Meshfree Methods in Conformation Dynamics. Marcus Weber

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1 Meshfree Methods in Conformation Dynamics Marcus Weber Zuse Institute Berlin Free University Berlin Berlin Center for Genom-based Bioinformatics DFG Research Center Matheon

2 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 Schütte, Wilhelm Huisinga, Alexander Fischer, Illja Horenko, Carsten Hartmann, Phillip Metzner, Eike Meerbach Berlin Center for Genome-based Bioinformatics (BCBio) DFG Research Center Matheon 2 Conformation Dynamics

Scientific Visualization Hans-Christian Hege, Daniel Baum, Johannes Schmidt-Ehrenberg, Timm Baumeister, Maro Bader Cooperation: FU

3 Outline 1. Motivation and Aims of Conformation Dynamics 2. Conformations 3. Robust Perron Cluster Analysis 4. Sampling Scheme 5. Meshless Basis Functions 3 Conformation Dynamics

4 1. Motivation and Aims of Conformation Dynamics 4 Conformation Dynamics

5 Motivation: Example N-pentane ZIBgridfree, amiramol 5 Conformation Dynamics

6 Motivation conformation dynamics identification of molecular conformations inter conformational dynamics sampling according to Boltzmann distr. 6 Conformation Dynamics

7 2. Conformations 7 Conformation Dynamics

8 Decomposition of Botzmann Distribution Conformations = Decomposition of the Boltzmann distribution into overlapping densities V Ω ZIBmol, amiramol 8 Conformation Dynamics

9 Approximation of Conformations V Ω 9 Conformation Dynamics

10 Approximation of Conformations V Ω 10 Conformation Dynamics

11 Approximation of Conformations V transition regions multiple minima Ω 11 Conformation Dynamics

12 Approximation of Conformations χ 1 0 Ω 12 Conformation Dynamics

13 Conformations as membership functions Conformations are membership functions (fuzzy sets): χ i : Ω [0, 1], i = 1,..., n C, n C i=1 Total spatial Boltzmann distribution: χ i (q) = 1, q Ω. π : Ω RI +, π(q) exp( β V (q)) Decomposition of the Boltzmann distribution (partition function w i ): π i (q) = 1 Ω χ i(q) π(q) dq χ i(q) π(q), q Ω. 13 Conformation Dynamics

.., n C, n C i=1 Total spatial Boltzmann distribution: χ i (q) = 1, q Ω.

14 3. Robust Perron Cluster Analysis 14 Conformation Dynamics

15 Function approximation Approximation via s positive, partition-of-unity basis functions Φ j, j = 1,..., s: χ i = s j=1 C(j, i) Φ j, 0 C RI s nc, i C(j, i) = 1. Φ Φ Φ Φ C(j,i) χ χ 1 2 j i 15 Conformation Dynamics

16 Function approximation Approximation via s positive, partition-of-unity basis functions Φ j, j = 1,..., s: χ i = s j=1 C(j, i) Φ j, 0 C RI s nc, i C(j, i) = 1. Further condition: More precisely (Schütte, 2000): χ i (q(τ)) χ i (q(0)). P τ χ i χ i, P τ : L 1,2 (π) L 1,2 (π). 16 Conformation Dynamics

17 Robust Perron Cluster Analysis (PCCA+) Perturbation theory, optimization approach: Deuflhard, W., 2005 Main idea: if where P C S C C = XA, X RI s n C, A RI n C n C P X = SXΛ, Λ I s. Problem: Find optimal A, such that C is row-stochastic. 17 Conformation Dynamics

18 Robust Perron Cluster Analysis (PCCA+) P X = SXΛ, Λ I s 18 Conformation Dynamics

19 Robust Perron Cluster Analysis (PCCA+) P X SX I S 19 Conformation Dynamics

20 Robust Perron Cluster Analysis (PCCA+) P X SX 20 Conformation Dynamics

21 Robust Perron Cluster Analysis (PCCA+) P XA SXA 21 Conformation Dynamics

22 Robust Perron Cluster Analysis (PCCA+) P C SC 22 Conformation Dynamics

23 Robust Perron Cluster Analysis (PCCA+) PCCA+ can be used for general cluster problems, too. Condition: P, S have a hidden block-diagonal structure. W., Rungsarityotin, W., Schliep, W., Kube, Survival fit, 2 clusters Proportion survived p= Time 23 Conformation Dynamics

24 4. Sampling Scheme 24 Conformation Dynamics

25 Rescaling Trick S kj = P kj = Ω Ω Φ k (q) Φ j (q) π(q) dq (P τ Φ k (q)) Φ j (q) π(q) dq 25 Conformation Dynamics

26 Rescaling Trick P, S are numerically not available. Instead of solve P X = SXΛ D 1 P X = D 1 SXΛ, where D is an (s, s)-diagonal matrix with d ii = Ω Φ i(q) π(q) dq. P = D 1 P, S = D 1 S are computable. Left eigenvector for λ 1 = 1 of P and S consists of the diagonal elements of D computation of thermodynamic weights is possible. 26 Conformation Dynamics

27 Umbrella sampling We have to compute (A= Φ k or A= P τ Φ k ): Ω S kj, P k,j = A Φ j(q) π(q) dq Ω Φ j(q) π(q) dq = Ω A Φ j (q) π(q) Ω Φ j(q) π(q) dq dq I.e. compute observable A via Monte Carlo methods for a density, which is known up to a normalization constant Metropolis-Hastings -type algorithm. 27 Conformation Dynamics

28 Umbrella sampling Unnormalized density: ( Φ j (q) exp( β V (q)) = exp β [V (q) 1 ) β ln(φ j(q))]. We use HMC with modified potentials (Torrie, Valleau, 1977): V j = V 1 β ln(φ j) 28 Conformation Dynamics

29 5. Meshless Basis Functions 29 Conformation Dynamics

30 Example: Cyclohexane W., Meyer, Conformation Dynamics

31 Ω vs. 3D-representation 31 Conformation Dynamics

32 Start Discretization 32 Conformation Dynamics

33 Start Discretization 33 Conformation Dynamics

34 Start Discretization 34 Conformation Dynamics

35 Partition of Unity Liu, Shepard, Some properties: Φ i (q) = For α Voronoi tesselation. exp( α dist 2 (q i, q)) s j=1 exp( α dist2 (q j, q)) With special choice of dist, Φ i strictly quasi-concave ( window property ). Block-structure of P and S is not sensitive wrt. α. In other words: Eigenvector data X, which is used for clustering, is not sensitive wrt. α. 35 Conformation Dynamics

36 Summary Conformations: Overlapping decomposition of the Boltzmann distribution. Complexity reduction: Membership functions. Robust Perron Cluster Analysis: Solution of an eigenvalue problem for the Schütte operator + optimization. Sampling: Rescaling trick. Meshless basis functions: Neccessary condition in order to avoid curse of dimensionality. Thank you for your attention!!! 36 Conformation Dynamics

37 References P. Deuflhard and M. Weber. (2005). Robust Perron Cluster Analysis in Conformation Dynamics. Lin. Alg. App., Vol 398C, pp H. Meyer. (2005). Die Implementierung und Analyse von HuMFree einer gitterfreien Methode zur Konformationsanalyse von Wirkstoffmolekülen. 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 Analysis with Qualified Support of Transition States and Thermodynamic Weights. Submitted to: CompLife05 in Konstanz (25th -27th September). 37 Conformation Dynamics

(Big Data) Approaches to Designing Functional Molecules. Christof Schütte. Research Center Matheon Mathematics for key technologies

(Big Data) Approaches to Designing Functional Molecules Christof Schütte Research Center Matheon Mathematics for key technologies Different visual representations of a protein Christof Schütte, ZIB + FU