Increasing the level of knowledge on the mechanics of biological structures such as protein microtubules is important because of two main reasons. The first reason is that the biological properties of living organisms are highly affected by their mechanical properties such as their elasticity modulus and resonance natural frequencies. Secondly, the mechanical properties and characteristics can be used as a clue in order to analyze the performance and functionality of a living organism. In this paper, the pioneering research studies on the vibrational behavior of protein microtubules as an important part of the cytoplasm are reviewed. Two widely used theoretical approaches, namely the pure nonlocal and nonlocal strain gradient models, are proposed. The natural frequencies of microtubules are strongly influenced by the stress nonlocality and the half wave number along the axial direction.

Keywords: protein microtubules, elasticity-based modelling, vibrational response, nonlocal elasticity, strain gradient

The mechanical behavior of biodegradable nanoscale/microscale materials^1–3 and natural ultrasmall biological structures such as protein microtubules (MTs)⁴ have attracted considerable attention in scientific communities due to their promising functions in microelectromechanical systems^5–8 and living organisms,^9–11 respectively. One of the most important part of a living cell is MTs, which have many crucial roles to play in many cellular processes such as protein transportation and cell division. Analysing the mechanics of these cytoplasm components would help us to better understand many vital functions inside living cells. Various continuum-based models have been developed for the vibration of ultra-small materials in recent years.^12–15 Wang et al.¹⁶ developed a classical shell theory in order to analyse the vibration of MTs; they used an orthotropic model since the shear and circumferential moduli of MTs are much lower than their axial modulus. In another work, Civalek and his co-workers¹⁷ developed a non-classical beam model for the free oscillation of these cytoplasm components via application of the Eringen model of stress nonlocality; they captured size effects in their model since the dimensional vibration characteristics of a structure is a function of its size at microscale levels. Xiang and Liew¹⁸ also proposed an atomistic-continuum model for the vibrational response of MTs incorporating the influence of boundary conditions; a mesh-free scheme is utilized for the vibration analysis as it exhibits intrinsic nonlocal properties. Farajpour et al.¹⁹ developed a surface elasticity-based model to predict the mechanics of MTs in living cells; they concluded that surface effects are of noticeable importance in the accurate modelling of protein MTs. Moreover, Demir & Civalek²⁰ used a nonlocal modelling approach to investigate the deformation of protein MTs subjected to a concentrated force. In another interesting work, the application of a piezoelectric microscale layer in the smart control of MTs has been investigated via a scale-dependent shell model.²¹

More specifically, Beni & Zeverdejani²² developed a scale-dependent shell model for analysing the MT vibrational behavior; they employed a modified model of the couple stress theory as a powerful tool to describe size dependency at microscale levels. In another research paper, Heireche et al.²³ investigated size influences on the vibrational behavior of MTs modeled via a nonlocal Timoshenko theory of beams. Li and his co-workers²⁴ analyzed the three-dimensional vibrational response of these living structures by developing a theoretical model based on the molecular structural mechanics. Moreover, Farajpour & Rastgoo²⁵ have analyzed the influence of added carbon nanotubes on the mechanical instability of a system of multiple MTs using a refined scale-dependent modelling. More recently, Taj et al.²⁶ have developed a shell model incorporating stress nonlocality for the vibration of MTs; they took into account orthotropic elastic properties for the protein MT in the shell model. In addition, a scale-dependent model has been developed in the literature to study the effect of taxol, a well-known stabilizing agent, on the mechanical stability of MTs.²⁷ In this paper, a brief review is performed on the available works on the vibrational response of MTs via utilizing scale-dependent elasticity-based modelling.

Scale-dependent modeling of MTs

In Fig. 1, the structure of a single MT is shown. MTs are rigid cylindrical filaments inside living cells, which are composed of two main tubulin types: 1) alpha type, and 2) beta type. Since the diameter of protein MTs is in the range of several nanometers, the nonlocal elasticity theory.^28–34 can be utilized as an appropriate theoretical tool for the vibrational analysis of these cytoplasm components. According to this size-dependent theory, the MT constitutive equation is mathematically described as:

$\left[1-{\left(e_{0}a\right)}^2{\nabla }^2\right]{\sigma }_{ij}^{nl}=C_{ijkl}{\epsilon }_{kl},$      (1)

where $e_0$  is a constant coefficient used to calibrate the model.^35–37 This coefficient is calculated by comparing the results of the continuum-based simulation with those of experimental measurements or molecular dynamics. a is a characteristics length associated with the internal molecular configuration of the protein MT. For instance, this parameter can be set to an internal bond length. ${\epsilon }_{kl}$ , $C_{ijkl}$  and ${\sigma }_{ij}^{nl}$  denote the MT strain, elastic coefficients and nonlocal stress, respectively. In the above equation, ${\nabla }^2$  is a mathematical operator, which is defined by the following relation for the nonlocal beam-based modelling of protein MTs

${\nabla }^2=\frac{{\partial }^2}{\partial x^2},$     (2)

Here x is the axial coordinate. For MT nonlocal shell models, the mathematical operator ${\nabla }^2$  is also expressed as

${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{1}{R^2}\frac{{\partial }^2}{\partial {\theta }^2},$     (3)

where $\theta$   and R are the circumferential coordinate and the average MT radius, respectively (Figure 1). In addition to the nonlocal elasticity-based modelling, a refined continuum-based modelling has been proposed for the vibration analysis of ultrasmall materials such as graphene sheets, carbon nanotubes, and protein MTs. This refined modelling is established by Lim et al.³⁸ in 2015 via utilising the nonlocal and strain gradient theories. Based on this modelling, the modified constitutive equation of MTs is:

$\left[1-{\left(e_{1}a\right)}^2{\nabla }^2\right]\left[1-{\left(e_{0}a\right)}^2{\nabla }^2\right]{\sigma }_{ij}^t=$

$\left\{1-{\left(e_{1}a\right)}^2{\nabla }^2-{\ell }_s^2\left[1-{\left(e_{0}a\right)}^2{\nabla }^2\right]{\nabla }^2\right\}{C}_{ijkl}{\epsilon }_{kl},$     (4)

Figure 1 A protein MT composed of alpha and beta tubulins.

where $e_0$ , $e_1$  and ${\ell }_s$  represent the lower-order calibration coefficient, higher-order calibration coefficient and strain gradient parameter, respectively. Since this type of continuum modeling incorporates more scale parameters, it can better describe the size dependency at microscale levels. However, the computational cost of the modeling would significantly increase compared to the classical nonlocal modelling of protein MTs. The influence of the nonlocal parameter on the frequencies of a MT controlled by a piezoelectric nanoshell is depicted in Figure 2 for various half wave numbers in the axial direction. For more information about the mechanical properties of this system, an interested reader is referred to Ref.²¹ From Figure 2, it is seen that the stress nonlocality in the constitutive relation of protein MTs leads to a significant reduction in the system frequency. This conclusion is based on the fact that the structure stiffness of MTs has a decreasing relation with the nonlocal parameter. Furthermore, it is noticeable that the influence of stress nonlocality is more significant at higher wave numbers.

Figure 2 Frequency of MTs controlled by a piezoelectric nanoshell for various half wave numbers [21]; reproduced by permission from Elsevier.

In this article, a number of important research works on the vibration of protein MTs have been reviewed and discussed briefly. The main focus was on the scale-dependent elasticity-based modelling of the vibrational behavior of MTs. The proposed constitutive equations of MTs using the pure nonlocal elasticity and nonlocal strain gradient elasticity were elaborated. The influences of stress nonlocality and axial half wave number on the MT vibration were discussed. As the stress nonlocality enhances, the protein MT undergoes lower resonance frequencies. In addition, the stress nonlocality influence is more pronounced when the half wave number along the axial direction of the MT is increased. The stress nonlocality influence is strongly related to a significant decrease in the MT structural stiffness.

None.

The author declares that there is no conflict of interest.

1. Rohani Rad E, Vahabi H, Formela K, et al. Injectable poloxamer/graphene oxide hydrogels with well‐controlled mechanical and rheological properties. Polymers for Advanced Technologies. 2019;30:2250–2260.
2. Vahabi H, Rad ER, Parpaite T, et al. Biodegradable polyester thin films and coatings in the line of fire: the time of polyhydroxyalkanoate (PHA)? Progress in Organic Coatings. 2019;133:85–89.
3. Rad ER, Vahabi H, de Anda AR, et al. Bio-epoxy resins with inherent flame retardancy. Progress in Organic Coatings. 2019;135:608–612.
4. Mandelkow E, Mandelkow EM. Microtubules and microtubule-associated proteins. Current opinion in cell biology. 1995;7:72–81.
5. Chen C, Lee S, Deshpande VV, et al. Graphene mechanical oscillators with tunable frequency. Nature nanotechnology. 2013;8:923–927.
6. Salmanzadeh A, Romero L, Shafiee H, et al. Isolation of prostate tumor initiating cells (TICs) through their dielectrophoretic signature. Lab on a Chip. 2012;12:182–189.
7. Rohani Rad E, Farajpour MR. Dynamics analysis of microparticles in inertial microfluidics for biomedical applications. Journal of Computational Applied Mechanics. 2019;50:157–164.
8. Kaliyannan G, Rathanasamy R, Palaniappan S. A review on the effect of carbon based nanofillers on the properties of elastomers. Material Sci & Eng. 2019;3:89–101.
9. Goodson HV, Jonasson EM. Microtubules and microtubule-associated proteins. Cold Spring Harbor Perspectives in Biology. 2018;10:a022608.
10. Villegas MR, Baeza A, Vallet Regí M. Proteolytic Enzymes and Cavitation as Strategies to Enhanced Penetration of Drug Nanocarriers. Material Science & Engineering International Journal. 2018;22–24.
11. Silver F. A matter of gravity-mechanotransduction: how mechanical force s influence biological materials. Material Sci & Eng Int J. 2017;1:00012.
12. Farajpour M, Shahidi A, Farajpour A. Elastic waves in fluid-conveying carbon nanotubes under magneto-hygro-mechanical loads via a two-phase local/nonlocal mixture model. Materials Research Express. 2019;6:0850a8.
13. Ma H, Gao XL, Reddy J. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids. 2008;56:3379–91.
14. Farajpour MR, Shahidi AR, Farajpour A. Frequency behavior of ultrasmall sensors using vibrating SMA nanowire-reinforced sheets under a non-uniform biaxial preload. Materials Research Express. 2019;6:065047.
15. Farajpour M, Shahidi A, Farajpour A. Influences of non-uniform initial stresses on vibration of small-scale sheets reinforced by shape memory alloy nanofibers. The European Physical Journal Plus. 2019;134:218.
16. Wang C, Ru C, Mioduchowski A. Vibration of microtubules as orthotropic elastic shells. Physica E: Low-dimensional Systems and Nanostructures. 2006;35:48–56.
17. Civalek Ö, Demir Ç, Akgöz B. Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Mathematical and Computational Applications. 2010;15:289–298.
18. Xiang P, Liew KM. Free vibration analysis of microtubules based on an atomistic-continuum model. Journal of Sound and Vibration. 2012;331:213–30.
19. Farajpour A, Rastgoo A, Mohammadi M. Surface effects on the mechanical characteristics of microtubule networks in living cells. Mechanics Research Communications. 2014;57:18–26.
20. Demir Ç, Civalek Ö. Nonlocal deflection of microtubules under point load. International Journal of Engineering and Applied Sciences. 2015;7:33–39.
21. Farajpour A, Rastgoo A, Mohammadi M. Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment. Physica B: Condensed Matter. 2017;509:100–114.
22. Beni YT, Zeverdejani MK. Free vibration of microtubules as elastic shell model based on modified couple stress theory. Journal of Mechanics in Medicine and Biology. 2015;15:1550037.
23. Heireche H, Tounsi A, Benhassaini H, et al. Nonlocal elasticity effect on vibration characteristics of protein microtubules. Physica E: Low-dimensional Systems and Nanostructures. 2010;42:2375–2379.
24. Li S, Wang C, Nithiarasu P. Three-dimensional transverse vibration of microtubules. Journal of Applied Physics. 2017;121:234301.
25. Farajpour A, Rastgoo A. Influence of carbon nanotubes on the buckling of microtubule bundles in viscoelastic cytoplasm using nonlocal strain gradient theory. Results in Physics. 2017;7:1367–1375.
26. Taj M, Majeed A, Hussain M, et al. Non-local orthotropic elastic shell model for vibration analysis of protein microtubules. Computers and Concrete. 2020;25:245–253.
27. Rohani Rad E, Farajpour MR. Influence of taxol and CNTs on the stability analysis of protein microtubules. Journal of Computational Applied Mechanics. 2019;50:140–147.
28. Eringen AC. Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science. 1972;10:425–435.
29. Reddy J. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science. 2007;45:288–307.
30. Malekzadeh P, Farajpour A. Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium. Acta Mechanica. 2012;223:2311–2330.
31. Mohammadi M, Moradi A, Ghayour M, et al. Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Latin American Journal of Solids and Structures. 2014;11:437–1458.
32. Asemi SR, Farajpour A, Borghei M, Hassani AH. Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics. Latin American Journal of Solids and Structures. 2014;11:704-724.
33. Farajpour M, Shahidi A, Farajpour A. Influence of shear preload on wave propagation in small-scale plates with nanofibers. Structural Engineering and Mechanics. 2019;70:407–420.
34. Farajpour MR, Shahidi A, Farajpour A. Resonant frequency tuning of nanobeams by piezoelectric nanowires under thermo-electro-magnetic field: a theoretical study. Micro & Nano Letters. 2018;13:1627–1632.
35. Mohammadi M, Farajpour A, Goodarzi M, et al. Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium. Latin American Journal of Solids and Structures. 2014;11:659–682.
36. Mohammadi M, Farajpour A, Goodarzi M, et al. Levy type solution for nonlocal thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium. Journal of Solid Mechanics. 2013;5:116–132.
37. Asemi SR, Mohammadi M, Farajpour A. A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory. Latin American Journal of Solids and Structures. 2014;11:1515–1540.
38. Lim C, Zhang G, Reddy J. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids. 2015;78:298–313.