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M. V. Sataric{l} , J.A. Tuszyllski(2) and R.B.

Zakula (3)

(1) Faculty of Technical Sciences,21000 Novi Sad

F .R. Yugoslavia

(2) Department of Physics, University of Alberta,

Edmonton, Canada, T6G 2Jl

(3) Institute of Nuclear Sciences "Boris Kidric",

Belgrade, F.R. Yugoslavia

Abstract An attempt is made to provide physical picture of transfer of in-

formations in cell microtubules. The quantitative model adopted for this purpose is

classical u4-model in the presence of a constant intrinsic electric field. It is demon-

strated that soliton formation in the form of kinks may be energeticaly favorable

under realistic conditions of physical parameter values.

Introduction

Of the various filamentary structures which comprise the cytoskeleton, rnicrotubules

(MT's) are the most promi,nent ones. Their structure and function is best charac-

terized and they appear to be very suited for dynamic information processing[l].

MT's represent hollow cylinders formed by protofilaments aligned along their

axes (see Fig.l) and whose lengths may span macroscopic dimensions.

Figure 1. Left: MT structure from x-ray diffraction

crystallography. Right top: MT -tubulin dimmer subunit,s

composed of a- and P-monomers.

In vivo, the cylindrical walls of MT's are assembliesof 13 longitudinal protofila-

rnents each of which is a series of subunit proteins known as tubulin dimmers.Each

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tubulin subunit is a polar, 8-nfi difier. It consists of two, slightly different 4-nm

monomers with molecular weight of 55 kilodaltons. Each dimmermay be viewed as an

electric dipole p which arises from the fact that 18 calcium ions (Ca++) are bound

within each dimmer.Thus, MT's can be identified as "electrets" or oriented assem-

blies of dipoles. Barnett[2] proposed that filamentary cytosceletal structures may

operate much like information strings analogous to semiconductor word processors.

He conjectured that MT's are processing channels along which strings of bits of

information can move.

2

In the theoretical model that is put forward here the basic assumption is that the

dipoles within protofilaments form a system of oscillators with only one degre of

freedom (DF) collinear with axys or MT. This DF is the longitudinal displacement

of center of mass of dimmers t the position n denoted by Un SO hat we have model

Hamiltonian as follows

N 1

H = L[2M

n=l

1~)2 + 4k(Un+

2 A 2 B 4

]U ) --U + -U -CUn 2 n 4 n n (1

The first term on the right hand side (1) represents the kinetic energy of the longi-

tudinal displacement of one dirner with effective mass M. If the stiffnes parameter

k is sufficiently large long wavelength excitations of the displacement field will be

formed. The parameters 1'1and B involved in double-well potential have the follow-ing meanings; A is usually assumed to be a linear function of temperature and B is

a positive, temperature-independent crystalline-field quartic coefficient.

The last term in eq.(l) arises from the experimental fact that the cylinder of a

MT taken as a whole represents one giant dipole. Together with the polarized water

surronding it, MT generates a nearly uniform intrinsic electric field (IEF) with the

magnitude E parallel to its axis. So it is legitimate that the aditional potential

energy of a dipole due to the electric field is

Vel qejjE (2)CUn

where qeff denotes the effective charge of a single dirner. If we finaly consider the

viscosity of the solvent taking into account the ViSCOU5orce acting on the dimmer's

motion

alLnF = -I at \-)

where I represents the da.mping coefficient (DC), the equation of motion for system

(1) in the continuum approximation becomes

(?)

auAu + Bu3 + I at -qeJJE = 0 (4)

-)

at

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where Ro represents the equilibrium separation between centers of adjacent dirners,

and x-axis is alined along the MT's axis. The equation (4) has a unique bounded

kink-like excitation (KLE)

~ )1/2{!u(t)

where we use the set of denotations

~ = (~)1/2(X -vt)

0' = qcffBl/2(IAI)-3/2 .E

Vo

The main point is that the above bounded solution (5) propagates along the

protofilament with a fixed terminal velocity v which depends on the IEF. We now

assess he magnitude E of IEF at least semi-quantitative. We take into account thatMT's length L is much greater than the diameter of its cylinder which is physicaly

quite reasonable. Hence, for the positions along the MT which are far enogh from

its ends, we simply have

QeffE- -47rt:or2 \'1

where Qeff represents the effective charge on the MT ends while r denotes the

distance from one end to the relcva.nt point along MT. If we suppose that one MT is

moderately long L = 10-6m, the effective charge consists then of 2 x 13 protofilament

ends each of which has a charge of 18 X 2e ( e = 1,.6 .10-190). Consequently

Qeff ~ 103e so that we estimate E"", 106~.

In the other hand, at present \ve do not have the exact values of the crystalline-

field coefficients A and B for MT but we will do a rough assess taking A ,.., 500Jm-2

for T = 3000 ( and B ,.., 1024 m-4.

Using these estimations the dimensionless parameter 0- from eq.(6) has the fol-

lowing order of magnitude

10-1° E~5 (8)

It is therefore clear that even for strong fields the inequality 0- < < 1 holds.

It implies that the travelling terminal velocity of KLE is small in comparison

with the sound velocity (v « vo). This brings about the simple relation between

terminal velocity and IEF as follows

3vo

IIAI

~ )1/2qe!!

2

E (9)=

In other words, we have obtained a linear response relationship. Then, the corre.

sponding KLE mobility It ma.)' he introduced as follows

~

B

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.0, 'at ' J

The absolute viscosity of water for physiological temperature (3000 K) has the

following value 77 = 7. 10-4kgm-]8-1. Inserting R = 4. 10-9m into eq.(II. one

obtains, = 5,6. 10-1lkg8-1.

The final quantity to estimate is the sound velocity vo. Hakim et al's [3] experi-

mental measurements of the sound velocity in DNA give the value Vo = 1,7 .103m8-1.

Finally then, putting M = 55 .103 x 2 .10-27kg ~ I, I. 10-22kg and qeJJ =

18 X 2 X 1,6. 10-19C, formula (10) gives approximately

f.L~ 3. 10-6m2V-18-1 (12)

If the intrinsic field has the value E = 105,lm-l the KLE velocity is on the order of

v ~ 0, 3ms-l. The time of propagation of one KJ.JE hrough one MT (L "' 10-6m) is

thus T = LV-l "' 3 .10-6s. It is obvious then that increasing MT's length the time

of information propagation as carried by KLE increases due to the following two

reasons. First, the magnitude or the TEF decreases which results in decreasing the

KLE velocity. Second, the length or the path increases. A very important physical

parameter characterizing the MT's system is the polarization switching time T8 .A

crude estimate gives T8 "' (nov)-l, where no represents the number of KLE's per

unit length. For typical ferroelectrics no is on the order of 10-5m and its value is

almost temperature independent. Under these circumstances the switching time is

In a JLsec rang.

3 Conclusion

In this paper biophysica-l picture regarding the structure and function of MT's has

been presented in order to moti\'ate the proposed physical model of their nonlinear

dipolar excitations. Model pa.rametershave been estimated with the use of available

experimental data. It was found that a unique bound solution exists which possesses

a unique velocity of propa.gation proportional to the magnitude of the electric field.

In adition to the intrinsic constant electric field one may also consider an addi-

tional externally applied electric field which could be seen as a significant control

mechanism in KLE dynamics. For example, applying an external electric field to a

microtubule ma)' ha.lt the KLE's motion a.nd "freeze" the information carried by it.

References

[I] Dustin, P. (1984) Microtubulcs, Springer, Berlin;

Hameroff, S.R. (1987) Ultimate Computing, North-Holland, Amsterdam;

In order to estimate KLE mobility it is necessary to assess he DC (,) using simple

considerations from fluid mechanics. First of all, each dimmerould be approximated

by a sphere with mass M. The drag force exerted by the fluid on the sphere is thus

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Rasmussen, S. Kaoramporsala, H. VajOdyanath, R. Jensen, K. and Hameroff, S.

(1990) ~~, 42,428-449.

[2] Barnett, M.P. (1987) Molecular systems to process analog and digital dataassociatively in Proceedings of the Thirs Molecular Electronic

Device Conference (F. Carter, Ed.) Resea.rch Laboratory, Washington, D.C.

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