’ WI%XXJS EFFECTS ON POLAR VIBRATIONS IN MICROTUriULES

institute of Radio Engiineeting and Electmnics, Acudenay of &iences of the Czech Republic, Chaberskd 57, 18% bi- Prague 8, Czech Republic

E-mail: pokorny@ure. cas. t%

ABSTRACT

Microtubules form organization structure of cytoskeleton of eucariotic cells. Important physical properties of microtubules are electrical polarity and high deformability at low stress. Electrical dipoles in tubulin het- erodimers attract ions from eytosol. There is a charge layer around microtubules. Slip boundary conditions at the charge layer and in the microtubule surface layers minimize damping effects of cytosol viscosity on vibra-

. tions. Relaxation time is more than 100 times greater. than oscillation period (at 10 MHz). Energy supplied by hydrolysis of GTP (guanosine triphosphate) to GDP (guanosine diphosphate) may, therefore, excite vibrations in microtubultis. E$tern+l electromagnetic field can contribute to excitations. .

IN’SRODUCTION

Eucariotic cells are structurally and dynamically organized by a protein polymer network called the cytoskele- ton-a highly dynamic structure which reorganizes continually as the cell changes its shape, divides, and responds to its environment [l]. But we do not know principles and mechanisms of organization and maintenace, The or- ganization mechanisms in biological systems may be of electromagnetic nature as is claimed in [2]. MicroJubules ’ form a fundamental structure of the cytoskeleton with extraordinary electrical properties capable to generate oscillating electric fields and interact with ixternal electromagnetic fields. The microtubule is a cylidrical struc- ture formed from tubulin heterodimers (Fig.la) [l] composed of a- and P-tubulins (Fig.lb). Each heterodimer has a net mobile negative charge and binds f8 calcium ions ‘[3, 41. The electric dipole moment might, be of 102-lo3 Debye (lo- 27-10-2” Cm). In the conformational cy state of a heterodimer the net negative charge is localized predominantly towards the a-monomer (Fig.&). Aft& hydrolysis of GTP (guanosine triphosphate) to GDP (guanosine diphosphate) in ,&monomer conformation of the heterodimer is changed and orientation ‘of the dipole moment is reversed (the negative charge is localized towards P-monomer). Ionic concentration inside, a cell is about 150mM of cations (K +, Na+) and the same concentration of anions (e.g. Cl-) [I]. The ions from the cytosoi form a charge layer around the microtubul*Fig.2 [5] (the small and the large circles in the upper part of Fig.2 denote water molecules and ions of cytosol, respectively, the circles in thick lines denote atoms in the microtubule wall). Thickness of the charge layer (the Debye thickness) may be of the order of 1 nm [6]. Due to hydration water molecules are bound to the ions in the layer and, therefore, the layer viscous properties are fundamentally different from those of water. The ions adjacent to the microtubule surface are strongly eoq&d by the electrical forces and form a buffer layer with respect to $ransfer of energy of vibrations. Elasticity is a particular property of microtubules too. The microtubules are highly deformable polymer struc- tures. In contrast with actin and intermediate filaments microtubular networks can be easily deformed and begin to flow without limit when the strain exceeds 50% (Fig.3 [l, 71). Dynamic shear modulus G is about 5 NmT2 [7].

Energy is supplied to microtubules from hydrolysis of GTP bound to P-tubulin in the tubulin heterodimer [8, 91. We may assess the energy supply to microtubules.from the rate of growth of individual microtubules, from fluorescent measurements indicating the rate of replacement of microtubules by new ones (formed from heterodimers from the soluble pool with nonhydrolyzed GTP), from the dynamic instability (growth and shrink- age of the microtubules), and from the rate of treadmilling (heterodimers are added continually to one end and

Fig.la. A schematic picture of a microtubule.

fYf)g a state p state

Fig.lb. Conformatiodof a heterodimer.

Cytosol

ionic layer

Microtubule wall

0 00 0

090 0 Oo

0 :o,o o”

0 o”oo 0

0 O 1;;

0 10

O OoO 0 00;

oo,,go” 0 o”O o”o (-Jo

00000b0°0~0”0080

00000000

00000000

00000000 vibrations

0 -I 0 1 i 3 i ; 6 i

Stress (Nm”)

Fig.2. A schematic illustration of a mi- Fig.3. Mechanical properties of actin, crotubular surface with ionic layer. vimentin, and tubulin filaments.

80

lost from the other end) [l]. Dynamic instability and treadmilling are important processes for continual rebuild- ing of the microtubular structure in the cell. The microtubules satisfy the basic requirements for excitation of vibrations and generation of endogenous oscillating electric field [6]. Coherent vibrations in biological structures were theoretically predicted by FrGhlich in’[lO]. Energy level of excited Frohlich’s systems does not depend on the manner of energy supply Ill].

Viscosity effects on damping of vibrations in microtubules were analysed in [12]. A model with an infinite plane with longitudinal vibrations in the direction parallel with the surface of the plane was used. Properties of the medium below the plane surface were assumed to be those of a microtubule and above the surface those of liquid medium (e.g: of water) with an ionic layer in between. Theoretical analysis in [12] disclosed that slip boundary conditions enable excitation of vibrations in microtubules. Cylindrical model with circular cross+ection was used ip 1131 but slip boundary conditions were neglected. The authors claimed that the relaxation time is several orders of magnitude shorter than period of oscillations in microtubules and that the damping is so strong that it is not meaningful to speak of ‘resonance’ at all. We will analyze effects of viscous damping on vibrations in microtubules using a cylindrical model and taking into account slip boundary conditions.

VISCOUS DAMPING AND RELAXATION TIME

A microtubule may be approximated by a cylindrical layer with circular cross section and with rotation symmetry with respect to the microtubule axis. We will assume longitudinal harmonic vibrations along the axis of the cylinder (but independent of the distance along the axis). Due to viscosity oscillations in the microtubule excite oscillations in the liquid medium (in the cytosol) around the microtubule. The viscous force exerted by Newtonian liquid per unit area of the ionic layer or of the microtubule surface (in the case that effects of the ionic layer are neglected) is given by

8V F:=q 5 >

( > (1)’

T=R where Q = pv is the dynamic viscosity of the cytosol (p is the density of mass and v is the kinematic viscosity), and v is the velocity of vibrations in the direction ofthe microtubule axis. The momentum equation is given by the Navier-Stokes equation [14]) which can be simplified if we neglect external forces and gradient of pressure. We get the diffusion equation &/& = VVV. If the time dependence of velocity is given by the function exp (iwt) we get from the diffusion equation in cylindrical coordinates

2a% av T p+T

- - &.3) Or v

=o. (2)

Solution of (2) can be easily found using Kelvin’s functions [15] which can be expressed by Macdonald’s functions KP (pm), where p denotes order of the function (the solution has to fulfil the requirements at infinity lim.,, v(r) = 0). We get [5]

where ‘= Ko(;fJ ’ (3)

, a is th!e amplitude determined from the boundary conditions at the outer surface of the ionic layer r = R, and the velocity is given by the relation v = wo exp (iwt). From (1) and (3) we get for the viscous force per unit

\ length

F, = -flu0 exp (iwt) where (4)

and St = 2nR is the area of the cylindrical surface of unit length. We assume that the thin ionic layer around a microtubule form a slip region between the microtubule and the cytosol. As a first order approximation we assume that the oscillations in a microtubule act on the damped upper surface of the ionic layer by the force

Fi = fYo exp [i(wt + #J)] where GSi

f = - , AR (5)

Yo is the amplitude of oscillations at the surface of the microtubule, f is the elastic force constant, and AR is the thickness of the ionic layer. The force Fi acts in the direction of the microtubule axis. Equation of motion ‘for forced vibrations at the upper surface of the ionic layer

(where wc is the characteristic angular frequency, y is the displacement along the microtubule axis, and m is the mass of the ionic layer) together with (4) and (5) yields the steady state solution

Yo ‘= f

m (w,” - w2) + i/?w Yo exp ($1

(ys is the amplitude). For w = we we get from (7)

YO

g-=&J

where I/?] = @(p2) + Q(p2). The same relation is valid for wo/Vo where Vi is amplitude of velocity in the microtubule. The average kinetic energy stored in the microtubule per unit length and the average rate of energy loss per unit length caused by viscous damping (as follows from (4)) are given by

E=$r(e-R&,)pl$ and E’ = +I$,

respectively. R, and R me in (9) are the outer and the inner radius of the microtubule. Using (9) the relaxation time r and its relative value ~~~1 (in the number of periods of oscillations) may be evaluated from [13]

Fig.4 shows the relative relaxation time as a function of frequency for q = 5 x 10e4 (the dashed lines) and 10m3 Pa.s (the full lines). We assume that viscosity of cytosol is equal to the viscosity of water and, there- fore, greater than 5~10~~ Pas for temperatures smaller that about 40°C. Rm = 12.5nm, Rmo = 8.5nm [l], R = & + AR, and the shear modulus G = 5Nms2 [7])., Th e relative relaxation time rrer for AR = 0.75nm is about lo2 at 10MHz. If the buffer effects of the ionic layer at the microtubular surface are neglected and the same procedure ti’ in the previous case applied we can evaluate the relaxation time for a slip between microtubule surface layers (the curves for AR = 0.25nm). ~~~1 is greater than 10 at 10MHz. The slip boundary conditions inside a microtubule considerably decrease the effect of viscosity damping too. Fig.5 visualizes the effect of deformability of microtubules on the relaxation time for 77 = 5 x low4 Pa.s and AR = 1 nm [5]. The shear modulus G ,= 5 Nmw2 is small enough to produce significant slip.

The average rate of energy loss is given by (9). We determine vo from (8) for amplitude of oscillations in the microtubule 0.5nm. At 10MHz power lost per 1 m length is about 3.3~10~~~ W/m (77 = 5 x 1OT4 Pa.s). This power is of the same order of magnitude as the power P,,, supplied by hydrolysis of GTP to GDP.

. 0 5 10 15 ,20

Frequency (MHz)

Fig.4. Relative relaxation time rre. ver- sus frequency. AR is a parameter.

CONCLUSIONS

0’ 5 10 15 20

Frequency (MHz)

Fig.5. Relative relaxation time rrer ver- sus frequency. G is a parameter.

Large deformability of microtubules at low stress and formation of the ionic layer at their surface are important physical properties for slip boundary conditions that minimize damping effets of cytosol viscosity. Theory based on no-slip (stick) boundary conditions [13] neglected the physical properties of microtubules, did not adequately represent conditions in which microtubules exist in the cell, and, therefore, claimed that excitation of vibrations is not possible. Slip boundary conditions enable excitation of vibrations at least in the frequency domain l- 100MHz. Excited vibrations and generated electromagnetic field may have fundamental role in organization of a cell. Exogenous electromagnetic fields can interact with microtubules and contribute to their excitation.

REFERENCES

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B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J.D. Watson, Molecular Biology of the Cell, 3rd ed., New York & London, Garland Publishing, 1994. H. Frauenfelder, P.G. Wolynes, and R.H. Austin, “Biological‘Physics”, Rev. Mod. Phys., Centenary 1999, vol. 71 (2), pp. s419-s430, 1999. J.A. Tuszyriski, S. Hameroff, M.V. Sat&C, B. Trpisova, and M.L.A. Nip, “Ferroelectric Behavior in Micro- tubule Dipole Lattices: Implications for Information Processing, Signaling and Assembly/Disassembly”, J. theor. Biol., vol. 174, pp. 371-380, 1995. J.A. Tuszynski, and J.A. Brown, “Models of Dielectric and Conduction Properties of Microtubules”, In Abstract Book, pp. 3-4, Int. Symp. Electromagnetic Aspects of Selforganization in Biology, Prague, Czech Republic, July 9-12, 2000. J. Pokornjr, “Viscous Effects on Polar Vibrations in Microtubules”, unpublished. J. Pokorny, and T.-M. WU., Biophysical Aspects of Coherence and Biological Order, Praha, Academia, Berlin, Springer, 1998. P.A. Janmey, U. Euteneuer, P. Traub, and M. Schliwa, “Viscoelastic Properties of Vimentin Compared with Other Filamentous Biopolymer Networks”, J. Cell Biol., vol. 113, pp. 155-160, 1991. M. Caplow, R.L. Ruhlen, and J. Shanks, “The Free Energy for Hydrolysis of a Microtubule-Bound Nu- cleotide Triphosphate Is Near Zero: All of the Free Energy for Hydrolysis Is Stored in the Microtubule Lattice”, J. Cell Biol., vol. 127, pp. 779-788, 1994. M. Caplow, and J. Shanks, “Evidence that a Single Monolayer Tubulin-GTP Cap Is Both Necessary and Sufficient to Stabilize Microtubules”, Molec. Biol. Cell, vol. 7, pp. 663-675, 1996. H. Friihlich, “The Biological Effects of Microwaves and Related Questions”, In Advances in Electronics and Electron Phys., vol. 53, pp. 85-152, 1980.

PI I

-I

fl i PI PI PI

PO1

Pll

PI

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F. Srob&r, “An Equifinality Property of the FrShlich Equations Describing Electromagnetic Activity of the Living Cells”, Book of Abstracts, p. 167, XVIth Int. Symp. Bioelectrochem. Bioenerg., Bratislava, Slovakia, June 1-6, 2001. J. Pokorny, J. HaSek, F. Jelinek, J. Saroch, and B. Palan, “Electromagnetic Field Generated by Mi- crotubules”, Workshop Risk Assessment of Electromagnetic Pollution from Mobile Phone Networks, Int. School of Plasma Phys., Varenna, Italy, May 10, 2000. K.R. Foster, and J.W. Baish, “Viscous Damping of Vibrations in Microtubules”, J. Biol. Phys., vol. 26, pp. 255-260, 2000. I.G. Currie, Fundamental Mechanics of &ids, New York, USA, McGraw-Hill, 1974. G.N. Watson, A Treatise on the Theory of Bessel I”unctions, Cambridge, Great Britain, Cambridge Univ. Press, 1958. - Janke-Emde-Losch. Tafeln HGherer finktionen, Stuttgart, Germany, Teubner, 1960.