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

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

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

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

Sensitivity of MLE Results

“With these issues in mind, the calculated rate is wellwithin an order of magnitude of expeirmental measurements.”

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

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

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

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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