1. 22/5/2006 EMBIO Meeting 1 EMBIO Meeting Vienna, 2006 Heidelberg Group IWR, Computational Molecular Biophysics, University of Heidelberg Kei Moritsugu MD simulation analysis of interprotein vibrations and boson peak Kinetic characterization of temperature-dependent protein internal motion by essential dynamics Langevin model of protein dynamics

2. Langevin Model of Protein Dynamics EMBIO Meeting Vienna, May 22, 2006 IWR, University of Heidelberg Kei Moritsugu and Jeremy C. Smith -Introduction Dynamical model for understanding protein dynamics Langevin equation -Direct application of Langevin dynamics: Velocity autocorrelation function model -Extension of the Langevin model: Coordinate autocorrelation function model

3. 22/5/2006 EMBIO Meeting 3 Physical interest:  multi-body (> ~1000 atoms)  inhomogeneous system Why Protein Dynamics? Anharmonic motion on rough potential energy surface Understand a “molecular machine” from physical point of view Biological/chemical interest:  expression and regulation of function  mediated by anharmonic protein dynamics conformational transition

4. 22/5/2006 EMBIO Meeting 4 Protein Dynamics: How to Analyze? Molecular Dynamics Simulation Neutron Scattering Experiment - low resolution - large, complex system with surrounding environments Dynamical Model Data Analysis Simplification - harmonic approximation - two-state jump model - Langevin model …. - atomic motions with fs-ns timescales - limited time <  s, system size < ~100 Å Settles et.al., Faraday Discussion 193, 269 (1996) Model Parameters Protein Dynamics

5. 22/5/2006 EMBIO Meeting 5 Dynamical Model Langevin Equation Random forceFriction PES roughness = Friction curvature = Frequency Harmonic Approximation of Potential Energy

6. 22/5/2006 EMBIO Meeting 6 Mode Analysis Simplifying Protein Dynamics Normal Mode/Principal Component Apply Dynamical Model for Each Mode collective motionhigh frequency vibration

7. 22/5/2006 EMBIO Meeting 7 Calculations of Langevin Parameters MD SimulationsNormal Mode Analysis 120 K in vacuum 300 K in solution Temperature dependence Solvent effects Velocity Autocorrelation Function (VACF) Model  n,  nn by each normal mode, n Langevin Parameters

8. 22/5/2006 EMBIO Meeting 8 Computations 1 Molecular Dynamics Simulations Normal Mode Analysis myoglobin (1A6G, 2512 atoms, 153 residues) equilibrium conditions at 120K and 300K 1-ns MD simulation with CHARMM vacuum: microcanonical MD solution: rectangular box with 3090 TIP3P waters, NPT, PME vacuum force field minimization of 1-ns average structure in vacuum calculate the Hessian matrix and its diagonalization independent atomic motion, with vibrational frequency,  n

9. 22/5/2006 EMBIO Meeting 9 Langevin Friction in water > in vacuum 300K > 120 K 300K water 300K vacuum 120K water 120K vacuum

10. 22/5/2006 EMBIO Meeting 10 Langevin Frequency  (anharmonicity) < 0 : low  >  high  300 K > 120 K  (solvation) > 0 : low  >  high  300 K = 120 K

11. 22/5/2006 EMBIO Meeting 11 Potential Energy Surface via Langevin Model NMA vacuum solution  : roughness  (anharmonicity) < 0 intra-protein interaction solvation: collisions with waters suppress protein vibrations increase of  : increased roughness  (solvation) > 0, independent of T Normal ModeWater MDVacuum MD

12. 22/5/2006 EMBIO Meeting 12 Dynamic Structure Factors MD Trajectory Langevin Model Langevin Model + Diffusion 120K water 120K vacuum 300K water 300K vacuum q = 2Å -1

13. 22/5/2006 EMBIO Meeting 13 Conclusion 1 Langevin model via VACF Protein vibrational dynamics Friction:  anharmonicity low  > high  high T > low T increase via solvation Frequency shift:  (anharmonicity) < 0  (solvation) > 0  S vib (q,  )

14. 22/5/2006 EMBIO Meeting 14 Modified Model for Diffusion Extended Langevin model 1) CACF model 2) Add diffusional contribution vibration t x(t)x(t) v(t)v(t) diffusion PCA mode 1PCA mode 100 PCA mode 1PCA mode 100 300K water

15. 22/5/2006 EMBIO Meeting 15 Probabilistic Vibration/Diffusion Model diffusion Langevin vibration   Coordinate Autocorrelation Function (CACF) Model PCA mode 1PCA mode 100 MD model MD model

16. 22/5/2006 EMBIO Meeting 16 Computations 2 Molecular Dynamics Simulations myoglobin (1A6G, 2512 atoms, 153 residues) in solution: rectangular box with 3090 TIP3P waters equilibrium conditions under NPT ensemble T = 120, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 280, 300 K 1-ns MD simulation with CHARMM PME Principal Component Analysis Fitting: Calculation of model parameters variance-covariance matrix: independent atomic motion, with square fluctuation, n MD trajectories least square fit to model function t = 0 ~ 5, 10, 20 ps diagonalization

17. 22/5/2006 EMBIO Meeting 17 Mean Square Fluctuations: Decomposition n : eigenvalue of PCA  : model parameter

18. 22/5/2006 EMBIO Meeting 18 Temperature Dependence: Dynamical Transition Vibrational FrictionVibrational FrequencyRatio of Vibration

19. 22/5/2006 EMBIO Meeting 19 Height of Vibrational Potential Wells via Model 230 K 250 K 280 K 300 K for  < 1

20. 22/5/2006 EMBIO Meeting 20 Diffusion Constant via Model k Kramers Rate Theory MD Kramers theory : diffusion on 1D lattice kkk ~~

21. 22/5/2006 EMBIO Meeting 21 S(q,w) MD CACF model VACF model q = 2Å -1 300 K in water

22. 22/5/2006 EMBIO Meeting 22 Conclusion 2 Langevin-vibration&diffusion model via CACF Protein dynamics Simulation-based probabilistic description Vibration: linear scheme with T  v  Diffusion: nonlinear scheme with T ,  v,  Diffusion constant via the present model using Kramers theory S(q,  )

23. 22/5/2006 EMBIO Meeting 23 Acknowledgement Thanks for your attention! Vandana Kurkal-Siebert Fellowship by JSPS