Ion Pumps’ Compensatory Role: Do Ion Pumps Counter Quantum Tunnelling and Membrane Leakage in VGICs?

Script

Hello, my name is Celeste Marcon, and I want you to imagine a disease or disorder you know caused by uncontrollable electrical signalling inside your cells. For example, Epilepsy may come to mind. But Epilepsy is also known as a channelopathy, diseases and disorders caused by dysfunction in ion channels which is a protein in your cell membrane that will open or close to let ions pass in and out of the cell. My project addresses the issue of channelopathies by proposing that ion pumps, another cell membrane protein which work to maintain the resting membrane potential, also work to counter the quantum tunnelling of ions through voltage-gated ion channels. Quantum tunnelling is the process where a particle is appears to pass through an energy barrier despite having insufficient energy itself, while a voltage-gated ion channel is a specific type of ion channel that will open or close in response to changes of voltage across the cell membrane.

Background

Quantum neurobiology is an emerging field which investigates how concepts in quantum mechanics, such as quantum tunnelling and entanglement, influence our brain and body, providing explanations where classical physics fails. The classical explanation for ion transportation across the cell membrane is limited to the idea that when an ion channels gate closes, it is impermeable. Yet, this fails to completely explain the membrane leak ratio experimentally observed in ion pumps pumping ions in and out of the cell to maintain equilibrium. My project proposes the quantum tunnelling of ions through a closed voltage-gated ion channel (VGIC, figure 2) results in ion pumps acting as a mechanism to counter that process while also maintaining the resting membrane potential.

Hypothesis

If ion pumps have a membrane transport leakage ratio of 2 potassium ions out for 3 sodium ions in to compensate for the quantum tunnelling of ions through closed VGICs, then the ratio for sodium and potassium ion tunnelling probability should be a similar number to that ratio.

Objectives

Mathematically assess the quantum tunnelling of ions through VGICs

Evaluate ion pumps' as a compensatory mechanism through the experimentally observed membrane leakage ratio

Analyze the quantum tunnelling of ions through VGICs as a potential cause for channelopathies

Applications

Dysfunction of ion channels results in diseases called channelopathies, such as epilepsy and LQTS. My research suggests researchers should view the quantum tunnelling of ions through VGICs leading to excessive depolarization or hyperpolarization in the cell as a potential cause of channelopathies.

Procedure

My procedure is illustrated in Figure 3.

Mathematical Model

Tunnelling Probability can be given by deriving Shrodinger's equation based on the potential barrier type (Eckart or rectangular potential shown in figure 3 and 4), and depends on the potentials barrier height (energy) V0, its width L, the particles mass m, and the particles kinetic energy E.

VGIC Gate Potential Barrier Types

A rectangular potential has a constant and finite height, and a finite width. There are 3 regions which can be used to describe the wave behaviour as it passes through the barrier. Region 1 is where the incident wave packet (incident particle) coexists with a reflected wave packet. In region 2, part of the incident wave is transmitted through the barrier, tunnelling through to region 3. Region 3 is where the transmitted wave packet (transmitted particle) moves in a potential-free zone.

The Eckart potential has a peak height and can be represented by a single equation:.

The Eckart potential is a more realistic model as it is a more realistic shape for the gate of a VGIC.

Equations

Equation 1 is based off the Rectangular potential, While Equation 2 is based of the Eckart potential barrier.

1.

2.

Variables Required

The 4 variables mentioned before can be found through the following equations.

1.V0 = (membrane potential) x (1.6022 x 10-19)

The sodium channel gate barrier height is V0Na= 1.12154 X 10-20J potassium gate height is V0K = 1.44198 x 10-20J.

2.E = 3/2 KBT

KB = 1.38 X 10-23, and T = 310K

E = 6.417 X 10-21

3.L and m

It was found through research LNa = 5.4 x 10-11m and LK = 4.4 x 10-11m, while 2 ion isotopes were used: K-39 = 6.5 x 10-26 Kg, Na-23 = 3.8 x 10-26 Kg.

The Quantum Tunnelling of The Sodium ions Through the Closed Rectangular Sodium VGIC Potential Barrier

Equation 1 with the variables subbed in becomes:

Where m = 3.8 x 10-26 Kg, = 1.05 x 10-34J, V0 = 1.12154 X 10-20J, and L = 5.4 x 10-11m.

T = 1.158 x 10-8.

The Quantum Tunnelling of The Potassium ions Through the Rectangular Potassium VGIC Potential Barrier

Equation 1 with the with the variables subbed in becomes:

Where m = 6.5 x 10-26 Kg, V0 = 1.44198 x 10-21J, and L = 4.4 x 10-11m.

T = 7.194 X 10-12.

The Quantum Tunnelling of Sodium ions Through the Closed Eckart Sodium VGIC Potential Barrier

Equation 2 with the variables subbed in becomes:

Where m = 3.8 X 10-26 Kg, V0 = 1.12154 X 10-20 and L = 5.4 x 10-11m, and h = 6.62607015 x 10-34.

T = 9.999880 x 10-1.

The Quantum Tunnelling of Potassium ions Through the Closed Eckart Sodium VGIC Potential Barrier

Where m = 6.5 x 10-26 Kg, V0 = 1.44198 X 10-20, and L = 5.4 x 10-11m.

T = 9.999630 x 10-1.

Comparison of the Rectangular and Eckart Potential Tunnelling Probabilities to The Membrane Leak Ratio

The previous results found of ions' quantum tunnelling probability through the rectangular and Eckart potential barrier can be used to address the concept of ion pumps acting as a compensatory mechanism for the quantum tunnelling of VGICs via the membrane leak transport ratio by comparing the quantum tunnelling probability to the ratio.

The ratio of ions tunnelling through the rectangular VGIC Potential to the leak ratio using

resulted in: 1609.67247289408 = 1.5

The ratio of ions tunnelling through the Eckart VGIC to the leak ratio using the previous equation resulted in: 1.00002461109 = 1.5

Simulations

Simulations were made by coding using Google Colab, showing how tunnelling probability changes over varying widths (L), as the width of the channel affects tunnelling probability the most. The simulations aligned with my results, and showed as the barrier width decreased, tunnelling probability increased. The reason for the straight line versus a curved line is likely because of the logarithmic scale.

Discussion

The previous results mathematically confirm that sodium and potassium ions can quantum tunnel through sodium and potassium VGICs, with the Eckart potential models having a much higher probability. The results of the Eckart should be treated as more reliable as the potential barrier model is more realistic for the gate of a VGIC. Sodium ions had a higher tunnelling probability compared to potassium ions, likely due to their lighter mass. The results of the comparison of rectangular potential sodium and potassium ion tunnelling probability to the membrane leak ratio suggests the quantum tunnelling of ions makes up a very small amount, whereas the Eckart potential heavily suggests quantum tunnelling makes up or is responsible for quite a bit, or at least some of the membrane leak ratio. This is likely because of the difference in exponents in the tunnelling probability answers.

Channelopathies are a disorder caused by dysfunction in ion channels, such as Epilepsy and LQTS. Currently, only 10% of FDA approved drugs target Ion channels. Future medications for channelopathies should consider the quantum tunnelling of ions through VGICs as a contributor to disruption of the resting membrane potential. Drugs could be made to target the closed state of the channel, situations where the channels energy is lowered, or the quantum tunnelling of ions. Specifically in hyperexcitability related channelopathies, future researchers can consider hyperexcitability, or reduced excitability as a result of sodium or potassium ions quantum tunnelling through closed VGICs.

Conclusion

This project should help guide future researchers investigating the physiology of VGICs and channelopathies. Key points that can be taken from this research are:

The results of these calculations suggest Na-23 and K-39 can quantum tunnel through VGICs.

The results of these calculations suggest that the quantum tunnelling of ions through VGICs does make up part of the membrane leak ratio, meaning ion pumps do counter the quantum tunnelling of ions through VGICs.

The quantum tunnelling of sodium and potassium ions should be considered when treating channelopathies related to changes in the resting membrane potential and hyperexcitability.

Tunnelling probability is heavily influenced by the shape of the barrier.

For future studies, this work could be improved by:

Consideration of other ion channels and ions which could give different tunnelling probabilities.

Different barrier shapes could be used to more accurately represent the conformational changes a ion channel's gate goes through, as seen in the results, there was a big difference between the 2 potential barrier types, so a different potential barrier may provide more accurate or different results.

The next steps of my research are to look deeper into ion dynamics, the effect of tunnelling on the resting membrane potential, and its possible cause for dysfunction resulting in channelopathies.

Thank you to Dr. Mawhinney for mentoring me, providing me guidance, resources, support and answering my questions throughout my research in this project.

Thank you to Emily Cross who provided me guidance and mentored me.

I would also like to thank the NWORSF organizers for their support and efforts to get me here, as well as my family and friends for their continuous support and encouragement.

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Figures 1, 3, 4 were made using Canva and BioRender. All graphs were made using Google Colab and Excel. Figures 2 and 5 are not my own work and were adapted or taken from the cited sources on the diagram and in the following references.