Moreover, the depolarization effect of magnesium ions might be responsible for their fetal neuroprotection action [ 8 10 ] and for a mood stabilizing action similar to lithium in treating bipolar patients [ 11 ], because it has been found that depolarization of the neuronal membrane activates DNA synthesis and mitosis in arrested neurons that exhibit neuroprotection actions [ 12 16 ]. Additionally, it has been postulated that lithium stabilizes the mood of bipolar patients by depolarizing the hyperpolarized membrane of their cells [ 6 ]. Thus, identifying the exact mechanism behind the depolarization effect of magnesium may offer better therapeutic solutions to treat several disorders and diseases.

A model of quantum tunneling of ions through closed channels has been postulated [ 4 ] and used to explain and understand different phenomena and actions that occur in biological systems such as referred pain [ 5 ], the action of lithium to stabilize the mood of bipolar patients [ 6 ], and myelin function in spatiotemporally confining action potentials and limiting hyperexcitability [ 7 ]. Therefore, this model is used in this study to explore how magnesium ions could depolarize the neuronal membrane.

Several hypotheses have been proposed to explain the depolarization effect of magnesium ions, including the inhibition of the activity of certain types of potassium channels that favor a depolarized membrane and the increase of the activity of the 1 Na/1 Mgantiport that depolarizes membranes [ 2 3 ]. However, in the present study, a quantum mechanical approach is used to explain the depolarization effect of magnesium ions.

Magnesium has several uses in the medical field. It is used to treat eclampsia, asthmatic patients, and certain cardiac arrhythmias. Additionally, magnesium is a crucial cofactor for enzymes, and it is vital for ATP production [ 1 ]. Regarding its electrophysiological features, magnesium depolarizes the action potential threshold, decreases the frequency of action potentials, and consequently suppresses the overall activity and excitability of neurons. However, one contradictory feature of magnesium has been documented. This feature is the ability of magnesium ions to depolarize resting membrane potential, which increases the excitability of neurons in contrast to the actual actions of magnesium, but the overall net effect is to suppress excitability [ 2 3 ].

2. Methods

The proposed mechanism is that magnesium ions tunnel through the closed channels of the neuronal membrane. The closed voltage-gated channels block the permeation of ions by forming a hydrophobic gate at the intracellular end of the membrane [ 17 ]. Therefore, the quantum tunneling phenomenon is applied on this gate [ 4 ]. The hydrophobic gate has been illustrated as an electric field in the space of parallel capacitor that resists ion movement [ 4 ]. This illustration is done to better determine how the barrier energy of the gate changes with ion’s position while passing through the closed gate. The model of quantum tunneling is applied on the sodium voltage-gated channel Nav1.2.

T Q = e − 8 m ℏ ∫ X 1 X 2 ( q E x ) g a t e − E K d x (1) T Q is the tunneling probability, m is the mass of magnesium ion (4.04 × 10−26 Kg), ℏ is the reduced Planck constant (1.05 × 10−34 Js), q is the charge of magnesium ion(3.2 × 10−19 C), E gate is the electric field required to prevent the ion from passing the gate, x is the position of the ion through the gate, E K is the kinetic energy of magnesium ion, and X 1 –X 2 is the forbidden region where magnesium ions cannot pass the gate. The tunneling probability of magnesium ions through the closed sodium voltage-gated channels can be calculated by the following equation [ 4 18 ]:whereis the tunneling probability,is the mass of magnesium ion (4.04 × 10Kg),is the reduced Planck constant (1.05 × 10Js),is the charge of magnesium ion(3.2 × 10C),is the electric field required to prevent the ion from passing the gate,is the position of the ion through the gate,is the kinetic energy of magnesium ion, and X–Xis the forbidden region where magnesium ions cannot pass the gate.

E gate is calculated by the following equation [ E g a t e = U q L (2) U is the energy that is needed to open the closed gate of the channels, q is the charge of magnesium ion, and L is the length of the sodium channel gate 5.4 × 10−11 m [ The electric fieldis calculated by the following equation [ 4 ]:whereis the energy that is needed to open the closed gate of the channels,is the charge of magnesium ion, andis the length of the sodium channel gate 5.4 × 10m [ 4 ].

U , the following equation can be used [ U = q g a t e e V 1 / 2 (3) q g a t e is the gating charge, e is the electron charge (−1.6 × 10−19 C), and V 1 / 2 is the membrane voltage at which half of channels are open. To calculate, the following equation can be used [ 19 ]:whereis the gating charge,is the electron charge (−1.6 × 10C), andis the membrane voltage at which half of channels are open.

E K ( o ) = q V m + 1 2 K B T (4) q is the magnesium ion charge, V m is the neuronal membrane voltage of −90 mV [ K B is the Boltzmann constant (1.38 × 10−23 J/K), and T is the absolute body temperature 310 K. On the other hand, when extracellular magnesium ions pass through the channels and reach the intracellular hydrophobic gate, they get kinetic energy from the neuronal membrane voltage of −90 mV [ 20 ]. In addition, they get kinetic energy from the thermal source of the body temperature; hence, their total kinetic energy can be calculated by the following equation:whereis the magnesium ion charge,is the neuronal membrane voltage of −90 mV [ 20 ],is the Boltzmann constant (1.38 × 10J/K), andis the absolute body temperature 310 K.

E K ( i ) = 1 2 K B T (5) Regarding intracellular magnesium ions, the neuronal membrane voltage does not contribute to their kinetic energy because the gate is located at the intracellular end so that intracellular magnesium ions hit the gate before going through the membrane voltage. Therefore, their kinetic energy is due to the thermal source of the body temperature, and it can be calculated by the following equation:

The forbidden region X 1 –X 2 is where the barrier energy is equal or higher than the kinetic energy of magnesium ion: qEx ≥ E K .

To investigate the effect of the quantum tunneling of magnesium ions on resting membrane potential, the quantum conductance of single channel and the quantum membrane conductance of magnesium ion must be calculated.

C Q M g can be calculated by the following equation [18, C Q M g = q 2 h T Q ( P M g P N a ) (6) q is the charge of magnesium ion, h is the Planck constant (6.6 × 10−34 J S ), T Q is the tunneling probability, and ( P M g P N a ) represents the degree of sodium channels’ selectivity to magnesium ions in comparison to sodium ions. The quantum conductance of single channel for magnesium ioncan be calculated by the following equation [ 4 21 ]:whereis the charge of magnesium ion,is the Planck constant (6.6 × 10),is the tunneling probability, andrepresents the degree of sodium channels’ selectivity to magnesium ions in comparison to sodium ions.

C Q M ( M g ) = C Q M g × D (7) D is the density of sodium channels in the neuronal membrane. Additionally, the quantum membrane conductance can be calculated by the following equation:whereis the density of sodium channels in the neuronal membrane.

Magnesium ions are divalent ions. Therefore, to determine the effect of magnesium ions on the resting membrane potential, the Goldman–Hodgkin–Katz equation of monovalent ions must be modified as in the following [ 22 ]:

0 = J N a + + J K + + J M g + 2 (8) J is the current density (A/m2). 0 = Y N a ( w [ N a ] i − [ N a ] o ) w − 1 + Y K ( w [ K ] i − [ K ] o ) w − 1 + Y M g ( w [ M g ] i 2 − [ M g ] o ) w 2 − 1 Y i o n = P i o n z V m 2 F 2 R T and w = e F V m R T , where P i o n is the membrane permeability to the ion (m/s), z is the valence of the ion, V m is the membrane potential, F is the Faraday constant, R is the gas constant, and T is body temperature. Y i o n values can be replaced by any proportional values that satisfy Equation (8). Therefore, they are replaced by the values of membrane conductance of ions C i o n (mS/cm2). Taking into consideration the quantum conductance of magnesium ions and that conductance is not the same for extracellular and intracellular magnesium, as is shown further in the discussion, the equation becomes: 0 = C N a ( w [ N a ] i − [ N a ] o ) w − 1 + C K ( w [ K ] i − [ K ] o ) w − 1 + ( C Q M ( M g ) i w [ M g ] i 2 − C Q M ( M g ) o [ M g ] o ) w 2 − 1 At equilibrium, the net ions movement across the neuronal membrane is zero:whereis the current density (A/m).whereand, whereis the membrane permeability to the ion (m/s),is the valence of the ion,is the membrane potential,is the Faraday constant,is the gas constant, andis body temperature.values can be replaced by any proportional values that satisfy Equation (8). Therefore, they are replaced by the values of membrane conductance of ions(mS/cm). Taking into consideration the quantum conductance of magnesium ions and that conductance is not the same for extracellular and intracellular magnesium, as is shown further in the discussion, the equation becomes:

w − 1) and multiplying by ( w + 1), the equation becomes: 0 = ( w + 1 ) C N a ( w [ N a ] i − [ N a ] o ) + ( w + 1 ) C K ( w [ K ] i − [ K ] o ) + ( C Q M ( M g ) i w [ M g ] i 2 − C Q M ( M g ) o [ M g ] o ) w 2 ( S 2 + M 2 ) + w ( S 2 − S 1 ) = S 1 + M 1 (9) S 1 = C N a [ N a ] o + C K [ K ] o (10) S 2 = C N a [ N a ] i + C K [ K ] i (11) M 1 = C Q M ( M g ) o [ M g ] o (12) M 2 = C Q M ( M g ) i [ M g ] i (13) By removing the factor (− 1) and multiplying by (+ 1), the equation becomes:By ordering and extending:where:

Equation (9) is a simple quadratic equation that can be solved after some rearranging:

w 2 + A w = B where A = S 2 − S 1 S 2 + M 2 and B = S 1 + M 1 S 2 + M 2

( w + A 2 ) 2 − A 2 4 = B w = − A 2 ± B + A 2 4 Completing the square:

w ) is chosen because the negative value is not compatible with exponential function. Therefore: w = − A 2 + B + A 2 4 w = S 1 − S 2 2 ( S 2 + M 2 ) + S 1 + M 1 S 2 + M 2 + ( S 2 − S 1 ) 2 4 ( S 2 + M 2 ) 2 w = S 1 − S 2 2 ( S 2 + M 2 ) + 4 ( S 1 + M 1 ) ( S 2 + M 2 ) + ( S 2 − S 1 ) 2 4 ( S 2 + M 2 ) 2 w = 1 2 ( S 2 + M 2 ) [ ( S 1 − S 2 ) + 4 S 1 S 2 + 4 S 1 M 2 + 4 S 2 M 1 + 4 M 1 M 2 + S 2 1 − 2 S 1 S 2 + S 2 2 ] The positive value of () is chosen because the negative value is not compatible with exponential function. Therefore: