The Translation-Libration-Screw-rotation () 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 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 formulae from basic principles is also provided. This manuscript describes the principles of 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.

Early on, crystallography was a domain of mineralogy and mathematics and dealt mostly with symmetry properties and imaginary crystal lattices. This changed when Wilhelm Conrad Röntgen discovered X-rays in 1895, and in 1912 Max von Laue and his associates discovered X-ray irradiated salt crystals would produce diffraction patterns that could reveal the internal atomic periodicity of the crystals. In the same year the father-and-son team, Henry and Lawrence Bragg successfully solved the first crystal structure of sodium chloride and the era of modern crystallography began. Protein crystallography (PX) started some 20 years later with the pioneering work of British crystallographers. In the past 50-60 years, the achievements of modern crystallography and particularly those in protein crystallography have been due to breakthroughs in theoretical and technical advancements such as phasing and direct methods; to more powerful X-ray sources such as synchrotron radiation (SR); to more sensitive and efficient X-ray detectors; to ever faster computers and to improvements in software. The exponential development of protein crystallography has been accelerated by the invention and applications of recombinant DNA technology that can yield nearly any protein of interest in large amounts and with relative ease. Novel methods, informatics platforms, and technologies for automation and high-throughput have allowed the development of large-scale, high efficiency macromolecular crystallography efforts in the field of structural genomics (SG). Very recently, the X-ray free-electron laser (XFEL) sources and its applications in protein crystallography have shown great potential for revolutionizing the whole field again in the near future.

Refinement of macromolecular X-ray crystal structures involves using complex software with hundreds of different settings. The complexity of underlying concepts and the sheer amount sof instructions may make it difficult for less experienced crystallographers to achieve optimal results in their refinements. This tutorial review offers guidelines for choosing the best settings for the reciprocal-space refinement of macromolecular models and provides practical tips for manual model correction. To help aspiring crystallographers navigate the process, some of the most practically important concepts of protein structure refinement are described. Among the topics covered are the use and purpose of R-free, geometrical restraints, restraints on atomic displacement parameters (ADPs), refinement weights, various parametrizations of ADPs (full anisotropic refinement and TLS), and omit maps. We also give practical tips for manual model correction in Coot, modelling of side-chains with poor or missing density, and ligand identification, fitting, and refinement.

The procedure for phase extension (PX) involves gradually extending the initial phases from low resolution (e.g., ~8Å) to the high-resolution limit of a diffraction data set. Structural redundancy present in the viral capsids that display icosahedral symmetry results in a high degree of non-crystallographic symmetry (NCS), which in turn translates into higher phasing power and is critical for improving and extending phases to higher resolution. Greater completeness of the diffraction data and determination of a molecular replacement solution, which entails accurately identifying the virus particle orientation(s) and position(s), are important for the smooth progression of the PX procedure. In addition, proper definition of a molecular mask (envelope) around the NCS-asymmetric unit has been found to be important for the success of density modification procedures, such as density averaging and solvent flattening. Regardless of the degree of NCS, the PX method appears to work well in all space groups, provided an accurate molecular mask is used along with reasonable initial phases. However, in the cases with space group P1, in addition to requiring a molecular mask, starting the phase extension at a higher resolution (e.g., 6Å) overcame the previously reported problems due to Babinet phases and phase flipping errors.