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On Updating Torsion Angles On Updating Torsion Angles of Molecular Conformationsof Molecular Conformations
Vicky ChoiDepartment 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 Coordinatese.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 fixedA 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 = q0 + i ¢ qx + j ¢ qy + k ¢ qz
Multiplication rules: i2=j2=k2=-1ij=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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In quaternion-vector form:
The rigid motion:
Rotation part
Translation part
b’ = Ri(b-Qi) + Qi
= Ri(b) + Qi – Ri(Qi)
In homogenous matrix form:
Representation of Bond Rotation
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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 – qiQ
iq
i]
- 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)
- 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.
Attach a local frame to each rotatable bond:
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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 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 (don’t need to compute ancestor atoms, but need to compute a sequence of local frames relational matrices)
- Conformer generator