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On Updating Torsion Angles On Updating Torsion Angles of Molecular Conformationsof Molecular Conformations

Vicky ChoiDepartment of Computer Science

Virginia Tech

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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 AnglesThe 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 QuaternionHypercomplex 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 FragmentationA 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 – qi

Qiq

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(n

a) (update positions of all atoms)

Zhang & 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 MatrixTo 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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Comparisonnrb – the number of rotatable bonds

Simple rotations implemented by Zhang & Kavraki

Local Frames by Zhang & 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 evaluation (don’t need to compute ancestor atoms, but need to compute a sequence of local frames relational matrices)

- Conformer generator