1 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 c onstant intrinsic electric field. It is demon- strated that soliton formation in the form of k inks may be energeticaly f avorable under realistic conditions of phy sical parameter values. Introduction Of the various filamen tary structures which comprise the cyt oskeleton, rnicrotubules (MT's) are the mos t 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 macrosc opic 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 assemblies of 13 longitudinal protofila- rnents each of which is a series of subunit proteins known as tubulin dimmers. Each
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M.V. Sataric, J.A. Tuszynski and R.B. Zakula: A Solitonic Model for the "Information Strings"
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8/3/2019 M.V. Sataric, J.A. Tuszynski and R.B. Zakula: A Solitonic Model for the "Information Strings"
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
8/3/2019 M.V. Sataric, J.A. Tuszynski and R.B. Zakula: A Solitonic Model for the "Information Strings"
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
8/3/2019 M.V. Sataric, J.A. Tuszynski and R.B. Zakula: A Solitonic Model for the "Information Strings"