EMBIO Meeting Vienna, 2006
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Transcript of EMBIO Meeting Vienna, 2006
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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
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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
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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
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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
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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
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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
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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
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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
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Langevin Friction
in water > in vacuum 300K > 120 K
300K water300K vacuum 120K water120K vacuum
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Langevin Frequency
(anharmonicity) < 0 : low >high 300 K > 120 K
vacuum NMA
NMA
water vacuum
NMA
(solvation) > 0 : low >high 300 K = 120 K
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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
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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
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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,)
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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
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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
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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
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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
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300
1
/ 300nn
300
,1
/ 300v v nn
Temperature Dependence: Dynamical Transition
0.375v v
Vibrational FrictionVibrational Frequency Ratio of Vibration
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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
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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
~ ~
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S(q,w)
MDCACF modelVACF model
q = 2Å-1
300 K in water
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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,)
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Acknowledgement
Thanks for your attention!
Vandana Kurkal-Siebert
Fellowship by JSPS