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Towards effective dynamics in complex systemsby Markov kernel approximation

Published online by Cambridge University Press:  08 July 2009

Christof Schütte  and
Tobias Jahnke
Christof Schütte
Affiliation:
Institut für Mathematik II, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany. [email protected]
Tobias Jahnke
Affiliation:
Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe (TH), Kaiserstr. 93, 76133 Karlsruhe, Germany. [email protected]
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Abstract

Many complex systems occurring in various application share the property that theunderlying 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 transferoperator 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 eigenvaluesis 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.

Keywords

Effective dynamicscomplex systemsMarkov processmetastabilitytransfer operatorsmodel reductionmixture models.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 721 - 742
Copyright
© EDP Sciences, SMAI, 2009

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References

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.
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).
Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M., Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Rel. Fields 119 (2001) 99–161. CrossRef
Chodera, J., Singhal, N., Pande, V., Dill, K. and Swope, W., Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. J. Comp. Chem. 126 (2007) 155101.
Davies, E.B., Metastable states of symmetric Markov semigroups I. Proc. London Math. Soc. 45 (1982) 133–150. CrossRef
Dellnitz, M. and Junge, O., On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (1999) 491–515. CrossRef
Dempster, A.P., Laird, N.M. and Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. B 39 (1977) 1–38.
Deuflhard, P. and Weber, M., Robust Perron cluster analysis in conformation dynamics. Lin. Alg. App. 398 (2005) 161–184. CrossRef
Deuflhard, P., Huisinga, W., Fischer, A. and Schütte, Ch., Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl. 315 (2000) 39–59. CrossRef
R. Duda, P. Hart and D. Stork, Pattern Classification. Wiley (2001).
Elsner, L. and Friedland, S., Variation of the Discrete Eigenvalues of Normal Operators. P. Am. Math. Soc. 123 (1995) 2511–2517. CrossRef
Fischer, S., Windshügel, B., Horak, D., Holmes, K.C. and Smith, J.C., Structural mechanism of the recovery stroke in the myosin molecular motor. Proc. Natl. Acad. Sci. USA 102 (2005) 6873–6878. CrossRef
Fischer, A., Waldhausen, S., Horenko, I., Meerbach, E. and Schütte, Ch., Identification of biomolecular conformations from incomplete torsion angle observations by Hidden Markov Models. J. Comp. Chem. 28 (2007) 1384–1399. CrossRef
Frauenfelder, H., Sligar, S.G. and Wolynes, P.G., The energy landscapes and motions of proteins. Science 254 (1991) 1598–1603. CrossRef
Hartley, H.O., Maximum likelihood estimation from incomplete data. Biometrics 14 (1958) 174–194. CrossRef
Horenko, I. and Schütte, Ch., Likelihood-based estimation of multidimensional Langevin models and its application to biomolecular dynamics. Multiscale Model. Simul. 7 (2008) 731–773. CrossRef
Horenko, I., Dittmer, E., Fischer, A. and Schütte, Ch., Automated model reduction for complex systems exhibiting metastability. Mult. Mod. Sim. 5 (2006) 802–827. CrossRef
Horenko, I., Hartmann, C., Schuette, Ch. and Noé, F., Data-based parameter estimation of generalized multidimensional Langevin processes. Phys. Rev. E 76 (2007) 016706. CrossRef
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.
Huisinga, W., Meyn, S. and Schütte, Ch., Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. 14 (2004) 419–458.
Jäger, M., Zhang, Y., Bieschke, J., Nguyen, H., Dendle, M., Bowman, M.E., Noel, J., Gruebele, M. and Kelly, J., Structure-function-folding relationship in a ww domain. Proc. Natl. Acad. Sci. USA 103 (2006) 10648–10653. CrossRef
Lafon, S. and Lee, A.B., 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. CrossRef
Laird, B.B. and Leimkuhler, B.J., Generalized dynamical thermostating technique. Phys. Rev. E 68 (2003) 016704. CrossRef
Nadler, B., Lafon, S., Coifman, R.R. and Kevrekidis, I.G., Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmon. Anal. 21 (2006) 113–127. CrossRef
Noé, F., Krachtus, D., Smith, J.C. and Fischer, S., Transition networks for the comprehensive characterization of complex conformational change in proteins. J. Chem. Theory Comput. 2 (2006) 840–857. CrossRef
Ostermann, A., Waschipky, R., Parak, F.G. and Nienhaus, G.U., Ligand binding and conformational motions in myoglobin. Nature 404 (2000) 205–208. CrossRef
L.R. Rabiner, A tutorial on HMMs and selected applications in speech recognition. Proc. IEEE 77 (1989).
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.
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.
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.
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.
C. Schütte, F. Noe, E. Meerbach, P. Metzner and C. Hartmann, Conformations dynamics, in Proceedings of ICIAM 2007, Section on Public Talks (to appear).
Singleton, G., Asymptotically exact estimates for metastable Markov semigroups. Quart. J. Math. Oxford 35 (1984) 321–329. CrossRef
D. Wales, Energy Landscapes. Cambridge University Press, Cambridge (2003).