T cell receptor structural repertoire database

TCR complex geometrical parameters (r, θ, and φ)

The binding mode of the T cell receptor (TCR) to peptide-major histocompatibility complex (pMHC) is described in a spherical coordinate system by three parameters (r, θ, and φ). The center of mass (CoM) of the α carbon atoms in the MHC peptide-binding groove (MHC CoM) is determined from residues 1-180 for class I proteins and from α chain residues 1-83 and β chain residues 1-93 for class II proteins. The origin of the coordinate system is assigned to the MHC CoM. A y axis is defined as a vector along the peptide using singular value decomposition of the α carbon atoms of the α helices delineating the MHC binding groove. The x-axis is defined normal to the y-axis, and the z-axis normal to the xy plane.


The magnitude of the vector connecting the CoM of the Vα/Vβ domain of the TCR and the MHC CoM is defined as r, and is given as:

theta (θ)

The θ angle is the angle between the x-axis and the projection of the TCR CoM – MHC CoM vector onto the xy plane and is given as:

phi (ϕ)

The ϕ angle is the angle between the z axis and TCR CoM – MHC CoM vector and is given as:

Calculate Geometrical parameters:


Source code and Pymol plugin

Singh, NK, Abualrous, ET, Ayres, CM, et al. Geometrical characterization of T cell receptor binding modes reveals class-specific binding to maximize access to antigen. Proteins. 2019; 1– 11.