TLS from fundamentals to practice.

Area of Science:

• Crystallography
• Structural Biology
• Biophysics

Background:

• The Translation-Libration-Screw-rotation (TLS) model, introduced in 1968, is a standard method in crystallography.
• It is widely used for analyzing rigid-body harmonic displacements in macromolecular structures.
• TLS is integrated into most modern crystallographic structure refinement software.

Purpose of the Study:

• To review the fundamental principles of the TLS model.
• To illustrate key features of TLS through simple examples.
• To provide simplified formulas for special cases and general derivations.

Main Methods:

• Review of existing literature and foundational principles of TLS.
• Illustrative examples demonstrating TLS model behavior.
• Derivation of general and simplified TLS formulas.

Main Results:

• Simplified formulas for specific scenarios in structure modeling and refinement.
• A clear derivation of general TLS formulas from basic principles.
• An extensive list of references showcasing TLS applications in macromolecular crystallography.

Conclusions:

• The TLS model is a crucial tool for understanding molecular motion in crystallography.
• This review provides practical insights and mathematical foundations for TLS application.
• The manuscript serves as a guide for utilizing TLS in macromolecular structure refinement.