j.a. tuszynski, j.a. brown and b. trpisova: energy and information processing in microtubules

8/3/2019 J.A. Tuszynski, J.A. Brown and B. Trpisova: Energy and Information Processing in Microtubules

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METMBS '00 lnternalional Conference -

J.A. ThszyIiski**J.A. Brown and B. TrpisovaDepartment of PhysicsUniversity of Alberta

Edmonton, Alberta, T6G 2JlCanada

Abstract The possibility of signalling via micro-tubtde filaments of the cytoskeleton is investigatedusing two different models based on tubulin's dipolemoment. The first model involves dipole flips in thetubtdin dimer which may propagate either sponta-neously or by electric field guidance. The secondmodel of signalling studied is based on moving do-main walls of dielectric polarization. We find inthis latter case, however, that thermal fluctuationssignificantly disturb the propagation of such solitarywaves unless strong electric fields are present.

advantage is great speed of propagation. [3]Microtubules (MT's) are cytoskeletal pro-tein filaments which have been shown to re-spond to both electric and magnetic fieldsaligning themselves to field lines [4]. Each ofthe MT's subunits, dimmers f tubulin, has anelectric dipole moment that contributes to theoverall polarity of the structure. The magni-tude of this dipole moment p has been esti-mated as the product of an elementary chargee with a spacing of 8nm corresponding to thesize of the dimmer.This gives a value of Po =1.28.10-27Cm (=384 debye) which is very closeto the value obtained from structural calcu-lations based on the tubulin crystallography[5]. The latter gives 337 debye for the axialprotofilament component of the tubulin dipolemoment [6]. Our earlier Monte Carlo simula-tion of the dipole-dipole interactions betweenthe dimmersf a microtubule demonstrated thatsuch a lattice of dipoles may be ordered at orbelow physiological temperatures [7]. The or-dering can be either ferroelectric (for the so-called 13A lattice) or antiferroelectric (for the13B lattice) It is tantalizing that it might bethis net polarization of the MT lattice which isresponsible for guiding the motor proteins suchas kinesin and dynein which travel in oppositedirections along the MT .

microtubules, biological signalling,waves, polarization solitary

1 Introd uctionSignalling by various means is necessary n or-der to regulate the complex behavior of livingsystems. The use of signaling molecules whichare first packaged, then dumped outside thecell where they spread diffusively facilitates cel-lular communication.[I] Synaptic signalling is arefined version of paracine signalling in whichneurotransmitter molecules are released fromnerve cell's synapses. 2] Endocrine signallingUSes ormones which are reverted into the cir-

system where they travel over longWithin the cell, the mechanismssignal transduction are less clear. The cy-has been implicated in intracellularby many researchers. Electromag-signalling has also been involved and its

2 Dipole Flips and Signallingvia Microtubules

00 also at: Starlab NV /SA, Boulevard St. Michel1040, Brussels, Belgium. We propose that the orientation of the indi-vidual dipoles may be flipped due to a confor-


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mational change of the tubulin dimmeresultingfrom a GTP hydrolysis or from physical inter-actions (e.g. dipole-dipole forces) [8]. We thenstudy the propagation of signals in the formof dipole patterns on the background of an or-dered lattice of a ferroelectric or antiferroelec-tric type [9]. We have used Monte Carlo simu-lations [10] to model the problem (see Fig. 1).Our model was inspired by Hameroff et al [11]who developed a cellular automaton approachwhereby a discrete charge on tubulin was ableto hop. In our model, a dipole moment was in-stead assigned to each dimmer hose orientationwas allowed to change. The interaction energybetween two neighboring dipoles is [121

along the MT unless some additional mecha-nism is added. Since there is nothing to di-rect the propagation of the defect, it takes arandom walk about the MT and its energyis gradually dissipated to the rest of the MT.The efficient propagation of signals could be re-stored by (i) the application of an external fieldwhich would bias signal propagation; (ii) anasymmetry in the dipole structure which sim-ply makes t more favour able to propagate in aparticular direction; or (iii) mechanical stressif dipoles are coupled to a lattice distortion.The third mechanism would be the result of apiezo-electric effect which was claimed to be aproperty of MT's a long time ago.

1 PI .p2 -3(PI .ii)(P2 .ii)~ r3Eint (I) 14A138 138 14Awhere ~ is the relative permittivity of themedium, ~o s the permittivity of free space, ~is the ",th tubulin dimer's electric dipole mo-ment. n is the normal vector pointing from theposition of the first dipole to the second dipoleand r is the distance separating the dipoles.It is known that MT's are assembled fromGTP-rich tubulin dimmers nd that this GTP

is hydrolyzed rapidly after the addition of thetubulin subunit. What is not yet known iswhat happens to this energy. We are propos-ing that some of the energy is stored in thelattice through a conformational change of theprotein dimer. The hydrolysis of GTP releasesabout 4.6 kcal/mol of energy which the latticecould use to flip conformational states of indi-vidual tubulin dimmers, ach flip would requireup to 2.0 kcal/mol of energy [9]. The energymight then propagate along the MT through asequenceof dipole flips as the lattice reorientsto accommodate the additional energy. Theseconformational changes or flips are believed tobe the result of a mobile electron. It may be lo-calized at one of two binding sites in the tubu-liD molecule. Movement of the electron fromone binding site to the other causes he tubu-lin dimmer nd its electric dipole to re-orient.

Unlike the Hameroff model [111, we do notobserve the smooth propagation of signals

(c) (d)a) (b)

Figure I. Portions of different microtubule lat-tices with dipole order illustrated. Light boxes rep-resent the dipole "up" state while dark boxes rep-resent the dipole "down" state. (a) The I3B latticeabove its critical temperature is characterized bydipole disorder, (b) the I3B lattice below its criti-cal temperature is antiferroelectrically ordered; (c)The I4A lattice is ferroelectrically ordered belowthe critical temperature (d) but it also supports do-main wall patterns (e) The I4A lattice just belowthe critical temperaure also supports large defectstructures. Notice that smaller thermal defects dotthe structure.


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The existence of an ordered phase is cru-cial if the MT is to be able to process infor-mation. When the lattice is not organized,this signifies that thermal fluctuations dom-inate over the dipole-dipole interactions andthat entropy dominates the lattice. Upon sucha background, any signals introduced to thelattice rapidly vanish (thermalize). We havealso been able to demonstrate that the applica-tion of a large axial field of 106V m along thecauses nearly all dipoles to orient them-

in the direction which most closely fol-that of the field. Thus, a wave of dipolealong the MT as the field is

an action potential movesan axon. The large field is felt by MTs inof the cell membrane. Suppose theoriented in a direction which favours anfor the lattice, such as is then MT, these dipoles will re-orient them-field acts like a pump and stores

in the lattice of dimmers.Once the fieldmay return to theirand release their storedA weaker field does not actually createby changing the orientation of dipoles

the movement of any existing defects.Our simulations place a firm limit on thethe dipoles required for self-The required dipole strength of10-21 C.m is comparable but slightly

than the value which we estimated forIf the angle between the dipole direc-tubulin's conformational states

than predicted, an even larger dipolebe required. Should the dipolethan this, informa-processing via dipole flips must be ruled

tric phase of MT dipoles. We also believe thatdipole flips are coupled linearly to a conforma-tional charge in the MT lattice due to MT'spiezoelectricity [16]. Based on the above as-sumptions, it has been demonstrated elsewherethat the Hamiltonian associated with the dy-namics of a dipole placed at site n in a MTprotofilament with N tubulin dimmers an berepresented by the following equation [13]

Propagation of Dipolar Do-main Wails

seen in our simulations above thatwalls occur naturally in the ferroelec-

In eq. (2) the variable Un represents the pro-jection on the protofilament axis of the distor-tion of the dimmer hich is in the {3 state withrespect to the Q state, 4M(dun/dt)2 is the ki-netic energy of the tubulin molecule of massM, 4K(Un+l -Un)2 represents the elastic en-ergy that originates from the restoring elasticforce$ characterized by a constant K that actbetween each two dimmers. he quartic double-well potential energy V(Un) = -~u~ + Tu~approximates the average effect of the sur-rounding dipoles on the dipole at site n. V(Un)can be viewed as a Landau n.ee energy expan-sion where Un corresponds to the order param-eter and the coefficients Q2 and Q4 are char-acteristics of the physical system. Assumingthat the phase transition from a ferroelectricto a paraelectric phase n a MT is a second or-der phase transition, Q4 is a positive constantand Q2 = a2(Tc -T), where a2 > 0. The coeffi-cient c represents a bias field ( electric or pres-sure gradient) acting on the dynamic variableuno Also, Ro is the equilibrium distance be-tween two neighbouring tubulin dimmers. asedon the Hamiltonian in eq. (2) we obtain thefollowing equation of motion in the continuumlimit and with the addition of a friction termproportional to 1


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74 METMBS '00 International Conference-

and

t/I(f.) = ~, Uo = (~)1/2. (7)Uo Q4The travelling kink wave solution of eq. (5)

is listed in (151. The solution that correspondsto the kink wave moving with the velocity v >O is (see fig. 2)

) 1/J1 1/J21/J(f. = 1/J2 r 1 + ~("'I-1/12)/h'where 1/J1and 1/J2satisfy the cubic equation

(8)

(1/1 1/Jl)(1/J 1/J2)(1/J 1/J3) 1/1" 1/1- a. (9)It can be shown that the velocity of propaga-tion of this domain wall structure follows anapproximate formula given by

v ~ ~(~)1/2qE (10)'YO.2 2The maximum electric field along the MThas been estimated at Emax ~ 2 .6 106 V /mwhich gives the corresponding kink velocity ofv ~ 1 .2 m/s.The velocity v depends linearly on the elec-tric field E. This means that larger electric

fields on MTs will generate kinks moving atlarger velocities and vice versa. The magni-tude of the intrinsic electric field of a MT canbe altered if the MT is subjected to an externalelectric field generated, for example, by a mem-brane or by other MTs. If the external electricfield is oriented in the same direction as theintrinsic field of the MT, this will result in afaster propagation of kinks. If the external andintrinsic electric fields are oriented in the oppo-site direction, the propagation of kinks will beslowed down or reoriented when the magnitudeof the external field is larger than the magni-tude of the intrinsic field of the MT. In theprevious section it was shown that sufficientlystrong external fields can alter the direction inwhich the dipoles are aligned when the MT is inthe ferroelectric phase. Then the external fieldand the intrinsic MT field would be orientedin the same direction. This would change the

The values of the constants in eq. (3) canbe determined as follows [13): The mass ofthe tubulin dimmer was calculated as M =1.83 x 10-22kg. The constants K and Ro arerelated to the velocity of longitudinal soundwaves through the formula K m, where Vois the sound velocity. Due to lack of directmeasurement in MT's we take the velocity ofsound in DNA, Vo = 1700 ms-l [14), to applyfor MT's. The coefficients a2 = a2(Tc -T)and a4 for a MT are also not known. How-ever, inorganic crystals exist in which ferrodis-tortive domain walls can be formed. For ex-ample, for the crystal PbsGe3Gll below thecritical temperature, a4 = 1.6 x 1024Jm-4 anda2 = 10Jm-2 K-t. If the MT is in the fer-roelectric phase then the critical temperatureTc can be approximately taken as 350K and Tis body temperature 310K. This gives for thecoefficient a2 ~ 400Jm-2.

To estimate the damping coefficient 'Y thetubulin dimmer can be considered a sphere ofradius R = 4 x 10-9m and mass M thatis moving in a fluid of viscosity 1]. Assum-ing that a MT is mainly surrounded by wa-ter molecules, 1] can be taken as the viscos-ity of water. At body temperature, 1] =l]water = 6.9 X 10-4kgm-ls-l. Consequently,'Y = 61rRl] = 5.2 x 10-llkgs-l.

For a constant electric field E, eq. (3) canbe solved analytically [13) and takes the formof a wave that travels at a constant velocity v[10), with a moving coordinate f. = x -vt)introduced as follows

la21~

- 1 1/2 (2) X= vt) '4The partial differential equation (3) then re-duces to an ordinary differential equation

lfl;'!/J dtt~+P~ Q (5)/J3 + 1/J + q

where

V'Y~(~)1/2Q2 Q2

p= a[Ma2(V~ -V2)]1/2'(6)


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~ETMBS '00 I~~nal Conference 75

Acknow ledgementsirection of propagation and increase the ve-locity of the kink waves. When the total elec-tric field to which the dipoles on the tubulindimmers re subjected is zero the kinks will notpropagate at all. According to the approxima-tion of the charge distribution in a MT chosenhere the magnitude of the intrinsic field of aMT also depends on its length. This meansthat MTs with different sizesmay support kinkwaves travelling at different velocities.

This work was supported by grants fromNSERC and the Consciousness Studies pro-gram administered by the University of Ari-zona.

References

We have also investigated the effects of im-Purity potentials on the propagation of kinksin the MT lattice. Both potential wells andbumps slow down moving domain walls dueto the force generated by their gradients. Acritical amplitude of the potential was foundat which the kink is stopped completely. Itsvalue was determined numerically to be 10-21J which is on the order of thermal noise atphysiological temperatures. This result indi-cates that under more realistic conditions oftemperature, electric fields and pressures, thepropagation of signals in the form of dipolardomain walls in a stochastic process stronglydependent on thermal fluctuations and hencethe model presented in the first part of the pa-per is quite realistic.

t t t t t t t~ 000.6

06.07 ..c..o7 -07 0.2.-01 -.~7x(mj

[1] B. Alberts, D. Bray, J. Lewis, M. Raff, K.RA:>berts,nd J.D. Watson, Molecular biology ofthe cell, Garland Publishing, London, 1994.[2] M. V. Volkentstein, General Biophysics,Academic Press, New York, 1983.[31 G. Albrecht-Buehler, The Centrosome,chapter Speculation about the function andformation of centrioles and basal bodies. pages69-102, Academic Press, San Diego, 1992.[4] P.M. Vassilev, R.T. Dronzine, M.P. Vas-sileva, and G.A. Georgiev, Parallel Arrays ofMicrotubules Formed in Electric and MagneticFields, Bioscience Reports 2, p. 1025-1029(1982).[5] E. Nogales, S.G. Wolf and K.H. Down-ing, Structure of the Alpha-Beta Thbulin dimmerby Electron Crystallography, Nature 39, 199(1998).[61 J.A. Brown, Ph.D. thesis, UniversityofAlberta, Edmonton, 1999.[71quad J .A. Thszynski, S. Hameroff, M. V. Sa-taric, B. Thpisova, and M.L.A. Nip, Ferroelec-tric behavior in microtubule dipole lattices: Im-plications for information processing, signal-ing and assembly/disassembly, J. Theor. BioI.174,371-380 (1995).[81 B. Thpisova and J .A. Thszynski, Phys.Rev. E 55, 3288 (1997).[91 J.A. Brown and J.A. Thszynski, Phys.Rev. E 56, 5834 (1997).[11 K. Binder, Monte Carlo simulation instatistical physics: an introduction, Springer-Verlag, New York, 1988.[111 S. Hameroff, S.A. Smith and R.C. Watt,Automaton model of dynamic organization inmicrotubules, Ann. N.Y. Acad. Sci. 466, 949-952 (1986).

Figure 2. A domain wall between two subchainsof a MT protofilament in which the tubulin dirnersare in two different states.


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{121 J.D. Jackson, ClassicalElectrodynamics,John Wiley and Sons, Toronto, 1975.{131 M. V. Sataric, R.B. Zakula, J .A. Thszyn-ski, A model of the energy transfer mecha-nism in microtubules involving domain-wall-

type solitons, Physical Review E 48, p. 589-597 (1993).{141 M.B. Hakim, S.M. Lindsay and J. pow-ell, The speed of sound in DNA, Biopolymers23, p. 1185-1192 (1984).{151 J.M. Dixon, J.A. Thszynski and M.Otwinowski, Special analytical solution of thedamped-anharmonic-oscillator equation, Phys-ical Review A 44, p. 3484-3491 (1991).{161 H. Athenstaedt, Pyroelectric and Piezo-electric properties o Vertebrates, Ann. N. Y.Acad. Sci. 238, 68--94 1974).

j.a. tuszynski, j.a. brown and b. trpisova: energy and information processing in microtubules

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