Abstract
We report on a novel approach to the automatic identification of metastablestates from long term simulation of complex molecular systems. The new approachis based on a hierarchical concept of metastability: metastable statesare understood as subsets of state or configuration space from which the dynamicsexits only very rarely; subsets with the smallest exit probabilities areof most interest, their further decomposition then may reveal subsets fromwhich exiting is less but comparably difficult for the system under investigation.The article gives a survey of the theoretical foundation of the approachand its algorithmic realization that generalizes the well-known concept of HiddenMarkov Models. The performance of the resulting algorithm is illustratedby an application to a 100 ns simulation of penta-alanine with explicit water.We demonstrate that the resulting metastable states allow to reveal theconformation dynamics of the molecule.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C. Schütte, A. Fischer, W. Huisinga, and P. Deuflhard (1999) A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. Special Issue on Computational Biophysics 151, pp. 146–168
C. Schütte and W. Huisinga (2003) Biomolecular conformations can be identified as metastable sets of molecular dynamics. In Handbook of Numerical Analysis (P. G. Ciaret and J.-L. Lions, eds.), Computational Chemistry, North-Holland
D. Chandler (1998) Finding transition pathways: Throwing ropes over rough montain passes, in the dark, in Classical and Quantum Dynamics in Condensed Phase Simulations (B. Berne, G. Ciccotti, and D. Coker, eds.), Singapure:World Scientific, pp. 51–66
W. E., W. Ren, and E. Vanden-Eijnden (2002) String method for the study of rare events. Phys. Rev. B 66, p. 052301
W. E., W. Ren, and E. Vanden-Eijnden (2005) Finite temperature string method for the study of rare events. J. Phys. Chem. B 109, pp. 6688–6693
A. Laio and M. Parrinello (2002) Escaping free-energy minima. Proceedings of the National Academy of the United States of America 99, pp. 12562–23566
R. Elber and M. Karplus (1987) Multiple conformational states of proteins: A molecular dynamics analysis of Myoglobin. Science 235, pp. 318–321
P. Deuflhard, W. Huisinga, A. Fischer, and C. Schütte (2000) Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl. 315, pp. 39–59
M. Dellnitz and O. Junge (1999) On the approximation of complicated dynamical behavior. SIAM J. Num. Anal. 36(2), pp. 491–515
P. Deuflhard and M. Weber (2005) Robust Perron cluster analysis in conformation dynamics. Lin. Alg. Appl. 398, pp. 161–184
I. Horenko, E. Dittmer, A. Fischer, and C. Schütte, Automated model reduction for complex systems exhibiting metastability. Submitted to Multiscale Modeling and Simulation
I. Horenko, E. Dittmer, and C. Schütte (2005) Reduced stochastic models for complex molecular systems. Computing and Visualization in Science 9, pp. 89–102
A. Fischer, S. Waldhausen, I. Horenko, E. Meerbach, and C. Schütte (2004) Identification of biomolecular conformations from incomplete torsion angle observations by hidden Markov models. Journal of computational Physics (submitted)
I. Horenko, E. Dittmer, F. Lankas, J. Maddocks, P. Metzner, and C. Schütte (2005) Macroscopic dynamics of complex metastable systems: Theory, algorithms, and application to b-dna. J. Appl. Dyn. Syst., submitted
S. D. Bond and B. B. L. Benedict J. Leimkuhler (1999) The Nosé–Poincaré method for constant temperature molecular dynamics. JCP 151(1), pp. 114– 134
A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein (2001) Metastability in stochastic dynamics of disordered mean–field models. Probab. Theor. Rel. Fields 119, pp. 99–161
E. B. Davies (1982) Metastable states of symmetric Markov semigroups I. Proc. London Math. Soc. 45(3), pp. 133–150
C. Schütte, W. Huisinga, and P. Deuflhard (2001) Transfer operator approach to conformational dynamics in biomolecular systems. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (B. Fiedler, ed.), Springer Berlin Heidelberg, pp. 191–223
G. Singleton (1984) Asymptotically exact estimates for metatstable Markov semigroups. Quart. J. Math. Oxford 35(2), pp. 321–329
W. E and E. Vanden-Eijnden (2005) Metastability, conformation dynamics, and transition pathways in complex systems preprint
W. Huisinga, S. Meyn, and C. Schütte (2004) Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. 14(1), pp. 419–458
C. Schütte and W. Huisinga (2000) On conformational dynamics induced by Langevin processes. In EQUADIFF 99 – International Conference on differential Equations (B. Fiedler, K. Gröger, and J. Sprekels, eds.), vol. 2, (Singapore), pp. 1247–1262, World Scientific
W. Huisinga and B. Schmidt (2005) Metastability and dominant eigenvalues of transfer operators. In New Algorithms for Macromolecular Simulation (C. Chipot, R. Elber, A. Laaksonen, B. Leimkuhler, A. Mark, T. Schlick, C. Schütte, and R. Skeel, eds.), vol. 49 of Lecture Notes in Computational Science and Engineering, Springer, to appear
M.Weber (2004) Improved Perron cluster analysis. ZIB-Report, (Zuse Institute, Berlin, pp. 03–04
P. Lezaud (2001) Chernoff and Berry–Esséen inequalities for Markov processes. ESIAM: P & S 5, pp. 183–201
B. J. Berne and J. E. Straub (1997) Novel methods of sampling phase space in the simulation of biological systems. Curr. Opinion in Struct. Biol. 7, pp. 181– 189
D. M. Ferguson, J. I. Siepmann, and D. G. Truhlar, eds. (1999) Monte Carlo Methods in Chemical Physics, vol. 105 of Advances in Chemical Physics. New York: Wiley
A. Fischer, C. Schütte, P. Deuflhard, and F. Cordes (2002) Hierarchical uncoupling-coupling of metastable conformations. In Computational Methods for Macromolecules: Challenges and Applications (T. Schlick and H. H. Gan, eds.), vol. 24 of Lecture Notes in Computational Science and Engineering, Springer Berlin Heidelberg, pp. 235–259
J. A. Bilmes (1998) A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and Hidden Markov Models. tech. rep., International Computer Science Institute, Berkeley
A. J. Viterbi (1967) Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Informat. Theory IT-13, pp. 260– 269
Y. Mu, P. H. Nguyen, and G. Stock (2004) Energy landscape of a small peptide revealed by dihedral angle principal component analysis. Proteins: Structure, Function, and Bioinformatics 58(1), pp. 45–52
G. N. Ramachandran and V. Sasiskharan (1968) Conformations of polypeptides and proteins. Advan. Prot. Chem. 23, pp. 283–427
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this chapter
Cite this chapter
Meerbach, E., Dittmer, E., Horenko, I., Schütte, C. (2006). Multiscale Modelling in Molecular Dynamics: Biomolecular Conformations as Metastable States. In: Ferrario, M., Ciccotti, G., Binder, K. (eds) Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 1. Lecture Notes in Physics, vol 703. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-35273-2_14
Download citation
DOI: https://doi.org/10.1007/3-540-35273-2_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35270-9
Online ISBN: 978-3-540-35273-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)