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PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014

An Improved Electrical Model of Microtubule as Biomolecular Nonlinear Transmission Line

Dalibor L. Sekulić, and Miljko V. Satarić

Abstract – The manner in which microtubules, the essential cellular biopolymers, handle and process electrical signals is still uncompleted puzzle. In this paper, we have elaborated some new electrodynamic properties of these protein–based nanotubes, specifically, their ability to conduct ionic currents. In that context, it has been established an improved electrical model of microtubule as biomolecular nonlinear transmission line. We described the basic nanoscale electric elements of model and estimated the corresponding parameters, stressing the particular importance of tubulin C–termini. The properties of the localized electric nanocurrent of positive ions and accompanying voltage along a microtubule are analytically and numerically analyzed here.

I. INTRODUCTION

Microtubules (MTs) are cytoskeletal biopolymers made up of guanosine triphosphate–dependent heterodimer tubulin proteins, each composed of α and βtubulin monomers [1], see Fig. 1a. Each tubulin monomer possesses a highly exible C–terminal tail region that can extend up to 4.5 nm from the surface. In vivo MT is generally made up of 13 parallel, loosely connected protofilaments, formed by polymerization of the tubulin heterodimer. This results in a hollow tube–like filament with external and internal diameters of 25 nm and 15nm, respectively, see Fig. 1b, while length typically varies from less than 200 nm to several micrometers. Interestingly, MTs are self–assembling protein–based nanotubes with mechanical properties similar to carbon nanotubes despite their very different chemical composition: proteins and non–covalent interactions in the case of MTs; carbon and covalent bonds in the case of carbon nanotubes [2].

The initial motivation for the present electrical model was based on the experimental evidences [3] revealed that MTs are good conductors of electrical signals. The conductivity of these protein structures is the result of their electrostatic and structural properties. In particular, the outer surface of tubulin heterodimer and C–termini are mostly electrostatically negative charged as one can see from Fig. 2. As a result, overall surface of MT is negatively

D. L. Sekulić is with the Department of Electronics, Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia, E-mails: dalsek@yahoo.com or dalsek@uns.ac.rs

M. V. Satarić is with the Department of Physics, Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia, E-mail: bomisat@neobee.net

Fig. 1. (a) The topology of a tubulin heterodimer with dimensions of the C–termini. (b) Cross–sectional view of microtubule. (c) Microtubule hollow cylinder with marked protofilament. charged with an average linear charge density of 52.2 e per dimer, or approximately 85 e/nm [4]. Also, it was found that the conductivity of MT is on the order of 10 nS [5], indicating a high level of ionic conductivity along this biopolymer. In the context of these findings, it is suggested [6] that MTs act as electrical transmission lines for ion flows along their lengths. These lines in general could be sequenced in ladders of repeated identical electric elementary units which possess specic values of corresponding electrical parameters of model.

In this paper, we establish an improved electrical model of MTs in terms the polyelectrolyte concept applied to the molecular structure and geometry of these biological nanotubes. Based on the properties of the analytical and numerical determined electric nanocurrent, it will be analyzed the ability of MTs to be the pathways for efficient and fast transport of positive ions.

112

Fig. 2. Electrostatic charge distribution on the surface of the tubulin heterodimer and segment of the MT’s lattice.

II. CHARACTERIZATION OF MT

Since sufficiently high linear electric charge density is present on outer surface of MTs, these biopolymers can be considered as true polyelectrolytes. In solution, positive ions are attracted to the negative charged surface of the MT, while negative ions are repelled, creating a positive condensed ionic cloud (CIC) localized around the MT’s landscape. In the presence of an applied voltage, the loosely held CIC is free to migrate along MT creating an ionic flow and an associated electrical current [3, 4]. Based on Manning’s theory of polyelectrolytes [7], negative ions of cytosol are repelled at the distance called the Bjerrum length (lB), which is defined by the balance between Coulomb attraction energy of the ions and the pertaining thermal energy [6]:

. π4 1B0r

2B

Tkεεzel (1) For divalent ions (z = 2) such as Ca2+ and Mg2+, width of the “depleted layer” sandwiched between these two charged regions is lB = 1.34 nm at physiological tempera-ture T = 310 K. Other parameters used in this estimate of the Bjerrum length are e = 1.6×10-19 C, εr = 80, ε0 = 8.85×10-12 F/m and kB = 1.38×10-23 J/K.

In accordance with Manning approach, the thickness of CIC around the rodlike filamentous polyon of radius r is given by the expression:

.2/1

B1

Db2/1

Db 8 ,1 , nllrl AA (2) Thus, the corresponding values for tubulin heterodimer (TD) and C–termini (CT) are λTD = 2.96 nm and λCT = 0.92 nm, respectively, for equilibrium ionic concentration n =

1.5×1023 m-3 and A = 0.5. The depleted layer between CIC and repelled anions plays the role of a dielectric between charged plates in “coaxial cable”, see Fig. 3. It provides resistive and capacitive components for the behavior of the tubulin dimers that make up the MT.

Based on the above, it is possible to establish an electrical model of MT. Due to symmetry of these biopolymers, it is plausible to consider just one of thirteen MT’s protofilaments and to introduce the so called electric elementary unit, which is a single tubulin dimer with its two C–termini and two nano–pores. The particular attention should be paid to the role of these two types of the nano–pores, which exist between neighboring dimmers within a MT wall. They exhibit properties like ionic channels [8].

In order to estimate the resistance of electric elementary unit, we rely on the numerical estimations of respective resistivity published in ref. [8]. The resistivity of the parallel flow of ions along elementary unit is R0 = 6.2×107 Ω, neglecting the ionic current which leaks through the depleted layer. In the order hand, the conductance of both nano–pores is G0 = 10.7 nS, which is determined as a sum of pretty different components reflecting difference in two types of nano–pores [6, 9].

In order to estimate the total static capacitance of electric elementary unit, we firstly consider the contribution of a tubulin dimer as half–cylindrical capacitor which has the value:

,F 1075.01ln

π 16

B

0TD

RllC r

(3)

where the parameter is R = rTD + λTD = 5.46 nm, representing the outer radius of respective CIC, see Fig. 3, while the other parameters are already mentioned. Similarly, taking an extended C–terminal as a smaller cylinder with r = rCT + λCT = 1.42 nm, and calculating the corresponding effective length of the part of C–terminal not plunged in λTD as leff = lCT - λTD = 1.54 nm; one obtains its capacitance as follows

F. 1014.01ln

2π 16

B

eff0

CT

rllC r

(4)

Since two C–termini pertain to each tubulin heterodimer, and keeping in mind that the capacitances of tubulin dimer and C–termini are in parallel arrangement [9], it implies that total maximal static capacitance of the elementary unit is readily estimated as the simple sum of above estimated components:

F. 1003.12 16CTTD0

CCC (5)

113

Fig. 3. Schematic illustration of a tubulin dimer with its condensed ionic cloud (CIC), depleted layer and the geometry of a “coaxial cable” with the dimensions. As can be seen the capacitance in our model represents the charge distribution in the region from the surface of the MT to approximately one λ away perpendicularly to it. Since the ions in this layer are assumed to be condensed, the charge on this capacitor varies in a nonlinear way with voltage as follows:

. )(Γ1 000 nnn vbvttCQ (6) This expression is derived and explained in detail in refs. [6, 10]. The Γ0Ω describes the effect of the shrinking or stretching, and oscillation of C–termini. The parameter b represents the change of capacitance of electric elementary unit with an increasing concentration of condensed cations.

In a previous paper [11], we found that corresponding inductivity of electric elementary unit is small enough that can be safely ignored in this model.

III. THE MODEL OF MT AS BIOMOLECULAR NONLINEAR TRANSMISSION LINE

On the basis of above estimations for the components

of electric elementary unit of the MT, we are now in the position to establish the corresponding electrical model. A typical section scheme is shown in Fig. 4.

Fig. 4. A typical section scheme of the corresponding electrical model of MT.

In order to derive the voltage equation of model, we begin by introducing the discrete potential in one section of the MT where Kirchhoff’s laws for the currents and voltages are applied

.

,

01

01

nnn

nn

nn

iRvv

vGt

Qii (7)

Introducing the characteristic impedance Z of electric elementary unit corresponding to characteristic frequency ω = 2π/(R0C0)

,11

2/1

20

220

20

220

10

0

GC

GCGRZ

(8)

and the function u(x; t) unifying the voltage and ionic current along a MT un(x; t) = Z1/2in = Z-1/2un; going over to the continuum approximation with respect to space variable x, we get the following characteristic voltage equation [6]

.036

323

0001

00

2/3

0000

3

3

uΓZCRZZGuuT

sbCZ

uΓZCuT

sZCu

(9) Here, the characteristic charging (discharging) time of the elementary capacitor C0 through the elementary resistance R0 is T = R0C0. In characteristic voltage equation of model (9), the dimensionless space and time variables are introduced as follows [10]:

, v,vv , , 00 T

lsTts

lx

(10)

where l = 8 nm is the length of one tubulin heterodimer.

One of possible numerical solutions of the voltage equation of MT’s electrical model is shown in Fig. 5. Numerical simulation is based on a nite–difference time–domain method applied to the differential equation (9) governing the voltages of circuit. The average velocity of localized voltage pulse can be estimated from the obtained graphic shown in Fig. 5 as follows:

,s

cm 21000

180vCT

Tl

(11)

where l is length of electric elementary unit that is one tubulin heterodimer. The TCT represents a swing period of C–termini which is given by the following expression [10]

114

Fig. 5. Numerical solution of the voltage function u(ξ, τ) for the specific set of parameters of the established model.

s.1072.03π2 7

01

0

0CT

RZZGZCT (12)

The obtained velocity of propagation of voltage pulse is the order of cm/s and it depends on the characteristic properties of the electrical circuit model of MT. It is also possible to estimate that the pulse width is of the order of ten elementary units. This fact supports the validity of continuum approximation. The range of this soliton-like e localized pulses is 180 × l ≈ 1.45 μm, which is of the order of cell’s diameter.

III. DISCUSSION AND CONCLUSION

Motivated by several intriguing experiments [3, 5], we offered the improved electrical model of MT that provides a framework for the analysis of generation and propagation of electric nanocurrents along this biopolymer. In summary, the geometric symmetry of MT provided the opportunity that one MT’s protofilament can be approximately seen as a “coaxial cable” with the “charged plates” composed of cytosolic counterions with a depleted layer of water molecules in between. It enabled that the series of tubulin dimer with pertaining C–termini could be viewed as series of identical elementary units with estimated capacitance and resistance, including the conductance of two nano–pores in MTs wall.

Numerical solution of characteristic equation of model showed that the localized voltage waves propagating along MT are stable pulses that cross distance of cell size despite ohmic dissipation. Velocities of these pulses are the order of cm/s, depending on the amount of the injected ions and valence of the ions. Under physiological conditions, this effect is much faster than process of propagation of the ions by pure diffusion. Interestingly, if we compare these velocities with the drift velocity of electrons in a typical

semiconductor, we get the same order of magnitude at current density of order of A/mm2. However, the velocity of establishing a local field around the protein biopolymers is much lower due to heaviness of the ions, which are much more massive than electrons.

We believe that these results may have important consequences for understanding of MTs ability to conduct electrical signals, which may affect the neuronal computational capabilities, among other known functions. Also, our findings could encourage experimentalists to conduct more subtle assays in an attempt to elucidate this important aspect of cellular activities.

ACKNOWLEDGEMENT

This research was supported by the Ministry of Education and Science of Serbia, Grant III43008.

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