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Bioinformatics: Practical Application of Simulation and Data Mining Markov Modeling I

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Title: Bioinformatics: Practical Application of Simulation and Data Mining Markov Modeling I


1
Bioinformatics Practical Application of
Simulation and Data MiningMarkov Modeling I
  • Prof. Corey OHern
  • Department of Mechanical Engineering
  • Department of Physics
  • Yale University

2
Protein 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

3
Markov 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.
4
Additional 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.

5
Markov 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

6
Lumping 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?

7
Mathematical 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

8
Toy Model
9
Transition Matrix
Tij
1
Lumped Transition Matrix
L
1
10
State Probabilities
11
Results from Toy Model
9 microstates
L(Tn)
Larger deviations practical
(L(T))n
12
(No Transcript)
13
Eigenvalue Spectra
?i
log ?i
Small deviations
log ?i
14
(No Transcript)
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