Mathematical Modeling of Molecular ``nano-machines''

V. A. Avetisov ; A. Kh. Bikulov ; A. P. Zubarev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2011), pp. 9-15.

Résumé

A new approach to mathematical modeling of “molecular machines”, e.g. macromolecular structures which functional prototypes are the proteins, is presented. In the center of the approach lies the description of multi-scale fluctuation induced mobility of proteins by the ultrametric random processes. In order todemonstrate how $p$-adic equations of the reaction–diffusion type are described the molecular machine operation, a heuristic model is constructed in this article. It is shown that such multi-scale modeling allows to have an insight into unexpected resources that can be used in order to control the functional cycle.

Détail

Keywords: mathematical modeling, proteins, molecular nano-machines, ultrametricity
Mots-clés : reaction-diffusion processes, $p$–adic equations.

Bibliographie

[1] Jencks W. P., Catalysis in Chemistry and Enzymology, McGraw-Hill, New York, 1969; Dzhenke V. P., Kataliz v khimii i enzimologii, Mir, M., 1972, 463 pp.

[2] Shaytan K. V., Rubin A. B., “Conformational dynamics of proteins and the simplest molecular “machines””, Biofizika, 27:3 (1982), 386–390

[3] Blumenfeld L. A., Problems of Biological Physics, v. 7, Springer Series in Synergetics, Springer Verlag, Berlin – Heidelberg – New York, 1981, 224 pp. | DOI | MR

[4] Fenimore P. W., Frauenfelder H., McMahom B. H., Young R. D., “Proteins are paradigms of stochastic complexity”, Physica A, 351:1 (2005), 1–13 | DOI | MR

[5] Frauenfelder H., Wolynes P., “Biomolecules: Where the physics of complexity and simplicity meet”, Physics Today, 47:2 (1994), 58–64 | DOI

[6] Avetisov V. A., Bikulov A. Kh., Kozyrev S. V., “Application of $p$-adic analysis to models of breaking of replica symmetry”, J. Phys. A, Math. Gen., 32:50 (1999), 8785–8791 | DOI | MR | Zbl

[7] Avetisov V. A., Bikulov A. Kh., Kozyrev S. V., Osipov V. A., “$p$-adic models of ultrametric diffusion constrained by hierarchical energy landscapes”, J. Phys. A, Math. Gen., 35:2 (2002), 177–189 | DOI | MR | Zbl

[8] Avetisov V. A., Bikulov A. Kh. Osipov V. A., “$p$-Adic description of characteristic relaxation in complex systems”, J. Phys. A, Math. Gen., 36:15 (2003), 4239–4246 | DOI | MR | Zbl

[9] Avetisov V. A., Bikulov A. Kh., “Protein ultrametricity and spectral diffusion in deeply frozen proteins”, Biophys. Rev. and Lett., 3:3 (2008), 387–396 | DOI

[10] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-Adic analysis and mathematical physics, v. 1, Series on Soviet and East European Mathematics, World Scientific Publishing Co., Inc., River Edge, NJ, 1994, 319 pp. | MR | MR | Zbl | Zbl

[11] Avetisov V. A., Bikulov A. Kh., Zubarev A. P., “First passage time distribution and the number of returns for ultrametric random walks”, J. Phys. A, Math. Theor., 42:8 (2009), 085003–08521 | DOI | MR