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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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).
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.
V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev (1981).
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).
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.
A. L. Medvedskii, “Dynamics of an inhomogeneous transversely isotropic sphere in acoustic media,” Vestn. Mosk. Aviats. Inst., 17, No. 1, 181–186 (2010).
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.
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.
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.
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.
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.
V. S. Chernina, Statics of Thin-Walled Shells of Revolution [in Russian], Nauka, Moscow (1968).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
W. Nowacki, Teoria Sprężystości, PWN, Warszawa (1970).
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.
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.
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.
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.
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford (1948).
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.
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.
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.
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.
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.
Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 60, No. 2, pp. 92–104, April–June, 2017.
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Zhuravlova, Z., Nerukh, D., Reut, V. et al. Investigation of an Idealized Virus Capsid Model by the Dynamic Elasticity Apparatus. J Math Sci 243, 111–127 (2019). https://doi.org/10.1007/s10958-019-04530-4