Bioinformatics: Practical Application of Simulation and Data Mining Markov Modeling III
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Bioinformatics: Practical Application of Simulation and Data
Mining
Markov Modeling III
Prof. Corey O’HernDepartment of Mechanical Engineering & Materials
ScienceDepartment of Physics
Yale University
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“Using massively parallel simulation and Markovianmodels to study protein folding: Examining the dynamics
of the villin headpiece,” J. Chem. Phys. 124 (2006) 164902.
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Villin headpiece-HP-36
MLSDEDFKAVFGMTRSAFANLPLWKQQNLKKEKGLF: PDB 1 VII
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50,000 trajectories *10ns/trajectory = 500 s
•Gromacs with explicit solvent (5000 water molecules)and eight counterions; Amber + bond constraints; T=300K
Simulation Details
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50,000 trajectories*10 ns/trajectory*1 conformation/100 ps = 4,509,355 conformations
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I. Native State Ensemble
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II. Unfolded State Ensemble•10,000 trajectories equilibrated at T=1000K for 1 ns•Remove all structure•Random walk statistics
P R( )=4πR 2
23π R 2
( )3/2 exπ −
3R 2
2 R 2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
end-to-end distance
R2 ~N1/2
N 3/5
⎧⎨⎩
idealexcludedvolume
N= # of amino acids
chaincrossing
•Each trajectory quenched from 1000K to 300K; run for 25 ns 6
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Estimation of Folding Time: Including Unfolded Events
τ −1 =N f
Ntrajectoriest fi + tu
i
i∈U∑
i∈F∑⎡⎣⎢
⎤⎦⎥
−1
Initially unfolded
states
F: folded
U: unfolded
first passagetime: tf
tu
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determined bydRMSD
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Floppy Residues
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Maximum Likelihood Estimator (MLE)
τF 4.3-10 s from laser-jump and other experiments
τF 8 s from MLE
τF 24 s from MLE + correction of water diffusioncoefficient
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Sensitivity of MLE Results
“With these issues in mind, the calculated rate is wellwithin an order of magnitude of expeirmental measurements.”
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III. Transition State Ensemble: Effect of Perturbations
PX ,Y =N X( )
N X( )+ N Y( )
PX ,Y s( )=PX,Y s'( ) s’: perturbed state after 500 pss: unperturbed state
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N(X)= # of trajectories that meet condition X before Y
Water does notaffect dynamics
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Markov States
• 4,509,355 conformations 2454 Markov statesbased on clustering of C dRMSD
sf
•No dead ends
s1
s4
s3
s2
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C dRMSD
dRMSDij =
0 dRMSD12 dRMSD13 dRMSD14
dRMSD12 0 dRMSD 23 dRMSD 24
dRMSD13 dRMSD 23 0 dRMSD 34
dRMSD14 dRMSD 24 dRMSD 34 0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
dRMSDij =
1N
rrki −
rrkj( )
2
k∑⎡
⎣⎢⎤⎦⎥1/2
k=sum over amino acidsi,j=configurations
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Transition Probabilites and Mean First Passage Time
P sa , sb( )=T s,sb( )T s,si( )
i∑
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stableMFPT
MFPT=3 s
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MSM: single exponential
Comparison of Short and Long Times
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QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
First Passage Time in Random Processes
foldedunfolded
unfolded folded partially unfolded
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Dx
P(Dx)
Gaussian
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Survival Probability for Two Particles
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Protein Aggregation
“Molecular simulation of protein aggregation,”Biotechnology & Bioengineering 96 (2007) 1.
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