Towards effective dynamics in complex systems by Markov kernel approximation

Schütte, Christof; Jahnke, Tobias

ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 721-742.

Many complex systems occurring in various application share the property that the underlying Markov process remains in certain regions of the state space for long times, and that transitions between such metastable sets occur only rarely. Often the dynamics within each metastable set is of minor importance, but the transitions between these sets are crucial for the behavior and the understanding of the system. Since simulations of the original process are usually prohibitively expensive, the effective dynamics of the system, i.e. the switching between metastable sets, has to be approximated in a reliable way. This is usually done by computing the dominant eigenvectors and eigenvalues of the transfer operator associated to the Markov process. In many real applications, however, the matrix representing the spatially discretized transfer operator can be extremely large, such that approximating eigenvectors and eigenvalues is a computationally critical problem. In this article we present a novel method to determine the effective dynamics via the transfer operator without computing its dominant spectral elements. The main idea is that a time series of the process allows to approximate the sampling kernel of the process, which is an integral kernel closely related to the transition function of the transfer operator. Metastability is taken into account by representing the approximative sampling kernel by a linear combination of kernels each of which represents the process on one of the metastable sets. The effect of the approximation error on the dynamics of the system is discussed, and the potential of the new approach is illustrated by numerical examples.

[1] M. Belkin and P. Niyogi, Laplacian eigenmaps and spectral techniques for embedding and clustering, in Advances in Neural Information Processing Systems 14, T.G. Diettrich, S. Becker and Z. Ghahramani Eds., MIT Press (2002) 585-591.

[2] J. Bilmes, A Gentle Tutorial on the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models. ICSI-TR-97-021 (1997).

[3] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Rel. Fields 119 (2001) 99-161. | MR | Zbl

[4] J. Chodera, N. Singhal, V. Pande, K. Dill and W. Swope, Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. J. Comp. Chem. 126 (2007) 155101.

[5] E.B. Davies, Metastable states of symmetric Markov semigroups I. Proc. London Math. Soc. 45 (1982) 133-150. | MR | Zbl

[6] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (1999) 491-515. | MR | Zbl

[7] A.P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. B 39 (1977) 1-38. | MR | Zbl

[8] P. Deuflhard and M. Weber, Robust Perron cluster analysis in conformation dynamics. Lin. Alg. App. 398 (2005) 161-184. | MR | Zbl

[9] P. Deuflhard, W. Huisinga, A. Fischer and Ch. Schütte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl. 315 (2000) 39-59. | MR | Zbl

[10] R. Duda, P. Hart and D. Stork, Pattern Classification. Wiley (2001). | MR | Zbl

[11] L. Elsner and S. Friedland, Variation of the Discrete Eigenvalues of Normal Operators. P. Am. Math. Soc. 123 (1995) 2511-2517. | MR | Zbl

[12] S. Fischer, B. Windshügel, D. Horak, K.C. Holmes and J.C. Smith, Structural mechanism of the recovery stroke in the myosin molecular motor. Proc. Natl. Acad. Sci. USA 102 (2005) 6873-6878.

[13] A. Fischer, S. Waldhausen, I. Horenko, E. Meerbach and Ch. Schütte, Identification of biomolecular conformations from incomplete torsion angle observations by Hidden Markov Models. J. Comp. Chem. 28 (2007) 1384-1399.

[14] H. Frauenfelder, S.G. Sligar and P.G. Wolynes, The energy landscapes and motions of proteins. Science 254 (1991) 1598-1603.

[15] H.O. Hartley, Maximum likelihood estimation from incomplete data. Biometrics 14 (1958) 174-194. | Zbl

[16] I. Horenko and Ch. Schütte, Likelihood-based estimation of multidimensional Langevin models and its application to biomolecular dynamics. Multiscale Model. Simul. 7 (2008) 731-773. | MR

[17] I. Horenko, E. Dittmer, A. Fischer and Ch. Schütte, Automated model reduction for complex systems exhibiting metastability. Mult. Mod. Sim. 5 (2006) 802-827. | MR | Zbl

[18] I. Horenko, C. Hartmann, Ch. Schuette and F. Noé, Data-based parameter estimation of generalized multidimensional Langevin processes. Phys. Rev. E 76 (2007) 016706.

[19] W. Huisinga and B. Schmidt, Metastability and dominant eigenvalues of transfer operators, in Advances in Algorithms for Macromolecular Simulation, C. Chipot, R. Elber, A. Laaksonen, B. Leimkuhler, A. Mark, T. Schlick, C. Schütte and R. Skeel Eds., Lect. Notes Comput. Sci. Eng. 49, Springer (2005) 167-182.

[20] W. Huisinga, S. Meyn and Ch. Schütte, Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. 14 (2004) 419-458. | MR | Zbl

[21] M. Jäger, Y. Zhang, J. Bieschke, H. Nguyen, M. Dendle, M.E. Bowman, J. Noel, M. Gruebele and J. Kelly, Structure-function-folding relationship in a ww domain. Proc. Natl. Acad. Sci. USA 103 (2006) 10648-10653.

[22] S. Lafon and A.B. Lee, Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning and data set parameterization. IEEE Trans. Pattern Anal. Mach. Intell. 28 (2006) 1393-1403.

[23] B.B. Laird and B.J. Leimkuhler, Generalized dynamical thermostating technique. Phys. Rev. E 68 (2003) 016704.

[24] B. Nadler, S. Lafon, R.R. Coifman and I.G. Kevrekidis, Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmon. Anal. 21 (2006) 113-127. | MR | Zbl

[25] F. Noé, D. Krachtus, J.C. Smith and S. Fischer, Transition networks for the comprehensive characterization of complex conformational change in proteins. J. Chem. Theory Comput. 2 (2006) 840-857.

[26] A. Ostermann, R. Waschipky, F.G. Parak and G.U. Nienhaus, Ligand binding and conformational motions in myoglobin. Nature 404 (2000) 205-208.

[27] L.R. Rabiner, A tutorial on HMMs and selected applications in speech recognition. Proc. IEEE 77 (1989).

[28] Ch. Schütte and W. Huisinga, On conformational dynamics induced by Langevin processes, in EQUADIFF 99 - International Conference on Differential Equations 2, B. Fiedler, K. Gröger and J. Sprekels Eds., World Scientific (2000) 1247-1262. | MR | Zbl

[29] Ch. Schütte and W. Huisinga, Biomolecular conformations can be identified as metastable sets of molecular dynamics, in Handbook of Numerical Analysis X, P.G. Ciarlet and C. Le Bris Eds., Elsevier (2003) 699-744. | MR | Zbl

[30] Ch. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys., Special Issue on Computational Biophysics 151 (1999) 146-168. | MR | Zbl

[31] Ch. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fielder Ed., Springer (2001) 191-223. | MR | Zbl

[32] C. Schütte, F. Noe, E. Meerbach, P. Metzner and C. Hartmann, Conformations dynamics, in Proceedings of ICIAM 2007, Section on Public Talks (to appear).

[33] G. Singleton, Asymptotically exact estimates for metastable Markov semigroups. Quart. J. Math. Oxford 35 (1984) 321-329. | MR | Zbl

[34] D. Wales, Energy Landscapes. Cambridge University Press, Cambridge (2003).