On Updating Torsion Angles of Molecular Conformations. Vicky Choi Department of Computer Science Virginia Tech (with Xiaoyan Yu, Wenjie Zheng). Molecular Conformation. Conformation : the relative positions of atoms in the 3D structure of a molecule.
Department of Computer Science
(with Xiaoyan Yu, Wenjie Zheng)
Conformation: the relative positions of atoms in the 3D
structure of a molecule.
2 different conformations of a molecule
e.g. PDB, Mol2
The dihedral angle between planes generated by ABC & BCD
Change torsion angles -> new Cartesian Coordinates of atoms?
A ligand bond is considered rotatable if it is single, acyclic and not to a terminal atom. This therefore includes, e.g., bonds to methyl groups but not to chloro substituents. It also includes bonds which, although single and acyclic, have highly restricted rotation, e.g. ester linkages. Finally, it incorrectly include bonds to linear groups, e.g. the bond between the methyl and cyanide carbons in CH3-CN.
v is the eigenvector
corresponding to the
eigenvalue +1 (Rv=v)
q = q0 + i¢ qx + j¢ qy + k ¢ qz
Multiplication rules: i2=j2=k2=-1
ij=k, ji=-k, jk=I, kj=-i, ki=j, ik=-j
Rotatable bond bi is not necessarily going through the origin
Translation partRepresentation of Bond Rotation
b’ = Ri(b-Qi) + Qi
= Ri(b) + Qi – Ri(Qi)
In quaternion-vector form:
In homogenous matrix form:
A molecule can be represented as a tree with rigid fragments as nodes and rotatable bonds as edges.
- Rotatable bonds: b1, b2, …, bi
(corresponding to rotations about rotatable bonds).
bond bi by angle i):
(x’,y’,z’,1)T = MiMi-1…M1(x,y,z,1)T
Zheng & Kavraki: A new method for fast and accurate
derivation of molecular conformations.
Journal of Chemical Information and Computer Sciences, 42, 2002.
# of multiplications: 75nrb + 9 na (using homogenous matrices)
# multiplications : 50nrb+9na
Attach a local frame to each rotatable bond:
To transform (xi,yi,zi) in Fi to (xi-1 yi-1 zi-1) in Fi-1:
Pi is rigid motion invariant and can be precomputed!
The coordinates of an atom in local frame Fi can be represented in global frame after a series of transformations:
(x', y', z', 1)T = M1M2… Mi (x, y, z, 1)T
(x’, y’, z’, 1)T = MiMi-1…M1(x, y, z, 1)T
- Local Frames:
(x', y', z', 1)T = M1M2 … Mi (x, y, z, 1)T
nrb – the number of rotatable bonds