# 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)

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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 -> new Cartesian Coordinates of atoms?

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Rotatable bonds

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

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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 (Rv=v)
-   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)
-   q0=cos(/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 = 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

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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)
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.

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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-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)
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Our Improvement
- Simple Rotations

where
- Improved Simple Rotations

# multiplications : 50nrb+9na
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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-1…M1(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       Local       Improved simple
implemented by Zheng   Frames by   rotations in unit
& Kavraki              Zheng &     quaternion
Kavraki
#                        75                48               50
multiplicatio
ns (nrb)

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

- Conformer generator

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