Normal Mode Analysis Jun Wan and Willy Wriggers School of Health Information Sciences & Institute of Molecular Medicine University of Teas Houston
Contents Introduction --- protein dynamics (lowfrequency vibration and variables reduction) Normal mode analysis (NMA) Application and limitation Conclusion and future wor
Protein Dynamics is Hierarchical Vibration of bonds: 0-5 s Protein folding/unfolding Large-scale functional motions 0-6 s, 0-3 s, s and even longer
Collective Coordinates and Dimensionality Reduction 0 z= + y 8 6 y 4 0 0 4 6 8 0 Zhiyong Zhang
Find a Model To investigate low-frequency movement (vibration) To reduce the number of degrees of freedom
Normal Mode Analysis Theory of vibration Harmonic potential Close to the potential minimum Orthogonal normal modes Conformational fluctuation = a superposition of normal modes.
Harmonic Approimation E E = E 0 + i + E i! i i j + j 3! 3 E i j i j +... Approimation: Potential energy => harmonic E E 0 + i E i i + i, j E i j i j
m m t A dt d m dt d m / ) cos( 0 = + = = + = ω δ ω Harmonic Oscillator Newton / Hooe Atsushi Matsumoto
m m ) ( ) ( dt d m dt d m = + = 0 = + dt d m ) cos( ) cos( δ ω δ ω + = + = t A u t A u ω = / m ω = 3 / m = u u 0 3 0 0 = + u u u u dt d m 0 3 0 = + = + u dt u d m u dt u d m Matri Diagonalization Coupled oscillators Atsushi Matsumoto
0 = + dt d m = u u F U = = 0 0 0 0 U U FU U t t λ λ U is chosen so that it satisfies the following conditions. Eigenvalue and Eigenvector Problem Matri Diagonalization Atsushi Matsumoto
) cos( ) cos( δ ω δ ω + + + = t A t A Lower frequency mode Higher frequency mode is given by a superposition of normal modes: ω = / m ω = 3 / m Conformational Fluctuation Atsushi Matsumoto
Atsushi Matsumoto Two-Atomic Molecule
Atsushi Matsumoto Three-Atomic Molecule
Multi-Atom Molecule Low frequencies Large collective motions High frequencies Localized motions Florence Tama
Success Story
Success Story
Success Story One mode can represent 70-90% of functionally relevant motion. For many observed movements, the first normal modes contain the relevant degrees of freedom
NMA using Molecular Mechanics Full atomic representation and MM interactions require: energy minimization diagonalization of the nd derivative of the potential energy (3N 3N Hessian matri)
Computational Challenges NMA requires: Problems for large systems: minimization epensive, cumbersome (MM) diagonalization of the Hessian matri memory requirements
Memory-Efficient Diagonalization DIMB => Diagonalization in mied basis (Perahia & Mouawad, 995, J. Comp. Chem. 9, 4) Group theory => Use symmetrical properties of viruses (Rou & Karplus, 988, Biophys. J, 53, 97; Simonson & Perahia, 99, Biophys. J., 6, 40; van Vlijmen & Karplus, 00, J.Chem. Phys, 5, 69) RTB => Rotation Translation Blocs method gives approimate lowfrequency NM ( Tama et al. 000, Proteins: Struc. Funct. Genet., 4, ) bloc = or several residues rotation + translation of bloc => new basis epression of Hessian in this new basis diagonalization of a matri 6n B *6n B Florence Tama
Reducing the Number of Variables Cartesian coordinate space 3N-6 variables are necessary N : number of atoms Torsion angle space Bond angles and bond lengths are fied, and only torsion angles are allowed to vary. Number of variables: ~/0 Atsushi Matsumoto
E Elastic Networ Model Monique M Tirion (996) Phys Rev Lett. 77, 905-908 Simplified force-field: no MM, already minimized C ( ) ( ) 0 r r r r a, b a, b a, b E = E r, r ( ) = p a b a, b Possibility to reduce level of detail (up to point for 40 residue) Florence Tama
R d =3 Vector Quantization { } Encode data (in ) using a finite set (j=,,) of codeboo vectors. R 3 Delaunay triangulation divides into Voronoi polyhedra ( receptive fields ): w j w 4 w w w 3 Encoding Distortion Error: E = i (atoms, voels) v i w j( i) m i Linde, Buzo, & Gray (980): Gradient descent finds nearest local minimum of E. Martinetz & Schulten (993): Global search with topology-representing neural nets.
Choice of Cut-off Eample: Adenylate inase, 4 residues codeboo vector residue 0- Å cut-off OK Reducing number of codeboo vectors too sparse connectivity.4. 0.8 0.6 0.4 0. 0 4 0 5 0 5 0 5 30 Inspect the pair-distance distribution of codeboo vectors and increase cutoff beyond first pea. Florence Tama 0. 0.08 0.06 0.04 0.0 0 50 0 5 0 5 0 5 30
Level of Detail not Important X-ray structure Open-Close X-ray 4 codeboo vectors Projection onto atomic normal modes for the first few modes Low frequency NM are similar to atomic NM 50 codeboo vectors Models can reproduce functional rearrangements even at 30Å resolution Florence Tama
Application to EM Data RNA Polymerase, S. Darst et al. Deposition of Density Map
Application to EM Data RNA Polymerase, S. Darst et al. Deposition of Density Map Vector Quantization
Application to EM Data RNA Polymerase, S. Darst et al. Deposition of Density Map Vector Quantization Choice of Cut-off
Application to EM Data RNA Polymerase, S. Darst et al. Deposition of Density Map Vector Quantization Choice of Cut-off NMA
Application to EM Data RNA Polymerase, S. Darst et al. Deposition of Density Map Vector Quantization Choice of Cut-off NMA Apply Displacements
Eamples Ribosome RNA Polymerase
What are the Limitations of NMA (I)? We do not now a priori which is the relevant mode, but the first low-frequency modes are probable candidates. The amplitude of the motion is unnown. NMA requires additional standards for parameterization, i.e. a screening against complementary eperimental data to select the relevant modes and amplitude. Epert user input / evaluation required Not based on first principles of physics (lie MD).
Solution : Annotation of Modes (Essam Metwally: emotion.biomachina.org) Annotation/ Comments Database Query Results Mode View: 3D Movie (VRML) Mode View: D Movie & Comments
What are the Limitations of NMA (II)? Normal modes may brea the symmetry of structures due to forced orthogonalization: Global representation Local description of dynamic feature
Local Feature Analysis (Zhiyong Zhang) Global modes replaced by 3N LFA highly correlated output functions Sparsification: n seed atoms + their neighboring correlated regions T4 Lysozyme: n=4 Cα 09 Cα 5 Cα 6 C α
Conclusion Normal mode analysis is an alternative method to study dynamics of molecules. Normal mode analysis does not require trajectory, woring with single structure. Conformational fluctuation is given by a superposition of normal modes. We are using normal mode analysis to refine small-angle X-ray scattering profiles.
Acnowledgement Dr. Zhiyong Zhang Dr. Pablo Chacon Dr. Florence Tama Dr. Atsushi Matsumoto at Wright-Rieman Laboratory, Rutgers University