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Journal of Mathematical Sciences

, Volume 243, Issue 1, pp 111–127 | Cite as

Investigation of an Idealized Virus Capsid Model by the Dynamic Elasticity Apparatus

  • Z. Zhuravlova
  • D. Nerukh
  • V. Reut
  • N. Vaysfel’d
Article
  • 4 Downloads

The three-dimensional dynamic theory of elasticity is applied to investigate the mechanical properties of the virus capsid. An idealized model of viruses is based on the 3D boundary-value problem of mathematical physics formulated in a spherical coordinate system for the steady-state oscillation process. The virus is modeled by a hollow elastic sphere filled with an acoustic medium and located in a different acoustic medium. The stated boundary-value problem is solved with the help of the method of integral transforms and the method of the discontinuous solutions. As a result, the exact solution of the problem is obtained. The numerical calculations of the elastic characteristics of the virus are carried out.

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References

  1. 1.
    N. D. Vaisfel’d and G. Ya. Popov, “Nonstationary dynamic problems of elastic stress concentration near a spherical imperfection,” Izv. Ros. Akad. Nauk, Mekh. Tverd. Tela, No. 3, 90–102 (2002); English translation:Mech. Solids, 37, No. 3, 77–88 (2002).Google Scholar
  2. 2.
    N. O. Horechko and R. M. Kushnir, “Thermostressed state of a composite plate with heat exchange under the action of a uniformly distributed heat source,” Mat. Met. Fiz.-Mekh. Polya, 54, No. 1, 153–162 (2011); English translation:J. Math. Sci., 183, No. 2, 177–189 (2012); https://doi.org/ https://doi.org/10.1007/s10958-012-0805-4.MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev (1981).zbMATHGoogle Scholar
  4. 4.
    А. N. Guz’ and V. D. Kubenko, “Theory of Nonstationary Aerohydroelasticity of the Shells,” in: A. N. Guz’ (editor), Methods of Calculation of Shells [in Russian], Vol. 5, Naukova Dumka, Kiev (1982).Google Scholar
  5. 5.
    H. S. Kit and V. A. Halazyuk, “Axisymmetric stress-strain state of a body with thin rigid disk-shaped heat-resistant inclusion,” Mat. Met. Fiz.-Mekh. Polya, 56, No. 3, 95–109 (2013); English translation:J. Math. Sci., 205, No. 4, 602–620 (2015); https://doi.org/ https://doi.org/10.1007/s10958-015-2269-9.MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. L. Medvedskii, “Dynamics of an inhomogeneous transversely isotropic sphere in acoustic media,” Vestn. Mosk. Aviats. Inst., 17, No. 1, 181–186 (2010).Google Scholar
  7. 7.
    Z. T. Nazarchuk, D. B. Kuryliak, M. V. Voytko, and Ya. P. Kulynych, “On the interaction of an elastic SH-wave with an interface crack in the perfectly rigid joint of a plate with a half space,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 2, 107–118 (2012); English translation:J. Math. Sci., 192, No. 6, 609–622 (2013); https://doi.org/ https://doi.org/10.1007/s10958-013-1420-8.MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. V. Panasyuk and M. P. Savruk, “On the determination of stress concentration in a stretched plate with two holes,” Mat. Met. Fiz.- Mekh. Polya, 51, No. 2, 112–123 (2008); English translation:J. Math. Sci., 162, No. 1, 132–148 (2009); https://doi.org/ https://doi.org/10.1007/s10958-009-9626-5.CrossRefGoogle Scholar
  9. 9.
    A. F. Ulitko, “Stress state of a hollow sphere loaded by concentrated forces,” Prikl. Mekh, 4, No. 5, 38–45 (1968); English translation:Int. Soviet Appl. Mech., 4, No. 5, 25–29 (1968); https://doi.org/ https://doi.org/10.1007/BF00886782.CrossRefGoogle Scholar
  10. 10.
    M. V. Khai and M. D. Hrylyts’kyi, “Mathematical statement of boundary conditions for problems of three-dimensional deformation of plates,” Mat. Met. Fiz.-Mekh. Polya, 42, No. 1, 55–61 (1999); English translation:J. Math. Sci., 109, No. 1, 1221–1228 (2002); https://doi.org/ https://doi.org/10.1023/A:1013744627572.CrossRefGoogle Scholar
  11. 11.
    S. M. Hasheminejad and M. Maleki, “Acoustic resonance scattering from a submerged anisotropic sphere,” Akustich. Zh., 54, No. 2, 205–218 (2008); English translation:Acoust. Phys., 54, No. 2, 168–179 (2008); https://doi.org/ https://doi.org/10.1134/S1063771008020048.CrossRefGoogle Scholar
  12. 12.
    V. S. Chernina, Statics of Thin-Walled Shells of Revolution [in Russian], Nauka, Moscow (1968).Google Scholar
  13. 13.
    А. V. Sheptilevskiy, V. М. Коsenkov, and I. T. Selezov, “Three-dimensional model of a hydroelastic system bounded by a spherical shell,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 1, 159–167 (2012); English translation:J. Math. Sci., 190, No. 6, 823–834 (2013); https://doi.org/https://doi.org/10.1007/s10958–013–1291-z.Google Scholar
  14. 14.
    S. R. Aglyamov, A. B. Karpiouk, Yu. A. Ilinskii, E. A. Zabolotskaya, and S. Y. Emelianov, “Motion of a solid sphere in a viscoelastic medium in response to applied acoustic radiation force: Theoretical analysis and experimental verification,” J. Acoust. Soc. Amer., 122, No. 4, 1927–1936 (2007); https://doi.org/ https://doi.org/10.1121/1.2774754.CrossRefGoogle Scholar
  15. 15.
    M. Buenemann and P. Lenz, “Elastic properties and mechanical stability of chiral and filled viral capsids,” Phys. Rev. E, 78, No. 5 (2008); https://doi.org/ https://doi.org/10.1103/PhysRevE.78.051924.
  16. 16.
    M. Buenemann and P. Lenz, “Mechanical limits of viral capsids,” Proc. Nat. Acad. Sci. USA (PNAS), 104, No. 24, 9925–9930 (2007); https://doi.org/ https://doi.org/10.1073/pnas.0611472104.CrossRefGoogle Scholar
  17. 17.
    C. Carrasco, A. Carreira, I. A. T. Schaap, P. A. Serena, J. Gómez-Herrero, M. G. Mateu, and P. J. de Pablo, “DNA-mediated anisotropic mechanical reinforcement of a virus,” Proc. Nat. Acad. Sci. USA (PNAS), 103, No. 37, 13706–13711 (2006); https://doi.org/ https://doi.org/10.1073/pnas.0601881103.CrossRefGoogle Scholar
  18. 18.
    M. M. Gibbons and W. S. Klug, “Nonlinear finite-element analysis of nanoindentation of viral capsids,” Phys. Rev. E, 75, No. 3 (2007); https://doi.org/ https://doi.org/10.1103/PhysRevE.75.031901.
  19. 19.
    G. M. Grason, “Perspective: Geometrically frustrated assemblies,” J. Chem. Phys., 145, 110901-1–110901-17 (2016); https://doi.org/ https://doi.org/10.1063/1.4962629.CrossRefGoogle Scholar
  20. 20.
    N. N. Kiselyova and G. Ch. Shushkevich, “Acoustic scattering by spherical shell and sphere,” in: Computer Algebra Systems in Teaching and Research: Mathematical Modeling in Physics, Civil Engineering, Economics, and Finance, Wyd. Collegium Mazovia, Siedlce, 91–99 (2011); https://elib.grsu.by/katalog/161816-348205.pdf.
  21. 21.
    I. Korotkin, D. Nerukh, E. Tarasova, V. Farafonov, and S. Karabasov, “Two-phase flow analogy as an effective boundary condition for modeling liquids at atomistic resolution,” J. Comput. Sci., 17, part 2, 446–456 (2016); https://doi.org/ https://doi.org/10.1016/j.jocs.2016.03.012.MathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Lidmar, L. Mirny, and D. R. Nelson, “Virus shapes and buckling transitions in spherical shells,” Phys. Rev. E, 68, No. 5 (2003); https://doi.org/ https://doi.org/10.1103/PhysRevE.68.051910.
  23. 23.
    A. Markesteijn, S. Karabasov, A. Scukins, D. Nerukh, V. Glotov, and V. Goloviznin, “Concurrent multiscale modelling of atomistic and hydrodynamic processes in liquids,” Phil. Trans. R. Soc. A. Math. Phys. Eng. Sci., 372, No. 2021 (2014); https://doi: https://doi.org/10.1098/rsta.2013.0379.MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. R. May and C. L. Brooks (3rd), “On the morphology of viral capsids: Elastic properties and buckling transitions,” J. Phys. Chem. B, 116, No. 29, 8604–8609 (2012); https://doi: https://doi.org/10.1021/jp300005g.CrossRefGoogle Scholar
  25. 25.
    V. V. Mykhas’kiv, I. Ya. Zhbadynskyi, and Ch. Zhang, “Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks,” Mat. Met. Fiz.-Mekh. Polya, 56, No. 2, 94–101 (2013); English translation:J. Math. Sci., 203, No. 1, 114–122 (2014); https://doi.org/ https://doi.org/10.1007/s10958-014-2094-6.MathSciNetCrossRefGoogle Scholar
  26. 26.
    W. Nowacki, Teoria Sprężystości, PWN, Warszawa (1970).zbMATHGoogle Scholar
  27. 27.
    R. Phillips, M. Dittrich, and K. Schulten, “Quasicontinuum representations of atomic-scale mechanics: from proteins to dislocations,” Ann. Rev. Mater. Res., 32, 219–233 (2002); https://doi.org/https://doi.org/10.1146/annurev.matsci. 32.122001.102202.Google Scholar
  28. 28.
    G. Polles, G. Indelicato, R. Potestio, P. Cermelli, R. Twarock, and C. Micheletti, “Mechanical and assembly units of viral capsids identified via quasirigid domain decomposition,” PLOS Comput. Biol., 9, No. 11, e1003331 (2013); https://doi.org/ https://doi.org/10.1371/journal.pcbi.1003331.CrossRefGoogle Scholar
  29. 29.
    W. H. Roos, M. M. Gibbons, A. Arkhipov, C. Uetrecht, N. R. Watts, P. T. Wingfield, A. C. Steven, A. J. Heck, K. Schulten, W. S. Klug, and G. J. Wuite, “Squeezing protein shells: How continuum elastic models, molecular dynamics simulations, and experiments coalesce at the nanoscale,” Biophys. J., 99, No. 4, 1175–1181 (2010); https://doi.org/ https://doi.org/10.1016/j.bpj.2010.05.033.CrossRefGoogle Scholar
  30. 30.
    A. Scukins, D. Nerukh, E. Pavlov, S. Karabasov, and A. Markesteijn, “Multiscale molecular dynamics/hydrodynamics implementation of two dimensional “Mercedes Benz” water model,” Eur. Phys. J. Spec. Topics, 224, No. 12, 2217–2238 (2015); https://dx.doi.org/ https://doi.org/10.1140/epjst/e2015-02409-8.CrossRefGoogle Scholar
  31. 31.
    E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford (1948).Google Scholar
  32. 32.
    G. A. Vliegenthart and G. Gompper, “Mechanical deformation of spherical viruses with icosahedral symmetry,” Biophys. J., 91, No. 3, 834–841 (2006); https://doi.org/ https://doi.org/10.1529/biophysj.106.081422.CrossRefGoogle Scholar
  33. 33.
    J. H. Wu, A. Q. Liu, H. L. Chen, and T. N. Chen, “Multiple scattering of a spherical acoustic wave from fluid spheres,” J. Sound Vibrat., 290, No. 1–2, 17–33 (2006); https://doi.org/ https://doi.org/10.1016/j.jsv.2005.03.015.CrossRefGoogle Scholar
  34. 34.
    R. Zandi and D. Reguera, “Mechanical properties of viral capsids,” Phys. Rev. E, 72, No. 2 (2005); https://doi.org/ https://doi.org/10.1103/PhysRevE.72.021917.
  35. 35.
    Z. Zhuravlova, D. Kozachkov, D. Pliusnov, V. Radzivil, V. Reut, O. Shpynarov, E. Tarasova, D. Nerukh, and N. Vaysfel’d, “Modeling of virus vibration with 3-D dynamic elasticity theory,” in: 23rd Internat. Conf. “Engineering Mechanics 2017”, 15–18 May, 2017, Svratka, Czech Republic, (2017), pp. 1126–1129.Google Scholar
  36. 36.
    M. Zink and H. Grubmüller, “Mechanical properties of the icosahedral shell of southern bean mosaic virus: A molecular dynamics study,” Biophys. J., 96, No. 4, 1350–1363 (2009); https://doi.org/ https://doi.org/10.1016/j.bpj.2008.11.028.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Z. Zhuravlova
    • 1
  • D. Nerukh
    • 2
  • V. Reut
    • 1
  • N. Vaysfel’d
    • 1
  1. 1.Mechnikov Odessa National UniversityOdessaUkraine
  2. 2.Aston UniversityBirminghamUK

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