VH/VL orientation and RMSD¶

antibody_utils.geometry compares antibody structures: it superposes them, measures their RMSD over chosen regions, and describes how an Fv’s VH and VL domains are oriented relative to each other.

The examples on this page use SAbDab’s IMGT-numbered file for the PDB entry 12E8, which holds two copies of the 2E8 Fab: chains H and L, and chains P and M.

>>> from antibody_utils.structure import Fv, read_structure
>>> structure = read_structure(
...     "pdb_000012e8_sabdab.cif.gz",
...     chains={"H": "H", "L": "L", "P": "H", "M": "L"},
... )
>>> first = Fv(structure["H"], structure["L"])
>>> second = Fv(structure["P"], structure["M"])

ABangle¶

ABangle describes the orientation of the VH and VL domains with five angles and a distance (Dunbar J, Fuchs A, Shi J and Deane CM, ABangle: characterising the VH–VL orientation in antibodies, Protein Eng Des Sel 26:611–620, 2013).

Each domain is given a coordinate frame by superposing a consensus domain on to the CA atoms of a core set of framework positions, which vary least in structure. That places, on each domain, a fixed point and two vectors in a plane fitted to the domain: H1 and H2 on VH, and L1 and L2 on VL. The vector C joins the two points. Then:

Measure	Meaning
HL	The torsion angle between H1 and L1, looking along C.
HC1, LC1	The angles between C and H1, and C and L1: how each domain bends towards or away from the other.
HC2, LC2	The angles between C and H2, and C and L2: how each domain twists.
dc	The length of C, in ångströms: how far apart the domains sit.

>>> from antibody_utils.geometry import abangle
>>> angles = abangle(first)
>>> round(angles.hl, 1), round(angles.dc, 1)
(-71.0, 16.0)
>>> {name: round(value, 1) for name, value in abangle(second).as_dict().items()}
{'HL': -62.4, 'HC1': 74.1, 'LC1': 118.8, 'HC2': 113.2, 'LC2': 81.6, 'dc': 15.9}

The two copies of the Fab in this crystal differ in HL by almost 9°.

ABangle’s core sets are defined in Chothia numbering, and have exact equivalents in IMGT numbering, so abangle needs an Fv numbered in either of those schemes.

RMSD¶

region_rmsd superposes two antibody chains or Fvs over chosen regions and atoms, and returns the RMSD in ångströms. Atoms are matched by chain type, numbered position and atom name, so both structures must be numbered in the same scheme, and only positions present in both are compared:

>>> from antibody_utils.geometry import region_rmsd
>>> round(region_rmsd(first, second, definition="imgt"), 2)
0.93
>>> round(region_rmsd(first.heavy, second.heavy, ["cdrh3"], definition="imgt"), 2)
0.28
>>> round(
...     region_rmsd(
...         first, second, ["cdrs"], definition="north", atoms=["N", "CA", "C", "O"]
...     ),
...     2,
... )
0.66

By default, CA atoms are compared. Pass superposed=False to compare the coordinates as they are, for structures that are already superposed.

orientation_rmsd measures how differently two Fvs place the VL domain relative to the VH domain. It superposes the VH frameworks, and separately the VL frameworks, and gives the RMSD between the two resulting placements of the first Fv’s VL framework. It isn’t quite symmetric, so consider averaging both directions:

>>> from antibody_utils.geometry import orientation_rmsd
>>> round(orientation_rmsd(first, second, definition="imgt"), 2)
3.14
>>> round(orientation_rmsd(second, first, definition="imgt"), 2)
3.13

Superposing coordinates¶

superpose and rmsd work on any arrays of equivalent points. superpose finds the rotation and translation that best fit one set of points on to another (the Kabsch algorithm), and never returns a reflection:

>>> import numpy as np
>>> from antibody_utils.geometry import superpose
>>> target = np.array([[0.0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]])
>>> mobile = target @ [[0, -1, 0], [1, 0, 0], [0, 0, 1]] + [5, 5, 5]
>>> superposition = superpose(mobile, target)
>>> np.allclose(superposition.apply(mobile), target)
True