Investigation of the Electrical Properties of Microtubule Ensembles under Cell-Like Conditions

Microtubules are hollow cylindrical polymers composed of the highly negatively-charged (~23e), high dipole moment (1750 D) protein α, β- tubulin. While the roles of microtubules in chromosomal segregation, macromolecular transport, and cell migration are relatively well-understood, studies on the electrical properties of microtubules have only recently gained strong interest. Here, we show that while microtubules at physiological concentrations increase solution capacitance, free tubulin has no appreciable effect. Further, we observed a decrease in electrical resistance of solution, with charge transport peaking between 20–60 Hz in the presence of microtubules, consistent with recent findings that microtubules exhibit electric oscillations at such low frequencies. We were able to quantify the capacitance and resistance of the microtubules (MT) network at physiological tubulin concentrations to be 1.27 × 10−5 F and 9.74 × 104 Ω. Our results show that in addition to macromolecular transport, microtubules also act as charge storage devices through counterionic condensation across a broad frequency spectrum. We conclude with a hypothesis of an electrically tunable cytoskeleton where the dielectric properties of tubulin are polymerisation-state dependent.


INTRODUCTION
Microtubules (MTs) are cylindrical polymers composed of the heterodimers of protein α, βtubulin that play a variety of well-recognised intracellular roles, such as maintaining the shape and rigidity of the cell, aiding in positioning and stabilisation of the mitotic spindle for allowing chromosomal segregation, acting as 'rails' for macromolecular transport and forming cilia and flagella for cell movement. Since the tubulin dimer possesses a high negative electric charge of ~23e and a large intrinsic high dipole moment of 1750 D 1-2 , MTs have been implicated in electrically-mediated biological roles [3][4][5][6] . They have been modelled as nanowires capable of enhancing ionic transport [7][8] , and simulated to receive and attenuate electrical oscillations 4,[9][10][11] . In solution, MTs have been shown to align with applied electric fields 2,[12][13][14][15][16] . Recently, MTs have also been modelled as the primary cellular targets for low-intensity (1-2 V), intermediate-frequency (100-300 kHz) electric fields that inhibit cancer cell proliferation, in particular glioma [17][18][19] . Indeed, MTs have been reported to decrease buffer solution resistance [12][13] , leading to a conductance peak at TTField-like frequencies 20 . While these studies show that MTs are highly sensitive to external electric fields, answers to the questions 'How do MTs effect a solution's capacitance?' and 'What is the capacitance of a single MT?' are still elusive and crucial to the determination of the dielectric properties of living cells. The tubulin concentration in mammalian cells varies in the micromolar range (~10-25 μM) [21][22] . In vitro, polymerizing tubulin at such high concentrations can lead to the formation of entangled meshworks, confounding quantification of the individual MT response to electric fields. Electro-rotation, dielectrophoresis and impedance spectroscopy are thus performed using low concentrations of tubulin, in the nanomolar regime, to enable robust observation of individual MTs.
MT formation and stability are known to be optimal in buffers with ionic strength between 80 and 100 mM [23][24] . A background of BRB80 (which consists of 80 mM PIPES, 2 mM MgCl2 and 0.5 mM EGTA), is thus used to study the dynamics and mechanical properties of MTs. To study their electrical properties however, the usage of such high ionic-strength solutions has historically been problematic because any dielectric attenuation caused by MTs is overwhelmed by the noise and high conductivity from the background. In the low-frequency regime (1 Hz-100 kHz), two approaches have thus far been used to estimate the dielectric properties of MTs and tubulin. One is to electrically observe low concentrations of MTs (tubulin concentration in the nanomolar regime) in the presence of low ionic strengths 12-13, 20, 25-26 . Such studies overlook the intrinsic ionic concentration of mammalian cytosol, which varies between 200 to 500 mM depending on the cell type [27][28] . Another approach to electrically interrogate MTs is to dry them: the conductivity of the buffer is nullified by evaporation, leaving polymeric tubulin behind [29][30] In a physiological situation however, MTs are solvated by the highly conductive and noisy cytosol.
In this paper, we report on our efforts overcome the barrier posed by a high ionic strength by performing electrochemical impedance spectroscopy (EIS) on cellular concentrations of tubulin.
We have been able to successfully observe differences in impedance using a background of BRB80, which contains within an order of magnitude of physiologically measured ionic concentrations. Surprisingly, we find that MTs increase the solution capacitance of BRB80 whereas free tubulin does not, implicating a difference in electrical properties based only on the morphology of this protein solute. We also report a 'reversal' in the resistive behaviour of MTs compared to BRB80, with a reduction in solution resistance peaking at 20-60 Hz, a finding consistent with recent reports showing that polymerized tubulin quasi-resonantly responds to electric oscillations at ~39 Hz 9-10 . Using an equivalent circuit model for MTs, we experimentally determine the capacitance and resistance of the MT meshwork to be 1.27 × 10 -5 F and 9.74 × 10 4 Ω at physiological concentrations of tubulin. Using approximations, we quantify the capacitance and resistance for an individual MT as 1.86 × 10 −12 F and 1.07 × 10 12 Ω. Our values are in agreement with previous calculations and indicate that the polymerization of tubulin into MTs alters spatial and temporal charge distribution, altering the electrical properties through charge storage in the cell. To determine the differences in the dielectric properties of solution caused by the presence of MTs, we aimed to create an electrode geometry that would be experimentally robust and easily modelled.

RESULTS
We fabricated an FTO-coated parallel-plate contact device ( Figure. 1a, Materials and Methods), which allowed EIS using a solution-exchange method.
We started by performing EIS on electrolytes found in the cytosol and observed a decrease in the imaginary component of impedance as a function of decreasing input frequency ( Figure. 1b). The total impedance of our system was given by: Here, Z is the impedance, ω is the angular frequency (given by 2 where is the input voltage frequency), C is the system capacitance, L c is the cable inductance, r s and r c are the solution and cable resistances respectively. We also observed a decrease in the real component of impedance as a function of decreasing input frequency (Figure 1c). Such a trend is expected from Warburg impedance [32][33] and follows the equation: where is complex impedance and is the Warburg coefficient. Our circuit simplifies to the equation below if we ignore the effect of cable inductance ωL c , at frequencies below 10 5 Hz: Our results emulated previous data [34][35][36] and validated the experimental setup for further analysis.  Interestingly, within this range, we also found that the addition of MTs lowered solution resistance compared to background buffer BRB80T.

MTs increase solution capacitance compared with background, while unpolymerized tubulin does not have a significant effect
A reversal frequency in resistance between MTs and tubulin has not been reported before and displays the utility of our 'cell-like' approach, because the extent of this reversal decreased with decreasing concentration and was not displayed in the imaginary component of impedance or when BRB80C and BRB80T were compared. It is worth noting that recent findings using this approach showed that MT bundles and tubulin sheets generate electrical signals at ~39 Hz 9-10 , and predicted an increase in solution conductance in the range that we observe, indicating that the conductance behaviour at such frequencies was due to MT-generated electrical oscillations.

The microtubule meshwork can be described as an RC circuit in parallel
Using our experiments, our next aim was to quantify the electrical properties of a single MT.
Impedance difference curves appeared linear on log-log plots with a slope of approximately negative unity, suggesting that the MT meshwork resulted in the addition of a capacitive element to the solution. We examined several combinations but a parallel RC circuit to represent the entire MT meshwork provided that best fit to observed curves ( Figure 5): Figure 5. The equivalent electrical circuit model representing the microtubule meshwork as a parallel RC circuit, with meshwork resistance ̃ and capacitance ̃. The external element has impedance 0 , while solution has impedance s . H is the small constant resistance that is ascribed to small fraction of unpolymerized tubulin that is present in MT containing solutions.
We modelled the impedance caused by external circuit elements and BRB80T as 0 and s respectively, as shown in Fig 5. The net impedance of the background BRB80T was thus given by: Denoting the impedance, resistance and capacitance of the entire MT meshwork by ̃M T , ̃ and ̃ respectively, the impedance for the circuit with MTs is given by: where, Additionally, the impedance differences between solutions with and without MTs are given by: where We subsequently fit experimental impedance difference curves shown in Figure 3 and Figure S7 to real and absolute value of imaginary parts of ∆ using H , ̃ and ̃ as our fit parameters.
Here, H is a resistance ascribed to the nominal fraction of unpolymerized tubulin present in MT containing solutions. The fitted curves are displayed in Figure 6 and the optimal fit parameters are listed in Table 1 (see Materials and Methods for details).   Figure 6.
Here, 0 is a Debye length (calculated to be ~10 nm) in a solution with ionic concentration 0 = 1 mM 13 . Based on the Debye length, the total capacitance of the MT cylinder has been calculated by previous models to be given by 8,44 : Substituting the value of from equation (7) and using the Taylor series expansion for ln(1 + ), we get: Fluorescence images (Figure 2 a) showed that MTs orient in the plane of glass surfaces at low concentrations, indicating that resistance to ionic charge transport in solution arose from flow across the porous MT cross section, rather than along their lengths. The lower limit of Ionic current through nanopores present between adjacent tubulin dimers in two-dimensional zinc-induced sheets is previously calculated to be 0.02 , at a driving voltage of 5 mV 10 . This leads to a nanopore resistance of = 2.50 × 10 14 Ω. We note that the conditions of our study vary significantly from those in this reference: Firstly, unlike two-dimensional sheets which present ions with a single resistance, cylindrical microtubules used in our study present both MT entering and MT exiting resistances (Figure 8 a). Secondly, the study utilizing sheets was performed at an ionic concentration of 140 mM, whereas our study was performed at a corresponding concentration of 80 mM. Thirdly and importantly, the resistance calculated above applies to resonant conditions, with the driving input frequency in the range of 10-100 Hz and would reduce as the input frequency is altered.
We calculated the total resistance for ions entering and exiting transversely through an MT as , = , = = 7.69 × 10 9 Ω where is the total number of nanopores (equal to the number of tubulin dimers) in a 20 µm long MT. We then addressed the first dissimilarity mentioned above, regarding MT entering and exiting impedances, by adding these resistances in series. This leads to total resistance = , + , = 1.54 × 10 10 Ω for ionic movement across the cross section. We also note that theoretically, the total ionic resistance across the cross section has been calculated by 48 : Here, the solution conductivity is proportional to the ionic concentration . Additionally, because depends inversely on the square root of ionic concentration, Ionic movement to enter the MT cylinder takes place against the potential gradient set up by the tubulin dipole in the MT wall, which could additionally account for the high resistance that we observe.

The physical underpinnings of an increased capacitance
Dense counterion condensation on the MT surface has been previously extensively predicted and simulated due to a variety of reasons 7-8, 44, 49-50 . Firstly, the negative charge of the tubulin dimer attracts counterions in solution, leading to the presence of a double layer and depletion region outside the microtubule surface 7-8, 46, 50 . In addition to this, the charge distribution in the MT protein wall is highly non-uniform, with the outer surface containing approximately four times the charge compared to the inner surface 49 . This asymmetry between the inner and outer electrostatic potentials serves to enhance capacitance and is responsible for the abnormally large dipole moment of the tubulin dimer 1 . The asymmetry also manifests through C-terminal 'tails' composed of 10-12 amino-acids, that can extend 4-5 nm outwards from each tubulin monomer. These slender Ctermini tails are highly negative, containing about 50 % of the charge of the tubulin dimer 51 . As they stretch outwards into the solution in a pH and ionic strength dependant manner, they increase the effective area of the tubulin dimer and significantly contribute to the overall MT capacitance [7][8] .
Coherent oscillations of these C-terminal tails are modelled to generate solitonic pulses of mobile charge along the outer surface of an MT, creating ionic currents along its' length 7,44,52 .
Ions from the bulk solution are also modelled to be pumped into the hollow MT lumen through nanopores in its' wall, resulting in charge accumulation inside the cylindrical MT over time 45 . A recent study using molecular dynamics simulations showed that the permeability of the MT lumen was significantly higher for Na + and K + as opposed to Ca 2+ , allowing for free movement of selective ions into the MT lumen across its porous surface 49 . To the best of our knowledge, our findings are the first to experimentally quantify this resistance encountered by charge flow across the MT cross section. These results implicate not only ionic movement along the microtubule axis, but also across and inside it, enhancing the modelled roles of MTs as complex subcellular nanowires. or Mn 2+ ions in the vicinity would lead to formation of two-dimensional tubulin polymers [56][57] .

Implications for the cell
Properties of the cytoplasm such as polarizability and relative permittivity would get severely

Future work
When compared to cells, the rates of MT nucleation and polymerization are significantly lower in BRB80. This difference can be attributed to the absence of MAPs and macromolecular crowding [63][64] . Mammalian cells contain high concentrations of K + ions (140-300 mM) [27][28] , which, in addition to MAPs and molecular crowding agents, would be included in a future study to attain physiological equivalence. Raw data from complex impedance will be used to derive properties of individual MTs such as the dielectric constant, refractive index, polarizability and Debye relaxation time through the Clausius-Mossotti equation. To improve the present two-contact parallel-plate device, a four-contact device will lower electrode polarization effects and enable quantification of the relative permittivity of MT and tubulin containing solutions. Our device geometry will also be used to perform DC (direct-current) measurements, determine the contribution of MTs to impedance relaxation time and evaluate the voltage dependence of capacitance on MT-containing solutions. Interestingly, this aspect has been discussed previously: the inductance of a single protofilament is calculated to be <1 fH 8 . Further investigation is required to experimentally confirm these predictions.

CONCLUSIONS
We used EIS to compare the complex impedance of MT-and tubulin-containing solutions. A high ionic strength buffer (BRB80) created a high noise, low impedance background, which was countered through the use of physiological concentrations of tubulin. While the presence of MTs increased solution capacitance, unpolymerized tubulin did not have any appreciable effect. We used an RC circuit to model the MT meshwork across three orders of concentrations, and found at physiological levels, the capacitance and resistance of the meshwork is to be 1.27 × 10 -5 F and 9.74 × 10 4 Ω . In a study that is the first of its' kind to the best of our knowledge, we determined the capacitance of a single MT to be 1.86 × 10 −12 F. We used our lowest, 0.222 M concentration, for this calculation and assumed that MTs were oriented in parallel to one another. We envision a dual electrical role for MTs in the cell: that of charge storage devices across a broad frequency spectrum (acting as storage locations for ions), and of charge transporters (bionanowires) at specific, low frequencies between 20 and 60 Hz. Our findings also indicate that the electrical properties of tubulin dimers change as they polymerize, revealing the potential impact of MT nucleation and polymerization on the cellular charge distribution. Our work shows, that by storing charge and attenuating local ion distributions, microtubules play a crucial role in governing the bioelectric properties of the cell.
Rhodamine-labelled tubulin solution was prepared by adding 4 μL of G-PEM buffer to labelled tubulin powder (Cytoskeleton Inc; TL590m) and subsequently adding 1 μL of MT cushion buffer.

Electrode design and device construction
Each 'plate' in the parallel-plate contact device was formed by FTO (Fluorine-doped Tin Oxide)coated glass slides (Sigma Aldrich, 735140). The slides were cleaved to dimensions of 1.5 mm x 10 mm x 50 mm for the upper contact and 1.5 mm × 27 mm × 50 mm for the lower contact. The cleaving dimensions were set using 3D printed devices that were placed as holders (The Shack, University of Alberta; Figure S1). The slides were sonicated and subjected to Reactive Ion Etching (RIE) using a 5-minute exposure to oxygen plasma (Oxford Instruments, NGP80) to remove surface particulate matter. 70 μm thick double-sided tape was used as a spacer, which formed a chamber of dimensions 3 mm × 1.25 cm × 70 μm. The top electrode was placed using a separate 3D-printed holder device ( Figure S1). Once the device was constructed using the above protocol, solution was perfused into the chamber using a pipette and a filter paper for suction (Video S1).
We used flat copper electrode clips in a three-electrode configuration to connect to our capacitor device. The counter electrode was connected to the lower electrode, and the working and reference electrodes were connected to the top electrode of our device.

Impedance measurements
Experiments were conducted using Electrochemical Impedance Spectroscopy (EIS) on a Zahner Zennium impedance analyzer. The parallel-plate contact device was placed into the 3D-printed holder for stabilisation ( Figure S1). The contacts from the machine were connected to the parallel-

Notes
The authors declare no competing financial interests.