Optimal control of molecular dynamics using Markov state models

Mathematical Programming

Volumn 134, Issue 1, 2012, Pages 259-282

  Schütte, Christof ^a   Winkelmann, Stefanie ^a   Hartmann, Carsten ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ALANINE DIPEPTIDES; COARSE-GRAINED; EIGENVALUE PROBLEM; HAMILTON JACOBI BELLMAN EQUATION; HIGH-DIMENSIONAL; INFINITE TIME HORIZON; LARGE DEVIATION PRINCIPLE; LOGARITHMIC TRANSFORMATIONS; MARKOV STATE MODELS; NONLINEAR EIGENVALUE PROBLEM; NUMERICAL EXAMPLE; OPTIMAL CONTROLS; PRINCIPAL EIGENVALUES; STOCHASTIC CONTROL;

AMINO ACIDS; MOLECULAR DYNAMICS; RANDOM PROCESSES;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84865692627     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-012-0547-6     Document Type: Conference Paper

References (47)

1. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)
2. Kushner, H.J., Dupuis, P.G.: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York (2001)
3. Barles, G., Souganidis, P.: Convergence of approximation schemes for fully nonlinear equations. Asymptot. Anal. 4, 271-283 (1991)
4. Barles, G., Jakobsen, E.: Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comput. 76, 1861-1893 (2007)
5. Carlini, E., Falcone, M., Ferretti, R.: An efficient algorithm for Hamilton Jacobi equations in high dimension. Comput. Vis. Sci. 7, 15-29 (2004)
6. Kappen, H.J.: Path integrals and symmetry breaking for optimal control theory. J. Stat. Mech. Theor. Exp. 2005(11), P11011 (2005)
7. Isralewitz, B., Gao, M., Schulten, K.: Steered molecular dynamics and mechanical functions of proteins. Curr. Opin. Struct. Biol. 11, 224-230 (2001)
8. Schlitter, J., Engels, M., Krüger, P., Jacoby, E., Wollmer, A.: Targeted molecular dynamics. Mol. Simul. 10, 291-309 (1993)
9. Rabitz, H., de Vivie-Riedle, R., Motzkus, M., Kompa, K.: Whither the future of controlling quantum phenomena? Science 288(5467), 824-828 (2000)
10. Kühn, O.,Wöste, L.: Analysis and control of ultrafast photoinduced reactions. In: Kühn, O.,Wöste, L. (eds.) Series in Chemical Physics, vol. 87. Springer, New York (2007)
11. Engel, A., Müller, D.J.: Observing single biomolecules at work with the atomic force microscope. Nat. Struct. Mol. Biol. 7, 715-718 (2000)
12. Fleming,W.H.,McEneaney,W.M.: Risk-sensitive control on an infinite time horizon. SIAM J. Control Opt. 33, 1881-1915 (1995)
13. Fleming, W.H., Sheu, S.J.: Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential. Ann. Probab. 25, 1953-1994 (1997)
14. Freidlin, M., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1998)
15. Jäger, M., Zhang, Y., Bieschke, J., Nguyen, H., Dendle, M., Bowman, M.E., Noel, J.P., Gruebele, M., Kelly, J.W.: Structure-function-folding relationship in a WW domain. Proc. Natl. Acad. Sci. USA. 103, 10648-10653 (2006)
16. Kobitski, A.Y., Nierth, A., Helm, M., Jäschke, A., Nienhaus, G.U.: Mg2+ dependent folding of a Diels-Alderase ribozyme probed by single-molecule FRET analysis. Nucleic Acids Res. 35, 2047-2059 (2007)
17. Fischer, S., Windshuegel, B., Horak, D., Holmes, K.C., Smith, J.C.: Structural mechanism of the recovery stroke in the myosin molecular motor. Proc. Natl. Acad. Sci. USA. 102, 6873-6878 (2005)
18. Noé, F., Krachtus, D., Smith, J.C., Fischer, S.: Transition networks for the comprehensive characterization of complex conformational change in proteins. J. Chem. Theor. Comp. 2, 840-857 (2006)
19. Ostermann, A., Waschipky, R., Parak, F.G., Nienhaus, U.G.: Ligand binding and conformational motions in myoglobin. Nature 404, 205-208 (2000)
20. Schaeffer, D.D., Fersht, A., Daggett, V.: Combining experiment and simulation in protein folding: Closing the gap for small model systems. Curr. Opin. Struct. Biol. 18(1), 4-9 (2008)
21. Noé, F., Schütte, C., Vanden-Eijnden, E., Reich, L., Weikl, T.R.: Constructing the full ensemble of folding pathways from short off-equilibrium simulations. Proc. Natl. Acad. Sci. USA 106, 19011-19016 (2009)
22. van Gunsteren, W., Dolenc, J., Mark, A.: Molecular simulation as an aid to experimentalists. Curr. Opin. Struct. Biol. 18(2), 149-153 (2008)
23. Schütte, C.: Conformational dynamics:modelling, theory, algorithm, and applications to biomolecules. Habilitation thesis, Fachbereich Mathematik und Informatik, FU Berlin (1999)
24. Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra Appl. 315, 39-59 (2000)
25. Schütte, C., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid MonteCarlo. J.Comp. Phys. (Special Issue onComput.Biophys.) 151, 146-168 (1999)
26. Schütte, C., Huisinga,W.: Biomolecular conformations can be identified as metastable sets of molecular dynamics. In: Le Bris, C. (ed.) Handbook of Numerical Analysis, pp. 699-744. Elsevier, New York (2003)
27. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible markov chains. Commun. Math. Phys. 228, 219-255 (2002)
28. Noé, F., Horenko, I., Schütte, C., Smith, J.: Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states. J. Chem. Phys. 126, 155102 (2007)
29. Chodera, J., Singhal, N., Pande, V.S., Dill, K., Swope, W.: Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. J. Chem. Phys. 126, 155101 (2007)
30. Buchete, N.V., Hummer, G.: Coarse master equations for peptide folding dynamics. J. Phys. Chem. B. 112, 6057-6069 (2008)
31. Pan, A.C., Roux, B.: Building Markov state models along pathways to determine free energies and rates of transitions. J. Chem. Phys. 129, 064107 (2008)
32. Sarich, M., Noé, F., Schütte, C.: On the approximation quality of Markov state models. Multiscale Model. Simul. 8(4), 1154-1177 (2010)
33. Voter, A.: Introduction to the kinetic Monte Carlo method. In: Sickafus, K., Kotomin, E., Uberuaga, B. (eds.) Radiation Effects in Solids. Springer, NATO Publishing Unit, Dordrecht (2005)
34. Huisinga,W.: Metastability of markovian systems a transfer operator based approach in application to molecular dynamics. Ph.D. thesis, Fachbereich Mathematik und Informatik, FU Berlin (2001)
35. Schütte, C., Noe, F., Lu, J., Sarich, M., Vanden-Eijnden, E.: Markov state models based on milestoning. J. Chem. Phys. 134, 204105-204120 (2011)
36. Borkar, V.S., Ghosh, M.K.: Ergodic control of multidimensional diffusions, II: Adaptive control. Appl. Math. Opt. 21, 191-220 (1990)
37. Bensoussan, A., Nagai, H.: An ergodic control problem arising from the principal eigenvalue of an elliptic operator. J. Math. Soc. Jpn. 43, 49-65 (1990)
38. Huisinga, W., Meyn, S.P., Schütte, C.: Phase transitions & metastability in Markovian and molecular systems. Ann. Appl. Probab. 14, 419-458 (2004)
39. James, M.: Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games. Math. Control Signals Syst. 5, 401-417 (1992)
40. Holland, C.J.: A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions. Commun. Pure Appl. Math. 31, 509-519 (1978)
41. Whittle, P.: Risk-sensitivity, large deviations and stochastic control. Eur. J. Oper. Res. 73, 295-303 (1994)
42. Donsker, M.D., Varadhan, S.S.: Asymptotic evaluation of certain Markov process expectations for large time, i. Commun. Pure Appl. Math. 28, 1-47 (1975)
43. Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. Springer, Berlin (1985)
44. Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A. Math.Gen. 31, 3719-3729 (1998)
45. Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333-365 (1999)
46. Pham, H.: Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009)
47. Robin, M.: Long-term average cost control problems for continuous timeMarkov processes: A survey. Acta Appl. Math. 1, 281-299 (1983)