble. All methods are implemented as part of the PHENIX project.

Keywords: TLS matrices; TLS model; correlated motion; diffuse scattering; ensemble of models; libration; model validation; molecular mobility; vibration.

Figure 1

General flowchart of the TLS decomposition into libration and vibration composite motions. Yellow ellipses indicate conditions to be verified. Green rectangles indicate the output parameters of the composite motions. The letters A–D indicate different steps of the procedure as described in the text.

Figure 2

The number of PDB entries (in thousands) as a function of various parameters. The blue histogram in (b), (c) and (d) is for the minimum eigenvalue and the red histogram is for the maximum eigenvalue. The leftmost and rightmost bins include all cases with values less than or greater than the limits given on the axis, respectively. The eigenvalues are given in rad² for L and in Ų for T. The total number of TLS groups is 203 261 for (a), (b) and (c) and about 70 000 for (d) when the matrix V could be calculated. (a) The number of TLS groups per entry; the largest is 283. (b) Distribution of eigenvalues of the matrix L; the minimum eigenvalue varies from −0.285 to 0.164 and the maximum eigenvalue varies from −0.001 to 0.409. (c) Distribution of eigenvalues of the matrix T; the minimum eigenvalue varies from −20.716 to 6.852 and the maximum eigenvalue varies from −1.551 to 28.676. (d) Distribution of eigenvalues of the matrix V (the S matrix optimized as described in the article); the minimum eigenvalue varies from −0.001 to 2.815 and the maximum eigenvalue varies from 0 to 5.950.

Figure 3

Examples of the vibration–libration ensembles. Red/salmon/magenta sticks indicate the principal vibration axes, with the origin in the centre of the group; blue/marine/black sticks are the libration axes. Yellow spheres in the 1dqv model show the reaction centres. (a) 1dqv model. (b) 1exr model; note pure vibrations for group 3 (the helix) and the absence of one of the libration axes for groups 1 and 2. (c) 4b3x model. Libration axes for the first group are not shown as they are too far from the molecule.

Figure 4

GpdQ TLS ensembles. The GpdQ TLS groups are projected onto the protein structure. The corresponding ensembles produced by phenix.tls_as_xyz are shown below. Each TLS PDB ensemble is shown as a single asymmetric unit outlined by the unit cell. An increase in overall motion is apparent going from left to right. The 20-member ensemble is shown for visual simplicity.

Figure 5

phenix.tls_as_xyz ensembles replicate TLS anisotropic motion. (a) GpdQ backbone with thermal ellipsoid representation of ‘entire molecule’ TLS anisotropic B factors. (b) phenix.tls_as_xyz ensemble backbones produced from ‘entire molecule’ TLS refinement. (c) Complete electron density predicted by ‘entire molecule’ TLS refinement. (d) Global correlation coefficient between experimental structure-factor amplitudes F _obs of the original GpdQ ‘entire motion’ refinement and phenix.tls_as_xyz ensembles of various sizes. Convergence values plateau at 0.935.