j.a. tuszynski, b. trpisova, d. sept, m.v. sataric and s.r. hameroff: the cell's microtubules:...

8/3/2019 J.A. Tuszynski, B. Trpisova, D. Sept, M.V. Sataric and S.R. Hameroff: The Cell's Microtubules: Self-Organization and I

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Les Douches School, May 30 to June 4, 1994.-,

i"

'

EditorM. Peyrard

Springer Les Editions de Physique

North AmericaPCG Inc.875-81 Massachusetts A venueCambridge MA 02139 U.5A

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FranceAvenue du HoggarZone Industrielle de CourtaboeufB.P. 11291944 Les Ulis cedex A


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LECTURE 29

JAT ' k .*'(I) BT .'* DS *.uszyns I , .rplSOVa , .ept ,M. V. Sataric** and S. Hameroft"'**

* Department of Physics. University of Alberta,Edmonton, AB. T6G 2J I, Canada* * Faculty of Technical Sciences.

21000 Novi Saa: Serbia, Yugoslavia*** Department of Anesthesiology.Univ. of Arizona, Tucson, AZ85721, US.A.

--

~

~t

I. BACKGROUND INFORMATION


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THE CELL'S MICROTUBULES.

389nation processing at as present within MT'sowing for cooperativenism could lead to theectrics. This, in turn,,,7].

/1A:1!

:;;I[ A

~

~

le an important 88-~tal structures mayessors. Thus MT'scan move transfer-el arrays which areproteins (MAP's).el-arrayed memory~ in the cell. Dur-,osomes [9]. In thevards the synapse.Icleus and the cell~ o the center and

T's are compatiblefominant physical'rties. In general,found to be: ( a)I-glass type. Eachectric field, MAP.hing mechanism.

\/

~}.I;\

at..lched 10U.""'omIn segnoonls ~ + IL

,at..lched1oU.'~N IIoaling bJlxdillla1Jts


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THE CELL'S MICROTL'BULES391s the rate of GTP

As MTs grow andow a critical valuein their assemblyJind GTP tubulinalllOJ claim thatd the rate of GTPJy the presence of~nd may be equal.dynamic MTs of1 simple model of

J1 = 5.77.10- JJ2 = -0.71.10-21 JJJ = 3.40 .10-21 J 8! = 0082 = 58.2

83 = 45.6

(2.1)Ids

3cos28-1 2411"(( r;l p (3.1).]

where (0 is the vacuum permitivity, ( the dielectric constant of the medium, rij is thedistance between sites i and j. The angle 8 is between the dipole axis and the direction

joining the two neighborng dipoles. Fg. 6 illustrates the relevant situaton used in ourcalculatons.In Fig. 6a the signs "+" and "-,, refer to dipole-dipole interactions that prefer either aparallel or an antiparallel arrangement of dipole moments, respectively. Above in Table 1, wepresent the numerical results for the constants J1, J2 and JJ and the corresponding angleswhich were found based on the known structural data [4]. With the known strong axialanisotropy of interactions we can map this situation onto an anisotropic two-dimensionalIsing mod el o n a t a ngul ar lat c e so that the a pproximate e fec ve Ham lto nian i s now

given by

Ei;=-- -(2.2)(2.3)

lment of a'oleof uninterrupt'edIf 3 generic types~, both ends grow \,e other shrinks.ote the strikingies of MT's tendllations (121.

regular ( trian-diate neighborsldergoes a con-" where elastic

H -~ J . sz sz--L.., 'J .J (3.2)and the effective spin variable sf = :f:1 denotes the dipole's projection on the vertical MTaxis. The exchange constants Jij take the values J1, J2, JJ depending on the choice of dipolepairs. Due to the fact that J2 < O and that there are an odd number (13) of protofilaments,the system exhibits frustratiQU 114] in its ground state. This means that for any closedpath, it is impossible to satisfy all bond requirements. Hence, there will always be a conflict(hence the word frustration) between satisfying the energetical requirements of "+"-bondsand "-"-bonds. The ensuing dipolar phase structure is known in the physical literature as a~in-g~ phase 115,16]. n a spin-glass (SG), spin orientations are locally "frozen" in randomdirections due to the fact that the ground state has a multitude of equivalent olientations.For example, for each triangle reversing the spin on one side with respect to the remainingtwo leads to an energetically equivalent configuration. Having the number of triangles onthe order of the number of lattice sites, i.e.. N "' 2 .104, yields the degeneracy of the groundstate on the order of 6N which is a very large number! This provides a very convenientproperty from the point of view of encoding information in such a highly degenerate dipolarlattice state. The basic structure of each of the three phases is schematically shown in Fig.7 where the different phases are contrasted.2.3).


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THE CELL'S MICROTUBULES39:J

fnergy

~

so~t poiarIzaIJoo~

the two electronicof a unit cell.l.egCIld:-PaneIedrIc (bieb-temp)

IlXndilader-femJelccIrk: (kIw-Iemp)llXnooler

t t t t t t 111111-Spin GI... (in"'-lial Ie~: fioile .Iza)lpin rrul~ "- t 1 t 1 t I... .:::=>

FIG.7. A comparison of the three distinct dipolar phases of a microtubule.4. DYNAMICS IN THE FERROELECTR1C PHASE

When a moderate external electric field is applied (or when the temperature is loweredfurther) a ferroelectric phase is favored which is characterized by long-range order manifestedby an alignment of spin directions along the axis. Obviously, from the point of view ofinformation processing potential, this is n2! a very useful phase. However, it can playamajor role in the assembly/disassembly processes as we discuss below.In the ferroelectric phase, the MT system ha..' a strong uniaxial dielectric anisotropy sothat the array of dipole oscillators can be effectively describcd in terms of only one degreeof freedom. In fact Athenstaedt [6] showed experimentally that a tubulin dimmer ndergoes aconformational change induced by the GTP-GDP hydrolysis in which one monomer shifts itsorientation by 29from the dimer's vertical axis. Thus it was deduced that the single degreemeaning.


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THE CELL'S MICROTUBULES395ler's displ~ement froms a giant dipole. WhenJuter surf~es of a MTJresents the orientatedined by the presence ofther with the polarizedtric field parallel to its

thus given by

: cun] . (4.1)

the longitudinal dis->nd term arises fromIt. The overall effectrely described by the8 are model param-Jly a linear..S-Ction, ie. A ~ a(T-Tc).such that2 V(O) =~O...in = -A /48 (Fig,e the average effect

due to the environment when all the neighboring dimmers ssume their equilibrium positions.The mobile electrons on e~h dimmer an be localized either more toward the Q-roonomeror more toward the 13-roonomer. The latter possibility is associated with a change in thedimer's conformation. Experimental evidence indicates that a conformational distortion of29 from vertical occurs in the 13state. Therefore, one can identify the variable u with theamount of 13-state distortion when the latter is projected on the vertical axis.In order to derive a realistic equation of motion for the system described, it is indis-

pensable to include the viscosity of the solvent and introduce the associated damping force.Assuming for simplicity that the solvent is made up of only water molecules the viscositycan be simply taken into account by adding the friction force to the equation of motion withdun~ -~~

(4.3)where T/J(.) represents the normalized displa{:ement field T/J(.) = ~ ' where Uo = ( ~ ) 112corresponds to the minimum of the double-well potential (Fig. Ba). The variable f. is themoving coordinate for the travelling-wave form of the solution

(x -vt) (4.4)

p = 1V [IAIM(v~ -V2)] -I/2(4.5)nd

q = qJB/A/-3/2E. (4.6}I c an b e s ho wn t he E q. ( 4. 4 } ha s a u ni q ue b ou n de d s ol ut o n w hi ch i s g i ve n b y t he

formula

u) -U2(f.) = U2 + ~( 4.7)as illustrated graphically in Fig. 9 where

A 1 3qE 3BUl = 2(-)1/2cos{-arccos[-(-)1/21}3B 3 2A A

A / 2 1r 1 3qE 3BU2 = -2(-)1 cos{- --arccos[-(-)1/21}3B 3 3 2A A

fJ = :f:~( ~ )1/2Vo 2M

(4.8)d is present; (b)

(4.9)

f/JF.F.+pf/JF.-f/J3+f/J+(1=O


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396 It. Tuszynskj et a[Thus the solution ( 4.7) is a kink giving the boundary between the two states uf. -+ +00 and u = U2 for f. -+ -00, if !3 > 0. The boundary moves with the unique

velocity given byu,

and it is gene~gation velociin this relatio1\V

T-*\=c (a)

.-~1-,-, SubstitutinglO3mls(for LJ

n-O n-3 n-2~ Thus takingvelocity of aI n n.1 n.2 n.3 n..

~ I-.1.-(b)

Assumirupropagation:

FIG.9. The foml of a kinklike excitation of eq. (4.7). (a) A graphic plot of the function of abounded kink. (b) The aITangement of dipoles involved in forming a kink.However, in~numerator IIelectric fieldwith L.It shoullignored he~example K~that thereMoreover,It is auBy addingmechanismpopulationhand, themotion mMT's as ~informatioConforbe coupl~'

:f:~(~A'Y M")'!2COS{ 3 '4.10)

and its width Ll is 2Mo ( )1/2 l '- ( ) B .Ul -U2For a long MT we can assume that for points that are sufficiently far from its ends themagnitude of the electric field can be approximated as E ~ Q( 47rfor2)-1 , where Q representsthe effective charge on the ends of a MT and r is the distance between the selected point andthe end of the MT. Taking as an example a moderately long MT consisting of approximately102 dimmerswith length L ~ 10-6m, the effective charge Q is estimated Q ~ 1.5. 10-16C sothat the intrinsic electric field in the vicinity of the middle point of a MT is found to beon the order of E ~ 106V/m. Accounting for the dielectric effects of the surrounding water

molecules, this value must be reduced to approximately E ~ 105V/m.As far as the potential coefficients A and B are concerned no reliable experimental dataexist so a crude estimate is made using come typical values for crystalline ferroelectrics know-ing that they do not vary substantially between different compounds. These are typicallyA~2.102Jm-2 B~1024Jm-4 (4.12)

~ i.ll )

which then yields and2.10-9.E 3)It is therefore clear that even for extremely strong electric fields (E "' lO8Vfm) the inequality a < 1 holds. The main consequence of the smallness of the electric field is that th,propagation velocity ( 4.9) can be safely approximated as

1 3qE 3B3 arccos(~( A )1/2]}


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THE CELL'S MICROTUBULES.397two states u : Ul forI the unique terminal 3Vo M Bv ~ -(-)1/2 qE (414 )YIAj 2 .

and it is generally much smaller than the sound velocity (v ~ Vo). Eq. (4.13) links the prop-agation velocity v and the magnitude of the electric field E. The coefficient of proportionalityin this relationship represents the kink mobility

Jl = ~ (~ ) 1/' I I q 2

(4,15)Substituting typical numerical data as; 'Y = 5,6. IO-llkgs-I(T = 3000K);vo = 1,7lQ3m/s(for DNA);M ~ 1.IO-22kg; and q ~ 6, IO-18C, yields

/l ~ 2. IO-5m2V-1 S-I,

~'I'i I

~

of the function off

(4.10)

(4.11)from its ends the1ere Q represents~Iected point andof approximately~ 1.5. 10-16C sor is found to berrounding waterperimental dataoe!ectrics know-~ are typically

(4.12)

(4.13)11 the inequaJ-~Id is that the

v ~ 2m/s. (4.17)Assumng a smooth journey from one end of the MT to the other, the average tme of

propagaton for a single kink should be

f = ; ~ 5. 10-7.'1


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THE CELL'S MICROTUBULES399-pU22 I (4.19) I(XX)I

800~Tc400

~

200(4.20)(4.21)

idering travellingj ODE's. Making11 erms j> and itI P. This tYpe ofg and chaos.

!'--'-~ N Tc (~~ ~~--20 304 400 31830 305 500 31340 296 600 30850 304 700 30460 302 800 31270 311 900 31480 301 1000 30690 299 1300 310

100 300 1500 303200 313

various phasesI by the Hamil- ~.the ferroelectric \it. The critical" 2J3k;r: --

(5.1) TABLE II. The dependenceof Tc on N for Q = 11.9 .10-56 C2 m2.12b) as two contrasting examples. In Table 2 we show how the transition temperature Tcchanges with N for the choice of model parameters which give Q = 11.9.10-56 C2 m2.We conclude that dynamic processes leading to the elongation of MT's could effectivelyremove the information processing capabilities of MT's by expelling the SG phase. Thesame can be achieved by raising the temperature above a characteristic value which is length

dependent.We have also examined the effect of external electric fields and MAP's (see Fig. 8

for typical MAP patterns) on the aforementioned transition. The electric field shifts thetransition region and makes it broader. A similar effect can be seen by incorporating MAP'sas "empty" (i.e. non-polar) lattice sites. The actual magnitude of the shift and broadeningdepends on the pattern of MAP's chosen and the ratio of MAP's to the total number oflattice sites. Taking the set of parameter values which yields Tc = (300:f: 15) K for theperfect lattice results in Tc = (250:f:20) K for the lattice with MAP's at a ratil>of 1:11 whileTc = (230:f: 20) K is obtained for a ratio of 1:8. This indicates that MAP's substantiallylower the transition temperature and make the SG-phase accessible to the MT system at

much lower temperatures that those required in the absence of MAP's.

of Tc dependsshown in Fig.the transitionociated phasesitive control)nal change),rrounding an)rder. Small.Ies, may re-se. Changes-phase effec-11 rocel;Sing

i:!;,1

,fonte-CarJo1licrotubuJeISS phase isis effect by5000 (Fig.


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THE CELL'S MICROTUBULES

~~ -.

..,..

~ .. -T-~(a)

~

~--

,.~~..lA

..0;; ,...; ~ -

,-~

(b)

~

rent conformation~tional states? Such'1 cognitive function.)rrelated with peakbegin their critical~urons in the visualction is drasticallyays old). Bensimon19colchicine causedoms of Alzheimer's

Lpi = 1 with O $ Pi $ 1i (6.2)

torage) stems from,r example, tubulinis very close to the3ers have maximalIlamically couplingve dipole states.

j:J,\

~I

identified.on asvon the

I = In Z -Q < p2 > +1 < p4 > (6.5)


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THE CELL'S MICROTUBULES. 403

mer for both the ferro-med. For the SG-phase,lin the local domain ofmain i we have a localsentially analogously toty distribution becomes[}of coherence

"I = LIi (6.9)i=l

where I; refers to eoch individual domain. Our numerical computation clearly indicates thatinformation capocity I is highest at the boundary between the spin-glass and the paraelectricphase (See. Fig. 14) and hence if MT's are to be effective as information processors, theyshould use this narrow "window of opportunity" at the border area between these two phases.Of course, the octuallocation of the border area depends on the magnitude of the electricfield applied and the concentration of MAP's present.

7. SUMMARY

~

(6.8)raelectric phases sinceI.rized while at T ==TAuce of (4.7) we obtain

600,500400300200100

Paraelectric

Spin Glassferroeleclric

-loot T. T.Temperature (K)FIG. 14. Plot of the infonnation capacity as a function of temperature.

.hases: (a) paraelec-


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j.a. tuszynski, b. trpisova, d. sept, m.v. sataric and s.r. hameroff: the cell's microtubules:...

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