On Updating Torsion Angles of Molecular Conformations

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On Updating Torsion Angles of Molecular
Conformations

• Vicky Choi
• Department of Computer Science
• Virginia Tech
• (with Xiaoyan Yu, Wenjie Zheng)

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

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

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Torsion Angles
The dihedral angle between planes generated by
ABC BCD
C
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Different Conformations
Change torsion angles -gt new Cartesian
Coordinates of atoms?
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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.
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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

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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 (Rvv)
• rotation angle ? arcos((Tr(R)-1)/2)

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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)
• q0cos(?/2), (qx,qy,qz)sin(?/2) v
• Let b 2 R3 and b' be the image of b after
  rotation q.

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Unit Quaternion

• Hypercomplex q(q0,qx,qy,qz)
• q q0 i qx j qy k qz
• Multiplication rules i2j2k2-1
• ijk, ji-k, jkI, kj-i, kij, ik-j

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Rigid Motion

• Represented by a rotation followed by a
  translation
• Representations
• 4x4 Homogenous matrix
• Quaternion-vector form q,t

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Representation of Bond Rotation

• Rotate about the rotatable bond bi, rotate by ?i

Rotatable bond bi is not necessarily going
through the origin

1. Translation (by Qi such that Qi becomes origin)
2. Rotation (unit vector along bi, rotation
   angle?i)
3. Translation back

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Representation of Bond Rotation
b Ri(b-Qi) Qi Ri(b) Qi Ri(Qi)
In quaternion-vector form
In homogenous matrix form
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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.

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Rigid Fragmentation
A molecule can be represented as a tree with
rigid fragments as nodes and rotatable bonds as
edges.
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Bond Rotations
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Bond Rotations
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(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-1M1(x,y,z,1)T
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Time Complexity

• Ni Mi Mi-1 M 1, Miqi, Qi qiQiqi
• Ni1 Mi1Ni
• It takes constant time to compute Mi1, and
  constant time to compute Ni1 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)
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Our Improvement

• Simple Rotations
• where
• Improved Simple Rotations

multiplications 50nrb9na
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(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.

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Local Frames Relational Matrix

• To transform (xi,yi,zi) in Fi to (xi-1 yi-1 zi-1)
  in Fi-1

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

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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

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Global Frame (Simple Rotations) vs Local Frames

• Global Frame
• (x, y, z, 1)T MiMi-1M1(x, y, z, 1)T
• - Local Frames
• (x', y', z', 1)T M1M2 Mi (x, y, z, 1)T

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Comparison
nrb the number of rotatable bonds
Simple rotations implemented by Zheng Kavraki Local Frames by Zheng Kavraki Improved simple rotations in unit quaternion
multiplications (nrb) 75 48 50
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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

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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 (dont need to compute
  ancestor atoms, but need to compute a sequence of
  local frames relational matrices)
• Conformer generator