Calculation of Vibration Modes of Mechanical Waves on Microtubules Presented like Strings and Bars
The study describes a physical model of vibrating microtubules in living cells, presented as strings and bars. Calculated are proper-frequencies of first four vibration modes of transverse and longitudinal waves on microtubules. For microtubules with length 1-30µm and shear modulus 5.0×106 N/m2 the proper-frequencies of standing transverse waves fall in diapason of 1×103 - 5×107 Hz. For microtubules with same length and Young’s modulus 108–109 N/m2 the proper-frequencies of standing longitudinal waves fall in diapason of 5×106 - 3×109 Hz. These calculated diapasons of frequencies overlap with experimentally registered diapasons of frequencies of mechanical and electric vibrations in bacteria, yeast cells, erythrocytes, infuzorii and soma cells. Some theoretical problems related to the present model are discussed.
Atanas Todorov Atanasov,
Calculation of Vibration Modes of Mechanical Waves on Microtubules Presented like Strings and Bars, American Journal of Modern Physics. Special Issue: High Energy Physics: Towards a New Synthesis of Fundamental Interactions.
Vol. 3, No. 6-1,
2014, pp. 1-11.
A. G. Gurwitsch and L. D. Gurwitsch, Die Mitogenetische Strahlung, Jena, Fischer, Germany, 1959.
R. VanWijk, “Bio-photons and biocommunication,” J. Sci. Explor., vol.15, pp.183-197, 2001.
J. Pokorný, J. Hašek, J. Vaniš and F. Jelínek, “Biophysical aspects of cancer-electromagnetic mechanism,” Indian. J. Exp. Biol., vol. 46, pp. 310-321, 2008.
G. Reguera, “When microbial conversations get physical,” Trends. Microbiol., vol. 19, pp.105-113, 2011.
G. Albrecht-Buehler,“Changes of cell behavior by near-infrared signals,” Cell. Mobil. Cytoskeleton,” vol.32, pp. 299-304, 1995.
J. Pokorný, J. Hašek, J. Vaniš and F. Jelínek. “Electromagnetic field of microtubules: effects on transfer of mass particles and electrons,” J. Biol. Phys., vol. 31, pp. 501-514, 2005.
H. Fröhlich,”Long-range coherence and energy storage in biological systems,” Int. J. Quantum Chem., vol. 2, pp. 641-649, 1968.
H. Fröhlich, “Evidence for coherent excitation in biological system,” Int. J. Quantum Chem., vol. 23, pp.1589-1595,1983.
C. H. Romanoff, Biological action of mechanical vibrations. Ed. ‘Nauka’, S. Peterburg, Russia, 1983.
B. Martinac, “Mechanosensitive ion channels: molecules of mechanotransduction,” J. Cell. Sci., vol. 117, pp.2449-2460, 2004.
L. V. Beloussov and I. Y. Volodyaev, “From molecular machines to macroscopic field: an accent to characteristic times,” Europ. J. Biophys., vol. 1, pp. 6-15, 2013.
W. F. Marshall and J. L. Rosenbaum, “Cell division: the renaissance of the centriole,” Curr. Biol., vol. 9, pp.R218-R220, 1999.
J. L. Salisbury, “A mechanistic view on the evolutionary origin for centrin-based control of centriole duplication,” J. Cell. Physiol., vol. 213, pp. 420-428, 2007.
M. Jibu, S. Hagan , S. R. Hameroff, K. H. Pribram and K. Yasue, “Quantum optical coherence in cytoskeletal microtubules: implications for brain function,” Bio-Systems vol.32:, pp.195-209, 1994.
N. J . Woolf and S. R. Hameroff, “ A quantum approach to visual consciousness,” TRENDS Cogn. Sci., vol. 5, pp. 472-478, 2001.
Y. M. Sirenko, M. A. Stroscio and K. W. Kim, “ Electric vibrations of microtubules in a field,” Phys. Rev. E-APS., vol. 53, pp. 1003-1010, 1996.
J. Pokorný, F. Jelinek, V. Trkal, I. Lamprecht and R. Hölzel, “Vibration in microtubules,” J. Biol. Phys., vol. 23, pp. 171-179, 1997.
S. Porter, J. A. Tuszyński, C. W. V. Hogue and J. M. Dixon, “Elastic vibrations in seamless microtubules,” Eur. Biophys. J., vol.34, pp. 912-920, 2005.
A. G. Pakhomov, Y. Akyel, O. N. Pakhomova, B. E. Stuck and M. R. Murphy, “Current state and implications of research on biological effects of millimetre waves: a review of the literature,” Bioelectromagnetics, vol.19, pp.393-413, 1998.
S. Banik, S. Bandyopadhay and S. Ganguly, “Bioeffects of microwaves-a brief review,” Bioresource technology, vol. 87, pp.155-159, 2003.
V. I. Fedorov, S. S. Popova and A. N. Pisarchik, “Dynamic effects of submillimeter wave radiation on biological objects of various levels of organization,” IRMMW, vol. 24, pp.235-1254, August 2003.
M. Matsuhashi, A. N. Pankrushina, S. Takeuchi, H. Ohshima, H. Miyoi, K. Endoh, K. Murayama, H. Watanabe, S. Endo, M. Tobi, Y. Mano, M. Hyodo, T. Kobayashi, T. Kaneko, S. Otani, S. Yoshimura, A. Harata and T. Sawada, “Production of sound waves by bacterial cells and the response of bacterial cells to sound,” J. Gen. Appl. Microbiol., vol. 44, pp. 49-55, 1998.
T. F. Anderson, S. Boggs and B. C. Winters, “The relative sensitivities of bacterial viruses to intense sonic vibration,” Science (NY), vol.108, pp.18, 1948.
A. E. Pelling, S. Sehati, E. B. Gralla, J. S. Valentine and L. K. Gimzewski, “Local nanomechanical motion of the cell wall of Saccharomyces cerevisiae,” Science, vol. 305, pp. 1147-1150, 2004.
M. Cifra, J. Vaniš, O. Kučera, J. Hašek, I. Frýdlová, F. Jelínek, J. Šaroch and J. Pokorný, “Electric vibrations of yeast cell membrane,” Bioelectrochemistry, vol. 51, pp. 142-148, 2007.
F. Jelínek, J. Šaroch, O. Kučera, J. Hašek, J. Pokorný, N. Jaffrezic-Renault and L. Ponsonnet, “Measurement of electromagnetic activity of yeast cells at 42 GHz,” Radioengineering, vol. 16, pp. 36-39, 2007.
A. Levin and R. Korenstein, “Membrane fluctuations in erythrocytes and linked to MgATP-dependent dynamic assembly of the membrane skeleton” Biophys. J., vol. 60, pp.733-737, 1991.
S. Tuvia, A. Bitler and R. Korenstein, “ Mechanical fluctuations of the membrane-skeleton and dependent on F-actin ATPase in human erythrocytes,” J. Cell. Biol., vol.141, pp. 1551-1561, 1998.
S. Tuvia, A. Moses, N. Gulayev, S. Levin and R. Korenstein, “ß-Adrenergic agonists regulate cell membrane fluctuation of human erythrocytes,” J. Physiol., vol. 516, pp.781-792, 1999.
A. Bitler and R. Korenstein, “Nano-scale fluctuations of red blood cell membrane reveal nonlinear dynamics,” Biophys. J., vol. 86, pp. 582, 2004.
S. Rowlands, The interaction of living red blood cells. In Biological Coherence and Response to External Stimuli, Fröhlich , H. Ed. Springer, Berlin, Heidelberg, New York, 1988, pp.171-191.
K. W. Ockelmann and O. Vahl, “On the biology of the polychaete Glycera alba, especially its burrowing and feeding,” Ophelia, vol. 8, pp. 275-294, 1970.
E. D. Kirson, Z. Gurvich, R. Schneiderman, E. Dekel, A. Itzhaki, Y. Wasserman, R. Schatzberger and Y. Palty, “Disruption of cancer cell replication by alternating electric fields,” Cancer Res., vol. 64, pp. 3288, 2004.
E. D. Kirson, V. Dbalý, F. Tovariš, J. Vymazal, J.F. Soustiet, A. Itzhaki, D. Mordechovich, S. Steinberg-Shapira, Z. Gurvich, R. Schneiderman, Y. Wasserman, M. Salzberg, B. Ryftel, D. Goldsher, E. Dekel and Y. Palti, “Alternating electric fields arrest cell proliferation in animal tumor models and human brain tu-mors,” Proc. Natl. Acad. Sci. USA, vol. 2104, pp. 10152, 2007.
L. Cucullo, G. Dini, K. L. Hallene, V. Fazio, E. V. Ilkanich, C. H. Igboechi, K. M. Kight, M. K. Agarval, M. Garrity-Moses and D. Janirgo, “Very low intensity alternating current decreases cell proliferation,” Glia, vol. 51, pp. 65, 2005.
I. Minoura and E. Muto, ”Dielectric measurement of individual microtubules using the electroorientation method,” Biophys. J., vol.90, pp.3739-3748, 2006.
M. Cifra, J. Pokorny, D. Havelka and D. Kucera, “Electric field generated by axial longitudinal vibration modes of microtubule,” BioSystems, vol.100, pp. 122-131, 2010.
D. Havelka, M. Cifra, O. Kučera, J. Pokorný and J. Vrba, “High-frequency electric field and radiation characteristics of cellular microtubule network,” J. Theoret. Biol., vol.286, pp. 31-40, 2011.
A. Mershin, A. A. Kolomenski, H. A. Schuessler and D. V. Nanopoulus, “Tubulin dipole moment, dielectric constant and quantum behaviour: computer simulations, experimental results and suggestions,” Biosystems, vol.77, pp. 73-85, 2004.
M. A. Rojavin and M. C. Ziskin, “Medical application of millimetre waves,” Q.J. Med., vol.91, pp. 57-66, 1998.
S. I. Alekseev and M. C. Zickin, “Millimeter microwave effect on ion transport across lipid bilayer,” Bioelectromagnetics, vol.16, pp. 124-131, 1995.
D. J. Panagopoulos, A. Karabarbounis and L. H. Margaritis,”Mechanism for action of electromagnetic fields on cells,” Biochem. Biophys. Res. Commun., vol. 298, pp. 95-102, 2002.
C.R. Cantor and P. R. Schimmel, Biophysical Chemistry, Part I: The Conformation of Biological Macromolecules, Ed. by W.H. Freeman and Company, San Francisco, 1980.
E. Nogales, S. C. Wolf and K. H. Dowing “Structure of the tubulin dimer by electron crystallography,” Nature, vol.391, pp. 199-203, 1998.
J. Lowe, H. Li, K. H. Downing and E. Nogales, “Refined structure of α/β tubulin of 3.5A resolution,” J. Mol. Biol., vol. 313, pp.1045-1057, 2001.
E. M. Mandelkow, R. Schulthesiss, R. Rapp, M. Muller and E. Mandelkow, “On the surface lattice of microtubules, helix starts, protofilaments number, seam and handedness,” J. Cell Biol., vol.102, pp. 1067-1073, 1986.
P. A. Janmey, U. Euteneuer, P. Traub and M. Schliwa, “Viscoelastic properties of vimentin compared with other filamentous biopolymer network,” J. Cell Biol., vol. 113, pp. 155-165, 1991.
A. Kis, S. Kasas, B. Babić, A. J.Kulik, W. Benoît, G.A.D. Briggs, C. Schönenberger, S. Catsicas and L. Forró, “Nanomechanics of microtubules,” Phys. Rev. Lett., vol.89, pp. 248101-248104, 2002.
J. Pokorný, “Excitation of vibrations in microtubules in living cells,” Bioelectrochemistry, vol. 63, pp. 321-326, 2004.
J. A. Tolomeo and M. C. Holley, “Mechanics of microtubule bundles in pillar cells from inner ear,” Biophys. J., vol. 73, pp. 2241-2247, 1997.
E. Mickey and J. Howard, “Rigidity of microtubules is increased by stabilizing agents,” J. Cell. Biol., vol. 130, pp. 909-917, 1995.
R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lecture on Physics. Addison-Wesley Publishing Co. Inc., Reading, MA, 1964.
P. J. Pablo, I. A. T. Schaap, F. C. MacKintosh and C. F. Schmidt, “Deformation and collapse of microtubules on nanometer scale,” Phys. Rev. Lett., vol.91, pp. 098101(1)-098101(4), 2003.
F. Pampaloni, G. Lattanzi, A. Jonas, T. Surrey, E. Frey and E.L. Florin. “Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistence length,” Proc. Natl. Acad. Sci. USA, vol.103, pp. 10248-10253, 2006.
M. Kikumoto, M. Kurachi, V. Tosa and H. Tashiro, “Flexural rigidity of individual microtubules measured by a buckling force with optical traps,” Biophys. J.,vol. 90, pp. 1687-1696, 2006.
W. Yu and P. W. Baas, “Changes in microtubule number and length during axon differentiation,” J. Neurosci., vol.14, pp. 2818-2829, 1994.
P. W. Baas, C. V. Nadar and A. M. Kenneth, “Axonal transport of microtubules: the long and short of it,” Traffic, vol.7, pp. 490-498, 2006.
D. Boal and D. H. Boal, Mechanics of the Cell. Cambridge Univ. Press © David Boal, 2012.
R. Sträcke, K. J. Böhm, L. Wollweber, J. A. Tuszynski and E. Under, “Analysis of the migration behaviour of single micr tubules in electric fields,” Biochem. Biophys. Res. Commun., vol. 292, pp. 602-609, 2002.
T. Kim, M. T. Kao, E. F. Hasselbrink and E. Mehöfer, “Nanomechanical model of microtubule translocation in the presence of electric field,” Biophys. J., vol. 94, pp. 3880-3892, May 2008.
A. E. Pelling, S. Sehati, E. B. Gralla and L. S. Gimzewski, “Time dependence of the frequency and amplitude of the local nanomechanical motion of yeast,” Nanomedicine, vol. 1, pp. 178-183, 2005.
M. Cifra, D. Havelka, O. Kučera and J. Pokorný, “Electric field generated by higher vibration modes of microtubule,” In 15th Conference on Microwave Techniques – COMITE 2010.
J. Tuszynski, J. Brown, E. Crawford, E. Carpenter, M. Nip, J. Dixon and M. Sataric, “Molecular dynamics simulations of tubulin structure and calculations of electrostatic properties of microtubules," Math. Comput. Model, vol.41, pp. 1055-1070, 2005.
M. Cifra, D. Havelka and M. A. Deriu “Electric field by longitudinal axial microtubule vibration modes with high spatial resolution microtubule model,” Journal of Physics: Conference Series 329: 012013-01226. 90th International Fröhlich’s Symposium, IOP Publishing Ltd, 2011.
L. A. Amos, “The microtubule lattice-20 years on,” Trends. Cell. Biol., vol. 5, pp. 48-51, 1995.
F. Metoz, I. Amal and R.H. Wade, “Tomography without Tilt: Tree-Dimensional Imaging of Microtubules/Motor Complexes,” J. Struct. Biol., vol. 118, pp.159-168, 1997.
K. R. Foster and M. H. Repacholi, “Biological effects of radiofrequency fields: does modulation matter?,” Radiation Research, vol. 162, pp. 219-225, August 2004.
A. A. Saha, T. J. A. Craddock and J. A. Tuszynski, “An investigation of the plausibility of stochastic resonance in tubulin dimmers,” BioSystems, vol.107, pp. 81-87, 2012.
K. F. Foster and J. W. Baish, “Viscous damping of vibrations in microtubules,” J. Biol. Phys., vol. 26, pp. 255-260, 2000.
F. Bueche, Principles of physics. Fourth Ed. McGraw-Hill International Book Company, UK, 1982.
J. L. Carminati and T. Stearns, “Microtubules orient the mitotic spindle in yeast through dynein-dependent interactions with the cell cortex,” J. Cell. Biol., vol. 138, pp. 629-641, 1997.
I. V. Savelev, Lecture of fundamental physics. Part 1-3, ‘Science’ Ed., Moscow, Russia, 1977.
C. B. Tyson,P. G. Lord and A. E. Wheals “Dependency of size of Saccharomyces cerevisiae cells on growth rate,” J. Bacteriol. Vol. 138, pp. 92-98, 1979.
P. A. Fantes, “Control of cell size and cycle time in Schizosaccharomyces pombe.” J. Cell Sci., vol. 24, pp.51-67, 1977.
A. T. Atanasov, “Method for tentative evaluation of membrane permeability coefficients for sodium and potassium ions in unicellular organisms,” Open J. of Biophysics, vol. 3, PP.91-98, 2013.
C. Goswami and T. Hucho, ”Novel aspects of submembraneous microtubule cytoskeleton,” FEBS Journal, vol. 275, pp. 4653, 2008.
I. B. Levitan, “Modulation of ion channels in neurons and other cells,” Ann. Rev. Neurosci. vol.11, pp. 119-136, 1988.
B. Hille, Endplate channels and other electrically inexitable channels. In: Ionic Channels of Excitable membranes. pp. 117-147. Sinauer Associated Inc., Sunderland MA, USA, 1984.
C. Miller, “Trapping single ions inside ion channels,” Biophys. J., vol. 52, pp. 123-126, 1987.