# Mathematical Modeling of Ion Quantum Tunneling Reveals Novel Properties of Voltage-Gated Channels and Quantum Aspects of Their Pathophysiology in Excitability-Related Disorders

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## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model

#### 2.1. The Conductance of the Voltage-Gated Channels According to the Laws of Thermodynamics

_{1/2}is the membrane voltage at which half of the channels are open, V

_{m}is the actual membrane voltage, q

_{g}is the gating’s charge, K

_{B}is the Boltzmann’s constant ($1.38\times {10}^{-23}$ J/K), and T is the absolute body temperature (310 K). The mathematical term ${q}_{g}{V}_{1/2}$ represents the gating free energy, which is the energy associated with the conversion from the ‘closed’ state to the ‘open’ state at ${V}_{m}=0$ [34]. Furthermore, the mathematical structure ${q}_{g}{V}_{1/2}-{q}_{g}{V}_{m}$ represents the energy required to switch from the ‘closed’ state to the ‘open’ state or the energy barrier that resists the passage of ions at a certain membrane potential V

_{m}.

_{m}, a fraction of this number of channels will be ‘open’ and be able to conduct ions. Therefore, the membrane conductance C

_{M}due to voltage-gated channels can be calculated by the following equation:

_{M}is the membrane’s conductance (S/m

^{2}), ${C}_{\mathrm{sin}gle}$ is the single channel conductance (S), and D is the channels’ density (channels/m

^{2}).

#### 2.2. The Conductance of the Voltage-Gated Channels According to Quantum Mechanics

_{Q}) through the hydrophobic gate, as solved from Schrodinger’s equation, can be calculated by using the following equation [12,31,32]:

_{1}-x

_{2}is the forbidden region of the gate where the barrier’s energy U(x) is higher than the kinetic energy of the ion KE.

_{ion}is the charge of the ion.

_{2}is at the end of the gate (${x}_{2}=L$), and x

_{1}is where $U({x}_{1})=\frac{{q}_{g}{V}_{1/2}-{q}_{g}{V}_{m}}{L}{x}_{1}=KE$. Thus, Equation (7) becomes:

_{m}and V

_{1/2}) are absolute values. This is made because the kinetic energy of the ions ${q}_{ion}{V}_{m}$ is a positive value and V

_{m}should be an absolute value of the actual membrane voltage. Additionally, when tunneling probability and its related equations are encountered, the membrane’s voltage is negative on the inside with regard to the outside, and the value of the membrane’s voltage is an absolute value.

^{2}), the quantum membrane conductance ${C}_{QM}$ can be calculated by this equation:

^{2}.

## 3. Results

#### 3.1. The Conductance of the Voltage-Gated Sodium Channels According to the Boltzmann Distribution

^{2}[1], and the single channel conductance of sodium channel ${C}_{\mathrm{sin}gle(Na)}$ is $15\times {10}^{-12}$ S [1].

_{m}and V

_{1/2}) are substituted with their negative sign when the Boltzmann distribution is applied on the voltage-gated channels.

#### 3.2. The Conductance of the Voltage-Gated Sodium Channels According to the Quantum Mechanics

#### 3.2.1. The Tunneling Probability of Sodium Ions through the Closed Intracellular Hydrophobic Gate

_{m}and V

_{1/2}) are substituted as the absolute values of their negative potentials. This is also will be valid wherever the quantum model is applied on the voltage-gated channels.

#### 3.2.2. The Quantum Conductance of Single Voltage-Gated Sodium Channel

#### 3.2.3. The Quantum Membrane Conductance of Sodium Ions

^{2}.

#### 3.3. The Conductance of the Voltage-Gated Potassium Channels According to the Boltzmann Distribution

_{V}1.2. These channels have gating charge ${q}_{g}=9.6e=15.36\times {10}^{-19}$ C [42,43] and a gating free energy ${q}_{g}{V}_{1/2}=5.35\times {10}^{-20}$ J [42,43]. Additionally, the density of potassium channels D will be substituted by $5\times {10}^{13}$ channels/m

^{2}[1] and the single channel conductance of potassium channel ${C}_{\mathrm{sin}gle(K)}=15\times {10}^{-12}$ S [1].

#### 3.4. The Conductance of the Voltage-Gated Potassium Channels According to the Quantum Mechanics

#### 3.4.1. The Tunneling Probability of Potassium Ions through the Intracellular Hydrophobic Gate

#### 3.4.2. The Quantum Conductance of Single Voltage-Gated Potassium Channel

#### 3.4.3. The Quantum Membrane Conductance of Potassium Ions

#### 3.5. The Influence of Quantum Tunneling of Ions on the Resting Membrane Potential

^{2}) [1,45], ${C}_{K}$ is the membrane conductance of potassium ions at the resting state due to leaky channels (5 S/m

^{2}) [1,45], F is Faraday’s constant (96,485.33 C/mol), R is the gas constant (8.31 J/Kmol), T is the body temperature (310 K), and V

_{m}is the resting membrane potential at the equilibrium. As it was explained before, the membrane potential V

_{m}is an absolute value of the negative potential of the membrane. Furthermore, the following values will be considered for the ions concentrations: ${\left[Na\right]}_{o}=142$ mmol/L [45], ${\left[Na\right]}_{i}=14$ mmol/L [45], ${\left[K\right]}_{o}=4$ mmol/L [45], and ${\left[K\right]}_{i}=140$ mmol/L [45].

## 4. Discussion

- Because voltage-gated channels and ions are a part of the biological system, it is expected that they could operate far from the thermal equilibrium, which indicates that ions may cool down through the hydrophobic gate even at higher body temperatures. This ‘cooling down’ can sustain the quantum coherence and make the quantum effects more apparent [17,74,75,76,77].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The relationship between the membrane potential and the common logarithm of the open probability of sodium channels at G = −6.33 J and over membrane potential range from −0.09 V to 0 V.

**Figure 2.**The relationship between gating free energy and the common logarithm of the open probability of the sodium channels at membrane potential −0.087 V and over gating free energy range from −12 J to −5 J.

**Figure 3.**The relationship between membrane potential and the common logarithm of the membrane conductance of sodium ions according to the Boltzmann distribution at G = −6.33 J and over membrane potential range from −0.09 V to 0 V.

**Figure 4.**The relationship between gating free energy and the common logarithm of the membrane conductance of sodium ions according to the Boltzmann distribution at membrane potential −0.087 V and over gating free energy range from −12 J to −5 J.

**Figure 5.**The relationship between the membrane potential and the common logarithm of tunneling probability of extracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.054 V to 0.09 V.

**Figure 6.**The relationship between membrane potential and the common logarithm of tunneling probability of intracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.048 V to 0.09 V.

**Figure 7.**The relationship between gating free energy and the common logarithm of tunneling probability of extracellular sodium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 10.77 J.

**Figure 8.**The relationship between gating free energy and the common logarithm of tunneling probability of intracellular sodium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 12.17 J.

**Figure 9.**The relationship between membrane potential and the common logarithm of quantum conductance of a single sodium channel for extracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.054 V to 0.09 V.

**Figure 10.**The relationship between membrane potential and the common logarithm of quantum conductance of a single sodium channel for the intracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.048 V to 0.09 V.

**Figure 11.**The relationship between gating free energy and the common logarithm of quantum conductance of a single sodium channel for the extracellular sodium ions at membrane potential 0.087 V and over a gating free energy range from 5 J to 10.77 J.

**Figure 12.**The relationship between gating free energy and the common logarithm of quantum conductance of a single channel for intracellular sodium ions at membrane potential 0.087 V and over a gating free energy range from 5 J to 12.17 J.

**Figure 13.**The relationship between membrane potential and the common logarithm of quantum membrane conductance of extracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.054 V to 0.09 V.

**Figure 14.**The relationship between membrane potential and the common logarithm of quantum membrane conductance of intracellular sodium ions at G = 6.33 J and over a membrane potential range from 0.048 V to 0.09 V.

**Figure 15.**The relationship between gating free energy and the common logarithm of quantum membrane conductance of extracellular sodium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 10.77 J.

**Figure 16.**The relationship between gating free energy and the common logarithm of quantum membrane conductance of intracellular sodium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 12.17 J.

**Figure 17.**The relationship between the membrane potential and the common logarithm of the open probability of potassium channels at G = −5.35 J and over a membrane potential range from −0.09 V to 0 V.

**Figure 18.**The relationship between gating free energy and the common logarithm of the open probability of potassium channels at membrane potential −0.087 V and over a gating free energy range from −12 J to −5 J.

**Figure 19.**The relationship between membrane potential and the common logarithm of the membrane conductance of potassium ions according to the Boltzmann distribution at G = −5.35 J and over a membrane potential range from −0.09 V to 0 V.

**Figure 20.**The relationship between gating free energy and the common logarithm of the membrane conductance of potassium ions according to the Boltzmann distribution at membrane potential of −0.087 V and over a gating free energy range from −12 J to −5 J.

**Figure 21.**The relationship between membrane potential and the common logarithm of tunneling probability of extracellular potassium ions at G = 5.35 J and over a membrane potential range from 0.044 V to 0.09 V.

**Figure 22.**The relationship between membrane potential and the common logarithm of tunneling probability of intracellular potassium at G = 5.35 J and over a membrane potential range from 0.039 V to 0.09 V.

**Figure 23.**The relationship between gating free energy and the common logarithm of tunneling probability of extracellular potassium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 11.33 J.

**Figure 24.**The relationship between gating free energy and the common logarithm of tunneling probability of intracellular potassium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 12.72 J.

**Figure 25.**The relationship between membrane potential and the common logarithm of quantum conductance of a single potassium channel for extracellular potassium ions at G = 5.35 J and over a membrane potential range from 0.044 V to 0.09 V.

**Figure 26.**The relationship between membrane potential and the common logarithm of quantum conductance of a single potassium channel for intracellular potassium ions at G = 5.35 and over a membrane potential range from 0.039 V to 0.09 V.

**Figure 27.**The relationship between gating free energy and the common logarithm of quantum conductance of a single potassium channel for extracellular potassium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 11.33 J.

**Figure 28.**The relationship between gating free energy and the common logarithm of quantum conductance of a single potassium channel for intracellular potassium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J to 12.72 J.

**Figure 29.**The relationship between membrane potential and the common logarithm of the quantum membrane conductance of extracellular potassium ions at G = 5.35 J and over a membrane potential range from 0.044 V to 0.09 V.

**Figure 30.**The relationship between membrane potential and the common logarithm of the quantum membrane conductance of intracellular potassium ions at G = 5.35 J and over a membrane potential range from 0.039 V to 0.09 V.

**Figure 31.**The relationship between gating free energy and the common logarithm of the quantum membrane conductance of extracellular potassium at membrane potential of 0.087 V and over a gating free energy range from 5 J to 11.33 J.

**Figure 32.**The relationship between gating free energy and the common logarithm of the quantum membrane conductance of intracellular potassium ions at membrane potential of 0.087 V and over a gating free energy range from 5 J and 12.72 J.

**Figure 33.**The relationship between gating free energy and the resting membrane potential under the influence of quantum tunneling of sodium ions.

**Figure 34.**The relationship between gating free energy and the resting membrane potential under the influence of quantum tunneling of potassium ions.

**Figure 35.**A schematic diagram represents the quantum wave behavior of an ion and the quantum tunneling through the intracellular gate (shown in red) indicated by the dashed line in the gate, which is formed by the crossing of the four S6 segments near the intracellular side (two of them are shown for simplicity). Figure (

**a**) represents quantum tunneling of an ion moving from the extracellular side to the intracellular side. Figure (

**b**) represents quantum tunneling of an ion moving from the intracellular side to the extracellular side. Notice that the extracellular ion, as in Figure (

**a**), has shorter wavelength (the distance from peak to peak) when it is compared with that of the intracellular ion, as in Figure (

**b**). This indicates that extracellular ions have higher kinetic energy since it is inversely correlated with the wavelength. Additionally, the wave amplitude of the extracellular ion passing after the gate, as in Figure (

**a**), is higher than that of the intracellular ion, as in Figure (

**b**). This indicates that extracellular ions have higher tunneling probability since it is proportional to the wave amplitude.

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**MDPI and ACS Style**

Qaswal, A.B.; Ababneh, O.; Khreesha, L.; Al-Ani, A.; Suleihat, A.; Abbad, M.
Mathematical Modeling of Ion Quantum Tunneling Reveals Novel Properties of Voltage-Gated Channels and Quantum Aspects of Their Pathophysiology in Excitability-Related Disorders. *Pathophysiology* **2021**, *28*, 116-154.
https://doi.org/10.3390/pathophysiology28010010

**AMA Style**

Qaswal AB, Ababneh O, Khreesha L, Al-Ani A, Suleihat A, Abbad M.
Mathematical Modeling of Ion Quantum Tunneling Reveals Novel Properties of Voltage-Gated Channels and Quantum Aspects of Their Pathophysiology in Excitability-Related Disorders. *Pathophysiology*. 2021; 28(1):116-154.
https://doi.org/10.3390/pathophysiology28010010

**Chicago/Turabian Style**

Qaswal, Abdallah Barjas, Omar Ababneh, Lubna Khreesha, Abdallah Al-Ani, Ahmad Suleihat, and Mutaz Abbad.
2021. "Mathematical Modeling of Ion Quantum Tunneling Reveals Novel Properties of Voltage-Gated Channels and Quantum Aspects of Their Pathophysiology in Excitability-Related Disorders" *Pathophysiology* 28, no. 1: 116-154.
https://doi.org/10.3390/pathophysiology28010010