TLS from fundamentals to practice

The translation-libration-screw-rotation (TLS) model of rigid-body harmonic displacements introduced in crystallography by Schomaker and Trueblood [On the rigid-body motion of molecules in crystals. Acta Cryst. 1968;B24:63-76] 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 modelling and refinement. The derivation of general TLS formulae from basic principles is also provided. This manuscript describes the principles of TLS modelling as well as some selected algorithmic details for practical application. An extensive list of applications references as examples of TLS in macromolecular crystallography refinement is provided.