Meshfree Methods in Conformation Dynamics. Marcus Weber

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Meshfree Methods in

Conformation Dynamics

Marcus Weber

Zuse Institute Berlin Free University Berlin

Berlin Center for Genom-based Bioinformatics

DFG Research Center “Matheon”

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ZIB Scientific Computing:

Dept. Computational Drug Design

Peter Deuflhard, Frank Cordes, Marcus Weber, Susanna Kube, Holger Meyer, Ulrich Nowak, Alexander Riemer

Dept. Scientific Visualization

Hans-Christian Hege, Daniel Baum, Johannes Schmidt-Ehrenberg, Timm Baumeister, Maro Bader

Cooperation:

FU Biocomputing:

Christof Sch¨utte, Wilhelm Huisinga, Alexander Fischer,

Illja Horenko, Carsten Hartmann, Phillip Metzner, Eike Meerbach Berlin Center for Genome-based Bioinformatics (BCBio)

DFG Research Center “Matheon”

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Outline

1. Motivation and Aims of Conformation Dynamics 2. Conformations

3. Robust Perron Cluster Analysis 4. Sampling Scheme

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1. Motivation and Aims of

Conformation Dynamics

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Motivation: Example N-pentane

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Motivation

conformation dynamics

identification of molecular conformations inter−conformational dynamics

sampling according to Boltzmann distr.

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Decomposition of Botzmann Distribution

Conformations =

Decomposition of the Boltzmann distribution into overlapping densities

Ω

V

ZIBmol, amiraMol

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Approximation of Conformations

Ω

V

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Approximation of Conformations

Ω

V

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Approximation of Conformations

Ω multiple minima

transition regions V

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Approximation of Conformations

0

Ω

χ 1 12 Conformation Dynamics

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Conformations as membership functions

Conformations are membership functions (fuzzy sets):

χi : Ω→[0,1], i= 1, . . . , nC, nC

X

i=1

χi(q) = 1, ∀q ∈Ω.

Total spatial Boltzmann distribution:

π: Ω→IR+, π(q)∝exp(−β V(q))

Decomposition of the Boltzmann distribution (partition function wi):

πi(q)=

1

R

Ωχi(q)π(q)dq

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3. Robust Perron Cluster Analysis

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Function approximation

Approximation viaspositive, partition-of-unity basis functions Φj, j = 1, . . . , s:

χi= s X j=1 C(j, i) Φj, 0≤C∈IRs×nC, X i C(j, i) = 1. i χ χ Φ Φ Φ Φ 1 1 2 2 3 4 C(j,i) j

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Function approximation

Approximation viaspositive, partition-of-unity basis functions Φj, j = 1, . . . , s:

χi= s X j=1 C(j, i) Φj, 0≤C∈IRs×nC, X i C(j, i) = 1. Further condition: χi(q(τ))≈χi(q(0)).

More precisely (Sch¨utte, 2000):

Pτχi≈χi, Pτ :L1,2(π)→L1,2(π).

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Robust Perron Cluster Analysis (PCCA+)

Perturbation theory, optimization approach:Deuflhard, W., 2005

Main idea: P C≈S C if C=XA, X∈IRs×nC_,_A_∈_I_RnC×nC where P X =SXΛ, Λ≈Is.

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Robust Perron Cluster Analysis (PCCA+)

P X

=

SX

Λ

,

Λ

≈

I

s

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Robust Perron Cluster Analysis (PCCA+)

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Robust Perron Cluster Analysis (PCCA+)

P X

≈

SX

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Robust Perron Cluster Analysis (PCCA+)

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Robust Perron Cluster Analysis (PCCA+)

P C

≈

SC

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Robust Perron Cluster Analysis (PCCA+)

PCCA+ can be used for general cluster problems, too. Condition: P , S have a hidden block-diagonal structure.

W., 2004. Rungsarityotin, W., Schliep, 2004. W., Kube, 2005.

0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0

Survival fit, 2 clusters

Time

Proportion survived

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4. Sampling Scheme

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Rescaling Trick

Skj = Z Ω Φk(q) Φj(q)π(q)dq Pkj = Z Ω (PτΦk(q)) Φj(q)π(q)dq

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Rescaling Trick

P , S are numerically not available. Instead of

P X =SXΛ

solve

D−1P X =D−1SXΛ,

whereDis an (s, s)-diagonal matrix with dii=

R

ΩΦi(q)π(q)dq.

• P =D−1P , S =D−1S are computable.

• Left eigenvector forλ1= 1 ofP andS consists of the diagonal elements of

D ⇒ computation of thermodynamic weights is possible.

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Umbrella sampling

We have to compute (A= Φk or A=PτΦk): Skj, Pk,j = R ΩAΦj(q)π(q)dq R ΩΦj(q)π(q)dq = Z Ω A R Φj(q)π(q) ΩΦj(q)π(q)dq dq

I.e. compute observableAvia Monte Carlo methods for adensity, which is known up to a normalization constant⇒ “Metropolis-Hastings”-type algorithm.

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Umbrella sampling

Unnormalized density: Φj(q) exp(−β V(q)) = exp −β[V(q)− 1 βln(Φj(q))] .

We use HMC with modified potentials (Torrie, Valleau, 1977):

Vj =V −

1 βln(Φj)

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Example: Cyclohexane

W., Meyer, 2005

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Start Discretization

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Start Discretization

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Partition of Unity

Liu, 2002. Shepard, 1968. Φi(q) = exp(−αdist2(qi, q)) Ps j=1exp(−αdist 2₍_q j, q)) Some properties:

• For α→ ∞ Voronoi tesselation.

• With special choice of “dist”,Φistrictly quasi-concave (“window property”).

• Block-structure of P and S is not sensitive wrt. α. In other words: Eigen-vector data X, which is used for clustering, is not sensitive wrt.α.

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Summary

• Conformations: Overlapping decomposition of the Boltzmann distribution. Complexity reduction: Membership functions.

• Robust Perron Cluster Analysis: Solution of an eigenvalue problem for the Sch¨utte operator + optimization.

• Sampling: Rescaling trick.

• Meshless basis functions: Neccessary condition in order to avoid “curse of dimensionality”.

Thank you for your attention!!!

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References

P. Deuflhard and M. Weber.(2005).Robust Perron Cluster Analysis in Confor-mation Dynamics.Lin. Alg. App., Vol 398C, pp. 161-184.

H. Meyer.(2005).Die Implementierung und Analyse von HuMFree – einer gitter-freien Methode zur Konformationsanalyse von Wirkstoffmolek¨ulen.Master Thesis, Free University Berlin.

M. Weber. (2005). Meshless Methods in Conformation Dynamics. PhD Thesis, Free University Berlin, in preparation.

M. Weber and S. Kube. (2005). Robust Perron Cluster Analysis for Various Applications in Computational Life Science.Submitted to: CompLife05 in Konstanz.

M. Weber and H. Meyer. (2005). ZIBgridfree – Adaptive Conformation Ana-lysis with Qualified Support of Transition States and Thermodynamic Weights. Submitted to: CompLife05 in Konstanz (25th -27th September).