Finite Markov Processes - Relation between Subsets and Invariant Spaces

Transcription

1 Finite Markov Processes - Relation between Subsets and Invariant Spaces M. Weber Zuse Institute Berlin (ZIB) Computational Molecular Design SON, Berlin, joint work with: K. Fackeldey, S. Röblitz, C. Schütte, L. Reinmiedl, RG A. Gleixner, N. Djurdjevac-Conrad

2 stochastic process -> finite Markov chain with transition matrix P

3 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

4 Durmaz, 216 optimizing dihydropterat-synthetase

5 dihydropterat-synthetase (catalytic reaction) DHPP paba

6 dihydropterat-synthetase (catalytic reaction) complex

7 dihydropterat-synthetase (catalytic reaction) complex

8 dihydropterat-synthetase (catalytic reaction) product (-> acid folique)

9 catalytic cycle

10 stationary distribution

11 NESS (non-equilibrium steady state)

12

13 detailed molecular description of the system (transitions P)

14 detailed molecular description of the system (transitions P) clustering

15 detailed molecular description of the system (transitions P) clustering/ projection efficiency of catalytic cylce

16 efficiency = non-reversibility of PC

17 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

18 P 1 1 = 1 1 Weber, Galliat, 22 Deuflhard, Weber, 25 Weber, 26 Röblitz, Weber, 213

19 P 1 1 = 1 1 Weber, Galliat, 22 Deuflhard, Weber, 25 Weber, 26 Röblitz, Weber, 213 Probability for a process starting here for ending up in that set in 1 step

20 P 1 1 = Röblitz, Weber, 213 Probability for a process starting here for ending up in that set in 1 step

21 P P P P P

22 P P P P P P c P c P c P c P c

23 membership vectors P P P P P P c P c P c P c P c linear coefficients of the membership vectors P c is a Galerkin discretization of P Membership vectors form an invariant subspace X of P starting point of P-chain inside the invariant subspace Weber, 211 Röblitz, Weber, 213 Fackeldey, Weber, 216

24 X orthogonal matrix with regard to the stationary distribution (separability = orthogonality of A)

25 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

26 X orthogonal matrix with regard to the stationary distribution

27 Reinmiedl, 216 Elowitz, Stanislas, Leibler, 2

28 X orthogonal matrix with regard to the stationary distribution diagonal matrix??

29 Schur decomposition

30 Schur decomposition

31 real Schur decomposition

32 real Schur decomposition

33 Schur = Eigen for reversible matrices Schur is well-conditioned Schur always exists Schur is not unique Schur values of P become Schur values of P C

34 P P C 1 1 1

35 Non-reversibility of P C is a consequence of the non-diagonality of the Schur matrix and the separability / orthogonality of the clusters. Djurdjevac-Conrad, Schütte, Weber,

36 Non-reversibility of P C is a consequence of the non-diagonality of the Schur matrix and the separability / orthogonality of the clusters. Djurdjevac-Conrad, Schütte, Weber,

37 SDE Hindemarsh-Rose (neuronal excitation) Reinmiedl, 216

38 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

39 Preparation of the invariant space: Pd=diag(sqrt(sd))*P*diag(1./sqrt(sd)); [Q, R]=schur(Pd); [Q, R]=SRSchur(Q, R); X=diag(1./sqrt(sd))*Q; X orthogonal with regard to the stationary distribution SRSchur sorts the eigenvalues function [val,pos] = select(r) %[val pos] = min(abs(1-r)); %Metastability [val, pos]=max(abs(r)); %Permutation Matrices %[val, pos]=max(abs(r-(real(r)<).*real(r))); % LOG %... Brandts, 21 Fackeldey, Weber, 216

40 Why taking the absolute value? sources/sinks = redundancy

41 Why taking the absolute value? sources/sinks = redundancy

42 momentum space low-friction Langevin dynamics Djurdjevac-Conrad, Schütte, Weber,

43 Non-reversibility of P C is a consequence of the non-diagonality of the Schur matrix and the separability / orthogonality of the clusters. Djurdjevac-Conrad, Schütte, Weber,

44 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

45 Find clustering ({,1}-entries of membership vectors) such that transitions are as non-reversible as possible between the clusters and coherent within the clusters. Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.)

46 Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.)

47 Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.)

48 SCIP is available in source-code ( and free for academic purposes to solve the MIP:

49 1) Cyclic behaviour in molecular processes 2) Robust Perron Cluster Analysis (PCCA+) 3) From Eigendecompositions to Schur Decompositions 4) Generalized PCCA for Dominant Cycles (function-based clustering) 5) Non-Dominant Cycles A MIP formulation (set-based clustering) 6) Comparative Example

50 A Synthetic Oscillatory Network of Transcriptional Regulators; Michael Elowitz and Stanislas Leibler; Nature. 2 Jan 2;43(6767):335-8.

51 A Synthetic Oscillatory Network of Transcriptional Regulators; Michael Elowitz and Stanislas Leibler; Nature. 2 Jan 2;43(6767):335-8.

52 Reinmiedl, 216

53 Generalized PCCA Lacl TetR cl Reinmiedl, 216 Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.)

54 Reinmiedl, 216 Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.) Generalized PCCA ( =.5)

55 Reinmiedl, 216 Beckenbach, Eifler, Fackeldey, Gleixner, Grever, Weber, Witzig, 216 (subm.) MIP formulation ( =.46)

56 Information to go membership vectors P P P P c P c P c efficiency = non-reversibility linear coefficients of the membership vectors