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

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## Abstract

**:**

^{−5}F and 9.74 × 10

^{4}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.

## 1. Introduction

_{2}and 0.5 mM EGTA, containing ~46 mM PIPES

^{2–}, ~36 mM PIPES

^{–}, ~68 mM Cl

^{–}, ~160 mM K

^{+}, and ~2 mM Mg

^{2+}[2]), 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 (1–100 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.

^{−5}F and 9.74 × 10

^{4}Ω respectively, at physiological concentrations of tubulin. Our values indicate that the polymerisation of tubulin into MTs alters the spatial and temporal charge distribution, altering the electrical properties through charge storage in the cell.

## 2. Materials and Methods

#### 2.1. Tubulin Reconstitution

#### 2.2. MT Polymerisation and Stabilisation

#### 2.3. Fluorescence Imaging of MTs

#### 2.4. Electrode Design and Device Construction

#### 2.5. Impedance Measurements

_{rms}was set to 5 mV. Solutions were perfused into the experimental chamber using a micropipette tip at one opening, and a filter paper at the other opening for suction, similar to protocols used for Total Internal Reflection Fluorescence (TIRF) microscopy [31]. The frequency range of the EIS measurement was set from 4 MHz to 1 Hz and data were subsequently collected.

#### 2.6. Data Analysis

^{5}F and 10

^{−5}Ω, respectively, based on visual inspection of raw data. The initial guess values for the nominal series resistor, R

_{H}, were set at 1.78, 0.6 and 0.4 Ω with tubulin concentrations of 0.222, 2.222 and 22.222 μM, respectively. The 95% confidence intervals were determined using the function nlparci. Error propagation was performed assuming no relationship between various days of data collection.

## 3. Results

#### 3.1. Validation of Parallel-Plate Contact Device to Measure Dielectric Properties of Physiologically Relevant Ionic Solutions

_{c}+ r

_{s}/(1 + (r

_{s}ωC)

^{2}) + j(ωL

_{c}– (r

_{s}

^{2}ωC)/(1 + (r

_{s}ωC)

^{2})),

_{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 input frequency (Figure 1c). Such a trend is expected from Warburg impedance [32,33] and is in accordance with the equation:

_{complex}= (A

_{ω})/√ω + (A

_{ω})/(j√ω),

_{complex}is the complex impedance and A

_{ω}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:

_{c}– j/ωC,

#### 3.2. The Effect of Microtubule Networks on Solution Capacitance at Physiologically Relevant Conditions

#### 3.3. The Effect of Microtubule Networks on Solution Resistance at Physiologically Relevant Conditions

#### 3.4. The Microtubule Network as an RC Circuit in Parallel

_{o}and Z

_{s}respectively, as shown in Figure 7. The net impedance of the background BRB80T was thus given by:

_{buffer}= Z

_{0}+ Z

_{s},

_{MT}, R

_{MT}and C

_{MT}respectively, the impedance for the circuit with MTs is given by:

_{MT+buffer}= Z

_{0}+ Z

_{s}+ R

_{H}+ Z

_{MT}

_{MT}= 1/R

_{MT}+ jωC

_{MT}

_{MT+buffer}– Z

_{buffer}= R

_{H}+ Z

_{MT},

_{MT}=R

_{MT}/(1 + (ωC

_{MT}R

_{MT})

^{2}) – j(ωC

_{MT}R

_{MT}

^{2})/(1 + (ωC

_{MT}R

_{MT})

^{2}),

_{H}, R

_{MT}and C

_{MT}as our fit parameters. Here, R

_{H}is a resistance ascribed to the nominal fraction of unpolymerised tubulin present in MT containing solutions. The fitted curves are displayed in Figure 8 and the optimal fit parameters are listed in Table 2 (see Materials and Methods for details).

## 4. Discussion

#### 4.1. The Physical Underpinnings of An Increased Capacitance

^{+}and K

^{+}as opposed to Ca

^{2+}, allowing for free movement of selective ions into the MT lumen across its porous surface [47]. 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.

^{−10}m [8,50]. Both actin filaments and MTs have been represented in these models by cable equations with effective real and imaginary impedance due to the viscosity of the solution-resisting ionic flows and the capacitive properties of the ionic double layers around the filaments, respectively [52,53]. The capacitance for a single ring of an MT including C-termini was calculated to be approximately 1.3 × 10

^{−15}F [8]. When extended to 20 µm, (representative of the length of a single MT for our measurements), the predicted value would be C = 3 × 10

^{−12}F, although an experimental confirmation of this prediction is not directly available through our measurements or in any previous work. We note the relatively weak dependence of network capacitance on MT concentration, and assign it to the random spatial locations and directional orientations of MTs in our solution. Indeed, the conductivity of randomly distributed RC networks has been shown to scale weakly with the number of elements in the network [54]. Additionally, qualitative similarities can be found in the models of random resistor and capacitor networks with a frequency-dependent crossover for both conductance and impedance in these networks due to percolation-type conduction [55]. We intend to develop a quantitative model for our experimental observations in a subsequent publication.

#### 4.2. Implications for the Cell

^{2+}ion storage/flow about an MT triggers its depolymerisation, whereas waves of Mg

^{2+}or lowering in the local pH (increasing H

^{+}) leads to MT stabilisation [57,58]. The attraction of Zn

^{2+}or Mn

^{2+}ions in the vicinity leads to the formation of two-dimensional tubulin polymers [59,60]. Properties of the cytoplasm such as polarisability and relative permittivity get severely attenuated because of the presence of MTs in the vicinity. Because of the polymerisation state of tubulin-altering solution capacitance, our findings implicate a temporal evolution of capacitance and ionic flows as the ratio of MTs to free unpolymerised tubulin changes [61,62,63]. MT lattice defects, which occur when a tubulin dimer is missing in an MT wall [64,65], cause a large ionic flux to develop at the defect site. Spatiotemporal charge distribution shifts are also critical at the MT end, where fluxes form because of sudden changes due to the polymerisation/depolymerisation of the MT. Free/polymerised tubulin hence regulates local and global electrical properties, creating spatially dynamic gradients of charge storage and flux. We envision a cytoskeleton that, in addition to transporting macromolecules, stores and transports ionic signals and electrical information across the cytoplasm (Figure 9a,b).

^{+}ions (140–300 mM) [27,28], which, in addition to MAPs and molecular crowding agents, will be included in a future study to attain physiological equivalence. We also note that the effect of PTMs (post-translational modifications) on the electrical properties of microtubules has not yet been explored. PTMs involve the addition of residues such as phosphate and glutamate that locally influence counterionic condensation around the outer microtubule surface.

## 5. Conclusions

^{−5}F and 9.74 × 10

^{4}Ω. These values correspond to an effective resistance per unit volume of 3.71 × 10

^{10}Ω/L and effective capacitance per unit volume of 7.65 F/L. 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) in the frequency region between 20 and 60 Hz. Our findings also indicate that the electrical properties of tubulin dimers change as they polymerise, revealing the potential impact of MT nucleation and polymerisation 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.

## Supplementary Materials

**a**) Top view (left) and (

**b**) side view (right) of holder for the parallel plate device used to perform impedance measurements. (

**c**) Top view (left) and side view (right) of slider used to position the double-sided tape exactly to fabricate the device. (

**d**) Top view (left) and side view (right) of the holder used to position the upper contact precisely on the lower contact. Figure S2. Validation of parallel-plate contact device using 0.5 mM electrolytic solutions. (

**a**) Imaginary component of impedance for electrolytic solutions at 0.5 mM and de-ionised water. (

**b**) Real component of impedance for electrolytic solutions at 100 mM and de-ionised water. Data display average values collected between 15 and 21 times. Error bars represent standard deviation. Figure S3. Example of microtubule and tubulin subtraction with backgrounds to display typical impedance. Figure S4. No ‘reversal’ in the resistive behaviour of microtubules is observed between 10 and 100 Hz. Graphs showing differences in the real component of impedance as a function of decreasing input AC frequency at total tubulin concentrations of (

**a**) 22.225 µM, (

**b**) 2.222 µM, (

**c**) 0.222 µM, (

**d**) comparison of the effect of paclitaxel and colchicine on impedance. Figure S5. One-sample t-tests were performed to determine if the impedance difference values were significantly above zero. This was carried out using the t-test function (‘ttest’) within MATLAB. Graphs showing the variation of obtained p-values for the imaginary components of impedance in (

**a**) tubulin and (

**b**) MT-containing solutions. Graphs showing the variation of obtained p-values for the real components of impedance in (a) tubulin and (b) MT-containing solutions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A parallel-plate contact device to measure the impedance properties of microtubules (MTs) compared to tubulin. The operation of the parallel plate device showing (

**a**) top view (left) and side view (right). The upper and lower contacts, double-sided tape and solution are labelled in green, grey and blue, respectively. (

**b**) Imaginary component of impedance for electrolytic solutions at 100 mM and de-ionised water. (

**c**) Real component of impedance for electrolytic solutions at 100 mM and de-ionised water. Data display average values collected between 15 and 21 times. Error bars represent standard deviation.

**Figure 2.**Microtubule imaging at different tubulin concentrations. Polymerisation was performed using 45 µM tubulin, and MTs were stabilised with 50 µM paclitaxel, and subsequently diluted to a final concentration of (

**a**) 0.222 µM tubulin (

**b**) 2.222 µM tubulin (

**c**) 22.225 µM tubulin, respectively. Scale bars represent 10 μm.

**Figure 3.**Examples of raw values of imaginary component of impedance in solutions containing (

**a**) MTs and (

**b**) tubulin, for the purpose of displaying typical impedance values. Mean differences in the imaginary component of impedance as a function of input AC (alternating current) frequency at total tubulin concentrations of (

**c**) 22.225 µM (n = 22 experiments for tubulin, n = 21 for MTs), (

**d**) 2.222 µM (n = 35 experiments for tubulin, n = 49 for MTs) (

**e**) 0.222 µM (n = 35 experiments for tubulin, n = 49 for MTs), (

**f**) comparison of the effects of paclitaxel (BRB80T) and colchicine (BRB80C, n = 49 experiments for BRB80T, n = 35 for BRB80C, n = 84 experiments for BRB80). Error-bars represent standard deviation.

**Figure 4.**Examples of raw values of real component of impedance in solutions containing (

**a**) MTs and (

**b**) tubulin, for the purpose of displaying typical impedance values. Mean differences in the real component of impedance as a function of input AC frequency at total tubulin concentrations of (

**c**) 22.225 µM (n = 22 experiments for tubulin, n = 21 for MTs), (

**d**) 2.222 µM (n = 35 experiments for tubulin, n = 49 for MTs) (

**e**) 0.222 µM (n = 35 experiments for tubulin, n = 49 for MTs), (

**f**) comparison of the effects of paclitaxel (BRB80T) and colchicine (BRB80C, n = 49 experiments for BRB80T, n = 35 for BRB80C, n = 84 experiments for BRB80). Error-bars represent standard deviation.

**Figure 5.**Graphs showing differences in the (

**a**) imaginary from Figure 3 and (

**b**) real component of impedance from Figure 4 as a function of tubulin concentration at input AC frequencies of 1 Hz, 10 Hz, 100 Hz, 1 kHz, 10 kHz and 86 kHz. Graphs display average values. Error-bars represent standard deviation.

**Figure 6.**Zoomed in view of the mean differences in the real component of impedance as a function of decreasing input AC frequency at total tubulin concentrations of (

**a**) 22.225 µM, (

**b**) 2.222 µM, (

**c**) 0.222 µM, (

**d**) comparison of the effect of paclitaxel and colchicine on impedance. (

**e**) A logarithmic plot obtained by translating the graphs (a), (b) and (c) upwards. The translation is performed by adding (1+minimum MT solution resistance) to the resistance of each MT concentration. A resistance reversal between 20–60 Hz is observed, with a peak at 39 Hz for the 22.225 µM concentration. Error-bars represent standard deviation.

**Figure 7.**The equivalent electrical circuit model representing the microtubule network as a parallel RC circuit, with network resistance R

_{MT}and capacitance C

_{MT}. The external element has impedance Z

_{0}, while solution has impedance Z

_{s}. R

_{H}is the small constant resistance that is ascribed to small fraction of unpolymerised tubulin that is present in MT containing solutions.

**Figure 8.**Mean differences of (

**a**) imaginary and (

**b**) real impedance curves for 0.222 µM, 2.222 µM and 22.222 µM, are fitted with the model described in Equation (6) and Figure 7. Fit parameters and confidence intervals are displayed in Table 2. The region between 1–100 Hz was not fit because of the negative differences in resistance from background BRB80T solutions.

**Figure 9.**Schematic of charge transport along and across an MT. (

**a**) A representation of charge flow across the MT cross section through nanopores present between adjacent protofilaments. (

**b**) A representation of charge flow through both inner and outer modes along an MT. Arrows depict charge flow via both mechanisms, enabling MT charge storage across a broad spectrum of frequencies, and charge transport at low AC frequencies in the cell. (

**c**) Side view (left) and top view (right) of the tubulin dimer, displaying distribution of electrostatic potential at different locations. The negatively charged C-termini face towards the solution and contains ~50% of the total negative charge on a tubulin dimer.

**Table 1.**Volumes of tubulin and buffer solution (BRB80T or BRB80C) used to stabilise microtubules (BRB80T) or free tubulin (BRB80C) in solution.

Tubulin Concentration (μM) | Volume of BRB80T or BRB80C (μL) | Tubulin Volume (μL) |
---|---|---|

0.222 | 99.5 | 0.5 |

2.222 | 95 | 5 |

22.225 | 5 | 5 |

**Table 2.**Fit parameters attained by fitting the real and imaginary components of impedance to Equation (7). Fit parameters represent effective capacitance C

_{MT}, and resistance R

_{MT}introduced into the solution through the addition of the MT network at different concentrations. R

_{H}is the small constant resistance that is ascribed to small fraction of unpolymerised tubulin that is present in MT-containing solutions. γ

_{R}and γ

_{C}describe the effective resistance and capacitance per unit volume introduced by the microtubule network. δ R

_{MT}, δ C

_{MT}and δ R

_{H}correspond to 95% confidence intervals for the fit parameters. The values δ γ

_{R}and δ γ

_{C}correspond to the uncertainties in the resistance and capacitance per unit volume. Corresponding graphs are displayed in Figure 8.

[Tub] (µM). | C_{MT} (F). | δC_{MT} (F) | R_{MT} (Ω) | δR_{MT} (Ω) | R_{H} (Ω) | δR_{H} (Ω) | γ_{R}(Ω/L) | γ_{C}(F/L) | δγ_{R}(Ω/L) | δγ_{C}(F/L) |
---|---|---|---|---|---|---|---|---|---|---|

22.222 | 1.27 × 10^{−5} | 1.48 × 10^{−7} | 9.74 × 10^{4} | 1.18 × 10^{4} | 2.12 | 40.61 | 3.71 × 10^{10} | 7.65 | 4.49 × 10^{9} | 0.056 |

2.222 | 1.25 × 10^{−5} | 1.67 × 10^{−7} | 1.00 × 10^{5} | 1.40 × 10^{4} | 0.61 | 34.79 | 3.81 × 10^{10} | 4.76 | 5.33 × 10^{9} | 0.063 |

0.222 | 2.01 × 10^{−5} | 3.38 × 10^{−7} | 9.97 × 10^{4} | 2.82 × 10^{4} | 0.41 | 31.95 | 3.80 × 10^{10} | 4.83 | 1.07 × 10^{10} | 0.12 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kalra, A.P.; Patel, S.D.; Bhuiyan, A.F.; Preto, J.; Scheuer, K.G.; Mohammed, U.; Lewis, J.D.; Rezania, V.; Shankar, K.; Tuszynski, J.A.
Investigation of the Electrical Properties of Microtubule Ensembles under Cell-Like Conditions. *Nanomaterials* **2020**, *10*, 265.
https://doi.org/10.3390/nano10020265

**AMA Style**

Kalra AP, Patel SD, Bhuiyan AF, Preto J, Scheuer KG, Mohammed U, Lewis JD, Rezania V, Shankar K, Tuszynski JA.
Investigation of the Electrical Properties of Microtubule Ensembles under Cell-Like Conditions. *Nanomaterials*. 2020; 10(2):265.
https://doi.org/10.3390/nano10020265

**Chicago/Turabian Style**

Kalra, Aarat P., Sahil D. Patel, Asadullah F. Bhuiyan, Jordane Preto, Kyle G. Scheuer, Usman Mohammed, John D. Lewis, Vahid Rezania, Karthik Shankar, and Jack A. Tuszynski.
2020. "Investigation of the Electrical Properties of Microtubule Ensembles under Cell-Like Conditions" *Nanomaterials* 10, no. 2: 265.
https://doi.org/10.3390/nano10020265