On Updating Torsion Angles of Molecular Conformations by f2gfdPJ4

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									On Updating Torsion Angles of
  Molecular Conformations


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

                                     1
Molecular Conformation

Conformation: the relative positions of atoms in the 3D
structure of a molecule.




2 different conformations of a molecule


                                                          2
Representations of Molecular Conformation
- Cartesian Coordinates
  e.g. PDB, Mol2

- Distance Matrix

- Internal Coordinates
   Bond length, bond angle, torsion angle
   E.g. Z-Matrix




                                             3
Torsion Angles
The dihedral angle between planes generated by ABC & BCD




                     C




                                                           4
Different Conformations




Change torsion angles -> new Cartesian Coordinates of atoms?

                                                           5
Rotatable bonds

- single bond
- acyclic (non-ring) bond
- not connects to a terminal atom




                                    6
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 (Rv=v)
-   rotation angle:  = arcos((Tr(R)-1)/2)

                                                       8
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.




                                                            9
Unit Quaternion
Hypercomplex q=(q0,qx,qy,qz)
 q = q 0 + i ¢ qx + j ¢ qy + k ¢ qz
Multiplication rules: i2=j2=k2=-1
ij=k, ji=-k, jk=I, kj=-i, ki=j, ik=-j




                                        10
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



                                                            12
Representation of Bond Rotation
       b’ = Ri(b-Qi) + Qi
          = Ri(b) + Qi – Ri(Qi)
The rigid motion:
                     Rotation     Translation
                       part          part

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.


                                     14
Rigid Fragmentation
A molecule can be represented as a tree with rigid
fragments as nodes and rotatable bonds as edges.




                                                     15
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-1…M1(x,y,z,1)T
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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)
                                                                   19
Our Improvement
   - Simple Rotations



   where
   - Improved Simple Rotations




# multiplications : 50nrb+9na
                                 20
(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:




                                                             22
Local Frames Relational Matrix




Pi is rigid motion invariant and can be precomputed!



                                                       23
 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




                                                               24
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

                                                      25
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




                                             26
Comparison
nrb – the number of rotatable bonds


                  Simple rotations       Local       Improved simple
                  implemented by Zheng   Frames by   rotations in unit
                  & Kavraki              Zheng &     quaternion
                                         Kavraki
  #                        75                48               50
  multiplicatio
  ns (nrb)




                                                                         27
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 (don’t need to
  compute ancestor atoms, but need to compute a
  sequence of local frames relational matrices)

- Conformer generator

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