Self-survival of Quantum Vibrations of a Tubulin Protein and Microtubule: Quantum Conductance and Quantum Capacitance

Abstract

Quantum capacitance and quantum inductance, two well-known signatures of quantum properties detect here subtle changes in the resonance frequencies as instant quantum markers of cancerous mutation of proteins. We find that any quantum property that is destroyed by measurement is true only if singular wave function is measured. Here, as we image the three magnetic wave functions of a protein complex, in three layers of Schrödinger’s wave functions packed one inside another, we find their geometric phase (Zak phase) rebuilds each other from nano-to-micro scale. Using the difference signal between magnetic and thermal nano-sensors located closely at the atomic edge of a probe, our interference-based sensing mapped cancerous microtubule’s local structural changes at the very onset of cancer in the noisy environments.

Appendices

Supplementary Materials

Experimental Details

2.1 Development of Fluxgate System for “Measurement of Ultra-Low Magnetic Flux (<10⁻¹³ T)”

Fluxgate system (1930, Victor Vacquier) is an inductive coil that resonates with the external magnetic flux, and sends a small current proportional to the magnetic flux. With Keithley 6430 sub-femto-ampere connected pre-amplifier, we can measure 0.1 × 10^–16 A current. A 60 turn coil with LM32N or LM324N, or LM386 amplifier can amplify current by amplifying nA current to µA and integrate the current to measure 10⁻⁶ T. With Keithley 6430 and pre-amp, we can measure in principle 10 orders lower, ~10⁻¹⁶ T, where is the noise level of SQUID currently. 10⁻¹³ T sensors are available in the market, but not good for microtubule. First, one can avoid the use of amplifier by connecting two identical 99.999% Cu coil made inductors (4 mH to 4 pH), triggering self-resonance, and measuring the deflection from its self-resonance frequency, due to the presence of an external magnetic field. We used this idea to engage bundles of nanowires acting as an array of fluxgate sensors at the atomic tip. It increases noise reduction when we add additional ac signal as described below. We used self-resonance and amplifier together for the microtubule measurement. Then we had coaxial atom probe with a hole at the top, filled with CuInSe₂ helical nano-coil with a pitch similar to the dimension of microtubule. Millions of helical nanowires were oriented to the axis of coaxial cavity by applying electric field during deposition of the nanowire on the tip. CuInSe₂ helical nano-coil is like a toroid shape structure. Thus, toroid shape, self-resonance, pre-amplifier were three keys to measure around pico-tesla signals from microtubule at an ambient atmosphere.

Matching the SQUID performance at an ambient condition requires two major challenges to be addressed further. First, eliminating the earth magnetic field 10⁻⁶ T and second, stopping the saturation of nano inductor connected to the tip. The first one we have addressed by making a layer of several magnetic shields of various kinds, in a two layered faraday cage inside a faraday cage oriented towards the earth magnetic field. The second problem is addressed by pumping a suitable rectangular pulse stream to the sensing inductor to charge and discharge it, keeping it fresh to detect the magnetic flux. The input pulse stream frequency changes as it senses the magnetic field, so a secondary coil is added with Phase locked loop (PLL) amplifier to fix the frequency. The frequency is calibrated; with a GHz spectrum analyser with 5–20 kHz pulse stream we measure 10⁻¹² T accurately. 10⁻¹² T chips are available in the market, but they cannot measure microtubules at nano-scale with CuInSe₂ sensor. Coaxial atom probe cavity is designed such that it resonates far above THz, using design simulators we can make such changes.

Electromagnetic and magnetic shield We used various layers of the following seven kinds of materials to create our own ultra-low noise zero Gauss chamber. Chips are kept inside zero Gauss chamber, inside a 3 layered Faraday cage. A 2D inductive layer is added on the chamber to trap the ac noise.

MCF7 layer (LF magnetic field, 30 dB (97%)); Attenuation HF: 40 dB at 1 GHz, thickness 0.02 mm, Permeability: µ 4 = 25,000; µ max = 100,000; saturation polarization: 0.55 T; Composition: Co69, Fe4, Mo4, Nb1, Si16, B7. The surface is grounded, it is conducting.

Mu layer Nickel permalloy foil.

MCL61 layer Permeability µ 2 = 10,000; µ 4 = 25,000; µ max = 100,000; saturation polarization: 0.55 T; Hc = 0.5 A/m; Remnant Br/Bs = 0.7; Curie temperature Tc = 225 °C.

EMF-RF, Ni-Cu Ripstop shielding from Faraday defense. This is an electromagnetic shielding.

Pure Cu shield from Faraday defense.

Brass and Aluminum metal sheet was used as core structure of the primary faraday cage inside which a secondary cage was built. Ar gas 99.99% dry, atmosphere was created between primary and secondary cage.

ESD/EMP 7.0MIL Material from Faraday defense was custom made at home from DIY kits, using multi-layered alternate Aluminum and polyester/polyethylene coating as part of primary cage. 2 Aluminum metal layers in this 5-layer bag provide maximum EMP bag protection,—33 lb puncture resistance (FTMS 101-C, 2065.1),—>40db EMI Attenuation (MIL-PFR-81705-REV.D),—Surface Resistivity 12 Ohms/Sq. In. (ASTM 1-257),—7 mil Thickness (MIL-STD-3010C Method 1003),—Moisture Barrier (MIL-STD-3010C Method 3030), Heat Sealing Conditions:—Temperature (400 °F, 204 °C);—Time (0.6 to 4.5 s);—Pressure (30–70 PSI, 206–482 kPa).

2.2 Microtubule Device Fabrication

Microtubule from 1 nM solution is injection sprayed on pre-fabricated e-beam lithography made Au/Cr 200 nm electrode Si/SiO₂ chips (1 cm × 1 cm) under 5 V/100 µm electric field (dielectrophoresis to fix an exact relative orientation). Water layer on microtubule surface is ~2–5 nm.

2.3 Regulating Noise for All Experiments

A dc bias charges a few proteins (2–3 at most) and then the entire structure is left deserted, as no formal current flows. The ac signal damps in a few loops (2–3 generally) of the helical array due to an inductive barrier. One can touch the end parts of a macrotubule to notice a signal burst. Noise at a very low bias roughly 500 times less than the charging voltage passes through all the way to the end of the device covering all proteins.

For the arrayed proteins it is like junk device for electronics, three noise factors are important to make it even more junk, take it to a super critical state.

• (i) Noise frequency. A noise could be created in different frequency domain. For example GHz noise, MHz noise or kHz noise. Noise frequency ranges we have not quoted here intentionally, because, depending on experimental setup it changes. An user has to use coaxial atom probe and connect multiple points of tubulin on microtubule to find at which noise frequency they are depleting the junctions.

• (ii) Type of noise. More random is the noise, better is the activation of proteins. The arrayed proteins store the charge as a group of elements, if there is an ordering in the noise. We edited sources from various origins.

• (iii) Amplitude of noise. If amplitude is less than threshold, then noise makes no effect, if very high, proteins cannot flow quasi particle wave. So, there is an upper and a lower limit of noise amplitude where we see quantum effect.

2.4 “Heat Pipe” Effect Measurement in Microtubule and the Crucial Role of Water Channel

In a “heat pipe” type effect (a hollow pipe filled with dielectric; R. S. Gaugler [25]; G. M. Grover [26]), physical motion of quanta by radiation and transmission is associated with the phase ripples where no physical objects move. Since phase part transmits in parallel, the conductivity could reach 100 times that of silver [27]. Capacitors are well studied exhibiting a heat pipe effect [28]. The phase transition of electrolyte work together to efficiently transfer heat between two neighboring capacitors and in its dielectric between two leads. Water channel increases the conductivity through microtubule by ~10³ times, the effect of capillary water is significant in heat pipe effect ((Measurement of thermal conductivity and heat pipe effect in hydrophilic and hydrophobic carbon papers; [29]). Joule heating of a capacitor is maximum during a dc current flow. However, under ripples, of an ac signal or noise, a capacitor rapidly heats up and cools down. Rapid heating/cooling leads to a rapid phase transition inside the capacitor, like a heat pipe. Rate of phase transition could modulate conductivity by ~10³ orders of magnitude [30]. The effects, phase regulation & automated heat stabilization [31] enable “heat pipe” capacitors as noise harvesting system. In a microtubule, the arrayed proteins are rolled on a water crystal, it enhances the magnetic flux condensation, depicting the phase transfer based transmission. Microtubule could act as a heat pipe under noise of particular (10–200 MHz; 5–70 GHz) frequency bandwidth, where water crystal activates. So we try to select a small part of noise that triggers the water channels. When we use coaxial atom probe, we directly touch the water core.

2.5 Non-classical Negative Resonance Properties of a Microtubule

Using vector network analyzer (VNA), at certain resonance frequencies of microtubule, we observed dc current burst from resonance goes either negative or positive, taking reflectivity in the Smith chart R >> 1 or R << 1. So, we reported earlier “negative resonance” [20]. This effect is an evidence of wave regulated origin of resonance that we observe in a microtubule. Tubulin monomer, dimer and microtubule change their localized density of states (ldos) uniquely for each resonance frequency, not together as in a magnetic wave. Using coaxial probe, we switch lattice magnetic wave frequency ω (2πν) to find the transition energy $\hbar \omega$ required for a magnetic or L_q wave. However, a tubulin located on a microtubule cannot be switched like it’s isolated ldos. Still, we could measure magnetic flux fluctuation $\Delta E$ during attempts to switching tubulin for all visible 300 out of 1650 tubulins on a 1 μm microtubule surface. For both R >> 1 or R << 1, we found $\Delta E\ll$ $\hbar \omega$, so, the available free energy k_BT cannot break symmetry and dephase intra-layer phase-lock. Thus, interference of quasi waves leading to negative and positive resonance has various impacts on the stability of holding quantum effects under a massive noise.

2.6 Beating of “Beats”: Hierarchical Phase in Magnetic Interference Seen in Fig. 3a

Axial thermal conductivity of a protein wall is the so-called “heat pipe” effect, which is an augmented magnetic transfer due to the phase change of the electrolyte. The charge Q that inner pair of electrodes measure is not a dynamic phase, but the geometric phase of the charge flow that regulates the magnetic flux production. Hence, the geometric phase ${e}^{-\Upsilon t}$ evolves H in a periodic oscillatory manner [32]. Quantization of magnetic and electric flux is derived from Hamiltonian, depending on geometric parameter, i.e. ratio of perimeter to its diameter [22].

2.7 Live Recording of Phase Growth as Shown in Fig. 3a Experimental Set Up

In a monomer, as the noise is pumped, phase $\theta$ of a magnetic wave between two localized density of states grow by $\zeta = \theta \left( {1 - e^{{ - i\frac{\partial \psi }{{\partial Q}}t}} } \right)$. In a dimer, simultaneously, phase $\theta$ of a magnetic wave between two monomers grows by $\theta =\delta (1-{e}^{-i\frac{\partial \psi }{\partial Q}t})$, we measure it by locking phase with the THz emission in the local area of tubulin lattice site a, b, then, the phase $\theta$ of condensed flux $\Psi$ spread over entire microtubule length d saturates to ${\theta }_{0}$ at rate $\delta ={\theta }_{0}(1-{e}^{-i\frac{\partial \psi }{\partial Q}t})$, we get only one saturation time τ_i ($\frac{\partial \psi }{\partial Q}=H\sim {10}^{-3}$).

An excited state wave function at $t=0$, $\Psi \left(0\right)={c}_{1}{\Psi }_{1}\left(0\right)+{c}_{2}{\Psi }_{2}\left(0\right)$. At time t, the excited state is given as $\Psi \left({t}\right)={c}_{1}{\Psi }_{1}\left(0\right){e}^{-i\left(\frac{{H}_{1}}{\hslash }+\frac{\Upsilon }{2}\right)t}+{c}_{2}{\Psi }_{2}\left(0\right){e}^{-i\left(\frac{{H}_{2}}{\hslash }+\frac{\Upsilon }{2}\right)t}$. Since the intensity of output signal $I\left(t\right)$ is proportional to the dipole moment matrix, $I\left(t\right)=C{\left|\langle \Psi \left({t}\right)\left|e.d\right|{\Psi }_{i}\rangle \right|}^{2}$, we get the intensity of interference due to lattice induced split, $I\left(t\right)=C{e}^{-\Upsilon t}\left(A+B\cos{\omega }_{12}t\right)$, where ${\omega }_{12}=\frac{{H}_{12}}{\hslash }$. This interference is local; $I\left(t\right)$ sources generated all over the lattice interfere again due to another periodicity in microtubule generated by proto-filament offset.

$$\Phi \left({x},{y},{t}\right)={c}_{1}{{I}}_{1}\left(0\right){e}^{-i{e}^{-\Upsilon t}\left(P\sin{\omega }_{12}t+Q\cos{\omega }_{12}t\right)}+{c}_{2}{{I}}_{2}\left(0\right){e}^{-i{e}^{-\Upsilon t}\left(P\sin{\omega }_{12}t-Q\cos{\omega }_{12}t\right)}$$

(1)

We have carried out imaging of $\Phi \left(x,y,t\right)$ for 12 h in multiple microtubule devices to derive the solution below, the Eq. (1) has two inherent imaginary parts, but as power (Like iⁱ).

$$\Phi \left({x},{y}\right)={K}{e}^{(L-{\upsigma )}^{2}/2\left(A\sin\left(\frac{D-nb}{b}\right)+B\cos\left(\frac{P-na}{a}\right)\right)}$$

(2)

here a, b are lattice parameters, L is length, D is diameter, P is pitch, n is an integer.

Both the $I\left(t\right)$ states are nested with an additional phase state in the Eq. (1), which changes the discrete beating peaks as an integral part of a common wave form, which is a standing wave (Eq. 2). Resultant beating wave is a condensate, forms a standing wave that we map using ultra-low magnetic sensors ($\frac{\partial\Phi }{\partial t}=0$, Eq. 2). These fractal interference would be discussed later in greater details, as the theory would change if we consider clocking moment and 3D effects.

2.8 How to Check the Existence of Super Non-conductivity

We need to find a material that is non-conducting first, greater than several hundreds of Giga-Ohms. Then, we need to pump noise and check current output with Keithley 6430 connected to resistance bridge that measures 10 autto-ampere resolution (note that one reading may take tens of hours). By trial and error, one can find a noise domain where current goes down from pico-ampere to less than a femto-ampere. Then, above certain amplitude and frequency range current output increases again. So, we know that in this particular operational region, the electrical resistance is tera-Ohm to peta-Ohm and even lower. Such a device is suitable to check for the properties of super non-conductivity. This phenomenon is inverse of superconductivity.

2.9 Measuring Fermi Velocity, Effective Mass, Quantum Conductance for Fractal Phase Scenario, as a Recipe to Calculate Quantum Capacitance and Quantum Inductance from Fig. 3b

The reflected wave from dos walls or quantum well is considered a virtual or imaginary or quasi particle scattered by perturbation (δΔ) in an energy gap, that changes the dos i.e. well configuration. The change in dos is given $\delta N(\omega )\propto \frac{\omega \Delta }{{\omega }^{2}-{\Delta }^{2}}\delta \Delta Si\left[\frac{2Zd{({\omega }^{2}-{\Delta }^{2})}^{1/2}}{\hslash {v}_{F}}\right]$, where $Si\left(x\right)\equiv {\int }_{x}^{\infty }(\sin y/y)dy$, its maxima occurs at $\frac{Zd{\left({\omega }^{2}-{\Delta }^{2}\right)}^\frac{1}{2}}{\pi \hslash {v}_{F}}=n$, $n=0,1,2, 3\dots$ $Z(\omega )$ is the usual re-normalization function in Green function, d is diameter. Phase velocity ${v}_{F}$ nears velocity of light, but it is reduced as $\omega \to \Delta$, i.e. more we fine tune resonance frequency of monomer, dimer, microtubule. For electronic dos, $\frac{{d}^{2}V}{d{I}^{2}}$ is measured, here, instead, we measure $\frac{{d}^{2}\psi }{d{Q}^{2}}$ using coaxial atom probe to get magnetic dos $\delta N\left(\omega \right).$ Thus, we determine ${v}_{F}$, for electron and magnetic wave, which also gives electron dwell time ${\tau }_{i}$ in a quantum well. Conductance is $\frac{\partial Q}{\partial \psi }$ instead of $\frac{\partial I}{\partial V}$, quantum conductance is given by not e²/h.

Supporting Online Figures

3.1 Figure 5: Pitch and Diameter Variation of Microtubule Capacitance and Inductance

See Fig. 5.

Fig. 5

Tomasch oscillations in the room temperature microtubule when inductance, capacitance are measured using a bridge circuit using chip depicted in Fig. 1a. Peaks and crests are equidistance in length (panel b), pitch (panel c) and diameter (panel a). The pitch data in the panel c has less number of Tomasch ripples because of experimental constraints. Capacitance were measured at 1 kHz

3.2 Figure 6: Variation of C_q and L_q as a function m* and g_v

See Fig. 6.

Fig. 6

Four plots using set up of panel a, two additional STM tips applied noise directly to the water crystal core of microtubule. Effective mass (blue) and degree of degeneracy (red) are plotted for quantum inductance L_q (fF = femto Henry = 10⁻¹⁵ H; filled square) and quantum capacitance C_q (aF = autto Farad, 10⁻¹⁸ F; open circle). These are experimental results only