# Microtubules as One-Dimensional Crystals: Is Crystal-Like Structure the Key to the Information Processing of Living Systems?

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Crystal Model

**RSI**(I = β) and parallel resistance

**R**

_{pi}and to keep the model as simple as possible neither capacitance nor inductance effects are taken into account.

_{0α}) and beta-MT unit (effective impedance Z

_{0β}). We can repeat this unit infinitely. The expressions for ${Z}_{0\alpha}$ and ${Z}_{0\beta}$ are as follows:

_{k}(k = 1,2 … N) for each frequency $\omega $, multiply all the matrices shown in Equation (8) to obtain M, then use the element A to calculate the propagation constant $K$.

#### 2.2. Finite Element Analysis

^{−6}was the numerical tolerance. The simulation results are the normal component of the electric field and in-plane magnetic field components. We used TE polarization configuration to solve the Helmholtz equation given as

## 3. Discussion

## 4. Conclusions

## 5. Materials and Methods

#### 5.1. Microtubules Preparation

_{2}, 60% v/v glycerol, 1 mm EGTA, 80 mm PIPES pH 6.8 and 2 mM MgCl

_{2}, 80 mM PIPES pH 7, 1 mM EGTA, respectively.

#### 5.2. Protein Preparation

#### 5.3. Electrical Transport Measurements

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sketch of a microtubule. (

**b**) Simple drawing of a single photonic crystal layered with 18° tilt.

**Figure 5.**Finite element simulated microtubule (MT) structure. Black layers represent $\alpha $-tubulin monomers (${n}_{Low}=1.07402$), white, red, and brown layers are $\beta $-tubulin monomers with ${n}_{High}=7.3$, ${n}_{d1}=3.3$, and ${n}_{d2}=2$ respectively.

**Figure 7.**Finite element simulations of $\left({H}_{x}+{H}_{y}\right)/\Vert {H}_{x}+{H}_{y}\Vert $ compared with experimental current distribution measurements at different frequencies [37]. (

**a**) 30 MHz, (

**b**) 101 MHz, (

**c**) 113MHz, and (

**d**) 185 MHz. The simulation shows similar patterns as the experimental current distributions for frequencies up to 113 MHz. Nonetheless, for the frequency of 185 MHz the experimental pattern is replicated only in two layers (shown by asterisks). Microtubule’s STM images are taken while pumping materials using a GHz antenna. We have described the experimental details in earlier works [21,22,29,37].

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**MDPI and ACS Style**

Sanchez-Castro, N.; Palomino-Ovando, M.A.; Singh, P.; Sahu, S.; Toledo-Solano, M.; Faubert, J.; Lugo, J.E.; Bandyopadhyay, A.; Ray, K.
Microtubules as One-Dimensional Crystals: Is Crystal-Like Structure the Key to the Information Processing of Living Systems? *Crystals* **2021**, *11*, 318.
https://doi.org/10.3390/cryst11030318

**AMA Style**

Sanchez-Castro N, Palomino-Ovando MA, Singh P, Sahu S, Toledo-Solano M, Faubert J, Lugo JE, Bandyopadhyay A, Ray K.
Microtubules as One-Dimensional Crystals: Is Crystal-Like Structure the Key to the Information Processing of Living Systems? *Crystals*. 2021; 11(3):318.
https://doi.org/10.3390/cryst11030318

**Chicago/Turabian Style**

Sanchez-Castro, Noemí, Martha Alicia Palomino-Ovando, Pushpendra Singh, Satyajit Sahu, Miller Toledo-Solano, Jocelyn Faubert, J. Eduardo Lugo, Anirban Bandyopadhyay, and Kanad Ray.
2021. "Microtubules as One-Dimensional Crystals: Is Crystal-Like Structure the Key to the Information Processing of Living Systems?" *Crystals* 11, no. 3: 318.
https://doi.org/10.3390/cryst11030318