J.A. Tuszynski, B. Trpisova, D. Sept, M.V. Sataric and S.R. Hameroff: "Microtubular Self-Organization and Information Processing Capabilities"

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(parallel-aligned), and of spin-glass type. Each of the above structuresexists under different conditions of temperature, electric field, MAP distri-

bution, and MT length giving rise to a possible sensitive state-switching

mechanism. The ferroelectric state appears to be most suitable for assem-

bly I disassembly processeswhile the spin-glass state is ideally suited for

information processing and thus can be seenas providing the substrate or

consciousness) s will be argued in this paper.

THE EMERGENCEOF DIPOLAR PHASES

Our basic premise is that the entire MT may be physically viewed as a reg-

ular (triangular) array of coupled local dipole moments that interact with

their immediate neighbors via dipole-dipole forces. Although Melki et al.

(1989) showed that tubulin undergoes a conformational change (Figure

30.2a),we will tentatively adopt a simplified view where elastic degreesof

freedom are not explicitly incJuded n the description. However, an appro-

priate generalization of the physical model poses no technical difficultiesand we comment on it in the last section.

The starting point in the analysis s to adopt a triangular lattice Structure

(located on the surface of the MT) with the dimensions and orientations asshown in Figure 30.2b and 30.2c.Each attice site is assumed to possessadipole moment p = Q .dwhere Q = 2 e and d = 4 nm and its projection on the

japted from Amos and

I1gperiod an MT sud-

alled a rescue).

:1ized which leads o

1d Mandelkow 1992).

:ormation processing

nett (1987)suggested

IS nformation strings

structural similarity

tecture is quite strik-

r""-.d: -lpha e

State ~

Beta

State

(a)

~#e

~

4.9nm

-..,ansferring

1tS of the cell interior

rallel arrays

8nmJz

5ioned as playing the

c.~e r

1- .rij

-

I--f

5nme emergence of con-

s ground-state prop-nts of dipoles we see

ectric), ferroelectric

(b) (c)Figure 30.2 A graphical description of the structural units in a microtubule: (a)

the two electronic states of a dimmer, b) the dimensions and angles of a unit cell,

and (c) a schematic illustration of the model parameters used and their meaning.

409 Tuszyrtski et al.: Microtubular Self-Organization



vertical axis can only be + p or -p. The nteraction (dipole-dipole) energy E..

between two neighboring lattice sites (labelled and ). is, therefore, 'I

1 3cos2(j-l

-

Ejj= 347rEE O r ..

'1

where EOs the vacuum permitivity, e he dielectric constant of the medium,

r j is the distance between sites and . The angle 6 s between the dipole axis

and the direction joining the two neighboring dipoles. Figure 30.3 llus-

trates the relevant situation used in our calculations.In Figure 30.3a he signs " + " and" -" refer to dipole-dipole interac-

tions that prefer either a parallel or an antiparallel arrangement of dipole

moments, respectively. The numerical results for the constants 1' /2' and /3and the corresponding angles that were found based on the known struc-

tural data (Rasmussenet al. 1990)are:h = 5,77 .10-21/, 2 = --0. 71 .10-21/,

!3 = 3.40.10-21 ,61 = 0°,62 = 58.2°, and 6; = 45.6°.With the known strong axial anisotropy of interactions we can map this

situation onto an anisotropic two--dimensional Ising model on a triangular

lattice so that the approximate effective Hamiltonian is now given by

p2 (1)

(2)

and the effective spin variable Siz= :t1 denotes the dipole's projection on

the vertical MT axis. The exchange constants ij take the values /1, 12,13

depending on the choice of dipole pairs.

Due to the fact that 12 < O and that there are an odd number (13) of

protofilaments, the system exhibits a certain amount of frustration (Suzuki1977).This means that for a closed path along the direction corresponding

to 12, t is impossible to satisfy all bond requirements. Hence, there will

always be a conflict (hence the word frustration) between satisfying theenergetical requirements of " + " bonds and " -" bonds. The ensuing dipo-

lar phase structure is known in the physical literature as a spin-glassphase

(Fischer 1983, 1985). In a spin-glass (SG), spin orientations are locally"frozen" in random directions due to the fact that the ground state has a

multitude of equivalent orientations. For example, for each riangle revers-

ing the spin on one side with respect o the remaining two, leads o an ener-

getically equivalent configuration. Having the number of triangles on theorder of the number of lattice sites, hat is, N -2 .104,yields the degener-

acy of the ground state on the order of ~ which is a very large number!This provides a very convenient property from the point of view of encod-

ing information in such a highly degenerate dipolar lattice state. Note,

however, that the other two directions do not exhibit frustration and thislimits the extent of the SG phase.

Relatively small potential barriers separating the various equivalent

arrangements of spins in the SG phase may playa very useful role sincerelaxation times are very long for the various accessible tates.Someother

properties of the spin-glass phase are the absence of long-range order

410 Tuszyliski et al.: Microtubular Self-Organization411







:i is long-

le state o

the small

ion of the

eld along

sufficient:hat fields

quite fea-

ature willtlso sensi-

'presence

the tem-

"acterizedons along

.rocessingnajor role

-nski, and

in the F

! range of

mation

perature lies between 200 K and 400 K indicating the possibility of the asso-

ciated phase ransition close to room temperature, that is, at physiological

conditions.

Many important factors may affect the value of Tc and thus provide sen-

sitive control mechanisms for phase selection. Through a coupling to the

elastic degreesof freedom (conformational change), he dielectric constant

E may be altered by the presence of water molecules surrounding an MT

structure thereby decreasing the value of Tc and introducing dipolar dis-

order. Small structural changes, n particular shifts in the angles betweenthe diller dipoles, may remove the frustration mechanism effectively pre-

venting the onset of the SG phase. Changes n the opposite direction can

enhance rustration favoring the SG over the F-phaseeffectively switching

from the growth mode of operation of MTs to their information-process-

ing behavior.

In order to obtain some insight into the above questions, we have per-

formed Monte--Carlo simulations for finite lattices with dimension 13 X N

(N is the length of the microtubule in terms of the number of layers). It is

clear that as N increases, he SG phase s gradually removed. We see thiseffect by directly plotting the mean polarization per site for N = 26 (Figure

30.3a)and N = 5000 (Figure 30.3b) as two contrasting examples.

We conclude that dynamic processes eading to the elongation of MTscould effectively remove the information processing capabilities of MTs by

expelling the SG phase. The same can be achieved by raising the tempera-

ture above a characteristic value that is length dependent.

We have also examined the effect of external electric fields and MAPs on

the aforementioned transition. The electric field shifts the transition region

and makes t broader. A similar effect can be seenby incorporating MAPs

as "empty" (that is, non-polar) lattice sites. The actual magnitude of the

shift and broadening depends on the pattern of MAPs chosen and the ratio

of MAPs to the total number of lattice sites. Taking the set of parametervalues which yields Tc = (300 :!:15) K for the perfect lattice results in Tc =

(250:!:20)K for the lattice with MAPs at a ratio of 1:11while Tc = (230:!:20)

K is obtained for a ratio of 1:8. This indicates that MAPs substantially

lower the transition temperature and make the SG-phaseaccessible o the

MT system at much lower temperatures than those required in the absence

of MAPs.

Nards

;ation of a

-erred end

the MT at

he various

n the con-

lIning that

5 the most

nce to the

nfinite tri-

only two

below Tc),

lat the crit-

the combi-le realistic

,itir !n tem-

INFORMATION CAPACITY ESTIMATES

In order to examine the usefulness of MTs as the cell's information

processorswe must first evaluate the information capacity within each of

the three phases dentified. These results can be used to find the optimal

conditions for the MTs to function as the substrate for consciousness-related activities. We base the calculations that follow on the standard

413 Tuszyriski et al.: Microtubular Self-Organization



(Shannon) definition of information I of a statistical system where (Haken

1990)

KI = -2 p;ln{p;) {3)

;=1

Here, Pi stands for a probability value in state i and, obviously, the proba-

bility distribution must satisfy:

~

I Pi = 1 with O ~ Pi ~ 1. (4)

For the ferroelectric and paraelectricphaseswe adopt the mean-field approx-

imation where eachstate s characterizedby the continuous variable p (mean

polarization per site). The energy functional is taken n the Landau form as

E = (~p2 + ~p4 )N2 4 ° (5)

where No is the total number of sites in the lattice, A = a(T T J and B > 0.

As is well-known above the critical temperature, that is, for T > T c' E is min-imized by P = 0 while below the critical temperature, that is, for T < Tc by

P:!: = :t(-A!B)l/2. The associated continuous probability distribution f(P)

that replaces Pi of Equation 3 is the Boltzmann-weighted distribution func-

tion in the form:

.f(P) = 2-1exp( -{3E) = foexp(aP2 yp4) (6)

where fo = 2-1 is the normalization, ~1 = kB T, a = -A/(2 kB T) and y =

B/(4 kB T). Hence, for T > Tc' (P) is single-peaked at P = 0 while for T < Tc

it is double-peaked at P = Pt.

Following Haken (1990) we calculate the information capacity in theparaelectric (P = 0) and ferroelectric (P * 0) phasesas

I = In(2) -a(P2) + y(p4) (7)

where the averagesare obtained using:

(pn) = fOO f(p)pndP (8)-00

We carry out the requisite calculations n a straightforward manner for both

the ferroelectric and paraelectric phaseswhere analytical calculations can

be performed. For the SG-phase,however, we assume hat the above pre-scription is valid only within the local domain of coherenceor within the

correlation length. Hence, or eachdomain i we have a local polarization p

and the associatedprobability distribution fi(P ) essentially analogously to

those of Equations 5 and 6. Thus, for the total system he probability distri-

bution becomesa product of local distributions eachof which characterizes

a domain of coherence

415

414 Tuszyriski et al.: Microtubular Self-Organization



nf = lltj(Fj)

j=l

-

(9)ere (Haken

where n is the number of domains.Note that n depends on temperature and we assume or simplicity that

(3)

the proba-

n = 1 +~o- l)(T- T B) (10)

TA-TB

in order to interpolate continuously between the ferroelectric and para-

electric phases since T B :s; T:s; TAoAt T = T B' virtually the entire system is

uniformly polarized while at T = TA it is completely depolarized and inco-

herent. Note, that as a consequence of Equation 9 we obtain for the infor-

mation capacity in the SG-phase

(4)

celd approx-

ble p (mean

iau form as

(5)

nI = I Ii (11)

i=l

where 4 refers to each individual domain. Our numerical computation

clearly indicates that information capacity I is highest at the boundary

between he SGand the paraelectricphase seeFigure 30.4)and hence f MTs

are to be effective as information processors, hey should use this.narrow"window of opportunity" at the border area between these two phases.Of

course, he actual location of the border area depends on the magnitude ofthe electric field applied and the concentration of MAPs present.

) and B > 0.

Tc'Eis min-

)rT <Tcby

"ibution j(P)

Jution func-

SUMMARY AND OUTLOOK(6)

.T) and 'Y =

le for T < T c

We have argued in this paper that the spatial arrangement of dipole

moments of a MT is crucial to its functioning as a dynamic self-organizing

lacity in the

600((7)

500 Paraelectric

d 400o.=tIS 300

§:B 200 Spin Glass

100

Ferroelectric40

TATBl00L

Temperature (K)

Figure 30.4 Plot of the information capacity as a function of temperature.

415 Tu8zynski et al.: Microtubular Self-Organization



system. Due to the presenceof frustration in the dipole-dipole interactions,a SG phase s predicted to arise at low enough temperatures and electric

fields. The presenceof MAPs will lower the temperature values requiredfor SG-formation. The transition temperature tself decreasesn proportion

to the MAP ratio. The attractiveness of the SG phase has been recognized

earlier (Stein 1992)and in the present context it lies in its maximum com-

putational capabilities offered by a highly degenerateground state. More-

over, long relaxation times give relative stability to short-range correlated

dipole patterns. Each pattern can be seenas containing binary information

encoded n the lattice.

The other ordered state hat is possible to exist s a F phasewith an almost

perfect alignment of dipole moments along the protofilament axis. It is

characterizedby long-range order and hence ts usefulness or information

transfer and processing s dubious. However, it eagerly supports the for-

mation of domain walls between the two stable orientations of dipolemoments. The application of an external electric field preferentially directs

kink-like excitation towards the properly aligned end causing a disassem-

bly of the protofilament due to the energy releasedby the kink.

We have recently performed preliminary calculations ncluding the pres-

ence of a conformational change associatedwith a f3-state.We have found

using Monte Carlo simulations that quite a different picture arises. nsteadof three dielectric phases,only two exist: a low-temperature F phase and a

high-temperature ferri-electric phase. The latter phase is characterizedbythe formation of local domains of polarization in two possible directions:

vertically up the MTs axis or downwards and at 29 o off the axis. However,

net polarization appears to persist well above room temperature. The

polarization may be a key physical factor in the hypothesized communica-tion between microtubules in sufficiently concentrated assemblies.

We have discussed he various possible control mechanisms (field, dis-

tortion, temperature, MAP patterns) that could provide a means by whichthe MT could select an operating mode between information processing

and assembly disassembly types. This could shed some ight on why the

MT formation rate is enhanced in particularly important stages of the

organism's history and development (learning, division, growth).

ACKNOWLEDGMENTS

Th~s esearchwas supported by NSERC (Canada),DAAD, and the Alexan-

der yon Humboldt Foundation.

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416 Tuszyrlski et al.: Microtubular Self-Organization



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J.A. Tuszynski, B. Trpisova, D. Sept, M.V. Sataric and S.R. Hameroff: "Microtubular Self-Organization and Information Processing Capabilities"