Lupine Publishers| Journal of Robotics and Mechanical Engineering
Abstract
Protein microtubules
take part in several cellular activities including mitosis, cell movement and
migration. During these cellular activities, they can be subject to various
types of external loading and pressure. In this study, the bucking of protein microtubules
obtained via scale-dependent continuum models are investigated. Several
continuum-based formulations, which have been proposed for the buckling of
protein microtubules, are reviewed briefly. Finally, the effects of surface
elastic properties on the growth rate of buckling in protein microtubules are
studied.
Keywords: Protein
microtubules; Buckling; Axial loading; Size effects
Introduction
Size effects have a
crucial role to play in the statics and dynamics of various ultra-small
structures [1-6]. On the other hand, the mechanics of nanostructures [7-14] and
microstructures [15-26] is of high importance due to their applications in
different nanomechanical and micromechanical systems such as Nano sensors and
nanoactuators. Therefore, developing sizedependent mathematical frameworks for
analyzing the statics and dynamics of both nanostructures and microstructures
would provide a useful tool in nanoengineering and microengineering. Protein
microtubules are one of the most important parts of living cells, which
participate in many processes inside cells [27,28]. For instance, in the
process of mitosis, microtubules help chromosomes to separate and migrate into
two opposite positions. In addition, these filaments provide a reliable pathway
for protein transportation inside cells. In these processes, microtubules are
likely to be subject to various loads such as axial compression. In this study,
the buckling instability of protein microtubules under axial compressive loads
is investigated. Different size-dependent models of these small-scale
structures are also reviewed.
Buckling of Microtubules
Let us consider a single
microtubule of length L, inner radius Ri and outer radius Ro. The microtubule
has a hollow cylindrical geometry and consists of α and β tubulins, as shown in
(Figure 1). It has been proven that size influences have a significant impact
on the mechanica0000000l behavior at small-scales [29-36]. Since the inner and
outer radii of microtubules are of several nanometers, the nonlocal theory is
mostly used to describe size influences. The nonlocal theory is an
elasticity-based theoretical tool, which was first utilized by Peddieson et al.
[37] for the deformation of nanostructures. According to this theory, we have
the following differential equation for the constitutive response of
microtubules.
Figure 1: The structure of a protein microtubule.
In which σ , C and ε are, respectively, the stress, elasticity and strain tensors; moreover, ∇2 and e0lc stand for the Laplace operator and nonlocal constant, respectively; also, lc and e0 are symbols, which are used for calibrating the model and incorporating the effects of the internal configuration of the structure [38,39]. In addition to nonlocal effects, surface influences have a crucial role to play in the mechanics of ultra small structures such as microtubules. At nanoscales, surface influences become important since the ratio of the surface energy to its bulk counterpart substantially increases. For the microtubule, there are two different surface layers (i.e. outer and inner surface layers). To incorporate surface influences, the following equations can be utilized [40,41].
Here “sur” is employed to indicate “surface”. λsur is the residual
stress in surface layers [42], and ∧ represents the microtubule surface energy
density. Figure 2 depicts the dimensionless growth rate of buckling in protein
microtubules [43] subject to axial compression. Calculations are conducted for
various surface elastic constants [40]. The horizontal axis of the figure
denotes the instability wave number. It is concluded that the growth rate of
buckling in microtubules decreases when the elastic constant of surface layers
increases. This is because of the fact that the surface elastic constant is
associated with an increase in the microtubule stiffness.
Figure 2: Buckling behaviour of microtubules for different surface elastic constants [40].
Conclusion
The buckling instability
of microtubules in human cells has been investigated via scale-dependent
theoretical models. Two main scale-dependent theories for the statics and
dynamics of microtubules (i.e., surface and nonlocal theories of elasticity)
were reviewed briefly. Finally, the influences of buckling wave number and
surface elastic constant on the buckling behaviour were studied. It was
concluded that higher surface elastic constants substantially reduce the growth
rate of buckling in the protein microtubule.
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