Keywords:-
Keywords:
Markovv-states model; two-states systems; calculation of eigenvalues; molecular dynamics; protein folding.
Article Content:-
Abstract
The two-states Markovv-states-model of molecular dynamics is newly analytically studied. The total reward of the path integral of the reaction within a
crisp Markovv landscape is proven to be expressed as a Laplace integral (kernel) after the opportune Radon measure. The evolution of the eigenvalues is newly
exactly analytical calculated; the corresponding relative error is newly analytically calculated. The problem of an m-states model is established within this
framework.
References:-
References
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11. P. K. Pollett, V. T. Stefanov, Path Integrals for Continuous-Time Markov Chains, Journal of Applied Probability 39, 901 (2002).
12. J. H. Prinz et al., Markov models of molecular kinetics: Generation and validation, The Journal of Chemical Physics 134, 174105 (2011).
13. N. Djurdjevac, M. Sarich, C. Schuette, Estimating the Eigenvalue Error of Markov State Models, Multiscale Modeling & Simulation 10, 48 (2012).
14. Constructing the equilibrium ensemble of folding pathways from short offequilibrium simulations, PNAS 106, 19011 (2009).
15. M. Sarich, C. Schuette, Approximating selected non-dominant timescales by Markov state models, Communications in Mathematical Sciences 10 , 1001 (2012).
16. J. D. Chodera, F. Noe’, Probability distributions of molecular observables computed from Markov models. II. Uncertainties in observables and their time-evolution, J. Chem. Phys. 133, 105102 (2010).
17. J. D. Chodera, W. C. Swope, J. W. Pitera, K. A. Dill, Long-Time Protein Folding Dynamics from Short-Time Molecular Dynamics Simulations, Multiscale Model. Simul. 5, 1214 (2006).
18. P. Metzner, F. Noe’, C. Schuette, Estimating the sampling error: Distribution of transition matrices and functions of transition matrices for given trajectory data, Phys. Rev. E 80, 021106 (2009).
19. R. D. Reiss, A course on point processes, Springer series in statistics, Springer, New York, USA (1993).
20. Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, Lecture Notes in Mathematics ( LNM, volume 1766), Chapter IX Stochastic Properties Of Dynamical Systems Theorems, Springer, Berlin, Heidelberg (2001).
21. J. P. Conze, A. Raugi, Convergence of iterates of a transfer operator, application to dynaical systems and Markovv chains, ESAIM: Probability and Statistics 7, 115 (2003).
22. Ya. G. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surv. 27, 21 1972.
23. V. Baladi, Advanced Series in Nonlinear Dynamics: Volume 16: Positive Transfer Operators and Decay of Correlations, World Scientific (2000).
24. G. Rubino, B. Sericola, Sojourn times in finite Markov processes, J. Appl. Prob. 27, 744 (1989).
25. R. Syski, Passage Times for Markov Chains, IOS Press, Amsterdam, The Netherlands (1992).
26. VEDI P. Collet, S. Isola, On the Essential Spectrum of the Transfer Operator for Expanding Markov Maps, Communications in Mathematical Physics 139, 551 (1991).
27. B. Trendelkamp-Schroer, F. Noe’, Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution, J. Chem. Phys. 138, 164113 (2013).
28. J. Ito, Transactions of the American Mathematical Society 110, 152 (1964).
2. T. Hempel, Independent Markov decomposition: Toward modeling kinetics of biomolecular complexes, PNAS 118, 2021 (2021).
3. C. R. Schwantes, R. T. McGibbon, V. S. Pande, Perspective: Markov models for long-timescale biomolecular dynamics, J. Chem. Phys. 141, 090901 (2014).
4. B. E. Husic, V. S. Pande, Markov State Models: From an Art to a Science, J. Am. Chem. Soc. 2018, 140, 2386 (2018).
5. M. Jaeger, H. Nguyen, J. C. Crane, J. W. Kelly, M. Gruebele, The folding mechanism of a beta-sheet: the WW domain, J. Mol. Biol. 311, 373 (2001).
6. C.-K. Chan, Y. Hu, S. Takahashi, D. L. Rousseau, W. A. Eaton, J.
7. Hofrichter, Submillisecond protein folding kinetics studied by ultrarapid mixing, Proc. Natl. Acad. Sci. U.S.A. 94, 1779 (1997).
8. O. Bieri, J. Wirz, B. Hellrung, M. Schutkowski, M. Drewello, T. Kiefhaber, The speed limit for protein folding measured by triplet–triplet energy transfer, Proc. Natl. Acad. Sci. U.S.A. 96, 9597 (1999).
9. H. Neuweiler, S. Doose, M. Sauer, A microscopic view of miniprotein folding: enhanced folding efficiency through formation of an intermediate, Proc. Natl. Acad. Sci. U.S.A. 102, 16650 (2005).
10. C. R. Schwantes, R. T. McGibbon, V. S. Pande, Perspective: Markov models for long-timescale biomolecular dynamics, J. Chem. Phys. 141, 090901 (2014).
11. P. K. Pollett, V. T. Stefanov, Path Integrals for Continuous-Time Markov Chains, Journal of Applied Probability 39, 901 (2002).
12. J. H. Prinz et al., Markov models of molecular kinetics: Generation and validation, The Journal of Chemical Physics 134, 174105 (2011).
13. N. Djurdjevac, M. Sarich, C. Schuette, Estimating the Eigenvalue Error of Markov State Models, Multiscale Modeling & Simulation 10, 48 (2012).
14. Constructing the equilibrium ensemble of folding pathways from short offequilibrium simulations, PNAS 106, 19011 (2009).
15. M. Sarich, C. Schuette, Approximating selected non-dominant timescales by Markov state models, Communications in Mathematical Sciences 10 , 1001 (2012).
16. J. D. Chodera, F. Noe’, Probability distributions of molecular observables computed from Markov models. II. Uncertainties in observables and their time-evolution, J. Chem. Phys. 133, 105102 (2010).
17. J. D. Chodera, W. C. Swope, J. W. Pitera, K. A. Dill, Long-Time Protein Folding Dynamics from Short-Time Molecular Dynamics Simulations, Multiscale Model. Simul. 5, 1214 (2006).
18. P. Metzner, F. Noe’, C. Schuette, Estimating the sampling error: Distribution of transition matrices and functions of transition matrices for given trajectory data, Phys. Rev. E 80, 021106 (2009).
19. R. D. Reiss, A course on point processes, Springer series in statistics, Springer, New York, USA (1993).
20. Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, Lecture Notes in Mathematics ( LNM, volume 1766), Chapter IX Stochastic Properties Of Dynamical Systems Theorems, Springer, Berlin, Heidelberg (2001).
21. J. P. Conze, A. Raugi, Convergence of iterates of a transfer operator, application to dynaical systems and Markovv chains, ESAIM: Probability and Statistics 7, 115 (2003).
22. Ya. G. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surv. 27, 21 1972.
23. V. Baladi, Advanced Series in Nonlinear Dynamics: Volume 16: Positive Transfer Operators and Decay of Correlations, World Scientific (2000).
24. G. Rubino, B. Sericola, Sojourn times in finite Markov processes, J. Appl. Prob. 27, 744 (1989).
25. R. Syski, Passage Times for Markov Chains, IOS Press, Amsterdam, The Netherlands (1992).
26. VEDI P. Collet, S. Isola, On the Essential Spectrum of the Transfer Operator for Expanding Markov Maps, Communications in Mathematical Physics 139, 551 (1991).
27. B. Trendelkamp-Schroer, F. Noe’, Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution, J. Chem. Phys. 138, 164113 (2013).
28. J. Ito, Transactions of the American Mathematical Society 110, 152 (1964).
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Lecian, O. M. (2023). Analytical results from the two-states Markovv-states model and applications to validation of molecular dynamics. International Journal Of Mathematics And Computer Research, 11(9), 3746-3754. https://doi.org/10.47191/ijmcr/v11i9.08
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