Nanotubes between cells

Apart from long-distance neuro-hormonal signaling, direct cell-to-cell communication is of crucial importance in multicellular systems. Direct communication is required to provide simultaneity and continuity of fundamental biological processes while neuro-hormonal signaling provides a higher-level central control. Very likely the most significant intercellular connecting structures in mammals are tunneling nanotubes (TNTs) as described by Rustom et al. and Scholkmann [Reference Rustom, Saffrich, Markovic, Walther and Gerdes15, Reference Scholkmann16]. A schematic layout of a TNT is shown in Fig. 1. TNTs are also called membrane nanotubes by Zhang and Zhang [Reference Zhang and Zhang17] or intercellular nanotubes by Hurtig et al. [Reference Hurtig, Chiu and Onfelt18]. The diameters of TNTs range from 50 to 200 nm and their lengths are in most cases up to several cell diameters. As an extreme case, the primary cell cultures prepared from human laryngeal squamous cell carcinoma samples exhibited cell-to-cell communication over a distance up to 1 mm through TNTs as described by Antanavičiūtė et al. [Reference Antanavičiūtė, Rysevaite, Liutkevicius, Marandykina, Rimkutė, Sveikatienė, Uloza and Skeberdis19]. The TNTs have been observed in many cell types in vitro and also in developing embryos of different species in vivo by Gerdes et al. [Reference Gerdes, Rustom and Wang20]. Cells can actively exchange small membrane carriers through the channels which provides evidence for a new mechanism of cell-to-cell communication as was observed by Rustom et al. [Reference Rustom, Saffrich, Markovic, Walther and Gerdes15]. TNTs are ultrafine membranous channels containing F-actin and microtubules, formed between cells that were previously in contact or developed from extending protrusion of one of the cells [Reference Zhang and Zhang17]. TNTs transport different material objects such as endocytic vesicles, mitochondria, plasma membrane proteins, cytoplasmic molecules, etc. An overview of the cell-to-cell communication between cells is provided by Gerdes and Pepperkok [Reference Gerdes and Pepperkok21] and Bloemendal and Kück [Reference Bloemendal and Kück22].

Fig. 1.

A schematic picture of cells connected by TNTs [Reference Pokorný, Pokorný and Vrba14].

Physiological and pathological implications of TNT formation between cells are analyzed, and the function of these communication bridges is described by Sisakhtnezhad and Khosravi and Lu et al. [Reference Sisakhtnezhad and Khosravi23, Reference Lu, Zheng, Li, Yu, Chen, Liu, Wang, Xu and Yang24]. TNTs promote intercellular microbial transport to T24 bladder cancer cells followed by an increase of their invasiveness as described by Lu et al. [Reference Lu, Zheng, Li, Yu, Chen, Liu, Wang, Xu and Yang24]. Measurement of electrical activity across a cell is described by Bathany et al. [Reference Bathany, Beahm, Besch, Sachs and Hua25]. Mitochondrial transfer from mesenchymal stromal cells (MSC) to innate immune cells leads to enhancement in phagocytic activity (which is an antimicrobial effect of MSC) [Reference Jackson, Morrison, Doherty, McAuley, Matthay, Kissenpfennig, O'Kane and Krasnodembskaya26]. TNT cell-to-cell transport of proteins and organelles such as mitochondria affects metabolism. Mesenchymal stem cells recruited to the tumor environment and their important role in tumor progression and resistance to therapy are described by Vignais et al. [Reference Vignais, Caicedo, Brondello and Jorgensen27].

Evaluation of the function of TNTs disclosed the possibility of neuro-hormonal independent, non-diffusible cell-to-cell signaling as was observed by Chaban et al. [Reference Chaban, Cho, Reid and Norris28]. Physical modes of communication could be widespread in nature, i.e. communication by sound waves, electromagnetic radiation, and electric current. These signals propagate rapidly and can be efficient even at a low intensity which was assessed by Reguera [Reference Reguera29].

TNTs are expected to have important significance for intercellular communication, namely a transfer of electrical signals between remote cells which was described by Wang and Gerdes [Reference Wang and Gerdes30]. The electric coupling depends also on the connections interposed at the membrane interface between TNT and the connected cell. A hypothesis that two modes of energy and signal transmission may occur, i.e. electrical/electrochemical and electromagnetic, was published by Scholkmann [Reference Scholkmann31]. TNTs mediate a bidirectional spread of electrical signals between TNT-connected normal rat kidney cells over distances of 10–70 μm [Reference Wang, Veruki, Bukoreshtliev, Hartveit and Gerdes32]. Long-range cell-to-cell communication by electric signals can be facilitated by TNTs. The significance of electrical cell-to-cell coupling through TNT for neuronal communication has been observed by Scholkmann [Reference Scholkmann31]. His short overview indicates that TNTs are important communication channels for the transfer of chemical, electrical, and electromagnetic signals.

It is reasonable to assume that connections between cells can be mediated by electric current if the electric conductivity of TNTs is increased, by dipole and multipole oscillations of the TNT membrane, and by electromagnetic excitation of the cavities of cells and TNTs by the field generated by microtubules and connected structures. Coherent excitation of the cavities seems to be important for biological functions. TNTs and cell cavities may form a united cavity system capable to provide a sufficient level of electromagnetic activity for all cells, even for the “weak” or “injured” ones. The cavity system thus should stabilize the function of the whole tissue. Our analysis of the system includes a theory of cavity waveguides and resonators and transfer of electromagnetic energy between different frequency modes. The microtubules are oscillators in a wide frequency spectrum from classical frequency bands up to UV spectrum which follows from the measurements of Sahu et al. [Reference Sahu, Ghosh, Ghosh, Aswani, Hirata, Fujita and Bandyopadhyay9, Reference Sahu, Ghosh, Fujita and Bandyopadhyay10]. Scholkmann [Reference Scholkmann16] emphasizes a large amount of mitochondria inside TNTs and suggests their role as the sources of energy for electric currents and electromagnetic fields. TNTs contain cytoskeleton components; most TNTs contain F-actin, some TNTs also contain microtubules. The most widely expressed gap-junction protein, connexin 43, is also present in some TNTs [Reference Zhang and Zhang17].

Electromagnetic field in living systems

Electric polarity depends on charge distribution in the macromolecular structures. It is well known that a peptide bond forms a large electric dipole moment whose inner electric field is oriented from nitrogen to oxygen explained by Pauling resonance between two hybrid electronic structures [Reference Schulz and Schirmer33]. Other parts of the polypeptide structure are electrically polar too. One type of polarity of amino acids is formed by ionized side chains (positively and negatively charged), another type by partial atomic charges dependent on the electronegativity of the atom. Polypeptide chains and other chain compounds can contain short regions behaving like solid-state periodic systems with electrons moving in the “conduction” band or strictly bound in the “valence” band. The electrons in the “conduction” band can oscillate at high frequencies in the UV region and absorb or release photons.

An important protein structure in eukaryotic cells is formed by microtubules built from heterodimers consisting of two monomers called α and β tubulin. Each heterodimer represents an electric dipole originating from 18 calcium ions bound approximately between α and β monomers as described by Satarić et al. and Tuszyński et al. [Reference Satarić, Tuszyński and Žakula34, Reference Tuszyński, Hameroff, Satarić, Trpisova and Nip35]. An equal number of negative charges (electrons) is localized in the neighboring α monomer. After polymerization, the main component of the electric dipole vector is oriented along the axis of microtubule (dipole electric polarity corresponds to the direction fast–slow growing end). Hydrolysis of GTP to GDP in the β tubulin changes its conformation and position of the β tubulin is tilted (about 28°) with respect to α tubulin [Reference Satarić, Tuszyński and Žakula34–Reference Melki, Carlier, Pantaloni and Timasheff37]. After hydrolysis of ATP in the β tubulin, the main component of the dipole moment has a reversed orientation.

The described dipole orientation corresponds to microtubules in cells. Heterodimers have the largest component of the dipole moment in the normal direction to their axes (1669 Debye, i.e. 5.567 × 10⁻²⁷ Cm). The C-termini extended outwardly from the microtubule carry a significant amount of negative electric charge (as much as 40%) of the total monomer amount. Oscillating high-frequency electric field aligns the microtubules parallel to the field direction [Reference Böhm, Mavromatos, Michette, Stracke and Unger38]. The normal components of the heterodimer dipole moment may be compensated along the spiral turn of a microtubule and additon of heterodimer axial components forms a dipole moment along the microtubule axix (ferroelectric state) [Reference Sataric and Tuszynski39] or the electron charge forming the electric dipole may change their position in the microtubule (a redistribution of electrons may enhance or reduce the transverse component of the dipole moment in heterodimers). The kink excitation and propagation along the microtubule axis [Reference Sataric and Tuszynski39], and electromagnetic excitation of the inner microtubule cavity in the UV region may be important factors for the dipole orientation along the microtubule axis (after the release of water from the inner cavity, microtubule oscillations disappear [Reference Sahu, Ghosh, Ghosh, Aswani, Hirata, Fujita and Bandyopadhyay9]).

The energy state far from thermodynamic equilibrium is excited and maintained by energy supply. Important life processes and biological functions can be provided by a coherent electromagnetic field as suggested by Pokorný et al. [Reference Pokorný, Pokorný and Kobilková12, Reference Pokorný, Pokorný, Foletti, Kobilková, Vrba and Vrba13, Reference Pokorný, Pokorný, Kobilková, Jandová, Vrba and Vrba40]. A considerable part of the power supply to a cell (of the order of magnitude of 0.1 pW, as estimated from the metabolic turnover of a body and a number of cells) should excite coherent vibrations and coherent electromagnetic field. A microtubule is excited by about 0.1 fW and this power is distributed to the dipoles in all heterodimers which generate the electromagnetic field in the frequency spectrum up to and in the UV region. Some spectral lines can be selectively excited as follows from the excitation–emission measurements. Figure 2 displays resonant frequencies of microtubules measured in classical, far infrared, and UV regions by Sahu et al. [Reference Sahu, Ghosh, Ghosh, Aswani, Hirata, Fujita and Bandyopadhyay9, Reference Sahu, Ghosh, Fujita and Bandyopadhyay10]. Power of the energy supply to a heterodimer dipole is of the order of magnitude of zW. Microtubule vibrational peaks condensed in a single mode and noise alleviation disappear when water is released from the inner microtubule cavity which was proved by Sahu et al. [Reference Sahu, Ghosh, Ghosh, Aswani, Hirata, Fujita and Bandyopadhyay9]. The UV spectrum has a specific property – excitation and emission lines exhibit different wavelengths: the excitation line 274 nm corresponds to emission lines 330 and 671 nm, and the excitation line 561 nm corresponds to the emission line 330 nm (the first excitation and emission lines are plotted in Fig. 2).

Fig. 2.

Spectrum of microtubule resonant frequencies in the classical range below 20 THz, in the far infrared, and UV regions. The dashed line in the classical frequency range is evaluated from the Fröhlich rate equation. The dotted line denotes estimated excitation power. Significant Raman spectral lines are at about 526 and 686 cm⁻¹. Excitation and emission spectral lines in the UV range are 274 and 330 nm, respectively; the other spectral lines exhibit wavelengths: excitation 274, 561 nm and emission lines 671, 330 nm (not plotted) – measured by Sahu et al. [Reference Sahu, Ghosh, Ghosh, Aswani, Hirata, Fujita and Bandyopadhyay9, Reference Sahu, Ghosh, Fujita and Bandyopadhyay10].

The electromagnetic spectrum in the UV region may have components in the short-wavelength UV part. Gurwitsch and Frank and Gurwitsch observed a weak radiation from onion roots in the UV region 193–237 nm and assumed their mitogenic function [Reference Gurwitsch41, Reference Frank and Gurwitsch42]. Microtubule inner circular cavity with a diameter of 17 nm has properties of a resonator and/or of a waveguide which was analyzed by Jelínek and Pokorný [Reference Jelínek and Pokorný43]. Calculated cutoff frequencies of the transverse magnetic (TM) and transverse electric (TE) modes in a circular cavity with reflecting walls are presented under ‘Microtubules and nanotube energy transfer’. The calculated cutoff frequencies of TNTs are lower but also in the short-wavelength part of the UV region.

Periodic structures

Frequency of resonance and propagation of the electromagnetic field in the cavities depend on geometrical dimensions of the cavity and permittivity and permeability of the medium inside it. Under the cutoff frequency, there is a region of evanescent modes. Transfer of electromagnetic signals in this frequency range is governed by structures inside the cavity. For example, if the cavity walls are not physically smooth, an effective increase of the shunt capacitance per unit length can be achieved without the decrease of a corresponding series inductance. The phase velocity of the propagating field is decreased compared to a plain waveguide. The lumped capacitance at periodic distances smaller than the wavelength of the electromagnetic field is used. An example of such a periodic structure shown in Fig. 3 is described by Collin in [Reference Collin44]. If the dimension a is large compared to b and this in turn is small compared to the wavelength of the electromagnetic field, the transfer is optimal.

Fig. 3.

Corrugated plane periodic structure with thick teeth [Reference Collin44]: a – tooth length, b – spacing between waveguide walls, d – teeth periodicity, s – teeth gap.

The electromagnetic field can propagate in a limited space region to prevent damping at the cavity walls and be transferred along a guiding way. Propagation of the electromagnetic signals can also occur along a helix (or a sheath helix) which is another method of realization of a periodic structure. A conductive tape with helical structure is separated by a thin insulated slit. Conductivity along the tape is infinite (and, consequently, the current too) and zero in the perpendicular direction. However, all methods have passband–stopband characteristics. There are frequency bands throughout which a wave propagates unattenuated and bands through which the wave does not propagate [Reference Collin44].

Microtubules are helical structures while in TNTs the periodic structure may be formed by extensions of the membrane into the inner TNT cavity, by mitochondria which can be periodically arranged along the TNT and similarly by lumped macromolecules and structures such as microtubules. The helical periodicity of microtubules is convenient for the generation of the electromagnetic field at distinct frequencies. Mitochondria form a layer of hydrogen ions in the cytosol space with an attached layer of ordered water [Reference Pokorný, Pokorný and Borodavka45]. The induced changes of series capacitance C per unit length (as well as inductance L per unit length) result in changes of the energy storage and transfer, i.e. changes of phase velocity of the propagation of the electromagnetic field (the phase velocity is given by the relation v_f = (LC)^−1/2 = (μμ ₀ɛɛ ₀)^−1/2 where μ and ɛ are the relative permeability and permittivity, respectively). The wavelengths in the UV frequency range may be comparable or even shorter than dimensions of the reflecting lumps, and the propagation of the electromagnetic wave may be described by short time and space periods. The narrow ordered water layer at the upper surface of mitochondria proton layer forms a channel for the transfer of electromagnetic signals. Ordered water forms a dielectric or a conductive material (by releasing electrons) for different pH values [Reference Zheng and Pollack46]. The phase velocity depends on permittivity and permeability of the ordered water which seem to be different from those of the normal (bulk) water. The layers of ordered water in the TNT may form convenient conditions for the propagation of the electromagnetic field.

Localization of mitochondria inside a TNT indicates that their function is connected with energy supply [Reference Scholkmann16]. The additional energy supply may amplify electric or electromagnetic signals entering the TNT. Two possible amplification concepts are introduced: parametric amplification in a wave that has multiple frequencies, and Fröhlich's approach involving interaction with a heat bath.

Parametric amplification

Parametric amplifiers utilize non-linear properties of the medium or properties that can be varied as a function of time. In the technological region of electrical engineering, it concerns a non-linear reactance or a reactance oscillating as a function of time after the application of a suitable pump signal. Original description of Manley and Rowe [Reference Manley and Rowe47] is further analyzed in [Reference Collin44] and [Reference Pokorný and Wu48]. Parametric amplification is mainly based on the time variation of a reactive parameter. The most notable variable reactance was produced by a semiconductor diode called a varactor. The analysis of these devices is based on the Manley–Rowe power relations

(1)$$\sum\limits_n {\sum\limits_m {\displaystyle{{nP_{n\comma m}} \over {n\omega _1 + m\omega _2}}} } = 0\comma \;\sum\limits_m {\sum\limits_n {\displaystyle{{mP_{n\comma m}} \over {n\omega _1 + m\omega _2}}} } = 0\comma \;$$

where n and m are the integers; in the first equation, n and m (in the second equation m and n) vary from 0 to +∞ and from −∞ to +∞, respectively. There are two important applications called negative resistance parametric amplifier and parametric up-converter. The negative-resistance systems use the pump frequency ɷ_P, the idler frequency ɷ_I = ɷ_P−ɷ_S and the input and output signal frequencies ɷ_S are equal. For the signal frequency ɷ_S, the indices are m = 1 and n = 0, for the pump frequency ɷ_P, they are m = 0 and n = 1, for the idler frequency ɷ_I, they are m = − 1, n = 1, and the corresponding powers are P_S = P _1,0, P_P = P _0,1, P_I = P _–1,1. (Power supplied to the non-linear element is positive, the delivered power is negative.) From the Manley–Rowe relations we get

(2)$$\displaystyle{{P_{1\comma 0}} \over {\omega _S}}- \displaystyle{{P_{- 1\comma 1}} \over {- \omega _S + \omega _P}} = 0\comma \;\displaystyle{{P_{0\comma 1}} \over {\omega _P}} + \displaystyle{{P_{- 1\comma 1}} \over {- \omega _S + \omega _P}} = 0$$

and a maximum signal gain

(3)$$\displaystyle{{- P_{0\comma 1}} \over {P_{1\comma 0}}} = \displaystyle{{\omega _P} \over {\omega _S}}.$$

The up-converter uses the input signal at the frequency ɷ_S (m = 1, n = 0), pump signal at the frequency ɷ_P (m = 0, n = 1), and the amplified and up-converted output signal exhibits the frequency ɷ_S + ɷ_P (m = 1, n = 1). For the up-converter Ps = P _1,0, P_P = P _0,1, and the up-converted power is P _1,1. From the Manley–Rowe relations we get

(4)$$\displaystyle{{P_{1\comma 0}} \over {\omega _S}} + \displaystyle{{P_{1\comma 1}} \over {\omega _S + \omega _P}} = 0\comma \;\displaystyle{{P_{0\comma 1}} \over {\omega _P}} + \displaystyle{{P_{1\comma 1}} \over {\omega _S + \omega _P}} = 0$$

and a maximum signal gain

(5)$$\displaystyle{{P_{0\comma 1}} \over {P_{1\comma 0}}} = \displaystyle{{\omega _P} \over {\omega _S}}.$$

The maximum signal gain depends on frequency relations of the input and pump signals. Parametric amplifiers can be important for the amplification of electromagnetic signals propagating in the TNT as in a cavity waveguide. The negative-resistance parametric amplifier could mediate communication between cells with similar electromagnetic activities. Both connected cells would then generate signals with the same frequencies. The up-converter system could connect different frequency components of the generated electromagnetic field.

Fröhlich relation

Fröhlich analyzed basic properties of a biological energy system excited by its energy supply and proposed spectral energy transfer based on emission–absorption interaction with a heat bath [Reference Fröhlich1–Reference Fröhlich, Marton and Marton5]. He assumed energy supply to electric polar vibrations resulting in coherence and a strong excitation of one or a few modes of motion. The vibration system is stabilized due to low emission and friction losses, phases correlated over macroscopic regions, and superimposed on random thermal fluctuations. He assumed oscillating properties of the heat bath in biological systems formed by macromolecules and macromolecular structures which contain an extremely wide quasi-continuous spectrum of random electric polar vibrations. Fröhlich analyzed extraordinary dielectric properties, derived relation for the metastable ferroelectric state of biological systems [Reference Fröhlich4], and suggested that cancer cells have shifted frequencies of their electromagnetic activity disturbing interactions with healthy cells [Reference Fröhlich49]. Fröhlich formulated the main properties of the cellular electromagnetic activity and the rate equation in the form

(6)$$\eqalign{{\dot{n}}_i& = s_i- \phi _i [n_i\exp \lpar \beta \nu _i\rpar - \lpar {n_i + 1} \rpar ] \cr & - \sum\limits_j {\chi _{ij} [n_i\lpar n_j + 1\rpar \exp \lpar \beta \nu _i\rpar } - \lpar {n_i + 1} \rpar n_j\exp \lpar \beta \nu _j\rpar ] \comma \;} $$

where n_i and n_j are the numbers of energy quanta in the modes with frequency ν_i and ν_j, respectively, ṅ_i is the first derivative with respect to time, ϕ_i and χ_ij are the coefficients of energy transfer, and s_i is the energy supply (in the number of energy quanta).

The Fröhlich rate equation describes energy transfer between oscillation modes in a frequency spectrum of a non-linear system. Energy condenses in the lowest frequency mode but condensation depends also on the coupling to the heat bath, energy transfer between modes (parameter χ_ij) and energy supply (s_i). The analysis of energy condensation for various conditions is evaluated in Pokorný and Wu [Reference Pokorný and Wu48].

Microtubules and nanotube energy transfer

In eukaryotic cells, microtubules form a well-organized filamentous structure which is a basis of the cytoskeleton. Microtubules are hollow tubes of a circular cross-section with the inner and outer diameter 17 and 25 nm, respectively, described by Amos and Klug [Reference Amos and Klug50]. In the interphase, they grow outward from the centrosome, a spherical structure in the center of the cell, and form radial fibers. Long-living microtubules (about 18 h) are bonded to structures at the membrane. In the M phase (cell division), the microtubules of the mitotic spindle emanate from two centrosomes.

Microtubule physical characteristics correspond to requirements for the generation of the electromagnetic field: they are electrically polar, non-linear, and excited by energy supply. Microtubules in cells are electric dipoles whose dipole moments differ from those in free microtubules (measured by Böhm et al. [Reference Böhm, Mavromatos, Michette, Stracke and Unger38] and Schoutens [Reference Schoutens51]) by space orientation. Several mechanisms supply energy for the excitation of polar vibrations. Energy is supplied by the hydrolysis of GTP to GDP in tubulin after polymerization [Reference Pokorný, Jelínek, Trkal, Lamprecht and Hölzel52, Reference Pokorný53], energy losses caused by the motion of motor proteins along microtubules [Reference Pelling, Sehati, Gralla, Valentine and Gimzewski54], and very likely also by non-utilized energy liberated from mitochondria [Reference Pokorný55]. Photons released from chemical reactions may supply energy in the UV and visible wavelength region. In the M phase, energy is supplied by treadmilling.

Excitation of the electromagnetic field in TNTs may be a fundamental process of their function. We suggest a possible mechanism. A TNT may be considered as a circular waveguide connecting cells. If the axis of a microtubule coincides with the TNT axis, then microtubule field can excite signal in the TNT (z direction). In this case, the electromagnetic excitation corresponds to a TM mode. The theory of waveguides with conductive walls can be used for the description of excitation of the electromagnetic field as a zero-order approximation. The intensity of the electric field transferred from a microtubule into the TNT waveguide may be amplified if its cavity is excited by energy supply. In the TM mode, the electric field along the axis of the waveguide (in the z direction) may be described by

(7)$$E_z = \gamma ^2J_n\lpar \gamma r\rpar \cos \lpar n\varphi \rpar [C_1\exp \lpar j\Gamma z\rpar + C_2\exp \lpar {-}j\Gamma z\rpar ] \comma \;$$

where J_n is the Bessel function for n = 0, 1, 2…, C ₁, C ₂ are constants, and j is the imaginary unit. The propagation constants (eigenvalues) Γ, γ, the wavenumber k, and the cutoff frequency ɷ_c are

(8)$$\Gamma ^2 = k^2-\gamma ^2\comma \;\quad \gamma = p_{nm}/a\comma \;\quad k^2 = \omega ^2\mu \varepsilon \comma \;\quad \omega _c = \displaystyle{\gamma \over {\sqrt {\mu \varepsilon } }}\comma$$

where p_nm is the m root of the Bessel function J_n (γa), a is the radius of the circular waveguide (of the nanotube), μ and ɛ are the permeability and permittivity of the inner medium, respectively. (Technical parameters of circular waveguides and resonators are described in [Reference Collin44].) For a diameter of the circular waveguide 50, 200, and 700 nm, the lowest cutoff frequencies of TM modes (and wavelengths) are 4.6 × 10¹⁵ Hz (65 nm), 1.1 × 10¹⁵ Hz (261 nm), and 3.3 × 10¹⁴ Hz (914 nm), respectively. The calculated cutoff frequencies of TM and TE modes of microtubule inner cavity and TNTs are in Fig. 4. A total reflection from the walls is assumed. In real biological systems, a part of the energy should penetrate into the surrounding medium. Except for this, the values of permittivity and of permeability used in evaluation correspond to vacuum. Therefore, the evaluated cutoff frequencies are only a rough approximation of the actual values. It should be noted that the photon energy of signals with frequencies in this region may correspond to the energy of electrons in atoms.

Fig. 4.

The calculated cutoff frequencies of circular waveguides with conducting walls corresponding to a microtubule (MT) and a nanotube (TNT) as a function of the sequence number of roots m with parameter n (the order of the Bessel function). m determines the distribution of the electromagnetic field along the radius – radial variations, and n settles dependence on the rotation angle along the longitudinal axis – circumferential variations. TM and TE – transversal magnetic and transversal electric field, respectively [Reference Pokorný, Pokorný and Vrba14].

The wave excited in the waveguide (TNT) may amplify electric signals transferred between cells. Generally, TNTs may be parts of a unified cavity system which sustains coherence in the whole space, supplies energy to weaker parts, and provides functional unity.