Title: Bioinformatics: Practical Application of Simulation and Data Mining Markov Modeling I
1Bioinformatics Practical Application of
Simulation and Data MiningMarkov Modeling I
- Prof. Corey OHern
- Department of Mechanical Engineering
- Department of Physics
- Yale University
2Protein folding kinetics
basin of attraction
native state
configuration space
initial state
transition
energy minimum
- eN local energy minima (configurations) connected
- via transitions
- Random walk on network from initial to native
state - States and transition probabilities obtained from
- simulations
3Markov Modeling of Proteins
Describing protein folding kinetics by Molecular
Dynamics Simulations. 1. Theory W. C. Swope, J.
W. Pitera, and F. Suits, J. Phys. Chem. B 108
(2004) 6571.
Describing protein folding kinetics by Molecular
Dynamics Simulations. 2. Example applications to
Alanine Dipeptide and a ?-hairpin peptide W. C.
Swope, J. W. Pitera, et al., J. Phys. Chem. B
108 (2004) 6582.
4Additional Reading
- Molecular simulation of ab Initio protein
folding for a Millisecond folder - NTL9(1-39), JACS 132 (2010) 1526.
- 2. Using massively parallel simulation and
Markovian models to study - protein folding Examining the dynamics of the
Villin headpiece, - J. Chem. Phys. 124 (2006) 164902.
- 3. Progress and challenges in the automated
construction of Markov - state models for full protein systems, J. Chem.
Phys. 131 (2009) 124101. - 4. Using generalized ensemble simulations and
Markov state models to - identify conformational states, Methods 49
(2009) 197. - 5. Stochastic dynamics of model proteins on a
directed graph, Phys. Rev. - Lett. 79 (2009) 061925.
5Markov Modeling
- Describes temporal evolution of state of the
system - No memory transition probabilities only depend
on - current state satisfied by MD trajectories
- Time domain (continuous or discrete) state space
- (continuous or discrete)
- Statistical description What is probability that
member - of the ensemble of systems will be in a given
state at time t? - How does one choose set of states for Markov
model - of protein dynamics---continuous degrees of
freedom yields - infinite number of states? Number of native
contactsbut not - specific enough
6Lumping of States From 11 to 3
C
A
native state
initial state
B
- Are transitions among aggregated states (A, B, C)
Markovian? - Yes, at sufficiently long time scales.
- How does one decide on lumping scheme?
7Mathematical Description
Tto, from
i,j1,Ns
- Elements non-negative
- Columns sum to 1
- Eigenvalues ?i 1 T ???
- ?(?1) gives steady-state probability
- distribrution
Detailed balance (no net flow)
- Eigenvectors form complete set
- Ns-1 eigenvalues determine relaxation
- rates
8Toy Model
9Transition Matrix
Tij
1
Lumped Transition Matrix
L
1
10State Probabilities
11Results from Toy Model
9 microstates
L(Tn)
Larger deviations practical
(L(T))n
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13Eigenvalue Spectra
?i
log ?i
Small deviations
log ?i
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