Skip to main content
SpringerLink
Log in

Nonlocal Elasticity Theory for the Mechanical Behavior of Protein Microtubules

  • Published:
Physical Mesomechanics Aims and scope Submit manuscript

Abstract

In this work, Reddy beam theory is used with the nonlocal differential constitutive relations of Eringen, and the equations of motion of the protein microtubules in terms of the generalized displacements are presented. Analytical solution of bending the protein microtubules is presented to bring out the effect of the nonlocal behavior on deflections. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of the protein microtubules or beams.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

REFERENCES

  1. Ingber, D.E. and Tensegrity, I., Cell Structure and Hierarchical Systems Biology, J. Cell. Sci., 2003, vol. 116, pp. 1157–1173.

    Article  Google Scholar 

  2. Baudriller, H., Maurin, B., Cañadas, P. Montcourrier, P., Parmeggiani, A., and Bettache, N., Form-Finding of Complex Tensegrity Structures: Application to Cell Cytoskeleton Modelling, Comptes Rendus Mécanique, 2006, vol. 334, pp. 662–668.

    Article  ADS  Google Scholar 

  3. Heireche, H., Tounsi, A., Benhassaini, H., Benzair, A., Bendahmane, M., Missouri, M., and Mokadem, S., Nonlocal Elasticity Effect on Vibration Characteristics of Protein Microtubules, Physica E, 2010, vol. 42, no. 9, pp. 2375–2379.

    Article  ADS  Google Scholar 

  4. Tounsi, A., Heireche, H., Benhassaini, H., and Missouri, M., Vibration and Length-Dependent Flexural Rigidity of Protein Microtubules Using Higher-Order Shear Deformation Theory, J. Theor. Biol., 2010, vol. 266, pp. 250–255.

    Article  Google Scholar 

  5. Benmansour, D.L., Kaci, A., Bousahla, A.A., Heireche, H., Tounsi, A., Alwabli, A.S., Alhebshi, A.M., Al-ghmady, K., and Mahmoud, S.R., The Nano Scale Bending and Dynamic Properties of Isolated Protein Microtubules Based on Modified Strain Gradient Theory, Advanc. Nano Res., 2019, vol. 7, no. 6, pp. 443–457.

    Google Scholar 

  6. Alwabli, A.S., Kaci, A., Bellifa, H., Bousahla, A.A., Tounsi, A., Alzahrani, D.A., Abulfaraj, A.A., Bourada, F., Benrahou, K.H., Tounsi, A., Mahmoud, S.R., and Hussain, M., The Nano Scale Buckling Properties of Isolated Protein Microtubules Based on Modified Strain Gradient Theory and a New Single Variable Trigonometric Beam Theory, Adv. Nano Res., 2021, vol. 10, no. 1.

  7. Civalek, O. and Demir, C., Bending Analysis of Microtubules Using Nonlocal Euler–Bernoulli Beam Theory, Appl. Math. Model., 2011, vol. 35, no. 5, pp. 2053–2067.

    Article  MathSciNet  Google Scholar 

  8. Demir, C. and Civalek, O., Torsional and Longitudinal Frequency and Wave Response of Microtubules Based on the Nonlocal Continuum and Nonlocal Discrete Models, App. Math. Model., 2013, vol. 37, no. 22, pp. 9355–9367.

    Article  Google Scholar 

  9. Schaap, I.A.T., Carrasco, C., de Pablo, P.J., MacKintosh, F.C., and Schmidt, C.F., Elastic Response, Buckling, and Instability of Microtubules under Radial Indentation, Biophys. J., 2006, vol. 91, pp. 1521–1531.

    Article  ADS  Google Scholar 

  10. Wang, N., Naruse, K., Stamenovic, D., Fredberg, J.J., Mijailovich, S.M., Tolic-Norrelykke, I.M., Polte, T., Mannix, R., and Ingber, D.E., Mechanical Behavior in Living Cells Consistent with the Tensegrity Model, PNAS, 2001, vol. 98, no. 14, pp. 7765–7770.

    Article  ADS  Google Scholar 

  11. Pirentis, A. and Lazopoulos, K., On the Singularities of a Constrained (Incompressible-Like) Tensegrity-Cytoskeleton Model under Equitriaxial Loading, Int. J. Solids Struct., 2010, vol. 47, no. 6, pp. 759–767.

    Article  Google Scholar 

  12. Hawkins, T., Mirigian, M., Selcuk, Y.M., and Ross, J.L., Mechanics of Microtubules, J. Biomech., 2010, vol. 43, pp. 23–30.

    Article  Google Scholar 

  13. Li, T., A Mechanics Model of Microtubule Buckling in Living Cells, J. Biomech., 2008, vol. 41, pp. 1722–1729.

    Article  Google Scholar 

  14. Shi, Y., Guo, W., and Ru, C., Relevance of Timoshenko-Beam Model to Microtubules of Low Shear Modulus, Physica E, 2008, vol. 41, pp. 213–219.

    Article  ADS  Google Scholar 

  15. Li, C. Ru, C.Q., and Mioduchowski, A., Length-Dependence of Flexural Rigidity as a Result of Aanisotropic Elastic Properties of Microtubules, Biochem. Biophys. Res. Commun., 2006, vol. 349, pp. 1145–1150.

    Article  Google Scholar 

  16. Kikumoto, M., Kurachi, M., Tosa, V., and Tashiro, H., Flexural Rigidity of Individual Microtubules Measured by a Buckling Force with Optical Traps, Biophys. J., 2006, vol. 90, pp. 1687–1696.

    Article  ADS  Google Scholar 

  17. Wagner, O.I., Rammensee, S., Korde, N., Wen, Q., Leterrier, J.-F., and Janmey, P.A., Softness, Strength and Self-Repair in Intermediate Filament Networks, Exp. Cell Res., 2007, vol. 313, pp. 2228–2235.

    Article  Google Scholar 

  18. Mehrbod, M. and Mofrad, M.R.K., On the Significance of Microtubule Flexural Behavior in Cytoskeletal Mechanics, PLOS ONE, 2011, vol. 6, no. 10, p. e25627.

  19. Mindlin, R.D. and Tiersten, H.F., Effects of Couple Stresses in Linear Elasticity, Arch. Ration. Mech. Anal., 1962, vol. 11, pp. 415–448.

    Article  Google Scholar 

  20. Koiter, W.T., Couple-Stresses in the Theory of Elasticity: I and II, Proc. Roy. Netherlands Acad. Sci. B, 1964, vol. 67, pp. 17–44.

    MathSciNet  Google Scholar 

  21. Toupin, R.A., Theory of Elasticity with Couple Stresses, Arch. Ration. Mech. Anal., 1964, vol. 17, pp. 85–112.

    Article  Google Scholar 

  22. Akbaş, Ş.D., Bending of a Cracked Functionally Graded Nanobeam, Adv. Nano Res. Int. J., 2018, vol. 6, no. 3, pp. 219–242.

    Google Scholar 

  23. Eringen, A.C., Theory of Micropolar Plates, Z. Angew Math. Phys., 1967, vol. 18, pp. 12–30.

    Article  Google Scholar 

  24. Eringen, A.C., Nonlocal Polar Elastic Continua, Int. J. Eng. Sci., 1972, vol. 10, pp. 1–16.

    Article  Google Scholar 

  25. Eringen, A.C., On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves, J. Appl. Phys., 1983, vol. 54, pp. 4703–4710.

    Article  ADS  Google Scholar 

  26. Balubaid, M., Tounsi, A., Dakhel, B., and Mahmoud, S.R., Free Vibration Investigation of FG Nanoscale Plate Using Nonlocal Two Variables Integral Refined Plate Theory, Comput. Concrete, 2019, vol. 24, no. 6, pp. 579–586.

    Google Scholar 

  27. Hussain, M., Naeem, M.N., Tounsi, A., and Taj, M., Nonlocal Effect on the Vibration of Armchair and Zigzag SWCNTs with Bending Rigidity, Adv. Nano Res., 2019, vol. 7, no. 6, pp. 431–442.

    Google Scholar 

  28. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R., and Tounsi, A., Dynamic Analysis of Nanosize FG Rectangular Plates Based on Simple Nonlocal Quasi 3D HSDT, Adv. Nano Res., 2019, vol. 7, no. 3, pp. 189–206.

    Google Scholar 

  29. Berghouti, H., Adda Bedia, E.A., Benkhedda, A., and Tounsi, A., Vibration Analysis of Nonlocal Porous Nanobeams Made of Functionally Graded Material, Adv. Nano Res., 2019, vol. 7, no. 5, pp. 351–364.

    Google Scholar 

  30. Fleck, N.A. and Hutchinson, J.W., A Phenomenological Theory for Strain Gradient Effects in Plasticity, J. Mech. Phys. Solids, 1993, vol. 41, pp. 1825–1857.

    Article  ADS  MathSciNet  Google Scholar 

  31. Vardoulakis, I. and Sulem, J., Bifurcation Analysis in Geomechanics, London: Blackie/Chapman & Hall, 1995.

  32. Aifantis, E.C., Gradient Deformation Models at Nano, Micro, and Macro Scales, J. Eng. Mater. Technol., 1999, vol. 121, pp. 189–202.

    Article  Google Scholar 

  33. Alimirzaei, S., Mohammadimehr, M., and Tounsi, A., Nonlinear Analysis of Viscoelastic Micro-Composite Beam with Geo-Metrical Imperfection Using FEM: MSGT Electro-Magneto-Elastic Bending, Buckling and Vibration Solutions, Struct. Eng. Mech., 2019, vol. 71, no. 5, pp. 485–502.

    Google Scholar 

  34. Karami, B., Janghorban, M., and Tounsi, A., Galerkin’s Approach for Buckling Analysis of Functionally Graded Anisotropic Nanoplates/Different Boundary Conditions, Eng. Comput., 2019, vol. 35, pp. 1297–1316.

    Article  Google Scholar 

  35. Karami, B. and Karami, S., Buckling Analysis of Nanoplate-Type Temperature-Dependent Heterogeneous Materials, Adv. Nano Res., 2019, vol. 7, no. 1, pp. 51–61.

    Google Scholar 

  36. Barati, M.R. and Shahverdi, H., Finite Element Forced Vibration Analysis of Refined Shear Deformable Nanocomposite Graphene Platelet-Reinforced Beams, J Braz. Soc. Mech. Sci. Eng., 2020, vol. 42, p. 33.

    Article  Google Scholar 

  37. Bekkaye, T.H.L., Fahsi, B., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H., Tounsi, A., and Al-Zahrani, M.M., Porosity-Dependent Mechanical Behaviors of FG Plate Using Refined Trigonometric Shear Deformation Theory, Comp. Concret., 2020, vol. 26, no. 5, pp. 439–450.

    Google Scholar 

  38. Chikr, S.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.R., Benrahou, S.R., and Tounsi, A., A Novel Four-Unknown Integral Model for Buckling Response of FG Sandwich Plates Resting on Elastic Foundations under Various Boundary Conditions Using Galerkin’s Approach, Geomech. Eng., 2020, vol. 21, no. 5, pp. 471–487.

    Google Scholar 

  39. Ebrahimi, F., Barati, M.R., and Civalek, Ö., Application of Chebyshev–Ritz Method for Static Stability and Vibration Analysis of Nonlocal Microstructure-Dependent Nanostructures, Eng. Comput., 2019, vol. 36, pp. 953–964.

    Article  Google Scholar 

  40. Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A., and Tounsi, A., Investigation of Thermal Buckling Properties of Ceramic-Metal FGM Sandwich Plates Using 2D Integral Plate Model, Steel Compos. Struct., 2019, vol. 33, no. 6, pp. 805–822.

    Google Scholar 

  41. Sahla, F., Saidi, H., Draiche, K., Bousahla, A.A., Bourada, F., and Tounsi, A., Free Vibration Analysis of Angle-Ply Laminated Composite and Soft Core Sandwich Plates, Steel Compos. Struct., 2019, vol. 33, no. 5, pp. 663–679.

    Google Scholar 

  42. De Pablo, P.J., Schaap, L.A.T., MacKintosh, F.C., and Schmidt, C.F., Deformation and Collapse of Microtubules on the Nanometer Scale, Phys. Rev. Lett., 2003, vol. 91, p. 098101.

    Article  ADS  Google Scholar 

  43. Reddy, J.N. and Pang, S.D., Nonlocal Continuum Theories of Beams for the Analysis of Carbon Nanotubes, J. Appl. Phys., 2008, vol. 103, p. 023511.

    Article  Google Scholar 

  44. Civalek, Ö., Demir, Ç., and Akgöz, B., Free Vibration and Bending Analyses of Cantilever Microtubules Based on Nonlocal Continuum Model, Math. Comput. Appl., 2010, vol. 15, no. 2, pp. 289–298.

    MathSciNet  Google Scholar 

Download references

Funding

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. J-177-135-1440. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Author information

Authors and Affiliations

  1. Department of Nuclear Engineering, Faculty of Engineering, King Abdulaziz University, 21589, Jeddah, Saudi Arabia

    E. Ghandourah

Authors
  1. E. Ghandourah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Ghandourah.

Additional information

Translated from in Fizicheskaya Mezomekhanika, 2021, Vol. 24, No. 2, pp. 91–98.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghandourah, E. Nonlocal Elasticity Theory for the Mechanical Behavior of Protein Microtubules. Phys Mesomech 24, 319–325 (2021). https://doi.org/10.1134/S1029959921030103

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1029959921030103

Keywords:

Search

Navigation