e θ is the angle in the x-y plane, ε is the dielectric constant of microtubules & R0 is the radius of the microtubule structure. To deduce the numerical piezoelectric characteristics of microtubules, we consider the fastest buckling growth wavenumber
and the time value to calculate the traction force viz. t = 0, 0.5/α, 1.0/α, 1.5/α & 2/α (to be put in Equation (5)) from the proposed model in this work. Time values were chosen as multiples of α to avoid calculation complexity. To get the numerical trends from the proposed model, we chose the following parameters:
w0 = 1.0; R0 = 12.5 ´ 10−9 m; d15 = −0.3 ´ 10−12 C/N; d33 = 0 C/N; ε = 3.0; me = 1 ´ 103 Pa; L = 1 ´ 10−6 m; T (in vivo ΔGGTP-GDP) = 8.68 ´ 10−20 J; EI = 5 ´ 10−24 N∙m2.
Collagen and microtubule-associated protein tau have been demonstrated to share crystalline homology with each other  . Thus we assume that the piezoelectric potential generation due to similar forces would be analogous in the two proteins. Hence, the values of shear piezoelectric coefficient d15 & longitudinal piezoelectric coefficient d33 were assumed to be in close approximation with those of the collagen fibril  . We define r/R0 = γ & to get the following relations of piezoelectric potential distribution from Equation (10).
Based on the potential generated (as in Equation (12)) in the microtubules’ lattice, the energy stored (i.e. harvested) in it could be approximated to the energy stored in the capacitive electrical double layer (EDL) surrounding the microtubules in the ionic cytoplasmic environment. It is given by the following relation (Equation (13)) & holds valid only for values of γ < 1, for it is this range in which the piezoelectric potential generation actually takes place.
Intracellular communication model applicable to microtubules has been described in detail in the next section. Detailed mathematical analysis of the same has been discussed in reference 24.
3. Results and discussion
3.1. Piezoelectric Potential Distribution
Equation (12) shows the distribution of piezoelectric potential inside the microtubule filament as well as outside. The potential distribution follows a cubic trend within the microtubule structure w.r.t the distance from the centre. As we move out of the structure, the potential starts dropping as it is inversely proportional to the distance from the centre-radius ratio. The same can be seen graphically in Figure 3.
The values shown in Figure 3 are significant because we can infer that the potential generated reaches its maximum at the surface of the microtubule filament. This gives inherent electrical stability to the structure due to the potential generated causing minimum damage, if any, at the core of the protein filament. This study sheds light on the quantitative aspect of piezoelectric behaviour of microtubules.
3.2. Energy Harvesting & Intracellular Communication of Microtubules
We also propose a model via which the microtubules could harvest energy and also communicate with neighbouring microtubules in the cytoskeleton. As we know, microtubules tend to display dynamic instability i.e. random lengthening and shortening of plus-end because of either addition or secession of GTP-tubulin. When GTP-tubulin binds to the cylindrical framework, there develops a compressive strain in the structure due to the preferential axial orientation of GTP-tubulin (parallel to the cylindrical axis) and GDP-tubulin (tilted w.r.t. the cylindrical axis)  . This strain build-up causes a voltage bias generation along the axis owing to the piezoelectric behaviour. The number of times this development of strain happens in the microtubule’s cylindrical lattice, it accounts for the number of cycles that would harness the energy in terms of stored charge due to piezoelectricity. This would result in a perpetual device that harnesses energy from its own movement/developed strain. As shown in Figure 4, this strain production is caused by either GTP-tubulin adding to the cylindrical structure or falling off from it. During the process of dynamic instability in microtubules, GTP hydrolysis takes place. The reported value of in vivo free energy of GTP hydrolysis is ~12.5 kcal/mol  i.e. amounting to 131.055 THz electromagnetic radiation. As per the model  , the ZnO nanorods have been shown to behave as nanogenerators of energy which also consume energy due to inter-ZnO-nanorod communication via terahertz (THz) range electromagnetic communication. Earlier in this paper, we have cited the structural homology of ZnO nanorods synthesised chemically with microtubules. Microtubules have also been analysed as piezoelectric filaments which are very densely packed within the cell. Hence, we propose a mechanism in which microtubules behave as energy harvesters which also consume energy due to communication via terahertz (131.055 THz) electromagnetic radiation. Schematic of the proposed model is shown in Figure 4. Mathematical treatise of the model could be found in the seminal work by Jornet and Akyildiz  . Details of the model (within the scope of this work) have been discussed in the next few paragraphs.
Figure 3. Graphs show the gradual increase of piezoelectric potentials w.r.t. the ratio of distance from the centre-radius (γ < 1) and inversely proportional to the ratio (γ > 1). (a) Values of angles are fixed to π/2 and the time parameter was varied to get the different amplitudes of piezoelectric potential generated; (b) At the start of the vibrations along the filament axis due to critical force, the values are plotted for varying angles along the x-y plane as well as on the z-axis.
Figure 4. Schematic proposes the model indicating microtubules harvesting energy from dynamic instability process utilizing their piezoelectric property. While GTP hydrolysis happens during dynamic instability, the in vivo free energy of GTP hydrolysis is proposed to be used for intracellular communication between the microtubules’ network. The energy of such communications is stipulated to be ~131 THz.
Energy harvesting in microtubules could be done when electric potential is generated in its lattice due to buckling (cytoplasmic loading & streaming, during the formation of mitotic spindles). Due to its intracellular ionic environment and prevalent negative charge, microtubules have an EDL formed around it by the positive ions. When the buckling happens, some amount of electric potential is generated in the microtubule lattice owing to its piezoelectric behaviour as discussed in the previous section. Henceforth we propose that this electric potential gets stored as energy in capacitive form in the EDL formed by the positive ions around microtubules.
Intracellular communication of microtubules is proposed to happen via electromagnetic radiation. The frequency of such communication is believed to be 131.055 THz. It represents the exact amount of energy that is generated by the in vivo free energy of GTP hydrolysis during dynamic instability of microtubules. Terahertz band, despite been absorbed by the molecules in the vicinity of microtubules causing a significant distortion (i.e. generating noise), are proposed to support very high bit-rates (in the order of 102 terabits/sec) within the cellular dimensions. This process would enable microtubules to faithfully transfer the information between each other to coordinate their collective behaviour as cytoskeletal elements. Testing of such a communication model is reported in detail in reference 24.
In this paper, we have studied the piezoelectric effect in cytoskeletal microtubules using ZnO nanorods as a model system and for the first time we have generated a quantitative estimate of the buckling that happens due to thermal fluctuations during dynamic instability i.e. GTP hydrolysis. Thermal fluctuations, as it appears from Equation (8), have miniscule effects on the critical force required for buckling. Based on the values of various parameters assumed, the effect comes out to be ~0.066%. Due to paucity of experimental data for microtubules, we have used collagen type-1 fibril, which is homologous to microtubule-associated-protein tau to approximate the shear and longitudinal piezoelectric coefficients.
We also propose a new model that describes how microtubules can function as energy harvesters and communicate with each other inside the cell at the expense of the in vivo free energy of GTP hydrolysis. This model is an inspiration generated from a similar work  on a cluster of ZnO nanorods. Energy harvesting mechanism has been proposed via the energy storage in the capacitive EDL, due to the electrical potential generation by the piezoelectric response of microtubules due to buckling. Intracellular mechanism has been proposed to take place via electromagnetic radiation at a very high bit-rate. Terahertz band communication has been proposed to faithfully transfer the information via the cytosol, despite its absorption by different molecules in the vicinity of microtubules. It can shed some light into the hypothetical mechanisms by which intracellular components interact with each other and into the energy efficiency of living systems.
The author is thankful to CEN (Phase-II) project at IIT Bombay (Project Code: 11DIT005) funded by Department of Information Technology, Ministry of Communication and Information Technology, Government of India. The author acknowledges the kind efforts of Prof. V. Ramgopal Rao, Dept. of Electrical Engineering at IIT Bombay for giving valuable suggestions in writing the manuscript and plotting the data & Dr. Shivani Srivastava, a post-doctoral fellow in Dept. of Mechanical Engineering at IIT Madras, for proofreading the manuscript.
Cite this paper
ArindamKushagra,11, (2015) Thermal Fluctuation Induced Piezoelectric Effect in Cytoskeletal Microtubules: Model for Energy Harvesting and Their Intracellular Communication. Journal of Biomedical Science and Engineering,08,511-519. doi: 10.4236/jbise.2015.88048