Free Vibration Analysis of Microtubules as Orthotropic Elastic Shells Using Stress and Strain Gradient Elasticity Theory

Document Type: Research Paper


1 Faculty of Engineering, Shahrekord University

2 Nanotechnology Research Center, Shahrekord University


In this paper, vibration of the protein microtubule, one of the most important intracellular elements serving as one of the common components among nanotechnology, biotechnology and mechanics, is investigated using stress and strain gradient elasticity theory and orthotropic elastic shells model. Microtubules in the cell are influenced by internal and external stimulation and play a part in conveying protein substances and taking medications to the intended targets. Therefore, in order to control the biological cell functions, it is important to know the vibrational behavior of microtubules. For this purpose, using the cylindrical shell model which fully corresponds to microtubule geometry, and by considering it as orthotropic which is closer to reality, based on gradient elasticity theory, frequency analysis of the protein microtubule is carried out by considering Love’s thin shell theory and Navier solution. Also, the effect of size parameter and other variables on the results are investigated.


[1] Wada H., 2005, Biomechanics at Micro- and Nanoscale Levels, World Scientific Publishing Company.
[2] Alberts B., Bray D., Lewis J., Raff M., Roberts K., Watson J., 1994, Molecular Biology of the Cell, Garland Publishing, New York.
[3] Faber J., Portugal R., Ros L.P., 2006, Information processing in brain microtubules, Biosystems 83: 1-9.
[4] Hawkins T., Mirigian M., Yasar M.S., Ross J.L., 2010, Mechanics of microtubules, Journal of Biomechanics 43: 23-30.
[5] Pampaloni F., Florin E.L., 2008, Microtubule architecture: inspiration for novel carbon nanotube-based biomimetic materials, Trends in Biotechnology 26(6): 302-310.
[6] Yuanwen G., Ming L.F., 2009, Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory, Biochemical and Biophysical Research Communications 387: 467- 471.
[7] Tadi Beni Y., Abadyan M., 2013, Size-dependent pull-in instability of torsional nano-actuator, Physica Scripta 88(5): 055801.
[8] Tadi Beni Y., Abadyan M., 2013, Use of strain gradient theory for modeling the size-dependent pull-in of rotational nano-mirror in the presence of molecular force, International Journal of Modern Physics B 27: 1350083-1350101.
[9] TadiBeni Y., Koochi A., Abadyan M., 2011, Theoretical study of the effect of Casmir force, elastic boundary conditions and size dependency on the pull-in instability of beam-type NEMS, Physica E 43: 979-988.
[10] Tadi Beni Y., Koochi A., Kazemi A.S., Abadyan M., 2012, Modeling the influence of surface effect and molecular force on pull-in voltage of rotational Nano–micro mirror using 2-DOF model, Canadian Journal of Physics 90: 963-974.
[11] Sharma P., Zhang X., 2006, Impact of size-dependent non-local elastic strain on the electronic band structure of embedded quantum dots, Journal Nanoengineering and Nanosystems 220: 17403499.
[12] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[13] Gittes F., Mickey B., Nettleton J., Howard J., 1995, Flexural rigidity of microtubules and actin filaments measured from thermal fluctuation in shape, Journal of Cell Biology 120: 923-934.
[14] Venier P., Maggs A.C., Carlier M.F., Pantaloni D., 1994, Analysis of microtubule rigidity using hydrodynamic flow and thermal fluctuations, Journal of Biological Chemistry 269: 13353-13360.
[15] Vinckier A., Dumortier C., Engelborghs Y., Hellemans L., 1996, Dynamical and mechanical study of immobilized microtubule with atomic force microscopy, Journal of Vacuum Science & Technology B 14: 1427-1431.
[16] Sirenko Y.M., Stroscio M.A., Kim K.W., 1996, Elastic vibration of microtubules in a fluid, Physical Review E 53: 1003-1010.
[17] Wang C.Y., Ru C.Q., Mioduchowski A., 2006, Vibration of microtubules as orthotropic elastic shells, Physica E 35: 48-56.
[18] Wang C.Y., Zhang L.C., 2008, Circumferential vibration of microtubules with long axial wavelength, Journal of Biomechanics 41: 1892-1896.
[19] Shen H.S., 2011, Nonlinear vibration of microtubules in living cells, Current Applied Physics 11: 812-821.
[20] Civalek Ö., Akgöz B., 2010, Free vibration analysis of microtubules as cytoskeleton components: nonlocal euler-bernoulli beam modeling, Transaction B: Mechanical Engineering 17: 367-375.
[21] Heireche H., Tounsi A., Benhassaini H., Benzair A., Bendahmane M., Missouri M., Mokadem S., 2010, Nonlocal elasticity effect on vibration characteristics of protein microtubules, Physica E 42: 2375-2379.
[22] Xiang P., Liew K.M., 2011, Free vibration analysis of microtubules based on an atomistic-continuum model, Journal of Sound and Vibration 331: 213-230.
[23] Karimi Zeverdejani M., Tadi Beni Y., 2013, The nano scale vibration of protein microtubules based on modified strain gradient theory, Current Applied Physics 13: 1566-1576.
[24] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 1477-1508.
[25] Fleck N., Muller G., Ashby M., Hutchinson J., 1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42: 475-487.
[26] Stölken J., Evans A., 1998, A microbend test method for measuring the plasticity length scale, Acta Materialia 46: 5109-5115.
[27] McElhaney K., Vlassak J., Nix W., 1998, Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments, Journal of Materials Research 13: 1300-1306.
[28] Nix W.D., Gao H., 1998, Indentation size effects in crystalline materials: a law for strain gradient plasticity, Journal of the Mechanics and Physics of Solids 46: 411-425.
[29] Chong A., Lam D.C., 1999, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research 14: 4103-4110.
[30] Tadi Beni Y., Karimi Zeverdejani M., 2015, Free vibration of microtubules as elastic shell model based on modified couple stress theory, Journal of Mechanics in Medicine and Biology 15(3):1550037-1550060.
[31] Askes H., Aifantis E.C., 2011, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures 48: 1962-1990.
[32] Tuszynski J.A., Luchko T., Portet S., Dixon J.M., 2005, Anisotropic elastic properties of microtubules, The European Physical Journal E 17: 29-35.
[33] De Pablo P.J., Schaap I..A.T., Mackintosh F.C., Schmidt C.F., 2003, Deformational collapse of microtubules on the nanometer scale, Physical Review Letters 91: 098101.
[34] Sirenko Y. M., Stroscio M. A., Kim K. W., 1996, Elastic vibrations of microtubules in a fluid, Physical Review E 53: 1003.
[35] Askes H., Aifantis E.C., 2009, Gradient elasticity and flexural wave dispersion in carbon Nanotubes, Physical Review B 80: 195412.
[36] Bennett T., Gitman I., Askes H., 2007, Elasticity theories with higher-order gradients of inertia and stiffness for the modeling of wave dispersion in laminates, International Journal of Fracture 148: 185-193.
[37] Gitman I., Askes H., Aifantis E.C., 2005, The representative volume size in static and dynamic micro-macro transitions, International Journal of Fracture 135: 3-9.
[38] Wang Q., 2005, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, Journal of Applied Physics 98: 124301.
[39] Civalek O., Demir C., Akgoz B., 2010, Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications 15: 289-298.
[40] Heireche H., Tounsi A., Benhassaini H., Benzair A., Bendahmane M., Missouri M., Mokadem S., 2010, Nonlocal elasticity effect on vibration characteristics of protein microtubules, Physica E 42: 2375-2379.
[41] Shen H. S., 2010, Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium, Biomechanics and Modeling in Mechanobiology 9: 345-357.
[42] Shen H. S., 2010, Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model, Journal of Theoretical Biology 264: 386-394.
[43] Park S., Gao X., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 2355-2359.
[44] Maranganti R., Sharma P., 2007, A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (ir) relevance for nanotechnologies, Journal of the Mechanics and Physics of Solids 55: 1823-1852.
[45] Duan W., Wang C.M., Zhang Y., 2007, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics 101: 024305-024307.
[46] Chan K., Zhao Y., 2011, The dispersion characteristics of the waves propagating in a spinning single-walled carbon nanotube, Science China Physics, Mechanics & Astronomy 54: 1854-1865.
[47] Shi Y.J., Guo W.L., Ru C.Q., 2008, Relevance of timoshenko-beam model to microtubules of low shear modulus, Physica E 41: 213-219.