On Updating Torsion Angles of Molecular Conformations

1. On Updating Torsion Angles of Molecular Conformations Vicky Choi Department of Computer Science Virginia Tech (with Xiaoyan Yu, Wenjie Zheng)

2. Molecular Conformation Conformation: the relative positions of atoms in the 3D structure of a molecule. 2 different conformations of a molecule

3. Representations of Molecular Conformation • Cartesian Coordinates e.g. PDB, Mol2 • Distance Matrix • Internal Coordinates • Bond length, bond angle, torsion angle • E.g. Z-Matrix

4. Torsion Angles The dihedral angle between planes generated by ABC & BCD C

5. Different Conformations Change torsion angles -> new Cartesian Coordinates of atoms?

6. Rotatable bonds • single bond • acyclic (non-ring) bond • not connects to a terminal atom 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.

7. Rotation (Mathematical Definition) • Isometry: a transformation from R3 to R3 that preserves distances • Rotation: an orientation-preserving isometry with the ORIGIN fixed • A rotation in R3 can be expressed by an orthonormal matrix with determinant +1 – rotation matrix • Let b 2 R3 and b' be the image of b after rotation R • b’= Rb

8. Rotation • Geometrically, a rotation is performed by an angle  about a rotation axis ov through ORIGIN • R: rotation matrix • rotation axis ov : v is the eigenvector corresponding to the eigenvalue +1 (Rv=v) • rotation angle:  = arcos((Tr(R)-1)/2)

9. Unit Quaternion • q=(q0,qx,qy,qz) unit vector in R4 • rotation angle  • v=(vx,vy,xz) the unit vector along the rotation axis (through origin) • q0=cos(/2), (qx,qy,qz)=sin(/2) v • Let b 2 R3 and b' be the image of b after rotation q.

10. Unit Quaternion Hypercomplex q=(q0,qx,qy,qz) 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

11. Rigid Motion • Represented by a rotation followed by a translation • Representations: • 4x4 Homogenous matrix: • Quaternion-vector form : [q,t]

12. Representation of Bond Rotation • Rotate about the rotatable bond bi, rotate by i Rotatable bond bi is not necessarily going through the origin • Translation (by –Qi such that Qi becomes origin) • Rotation (unit vector along bi, rotation angle=i) • Translation back

13. The rigid motion: Rotation part Translation part Representation of Bond Rotation b’ = Ri(b-Qi) + Qi = Ri(b) + Qi – Ri(Qi) In quaternion-vector form: In homogenous matrix form:

14. Rigid Fragmentation • A molecule can be divided into a set of rigid fragments according to the rotatable bonds. • Rigid Fragments • Atoms in a RF are connected. • None of the bonds inside the RF is rotatable. • Bonds between two RFs are rotatable.

15. Rigid Fragmentation A molecule can be represented as a tree with rigid fragments as nodes and rotatable bonds as edges.

16. Bond Rotations

17. Bond Rotations

18. (1) Simple Rotations - Rotatable bonds: b1, b2, …, bi • Rotation angles: 1, 2, …, i • Atoms are updated by a series of rigid transformations (corresponding to rotations about rotatable bonds). • Let Mi be the ith rigid motion(rotate about bond bi by angle i): (x’,y’,z’,1)T = MiMi-1…M1(x,y,z,1)T

19. Time Complexity • Ni = Mi Mi-1 … M 1, Mi=[qi, Qi – qiQiqi] • Ni+1 = Mi+1Ni • It takes constant time to compute Mi+1, and constant time to compute Ni+1 from Ni • Let nrb be # of rotatable bonds; na be the # of atoms • Total time: O(nrb) (compute all the rigid motions) + O(na) (update positions of all atoms) 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)

20. Our Improvement • Simple Rotations where • Improved Simple Rotations # multiplications : 50nrb+9na

21. (2) Local Frames (Denavit-Hartenberg) Attach a local frame to each rotatable bond: • Fi = {Qi; ui, vi, wi}is attached to the rigid fragmentation gi. • wi is the unit vector along bond bi pointing to its parent RF gi-1 • ui are chosen arbitrary as long as it is perpendicular to wi. • vi is perpendicular to both wi and ui. • Qi is one end of the bond bi in RF gi.

22. Local Frames Relational Matrix To transform (xi,yi,zi) in Fi to (xi-1 yi-1 zi-1) in Fi-1:

23. Local Frames Relational Matrix Pi is rigid motion invariant and can be precomputed!

24. Local Frames Contd. • After D rotates around wi by i, it will move to the new position (xi’,yi’,zi’) in Fi, • We get the corresponding position of (xi’,yi’,zi’) in Fi-1

25. Local Frames 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

26. Global Frame (Simple Rotations) vs Local Frames • Global Frame: (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

27. Comparison nrb – the number of rotatable bonds

28. Example • 1aaq : 21 rotatable bonds • Average running time for 10,000 rounds of random rotations is 0.25ms for both local frames and improved simple rotations

29. Conclusions • Computational cost is almost the same but local frames require precomputations of a series of local frames relational matrices • Local Frames: Lazy look up (don’t need to compute ancestor atoms, but need to compute a sequence of local frames relational matrices) • Conformer generator