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A minimal 2D model of the free energy surface for a unidirectional natural molecular motor

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Abstract

A schematic model of a natural molecular motor is proposed. It uses the change of the free energy surface to an effective surface as long as the enzyme is active. This effective surface acts as a trapdoor and explains the power stroke in biomotors, as well as its unidirectional movement. Then a thermal relaxation can do the energy transduction of the motor. The model uses Newton trajectories to explain the movement of stationary points on the effective surface.

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References

  1. A.P. Davis, Angew. Chem. 110, 953 (1998)

    Article  Google Scholar 

  2. C. Bustamente, Y.R. Chemla, N.R. Forde, D. Izhaky, Ann. Rev. Biochem. 73, 705 (2004)

    Article  Google Scholar 

  3. M. Schliwa, Molecular Motors (Wiley-VCH, Weinheim, 2003)

    Google Scholar 

  4. Y.M. Romanovsky, A.N. Tikhonov, Phys. Uspekhi 53, 893 (2010)

    Article  CAS  Google Scholar 

  5. J. Walker, Biochem. Soc. Trans. 41, 1 (2013)

    Article  CAS  Google Scholar 

  6. S. Mukherjee, R.P. Bora, A. Warshel, Q. Rev. Biophys. 48(4), 395 (2015)

    Article  CAS  Google Scholar 

  7. A. Warshel, R.P. Bora, J. Chem. Phys. 144, 180901 (2016)

    Article  Google Scholar 

  8. B.L. Feringa, Acc. Chem. Res. 34, 504 (2001)

    Article  CAS  Google Scholar 

  9. R.D. Astumian, Sci. Am. 285, 56 (2001)

    Article  CAS  Google Scholar 

  10. W. Mock, K. Ochwat, J. Phys. Org. Chem. 16, 175 (2003)

    Article  CAS  Google Scholar 

  11. E. Kay, D. Leigh, F. Zerbetto, Angew. Chem. 119, 72 (2007)

    Article  Google Scholar 

  12. J. Michl, E. Sykes, ACS Nano 3, 1042 (2009)

    Article  CAS  Google Scholar 

  13. J.P. Sauvage, P. Gaspard (eds.), From Non-covalent Assemblies to Molecular Machines (Wiley-VCH, Weinheim, 2010)

    Google Scholar 

  14. M. Guentner, M. Schildhauer, S. Thumser, P. Mayer, D. Stephenson, P.J. Mayer, H. Dube, Nat. Commun. 6, 8460 (2015)

    Article  Google Scholar 

  15. L. Greb, A. Eichhöfer, J.M. Lehn, Angew. Chem. Int. Ed. 54, 14345 (2015)

    Article  CAS  Google Scholar 

  16. M.R. Wilson, J. Solà, A. Carlone, S.M. Goldup, N. Lebrasseur, D.A. Leigh, Nature 534, 235 (2016)

    Article  CAS  Google Scholar 

  17. C. Cheng, J.F. Stoddart, ChemPhysChem 17, 1780 (2016)

    Article  CAS  Google Scholar 

  18. C.R. Hall, J. Conyard, I.A. Heisler, G. Jones, J. Frost, W.R. Browne, B.L. Feringa, S.R. Meech, J. Am. Chem. Soc. 139, 7408 (2017)

    Article  CAS  Google Scholar 

  19. C. Pezzato, C. Cheng, J.F. Stoddart, R.D. Astumian, Chem. Soc. Rev. 46, 5491 (2017)

    Article  CAS  Google Scholar 

  20. H. Wang, G. Oster, Nature 396, 279 (1998)

    Article  CAS  Google Scholar 

  21. J. Ma, T.C. Flynn, Q. Cui, A.G. Leslie, J.E. Walker, M. Karplus, Structure 10, 921 (2002)

    Article  CAS  Google Scholar 

  22. A. Warshel, P.K. Sharma, M. Kato, Y. Xiang, H. Liu, M.H.M. Olsson, Chem. Rev. 106(8), 3210 (2006)

    Article  CAS  Google Scholar 

  23. S. Mukherjee, A. Warshel, Proc. Natl. Acad. Sci. USA 108(51), 20550 (2011)

    Article  CAS  Google Scholar 

  24. C.T. Liu, J.P. Layfield, R.J. Stewart III, J.B. French, P. Hanoian, J.B. Asbury, S. Hammes-Schiffer, S.J. Benkovic, J. Am. Chem. Soc. 136, 10349 (2014)

    Article  CAS  Google Scholar 

  25. S.D. Fried, S.G. Boxer, Ann. Rev. Biochem. 86(1), 387 (2017)

    Article  CAS  Google Scholar 

  26. F.A. Kiani, S. Fischer, J. Biol. Chem. 288, 35569 (2013)

    Article  CAS  Google Scholar 

  27. S.S.M. Konda, J.M. Brantley, C.W. Bielawski, D.E. Makarov, J. Chem. Phys. 135, 164103 (2011)

    Article  Google Scholar 

  28. S.S.M. Konda, S.M. Avdoshenko, D.E. Makarov, J. Chem. Phys. 140, 104114 (2014)

    Article  Google Scholar 

  29. S.M. Avdoshenko, D.E. Makarov, J. Phys. Chem. B 120, 1537 (2015)

    Article  Google Scholar 

  30. S.M. Avdoshenko, D.E. Makarov, J. Chem. Phys. 142, 174106 (2015)

    Article  Google Scholar 

  31. D.E. Makarov, J. Chem. Phys. 144, 030901 (2016)

    Article  Google Scholar 

  32. W. Quapp, J.M. Bofill, Theor. Chem. Acc. 135(4), 113 (2016)

    Article  Google Scholar 

  33. W. Quapp, J.M. Bofill, J. Ribas-Ariño, J. Phys. Chem. A 121, 2820 (2017)

    Article  Google Scholar 

  34. W. Quapp, J.M. Bofill, J. Comput. Chem. 37, 2467 (2016)

    Article  CAS  Google Scholar 

  35. K. Henzler-Wildman, V. Thai, M. Lei, M. Ott, M. Wolf-Watz, T. Fenn, E. Pozharski, M. Wilson, G.A. Petsko, M. Karplus, C. Hubner, D. Kern, Nature 450, 838 (2007)

    Article  CAS  Google Scholar 

  36. H. Eying, J. Walter, G.E. Rimball, Quantum Chemistry (Wiley, New York, 1944)

    Google Scholar 

  37. G.I. Bell, Science 200, 618 (1978)

    Article  CAS  Google Scholar 

  38. M.T. Ong, J. Leiding, H. Tao, A.M. Virshup, T.J. Martínez, J. Am. Chem. Soc. 131(18), 6377 (2009)

    Article  CAS  Google Scholar 

  39. J. Ribas-Ariño, M. Shiga, D. Marx, Angew. Chem. Int. Ed. 48, 4190 (2009)

    Article  Google Scholar 

  40. K. Wolinski, J. Baker, Mol. Phys. 107, 2403 (2009)

    Article  CAS  Google Scholar 

  41. J. Ribas-Ariño, D. Marx, Chem. Rev. 112, 5412 (2012)

    Article  Google Scholar 

  42. W. Quapp, J.M. Bofill, Int. J. Quantum Chem. https://doi.org/10.1002/qua.25522 (2017)

  43. W. Quapp, M. Hirsch, O. Imig, D. Heidrich, J. Comput. Chem. 19, 1087 (1998)

    Article  CAS  Google Scholar 

  44. W. Quapp, M. Hirsch, D. Heidrich, Theor. Chem. Acc. 100(5/6), 285 (1998)

    Article  CAS  Google Scholar 

  45. J.M. Bofill, J.M. Anglada, Theor. Chem. Acc. 105, 463 (2001)

    Article  CAS  Google Scholar 

  46. R. Crehuet, J.M. Bofill, J.M. Anglada, Theor. Chem. Acc. 107, 130 (2002)

    Article  CAS  Google Scholar 

  47. W. Quapp, J. Theor. Comput. Chem. 2, 385 (2003)

    Article  CAS  Google Scholar 

  48. H.S. Smalo, V.V. Rybkin, W. Klopper, T. Helgaker, E. Uggerud, J. Phys. Chem. A 118, 7683 (2014)

    Article  Google Scholar 

  49. T. Stauch, A. Dreuw, Chem. Rev. 116, 14137 (2016)

    Article  CAS  Google Scholar 

  50. K. Adachi, K. Oiwa, T. Nishizaka, S. Furuike, H. Noji, H. Itoh, M. Yoshida, K. Kinosita Jr., Cell 130, 309 (2007)

    Article  CAS  Google Scholar 

  51. S. Akbulatov, Y. Tian, Z. Huang, T.J. Kucharski, Q.Z. Yang, R. Boulatov, Science 357, 299 (2017)

    Article  CAS  Google Scholar 

  52. E. Evans, A. Leung, H. Volkmar, C. Zhu, Proc. Natl. Acad. Sci. 101, 11281 (2004)

    Article  CAS  Google Scholar 

  53. A. Garai, S. Mogurampelly, S. Bag, P.K. Maiti, J. Chem. Phys. 147, 225102 (2017)

    Article  Google Scholar 

  54. T.L. Amyes, J.P. Richard, Biochemistry 52(12), 2021 (2013)

    Article  CAS  Google Scholar 

  55. B.L. Grigorenko, I.A. Kaliman, A.V. Nemukhin, J. Mol. Graph. Model. 31, 1 (2011)

    Article  CAS  Google Scholar 

  56. L. Pauling, Chem. Eng. News 24, 1375 (1946)

    Article  CAS  Google Scholar 

  57. M. Štrajbl, A. Shurki, M. Kato, A. Warshel, J. Am. Chem. Soc. 125, 10228 (2003)

    Article  Google Scholar 

  58. S. Fischer, B. Windshügel, D. Horak, K.C. Holmes, J.C. Smith, Proc. Natl. Am. Soc. 102, 6873 (2005)

    Article  CAS  Google Scholar 

  59. M. Bier, BioSystems 93, 23 (2008)

    Article  CAS  Google Scholar 

  60. K.W. Müller, A.M. Birzle, W.A. Wall, Proc. R. Soc. A 472, 20150555 (2016)

    Article  Google Scholar 

  61. S. Kubo, W. Li, S. Takada, PLOS Comput. Bio. https://doi.org/10.1371/journal.pcbi.1005748 (2017)

  62. J.R. Moffitt, Y.R. Chemla, K. Aathavan, S. Grimes, P.J. Jardine, D.L. Anderson, C. Bustamante, Nature 457, 446 (2009)

    Article  CAS  Google Scholar 

  63. R.D. Astumian, S. Mukherjee, A. Warshel, ChemPhysChem 17, 1719 (2016)

    Article  CAS  Google Scholar 

  64. S. Koppole, J.C. Smith, S. Fischer, Structure 15, 825 (2007)

    Article  CAS  Google Scholar 

  65. K. Shiroguchi, H.F. Chin, D.E. Hannemann, E. Muneyuki, E.M. De La Cruz, K. Kinosita Jr., PLoS Biol. 9, e1001031 (2011)

    Article  CAS  Google Scholar 

  66. A. Baumketner, Proteins 80, 2701 (2012)

    Article  CAS  Google Scholar 

  67. R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr., H. Itoh, Nature 410, 898 (2001)

    Article  CAS  Google Scholar 

  68. K. Shimabukuro, R. Yasuda, E. Muneyuki, K.Y. Hara, K. Kinosita Jr., M. Yoshida, Proc. Natl. Am. Soc. 100(25), 14731 (2003)

    Article  CAS  Google Scholar 

  69. J. Pu, M. Karplus, Proc. Natl. Am. Soc. 105(4), 1192 (2008)

    Article  CAS  Google Scholar 

  70. S. Volkán-Kacsó, R.A. Marcus, Proc. Natl. Acad. Sci. USA 113(43), 12029 (2016)

    Article  Google Scholar 

  71. K. Nam, J. Pu, M. Karplus, Proc. Natl. Am. Soc. 111(50), 17851 (2014)

    Article  CAS  Google Scholar 

  72. R. Perez-Carrasco, J.M. Sancho, Europhys. Lett. 91(6), 60001 (2010)

    Article  Google Scholar 

  73. V. Serreli, C.F. Lee, E.R. Kay, D.A. Leigh, Nature 445(445), 523 (2007)

    Article  CAS  Google Scholar 

  74. M. Klok, M. Walko, E.M. Geertsema, N. Ruangsupapichat, J.C. Kistemaker, A. Meetsma, B.L. Feringa, Chem. Eur. J. 14, 11183 (2008)

    Article  CAS  Google Scholar 

  75. J.M. Lehn, Chem. Eur. J. 12, 5910 (2006)

    Article  CAS  Google Scholar 

  76. H. Itoh, A. Takahashi, K. Adachi, H. Noji, R. Yasuda, M. Yoshida, K. Kinosita Jr., Nature 427, 465 (2004)

    Article  CAS  Google Scholar 

  77. R. Watanabe, D. Okuno, S. Sakakihara, K. Shimabukuro, R. Iino, M. Yoshida, H. Noji, Nat. Chem. Biol. 8, 86 (2012)

    Article  CAS  Google Scholar 

  78. R. Elber, A. West, Proc. Natl. Acad. Sci. 107, 5001 (2010)

    Article  CAS  Google Scholar 

  79. R.V. Agafonov, I.V. Negrashov, Y.V. Tkachev, S.E. Blakely, M.A. Titus, D.D. Thomas, Y.E. Nesmelov, Proc. Natl. Am. Soc. 106, 21625 (2009)

    Article  CAS  Google Scholar 

  80. H. Yu, L. Ma, Y. Yang, Q. Cui, PLoS Comput. Biol. 3, e23 (2007)

    Article  Google Scholar 

  81. B.T. Sutcliffe, The mathematics of vibration-rotation calculations, chap. 2, in Methods in Computational Chemistry, ed. by S. Wilson (Plenum Press, New York and London, 1992), pp. 33–89

    Google Scholar 

  82. E.H. Ahmed, P. Qi, B. Beser, J. Bai, R.W. Field, J.P. Huennekens, A.M. Lyyra, Phys. Rev. A 77, 053414 (2008)

    Article  Google Scholar 

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Acknowledgements

I thank Profs. Josep Maria Bofill and Jordi Ribas-Ariño from Barcelona for many discussions.

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Correspondence to Wolfgang Quapp.

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Quapp, W. A minimal 2D model of the free energy surface for a unidirectional natural molecular motor. J Math Chem 56, 1339–1347 (2018). https://doi.org/10.1007/s10910-018-0861-1

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