embio meeting vienna, 2006

22/5/2006 EMBIO Meeting 1

EMBIO Meeting Vienna, 2006

Heidelberg GroupIWR, 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

Langevin Model of Protein Dynamics

EMBIO Meeting

Vienna, May 22, 2006

IWR, University of HeidelbergKei 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


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

conformationaltransition


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


Dynamical Model

Langevin Equation2

2

)( )

(ij j ii

ji j

jiijF

d Vm q q

dtq R t

q

q

1

( ) 0

(0) ( ) 2 ( )

i

i j ij

R t

R R t t

Random forceFriction

PES roughness = Friction curvature = Frequency

,

1( ) ( )

2( ) i j

i jijV FV V q q q 0q

Harmonic Approximation of Potential Energy


Mode Analysis Simplifying Protein Dynamics

Normal Mode/Principal Component

Apply Dynamical Model for Each Mode

collective motion high frequency vibration

1 1 1( ; , )x f t 2 2 2( ; , )x f t 3 3 3( ; , )x f t 4 4 4( ; , )x f t


Calculations of Langevin ParametersMD Simulations Normal Mode Analysis

2

(0) ( )exp( / 2)(cos sin )

2n n

n nnn

nnnn

v v tt t t

v

2 2 / 4nn n n

120 K in vacuum

300 K in solution ( )

( )i

i

r t

r t

( )

( )

n in ii

n in ii

x t u r

v t u r

2FU U

Temperature dependence

Solvent effects

Velocity Autocorrelation Function (VACF) Model

n , nn

by each normal mode, n

Langevin Parameters


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

1, 2, ,( , , )Tn n n N nu u u u


Langevin Friction

in water > in vacuum 300K > 120 K

300K water300K vacuum 120K water120K vacuum


Langevin Frequency

(anharmonicity) < 0 : low >high 300 K > 120 K

vacuum NMA

NMA

water vacuum

NMA

(solvation) > 0 : low >high 300 K = 120 K


Potential Energy Surface via Langevin Model

NMAvacuumsolution

: roughness (anharmonicity) < 0

intra-protein interaction solvation: collisions with waters suppress protein vibrations

increase of : increased roughness (solvation) > 0, independent of T

Normal Mode Water MDVacuum MD


Dynamic Structure Factors1

( , ) ( , ) e2

i tS F t dt

q q

(0) ( )2,

1

( , ) i i

Ni i t

MD inc ii

F t b e e

q r q rq

MD Trajectory

Langevin Model3 6

2 ( ) 2, 2

1 1

( , ) exp | [1 ( )]N N

iBL inc i n n

i n i n

k TF t b t

m

q | q u

/ 2

2

(0) ( )( ) (cos sin )

2nnn n t nn

n n nnn

x x tt e t t

x

Langevin Model + Diffusion

(q, ) (q, ) (q, )corr L DF t F t F t2(q, ) exp( )DF t Dq t

120K water120K vacuum 300K water300K vacuum

q = 2Å-1


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

Svib(q,)


Modified Model for Diffusion

Extended Langevin model1) CACF model2) Add diffusional contribution

vibration

t

x(t)

v(t)

diffusionPCA mode 1 PCA mode 100PCA mode 1 PCA mode 100300K

water


Probabilistic Vibration/Diffusion Model

20

0

exp( / 2)(co(0) ( )

1s1 exp(

))

2(sin )v

v v vv

t tx x t t

txt

diffusion0

Langevin vibration

,v v

Coordinate Autocorrelation Function (CACF) Model

2 2 / 4v v v

PCA mode 1 PCA mode 100

MDmodel

MDmodel


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: ij i jC x xindependent atomic motion,

with square fluctuation, n

1, 2, ,( , , )Tn n n N nu u u u

MD trajectories (0) ( )n n MDx x t

least square fit to model functiont = 0 ~ 5, 10, 20 ps

diagonalization


Mean Square Fluctuations: Decomposition

85% 85%

85% 85%

2 2

1 1

2 2

1 1

(1 )

N N

n n nvib vibn n

N N

n n ndiff diffn n

x x

x x

n: eigenvalue of PCA: model parameter


300

1

/ 300nn

300

,1

/ 300v v nn

Temperature Dependence: Dynamical Transition

0.375v v

Vibrational FrictionVibrational Frequency Ratio of Vibration


Height of Vibrational Potential Wells via Model

230 K250 K280 K300 K

E

v

2

vibx 2 2

2

( ) / 2

/ 2

v v vib

v

E x

for < 1


Diffusion Constant via Model

E ,v v

k Kramers Rate Theory

2 2/ 4 / 2exp[ ]

2v v vk E

2

vibx

MDKramers theory

2 2

vibD ka k x : diffusion on 1D lattice

a a a

kkk

v

~ ~


S(q,w)

MDCACF modelVACF model

q = 2Å-1

300 K in water


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

2

vibx

2

diffx

S(q,)


Acknowledgement

Thanks for your attention!

Vandana Kurkal-Siebert

Fellowship by JSPS

embio meeting vienna, 2006

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