m.v. sataric, j.a. tuszynski, s.r. hameroff and r.b. zakula: nonlinear mechanism of dipole dynamics...

8/3/2019 M.V. Sataric, J.A. Tuszynski, S.R. Hameroff and R.B. Zakula: Nonlinear Mechanism of Dipole Dynamics in Microtubul

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NONLINEAR MECHANISM OF DIPOLE DYNAMICSIN MICROTUBULF;}S; IMPLICATIONFOR INFORMATION PROCESSING,

SIGNALING AND ASSEMBLYM. V. Sataric 1, J.A. Tuszyriski 2, S. Hameroff 3, R.B. Zakala 4

1Faculty of Technical Scienc~s, Novi Sad,. Yugoslavia2 University of Alberta, Edmonton, Canada

3 Department of Anesthesiology, University of Arizona, Tuscon, Arizona, USA4 nstitute of Nuclear Sciences, Beograd, Vinca, Yugoslavia

AbstractInteriors of living cells are structurally and dynamically organized by networks of protein polymers

called cytoskeleton. Of these filamentous structures micro tubules are the best characterized and ap-pear to be most fundamental. Some evidence links the cytoskeleton with information processing andcognitive function.In this paper we address the question of how some functions of micro tubules may be related to theground state properties of the lattice of dipoles. In general, three kinds of geometrical arrangements ofdipoles are found: (a) random, (b) ferroelectric and ( c) spin-glass types. Each of these arrangementsmay exist under different conditions of temperature, electric field and microtubule length. The ferro-electric state appears most suitable for assembly and signaling processes whUe the spin glass state isideally suited for information processing.

Introduction1Ilteriors of living cells are structurally organized by networks of protein polymers called the cytoskeleton.Of these filamentous structures which include a-ctin and intermediate filaments, microtubules (MT's) arethe best characterized and appear to be the most fundamental (Dustin, 1984). MT's are hollow cylinders2.5 nanometers (nro) in diameter whose lengths may span macroscopic dimensions. MT's are assembliesof 13 longitudinal protofilaments, each of which is a series of subunit proteins known ~ tubulin (Figu.re1 . Each tubulin subunit is a ~olar, 8 nro-long dimmerwhich consists of two slightly different 4 nm:.longmonomers, each with a molecular weight of 55 kilodaltons. These two constituent parts are referredto as Q and 13 ubulin. Each dimmerhas a net mobile electronegative cha-rge due to an abundance ofacidic arnino acids, and binds 18 calcium ions. The net negative mobile charge is localized predomina-ntlytowards the Q monomer so that each tubulin dimmer ears an oriented dipole. Thus, MT's are an exampleof an electret substance, i.e., an assembly of oriented dipoles. In this paper we investigate the types ofspatial arrangements of these dipoles in detail and their implications on MT behavior.

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x -+-E .1111 41

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tp-t-:

;.p.Figure 1. An Illustration of the microtubular arrangement.

Conformational states of tubulin dimmerspresent within MT's have been suggested to be coupled tocharge or dipolar states, thereby allowing for cooperative interactions with neighboring tubulin dimmersites (Hameroff, 1987; Rasmussen et al, 1990; Hameroff and Watt, 1982). This coupling would also leadto the presence of piezoelectric properties which are very common in ferroelectrics which, in turn, couldprove very important in MT signaling, communication and assembly disassembly behavior (Athensteadt,1974; Margulis et al, 1978).

MT associated proteins (M.I\.Ps) which may be structural, contractile or enzymatic are attached toMT's at specific tubulin dimmer ites and can link MT's with other MT's, 9ther parallel arrayed cytoskeletalelements, membranes and organelles to form networks and scaffolding within living cells.

Some evidence links the cytoskeleton ',,:,ith nformation processing and cognitive function. For example;Mileusnic et al (1980) correlated production of MT subunit protein ("~ubulin") and MT activities withpeak learning, memory and experience in baby chick brains. Cronley-Dillon et al (1974) showed that whenbaby rats begin their critical learning phase for the visual system (when they first open their eyes , neuronsin the visual cortex begin producing vast quantities of tubulin. Tubulifi,production is drastically reducedwhen the critical learning phase is over (when the rats are 35 days old). Moshkov et al (1992) showedstructural and chemical changes in goldfish brain neuronal cytoskeleton..following sensory stimulation. Ingerbils e.xposed to cerebral ischemia, K udo et al ( 1990) correlated the amount of reduction in dendriticMAP2 with the degree of cognitive impajrment. Bensimon and Chernat (1991) found that selectivedestruction of brain MT's by the drug colchicine caused cognitive defects in learning and memory whichmimic the clinical symptoms of Alzheimer's disea"!;e,n which the cytoskeleton becomes entangled. Geertset al (1992) showed that sabeluzole, a memory-enhancing drug, increases fast axoplasmic transport.Matsuyama and Jarvik (1989) have proposed that Alzheimer's is a disease of MT and MAPs. In specifichippocampal regions of the brains of schizophrenic patients, Arnold et al (1991) fourid distorted neuronalarchitecture due to a lack of2 MAPs (MAP-2 and MAP-5).

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Models of learning in mammalian NMDA (n-roethyl-d-aspartate) hippocampal neurons (i.e. long termpotentiation: "LTP" ) involve pre- and post-synaptic enhancement of synaptic function. Silva et al (1992)have shown calcilim-calmodulin kinase to be essential for LTP. Aszodi et al (1991) have demonstrated pro-tein kinase A to be involved in learning. Halpain and Greengard (1990) have shown "that post-synapticactivation ofNMDA receptors induces rapid depho;sphorylation of dendrite-specific MAP2. Aoki andSiekevitz (1988) linked learning to protein kinase C mediated MAP2 phosphorylation/dephosphorylation,and Thelirkalif and Vallee (1983) demonstrated that MAP2 phosphorylation/dephosphorylation con-sumed a huge proportion of brain biochemical energy. Bigot and Hunt (1990) showed that NMDA andother excitatory amino acid neurotransmitters caused redistribution of intraneuronal MAPs, and Wangand Rasenick (1991) have shown that G proteins directly link with MT's. Surridge and Burns (1992)demonstrated that the lipid second messenger phosphatidylinositol binds specifically to MAP2, a cou-pling which may "constitute a link by which extracel1ularjactors influence the microtubule cytoskeleton".Kwak alld Matus (1988) showed that denervation caused long-lasting changes in MAP distribution, andOesmond and Levy ( 1988) showed that LTP involved cytoskeletal-roediated shape changes in dendriticspines. Lynch and Baudry ( 1987) have proposed that proteolytic digestion of the sub-synaptic cytoskele-ton, followed by its structural reorganization, correlate with learning and Friedrich (1990) has formalizeda learning model of synaptic cytoskeletal restructuring. Synaptic regulation, both in learning and steadystate, also depends on material (enzymes, receptors, neurotransmitters, etc:) synthesized in the cell bodyand transported along ax9ns and dendrites by contractile MAPs attached to MT ("axoplasmic trans-port"). Similar mechanisms involving, actin, synapsin, and other cytoskeletal polymers are involved inneurotransmitter release.Further suggestion for cytoskeletal computation and/or information storage stems from the spatialdistribution of discrete sites ( or states) in the cytoskeleton. For example, tlibulin subunits in closelyarrayed MT have a density of about 1017 per cm3, which is very close to the theoretical limit for chargeseparation (Gutroan1L, 1986). Thus cytaskeletal polymers have maximal density for information represen-tation by charge, and the capacity for dynamically coupling that information to mechanical and chemicalevents via cooperative dipole states.

The dipolar lattice and spin-glass phaseOur baSic premise physically views the entire MT as a regular array of coupled local dipole states whichiJlteract with their immediate neighbors. Although tubulin undergoes conformational changes (e.g. Melkiet al, 1989), we a;dopt a simplified view in which elastic degrees of freedom are not explicitly included inthe description. We will return to this question in the Discussion.

The starting point in the analysis is to adopt a hexagonal lattice structure with the dimensions andorientations observed by X-ray diffraction of MT and shown in Fig. 2. Each lattice site is assumed topossess a dipole moment p = Q .d where Q = 2e and d ~ 4 nm whose projections on the vertical a-Xiscan only be +p or -p. The interaction energy between two neighboring lattice sites is

3 COS2 J -1 23 pr. .'3 (2.1)ij = / 1

where 0 is the vacuum permitivitty and Tij is the distance between sites i and j. The angle O s betweenthe dipole axis and the direction between the two dipoles. Fig. 2 illustrates the situation resulting fromour calculations; .

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(b}(a)

Figure 2. A tubulin dimmer nd i ts nearest neighbors. ( a) Exchange constants (b) and their signs ( cwithin a unit hexagonal cell ( c . ( d ) A band of hexagons spanning the circumference of a microtubule.

In Fig. 2c the signs" + " and" -" refer to dipole-dipole interactions that prefer either a parallel orantiparallel arrangements of dipole moments, respectively. Below, we present the numerical results forthe constant J1, J2, J3 and the corresponding angles,

]1 = 5.77.10-21]]2 = -0.71.10-21]]3 = 3.40 .10-21 ]

el = 00e2 = 58.2e3 = 45.6.

We can, therefore, map this situation as an anisotropic two-dimensional Ising model on a triangularlattice so that the effective Hamiltonian is

H = -'\:""""' J. . S .Z S~L.., tJ t J (2.2)and the effective spin variable Si = :!:1 denotes the dipole's projection on the vertical axis. The exchangeconstants ]ij take the values ]1, ]2, ]3, depending on the choice of dipole pairs.

What, therefore, appears in Fig. 2b is that the system exhibits frustration (Suzuki, 1977) in its groundstate. This means that for any closed path it is impossible to satisfy all bond requirements. Hence, therewill al ways be a conflict between satisfying the energetical requirements of .', + " bonds and " -" bonds.The ensuing phase structure is known in the physicalllterature as a spin-glass phase (Fischer, 1983;1985 . In a spin glass (SG ), spin orientations are locally "frozen" in random directions due to the factthat the ground state has a multitude of equivalent orientations. For example, for each triangle reversingthe spin on one side with respect to the remaining two leads to an equivalent configuration. Having thenumber of triangles on the order of the number of lattice sites N "' 2 .104 yields the degeneracy of the

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ground state on the order of 6N which is a very large number! (namely"", lO1S,OOO)his is schematicallyshown in Fig. 3.

A schematic representation of the energy dependence on co:nformationa.l states.Figure 3.Small potential barriers separating the various equivalent arrangements of spins play another useful

role. Relaxation times are very long for the various accessible states giving them long-term stability.Some other properties of the spin glass phase are: ( a) absence of long-range order between spin sites; (b )the presence ofshort-range correlations This means that the system is very "soft" energetically permittinga number of spin arrangements which are relatively stable. However, there is no tendency towards theformation of large correlated spin clusters. Thus, a spin glass phase exists near a transition to chaos.LaJIgton (1990) has shown that such a quality is optimal for computational applications as discussedby Lcwin (1992). Spin glass phases also allow easy formation of local ordered states, each. of whichcarries an information content and is long-lived, i.e. relatively stable over time. On the other hand, itis not difficult to switch from one state to another ( e.g. f-rom spin-glass to ferroelectric) through variousphysical means. 'rhis can be achieved by: (a) an application of an electric fieln along the axis. Weestima.te that intracellular fields on the order of 104 V /m are sufficient to switch an MTfrom a spin glassto a ferroelectric state (F). (Note that fields up to 107 V /m are known to exist across cell membranes). (b)Raising the temperature above the spin-glass transitioRtemperature will drive the system to a disordered(paraelectric) phase. ( c) A spin-glass is also sensitive to boundary conditions. This means, for example,that the presence of domain walls will reduce the effective size of a correlated cluster and this will have aprofound effect on the SG phase. As will be discussed in the next section, domain walls may easily formin a microtubule under specific conditions.We mentioned above that another ordered phase is favored when a moderate external field is applied( or indeed when the temperature is lowered further ). This phase, called a ferroelectric phase, is charac-terized by long-range order manifested by an alignment of spin directions along the axis. This is shownin Fig. 4. Obviously, from the point of view of optimal information content, this is ll.Q1 a very usefulphase. However, the ferroelectric phase can playa major role in signaling and the assembly / disassemblyprocesses. This will be discussed at lenght in the next Section. Before we discuss this aspect, we inves-tigate its general properties. The ferroele


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fEf T < Tcf

r~igure 4. The dipole orientations in the ferroelectric state..The result we obtajned is Tc ~ 402 K for the charge of q = 2e, the permitivity E: = io and the dipoledisplacement d = 4 nm. Takjng q = le, E:= 5 and d = 4 nm yields Tc = 201 K. This very rough estimateof Tc indicates nevertheless that an interesting phase behavior can be expected in the physiologicaltemperature range around 300 K. In addition, it js well known that the size of a lattice affects Tc greatly.In general, for size N, T c( N) is related to that for an infinite lattice T c( 00) through (Binder et al, 1985).

(2.4)where a is, an approximate proportionality coefficient and l' is the correlation length exponent ( ypicallyclose to 1/2). Thus, for a.finite-size lattice the transition temperature is expected to be less than that foran infinite lattice. In other words, dipole ordering is more likely to occur when a critical size of the MTis achieved. In addition, other important factors may affect the value of Tc and thus provide sensitivecontrol mechanisms. Through a coupling to the elastic degrees of freedom (conformational change), thevalue of Tc may be shifted. The dielectric constant may be altered by the presence of water moleculessurrounding an MT structure. This would decrease the value of T c and introduce disorder. Small changesin the angles between the dimer dipoles may remove the frustration mechanism effectively preventing theonset o(the SG phase. Changes in the opposite direction can enhance frustration favoring the SG overthe F phase and effectively switching from the growth mode of operation to an information processingbehavior. Thus symmetry transitions or mechanical forces may effect switching between phases.

Once in the [erroelectric phase, the system is also very sensitive to the frequency of electromagneticoscillations. It is known that in organic ferroelectric materials a resonant frequell;cy exists Wo whichis inversely proportional to the length, stiffness constant and the square root of the density (Matthias,1973). Extrapolating the data given by Matthias (1973) one finds for microtubules a resonant frequencyrange exceeding 108 Hz and thus being rather close to the microwave range predicted by Frohlich (1980)to playa major role in living cells.In Figure 5 we have shown the results of our Monte Carlo computations intended to reveal the sizedependence of the phase behavior discussed above. Figure 5a) is a plot of the mean polarization persite when the microtubule is composed of only 26 layers (i.e. it is a 13 x 26 lattice) it is clear thatthree distinct ranges exist: ( a) the low- temperature ferroelectric range, (b) the intermediate spin-glassrange and ( c) the high-temperature paraelectric'"fange. Figure 5b) illustrates the same dependence for amicrotubule with 5000 layers (i.e. it is a 13 x 5000 lattice). We conclude that the intermediate spin-glassphase all but disappeared.

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Mean Polarization Per SiteMoon Pol81"iz8lion POI" Sitet. [

0. .

0.610.41

i

0.210.,200 300 400

T(K)

A plot of the mean polarization persite for a triangular lattice whose size is a) 13 x 26 andb) 13 x 5000.

Figure .5

FerrQelectricity, signaling and assembly /disassemblyIt is well known that ferroelectrics are on~ of the few types of materials whose description is most~dequately provided in terms of mean field models (Kretschmer and Binder, 1979). A typical approach isto postulate a double-well quarticon-site potential V(Sn) that gives the overall effect of the environm~nton a selected dipole site n. Thus,

V(Sn) = -~AS~+ lBS~ (3.1)where Sn is the effective spin variable at site n and the coefficients A ~nd B are model dependent. Itwill be subsequently assumed that over long distances the variable Sn cha:ngesvery gradu~y and can betreated as a continuous variable.In the F -phase, the MT cylinder taken as a whole represents a giant dipole. When the cross-section ofa MT is viewed using electron microscopy, the outer surfaces of a MT are surrounded by a "clear zone"of several nm which apparently represents the oriented molecules of cytoplasmic water and enzymes(Stebbings and Hunt, 1982). This could be explained by the presence of an electric field produced bya MT. Therefore, we assume that together with the polarized water surrounding it, a MT generatesa nearly uniform intrinsic electric field parallel to its axis (see Fig. 1). Consequently, the additionalpotential energy due to this electric field and associated with each dIpole is

(3.2)el = -CSn, .C = qE

where q denotes the effective cha.rge of a "single dimer and E is the magnitude of the intrinsic electric field.Within this picture, the model Hamiltonian for the dipole moments of a microtubule can be written as

{3.3}The first term above represents the kinetic energy associated with the longitudinal displacements ofconstituent dimmerseach of which has mass M. The second term arises from the restoring strain forcesbetween adjacent dimmersn the protofilament. 1f the stiffness parameter K is sufficiently Jarge, then itis expected that large-amplitude long-wave length excitations of the displacement field will be formedmanifested by a slowly varying modulation of Sn along the MT axis.

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In order to derive a realistic equation of motion for this system, the viscosity of the solvent is indis-pensable and the associated damping force must be introduced. Assuming for simplicity that the solventis made up of only water molecules we may infer that water will provide a viscous medium that willdamp out vibrations of dimmerdipoles, This can be taken 'into account by adding the viscous force to theequation of motion with F = -1'~ where l' is theda.mping coefficient.

Based on our assumption that the dipolar oscillations of dimers within a MT form a system thatcan be classified as displacive ferrodistortive, we can use the continuum approximation to obtain thecorresponding equation of motion as

by(3.5). = [~]1/2(X ~ Vt) = a(x -Vt)

with V denoting the propagation velocity and the coefficient a is, of course,- [ IAI ] 1/2a- M Vo2 -V2) ,

Consequently, eq. (3.4) is reduced to the ordinary differential equation belowM Q2(V2 -VO2)S'I -/QV Sf ~ AS + BS3 -qE = 0. (3.6)

Where S' = dS! df. This is an equation for an anharmonic oscillator with linear friction.For the sake of convenience we now introduce a normalized displacement field as: 'I/J(f.) = S(f.)! Sowhere the normalizing coefficient So = vrxrm corresponds to the minimum of the double well potentialin eq. (3.1).It has been shown by Collins et al (1979) that eq. (3.6) has a unique bounded solution which is given ,by the formula ( and illustrated graphically in Fig. 6

(3.7)

Figure 6. The function '!/J(.) at rest ( -) and moving at a terminal velocity V > O n the appliedelectric field E( ...).

8

-c'; c7*-~~'"-'--';';;~;--,!--,-~~ -,

Note that the x-coordinate is along the protofilament axis. Moreover, for longitudinal sound waves thedispersion relation I..J= ,jKJM can be used to identify the sound velocity with VO2 I..JRo= ,jKJMRo.We now seek solutIons of eq. (3.4) in the traveling-wave form where the moving coordinate 0; s given


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where ,8 = (b -a)/J2 and the parameters a, b and d satisfy the cubic equation(3.8)I/; -a )( 'I/; -b )( 'I/; -d) = '1/;3 'I/; -0"

where 0- = qVBrAI-3/2 E.Formula (3.7) describes a kink-like excitation (KLE) which takes the ,form of a domain wall separatingtwo distinct ferroelectric domains.It is also imp.ortant to note that the above kink-like soliton propagates along the protofilament witha fixed velocity

- 1[ 212 ] 1/2V -Vo 1 + 9d'iMV,' (3.9)which is ~ than the sound velocity Vo and decreases with an increase of the friction coefficient 1. It isclear that V depends on the magnitude of the electric field E through the parameter d.

For T well below Tcl V can be approximated by a linear relationship with E

where the coefficient of proportionality is called mobility and will be denoted by Jl. Using simple argu-ments from fluid dynamics, Jl can be estimated numerically as

J.L ~ 2 .lO-5m2V-1 S-1

Thus, for E = 105 Vm-l we obta-in V ~ 2 m/s which can increase to 102 m/s close to the MT's endswhere E increases dramatically.Assuming a smooth journey from one end of the protofilament to the other we estimate the averagetime of propagation T for a single KLE to be

To summarize, we showed that a unique kink-like excitation exists in the ferroelectric phase whichmay propagate along a MT with a velocity that is proportional to the electric field E. KLE's in MT'swhich ate stable may constitute signals transducing membrane events via second messengers to othercell regions. Energy for such KLE's may come from GTP hydrolysis or phosphorylation via MAPs. Inneurons, KLE's in MT's may signal error information retrograde (i.e. from axon to dendrite) whichwould be useful in synaptic regulation and learning as in artificial neural network "back-propagation".In MT's which are unstable (i.e. whose ends are not stabilized by MAPs or other structures), KLE'scan explain the fact that growth rates are different at the two ends of a MT. To this end we assumethat KLE formation is mainly due to the hydrolysis of GTP into GDP so that one act of hydrolysiscorresponds to conformational change resulting in the formation of a single KLE. However, a KLE ispreferentially oriented towards the direction of the intrinsic electric field. The propagation of a KLE willthen distribute the energy of hydrolysis at the prefereed end ofa MT. This energy :may then be used todetach dimers from the MT. This picture is in accordance with a hypothesis put forward by Kirschnerand Mitchi~on (1986) who stated that on-rate (growth) is limited in principle only by the rate of diffusionof the monomer subunits into MT polymers (i.e. by the concentration of the constituent monomers insolution). On the other hand, the off-rate ( depolymerization) can be extremely rapid and the net amountof MT polymer formed can be regulated independently by the hydrolysis of GTP.

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It is reasonable to expect that at the onset of the MT assembly process, when MT's are very short,the formation of KLE's is very unlikely ( or even impossible ), since lor a short chain the displacementfield gradient required would be very high and the associated energy to form it prohibitively large.However, when the length of a MT reaches a threshold value, kink propagation could become a realpossibility. Excitation of kinks being related to GTP hydrolysis would become dependent on the GTPconcentration. As MT's become longer, KLE propagation accelerates leading to a disassembly at the"-,, end. KLE propagation in stable MT's would depend on MAP phosphorylation or cytoplasmic kinaseactivity triggered by membrane events and second messengers.

Summary

References[1] Aoki C; and Siekevitz P., Sci Am 34-42, 1988.,[2] Arnold S.E., Lee V .M. Y ., Gur R.E. and Trojanowski J .Q., Proc Natl Acad Sci 88:10850-10854) 1991.[3] AthenstaedtH., Ann. N.Y. Acad. Sci. ~, 68 (1974).[4] Barnett M., in F. Carter (ed.), (Naval Research Laboratory) Washington) D.C., 1987).[5] Bensimon G. & Chernat R., Pharmacol Biochem Behavior38:141-145, 1991.[6] Bigot D. and Hunt S.P.) NeurosciLett 111:275-280,1990.[7I Binder K., Nauenberg M.) Privman V. and Young A.P., Phys. Rev. B.~, 1498 (1985).[8] Collins M.A., Elumen A., Currie J.F. and RossJ., Phys. Rev. EN, 3630 (1979).[9] Cronly-Dillon J ., Carden D. & Birks C., J Exp BioI 61:443-454, 1974.

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[12] Dustin p;, Microtubules (Springer, Berlin, 1984).[13] Fischer K.H., Phys. Stat. Sol. (6) ill., 357 (1983).[14] Fischer K.H., Phys. Stat. Sol. (6) liQ., 13 (1985).[15] Friedrich P., Neurosci 35:1-T, 1990.[16] Frohlich H., Adv. Electr. Electr. Phys. ~, 85 (1980).[17] Geerts H., Nuydens R., Cornelissen F., De Brabander M., Pauwels P., Janssen P.A.J., Song Y.H.

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177-197,1986.[19] Halpain S. and Greengard P.. Neuron 5:237-246, 1990.120] Hameroff S., Ultimate Computing (Elsevier North-Holland, Amsterdam, 1987).[21] Hameroff S.R., Dayhoff J .E., Lahoz-Beltra R., Samsonovich A., Rasmussen S., IEEE Computer

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Biology", pp. 12-20 (Karger, Basel, 1973).[31] Melki R., Carlier M.F., Pantaloni D. and Timasheff S.N., Biochem. 2.8.,9143 (1989).[32] Mileusnic R., Rose S.P. & Ti1lson P., Neur Chem 34:1007-1015, 1980.[33] Mithieux G., Chauvin F ., Roux B. and Rousset B., Biophysical Chem 22,307-316 (1985).[34] Moshkov D.A., 8a.valjeva.L.N ., Ya.njushina. G. V .,Funtikov V .A., Acta Histochemica Suppl-Band XLI,

8:241-247,1992.[35] Rasmussen S., Karampurwala H., Vaidyanath R., Jensen K. and Hameroff S.R:, Physicia D .4;..2.,428

(1990).[36] Silva A. V ., Stevens C.F ., Tonegawa S. and Wang, Science 257:201-206, 1992.[37] Silva A.V., PaylorR., Wehner J.M. and Tonegawa S., Science 257:206..211,1992.

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m.v. sataric, j.a. tuszynski, s.r. hameroff and r.b. zakula: nonlinear mechanism of dipole dynamics...

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