Mechanical Behavior and Physical Properties of Protein Microtubules in Living Cells Using the Nonlocal Beam Theory

A biomechanical model for vibrational analysis with characteristics of protein microtubules based on the nanobeam shape inside the cellular structure is presented. Young’s modulus of protein microtubules and unexplained length-dependent flexural rigidity are studied using a higher-order nonlocal shear deformation theory. The governing equations are provided by employing the principle of virtual work. The protein microtubules are considered simply supported for all numerical studies. The obtained results are critically discussed together with the theories as well as demonstrated in each case graphically.


INTRODUCTION
Experimental investigations are performed in laboratories to study physical and mechanical properties, elastic characteristics and mechanical behavior of protein microtubules (MTs) under external force. Diminishing the structural properties to the microscale leads to different mechanical response. Indeed, some experimental works showed that the scale effect plays a considerable role in the mechanical behavior of small-scale structures. Conventional elasticity theories contain no intrinsic size parameters and do not permit predicting size-dependent responses of nano-and microscale structures. Consequently, various nonclassical elasticity models were introduced to examine the mechanical behavior of structures on a reduced scale, such as the model of couple stresses.
The smooth muscle cell migration plays a vital role in forming tubes of hollow organs, like blood vessels and the developing airways. The motility of smooth muscle cells was also concerned in airway pathogenesis remodeling, an essential kind of asthma. Additionally, hypertrophy and hyperplasia, and airway smooth muscle cell migration donate the expansion of the airway renovation. Thickening of smooth muscles in the airways can stem from proliferation of the cell migration in muscle bundles or circulation of the recruitment of precursor cells in the flat muscle layer [1][2][3]. Microtubules (MTs), microfilaments, and intermediate filaments are three dissimilar kinds of filaments organized in the network of eukaryotic cells. Each of the filaments has particular physical properties and appropriate structures based on the cell function. MTs contain compressive forces in living cells more than other filaments, which leads to traction forces inside the cytoskeleton of eukaryotic cells [4]. Three types of living cells are MTs, intermediate filaments, and actin [5]. The two latter can tolerate only traction due to their minor cross sections, but MTs are generally under compression and behave like a rigid bar [6,7]. MTs provide support to the eukaryotic cell to preserve its shape [8][9][10]. MTs have essential roles in eukaryotic cell motility, meiosis, growth, and mitosis. MTs have highly dynamic structures based on rapid polymerization/ depolymerization and act within eukaryotic cells for motor proteins to transmit cargoes across the cytoplasm [11]. MTs have outer diameters 25 nm, inner diameters 17 nm, and length 10 nm-100 mm, as illustrated in Fig. 1. In the cytoskeleton, MTs do not need to exist directly; however, they keep the magnitude less than its perseverance length that can be 0.2 to 9 mm [11]. The perseverance length of protein MT filaments is more significant than their length inside eukaryotic cells.
The MT geometry is labeled by a pair of integers named N-S, which are protofilaments and start plexus numbers. MTs 13-3 are a common type of MTs composed in vivo, as protofilaments are taken as parallel to the longitudinal axis, i.e. the deviation angle tends to zero [12][13][14]. Many exceptions in various kinds and models of cells were observed and configured for typical MTs 13-3.
Recently, higher kinds of theories based on shear deformation have been used to investigate the mechanical performance of structures on different scales [15][16][17][18][19][20][21][22]. These theories can be applicable to the biomechanical response of MTs. It should be noted that nonlocal elasticity theory and strain gradient theory were also used [23][24][25].
This work presents a biomechanical analysis of MTs using the nonlocal higher-order shear deforma-tion model of beams. We deal with vibrational characteristics of MTs in the surrounding cytoplasm using a nonlocal continuum model. The higher-order shear deformation theory of beams is very rarely implemented and is still in an early stage. We compare between the previous and present nonlocal theories of simply supported thick MTs.

MATERIAL AND METHODS
In this section, the exact parameters of scaling effects are implemented [26]. The influence of the small scale on the vibrational behavior of MTs is also presented. The nonlocal Reddy beam system is investigated using vibrational features of MTs in the cytoplasm [26][27][28][29][30][31][32][33]. The fundamental relations for nonlocal protein MTs can be expressed in a one-dimensional form as [3,4] The strain effects are ignored at any point other than x, and the local/classical elasticity system is achieved by using e 0 = 0. The factors E and G indicate Young's modulus and shear modulus of MTs, respectively, γ shows the shear strain, σ(x) and τ(x) are the normal stress and the transverse shear stress, respectively. Hence, the scale coefficient e 0 a leads to a small-scale effect on the response of the nanosized structure.
Here τ = e 0 a/l is the material constant; a depicts an internal typical length, i.e. lattice parameter, length of the C-C bond, and granular distance. The term l is an individual external length, i.e. crack or wavelength. It is observed that the e 0 values need to be determined through experiments or overlapping dispersion plane wave curves with the dynamics of the atomic lattice.
To investigate small-scale effects on the dynamic characteristics of protein MTs in the surrounding viscoelastic cytoplasm, the nonlocal Reddy beam model is proposed, and the following governing equations for protein MTs based on constitutive relations (1) and (2) can be obtained as [13] 11 , 11 , , where u is the middle displacement surface component within MTs in the x direction, w is the middle surface displacement in the z directions, and u 1 is the cross-section rotation.

SOLUTION OF A VIBRATION PROBLEM IN PROTEIN MICROTUBULES
In the present research work, isolated protein MTs of length L subject to boundary conditions (BCs) are considered as 0 at 0 and , The following analytical solutions satisfy the BCs where K is the rigidity, Δ is the column vector of unknown coefficients, M is the inertia-based vibration of protein MTs, and λ is obtained by taking a zero determinant.

RESULTS AND DISCUSSION
This section provides details of the obtained results using vibrational features of MTs and the parametric study. The small-scale length impact, as well as the dependence of the shear modulus ratio on frequency, is investigated based on the above equations along with numerical measures. We consider a simply supported MT with Young's modulus E = 1 GPa, mass density ρ = 1.47 g/cm 3 , small radius r = 12.5 nm, Poisson's ratio ν = 0.3, and the shear modulus ratio β = G/E around 0.000004 and 0.001 [10,34,35]. In the literature, protein MTs with a circular cross section will be handled to the shape of the corresponding circular annular keeping the comparable thickness h = 2.7 nm [26,36,37]. To investigate MTs based on Young's modulus, flexural rigidity and to verify the presented reports, the obtained outcomes of the third-order shear deformation theory (Reddy beam theory (RBT)) are compared to those of the Timoshenko beam theory (TBT) and the experimental data [38] and illustrated in Fig. 2. The Reddy beam model is applied to examine dynamic features of protein MTs. Numerous experimental schemes were applied to the classical beam system to find the flexural rigidity of protein MTs. Hence the efficiency of our investigations using Reddy systems in respect to length-dependent flexural rigidity of protein MTs can be scrutinized in comparison with the experimental statistics [33,38]. The flexural rigidity of an isolated MT assumed for both systems (Reddy and Timoshenko ones, I = e 0 a = 0) is illustrated in Fig. 2 without viscoelastic effects. With reference to Fig. 2, the length-flexural rigidity dependence predicted by the Reddy nonlocal beam theory is in good agreement with the experimental statistics.  Comparison between the local and nonlocal outcomes reveals the impact of the small-scale factor I = α = e 0 a/L on free vibrations of MTs using the surrounding cytoplasm. It is assumed that E c = 1 KPa, L/r 0 = 40, and μ = 695 × 10 -6 Pa s [39]. The frequency of protein MTs from the local/nonlocal Reddy beam systems at different nonlocal parameters is illustrated in Fig. 3. The plotted outcomes for n = 1 and l = 0, 4, 7 nm with β = G/E is 0.00001. The diagrams show that the frequency enhances with increasing scale parameter. This indicates that the dynamic values (frequency) of protein MTs from the classical or local Reddy beam system are lower than the values predicted by the Reddy nonlocal beam system. It is also observed that the frequency is inversely related to the length of protein MTs. It can be confirmed that the impact of shear deformation becomes more significant because of the composite structure and anisotropic molecular architecture of MTs, particularly at smaller geometric ratios. The present approach requires further investigations into other shear deformation models and other types of materials [40][41][42][43][44][45]. Figure 4 illustrates the effects of the shear modulus ratio β on the frequency of protein MTs with the mode number m = 3 and I = 2. It can be observed from the diagram that the frequency has a direct correlation relation with the shear modulus ratio β.

CONCLUSIONS
The present work was related to the study of small-scale effects based on the dynamic analysis of protein MTs using the higher-order theory and nonlocal elasticity (RBT and TBT). The detail of MTs was provided along with their diameter and size. The general analytical form of vibrational investigations of protein MTs embedded into the neighboring cyto- plasm was expressed. Dynamic properties of protein MTs were examined and discussed in detail. The obtained outcomes were found to be different from the predicted results using classical elasticity TBT and RBT with nonlocal effects. Moreover, the length-flexural rigidity dependence for isolated protein MTs is compared between the local (TBT and RBT) and nonlocal theories (RBT). The frequency of protein MTs versus the length/size effect parameter corresponds to the Reddy beam model at n = 1 and l = 0, 4, 7 nm. The influence of the shear modulus ratio on the frequency for protein MTs at I = 2 is also provided in this study.