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Extracting Markov models of peptide conformational dynamics … · 2005. 11. 25. · Extracting...
Transcript of Extracting Markov models of peptide conformational dynamics … · 2005. 11. 25. · Extracting...
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Extracting Markov models of peptide
conformational dynamics from simulation data
Verena Schultheis, Thomas Hirschberger, and Paul Tavan
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p 2Extracting Markov models of peptide conformational dynamics from simulation data
Motivation
A complex example for
peptide conformational dynamics:
The cellular prion protein PrPC in solution
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p 3Extracting Markov models of peptide conformational dynamics from simulation data
Simulation system
• periodic orthorhombic dodecahedron(inner radius r = 52Å)
• PrPC (125-228)molecular mechanics (MM) force field: CHARMM22
M205R mutant obtained by remodeling the PrPC structure
• ~25800 H2O molecules(MM force field: TIP3P)
• ~150 Na and Cl ions(165 mM NaCl)
PrPC-structure: Zahn et al., PNAS 97, 145 (2000)
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p 4Extracting Markov models of peptide conformational dynamics from simulation data
MD methods
MD program EGO-MMII used for
• 10 ns simulations at T = 300 K, p = 1 atm
• computer time using six 1.6 GHz processors in parallel: ~ 20 weeks
• time step 1fs
Niedermeier, C, and P Tavan (1994). J. Chem. Phys. 101: 734-748.Niedermeier, C, and P Tavan (1996), Mol. Simul. 17: 57-66.Eichinger, M, H Grubmüller, H Heller, and P Tavan (1997). J. Comp. Chem. 18: 1729-1749.Mathias, G, B Egwolf, M Nonella, and P Tavan (2003). J. Chem. Phys. 118, 10847-10860.Mathias, G, and P Tavan (2004). J. Chem. Phys. 120, 4393-4403.
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p 5Extracting Markov models of peptide conformational dynamics from simulation data
Trajectories
wtPrP PrP-M205R
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p 6Extracting Markov models of peptide conformational dynamics from simulation data
General Problem
How can one gain insight into such processes
beyond showing movies?
?complex virtual reality simplified models
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p 7Extracting Markov models of peptide conformational dynamics from simulation data
A much more simple example
50 ns backbone dynamics of the
tripeptide Ac-Ser-Ser-Ser-NH2
fluctuating in water
is described by a time series
of six dihedral angles
64| 1,2, ,5 10 0,2tx t ps
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p 8Extracting Markov models of peptide conformational dynamics from simulation data
Trajectory
( , , , , , )1 1 2 2 3 3
xt t
t [ns]
1
1
2
2
3
3
0 50
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p 9Extracting Markov models of peptide conformational dynamics from simulation data
Look at 2
2
t [ns]500
2 randomly switches between two ranges and of values
3 angles 2³ = 8 combinations: r = , , ...,
into 8 disjoint partial volumes6
0, 2rV
discretization of the configuration space
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p 10Extracting Markov models of peptide conformational dynamics from simulation data
Conformations
The average peptide structures in the volumes Vr
1t r t r
trx V x V
x x
represent coarse-grained meta-stable states, the so-called
peptide conformations r = 1, ..., 8
the define a conformational dynamics
of transitions among the volumes Vr characterized by
the transfer operator'
'
'rr
transitions r r
all transitions f rT
rom
tx
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p 11Extracting Markov models of peptide conformational dynamics from simulation data
Markov processes
System switching randomly
between R states within
Transition probabilities Trr' depend only on present state r‘,
not on the past
A B
CTCA
Time discrete Markov process
of occupation probabilities pr(t) ( ) ( )' '' 1
Rp t pT tr rr r
r
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p 12Extracting Markov models of peptide conformational dynamics from simulation data
The question
The definition of conformations was natural and trivial for our tripeptide:
Suitable discretization from simple inspection of i (t)
Can a suitable discretization be determined
• automatically
• without inspection
• for arbitrary proteins sampled by MD?
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p 13Extracting Markov models of peptide conformational dynamics from simulation data
Naive construction of the transfer operator
Step 1: Grid partition
'
' '
transitions r r
rr all transitions f rT
rom
Step 2: Count transitions
between R grid cells
Problem: R ~ exp(D) "curse of dimensionality„
bad statistics for many Vr
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Alternative discretization
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p 15Extracting Markov models of peptide conformational dynamics from simulation data
Alternative discretization
Density of data points modeled as a mixture
Maximum likelihood principle parameters , | 1,...,r Rwr
Algorithms:
Kloppenburg & Tavan (1997). Phys. Rev. E 55, 2089-2092.
Albrecht et al. (2000), Neural Networks 13, 1075-1093.
( )p x
of univariate normal distributions
centered at points with identical widths and weights 1/R.
1
1
1( | , , , )ˆ | ,
R
r
R rp w w gR
x wx
rw
tx
| ,rxg w
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p 16Extracting Markov models of peptide conformational dynamics from simulation data
Alternative discretization
Bayesian association probabilities
1( | , )
ˆ ( | )ˆ ( | , , , )
1
g x wrRP r x
p x w wR
define fuzzy volumes Vr discretizing configuration space.
Advantages of this fuzzy partition:
number of Vr independent of D
load balance:
same statistics for all Vr
1
1ˆ: ( | , , , , )t Rr P r x w wR
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p 17Extracting Markov models of peptide conformational dynamics from simulation data
Alternative discretization
But otherwise it is still an abritrary discretization
How to construct a „natural“ discretization ?
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p 18Extracting Markov models of peptide conformational dynamics from simulation data
Alternative discretization
But otherwise it is still an abritrary discretization
How to construct a „natural“ discretization ?
like this:
Idea:
Construct „natural“ coarse-grained discretization by
• successive unification of originial fuzzy sets Vr
• guided by an analysis of the Markovian dynamics given by the Vr
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p 19Extracting Markov models of peptide conformational dynamics from simulation data
Transfer operator
Trajectory R-dim Markovian transfer matrix
1
'
( | ) ( ' | )
( ' | )
t t
rr
t
P r x P r xT
P r x
Choice of R dictated by statistics (R
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p 20Extracting Markov models of peptide conformational dynamics from simulation data
1-dim sample data
17 0 0
20 3 0%
0 3 2
80
80
0
0 0
8
80
7
0
1
exTdefine a 4-state Markov matrix
• associate a 1-dim normal distribution to each state
• generate a 1-dim trajectory xt , t = 1, 2, ...,in a two-stage stochastic process
17%3%
invariant density pinv(x) of the process estimated by R-bin histogram of {xt}or R-component Gaussian mixture
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p 21Extracting Markov models of peptide conformational dynamics from simulation data
Bayesian classification of the data
-1 0 1x
0
1
2
^ p
Model of pinv(x):
• coarse grained states are obvious
'nnTn'
n' nBayes transitions
all transitions from
Counting
24 0 0
28 3 0%
0 3 28
0 0 2
7
4
73
7
3
2
72
• and are obtained by Bayesian classification
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p 22Extracting Markov models of peptide conformational dynamics from simulation data
Bayesian classification of the data
but otherwise coarse graining is trivial in 1-dim
3 80 20
1
80
7 8
0 0
0
20 80
7
0%
0
3
0 0
1
exT
0 0
0%
0
0 0 24 7
3 73 2
72 24
8
28 73 3
2
and we get the typical Bayesian decision errorsfor overlapping classes(optimal result)
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p 23Extracting Markov models of peptide conformational dynamics from simulation data
Transfer operator
Original discretization R-dim Markovian transfer matrix
1
'
( | ) ( ' | )
( ' | )
t t
rr
t
P r x P r xT
P r xfor 1-dimexample
Alternative to Bayes:
Sequentially unify the fastest mixing Vr
until the four long-lived states remain!
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p 24Extracting Markov models of peptide conformational dynamics from simulation data
Unification algorithm
Join sequentially states i, j with fastest transitions i j
' | |' '
T P r x T P r xrr t r r t
Assume detailed balance:
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p 25Extracting Markov models of peptide conformational dynamics from simulation data
Unification algorithm
Join sequentially states i, j with fastest transitions i j
Assume detailed balance: ' '
| ' |
T Trr r r
P r x P r xt t
''
|
rrrr
t
TD
P r xLook at '
, 'max rrr r r
Dand choose i, j :
Unify partition functions | | , 1, , 1n
t t
r I
P n x P r x n R
and the transfer matrix , ' , 'n n r rT T
V. Schultheis, T. Hirschberger, H. Carstens, P. Tavan (2005) J. Chem. Theory Comput. 1, 515-526.
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p 26Extracting Markov models of peptide conformational dynamics from simulation data
Unification algorithm
Unification
T
level
D
eigenvalues r
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p 27Extracting Markov models of peptide conformational dynamics from simulation data
Unification algorithm
from smallest eigenvalue r
get fastest time scale at each level:min
1
1
4 and 2 state models are clearlydistingushed by jumps to slowertime scales of dynamics
„natural“ models
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p 28Extracting Markov models of peptide conformational dynamics from simulation data
Unification algorithm
3 71 32
2
69 26
6 68
0 0
0%
0
0
3 3
0
1 71TUnified transfer matrix at level = 4:
approximtely reconstructs optimal decision:3 73 28
28 73
0 0
0%
0
72 24
3
0 40 2 72
BayesT
'max ' |t t
nx n n P n xif
and the unified partition functions yield a classification of the data by
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p 29Extracting Markov models of peptide conformational dynamics from simulation data
Application to tripeptide trajectory
Discretize data by 25-component Gaussian mixture
Transfer operator:
1
'
ˆ ˆ( | ) ( ' | )
ˆ ( ' | )
t t
rr
t
P r x P r x
P rT
x
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Hierarchy of Markov models
R = 1
unification
R = 25
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Hierarchy of Markov models
R
lnR = 8
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8-state model
, 1, ...,8r
x rAssociated conformations
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Dimension reduction
R = 25 R = 8
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8-state model
t
1(t)
1(t)
2(t)
2(t)
3(t)
3(t)
classify to
8 conformational states
( )x t
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Results
Gibbs free energyMarkov model
lnn B nG k T P
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Summary
Tools for the analysis of simulation data:
• fuzzy partition Transfer operator at good statistics and moderate
dimension
• coarse graining hierarchy of Markov models
• most plausible model selected by certain observables
Simplified models
insights into the structures and conformational dynamics
of high-dimensional systems