TLS from fundamentals to practice

Abstract

The Translation-Libration-Screw-rotation (TLS) model of rigid-body harmonic displacements introduced in crystallography by Schomaker & Trueblood (1968) is now a routine tool in macromolecular studies and is a feature of most modern crystallographic structure refinement packages. In this review we consider a number of simple examples that illustrate important features of the TLS model. Based on these examples simplified formulae are given for several special cases that may occur in structure modeling and refinement. The derivation of general TLS formulae from basic principles is also provided. This manuscript describes the principles of TLS modeling, as well as some select algorithmic details for practical application. An extensive list of applications references as examples of TLS in macromolecular crystallography refinement is provided.

Keywords: ADP; TLS; atomic displacement parameter; rigid body motion; structure refinement; translation libration screw model.

Figure 1

Hierarchy of contributions to atomic displacement parameters and possible ways to model these contributions (bottom row).

Figure 2

Schematic illustration of the definition of a rotation axis l. (a) Axis parallel to k; its position is defined by 2 parameters, either by w_x and w_y, or by R and α. (b) Axis in an arbitrary orientation crossing the origin; its orientation is defined by 2 angles. (c) Axis in a general position; n is the vector parallel to l and crossing the origin; the plane normal to n and l is in grey. Similar to (b), two parameters are sufficient to define the orientation of n and l; to define the position of l two more parameters are required as in (a); they are coordinates of the intersection w of l with the grey plane. (d) Normalized Ow and l are the rotated i and k; p is the distance |Ow|.

Figure 3

(a) Schematic illustration of the instantaneous displacement v (red vectors) of a point r (blue vectors) for a libration around the axis k. Each displacement vector is in the plane normal to k and also normal to the corresponding vector r. (b) Schematic illustration of a point r_n librating around the axis k. The mean (apparent) position of the point (dashed arrow) is in the middle of the circular segment OAB and is closer to the origin than the point itself.

Figure 4

Three examples of possible combinations of a rotation (blue arrow) followed by a translation (red arrow) corresponding to the same transformation. The dashed lines show the vectors from the rotation axis, always perpendicular to the plane of the page, to the object. Left image: the axis crosses the origin. Middle image: the axis passes through the final position of the object; corresponding translation is smaller than in the left image. Right image: the same transformation performed as a pure rotation.

Figure 5

Libration around the axis k and a random translation parallel to the rotation axis. (a) For a random isotropic translational displacement, the atomic position is spherically distributed around the central position A; more frequent positions are shown as darker. (b) In the presence of an additional libration, the spherical electron clouds are distributed along the arc (red arrow) or, in a linear approximation, along a line in the rotation plane and normal to OA. (c,d) When the displacement u (due to translation) along k is correlated with the rotational displacement v, the line along which the clouds are distributed (blue arrow) is no longer in the rotation plane, and its slope depends on the relation between the rotation and translation shifts (parameter s_z in text). The total displacement becomes a screw rotation. See Section 5.1 for more detail.

Figure 6

Correlation of a libration around an axis k and a translation in the normal direction, i; the projection normal to k is shown. (a) For a pure libration around O, in a linear approximation all points (A,B,C,D in this case) move perpendicularly to the corresponding radius, shown as red arrows. The presence of a random isotropic translation displacement is illustrated by spherical clouds around each position. (b) For a pure translation in the direction i, displacements of all points are the same, black arrows. Overall clouds for A and C are the same as in (a). (c) An instantaneous displacement due to counter-clockwise libration (red arrow). Centers of electron clouds shift from their grey positions toward the blue ones. d) A translation component (black arrow) added to libration (c) and highly correlated with it, e.g. it is always equal to the displacement of the point C due to libration. The resulted displacements are shown by blue arrows. Point A is immobile. (e,f) are similar to (c,d) and show a clockwise instantaneous rotation around O and the corresponding translation. Movements in (d) and (f) are rotations around an apparent axis normal to the plane and crossing it in the point A. As for (a) and (b), the random isotropic translation is illustrated by a grey cloud for the initial position and by a blue cloud for a shifted position. See section 5.1 for more details.

Figure 7

Schematic illustration of a libration around a bond. a) The peptide group of the arginine is considered fixed. Angle δ is the parameter describing random oscillations around the bond C_αC_β. A shift of each atom is proportional to its distance to this rotation axis (conformations A, B, C). The bond C_αC_β is considered as a fixed rotation axis in Sections 7.1–7.2. If NεCς and C_αC_β are roughly parallel, the libration around C_αC_β translates the bond NεCς rather than changes its orientation (conformations A, B, C) (Sections 7.4–7.5). b) On the contrary, the rotation around C_βC_γ changes the orientation of NεCς (conformations A and D) (Section 7.6).