Oszillationen innerhalb von Konformation

1
The Effect of Fast Scales on Macroscopic Dynamics
of Biomolecular Systems
Christof Schütte Biocomputing Institute of
Mathematics II FU Berlin Pasadena, Nov. 2002
2
Alexander Fischer, Carsten Hartmann Burkhard
Schmidt, Illia Horenko, Christian Salzmann,
Elmar Diederichs, Sonja Waldhausen, Jessica
Walter, Regina Telgmann, Jens Antony Wilhelm
Huisinga
DFG priority program 1095 Multiscale problems
DFG Research Center Mathematics in key
technologies
F. Cordes, T. Galliat, D. Baum, AMIRA-Team _at_ ZIB
3
Introduction
4
Different Formulations of MD
5
Different Formulations of MD
6
Identification of metastable subsets
spectrum 1.00, 0.99, 0.75, 0.66, 0.63, , hence
two metastable subsets



-
idea exploit almost constant level structure for
identification
7
Identification of metastable subsets
three metastable subsets for moderate temperature
typical dynamics behavior
8
Identification of metastable subsets
three metastable subsets for moderate temperature
9
Mathematical justification
Assumption
reversible,
Theorem for arbitrary decomposition
upper bound
lower bound
with measures constantness of
w.r.t.
decomposition, and with
.
H. 01, H./Schmidt 02 (Davies 82, Singleton 84)
interpretation almost const. on
metastable subsets
10
Fast and Slow Scales
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Biomolecular Conformations as Metastable Sets
12
Multiple scale in MD
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Free energy landscape in biophysics
y
x
x1
x2
Conformational degree of freedom
14
Free energy landscape in biophysics
Fixman potential
or
conformational free energy landscape
15
Stochastic Effects of Fast DOFs
16
Diffusive process with fastslow variables
invariant measure for all e
inverse temperature
17
Averaging
m
y
fiber
x
invariant measure along each fiber for fixed x
m
x
18
Averaging and free energy landscape
Averaged force via averaged potential?
19
Illustrative Example
20
Fokker-Planck equation
motion of probability density in state space
21
Multiscale Asymptotics I
ansatz
conditioned expectation
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Eigenmodes of transfer operator
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Illustration
projected 2nd and 3rd eigenvector
full 2nd and 3rd eigenvector
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Multiscale Asymptotics II
25
Multiscale Asymptotics II
gt0 spectral gap of generator for
fixed x
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Multiscale Asymptotics II
27
Metastable fast variable I
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Metastable fast variable II
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Metastable fast variable III
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Stochastic Modelling of Fast
Metastable Scales
31
Extended averaging
idea averaging projection of dynamics to
subspace spanned by lowest eigenvalue
32
Results simplest case
m
y
fiber
x
m
m
1
2
x
x
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Conclusion

• averaging appropriate iff dominant
  eigenmodes are almost constant in direction
  of fast variables

• fast variables might be responsible for
  metastability

• extended averaging can appropriately model
  these effects

• algorithms from nonadiabatic quantum
  dynamics available

Sch., Walter, Hartmann, Huisinga (Nov. 2002)

On the effect of fast degrees of freedom on
macroscopic dynamical
behavior
34
Questions

• Applicability to realistic biomolecular
  systems ?

• Generalization to other dynamical processes ?

• Problems ?

35
Exit Rates and Eigenmodes
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Exit Time Approach
metastability of subset exit rate
(approximately decay rate of exit times
distribution) is close to zero
exit time
distribution of exit times
Proposition for V-uniformly ergodic Markov
processes
upper bound decays exponentially with rate .
problem dependence on initial state x versus set
oriented concept
37
Exit Rates
restriction to Markov processes with cont.
sampling paths, e.g.,
Smoluchowski system
definition of exit rates via
spectral radii of twisted processes
Theorem for with
define
open, connected subsets A,B, via zeros of h.
Then, under some technical conditions, we have
each subsets A,B, has exit rate
In this case, very close relation of to the
asymptotic decay rate of the distribution of exit
times.
H. /Meyn/Schütte 02
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Example three-well potential
Huisinga /Meyn/Schütte 02
Smoluchowski system
eigenvalue problem
potential function U
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Example three-well potential II
Smoluchowski system
potential function U
decay rate of exit time distribution
(numerically)
0.037(1)
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Example three-well potential III
Smoluchowski system
potential function U
decay rate of exit time distribution
(numerically)
41
Exit Rates
aim computable characterization of asymptotic
decay rate
restricted transfer operator on subset D
def. exit rate for a given subset D
def. an open, connected subset D is
metastable with exit rate ,if
for every open, connected proper subset
.
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Exit Rates and Asymptotic Decay
Consider propagator
of diffusion process
(Smoluchowski eq.)
Theorem under reasonable conditions, if
are metastable with exit rate
and for every
Huisinga /Meyn/Schütte 02
hence zeros of eigenfunctions define metastable
decomposition
43
Zwanzig-Mori Approach
44
The Zwanzig-Mori Approach I
Dynamical System with fast/slow DOFs
Only random initial conditions known (at least
for fast DOFs)
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The Zwanzig-Mori Approach II
projected equations of motion
projected RHS
noise-contribution of fast DOFs
memory-contribution of fast DOFs
46
Deterministic setting
47
Strong constraining potential
48
Homogenization result
49
Averaging Properties of Limit Dynamics
50
Full x-chain vs. Reduced q-chain I
Second eigenvectors of propagator
51
Full x-chain vs. Reduced q-chain II
Third eigenvectors of propagator
52
Ackno Further Information
Wilhelm Huisinga, Alexander Fischer, Carsten
Hartmann, Jessica Walter, Burkhard Schmidt,
Illia Horenko, Christian Salzmann, Jens
Antony, Sonja Waldhausen, Elmar Diederichs,